If it were truly infinite, in the context of decreasing in size, it would reach such a small size beyond what is possible due to the makeup of this world, ie. smaller than an atom. Or perhaps not as we're constantly talking with the implied context of "reality" ie. things as they are at the moment of observation. Before then even, once it becomes smaller than what allows a single molecule of paint, it would of course "fill up". Though it is interesting because assuming once we reach the point of smaller than a molecule, you could still paint it's exterior because even though it's girth or width may not be able to hold a single molecule of paint, perhaps it's length would... actually no. Molecules, as we're told are symmetrical. There is probably a single point after a molecule of paint would no longer fit, where due to the structure itself counting as at least one molecule or atom, there is a point where the exterior can be painted (one molecule of paint can reside on the outside past the point where one molecule of paint can no longer pass on the inside). I couldn't imagine in a million years where this "paradox" would ever come up or be relevant in.. literally anything would ever do or ever come across but, isn't free society fun. Lots of time for non-productive speculation.
Deleted UserFebruary 18, 2021 at 17:41#5010030 likes
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Quite. I must have. Paint however is not mathematics, it consists of molecules of a finite and measurable size, or else it is no longer paint. Though perhaps "paint" is implied to be simply the smallest unit of measure available, ie. an atom or the smallest known subatomic particle being a quark (which is still measurable). Or does it transcend even this? Basically, how can something be infinitely long if it's slowly decreasing from a measurable state or point? Writing off the world we live in for numbers that could in theory go on forever.. it's still kinda confusing lol. Almost seems kinda like arguing over a glass being half full or half empty. Which of course is actually simple. If it was an empty glass filled up, it is half full. If it was a full glass, with half removed, it's half empty. Unfortunately, this does seem to boggle the mind. Hopefully not by design.
I never liked math. Perhaps someone who does can shine some light on this for the both of us? Basically, what is the largest number that can be reached, and is it any closer to infinity than 1? What is the smallest unit of measurement to reach infinity? What is the smallest fraction that can be reached without there being nothing left, etc. The universe, at least the world we live in, seems to disprove the existence of 'infinity' beyond "oh look a solid figure".
The apparent paradox comes from equating volume as defined by an integral with volume as defined by a concrete, physical enclosed region. Minimally, the infinity of the shape breaks the correspondence between the two ideas.
There are other examples, like the Cantor set, which is dispersed throughout the interval (0,1) - like being completely dissolved in a fluid - but nevertheless has no volume whatsoever.
Physical volume and mathematical volume don't have to correspond!
But the surface area of the horn, itself being infinite, cannot be painted with a finite amount of paint.
No that's not true. Divide the infinite surface area into sections 1 unit long, as for example the positive real number line is partitioned by the integers 1, 2, 3, 4, ...
Between 1 and 2 you use 1/2 gallon of paint. Between 2 and 3 you use 1/4 gallon. Between 3 and 4 you use 1/8 gallon, and so forth. You will then cover the entire infinite surface of the cone with only one gallon of paint. Of course this is a-physical, since the thickness of the paint would soon be far less than the width of an atom. But mathematically you can indeed cover an infinite surface area with a finite amount of paint.
I didn't see the video, but it was referenced on Reddit this morning with the same misconception, so I wonder if Numberphile perhaps confused people on this point.
ps -- It's perhaps easier to see this in two dimensions. Imagine the graph of y = 1/x from 1 to infinity. We know from calculus that the area under the curve is infinite. But for any positive integer n, the area under the curve on the interval [n, n+1] is finite. So you just cover each such interval with a finite amount of paint according to some convergent infinite series such as 1/2 + 1/4 + 1/8 + ... = 1.
If you fill the horn, have you not essentially painted the surface with a finite amount of paint? So it would seem that the surface cannot be painted, yet in fact is paintable.
You painted the internal surface of the horn, but not the external one, which is bigger, even if just infinitesimally. How is this accounted for?
If you fill the horn, have you not essentially painted the surface with a finite amount of paint? So it would seem that the surface cannot be painted, yet in fact is paintable.
The solution lies in the meaning that we silently assume for the word 'paint', which is to cover an area with a layer of liquid paint with a constant, nonzero thickness. We cannot 'paint' the horn in that sense because the volume required would be the area (infinite) multiplied by the thickness (nonzero), which means an infinite volume.
In the filled horn, the internal surface is indeed covered by paint. The thickness of the covering layer at a particular place equals the radius of the horn at that place. Since the radius decreases towards zero, the thickness of the layer just keeps getting thinner. There is no nonzero thickness of layer with which we can say that filling the horn has 'painted' it in the above-defined sense.
This solution does not rely on any physical limits such as Planck lengths or diameters of paint molecules. It simply lies in the definition of 'paint' requiring a constant, nonzero thickness of layer. If we remove that requirement then we can 'paint' any infinite two-dimensional surface (including painting the horn externally, by the way*) with any nonzero volume of paint, however small. We might have to make some technical restrictions such as only allowing surfaces that can be smoothly embedded in 3D Euclidean space. But those restrictions would only interest mathematicians as they would not exclude any surfaces non-mathematicians might imagine.
The mathematical principle behind this is that the function f(x) = 1/x has no convergent integral from 0 to infinity whereas the function g(x) = 1/x^2 does.
Or, avoiding calculus, the sum 1/1 + 1/2 + 1/3 + ... diverges to infinity whereas the sum 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/ 36 + ....converges to a value around 1.645.
* As fishfry points out, the horn has no distinction between its 'internal' and 'external' surface. But we can meaningfully define 'painting the horn internally (externally)' as follows. First define the inside I of the horn as the set of points x such that the shortest line segment connecting x to the horn's axis does not intersect the horn's surface S. The outside O of the horn is all of 3-space excluding the inside and the surface.
Then an internal (external) painting of S is a subset A of Euclidean 3-space (which we think of as being 'full of paint'), such that, for every point P on S, there exists an open neighbourhood N(P) whose intersection with I (O) lies entirely within A.
Under this definition, which we note specifies no minimum thickness of paint, we can make both an internal and an external 'painting' of S using a finite volume of paint.
TheMadFoolFebruary 18, 2021 at 22:15#5011110 likes
Ziploc bags. An object with a given surface area can be collapsed and that can reduce the volume without affecting the surface are...if you do it just right, the volume can become zero but not the surface area.
Reply to tim wood
Reminds me of this: https://youtu.be/5LWfXhggC70
It's just semantic trickery of the mind in the end that creates these apparent paradoxes.
They are not very interesting (in my opinion).
Deleted UserFebruary 18, 2021 at 22:33#5011140 likes
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It would seem, then, that if we want to paint the interior surface, we need only pour in ? amount of paint
Correct, but the paint can't be uniform since it must be spread over an infinite surface area. The thickness of the paint has to decrease as some convergent infinite series: 1/2 gallon for the area between x = 1 and x = 2, 1/2 gallon for the next chunk, and so forth. So nothing you said is in conflict with the story, which is simply that 1/x has an infinite area but its solid of revolution has finite volume.
Deleted UserFebruary 19, 2021 at 16:15#5012230 likes
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Why? Why cannot it be a consistent thickness, mathematically thin?
Because if the thickness is constant its volume must be infinite. Think of the infinitely long real line and a rectangle above it with height 1/10. Width x height is infinite. Say the height is 1/million. Same thing. 1/zillion. Same thing. As long as the height is nonzero and uniform, the total area across an infinite surface must be infinite.
If I wish to paint my living room, it does not occur to me that I must fill the living room with paint or that the paint must vary in thickness.
Nothing to do with the previous sentence. And the whole idea of painting your living room is a red herring, since this is a mathematical and not a physical situation. It's like people complaining about Hilbert's hotel that there can't be any such hotel, or how would the maid make up all those beds, and so forth. The attempt to put the puzzle in "practical" terms ends up confusing the issue. But the point still stands that if the surface to be painted is infinite and the paint layer is uniform and of positive thickness, no matter how small, the volume of paint must be infinite.
I'll attempt a more rigorous description of the paint and painting. And for that purpose I'll borrow from the argument that the cardinality of the points on the number line between zero and one is the same as the cardinality of the points in a cube measuring one mile on a side.
Since that's true, that's a hint that a cardinality argument will be of no use here.
Let's consider the cardinality of the points that make up the inner and outer surfaces of the horn respectively.
There's a big problem, which is that there is no such thing as the "inner or outer surface" of a line. For example, what is the "upper surface" of the real line? There clearly isn't one because if the surface is any positive height above or below the line, it's not the "surface," and if it's directly on the line, it's the line.
Same argument with a circle. There is no inner and outer surface of a circle. Do you follow this point? It's crucial, since the "inner and outer surface" of a two-dimensional surface in 3-space is being bandied about in this thread and there is no such thing.
What are the inner and outer surfaces of a sphere? Can you name a point on the inner surface? No. Because if the point is zero distance from the sphere, it's on the sphere. And if it's a positive distance inside the sphere, it's not the "inner surface" because there are many points strictly between that point and the sphere.
In fact the "surface of a sphere" can only be the sphere itself. Like FDR said! The only surface of the sphere is sphere itself. Ok nevermind.
Bearing in mind that there is no such thing as the inner or outer surfaces of the horn, which is a tw- dimensional object living in 3-space, I'll play along. But remember the premise is false already.
Now, if we may, the cardinality of the points that make up the paint itself. By painting is meant an assignment of one point of paint to each point of surface.
Ok. But think of a circle in the plane. Can you paint it's "inner" or "outer" surfaces? No, because there's no such thing. If you paint the interior of a circle you either include the points on the circle or you don't. If you do, you've painted the circle itself. But if you don't, then there's always a space between your paint and the circle into which you could have put more paint; that is, gotten a little closer to the circile.
I hope you see this, if not please say so, because this point is essential. There are no inner or outer surfaces to a 1-D object living in 2-D, or a 2-D object living in 3-D.
And greater because for any cross section of the horn the inner surfaces never meet, and consequently there is always more paint in the cross section when the horn is filled than is needed to just paint the surface.
This doesn't make sense (because there are no inner and outer surfaces) and doesn't follow logically even if I granted you that there are such things. It's your own cardinality argument. Any continguous chunk of 3-space has the same cardinality as the unit interval on the line.
Thus on the assumption that the horn is filled with a finite amount of paint, understood to be proved, then the inner surface has been painted. Because the inner surface thus paintable, and the outer surface the same area, the outer surface must be paintable.
There are no inner or outer surfaces. But you haven't got an argument here even if there were. After all the interval []0,1] has the same cardinality as [0,2] which is twice as long. You already made that point earlier. I don't see that you've made an argument.
At least half a dozen so far. But more to the point you haven't made an argument, only claims that you yourself refuted at the beginning by noting that every contiguous subset of 3-space has the same cardinality as the real line or as a finite segment of the real line.
Which of the many errors I pointed out do you disagree with? Starting from the false concept of inner and outer surfaces of 2-D shapes living in 3-space? Just think a sphere in 3-space or (easier) a circle in the plane. Name a point on the "inner" or "outer" surface? You can't. If you pick a point on the circle that's on the circle. But if the point is any positive distance whatsoever off the circle, it's not the inner or outer surface because there are points strictly between your point and the circle.
If it depends on an if, then the if is that the horn can be filled in the first place and the rest "flows" from that, so it seems.
This I didn't understand. But imagine filling the inside of a circle in the plane. If you tell me you're including the circle in your painted region, then the points on the circle are painted. But if not, then there's some positive distance between painted region and the circle, so that's not the "inner or outer" surface after all.
In the end perhaps casting this problem in terms of paint causes more problems than it solves. But just saying, "1/x is square-integrable but not integrable" (a point made earlier by @andrewk) doesn't have quite the same ring to it.
Deleted UserFebruary 20, 2021 at 02:54#5013690 likes
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The video, and others, tells me the volume in the horn is finite, is in fact ? in appropriate units.
Yes. It's a finite volume with infinite surface area. It's a veridical paradox: "A veridical paradox produces a result that appears absurd, but is demonstrated to be true nonetheless." In other words it's a true fact (provable in freshman calculus) that's so counterintuitive it seems impossible. But there is no actual paradox. There's no statement that's both true and false. Math is filled with these kinds of things.
That it is fillable, with ? "mathematical paint" - whatever that means.
Whatever that means. It's a shape with finite volume and infinite surface area. That's all you can say. Yes it's "fillable with paint" if you want to view it that way. But if you push too hard on that visualization it gets confusing.
I take it to mean analogously the same as when a paint can is filled. If a paint can is filled with paint, then it seems fair to say that the inner surface of the can is painted.
I agree with that, about actual paint cans. But I don't agree with that about Gabriel's horn, because we're not filling anything with paint. And there is no inner surface. I went at this in detail in my previous post. You understand that for example a circle in the plane has no "inner surface." The horn is a two-dimensional surface in 3-space. It's not like a paint can. That's another source of bad intuition. A paint can has thickness, with an inner and outer surface. But the horn has no thickness and no inner or outer surfaces.
Analogously if the horn can be filled, then whatever it has that passes for a surface is "mathematically" painted. Any problem with this so far?
Well, to the extent that I accept that, and I sort of do, it's a veridical paradox. We have a finite volume that has infinite surface area, and it's a seeming common-sense paradox without being an actual logical contradiction.
The proposition of the paradox, as I get it from the video, is that the amount of "paint" is not enough to cover the outside of the horn because the area to be covered is infinite.
I didn't actually watch this particular video, I've seen this example in the past. But I disagree with that statement because we could paint the outside of the horn by using pi/2 gallons of paint on the segment between 1 and 2; and pi/4 gallons between 2 and 3; and pi/8 gallons between 3 and 4; and so forth. We'd use pi gallons of paint to cover the entire infinite surface area. Of course we can't paint it uniformly, unless we imagine paint of zero thickness, but then we can't sensibly have any meaningful volume of paint at all.
I agree with that. I saw this example years ago and it hasn't lost its force. It's a real puzzler. The best you can say is that it's a veridical paradox. There is no actual logical contradiction; only a violation of common sense. But the math is clear. The horn has finite volume and infinite surface area.
if the inside is covered by the "paint" inside, then why cannot the same volume of paint cover the outside? Is the area of the "outside" somehow different from the area "inside"?
I don't know the answer. It's a veridical paradox. How can an infinite surface area enclose a finite volume, or a finite volume have an infinite surface area? I don't know. It just does. I don't suppose that's satisfactory. That's why Numberphile got everyone talking about it. For what it's worth, Wiki has my solution and nothing better:
Since the horn has finite volume but infinite surface area, there is an apparent paradox that the horn could be filled with a finite quantity of paint and yet that paint would not be sufficient to coat its inner surface. The paradox is resolved by realizing that a finite amount of paint can in fact coat an infinite surface area?—?it simply needs to get thinner at a fast enough rate (much like the series 1/2^N gets smaller fast enough that its sum is finite). In the case where the horn is filled with paint, this thinning is accomplished by the increasing reduction in diameter of the throat of the horn.
https://en.wikipedia.org/wiki/Gabriel%27s_Horn
If you didn't find my explanation compelling you won't be satisfied by Wiki's identical explanation (except by virtue of Wiki's authority). And as I say, nobody on Wikipedia could come up with a better explanation. It's just one of those things in math that, as von Neumann said, you don't understand; you just get used to it.
ps -- Here's a tl;dr:
* It's just one of those things. It's counterintuitive but not a true paradox.
* The horn is two-dimensional whereas a paint can has thickness. That's throwing off our intuition. Paint cans have an inside and outside surface. The horn doesn't.
TheMadFoolFebruary 20, 2021 at 07:46#5014160 likes
There's another way to approach this puzzle. Whatever a Gabriel's horn is, it can be approximated as a cylinder.
Volume V of a cylinder = pi * r^2 * h
Surface area A of a cylinder = 2 * pi * r * h
for a cylinder with radius r and height h
Ratio of A to V = (2 * pi * r * h)/(pi * r^2 * h) = 2/r
As r approaches 0, V too approaches 0 but, oddly, A doesn't.
Deleted UserFebruary 20, 2021 at 11:37#5014610 likes
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Metaphysician UndercoverFebruary 20, 2021 at 12:15#5014640 likes
Reply to tim wood
If you would have listened to me in those other threads, where I explained the deficiencies in mathematical axioms, especially those which make what is mathematically indefinite into some thing definite, like the closure at the end of the infinitely long horn, you would have no problem with this issue.
It's quite obvious that the contradiction is in "infinitely long horn". If you pour paint in the top, and it is infinitely long, the paint never reaches the bottom. But if you assume the horn has a closure, a bottom, just like you might assume that .999... is 1, then the horn gets filled.
The film clip just demonstrates the inconsistencies in how "infinite" is dealt with by mathematical axioms. In one group of axioms, "infinite" is allowed to be a real mathematical object, so the infinite sutface requires an infinite amount of paint. But in cases like .999... and the volume of the horn, the infinite is rounded off and given closure.
But in cases like .999... and the volume of the horn, the infinite is rounded off and given closure.
Bullpucky. There is no "closure at the bottom of the horn." You just make stuff up and claim mathematicians said it when they didn't. That's called a strawman argument.
like the closure at the end of the infinitely long horn
Bullpucky bullpucky bullpucky. YOU said that, not any mathematician, ever. As always you take your own mathematical ignorance and project it onto mathematics itself. You wield your ignorance like a weapon.
ps -- Let me not be so ill mannered. Perhaps you could explain to me your version of calculus in which the area of 1/x from 1 to infinity isn't infinite, and the area under 1/x^2 from 1 to infinity isn't finite. If you could elucidate your version of calculus then I'd be enlightened. I seek your wisdom. Or perhaps you reject calculus entirely. I'd like to hear your perspective on this.
Do you happen to understand that there is no "closure at the bottom" of the cone? That this is NOT anything that any mathematician says or thinks? That is is entirely something you made up and then claim to mock? A strawman, as they say. Do you apprehend this point?
Deleted UserFebruary 20, 2021 at 20:00#5015380 likes
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Metaphysician UndercoverFebruary 21, 2021 at 02:14#5016650 likes
Do you happen to understand that there is no "closure at the bottom" of the cone?
One divided by infinity is not zero, it is indefinite. If you assume that one divided by infinity equals zero, you assume that the value for y reaches zero, therefore closure. It's very clearly stated in the YouTube video, he says we're taking the value of y to zero. However, this clearly contradicts the premise that the horn continues infinitely. The real issue is that integrals are approximations, and infinity has no place in an approximation. So that method of integration is simply not applicable to an infinitely long cylinder.
One divided by infinity is not zero, it is indefinite.
Absolutely correct. Also absolutely irrelevant, since nothing in this problem involves dividing one by infinity. I'm afraid this looks like yet another case of your misunderstanding the math, making up a story about what the math is saying, then arguing with your own story. A classic strawman argument.
But you know, you did say something mathematically correct, that one divided by infinity is not defined. I give you credit for that.
The cone or horn is NEVER closed. Pick any point on the real line and the cone is still open at the bottom. There is no "point at infinity" in this problem nor is there one in the real numbers. Again you misunderstand the math, make up a story, then argue against your own story. Your strawman has by now had all the straw beaten right out of it.
It's very clearly stated in the YouTube video, he says we're taking the value of y to zero.
Ok to be fair, as I've mentioned, I've seen Gabriel's horn before but didn't watch the video, so I don't know if perhaps he said something misleading. But it's true that as x gets arbitrarily large, y gets arbitrarily close to zero. That's what's meant by "going to zero." It's a technical phrase meaning that as x increases without bound, y gets as close as you like to zero. But x never "becomes infinite" nor does y ever become zero. The mathematical phrasing is a clever and subtle way of talking about these things WITHOUT saying that x becomes infinite or that y becomes zero. It's your mathematical ignorance of this terminology that's leading you into error. And since you repeatedly do this, when Wikipedia and other online sources could easily explain these things to you, I must assume at some point you choose not to learn the math, but rather to flail at strawmen of your own creation. I don't mean to sound uncharitable but if you have a better explanation I'm open to hearing it.
The tl;dr here is that as x increases without bound, but remains at all times finite; y gets as close as you like -- "arbitrarily close" as they say -- to zero. x does NOT "become infinite" nor does y ever become zero. That's the math. It's well-known. They teach it to college freshman. These days they even teach it to high school students. Not that it's taught well at that level or that anyone understands it. I'll stipulate that nobody comes out of calculus class with a clear understanding of the logical fine points. But it's all on Wiki.
However, this clearly contradicts the premise that the horn continues infinitely.
Since your beliefs are false, you haven't got a contradiction; only confusion caused by your lack of mathematical knowledge. But now that I've explained it to you, you can no longer claim to be ignorant.
The real issue is that integrals are approximations,
Integrals are exact. The Riemann sums that define them are approximations, but the integrals are the limits of the Riemann sums and they are perfectly exact. The volume of the horn is exactly pi and the surface area is "infinite," which has a technical meaning: It's greater than any real number you could name. It's not the metaphysical infinite. I think that's a point of confusion in these conversations.
Integrals are perfectly exact, that's the takeaway.
Well they're not approximations. And secondly, "infinity" is not a magic thing that you should allow to cloud your mind. Infinity in this context is nothing more than a shorthand for "increases without bound." We say "x goes to infinity" meaning that x increases without bound, but is at every point finite. Mathematical infinity is not the metaphysical infinity. Perhaps that needs to be said more often. It's more of a technical gadget or shorthand for a quantity that increases without bound, or is larger than any finite number.
So that method of integration is simply not applicable to an infinitely long cylinder.
Of course it is. It's freshman calculus. It's been working well since the days of Archimedes, formalized into a systematic scientific tool in the late 17th century, and put on a logically rigorous footing since the late 19th century.
Metaphysician UndercoverFebruary 21, 2021 at 11:55#5017960 likes
Ok to be fair, as I've mentioned, I've seen Gabriel's horn before but didn't watch the video, so I don't know if perhaps he said something misleading. But it's true that as x gets arbitrarily large, y gets arbitrarily close to zero. That's what's meant by "going to zero." It's a technical phrase meaning that as x increases without bound, y gets as close as you like to zero. But x never "becomes infinite" nor does y ever become zero. The mathematical phrasing is a clever and subtle way of talking about these things WITHOUT saying that x becomes infinite or that y becomes zero. It's your mathematical ignorance of this terminology that's leading you into error. And since you repeatedly do this, when Wikipedia and other online sources could easily explain these things to you, I must assume at some point you choose not to learn the math, but rather to flail at strawmen of your own creation. I don't mean to sound uncharitable but if you have a better explanation I'm open to hearing it.
It's not a matter of what is "meant by 'going to zero'", it's a question of what value is given in the calculation. Look at 6:25 in the video where he's doing the calculation, plugging the values into the formula. He says "well one over infinity that's zero, so you get nothing from that".
Absolutely correct. Also absolutely irrelevant, since nothing in this problem involves dividing one by infinity.
Clearly you are using a different calculation than the one in the video then. If you know of a method to figure out the volume of that horn, which avoids rounding off the infinitely small dimeter to zero, then maybe you should present it for us.
I can't speak for Numberphile, which I generally don't watch because the guy annoys me. Perhaps they were trying to simplify the fact that [math]\displaystyle \lim_{x \to \infty} \frac{1}{x} = 0[/math] and you were confused by their simplification. In which case it's on them and not on you. But now that I've explained it to you, it's up to you. Truly if they said that on Numberphile they shouldn't have exactly for this reason. It confuses people.
When we see the expression 1/infinity it's a shorthand for the limit and people generally know that. But if the video is intended (as it apparently is) for people who don't know that, they shouldn't have done it.
ps -- Ok I watched that section of the original video. That's a shorthand they do in integration problems, formally you'd replace it with a limit. I guess now I'd be in a position of trying to defend calculus pedagogy, that's hopeless. In general 1/infinity isn't defined but when it comes up in problems like this you can take it to be zero. I can't argue with you that there's a bit of flimflam to the whole enterprise. I can only say that the procedure can be made rigorous, but at the level of calculus problems, it rarely is. I don't expect that to be a satisfying answer, it's not to me either.
pps -- For integration problems we can consider ourselves working in the extended real numbers. These are the standard reals with special symbols [math]\pm \infty[/math].
One of the rules for these symbols is that if [math]a[/math] is a positive real number, the symbol [math]\frac{a}{\infty} = 0[/math].
That's actually the formally correct answer to your question and I probably should have thought of it earlier. The extended reals serve as a shorthand so that we don't have to use cumbersome limits to talk about expressions involving infinity.
Metaphysician UndercoverFebruary 22, 2021 at 03:51#5019950 likes
The extended reals serve as a shorthand so that we don't have to use cumbersome limits to talk about expressions involving infinity.
I think the real issue is that it's cumbersome to talk about limits when the subject is infinite, because it's contradictory. Is the horn closed (limited), or is it infinite (unlimited). Clearly the premise is that it is unlimited, infinite, and any mathematical axioms which deal with it by imposing a limit, are not truthfully adhering to the premise. And that's why the appearance of a paradox arises.
Is the horn closed (limited), or is it infinite (unlimited).
It's infinite. But let me ask you this. Are you familiar with the graph of the function [math]y = \frac{1}{x}[/math]? Isn't it infinite on the right? You can go out as far as you like, right? Did you need me to copy a picture from the web? No matter how far you go, like x = a zillion, there's a point on the graph at y = 1/zillion. Right? It's the cross-section of Gabriel's horn. We can do the integration to see that the area is infinite. And if you don't like calculus, there's a simple visual demonstration that I could provide.
Is that what you're objecting to? That the area under 1/x from 1 to infinity is infinite? Or what mathematical fact are you objecting to?
Clearly the premise is that it is unlimited, infinite, and any mathematical axioms which deal with it by imposing a limit, are not truthfully adhering to the premise.
You're equivocating the word limit, as in "limited," versus the mathematical theory of limits to infinity. A cheap rhetorical trick. How can you accept confusing yourself like this? Surely you know better. Or you could just ask.
Is that what you're objecting to? That the area under 1/x from 1 to infinity is infinite? Or what mathematical fact are you objecting to?
I'm objecting to the method employed by the person in the YouTube clip, which replaces the stated infinite limit, (approaches zero) with zero, as I referenced above. By that method, the equation for the volume of the horn resolves to pi, as stated in the op. If you have a method to figure out the volume of the horn without that substitution, then you might present it. If not, then we probably don't disagree, and we're just wasting our time talking past each other.
I suggest that the proper representation is that the volume is necessarily indefinite, rather than finite, and there is no paradox. This means that the amount of paint required to fill the horn cannot be determined. Therefore no act of pouring a determined amount of paint into the horn will fill it
Deleted UserFebruary 22, 2021 at 17:15#5021230 likes
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Metaphysician UndercoverFebruary 22, 2021 at 18:54#5021460 likes
I'm objecting to the method employed by the person in the YouTube clip, which replaces the stated infinite limit, (approaches zero) with zero, as I referenced above. By that method, the equation for the volume of the horn resolves to pi, as stated in the op.
If I'm not mistaken, the part you object to is when he's proving that the cross-sectional area is infinite; that is, the area under 1/x from 1 to infinity. I could be wrong, I only looked at a couple of seconds of the vid. Either way though, I already explained that he's implicitly using the extended real number system as a shorthand for the cumbersome use of limits.
If you have a method to figure out the volume of the horn without that substitution, then you might present it. If not, then we probably don't disagree, and we're just wasting our time talking past each other.
That was already clear two years ago. But I'm surprised to see you still complaining about something I already explained to you -- that we're working in the extended reals -- and that you're rejecting freshman calculus. I must admit you're making progress ... from being confused about 2 + 2 = 4 to being confused about integrating 1/x.
This means that the amount of paint required to fill the horn cannot be determined.
So you don't agree with the determination of the volumes of solids of revolution. Ok. Whatever. An intellectual nihilist. Throw out all the engineering, all the physics, all the physical science simply because you aren't willing to take the time to understand it.
Therefore no act of pouring a determined amount of paint into the horn will fill it
Yeah yeah.
ps -- You know, if you said, "Modern calculus does get the right answers and it's useful for physics and engineering; but the mystery of the ultimate nature of infinitesimal quantities, whether mathematical or physical, is not satisfactorily addressed by the formalism," that would be an intelligent criticism.
But to deny the mathematical result of computing the volume of the solid of revolution ... that's just ignorance for its own sake.
TheMadFoolFebruary 23, 2021 at 03:09#5022580 likes
If I'm not mistaken, the part you object to is when he's proving that the cross-sectional area is infinite; that is, the area under 1/x from 1 to infinity.
The part I am bringing your attention to (and I'm not objecting to it, I'm bringing your attention to it, as relevant to the so-called paradox), is where he determines the limits to the radius of the horn. The radius is given as represented by y, which is equal to 1/x. Then when he plugs the values into the equation, 1/x becomes 1/infinity, which he says is zero.
TheMadFoolFebruary 23, 2021 at 03:26#5022660 likes
Sorry timwood because right now I feel like a drunk driver being asked to conduct a sobriety test on himself.
As a civil libertarian I am always conflicted. On the one hand, sobriety checkpoints are unconstitutional as they are a search without probable cause. On the other hand as a driver I'm perfectly happy to get some drunks off the road.
The part I am bringing your attention to (and I'm not objecting to it, I'm bringing your attention to it, as relevant to the so-called paradox), is where he determines the limits to the radius of the horn. The radius is given as represented by y, which is equal to 1/x. Then when he plugs the values into the equation, 1/x becomes 1/infinity, which he says is zero.
[math]\frac{1}{\infty} = 0[/math] in the extended real number system, which is always the implicit domain of integration problems. The Numberphile guy didn't mention it since it's taken for granted: either glossed over, in the case of freshman calculus; or explicitly formalized, in the case of a more rigorous class in real analysis. Either way it's perfectly rigorous. We could live without it by substituting the phrase, "increases without bound" instead of infinity, and plugging in limits as needed. But that's far more messy and confusing than simply defining the extended real numbers as a notational shorthand.
And to be fair, and to hold you to your own words, you ARE objecting, because you have denied that the volume of the horn is pi, when in fact it is exactly pi.
TheMadFoolFebruary 23, 2021 at 03:45#5022710 likes
As a civil libertarian I am always conflicted. On the one hand, sobriety checkpoints are unconstitutional as they are a search without probable cause. On the other hand as a driver I'm perfectly happy to get some drunks off the road.
Get to the point if you don't mind me saying. Either tell me where I'm wrong or stop wasting your time. :smile:
:ok: :up: By the way, what's "normal programming"? Do you have one yourself? And also, you haven't gotten round to pointing out the error, if there's one, in my argument. Please focus on the issue.
1?=01?=0 in the extended real number system, which is always the implicit domain of integration problems.
So, as I said this implies a zero radius and therefore closure of the horn. That's the reason for the appearance of a paradox. To figure the volume of the horn requires that zero is taken as the limit, rather than the unlimited "infinity".
And to be fair, and to hold you to your own words, you ARE objecting, because you have denied that the volume of the horn is pi, when in fact it is exactly pi.
The volume of the horn is figured to be pi only when the infinitely small radius, as stipulated by the premise, is taken to be zero, as required for calculating the volume.
TheMadFoolFebruary 23, 2021 at 03:56#5022810 likes
So, as I said this implies a zero radius and therefore closure of the horn. That's the reason for the appearance of a paradox. To figure the volume of the horn requires that zero is taken as the limit, rather than the unlimited "infinity".
No, the horn is not closed. There are many online calculus tutorials and classes available that explain the theory of limits.
The volume of the horn is figured to be pi only when the infinitely small radius, as stipulated by the premise, is taken to be zero, as required for calculating the volume.
That's now how limits are defined. What's literally true here is that as [math]a[/math] gets arbitrarily large, the volume of the solid between 1 and [math]a[/math] gets arbitrarily close to [math]\pi[/math]. Since that is the case, we say that the limit is [math]\pi[/math]. That's the definition of a limit.
You mentioned sobriety checkpoints, and I went off-topic for a moment to express my ambivalence about them in my dual societal role as both a civil libertarian and a driver. Like the scene in Full Metal Jacket where Joker is confronted by a Colonel for wearing a peace sign and having "Born to Kill" written on his helmet, and he responds, "The duality of man. The Jungian thing Sir!" That was another little off-topic excursion.
And now that you mention it ... perhaps that's what Gabriel's horn is ultimately about. A shape that reflects aspects of both the finite and the infinite. I think Jung would approve.
So, as I said this implies a zero radius and therefore closure of the horn. That's the reason for the appearance of a paradox.
No, the reason for the appearance of a paradox is that the shape has finite volume and infinite area; that those are two completely different kinds of things; and that our intuitions about paint "connect" the two. We just "know" that if we buy a gallon of paint and start painting the walls, we'll run out at some point and cover a certain area of the walls. But the reason that is, as already mentioned by @andrewk early on in this thread (and apparently severely underappreciated), is that painting areas with paint requires some thickness of paint.
You make it sound like the problem is that Gabriel's horn stretches out for an infinite while, but that's actually a red herring. You get the same exact problem with any shape of finite inner volume/infinite area, such as extruding a Koch snowflake and giving it a bottom. The only reason Gabriel's horn is noteworthy is that it's a curiosity with a built in toy homework exercise for people taking a calculus class.
TheMadFoolFebruary 23, 2021 at 04:18#5022940 likes
No, the horn is not closed. There are many online calculus tutorials and classes available that explain the theory of limits.
It's quite simple. The radius is y, which is represented at the one limit as y=1/x=0. From someone who adamantly argues that = means "the same as", I don't see that you have anything to argue. The diameter is taken to be 0 at that point in the solution. This means that the horn is closed at that point, represented within that method of figuring out the volume. There's nothing to discuss, it's clear and obvious.
The nature of "a limit" is irrelevant and just a ruse. What matters is the values that are plugged into the equation, the values which bring about the apparent paradoxical conclusion. If y=0 then the radius is zero at that point, which is what is employed in that method. If the premise is that the radius never reaches zero, then the solution does not truthfully adhere to the premise.
No, the reason for the appearance of a paradox is that the shape has finite volume and infinite area...
The shape is only said to have finite volume because of the method employed to determine the volume. As explained above, this method assumes a point where the radius of the shape is zero. Therefore this method contradicts the premise of the problem, which states that the horn continues infinitely without reaching a zero radius.
If we were to properly consider that infinite extension of the horn, then despite the fact that it is an infinitely small circumference, being an infinitely long extension of that circumference gives us an infinitely large volume. That volume could not ever be filled. Quoting InPitzotl
But the reason that is, as already mentioned by andrewk early on in this thread (and apparently severely underappreciated), is that painting areas with paint requires some thickness of paint.
Andrewk does not provide a solution. The inside of the horn has a non-zero diameter with infinite extension. This means that there is an infinite surface area on the inside of that horn, to be covered with paint, just like there is on the outside.
If the argument is that the thickness of the paint prevents it from going into that tiny channel, then we're just arguing physical properties, which has already been dismissed, as not what is to be discussed. Therefore if we are assuming that the paint can go around the tiny surface on the outside, it can also go around the tiny surface on the inside, because we are not discussing the physical properties which would prevent the thought experiment. If that were the case, we'd just reject the whole idea of an infinitely long horn as ridiculous in the first place.
The shape is only said to have finite volume because of the method employed to determine the volume.
The questioning of the shape's volume is only said to be problematic because the questioner thinks that the questioner knows what he is talking about. Quoting Metaphysician Undercover
As explained above, this method assumes a point where the radius of the shape is zero. Therefore this method contradicts the premise of the problem, which states that the horn continues infinitely without reaching a zero radius.
The method makes no such assumption.
[math]\lim_\limits{x\rightarrow \infty}{f(x)}=L \Rightarrow \forall \epsilon >0, \exists M:x > M \Rightarrow L-\epsilon \leq f(x)\leq L+\epsilon[/math]
In other words:
[math]\lim_\limits{x\rightarrow \infty}{f(x)}=0 \Rightarrow \forall \epsilon >0, \exists M:x > M \Rightarrow -\epsilon \leq f(x)\leq \epsilon[/math]
Limits don't need points at infinity to work. Quoting Metaphysician Undercover
Andrewk does not provide a solution. The inside of the horn has a non-zero diameter with infinite extension. This means that there is an infinite surface area on the inside of that horn, to be covered with paint, just like there is on the outside.
So? Areas have no volume. The paint analogy tricks you into thinking they at least relate when, in fact, they don't. A cubic foot of paint is roughly 7.5 gallons. A gallon of paint can paint about 400 square feet of wall; thus our cubic foot can paint about 3000 square feet of wall. Such paint has a specific layer width of 1/3000 feet. The intuitive equating of a particular area of paint to a particular volume requires such a specific nonzero layer; in the case of actual paint, 1/3000 feet thick layer.
But that inside Gabriel's horn you refer to is less than 1/3000 units across beyond 3000; so if you fill it, you're filling it with less than a layer's worth. It's less than a thousandths of a layer's worth beyond 3,000,000; less than a billionth beyond 3,000,000,000,000. Quoting Metaphysician Undercover
If the argument is that the thickness of the paint prevents it from going into that tiny channel, then we're just arguing physical properties, which has already been dismissed, as not what is to be discussed.
The physical properties are implied by the intuitions, so they're imported by a back door; the paradox is always phrased about paint filling versus painting the horn... that's tricking you to use your intuitions of paint to compare a volume ("filling") to an area ("painting"). You don't need a shape extending into the infinite to make this paradox "work"; you just need a finite volume/infinite area, and to be tricked into thinking areas relate to volume. The actual thickness of the surface of Gabriel's horn is 0 units, so that infinite surface area actually doesn't consume any meaningful volume at all. Comparing the infinite surface area to the finite volume is simply a false comparison.
Deleted UserFebruary 23, 2021 at 15:26#5024140 likes
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Deleted UserFebruary 23, 2021 at 16:05#5024160 likes
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In terms of the so-called paradox, do you not agree that if the paint could fill the horn, then necessarily that same volume of paint paints the surface (boundary) of the horn?
Because filling the inside means the inside is painted,
If filling the inside means the inside is painted, then there's no positive minimal thickness of paint required to paint things; i.e., paint can be 0 thick (quick proof; assume there is such a thickness t... then if you go out 1/t on the horn, and toss the finite part, you're left with a horn whose insides are too thin to paint). At that point you're basically just mapping points to points, and there are plenty enough points in a tiny droplet of paint to map to the infinite surface area of the horn.
If you just think of this as whether or not you can spread a drop of mathematical paint indefinitely thin, then I don't quite see any intuitive conflict to build a paradox from.
Deleted UserFebruary 23, 2021 at 19:54#5024580 likes
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Metaphysician UndercoverFebruary 24, 2021 at 03:33#5025860 likes
But that inside Gabriel's horn you refer to is less than 1/3000 units across beyond 3000; so if you fill it, you're filling it with less than a layer's worth. It's less than a thousandths of a layer's worth beyond 3,000,000; less than a billionth beyond 3,000,000,000,000.
I don't see how this is relevant. Are you forgetting that it's infinitely long? It doesn't really matter how many time less than a layers worth of paint you're putting in there, it's infinitely long, so whatever the layer is, it will be multiplied by infinity, and therefore enough paint to make an infinite number of those "less than layers worth" layers on the inside of that horn..
I don't see how this is relevant. Are you forgetting that it's infinitely long?
You've got this backwards. I have 400 square feet of wall coated with 1/3000th of a foot of paint. How much paint is that? Well, given that the entire area is covered with a 1/3000th foot layer, we can just multiply 400 by 1/3000 and we get about 1/7.5. Now what's this you were saying about the horn? Quoting Metaphysician Undercover
It doesn't really matter how many time less than a layers worth of paint you're putting in there, it's infinitely long, so whatever the layer is, it will be multiplied by infinity,
...ah, yes. Tiny bit of a problem you have there, though. It doesn't matter how many times less than a layer's worth of paint you've got inside, the horn can only have an inner layer that thick if it's thick enough on the inside. And only a finite portion of the horn so qualifies. The rest of the infinite horn (the infinite portion) is too small to have an inner layer that thick. You can't just multiply your paint's thickness by infinity if it isn't covering the infinite area. So where are you getting this crazy notion that you can multiply your layer's thickness by infinity?
TheMadFoolFebruary 24, 2021 at 05:03#5025950 likes
It doesn't matter how many times less than a layer's worth of paint you've got inside, the horn can only have an inner layer that thick if it's thick enough on the inside. And only a finite portion of the horn so qualifies.
The physical properties of the paint being incompatible with an infinite horn, was already rejected as not the subject of this discussion. If we were discussing whether the molecules of paint could fit down inside an infinitely small tube, we might just as well reject the infinitely small tube as a nonsensical proposition in the first place.
The physical properties of the paint being incompatible with an infinite horn, was already rejected as not the subject of this discussion. If we were discussing whether the molecules of paint could fit down inside an infinitely small tube, we might just as well reject the infinitely small tube as a nonsensical proposition in the first place.
Andrewk does not provide a solution. The inside of the horn has a non-zero diameter with infinite extension. This means that there is an infinite surface area on the inside of that horn, to be covered with paint, just like there is on the outside.
so whatever the layer is, it will be multiplied by infinity,
That is what we're discussing. You made a claim that there's some number, you call it "whatever the layer is", that you multiply by infinity. The problem is, there's only a finite portion of the horn that can fit "whatever the layer is", and there's an infinite portion of the horn too thin to have "whatever the layer is" be the thickness of "whatever the layer is". Because of this, you cannot multiply "whatever the layer is" by infinity and get anything meaningful.
You're the one who keeps dragging physical properties of paint into this. I'm just talking about your wrongness; your "whatever the layer is" that you think you get to multiply by infinity.
We could be having the same argument about your ability to fit a y by y by x rectangular prism in the center of infinite volume (V=y*y*x, with x being "infinitely long"). There's no such prism you can fit in the horn; no matter what your y is, it will only go in so far. The only way you get to have an infinitely long prism in there is if y is 0; and 0*0*infinity is indeterminate.
Metaphysician UndercoverFebruary 24, 2021 at 23:39#5027930 likes
The problem is, there's only a finite portion of the horn that can fit "whatever the layer is", and there's an infinite portion of the horn too thin to have "whatever the layer is" be the thickness of "whatever the layer is". Because of this, you cannot multiply "whatever the layer is" by infinity and get anything meaningful.
Obviously, if we can assume that the horn is infinitely long, with an infinitely small diameter, we can also assume that the paint can go infinitely thin. That this is not an issue of the real physical properties of a real horn, or the real physical properties of real paint was determined in the first couple of replies in the thread. Do you agree that if the horn is allowed to go infinitely thin, then the paint must play by the same rules, and be allowed to go infinitely thin as well?
Obviously, if we can assume that the horn is infinitely long, with an infinitely small diameter, we can also assume that the paint can go infinitely thin.
You're kind of mixing two things in here. Imagine a mathematical bag; inside the bag, we'll put all finite numbers. All of them, mind you, but only the finite ones. There are an infinite number of numbers in this bag, but infinity isn't in the bag. So if we admit that the horn is infinitely long, we're not necessarily admitting that there's an infinity-point on the horn (a point where there's an infinitely small diameter).
But if you want to talk about an infinite point on the horn, you're free to do so (it's just not implied by admitting the horn is infinitely long). You can even say the horn is infinitely thin at that point. But that doesn't help you here: Quoting Metaphysician Undercover
so whatever the layer is, it will be multiplied by infinity,
...because now you're talking about an infinitely tiny quantity multiplied by infinity. And that's still not meaningful. Quoting Metaphysician Undercover
Do you agree that if the horn is allowed to go infinitely thin, then the paint must play by the same rules, and be allowed to go infinitely thin as well?
At that point you're basically just mapping points to points, and there are plenty enough points in a tiny droplet of paint to map to the infinite surface area of the horn.
In other words you want the paint to follow different rules for the inside of the horn, than for the outside of the horn, allowing a finite volume of paint to cover an infinite surface area on the inside, but not the outside. What's the point, if you can just make up whatever rules you want?
Why can't we just do the same thing with the infinite surface area on the outside of the horn as well? We can just say that any tiny amount of paint can be spread out over the infinite outside surface area, just like you say for the inside.
In other words you want the paint to follow different rules for the inside of the horn, than for the outside of the horn, allowing a finite volume of paint to cover an infinite surface area on the inside, but not the outside.
Wrong. You're only confusing yourself here. I haven't specified any rules for paint at all, much less different rules for the inside and outside. Rather, I've talked about three things:
(A) Intuitive paint, which is based on real life paint (not necessarily physics; a painter can get by just fine if he believes atomic theory is a conspiracy, so long as he follows the rules of intuitive paint whereby he calculates he needs 8 gallon cans to paint 3000 square feet).
(B) tim wood's paint, which he explicitly said wasn't intuitive paint and must meet particular conditions
(C) MU's paint, a strange kind of thing that supposedly lets us multiply some number by infinity and get some meaningful conclusion out of it
(A) is where all of the confusion sets in; it's why we think there's a paradox when there is in fact nothing here. When I use my gallon of paint to paint a 20 foot by 20 foot square, and I think that the thing I painted was an area, I've just tricked myself into thinking volumes (gallons) relate to areas (square feet); namely, that 1 gallon=1/7.5 cubic feet relates to 400 square feet. But that intuitive relation is illusory; the 20x20 swatch is actually 1/3000 foot thick, making that paint layer a volume not an area. Gabriel's horn has infinite surface area, but holds a finite volume. But as I've said repetitively, areas have no volume. "An infinite number of square feet" conveys no meaningful amount of cubic feet.
(B) is some new kind of paint tim wood was proposing; whatever that was, it's something that by filling the horn counts as painting its inside. That's the discussion I linked you to. But that sort of paint necessarily must count a width 0 layer of paint (or if you prefer, infinitely thin) as painting the inside surface. There's no separate rule for the outside; (B) kind of paint is just as good for the outside as the inside, and you don't even need pi paint to paint either... any tiny droplet would do. In fact, any volume of paint would paint any area, even if the volume is small and the area infinite.
Then we have C paint, whereby MU is trying to justify that multiplying some number by infinity means that volumes really do relate to areas, and/or demonstrate that he should not have to understand a conversation before he pretends he's contributing to it.
Metaphysician UndercoverFebruary 25, 2021 at 11:56#5029700 likes
Gabriel's horn has infinite surface area, but holds a finite volume.
The volume of the horn is only determined as finite when the infinite radius is rounded off to zero at some determinable length, as is demonstrated by the YouTube video. Otherwise the infinite length of the horn ensures that the volume is infinite.
The volume of the horn is only determined as finite when the infinite radius is rounded off to zero at some determinable length, as is demonstrated by the YouTube video.
Wrong. The video used limits (and integrals, which are built off of limits). Limits don't round off to zero.
Here's how a limit works:
[math]\lim_\limits{x \rightarrow \infty}{\frac{1}{x}}=0[/math]
...and it's equal to 0 exactly... not rounded off, because L=0 meets the conditions:
[math]\lim_\limits{x \rightarrow \infty}{\frac{1}{x}}=L \Rightarrow \forall \epsilon > 0, \exists M : x > M \Rightarrow L - \epsilon < \frac{1}{x} < L + \epsilon[/math]
...and that can be shown generically for any ?. For such ?, simply choose [math]M=\frac{1}{\epsilon}[/math].
No other number works for that limit; a billionth doesn't work for example; ?=two billionths betrays it, because for all x's greater than two billion, [math]\frac{1}{x} < \frac{1}{2,000,000,000}[/math] which is more than two billionths away from one billionth.
Likewise, every close to 0 integer doesn't work; only 0 exactly works.
Clearly you are using a different calculation than the one in the video then. If you know of a method to figure out the volume of that horn, which avoids rounding off the infinitely small dimeter to zero, then maybe you should present it for us.
...is just uninformed non-sense.
Metaphysician UndercoverFebruary 26, 2021 at 03:22#5031890 likes
Reply to InPitzotl
Sure chief, one over infinity is zero, and that's not a matter of rounding off. Tell me another.
Sure chief, one over infinity is zero, and that's not a matter of rounding off.
At this point, it's just denial, and you're unqualified to continue this discussion with. Hardly surprising, given this is the same exact thing you failed to grasp in the other thread.
Metaphysician UndercoverFebruary 26, 2021 at 13:52#5032680 likes
Reply to InPitzotl
LOL! Look who's in denial!
Saying that 1/infinity equals zero is obviously an instance of rounding off. There's nothing wrong with rounding off. We do it all the time with pi, square roots, etc.. That's how we get the job done by rounding off. If we couldn't round off, we couldn't get the job done in many instances. So you shouldn't be embarrassed by it. You should be embarrassed by insisting that it's not an instance of rounding off, when it clearly is, though.
Saying that 1/infinity equals zero is obviously an instance of rounding off.
It's right there under your nose and you can't see it. You read:
[math]\lim_\limits{x \rightarrow \infty}{\frac{1}{x}} = 0[/math]
...as "saying 1/infinity equals 0". But that's not what it says, and it's not what it means. I told you what it means, and showed you a link. Quoting Metaphysician Undercover
There's nothing wrong with rounding off. We do it all the time with pi, square roots, etc. That's how we get the job done by rounding off. If we couldn't round off, we couldn't get the job done in many instances. So you shouldn't be embarrassed by it.
Okay, so there's nothing wrong with rounding off. And okay, we do it all of the time. So what? It's cute and all that you're trying to "counsel" me so that I can "cope" with rounding off, but your projection of some imagined psychological trauma is a red herring. Limits still aren't rounding off. Quoting Metaphysician Undercover
You should be embarrassed by insisting that it's not an instance of rounding off, when it clearly is, though.
Clearly it's not rounding off, though. Clearly, you just don't understand what a limit is. And that's okay, MU. Not understanding something isn't the end of the world. It's nothing to be embarrassed about; there's a lot of knowledge in the world and not everyone knows everything. There's nothing to be embarrassed about by admitting that you don't know something. But you should be embarrassed by insisting that you understand when, clearly, you don't.
Rounding off implies that there's a stated answer a, and a real answer b, and that a is not b but is "close enough" to it.
But there's no such thing as an [math]L \neq 0[/math] such that:
[math]\forall \epsilon > 0, \exists M : x > M \Rightarrow L - \epsilon < \frac{1}{x} < L + \epsilon[/math]
There is, however, such a thing as an L such that this condition is met; namely, L=0.
So the limit here is met by the value 0 exactly. This is a binary thing; either something works, aka meets the definitive criteria, or it doesn't work, aka it doesn't meet the criteria. 0 falls in the "meets the criteria" camp; it's the real answer b. "0 approximately" falls in the doesn't meet the criteria camp. There's genuinely no rounding off here. There is only an infinite amount of MU confusion.
Metaphysician UndercoverFebruary 27, 2021 at 12:21#5036280 likes
t's right there under your nose and you can't see it. You read:
limx??1x=0limx??1x=0
...as "saying 1/infinity equals 0". But that's not what it says, and it's not what it means. I told you what it means, and showed you a link.
I saw it, and heard it distinctly stated, on the Youtube video at the referenced point, he distinctly says "well one over infinity that's zero, so you get nothing from that". And, you said: your method is the same as the one on the video. You can insist, as fishfry stated, that this is the convention in such procedures, to take one divided by infinity as zero. But that is to round off, so we need to respect the fact that the solution to the question of the volume of the horn, is a rounded off solution.
"Infinite" means unlimited. When you apply limits to the unlimited, you are either contradicting or rounding off. Which charge do you prefer? If you will not accept the fact that you are rounding off, then the paradox arises due to the contradiction.
Rounding off implies that there's a stated answer a, and a real answer b, and that a is not b but is "close enough" to it.
Yes, and that describes exactly what is the case with the volume of Gabriel's horn. The "real answer" is that the horn is, stipulated by the stated premise, as infinitely long. And, the volume of an infinitely long container cannot be determined. The "close enough" answer is produced by applying the limits of integration, which works with approximations. The appearance of a paradox arises, because "close enough" fails the task intended by the proposition "infinite". In other words, there is no such thing as "close enough" when we're talking about the infinite.
To impose a limit on the infinite is to contradict. So it is not the determined value, 0, which is wrong. That value is correct according to the terms of application, but the judgement that the situation is suited to the application, is wrong. The example stipulates that the horn is of infinite length. Therefore to impose a limit on its length is a mistaken procedure.
In common practice, this is not a problem. We don't often encounter infinitely long things which we want to figure the volume of. But when we encounter an infinitely long thing proposed in a thought experiment, it is of paramount importance that we stay true to the premises, and have one consistent way of interpreting "infinite", or else someone will claim that the results are paradoxical.
It's seems quite simple to take the easy way out and suggest the divine link to the infinite - archangel Gabriel trumpet to announce Judgement Day and the finite number that the divine will take or make it through Judgement...there's not a paradox as such that brings doubt and uncertainty, there is instead the finite within the infinite and vice versa the infinite in the finite...the concept of the divine is difficult to grasp or imagine because it goes beyond our imagination as finite humans...those who believe in a soul understand that this is a connection with the infinite within the finite...as for paint that's another substance that has a finite life cycle....
The mathematics others have presented ...still complex and fascinating and obviously I haven't watched the YouTube clip...
Sure. But remember the bag I was talking about earlier? That bag has all of the finite numbers in it, but no infinite numbers. This is the situation here. The bag has real numbers in it; there's no such thing as an infinite real number though. Nevertheless, "unlimited" describes the extent of such numbers on the number line in the positive direction. And that ? symbol when used in the limit is used to represent just that... that's not a number, it's just shorthand for representing the infinite extent of the numbers in my bag.
For example, a lower limit of 1 and an upper limit of 2 refers to "all of the numbers in the bag that are greater than or equal to 1, and less than or equal to 2". By contrast, a lower limit of 1 and an upper limit of ? simply means: "all of the numbers in the bag that are greater than or equal to 1". Quoting Metaphysician Undercover
When you apply limits to the unlimited, you are either contradicting or rounding off.
If you will not accept the fact that you are rounding off, then the paradox arises due to the contradiction.
Those things don't follow, there isn't a paradox here in the first place, and anyone trying to play the if-you're-wrong-that-means-I'm-right card should have their philosophy license revoked. Quoting Metaphysician Undercover
that describes exactly what is the case with the volume of Gabriel's horn
Wrong. The real issue is very simple... areas aren't volumes. "Paint" tricks you into thinking they are. It is very interesting to note that you never actually tried to address this explanation, just as you never commented on the extruded Koch snowflake with a bottom having the same "issue". It's no wonder you're trying to play the if-you're-wrong-that-means-I'm-right card. Quoting Metaphysician Undercover
To impose a limit on the infinite is to contradict.
That sounds like a confused equivocation. A 1x1x1 cube by definition is 1 cubic units of volume, but it has an infinite number of points. It's not a contradiction to say that it has an unlimited number of points but a limited volume. Gabriel's horn has an unlimited extent into the x axis in the positive direction; that means it is unlimited... in extent... along the x axis. And that's it. It doesn't mean that the horn surrounds an infinite volume, as your equivocation is apparently meant to imply, any more than the fact that the 1x1x1 cube contains an infinite number of points suggests it should have an infinite volume.
Metaphysician UndercoverFebruary 27, 2021 at 23:20#5037970 likes
Yes, apply a limit to what is stipulated by the premise, as without limit. That is the mistake. Can't you see that it is stipulated that there is no limit to the length of the horn, therefore to apply a limit is to contradict the premise?
For example, a lower limit of 1 and an upper limit of 2 refers to "all of the numbers in the bag that are greater than or equal to 1, and less than or equal to 2". By contrast, a lower limit of 1 and an upper limit of ? simply means: "all of the numbers in the bag that are greater than or equal to 1".
This is not relevant either. It is stipulated in the Gabriel's horn example, that there is no lower limit. It is stipulated that the horn continues infinitely. That means no limit. It is therefore not a case of having an upper limit and a lower limit, and to represent it as such is a mistake. To impose a lower limit (such as zero) is to contradict the premise of the example. We are not talking about how many numbers there are between two numbers here, we are talking about an unlimited length. To impose a limit on that length, a point where the diameter of the horn reaches zero, is to contradiction the premise of the example.
Wrong. The real issue is very simple... areas aren't volumes. "Paint" tricks you into thinking they are. It is very interesting to note that you never actually tried to address this explanation, just as you never commented on the extruded Koch snowflake with a bottom having the same "issue". It's no wonder you're trying to play the if-you're-wrong-that-means-I'm-right card.
I did addressed this. If the horn can go infinitely thin, then so can the paint. They must play by the same rules.
That sounds like a confused equivocation. A 1x1x1 cube by definition is 1 cubic units of volume, but it has an infinite number of points.
Again, we're not talking about an infinite number of points within a confined space, so that is irrelevant. The horn is infinitely long therefore there is no confined space. If someone were to say to you that there is an infinite extension of the universe, would you think that this implies a confined space? If someone says to you, take this line which continues infinitely, would you think that they were talking about a line segment which extends between two points?
This seems to be where your misunderstanding lies. You want to make this into an issue of a confined, "limited" space, but it is clearly stipulated that the horn is infinitely long, therefore there is no such confined space.
TheMadFoolFebruary 28, 2021 at 00:31#5038260 likes
Surface area of a cylinder A = 2 * pi * r * h where r is the radius and h is the height.
Volume of a cylinder V = pi* r^2 * h where r is the radius and h is height
r approaches zero and h approaches infinity
A = 2 * pi * r * h = 2 * pi * (r approaching zero) * (h approaching infinity)
V = pi * (r approaching zero) * (r approaching zero) * (h approaching infinity)
Yes, apply a limit to what is stipulated by the premise, as without limit. That is the mistake. Can't you see that it is stipulated that there is no limit to the length of the horn, therefore to apply a limit is to contradict the premise?
I can see that you're equivocating. You're confusing "limit" as a method with "limit" as a point beyond which you don't go. Quoting Metaphysician Undercover
You cannot put all the finite numbers in a bag, because there is an infinite quantity of them.
Of course it's relevant. The 1/? that you're whining about is the 1/? in the video at 6:20. Right? That 1/? is 1/x with ? substituted in it. That 1/x is the 1/x from [math]\pi \frac{-1}{x}[/math]. And that is from [math]\int_{1}^{\infty}{\pi \frac{1}{x^2}dx}[/math]. And that is a Riemann integral; it's applied over the range where we want to take this volume. For Gabriel's horn, the lower limit here is 1, and there is no upper limit. To simply plug in infinity here as if it's a number is to say something different than what you yourself said in the prior post... that "infinite" simply means unlimited.
According to the rules of the method being applied, this is an improper integral. For improper integrals of this type, the calculation for the infinite portion is performed using a limit, whose definition I gave earlier. Quoting Metaphysician Undercover
It is stipulated in the Gabriel's horn example, that there is no lower limit.
If there's no lower limit, then what's up with this big giant yellow arrow pointing to the lower limit during the segment where Gabriel's horn is defined?:
https://youtu.be/yZOi9HH5ueU?t=56
...not to mention Tom saying straight up, "and what we do first of all, is we're going to chop it here at 1"? Quoting Metaphysician Undercover
I did addressed this. If the horn can go infinitely thin, then so can the paint. They must play by the same rules.
No, you didn't address this, because if this were indeed the case... if you could paint infinitely thin, then you can paint an infinite area with a finite amount of paint. And if you can do that, then there's no paradox. And you can do that, and there is no real paradox. But such infinitely thin paint simply becomes an empty metaphor... it's equivalent to what I was saying here. But you replied to that post saying that it wasn't the "real answer", and as recently as here you were peddling this one: Quoting Metaphysician Undercover
The "real answer" is that the horn is, stipulated by the stated premise, as infinitely long
But your proposed "real answer" doesn't address the extruded Koch snowflake, because that isn't infinitely long. The thing you're proposing isn't the real answer does address the Koch snowflake, because the infinite area on its perimeter is not a volume. Quoting Metaphysician Undercover
Again, we're not talking about an infinite number of points within a confined space...This seems to be where your misunderstanding lies. You want to make this into an issue of a confined, "limited" space, but it is clearly stipulated that the horn is infinitely long, therefore there is no such confined space.
You're reasoning by equivocation; "the extent is infinite, therefore the volume is infinite" simply doesn't follow.
Metaphysician UndercoverFebruary 28, 2021 at 01:48#5038670 likes
No, you didn't address this, because if this were indeed the case... if you could paint infinitely thin, then you can paint an infinite area with a finite amount of paint.
That's not true at all. Your interpretation of "infinite" is dreadful. You cannot paint an infinite area regardless of how much paint you have, because no matter how much painting you do there is always more to be painted. That's the issue with Gabriel's horn. It's infinitely long, so no matter how much paint you pour in the top, it never reaches the bottom.
It doesn't matter what you propose as the volume of the horn, you still cannot fill it with paint . Suppose you conclude it's 3.1 gallons,. You pour that in, but you haven't filled the horn because it hasn't reached the bottom.
But your proposed "real answer" doesn't address the extruded Koch snowflake, because that isn't infinitely long. The thing you're proposing isn't the real answer does address the Koch snowflake, because the infinite area on its perimeter is not a volume.
We're discussing Gabriel's horn not snowflakes. How my answer relates to a snowflake is irrelevant. I don't see why you feel the need to bring up so many irrelevant issues.
Come on Pitzotl, you're reaching for straws. Get back to the subject.
But the subject is the paradox of Gabriel's horn; it's literally the title of this thread. Gabriel's horn is an object defined using algebraic geometry. Algebraic geometry defines points in a space using coordinates using number lines. Number lines are defined with real numbers.
And that's what this bag is; it's the real numbers.
Ironically, your accusation and your imperative to me to get back to the subject is itself an avoidance of discussing this subject. Quoting Metaphysician Undercover
You cannot paint an infinite area regardless of how much paint you have, because no matter how much painting you do there is always more to be painted. ... Suppose you conclude it's 3.1 gallons,. You pour that in, but you haven't filled the horn because it hasn't reached the bottom.
These are non-issues. In the paradox that is the subject of this thread, we're concerned primarily with the quantities not the length of the tasks. These are mathematical spaces; in the span of 30 seconds we define the entire infinite horn in an algebraic geometry space... doing infinite things is certainly not a problem here. Your imagined alleged problem has a lot more problems than you're letting on... the 3.1 gallons not going to the bottom is child's play. How are you upturning the horn in a gravitational field, and where do you put the planet? How does that planet manage to exert a field on the top of the horn anyway? But the biggest and most relevant question of all here is... do these really sound like mathematical questions? Quoting Metaphysician Undercover
We're discussing Gabriel's horn not snowflakes. How my answer relates to a snowflake is irrelevant.
We're discussing the presumed paradox of Gabriel's horn. The presumptions that appear to introduce the conflicts is the point of the thread. Those presumptions are based on the fact that the quantity of the surface are "on the outside" is infinite while the quantity of volume "on the inside" is finite... the "outside-ness" of the surface of which is really an irrelevant detail that is part of the intuitive distraction of using paint to compare areas to volume. You are presuming that the paradox is solved by questioning the volume, but you haven't even shown a good reason to doubt the volume much less the flaw in the presumptions leading to the paradox.
Gabriel's horn is an object defined using algebraic geometry. Algebraic geometry defines points in a space using coordinates using number lines. Number lines are defined with real numbers.
Nonsense. This has nothing to do with algebraic geometry. G's Horn is elementary calculus. :roll:
You guys should just let this go and get back to epistemological metaphysics where accuracy is optional.
Wolfram:
Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of polynomials, and the geometry is the set of zeros of polynomials, called an algebraic variety. For instance, the unit circle is the set of zeros of [math]x^2+y^2=1[/math] and is an algebraic variety, as are all of the conic sections.
Gabriel's horn is the algebraic variety defined by the polynomial z^2+y^2=(1/x)^2 starting at x=1. Quoting jgill
G's Horn is elementary calculus.
Sure, you use calculus to analyze the surface area of and volume surrounded by this object, as they did in the video. But that doesn't preclude the fact that you're studying geometric properties of an algebraic variety.
Metaphysician UndercoverFebruary 28, 2021 at 13:13#5040150 likes
These are non-issues. In the paradox that is the subject of this thread, we're concerned primarily with the quantities not the length of the tasks.
This is where your mistaken. The paradox assumes a spatial form, created from mathematical principles, and names this form Gabriel's horn. Therefore the subject is this form which is created by mathematics, not the quantities which are used to create it. You need to divorce yourself from the numbers, from the mathematics, look at the form described, directly, and ask how is it to be measured.
This might be why you seem to be having so much difficulty understanding the problem. You want to use the same numbers which create the form, to measure the form. But when infinity is induced, this represents a failing in the capacity of the numbering system, so it is impossible to measure the form produced with the same numbers which produce it. That's what happens with the square root of two, and pi. There is a failure in the numbering system's capacity to measure the spatial form produced because of an inherent incommensurability. So we have a simple solution, we round off. Or, if necessary we can move toward employing more complex numbering systems, real numbers for example.
Those presumptions are based on the fact that the quantity of the surface are "on the outside" is infinite while the quantity of volume "on the inside" is finite.
This is your false presumption, which is misleading you, that "the quantity of volume 'on the inside' is finite". If you would approach the problem with an open mind, rather than with what I see as a false presumption, we could probably make better progress in this discussion.
Here's what I propose. I'll justify my presumption, and you justify your presumption. I see that the spatial form which we are talking about, Gabriel's horn, is infinitely long. Therefore it is impossible, in theory, to precisely figure its volume. The volume therefore, in theory, is indefinite, being "infinite", unbounded, just like the extent of the natural numbers is "infinite", indefinite, or unbounded. We can however figure the volume of such forms, for practical purposes, by rounding off.
Will you justify your presumption that the volume is finite?
You are presuming that the paradox is solved by questioning the volume, but you haven't even shown a good reason to doubt the volume much less the flaw in the presumptions leading to the paradox.
I told you already, Gabriel's horn is a spatial form with an infinite length. That's very good reason to doubt the volume. All you said was "You're reasoning by equivocation; 'the extent is infinite, therefore the volume is infinite'. Clearly there is no equivocation. A spatial form which has an unlimited (infinite) extension in one of its dimensions, will have an unlimited (infinite) volume accordingly
The paradox assumes a spatial form, created from mathematical principles, and names this form Gabriel's horn.
The spatial form is given by Cartesian coordinates with three axes at right angles; that defines a space where the set of all points are (x, y, z) coordinates with x, y, z being reals.
Formatting for brevity.
"You want to use the same numbers which create the form, to measure the form." If it works.
"But when infinity is induced, this represents a failing in the capacity of the numbering system, so it is impossible to measure the form produced with the same numbers which produce it." That does not follow.
"That's what happens with the square root of two, and pi." The square root of two and pi are real numbers. See above. Also, square root of 2 is not infinite; it's between 1.4 and 1.5. Pi is not infinite; it's between 3.1 and 3.2.
This is your false presumption, which is misleading you, that "the quantity of volume 'on the inside' is finite". If you would approach the problem with an open mind, rather than with what I see as a false presumption, we could probably make better progress in this discussion.
It's not a presumption; it's the result of a calculation. The fundamental issue here is that you're critiquing the methods without understanding what they are or why they are employed. Take your critique of Tom at 6:20 in the video for example. There, Tom is calculating the results of an integral. Tom has an improper integral, and this describes the method for its evaluation:
Paul's online notes:
[math]\text{1. If }\lim_{a}^{t}{f(x)dx}\text{ exists for every }t > a\text{ then} \\ \hspace{2 em} \int_{a}^{\infty}{f(x)dx}=\lim_\limits{t \rightarrow \infty}{\int_{a}^{t}{f(x)dx}} \\ \text{provided the limit exists and is finite.} [/math]
When Tom says: "Well one over infinity, that's zero", that's okay, so long as we know it's a shortcut for:
[math]\hspace{2 em}\lim_\limits{x \rightarrow \infty}{\frac{1}{x} }[/math]
...which is exactly 0, as shown previously by definition of that limit. You have confused this with saying that 1/infinity=0. That's baseless; infinity is not a real number; the domain of the integral is the same domain as the x axis, and infinity isn't even in that domain. The method isn't "plug in infinity", and there's a reason it isn't. Quoting Metaphysician Undercover
I see that the spatial form which we are talking about, Gabriel's horn, is infinitely long. Therefore it is impossible, in theory, to precisely figure its volume.
Will you justify your presumption that the volume is finite?
That was already shown, and you're mischaracterizing the problem. The most fundamental problem here is that you're objecting to the efficacy of these methods without understanding what the methods are being employed or why they are employed. The other big problem is the obvious bias portrayed in objecting to the efficacy of the method before understanding these things. Quoting Metaphysician Undercover
A spatial form which has an unlimited (infinite) extension in one of its dimensions, will have an unlimited (infinite) volume accordingly
That does not follow.
Metaphysician UndercoverMarch 01, 2021 at 00:58#5041740 likes
It's not a presumption; it's the result of a calculation. The fundamental issue here is that you're critiquing the methods without understanding what they are or why they are employed. Take your critique of Tom at 6:20 in the video for example. There, Tom is calculating the results of an integral. Tom has an improper integral, and this describes the method for its evaluation:
1. If limtaf(x)dx exists for every t>a then??af(x)dx=limt???taf(x)dxprovided the limit exists and is finite.1. If limatf(x)dx exists for every t>a then?a?f(x)dx=limt???atf(x)dxprovided the limit exists and is finite.
— Paul's online notes
When Tom says: "Well one over infinity, that's zero", that's okay, so long as we know it's a shortcut for:
limx??1xlimx??1x
...which is exactly 0, as shown previously by definition of that limit. You have confused this with saying that 1/infinity=0. That's baseless; infinity is not a real number; the domain of the integral is the same domain as the x axis, and infinity isn't even in that domain. The method isn't "plug in infinity", and there's a reason it isn't.
Well you distinctly said it was a presumption. "Those presumptions are based on the fact that the quantity of the surface are "on the outside" is infinite while the quantity of volume "on the inside" is finite..."
We've been through all this. Your so-called calculation, "limx??1xlimx??1x...which is exactly zero by definition..." is nothing but a rounding off. See, it's zero by definition, not by calculation.
What do you mean this has nothing to do with algebraic geometry?
Whereas one can describe the collection of points in 3-space comprising GH with the zeros of
[math]P\left( x,y,z \right)={{z}^{2}}{{x}^{2}}+{{y}^{2}}{{x}^{2}}-1[/math], the paradox of GH does not emanate from that perspective, but from elementary calculus. Why even bring varieties up since it is irrelevant to the issue being discussed, and participants of the thread might well be familiar with the rudiments of calculus, but have little acquaintance with algebraic geometry?
Metaphysician UndercoverMarch 01, 2021 at 02:07#5041840 likes
I see that we have a fundamental difference of opinion concerning the logic of spatial areas. I think that it is illogical to believe that a 3d spatial form with an infinite extension in one dimension could have a finite volume. You disagree.
I see that we have a fundamental difference of opinion concerning the logic of spatial areas. I think that it is illogical to believe that a 3d spatial form with an infinite extension in one dimension could have a finite volume.
Well ok. What would you say is the volume of the solid of revolution of [math]y = \frac{1}{x}[/math] between 1 and [math]\infty[/math] when the curve is revolved around the x-axis? Here's the theory and formula, if you forgot.
https://en.wikipedia.org/wiki/Solid_of_revolution
Metaphysician UndercoverMarch 01, 2021 at 02:23#5041860 likes
Reply to fishfry
If the extension is infinite, the volume cannot be figured. You can only figure the volume by assuming that there is an end, a limit, and this is rounding off. But then you are not figuring the volume of an infinite extension.
Ryan O'ConnorMarch 01, 2021 at 02:25#5041870 likes
As everyone on this thread knows, the reason why calculus was reformulated based on limits was to avoid talking about infinitesimals (and actual infinity). Instead of actually computing the volume of an infinitely long horn, we break the calculation down into a potentially infinite process, where we compute the volumes of an endless sequence of horns of increasing length (all of which have a finite volume - and so there is nothing paradoxical about any of these horns). The problem is that we're not interested in any of these horns, we're interested in the horn of actually infinite length. And the paradox (re)surfaces because we're using limits (which were introduced to avoid the paradoxes of actual infinity) to describe something that is actually infinite. I'm inclined to believe that Gabriel's Horn doesn't exist any more than the "number" 1/?.
If the extension is infinite, the volume cannot be figured. You can only figure the volume by assuming that there is an end, a limit, and this is rounding off. But then you are not figuring the volume of an infinite extension.
LOL. Well then how do you know the area under the curve is infinite then?
Metaphysician UndercoverMarch 01, 2021 at 02:36#5041900 likes
And the paradox (re)surfaces because we're using limits (which were introduced to avoid the paradoxes of actual infinity) to describe something that is actually infinite.
Whereas one can describe the collection of points in 3-space comprising GH with the zeros of [math]P(x,y,z)=z^2x^2+y^2x^2?1[/math]
And the lazier definition of it typically given, such as the one in the video, is that we start with the curve 1/x starting at x=1 and rotate it about the x axis. Then the resulting object we call GH (with the same "starting at x=1" specification). But that is precisely this object. And we're talking about this object, so it's relevant for that reason.
So what are you on about jgill? And I mean that question literally; I have no idea what you're actually objecting to. Incidentally, no, calculus doesn't give us the GH paradox... broken intuitions do. I also find it a bit strange to claim that calculus is used to define the object; rather, it's used to analyze the object (surface area/volume in this case).
Because it cannot be measured. That's what infinite means.
soph·ist
/?säf?st/
* a paid teacher of philosophy and rhetoric in ancient Greece, associated in popular thought with moral skepticism and specious reasoning.
* a person who reasons with clever but fallacious arguments.
Ryan O'ConnorMarch 01, 2021 at 03:10#5041990 likes
Because it cannot be measured. That's what infinite means.
The surface area of the horn has no limit - as we consider larger and larger horns that area continues to increase, so we can loosely say that it has 'infinite' area. But as MU notes, I think it's more appropriate to say that we cannot measure the area (or I would argue that the horn simply doesn't exist). What's paradoxical is that the volume does have a limit. But that's not to say that the horn has a definite volume. We cannot complete the potentially infinite process associated with the limit any more than we can explicitly list all decimal digits of pi, so we cannot claim that it has a definite volume. The best anyone can do to offer a measure of the volume is to prematurely terminate the potentially infinite process and output a rational approximation of the volume.
(all of which have a finite volume - and so there is nothing paradoxical about any of these horns).
...is there? Back to "intuitive paint" based on real paint, 1 cubic foot of paint can paint 3000 square feet of wall. With that in mind, we can construct a Gabriel's horn with units of feet, chopped off at a finite length, such that it holds less than 3.15 cubic feet of paint, but takes over 1000 cubic feet to paint the outside.
Ryan O'ConnorMarch 01, 2021 at 03:14#5042010 likes
The best anyone can do to offer a measure of the volume
Let me clarify this...there's something better than giving a numerical measure....the best we can do is to stick with an algorithm which defines a potentially infinite process. And that's how we define pi.
Ryan O'ConnorMarch 01, 2021 at 03:18#5042020 likes
we can construct a Gabriel's horn with units of feet, chopped off at a finite length, such that it holds less than 3.15 gallons of paint, but takes over 1000 gallons to paint the outside.
I don't see the paradox - as long as we're allowed to play with the thickness we can come up any combination of numbers for the finite horns. What's important is that the numbers in your example are all finite.
I don't see the original paradox; a square foot of area has no meaningful volume.
But the same exact questions arise in the finite scenarios. "If it's holding about 3.15 cubic feet of paint, isn't that already painting the inside? If so how come it takes 1000 cubic feet then to paint the outside?"
For example, why compare an imaginary horn with real paint?
If it's a real horn, then there must be a certain limit beyond which the internal space cannot be diminished while still maintaining the coherence of molecular and atomic structure of what constitutes the physicality of anything (a horn).
So, why compare a metaphysical horn with actual paint?
If we postulate paint made up of a 'quantized super fluid with infinite tension/compression (whichever fits) capability' (e.g. the fundamental matter of cosmic space, just an idea), then we can paint infinity.
=> If 'infinite' is compared to 'finite' then obviously, ultimately, there would be incongruence - call it paradox, chaos, whatever...
So what are you on about jgill? And I mean that question literally; I have no idea what you're actually objecting to. Incidentally, no, calculus doesn't give us the GH paradox... broken intuitions do. I also find it a bit strange to claim that calculus is used to define the object; rather, it's used to analyze the object (surface area/volume in this case
Math 631 (Algebraic Geometry) (U of Mich):
"Intended Level: Graduate students past the alpha algebra (593/594) courses. Students should either already know or be concurrently taking commutative algebra (Math 614). Students should also know the basic definitions of topology — we won't be using any deep theorems, but we will use topological language all the time. Basic familiarity with smooth manifolds will be very helpful, as much as what we do is the hard version of things that are done more easily in a first course on manifolds. Undergraduate students intending to take this course should speak to me about your background during the first week of classes."
This description speaks for itself. Correct me if I am mistaken, but it appears you have tossed in AG to impress the readers of this thread. If you are indeed a mathematics professor and feel AG is necessary, then I would understand. Are you? I was one for many years and we never had an undergraduate course in AG, although some schools do. GH always came up in a standard calculus course. Tell me where you are coming from and why you found it essential to define GH this way.
By the way, you should now go to the Wikipedia article on GH and inject your considered opinion. It's a nice piece and never mentions AG. You apparently think it should. Again, if you are or were a professional math person and have strong feelings about this I will understand.
Jeez I'm with @jgill here. This is a standard example from freshman calculus. The integral of 1/x from 1 to infinity is infinite and the integral of 1/x^2 is finite. Or 1/x is square integrable but not integrable if you prefer. Algebraic geometry is a little high-powered in this context, it's not needed. Your judgment is off from letting yourself be trolled by @Metaphysician Undercover.
This is a standard example from freshman calculus.
I'm open to suggestions, but all I'm after here is a description of the space and the object. This is related to the conversation. This came up several times: Quoting Metaphysician Undercover
Saying that 1/infinity equals zero is obviously an instance of rounding off.
...referring to 6:20 in the video. So MU thinks the process is to substitute infinity in, as one would do in a proper integral. The point being made here is that in contrast to the proper integral limit, the ? in the improper integral with infinite limit isn't even a coordinate in the space.
Metaphysician UndercoverMarch 01, 2021 at 13:53#5043390 likes
Reply to InPitzotl
This is not really what I'm saying. What I mean is that if the value on the x axis, or y for that matter, is proposed as infinite, then it is wrong to assign a limit, such as zero, to the perpendicular axis. By saying that the one is infinite you say that there is no limit to how small the value of the other can be, and zero is an incorrect representation. In other words it is incorrect to speak of a limit here, and to say that infinity is a limit is nonsense.
What I think is that there is a fundamental incommensurability between two distinct dimensions of space, demonstrated by the irrationality of pi and the square root of two, which indicates that representing spatial forms with perpendicular axis is a sort of misrepresentation. Spatial forms, as we know them, cannot be accurately represented this way.
So if we look at the difference between a straight line and a curved line, we see that one requires two dimensions, fundamentally, while the other defines (or establishes the limits of) one dimension. The curved line, even at an infinitesimally small length requires two dimensions to be mapped by a straight-lined coordinate system. This would mean that even the most simple thing in a spatial reality (a point particle for example), would require a multi-dimensional representation. And, the multi-dimensional representation would get it wrong because of the incommensurability between two distinct dimensions.
Therefore the curved (real) line will never be commensurable with the straight (artificial) line. And, when we map the curved line with a straight-lined coordinate system, the incommensurability shows as infinity. Gabriel's horn is a curved line being mapped by a straight-lined coordinate system, and the incommensurability is evident.
The further question, which comes to my mind, is which is the proper way to represent space. Which way represents how space really is? Is the proposed "real" curvature just an illusion created by deficiencies of observation, and there is no such thing as a true arch, or perfect circle. Or does all of space consist only in multi-dimensional, non-straight relations, and our artificial dimensional straight-lined coordinate systems are incapable of giving precise representations of multidimensional existence? Or is the apparent incompatibility due to something else, which we have not yet grasped? Perhaps we ought not be representing multi-dimensional lines with perfect curves, archs, and circles. Maybe we need to banish this type of object from geometry as not properly representing reality.
But that is really what's going on at 6:20; the ? there is the ? symbol of the upper limit of the integral, and it is a sentinel; a placeholder meaning unlimited. There's a notation here that requires filling in a spot for the lower limit and a spot for the upper limit. "Usually" you would fill that in with something like 1 and 2. To show you're doing this same kind of thing, but there is no upper limit, you put ? there. The 1/x comes from that integral. Quoting Metaphysician Undercover
By saying that the one is infinite you say that there is no limit to how small the value of the other can be, and zero is an incorrect representation.
You told me that infinite just means unlimited. Try taking that seriously for a moment. Don't say it's infinite, just say it's unlimited. Quoting Metaphysician Undercover
By saying that the one is infinite you say that there is no limit to how small the value of the other can be, and zero is an incorrect representation.
But there is a limit to how small the value can be; for any real number > 1, 1/x cannot be less than 0, whereas it can be less than any other positive real number. The limit is saying something similar... that the farther out you go, the closer you get to 0 (and that you can get arbitrarily close). The limit in its definitive form can be used to show that this is only true for 0; it is not true to say that the farther out you go, the closer you get to 1 billionth. It is only true to say that the farther out you go, the closer you get to 0 (arbitrarily so). Quoting Metaphysician Undercover
What I think is that there is a fundamental incommensurability between two distinct dimensions of space
Again, pi and square root of 2 are real numbers. So the spaces involved have those as coordinates. Quoting Metaphysician Undercover
So if we look at the difference between a straight line and a curved line
From here down you're pontificating about physical space, which is not the space being used here.
Metaphysician UndercoverMarch 01, 2021 at 17:19#5043980 likes
But there is a limit to how small the value can be; for any real number > 1, 1/x cannot be less than 0, whereas it can be less than any other positive real number.
Let's remove this necessity of a "real number", maybe that's what's misleading you. Is there any limit to how small the value can be? No, that's what's meant by infinitely small, we can conceive that there is always a further value, smaller than any value which we put a number to. That's the same as what's meant when we say that the natural numbers are infinite, only in the inverse direction. We can conceive that there is a further value, larger than any value which we put any number to.
Therefore, what is meant is that there is no limit to how small the value can be, just like in the natural numbers there is no limit to how big the value can be. This, I think, is what's misleading you. You keep thinking that there must be a limit to how small the value can be, but what is indicated by "infinite" is that there is no such limit. If a limit was intended, we'd employ "infinitesimal", which indicates that there is a smallest possible. But "infinite" indicates that there is no limit to how small we can go.
The limit in its definitive form can be used to show that this is only true for 0; it is not true to say that the farther out you go, the closer you get to 1 billionth. It is only true to say that the farther out you go, the closer you get to 0 (arbitrarily so).
What is arbitrary is the choice of "0" here, as the representation of some non-existent limit. There is no limit, the value can keep getting smaller and smaller, always beyond any numerical representation which you might give it, that's what's indicated by "infinite". So there is no point in representing this value as getting close to some imaginary limit, "0". The value keeps changing without ever reaching that proposed limit, so in reality it never gets any closer to that limit. There is always infinite more values to cover before it gets there. The value really never gets any closer to the proposed limit, it's always infinitely far away, so the limit is completely irrelevant. Therefore "the farther out you go, the closer you get" is not an accurate representation at all. because the whole point in saying "infinite" is to say that there is no end, so it's impossible that the end is getting any closer.
Ryan O'ConnorMarch 01, 2021 at 17:51#5044190 likes
But the same exact questions arise in the finite scenarios. "If it's holding about 3.15 cubic feet of paint, isn't that already painting the inside? If so how come it takes 1000 cubic feet then to paint the outside?"
You are essentially saying that it takes <3.15 ft3 to paint the finite horn AND it takes >3.15 ft3 to paint the finite horn. If film thickness doesn't explain the apparent contradiction, then I would conclude that your problem definition is invalid. Perhaps you cannot claim that 1 ft3 can only paint 3000 ft2 of wall. I think focusing on physical paint is a distraction.
I don't see the original paradox; a square foot of area has no meaningful volume.
I agree that an area has no volume, so a single drop of mathematical paint could paint any surface of arbitrarily large size. However, we are not justified to claim that it can paint a surface of actually infinite area. That requires a leap of though which we are not in a position to make.
Ryan O'ConnorMarch 02, 2021 at 01:10#5045610 likes
1/x cannot be less than 0, whereas it can be less than any other positive real number.
Consider the Stern-Brocot tree. If L=1/2, LL=1/3, LLL=1/4, and so on, then L_repeated is a "real number" which it cannot be less than. I suspect you'll argue that L_repeated = 0, which brings us to the classical debate of whether 0.9_repeated=1. I would argue that 0.9_repeated describes a potentially infinite process and is not a number in the same sense that 1 is (dare I say that 0.9_repeated is not a rational number), but that's a debate for another time. In any case, consider this: if a number at the 'bottom' of the Stern-Brocot tree is equivalent to both of its neighbors, are any of the real numbers actually distinct from each other?
What is arbitrary is the choice of "0" here, as the representation of some non-existent limit.
Given that y continually approaches 0 as x increases, the limit is 0. We don't need a point at (?,0) for the limit to be equal to 0. I think the problem is that your definition of limit doesn't match the standard definition. With that said, I sympathize with your view and I think your argument would be stronger if you focused on the limit used to calculate the volume of GH. Such a limit is a whole different beast since it converges to pi - an irrational 'number'. Just because the interval corresponding to V can get arbitrarily small as a approaches infinity, it doesn't necessarily mean that V has a definite value. Just as it is impossible to explicitly compute all decimal digits of pi, the best anyone can do here is either (1) provide a small interval for V or better yet (2) leave V in algorithmic form (i.e. pick your favorite formula for pi and don't bother to compute it).
Let's remove this necessity of a "real number", maybe that's what's misleading you. Is there any limit to how small the value can be?
You seem to be imagining a hypothetical number "so big that" 1/x dips below 0. This sounds like speculative fantasy to me. Your reasoning that it dips below 0 could equally be applied to an imagined consequence that it levels off at and stays at 0, or that it rises again. There's basically no meaning to this. Quoting Ryan O'Connor
Perhaps you cannot claim that 1 ft3 can only paint 3000 ft2 of wall.
Those are loosely based on real numbers. A gallon of paint can paint approximately 400 square feet . A gallon is about 1/7.5 square feet.
But thinking like an engineer, our really long Gabriel's horn is "mostly" "essentially" a negligible sized needle... the first portion that's much bigger is essentially a fixed amount of error. If our paint is 1/3000 feet thick (as the volume to area-covered ratio implies), then most of our paint on the outside is a cylinder with radius 1/3000. We could imagine this as if we're squeezing paint out of a tiny hole like a roll of toothpaste; in that sense, there's no limit to the amount of "paste" we can squeeze out of the tube. Quoting Ryan O'Connor
Consider the Stern-Brocot tree.
Already had that thread with MU.
Ryan O'ConnorMarch 02, 2021 at 02:23#5045960 likes
But thinking like an engineer, our really long Gabriel's horn is "mostly" "essentially" a negligible sized needle... the first portion that's much bigger is essentially a fixed amount of error.
Most of the area is on the "needle" portion but most of the volume is enclosed by the "horn" portion. I find it odd that you're focusing on the needle portion. The typical area and volume calculations for GH are based on limits as a approaches infinity (not the other way around) - in other words, as the horn gets longer and longer. But sure...
We could imagine this as if we're squeezing paint out of a tiny hole like a roll of toothpaste; in that sense, there's no limit to the amount of "paste" we can squeeze out of the tube.
You lost me here. Are you saying that the paint on the outside is a cylinder of radius 1/3000 and infinite length and asking why that infinite amount of paint doesn't agree with the finite amount of paint needed to fill needle?
Are you saying that the paint on the outside is a cylinder of radius 1/3000 and infinite length and asking why that infinite amount of paint doesn't agree with the finite amount of paint needed to fill needle?
Almost; I'm talking about arbitrarily large but finite amounts of paint. And the only point here is that it's not really surprising this "outside" can be indefinitely long with a layer while the inside is limited.
Ryan O'ConnorMarch 02, 2021 at 03:17#5046310 likes
Almost; I'm talking about arbitrarily large but finite amounts of paint. And the only point here is that it's not really surprising this "outside" can be indefinitely long with a layer while the inside is limited.
It seems like you're implying that finite math is equally paradoxical by comparing the infinitely long needle with an infinitely long horn. If you want to say something about finite math you need to talk about a horn of finite length.
Metaphysician UndercoverMarch 02, 2021 at 04:12#5046490 likes
You seem to be imagining a hypothetical number "so big that" 1/x dips below 0.
I didn't say anything about dipping below zero. Where did you get that idea from? I said it doesn't ever get any closer to zero. There is always an infinitude of values between it and zero, so it's really not ever getting any closer to zero. Zero is off the scale, it's literally not part of the scale, as it is excluded by virtue of being impossible. So there is always an infinity of values between the value of y and zero. Since there is always that infinity of values between any given value of y, and zero, it makes no sense to say that it is getting closer to zero.
Given that y continually approaches 0 as x increases, the limit is 0.
As I explained y does not in any way approach zero. It is always infinitely far away from zero, no matter what value it has, therefore it never gets any closer to zero, and cannot be said to approach zero. Zero is imposed as an arbitrary limit, on something which, by definition, has no limit. If the line came to an end at a particular value, we could say that value is the limit. But it doesn't, the line continues onward without limit. It doesn't reach zero, yet continues infinitely, so zero is right off the scale, irrelevant as unobtainable.
Consider this example.
First proposition: The natural numbers are infinite, therefore there is no highest number.
Second proposition: 20 is closer to the highest number than 10 is.
Do you see how the second proposition contradicts the first? We have the very same type of contradiction when we say that x can increase infinitely without y ever reaching zero, yet we also say that it is getting closer to zero. Zero has been excluded from the scale as an impossibility, just like the highest natural number. So we can't say that one point is closer to zero than another. How does that make any sense? It's just like saying that one number is closer to the highest number than another when there is no highest number. Here it's the lowest number, we're talking about and that's not zero because there's an infinitude of numbers to go through, making zero impossible, just like the highest number is impossible.
That's not the proper way to speak, to say that one number is closer than another to the highest number. One number is higher than another, but it is not closer to the highest number, because there is no higest number. Likewise, with the value of y, one value is lower, and another higher, but we can't say that one value is closer to the lowest number because there is no lowest number, just like there is no highest natural number. There is just more and more numbers, and zero cannot be posited as the lowest of those numbers, because it is not one of them, as unobtainable, impossible, outside the bounds.
There is always an infinitude of values between it and zero, so it's really not ever getting any closer to zero.
That does not follow. 1 is closer to 0 than 2 is, despite the infinite number of points between 0 and 1. Quoting Metaphysician Undercover
Zero is off the scale, it's literally not part of the scale, as it is excluded by virtue of being impossible.
I assume by off the scale you mean 0 will never be reached. But that's not required for 0 to be the limit.Quoting Metaphysician Undercover
Since there is always that infinity of values between any given value of y, and zero, it makes no sense to say that it is getting closer to zero.
Sure it does; but it's a bit more precise than this. The limit specifies that it's possible to get arbitrarily close to 0. "Arbitrarily" here is used in a strong sense that includes all positive distances at once.
Metaphysician UndercoverMarch 02, 2021 at 12:44#5047640 likes
That does not follow. 1 is closer to 0 than 2 is, despite the infinite number of points between 0 and 1.
I told you, you need to stop thinking about the numbers which make up the coordinate system which produces the shape. They are outside the shape, irrelevant, and insufficient for measuring the shape. There is no zero point to start from in our measurement, nor is there a zero ending point. The value of one dimension is allowed to increase indefinitely such that there is no highest value, and correspondingly, the value of the other dimension is allowed to decrease indefinitely such that there is no lowest value. Zero does not enter the picture. It is excluded, (just like "highest value" is excluded, so is "lowest value", or zero) and therefore cannot be a part of the measurement scheme.
Sure it does; but it's a bit more precise than this. The limit specifies that it's possible to get arbitrarily close to 0. "Arbitrarily" here is used in a strong sense that includes all positive distances at once.
See, zero is irrelevant. At any point on the shape, the y value is "arbitrarily" close to zero. So it is false to say that it gets any closer, at any point, because it's always the same, "arbitrarily close". This arbitrariness indicates that zero is completely irrelevant to any valid measurement.
It appears to me, like you want to allow the x value to increase indefinitely, without limit, assuming no highest number, but you will not allow the y value to decrease in the corresponding way. You want to limit the y value's decrease with an imposed zero. This removes the symmetry from the shape, and is a false representation of it.
They are outside the shape, irrelevant, and insufficient for measuring the shape.
Your entire response is misguided. The limit is not describing a point in the shape. If it were, it would be an empty concept; the limit would just be a function evaluation. Take:
[math]\hspace{2 em}f(x)= \left\{ \begin{array}{ll} \frac{x^2-1}{x-1} & x \neq 1 \\ 8 & x=1\end{array} \right.[/math]
The value of this piecewise function at f(1) is 8, as explicitly denoted. But:
[math]\hspace{2 em}\lim_\limits{x \rightarrow 1}{f(x)}=2[/math]
Despite explicitly talking about 1, which is explicitly defined by the bottom piece, the limit is about that top piece, which doesn't even have a value at x=1, nor is there an f(x)=2.
Ryan O'ConnorMarch 03, 2021 at 01:11#5049260 likes
So it is false to say that it gets any closer, at any point, because it's always the same, "arbitrarily close". This arbitrariness indicates that zero is completely irrelevant to any valid measurement.
Your notion of 'close' that is based on the number of points between A and B can only have value in a number system which is not dense in the reals, such as the integers. For example, since there are 3 integers between 0 and 4 but 6 integers between 0 and 7, we can conclude that so 4 is closer to 0 than 7. If you want to restrict your mathematics to the integers then your notion of 'close' is suitable. However, such a mathematics is far less powerful than orthodox math so I suspect that you'd have a very tough time convincing anyone to adopt your view.
But as I said before, I sympathize with your position. In one breath we say that we use limits to avoid actual infinity but then in the next we say that the limit can be a number inseparably tied to actual infinity (e.g. pi). But I would argue that the resolution to such contradictory thinking is simple: just don't say that the limit is a number with actually infinite digits. Instead, keep the limit in algorithmic form - say that the limit is a potentially infinite process which is described by [pick your favorite] algorithm for pi, for example, 4(1-1/3+1/5-1/7+1/9-...)
Metaphysician UndercoverMarch 03, 2021 at 04:09#5049920 likes
Your notion of 'close' that is based on the number of points between A and B can only have value in a number system which is not dense in the reals, such as the integers. For example, since there are 3 integers between 0 and 4 but 6 integers between 0 and 7, we can conclude that so 4 is closer to 0 than 7. If you want to restrict your mathematics to the integers then your notion of 'close' is suitable. However, such a mathematics is far less powerful than orthodox math so I suspect that you'd have a very tough time convincing anyone to adopt your view.
You don't seem to quite grasp why I reject "closer". The line is known to never reach the limit, that's the point with "infinite". So it makes no sense to say that it is getting closer to the limit. It is impossible for the line to reach the limit. so it is impossible that one point is closer to the limit than another. The proposed limit is outside the parameters by which the line can be measured.
The issue is how to best measure the shape which is Gabriel's horn. Orthodox mathematics, while it is very good at other things, fails here, as is evident from the appearance of the paradox. The reason it fails, I believe, is because it describes a feature which very clearly cannot be described in terms of limits, as getting closer to a limit, and that's nonsensical.
So I don't agree with your resolution, because you still want to apply limits, where the shape denies the application of limits. If measurement necessarily involves the application of limits, then we regard this as having created a shape which cannot be precisely measured, just like a circle.
Ryan O'ConnorMarch 04, 2021 at 01:21#5053530 likes
You don't seem to quite grasp why I reject "closer".
Perhaps I don't grasp it, or perhaps I just don't agree with it. Let's assume it's the former. Please tell me whether the following points aligns with your view:
1) One can travel along y=1/x in the positive-x direction, without bound.
2) The limit of the journey corresponds to the final destination, which if anything would be (?,0).
3) The point (?,0) does not exist (since ? is not a number) therefore there is no limit.
Perhaps I don't grasp it, or perhaps I just don't agree with it. Let's assume it's the former. Please tell me whether the following points aligns with your view:
1) One can travel along y=1/x in the positive-x direction, without bound.
2) The limit of the journey corresponds to the final destination, which if anything would be (?,0).
3) The point (?,0) does not exist (since ? is not a number) therefore there is no limit.
I surely disagree. There is no "final destination." That's MU's error, why are you amplifying it?
Yes that's what I'm saying, there is no final destination, so to even produce any representation (such as ?,0), as if it is a final destination, is a misrepresentation amounting to contradiction.
There is a particular line we are talking about, and #1 ought to state that this line is extended "without bound", which means "there is no limit". So #3 ought to read "the point (?,0) does not exist because there is no limit". Now we could add #4: "?" is a description of the entirety of the line (not a point on the line), and 0 is completely unrelated to the line, therefore irrelevant.
Here's another way to look at the position of zero. It is an ideal, like infinite is an ideal. The two are opposing ideals, like hot and cold are opposing ideals. The line takes the characteristic of the one ideal, the infinite, therefore the opposing ideal, zero, is excluded. It's just like if we were talking about the absolute, ideal hot, cold would be completely excluded.
Ryan O'ConnorMarch 04, 2021 at 04:00#5054360 likes
Yes that's what I'm saying, there is no final destination
This is where the misunderstanding is. Nobody is saying that there is a final destination or that (?,0) exists. The standard definition of a limit does not require a final destination, it only requires one to approach a number as they advance along the journey. A limit is the process of approaching not the act of arriving. And you must admit that any workable definition of 'approach' will have one approach y=0 as they travel along y=1/x. [As mentioned before, your definition of approach involving looking at the number of intermediate numbers is not workable for number systems which are dense in the reals, e.g. the rational numbers].
In short, I believe our disagreement is simply the result of us having a different definition of limit.
Metaphysician UndercoverMarch 04, 2021 at 04:28#5054550 likes
Reply to Ryan O'Connor
Did you read the rest of my post? What I'm saying is that 0 is not even relevant. That's what I've been trying to explain, that to describe the value of y as approaching 0 is a false representation. Y is always infinitely far away from zero, because zero is impossible on that line. The value for y never "approaches 0". It is correct to say that the value gets lower and lower, but it is incorrect to say that it approaches 0, because no matter how low it gets it never approaches 0. 0 is not at all relevant to this line.
To say that y approaches zero is an inaccurate simplification, nothing but a rounding off in your description. The real description is that the value of y gets lower and lower without ever approaching zero. Of course the true description is "not workable", that's the nature of any infinity. The appearance of infinity is the result of something being not workable. To make an infinity into something workable is to provide a false representation.
In short, I believe our disagreement is simply the result of us having a different definition of limit.
I think where we disagree is in the role that zero can play in this measurement. You think 0 can play a role, as the value which y approaches. I think that since the line has no start nor end, 0 is not an applicable number.
Ryan O'ConnorMarch 05, 2021 at 02:37#5059150 likes
It is correct to say that the value gets lower and lower, but it is incorrect to say that it approaches 0, because no matter how low it gets it never approaches 0.
The real description is that the value of y gets lower and lower without ever approaching zero.
Our disagreement is also due to us having different definitions of approach. We both agree that y gets lower and lower (and perhaps you would even agree that the greatest value which y never reaches is 0) but I call that approach and you call that not approaching. Let us agree to disagree on definitions!
(and perhaps you would even agree that the greatest value which y never reaches is 0)
There is no such value. Might it be 100? Or 1000?. y=1/x, x>1 has a greatest lower bound, 0, which it never reaches. Your attempts at the philosophy of mathematics may never bear fruit if you consider this a cogent statement.
Ryan O'ConnorMarch 05, 2021 at 04:04#5059370 likes
We both agree that y gets lower and lower (and perhaps you would even agree that the greatest value which y never reaches is 0) but I call that approach and you call that not approaching. Let us agree to disagree on definitions!
As I explained, by way of example, to assume such a "greatest value", or "lowest value" is contradiction. When we say that the natural numbers are infinite, and therefore have no highest value, it's contradiction to say that 20 is closer to the highest value than 10. Likewise, when there is no lowest value, it's contradiction to say that .01 is closer to the lowest value than .02.
What is misleading in the example of Gabriel's horn, is that the x and y axes are set to converge at 0, at a right angle in relation to each other. This proposed point of convergence creates the illusion that 0 is a valid value, where x and y are 'the same". However, as I described earlier, the spatial representation of two dimensions at right angles to each other is actually a false representation, making the two dimensions incommensurable, as demonstrated by the irrationality of the square root of two. This incommensurability indicates that the two proposed lines, x and y, cannot actually be modeled as intersecting, and sharing a common point at 0.
So this false idea that x and y actually meet each other at that point, 0, is what misleads you into thinking that 0 is a valid measurement.
Metaphysician UndercoverMarch 05, 2021 at 14:06#5060610 likes
To put it simply, non-dimensional existence, which is represented by the point, 0, is incompatible with our representations of dimensional existence. So 0 cannot enter into our scales for measuring dimensional existence, as a valid measurement point until we establish commensurability between non-dimensional and dimensional existence. This problem with zero becomes very relevant when we start to consider motions, and acceleration from rest. An infinite acceleration is required to go from rest to moving. The problem is somewhat avoided with relativity theory which denies the reality of rest, making acceleration simply a change in direction. But what that does is make the mathematics extremely complex, still working with points and vectors, rather than resolving the problem of how the non-dimensional truly relates to the dimensional.
An infinite acceleration is required to go from rest to moving.
No it's not. When you get in your car and start driving to the store, do you experience infinite acceleration? What's that feel like, exactly? According to special relativity, you should be pressed against the back of your seat with infinite force. You'd be crushed before you drove a foot. What do you say?
ps -- Let's do the math. Say I'm at rest and start moving at 1 unit/second or whatever. In physics we need to give the units but in math we'll just say the velocity is 1. So at 1 second we've gone 1 unit, at 2 seconds we've gone two units, etc.
So our position function is p(t) = t; and our velocity is always 1, which is consistent with the first derivative of position being the derivative of t with respect to t, or 1.
Now this is tricky and this is where you got yourself confused. What was our instantaneous acceleration at 0? After all we weren't moving and then a tiny moment later we were. Well, the graph of our position look like this:
/
/
----------o
This function is not differentiable at zero. There is no instantaneous velocity at zero and no definite acceleration either. I agree that this is counterintuitive, and your intuition is not uncommon. But it's wrong. Clearly it's wrong. If you experienced infinite acceleration even for a moment, every atom in your body would be flattened like so many pancakes.
I actually agree with you about the intuition. If we're not moving, how do we start moving? It's a bit of a mystery actually, I'm not sure what physicists say about this. Well I guess I do know. If we're a steel ball in Newton's cradle, or we're a ball on a pool table, we start moving when we get smacked by another ball that transfers its momentum to us. But how does our velocity go instantaneously from zero to nonzero? The Newtonian physics works out, but not the intuition.
resolving the problem of how the non-dimensional truly relates to the dimensional.
A little woo-woo-y there @MU. By the way, how do the zero-dimensional, zero-length points in the unit interval make up the one-dimensional, length 1 unit interval? That's actually another mystery, despite the fact that we have a mathematical formalism that says [math]\int_0^1 dx = 1[/math]. The math works but we have no metaphysical explanation that I know of.
We cannot 'paint' the horn in that sense because the volume required would be the area (infinite) multiplied by the thickness (nonzero), which means an infinite volume.
Is the area infinitely large or merely infinite in the sense of 'unbounded'?
Metaphysician UndercoverMarch 06, 2021 at 01:23#5063160 likes
This function is not differentiable at zero. There is no instantaneous velocity at zero and no definite acceleration either. I agree that this is counterintuitive, and your intuition is not uncommon. But it's wrong. Clearly it's wrong. If you experienced infinite acceleration even for a moment, every atom in your body would be flattened like so many pancakes.
I actually agree with you about the intuition. If we're not moving, how do we start moving? It's a bit of a mystery actually, I'm not sure what physicists say about this. Well I guess I do know. If we're a steel ball in Newton's cradle, or we're a ball on a pool table, we start moving when we get smacked by another ball that transfers its momentum to us. But how does our velocity go instantaneously from zero to nonzero? The Newtonian physics works out, but not the intuition.
I addressed the issue in my post. There is only a need to conclude infinite acceleration if we assume absolute rest, zero velocity, but relativity denies absolute rest. If something had an absolute zero velocity, and changed from that zero velocity to having a positive velocity, this would imply a point in time (not a short duration) when the thing goes from zero velocity to a positive velocity. At that point in time, since there is no duration, but there is acceleration, the acceleration would be infinite. We avoid this problem with relativity theory which denies the reality of rest, and makes any supposed zero point in time into an extended duration.
As I explained, by way of example, to assume such a "greatest value", or "lowest value" is contradiction. When we say that the natural numbers are infinite, and therefore have no highest value, it's contradiction to say that 20 is closer to the highest value than 10. Likewise, when there is no lowest value, it's contradiction to say that .01 is closer to the lowest value than .02.
You are equating 'approaching' with 'arriving at'. If I could only tell you one piece of information about my trip I would tell you my destination. But if my trip never ends there are some situations where I could still give you some useful information since in some situations I could still tell you which direction I'm pointing (e.g. what I'm approaching). But I think we are both firm with our incompatible definitions and we've said what could be said. I think our words would be much better spent on the new topics of this thread.
No it's not. When you get in your car and start driving to the store, do you experience infinite acceleration? What's that feel like, exactly?
This reminds me of Diogenes the Cynic's rebuttal to Zeno's paradoxes. According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions. I'm not convinced that Diogenes appreciated that profundity of Zeno's paradoxes.
I actually agree with you about the intuition. If we're not moving, how do we start moving? It's a bit of a mystery actually, I'm not sure what physicists say about this.
Again, Zeno's Paradox. The issue is that we're looking at things with a 'whole-from-parts' view. We want to advance forward one point at a time and can't seem to get anywhere...our intuition is saying that it doesn't make sense how a line can be formed from points. But with a 'parts-from-whole' view it's easy. We start with the whole (the unobserved wave function of the universe spanning all of time) and then we make (quantum) measurements. At one measurement we're here and then at the next measurement we're there. Change doesn't happen at points (or instants in time). Instantaneous velocity makes no sense. Change it happens in between the points. And if we draw a graph using the 'parts-from-whole' view as I mentioned in the Have we really proved the existence of irrational numbers? thread, the change of a function happens in between the points...across the unmeasured curves.
I'm not convinced that Diogenes appreciated that profundity of Zeno's paradoxes.
If I point out to @Metaphysician Undercover that he can get in his car and drive to the store without being crushed before he drives the first inch; am I failing to appreciate the profundity of his beliefs? I think not!
I addressed the issue in my post. There is only a need to conclude infinite acceleration if we assume absolute rest,
I addressed this in my post. The position and velocity functions are not differentiable at time zero. So there's no well-defined acceleration. Nor as others pointed out does relativity bail us out. Relative to your own frame of reference, you are at zero velocity at time zero and nonzero velocity a short time afterward. You have to come to terms with that.
I actually agree with you about the intuition. If we're not moving, how do we start moving? It's a bit of a mystery actually, I'm not sure what physicists say about this. Well I guess I do know. If we're a steel ball in Newton's cradle, or we're a ball on a pool table, we start moving when we get smacked by another ball that transfers its momentum to us. But how does our velocity go instantaneously from zero to nonzero? The Newtonian physics works out, but not the intuition.
Collision is a notoriously messy scenario - both physically and mathematically. Better to think of a ball in Newton's cradle at its highest point: at that point it is instantaneously at rest, then it starts moving again. Voila, motion from rest. Or easier still, just pick up that ball, gently release your grip and let it fall to the ground. Same deal, and we even know pretty exactly what its acceleration is when it starts moving. This doesn't seem so unintuitive to me.
I remember struggling with the concept of acceleration when it was first introduced - in middle school, I guess. It started making sense after a while. But some people just can't come to grips with such abstract concepts. Most of them have the good sense to leave it alone and apply themselves to something they are better at. Those who can't leave it alone become lifetime cranks, like MU. Or philosophers :)
Yes that's what I'm saying, there is no final destination, so to even produce any representation (such as ?,0), as if it is a final destination, is a misrepresentation amounting to contradiction.
FWIW, and because no one has mentioned it yet, 'infinite limits' are taught in calculus as usefully specific ways to indicate divergence.
You can even write [math] f(x) = x [/math] and then [math]f(\infty) = \infty[/math], but only as a cute abbreviation for something more technical. Later there is the extended real line, represented as [math] [-\infty, \infty] [/math], but there's nothing magical about this, no more than there's anything magical about [math] 0! = 1 [/math]. It's all philosophically agnostic. Indeed, I know one mathematician who thinks the world is discrete and that continuity is a fiction, and then I know another who believes the reverse. Another dislikes philosophy altogether, and still another more has read Kant's CPR in German.
Metaphysician UndercoverMarch 06, 2021 at 13:58#5065610 likes
There's at rest in a given inertial frame. Which is to say, really, that any acceleration, on your theory, should involve instantaneous infinite acceleration. My hand is at rest on the table. I raise it to type. Space-time not locally crushed in the process.
It's "absolute rest" which I said is a problem, because this makes a point in time into a real situation rather than a perspective (reference frame) dependent designation. That's a point when no time passes relative to the thing at absolute rest.
There's at rest in a given inertial frame. Which is to say, really, that any acceleration, on your theory, should involve instantaneous infinite acceleration. My hand is at rest on the table. I raise it to type. Space-time not locally crushed in the process.
This "at rest" which you refer to isn't real, because the earth is moving. Your hand is never at rest. So the moving of your hand is just a change in the existing motion of your hand, it is not an act of acceleration from rest. Physicists might represent it as an acceleration from rest, but the point I am arguing is that this is really an incorrect representation, which serves the purpose, just like representing Gabriel's horn as approaching 0 is an incorrect representation, which serves a purpose.
You are equating 'approaching' with 'arriving at'.
No I'm not equating these two. If there is no such thing as the lowest point, then it is impossible to be "approaching" the lowest point. In the case of the natural numbers, do you see that there is no such thing as "approaching" the highest number? We recognize that there is no such thing as "the highest number", so it doesn't make sense to say that if a person is counting higher and higher, they are "approaching" the highest number. You can never approach the highest number. If you can apprehend this, then why can't you turn it around, and see that when infinity is at the low end, there is no such thing as "the lowest number", and it doesn't make any sense to say that someone counting lower and lower is "approaching" the lowest number?
But if my trip never ends there are some situations where I could still give you some useful information since in some situations I could still tell you which direction I'm pointing (e.g. what I'm approaching).
OK, this is a good point. The question here is what grounds or substantiates "direction". You imply that direction must be grounded by going toward something, but you forget that it might equally be substantiated by going away from something. In Gabriel's horn we have both, moving away from one axis, and moving toward the other. The axes are artificial confines, imposed as standards of measurement, and through the descriptive term of "infinite", the line of the form is stipulated as going beyond the capacity of the measuring scale. Therefore to understand that line we can no longer employ those measurement axes.
This is the problem we have here. Generally we assign infinite capacity to the measuring tool, and this allows us the capability to measure anything with that tool. The natural numbers are infinite for example, and this allows that we might count absolutely any multitude of objects. In Gabriel's horn, we have turned the table. We propose an infinite shape to be measured. Of course we cannot measure it, because it is defined as infinite, meaning that we cannot measure it, the thing is stipulated as going beyond the capacity of the measuring tool. So there is a trick hidden in the proposal, it's asking us to measure what cannot be measured by the tool. Then when we look at the shape, we see it getting further and further from the one axis, and we conclude, 'that's impossible to measure'. But we also see it getting closer and closer to the other axis, and intuition tells us, 'that's a finite distance which can be measured'. However, we must adhere to the principles of the construction, which state that the shape will appear to approach the axis, to a point beyond our capacity to measure the distance between them. Therefore we must resist our intuition and inclination to say that this distance is measurable.
So we must remove the axes as incapable of giving us the scale required for the measurement. The axes are what produced the form, which is by that construction, infinite and therefore immeasurable. Therefore the axes cannot be used to measure that form, because it has been constructed by them, as immeasurable. Now we have no basis for the terms of "farther from" or "closer to", because these values have been stipulated as going beyond our capacity to measure. What we are left with now, is just theoretical values, to be assumes as spatial distances, values which we acknowledge cannot actually be measured as spatial distances. Now we are really not talking about "farther from" or "closer to" any more, even though the numbering system employed was originally derived from that. We have explicitly gone beyond our capacity to determine farther from or closer to, and all we are talking about now is a higher value and a lower value. If we do not divorce the value from the spatial distance, we are just left with a spatial distance which is impossible to measure.
The point now, is that since we have done what the example requires, and taken the values beyond our capacity for making spatial measurements, we cannot use spatial references to ground or substantiate "direction". All we have now is a higher value and a lower value, and the stipulation that each of these may continue infinitely. The two directions (values) are actually defined in relation to each other. As the one gets a lot larger, the other gets a tiny bit tinier. And so long as we allow that the one can continue to get a lot larger, we must allow that the other can get a tiny bit tinier. But these "directions" must be thought of solely as numerical values, because we have gone beyond the relevance of spatial distances as dictated by the proposed example. So we cannot look at them as spatial directions of "farther" or "closer" or else we just fall back into the stipulated impossible to measure..
I addressed this in my post. The position and velocity functions are not differentiable at time zero. So there's no well-defined acceleration. Nor as others pointed out does relativity bail us out. Relative to your own frame of reference, you are at zero velocity at time zero and nonzero velocity a short time afterward. You have to come to terms with that.
So the point I'm making, is that zero is completely arbitrary, and represents nothing real, just like in the case of Gabriel's horn. That's why we must decline this idea of "approaching zero". It is extremely useful in practice, yes, for sure it serves the purpose. But this is an exercise in theory, and we need to be able to go beyond what works in practice to be able to see that the principles which we employ in practice mislead us in our metaphysical efforts to understand the true nature of reality. The existence of paradoxes such as Zeno's demonstrate an incompatibility between theory and practice, and these incompatibilities expose where we misunderstand the true nature of reality.
I know one mathematician who thinks the world is discrete and that continuity is a fiction, and then I know another who believes the reverse.
This is a different, but related issue, the difference between discrete and continuous. The issue is not whether the world is discrete or continuous, it is to find compatibility between the two. In practice the world is continuous (time passes continuously), but in theory the world is discrete (represented by distinct units, numbers). Simply modeling the world as discrete, or modeling the world as continuous, is fine either way, until someone approaches you with an example of the other, and makes a paradox jump out at you.
If I point out to Metaphysician Undercover that he can get in his car and drive to the store without being crushed before he drives the first inch; am I failing to appreciate the profundity of his beliefs? I think not!
I think he's touching on something important. If time can be broken down into a collection of instants and if at one instant we're stationary and the next instant we're not then in one sense it does appear that we have undergone infinite acceleration. Pointing to our physical reality and suggesting that it proves he's wrong is besides the point. The real issue is with our assumption: that time can be broken down into a collection of instants. Or more generally, that a line is composed of infinite points. Anyway, you've already mentioned that you are puzzled by this so the analogy to Diogenes does not exactly fit so don't put much weight on my 'throwaway' comment.
If you can apprehend this, then why can't you turn it around, and see that when infinity is at the low end, there is no such thing as "the lowest number", and it doesn't make any sense to say that someone counting lower and lower is "approaching" the lowest number?
I don't have a problem with saying 'x approaches infinity' in the context of a potentially infinite process. I interpret it as 'the value of x is continuously growing'. 'x approaches infinity provides some information about the journey, even if we never can arrive at some final destination.
Consider the graph linked here: https://imgur.com/FZANGZ8
This is not a typical graph in that it spans all possible values of x and y. Think of it topologically in that it is a single system which maintains its topological properties when undergoing continuous deformations. In this plot, there is a point at (1,1) and a pseudo-point at (?,0). In this context, it makes sense to say that we're starting at (1,1), travelling along y=1/x and heading towards the pseudo-point at (?,0). By plotting Gabriel's Horn like this, (?,0) is no longer out of sight, it's right there in front of us. And because of that we have the ability to use it in some contexts without requiring infinite measuring capacity.
In practice the world is continuous (time passes continuously), but in theory the world is discrete (represented by distinct units, numbers).
I would argue that objects are continuous but measurements are discrete. This allows us to use the richness of mathematics that calculus offers while avoiding the paradoxes of actual infinity.
If time can be broken down into a collection of instants and if at one instant we're stationary and the next instant we're not
Second clause does not follow from the first. A mathematical line is composed of points. But there is no "next" point after any given point. You are confused on this ... point.
A mathematical line is composed of points. But there is no "next" point after any given point.
How can we travel from one point to the another without traversing through the intermediate points in sequence? This is essentially Zeno's paradox and if you cannot offer a resolution to it then you are not justified to claim that a line is composed of points.
By the way, how do the zero-dimensional, zero-length points in the unit interval make up the one-dimensional, length 1 unit interval? That's actually another mystery...
And you even admit that your view is shrouded in mystery. Why not consider the alternative...that a line is not composed of points, but instead points emerge from lines? Why won't you consider my...line...of thought?
I would argue that objects are continuous but measurements are discrete. This allows us to use the richness of mathematics that calculus offers while avoiding the paradoxes of actual infinity.
One theory that I've toyed with is that we have intuitions of both the continuous and the discrete that don't play nice together. Measurements are clearly discrete, as you say, but we also can draw the unit square and its diagonal and try to measure it 'perfectly' or 'ideally' and discover irrational numbers. The arithmetization of analysis was maybe driven by epistemological concerns. We want proofs in a universal language, and pictures aren't computer-checkable. (?)
Ryan O'ConnorMarch 07, 2021 at 01:42#5069400 likes
What does [that a line is not composed of points, but instead points emerge from lines] mean?
Imagine that lines are fundamental, not composite objects. Take a string and mentally label the two endpoints -? and ?. In my world, this string only has two points - the endpoints (I'd actually call them pseudo-points but that's not important here). Cut the string somewhere and mentally label the cut (i.e. the gap) 42. This string now has an additional 'point'. It doesn't matter where this cut was along the string as long as you follow this rule: if you cut the string somewhere between 42 and ? then that cut will be mentally labelled with a number greater than 42 (and if you cut the string somewhere between 42 and -? then that cut will be mentally labelled with a number less than 42).
The only object in my example was a string, which I'm using to describe a line. There are only a finite number of points on the string and they correspond to the cuts...the absence of string. The points only emerge when you make cuts and the numbers and their proper ordering are contingent on you (a 'computer') actively thinking about them. If you forget about the mental labels, then the numbers no longer exist. You are left with a string...a line...a continuum....and nothing more.
This way of building the parts from the whole is not new. It was (loosely) how Aristotle resolved Zeno's Paradox. I believe that this construction is paradox-free, but I'd love to hear your criticisms.
The real number line is composed of real numbers. How can you disagree with that?
What if instead of 'the real number line', I defined the object that's constructed from all of the real numbers 'the real number point'? Does my definition make it so? It may seem like I'm giving a silly example but I truly think you'd have a tough time explaining why your definition is better than mine, and I think this because I believe that the real numbers are singular. But since you are not convinced by Wildberger's arguments, what are the chances that an untrained engineer could convince you with informal reasoning. Perhaps your time is better spent telling me why my 'parts-from-whole' view is wrong rather than hearing me informally complain about why I think your 'whole-from-parts' view is wrong...
we also can draw the unit square and its diagonal and try to measure it 'perfectly' or 'ideally' and discover irrational numbers.
We can perhaps use Newton's method or some other algorithm to produce better and better approximations of sqrt(2) but trying to measure a 'perfect' value doesn't imply that you've discovered it. Perhaps all that you've discovered is an algorithm...and not an irrational number.
Metaphysician UndercoverMarch 07, 2021 at 02:32#5069500 likes
You were dismissive of my question without answering it and apparently not even getting the substance of it.
I said your hand is not at absolute rest because the earth is moving. And you asked me how do I know this. So I answered that. How am I supposed to know that you were asking me something other than what you were asking me?
If I am at rest in any sense whatsoever, then on your account any acceleration I'm subject to must be in the instant infinite. And that is absurd.
It isn't absurd, it just shows that the idea of an "instant", a zero point in time is absurd. The conclusion of an infinite acceleration is only produced from the idea that there is a point in time when a thing goes from resting to moving. Obviously then, what is absurd is the idea of a point in time, not the idea of rest. And the falsity of this idea (of a point in time) is borne out by special relativity which describes simultaneity (being how we determine a point in time) as frame dependent. There are no real points in time, they are arbitrarily assigned according to a frame of reference.
I don't have a problem with saying 'x approaches infinity' in the context of a potentially infinite process. I interpret it as 'the value of x is continuously growing'. 'x approaches infinity provides some information about the journey, even if we never can arrive at some final destination.
I don't like this phrase "approaches infinity" because it reifies infinity as a thing which is approached. Either the process is designated as infinite, or it is not. Suppose we don't know whether it is or it is not infinite, then we might say that it is potentially infinite, because we don't know. Perhaps, for some reason we are not ready to commit to infinity, like if someone worked out pi to hundreds of decimal points, and was still not convinced that it would go forever. That person would say that it appears to approach infinity, and it is potentially infinite, but I think it might still reach an end at some point, so I won't admit that it's actually infinite.
Don't you think that we have enough evidence to make the judgement, it is infinite? Suppose we say that it appears to be infinite, but it's still possible that it is not. How is this compatible with "approaches infinity"? Then you'd be claiming that there is some actual thing called "infinity", and the values appear to be heading in that direction. But how does this make any sense? What sort of thing could that be, which is called "infinity"?
This is not a typical graph in that it spans all possible values of x and y. Think of it topologically in that it is a single system which maintains its topological properties when undergoing continuous deformations. In this plot, there is a point at (1,1) and a pseudo-point at (?,0). In this context, it makes sense to say that we're starting at (1,1), travelling along y=1/x and heading towards the pseudo-point at (?,0). By plotting Gabriel's Horn like this, (?,0) is no longer out of sight, it's right there in front of us. And because of that we have the ability to use it in some contexts without requiring infinite measuring capacity.
But what is the meaning of that point which you label as (?,0)? How can ? represent a point? You say it's a "pseudo-point". I assume that this means that it's not a valid point. What's the point in having a non-valid point? I can see how it's useful in practice, but this is an exercise in theory.
god must be atheistMarch 07, 2021 at 02:39#5069530 likes
an infinitely long horn, were you to pour paint into it, could be filled with a finite amount of paint
How do you figure this? It's impossible. The integral of any function that is between zero and plus infinity and the curve of which never reaches the X axis, is infinity. And if the curve reaches the X axis, but on the opening end it is constantly widening, as a horn would be, then the definite integral is still infinity. I am obviously wrong in this opinion.
Ryan O'ConnorMarch 07, 2021 at 03:03#5069590 likes
if someone worked out pi to hundreds of decimal points, and was still not convinced that it would go forever. That person would say that it appears to approach infinity, and it is potentially infinite, but I think it might still reach an end at some point, so I won't admit that it's actually infinite.
When I say potential infinite, I don't mean 'something that might be infinite'. I mean a process that certainly goes on to no end. We are certain that if you begin to write out the digits of pi that that process would never end.
But what is the meaning of that point which you label as (?,0)? How can ? represent a point? You say it's a "pseudo-point". I assume that this means that it's not a valid point. What's the point in having a non-valid point? I can see how it's useful in practice, but this is an exercise in theory.
I call it a pseudo-point because it's not an actual point on the graph but instead a potential point. It's a point that you can approach but never arrive at. By 'never arrive at' I mean you can't plug one of its coordinates into a calculator to compute the other.
We can perhaps use Newton's method or some other algorithm to produce better and better approximations of sqrt(2) but trying to measure a 'perfect' value doesn't imply that you've discovered it. Perhaps all that you've discovered is an algorithm...and not an irrational number.
Well, yes! I'm certainly open to that view. Irrational numbers are fictions, constructions. The Cauchy sequence construction sorta-kinda says that a real number just is a streaming approximation.
For instance: 1, 1.4, 1.41, 1.414, 1.4142, .... just 'is' the square root of 2, if only we could ignore the extra complexity of equivalence classes. We can't, because any subsequence of the one above also represents root(2), and that's just the beginning of equivalent sequences.
Have you looked into Dedekind cuts? Consider the set of all rational numbers q such that q < 0 or q^2 < 2 for q >= 0. That set of rational numbers just is root(2), and we don't have to worry about equivalence classes. There are something like 15 constructions of the real numbers that I've heard of and looked into (some quite briefly, because some are quite complex and strange.)
If you want to use algorithms (an idea I like), it seems you need to either use mainstream computability theory or rebuild that too. But the computable numbers have measure 0, so you'll have to rebuild measure theory or stick with early analysis.
But to your original point: I'm happy with the word 'invented.' My point is that people wanted to connect symbol math to pictorial math and discovered that our two basic forms of intuition (discrete and continuous) don't play well together. (Consider Zeno's paradoxes in this light.)
Metaphysician UndercoverMarch 07, 2021 at 13:05#5071070 likes
The north pole of the Riemann sphere. But carry on.
Amazing, the things which mathematicians will come up with, in an attempt to solve their problems, instead of simply recognizing that the dimensional representation of space is wrong.
Amazing, the things which mathematicians will come up with, in an attempt to solve their problems, instead of simply recognizing that the dimensional representation of space is wrong.
When I say potential infinite, I don't mean 'something that might be infinite'. I mean a process that certainly goes on to no end. We are certain that if you begin to write out the digits of pi that that process would never end.
Why call this "potential infinite" then? If you are certain that the process goes without end, then you are certain that it is actually infinite.
We might however use terms like potential and actual to distinguish between things which have real existence in the world, and things which are completely imaginary. In this case, there is no such infinite process actually going on, so we say that if someone endeavoured to carry out that process they would find it unending, therefore infinite,. But since they would never arrive at "the infinite", we'd say that such an infinity only exists potentially. One could never prove it to be infinite by reaching infinity.
This is why I didn't like your use of "infinity". You used it as if it signified something with actual existence, which one could be approaching. It seemed as if you thought that if you carried out such an activity, then after a designated point you could be said to be "approaching infinity", when in reality you know that you would never be approaching infinity, because you clearly recognize that the process would never end without ever reaching infinity. Therefore that is a deceptive us of words, when you claim to be approaching something which you clearly acknowledge you can never reach.
Metaphysician UndercoverMarch 07, 2021 at 14:32#5071270 likes
Incidentally, GR and SR use dimensional representations of space. This includes with GR the use of Penrose diagrams, which have multiple infinities.
Exactly, look at the problems which envelope modern cosmology and quantum physics, due to the use of inadequate spatial representations, and a massively complex and fractured numbering system with imaginary numbers etc., required to cope with these problems.
The problem with Platonism (Pythagorean Idealism), is that once we assign to mathematics the status of objects, the objects obtain the status of unchangeable, and therefore eternal truth. Then it appears impossible that these mathematical constructs could be wrong. So when a problem emerges, something which cannot be figured out under the current mathematical system, a fix must be created, axiomatized, and added to the system. These fixes are nothing other than exceptions to the fundamental rules, causing the whole mathematical structure to get extremely top heavy with fix after fix. The fundamental rules which ought to be seen as faulty if they require such exceptions, cannot be apprehended as such, because they are already axiomatized as objects, eternal truths of Platonism.
Some time ago, at this forum, there was a discussion concerning the validity of Euclid's fifth postulate, the parallel postulate. It was argued that the geometry required by GR rendered the parallel postulate invalid. To any rational human being, this would indicate that the fundamental postulates of dimensional space need to be revisited and recreated. But human laziness inclines us to avoid this task, and simply make exceptions to the rules in an attempt to establish compatibility between what science now tells us about spatial existence, and what the rules created thousands of years ago say about spatial existence. Then we rationalize our laziness by saying that mathematical principles are eternal Platonic objects, and couldn't be changed even if we wanted to change them.
Metaphysician UndercoverMarch 07, 2021 at 14:37#5071280 likes
Reply to InPitzotl In case you didn't quite get it, what you call "multiple infinities" is what I called "fix after fix". Instead of addressing the problem which is the question of what causes the appearance of an infinity, the mathematicians create a "fix" to deal with the infinity.
the mathematicians create a "fix" to deal with the infinity.
...and how are you fixing that? And why should we trust a guy whining about lack of reality when it's the same guy who claims it takes an infinite amount of acceleration to move an object at rest?
Perhaps your time is better spent telling me why my 'parts-from-whole' view is wrong rather than hearing me informally complain about why I think your 'whole-from-parts' view is wrong...
I don't understand your idea at all. Suppose the position of a particle at time [math]t[/math] is given by [math]f(t) = t^3 - 5 t^2 + 9t - 6[/math]. Find the acceleration of the particle at t = 47.
How do you do that problem after you've thrown out 350 years of calculus and our understanding of the real numbers? What happens to the whole of physics and physical science? Statistics and economics? Are you prepared to reformulate all of it according to your new principles? And what principles are those, exactly? That there aren't real numbers on the real number line?
Ryan O'ConnorMarch 08, 2021 at 00:27#5074230 likes
For instance: 1, 1.4, 1.41, 1.414, 1.4142, .... just 'is' the square root of 2
I find this example unsatisfying given that everything important is contained in the ellipses. You are no better just writing "For instance: ... just 'is' the square root of 2". And so that equivalence class could just as well correspond to 42. The only way to give it meaning is to state the algorithm used for generating the sequence, which is why I think non-computable numbers are questionable since there is no algorithm behind them.
Here's a dumb question for you: how can the rational numbers (of which there are only aleph-0) can be cut in c unique ways? For example, if there are 2 numbers, then there's only 1 unique cut. If there are 3 numbers, then there are only 2 unique cuts. If we approach the limit, how do we end up with more cuts than numbers?
If you want to use algorithms (an idea I like), it seems you need to either use mainstream computability theory or rebuild that too. But the computable numbers have measure 0, so you'll have to rebuild measure theory or stick with early analysis.
I think our problem is that we're using numbers to model a continuum. As I'm discussing with fishfry in this thread, I think we should do the opposite and instead use a continuum to model numbers. I think flipping this on its head avoids the paradoxes, allows objects to have non-zero measure, and does not require us to decide between the discrete and continuous because they actually do play well together.
I am still waiting for your explanation of your claim that you know that nothing is at absolute rest. And this not a claim there is such a thing or place, but instead how it is that you know that there is not.
What I said is that I know your hand is not at absolute rest. And, I also said that if there is something at absolute rest, that thing would have to go through infinite acceleration to start moving. You, and some others here maybe, are the ones claiming infinite acceleration is absurd. I suggest to you, that instead of thinking about the absurdity of infinite acceleration, you simply apprehend absolute rest as absurd.
Then there is no need to think about infinite acceleration because it is only the idea of absolute rest which produces that idea. Make sense?
That problem is, the deficiency in our capacity to measure. Obviously, if the thing appears to be infinite, this means that we do not have adequate capacity to measure it.
I told you already. We need to revisit our spatial representations, from the beginning. Go back to the fundamental principles, armed with what we now know about space and time, derived from modern science, and rework them all, from the very beginning, starting with the most basic relationship between the non-dimensional point, and the dimensional line. The fundamental (Euclidean) axioms of geometry provide us with inadequate modelling principles which incapacitates our attempt to understand a large portion of spatial existence. Ryan gets it:
The real issue is with our assumption: that time can be broken down into a collection of instants. Or more generally, that a line is composed of infinite points.
And you even admit that your view is shrouded in mystery. Why not consider the alternative...that a line is not composed of points, but instead points emerge from lines? Why won't you consider my...line...of thought?
I have a slightly different way of looking at this issue. I give priority to non-dimensional existence, represented as points. Points can be related to each other through lines, but these are still non-dimensional, meaning that these lines do not correspond to real physical, dimensional existence, they are ideals, by which we relate other ideals, points. To establish a correspondence between the non-dimensional, ideal, the point, and the dimensional, real physical existence, we need to bring time in as the first "dimension". Time must be the most fundamental mathematical representation as simply order. Order can be expressed without spatial reference. An understanding of the passing of time will determine the relationships which points can have with each other in real dimensional existence, and geometrical figures must be produced in accordance with the principles derived.
I don't understand your idea at all. Suppose the position of a particle at time tt is given by f(t)=t3?5t2+9t?6. Find the acceleration of the particle at t = 47.
How do you do that problem after you've thrown out 350 years of calculus and our understanding of the real numbers? What happens to the whole of physics and physical science? Statistics and economics? Are you prepared to reformulate all of it according to your new principles? And what principles are those, exactly? That there aren't real numbers on the real number line?
Because I'm lazy, I'm going find the velocity of the particle instead of acceleration. First, I want to show you three valid graphs of y=t3-5t2+9t-6 (valid in my construction, that is).
You have to remember that these plots are topological so even though it looks linear in the first image I could have just as well drawn it with squiggles. The key thing is that I've only made a finite number of 'cuts' (exactly like cutting the string in my last post to you, but this time I'm cutting a 2D continuum). All of the information contained in these three graphs is correct. It's just that the graphs have the potential to be cut infinitely more times. The more cuts I make, the more points will emerge and I (a 'computer') will make sure that the points have coordinate values consistent with the functions.
First, I need to point out that (in my view) velocity at an instant is meaningless. It's equivalent to 0/0. Velocity applies, not to points, but to curves. In the third graph I've highlighted a section of the function between x=47 and x=48. As normal, I can calculate the average velocity across this interval using the coordinates as follows (99498-93195)/(48-47)=6303.
But obviously we're not satisfied with that. We want to shrink this interval as much as possible. And we can do so by making cuts closer and closer to x=47 and finding the average velocity across those shrinking intervals. This is what the limit describes (in my construction), it is a potentially infinite process.
What calculus does is describe the potential of that process. And I believe that when calculus was made rigorous by going from numbers (infinitesimals) to processes (limits) some 'infinite-like' numbers (irrational numbers) were left behind. I believe that to complete the job, we need to reinterpret irrational numbers as irrational processes. Calculus is the study of potentially infinite processes. In my view, the math is the same, dy/dt=3t2-10t+9. It's just that the philosophy is different.
This may seem like a trivial difference, but I believe that with this continuum-based view (as opposed to the standard points-based view) many paradoxes are no longer paradoxical. In fact, I can't even think of a paradox with this view (especially given our refined intuitions developed through quantum-mechanics).
I hope I was clear in my explanation. I'd love to hear your feedback, especially if you have criticisms. Thanks!
I find this example unsatisfying given that everything important is contained in the ellipses. You are no better just writing "For instance: ... just 'is' the square root of 2". And so that equivalence class could just as well correspond to 42. The only way to give it meaning is to state the algorithm used for generating the sequence, which is why I think non-computable numbers are questionable since there is no algorithm behind them.
Consider though that ellipsis are just shorthand that lazy mathematicians use for one another. In this case, it should be obvious how the sequence proceeds. Lots of different algorithms can give the exact same sequence, and that's why equivalence classes are necessary.
Personally I think non-computable numbers are questionable, but root(2) is of course not one of them and is ultimately a finite object (since there's a finite Turing [actually many!] machine that gives arbitrarily good approximations. )
Just to be clear, I think metaphysics in general is a bust. To talk about numbers is (for me) to talk about the talk about numbers. So I'm just quoting some mainstream math, pointing out some of the issues.
Here's a dumb question for you: how can the rational numbers (of which there are only aleph-0) can be cut in c unique ways? For example, if there are 2 numbers, then there's only 1 unique cut. If there are 3 numbers, then there are only 2 unique cuts. If we approach the limit, how do we end up with more cuts than numbers?
Don't forget the jump between finite and infinite sets! A typical limit operation doesn't make sense here. I do like your question, because it highlights the strangeness of math. (The diagonal proofs, when I first saw them, were lots of what seduced me away from engineering toward pure math and proofs. I've come up with some of my own to prove smaller subsets of R uncountable, where the diagonal is not perfect...fun stuff, engineering with infinity, in pure thought.)
I think our problem is that we're using numbers to model a continuum. As I'm discussing with fishfry in this thread, I think we should do the opposite and instead use a continuum to model numbers. I think flipping this on its head avoids the paradoxes, allows objects to have non-zero measure, and does not require us to decide between the discrete and continuous because they actually do play well together.
I'm all for bold ideas. I don't know of any paradoxes. I think 'discomforts' is better. AFAIK, mainstream math works, is correct (even if we can't prove it.) The only problem is that it offends lots of peoples intuition here and there.
You have to remember that these plots are topological so even though it looks linear in the first image I could have just as well drawn it with squiggles.
I know what you mean here, but reading it makes me uneasy.
Imagine that lines are fundamental, not composite objects. Take a string and mentally label the two endpoints -? and ?. In my world, this string only has two points - the endpoints (I'd actually call them pseudo-points but that's not important here).
I like this string. It's maybe how we first think of the line. This idea would require a radical change in the foundations, sounds like even set theory is jettisoned. If you could rewrite a calculus textbook so that calculations come out the same (so as not to clash with mainsteam math in applications), it could be presented as a pedagogical alternative. The vibe I'm getting is pre-proofs applied mathematics, because there's no mention of axioms. You are offering a nice intuitive foundation. If you want a 'normal discourse' (an objective discipline where disputes can be resolved somehow) you'll have to come up with rules, something like axioms. It could be a calculus in the old sense of the world, with rules like arithmetic except in a calculus context. (I think non-math people experience calculus this way already.)
Epistemologically speaking, why should people trust the results? If you get mainstream results, all is well. You could just present Newton's calculus with different metaphors. What would you do with limits? Infinite sums?
To sum up, nice basic metaphor, but you'd probably have your hands full fleshing it out (which is not meant to be discouraging)
But obviously we're not satisfied with that. We want to shrink this interval as much as possible. And we can do so by making cuts closer and closer to x=47 and finding the average velocity across those shrinking intervals. This is what the limit describes (in my construction), it is a potentially infinite process.
What calculus does is describe the potential of that process.
If you look at the definition of a limit, it's actually timeless. For all epsilon > 0 , there exists a delta > 0 such that ETC. So there is a leap from the intuition of the potentially infinite approximation process. The fundamental question is something like: what are we approximating? A limit is a real number, a point, and not the process (in the mainstream view). Different processes can converge to the same point. (Subsequences make this obvious, but it's not only subsequences.)
We can draw the symbol root(2) confidently because we can prove that it exists from the axioms, (IVT) entirely without pictures. The desire to free math from pictures should perhaps be addressed here. Can your system free itself from pictures? A theory of continua would presumably have to be symbolically established. Would classical logic work? Would you still find the system charming if the pictures were secondary and only props for the intuition? (Just trying to ask productive questions. Hope they inspire you!)
Metaphysician UndercoverMarch 08, 2021 at 13:48#5076710 likes
What calculus does is describe the potential of that process. And I believe that when calculus was made rigorous by going from numbers (infinitesimals) to processes (limits) some 'infinite-like' numbers (irrational numbers) were left behind. I believe that to complete the job, we need to reinterpret irrational numbers as irrational processes. Calculus is the study of potentially infinite processes. In my view, the math is the same, dy/dt=3t2-10t+9. It's just that the philosophy is different.
This is very good insight. My belief is that we need to go one step further, and apprehend an infinite process, or "irrational process" as actually impossible. But since this process is a potential process, as you describe, this means that it is a possible process which is actually impossible. Therefore the infinite process must be rejected as logically invalid, because it's contradictory.
Rejection of the impossible needs to be recognized as the greatest epistemological tool which human beings possess. It is the basis for certainty in knowledge. In our quest for an understanding of "what is" we narrow the field of possibilities by rejecting any proposed possibility which can be determined as impossible.
In the case of the infinite process, we have a very difficult judgement. It appears to be a reasonable and logically valid possibility. But for some reason it's extremely repugnant to intuition, and the reason why it is repugnant cannot be properly identified, so as to prove logically that it is impossible. It appears as absolutely impossible to show, or demonstrate the impossibility of infinite possibility, because it cannot be done with an empirical demonstration. There is however a logical demonstration commonly known as the cosmological argument, in Aristotle's Metaphysics, and I assume that other forms may be produced.
This may seem like a trivial difference, but I believe that with this continuum-based view (as opposed to the standard points-based view) many paradoxes are no longer paradoxical. In fact, I can't even think of a paradox with this view (especially given our refined intuitions developed through quantum-mechanics).
I believe that this distinction between the continuum perspective and the points perspective is a very good start, but I don't think it's an either/or question. We need to allow for both. It is the application of both, the two being fundamentally incompatible, which leads to infinity, and the appearance of paradoxes. However, we cannot simply exclude one or the other as unreal, and unnecessary, because there is a very real need for both non-dimensional points, and dimensional lines. You cannot remove the points because this would invalidate all individual units, therefore all number applications would be arbitrary.
What I propose is a fundamental division between numerical arithmetic and geometry, which recognizes the incompatibility between these two. Numbers refer to discrete units, and geometrical constructs refer to a continuum. We ought to recognize that reality consists of both these aspects, and the metaphysical question which we are faced with is to determine in which circumstances each system is applicable. Of course the need to establish a system of correlation between the discrete and continuous will never go away, this is mathematics, but we need to get a firm handle on which aspects of reality are discrete, and which are continuous, before we can axiomatize that correlation in a way which might serve us adequately.
If you look at the definition of a limit, it's actually timeless. For all epsilon > 0 , there exists a delta > 0 such that ETC. So there is a leap from the intuition of the potentially infinite approximation process. The fundamental question is something like: what are we approximating? A limit is a real number, a point, and not the process (in the mainstream view). Different processes can converge to the same point. (Subsequences make this obvious, but it's not only subsequences.)
The problem I see here, is that a process is fundamentally continuous. If it were not continuous we would identify it as a number of different processes. We apply a point, to limit or individualize a process, marking a beginning or ending. But if these points are arbitrarily applied, i.e. not in reference to real points in the apparently continuous world, the numbering system will not provide us with truth (in the sense of correspondence).
When we look at the physical world empirically, it appears like we find true limits in spatial extensions, the boundaries of objects. The reality of these spatial limits have always justified individuals, units, numbers, and quantity, with distinct spatial objects forming discrete entities. However, now that we've come to look closer at these entities, the boundaries have become vague, and when we take the physical object right down to the micro level, those boundaries are not at all valid. So we find that the empirical evidence has really been misleading us, we think that there are spatial boundaries, and distinct entities when there really is not, spatial existence is continuous.
Likewise, if we look at time rationally, (because we cannot look at it empirically), we find the exact opposite situation. Our intuition tells us that time is continuous, that's how it seems to be, time is continually passing. However, we know that there is a big difference between past and future. Since there is such a difference, it is necessary to assume a boundary between past and future. If such a boundary exists, (and it is logically necessary that it does, to maintain a difference between past and future), then this boundary will provide the necessary principles for discrete units of time.
Yes, and now for the third time you have - I have to presume - deliberately evaded the question. Which is unfortunately par for the course for you. Which earns for you a change of tone. How the F do you know, you ******* ******?
Do you understand the principle, that if you have two premises which you do not know whether they are true or not, and they lead to a logical conclusion which is obviously false, then one or both of the premises must be false? The two premises are, rest may be absolute, and there is motion. These two premises lead to the conclusion of infinite acceleration. The conclusion is obviously false. The second premise, "there is motion" looks true, so the first must be false.
Small point. I have asked you how you now something, and you have just exhibited that you do not know it, but instead accept it as the consequence of an argument, which is neither an answer to my question nor, if the argument is otherwise flawed, defensible in that way either.
I give up. I've tried numerous times to answer your question, and it appears I have no idea what you're asking. It seems like you're having the same difficulty with the word "absolute", which you had the last time we discussed the use of that word. You just carry on as if the word isn't there, and insist that it has no meaning. So, onward with your diatribe if it makes you feel good, tim.
It seems like you're having the same difficulty with the word "absolute"
Actually, it seems you're having problems with the word "acceleration". If an object at rest remains at rest, it is ipso facto not accelerating. Likewise, if an object at rest accelerates, it ipso facto starts moving. But somehow in MU land, absolute means something can stay at rest and still accelerate, so long as it's not accelerating infinitely. Whatever that means.
Consider though that ellipsis are just shorthand that lazy mathematicians use for one another. In this case, it should be obvious how the sequence proceeds. Lots of different algorithms can give the exact same sequence, and that's why equivalence classes are necessary.
It's only obvious to me because I only know a handful of real numbers so I assume you're talking about sqrt(2). But it's not a matter of laziness, no finite amount of terms would have allowed me to eliminate any possibility. From this view (when there is no algorithm) it seems like the only important number in a Cauchy sequence is the last one...and there is no last one! Anyway, sorry for putting you in a position having to defend a position you don't support!
I'm all for bold ideas. I don't know of any paradoxes. I think 'discomforts' is better. AFAIK, mainstream math works, is correct (even if we can't prove it.) The only problem is that it offends lots of peoples intuition here and there.
There are a lot of infinity-related paradoxes which offend students' intuitions. Given that it drew you in, perhaps the strangeness is a strength (not a weakness) of infinity. I have no doubt that math works, it's 'why' that has puzzled me for many years.
This idea would require a radical change in the foundations, sounds like even set theory is jettisoned. If you could rewrite a calculus textbook so that calculations come out the same (so as not to clash with mainstream math in applications), it could be presented as a pedagogical alternative.
I haven't worked it out, but IF this is a valid perspective I don't think the math would change much. I think that (in a way) this is consistent with how we've been thinking all along (as I described to jgill below). We'd write the same equations and draw the same pictures, we'd just think about it differently. I think it's a matter of philosophy, not math. But you're totally correct, this is only an intuition...an idea...a formal theory is a whole different thing. And you're right...I'd have my hands full fleshing it out. And to be honest, I don't have the necessary skills to flesh it out.
I believe that my view is in agreement with limits and infinite sums as long as you think of them as descriptions of unending processes. For example, 1+1/2+1/4+1/8+... corresponds to the unending process of computing larger and larger partial sums which approach but never arrive at 2.
The fundamental question is something like: what are we approximating?
I think this question is very important. In my view, the topological graphs that I drew actually exist. The geometric graphs that we imagine imagining don't exist, but they are incredibly convenient approximations of what we could do in reality to topological graphs.
A limit is a real number, a point, and not the process (in the mainstream view).
True, but what if we reinterpret real numbers as real processes which describe continua, not points? Wouldn't we be able to keep the same math? Can't we just say that our algorithms for calculating the 'number' pi can never output the number completely and that pi actually corresponds to those (potentially infinite) algorithms? Why do we need the number pi anyway? We have never precisely used it as a number anyway.
We can draw the symbol root(2) confidently because we can prove that it exists from the axioms, (IVT) entirely without pictures. The desire to free math from pictures should perhaps be addressed here. Can your system free itself from pictures? A theory of continua would presumably have to be symbolically established. Would classical logic work? Would you still find the system charming if the pictures were secondary and only props for the intuition? (Just trying to ask productive questions. Hope they inspire you!)
These are good questions indeed, thanks!! I don't know the answers to them, but here are my thoughts. There's a convenience to the completeness of the real numbers. We get a one-to-one correspondence between our numbers/equations and our graphs. If in our graphs, lines cross at a point, then we can always find the coordinates of that point (and most often they'll be irrational numbers). If we calculate the derivative of a function at a point, then we can always draw a tangent at that point. With my view, we cannot always do these as they would amount to completing an infinite process. We need to be clever to avoid actual infinity and I think the way to do it is to use both numbers/equations and graphs. Sometimes numbers/equations are needed to describe a system, sometimes graphs are. We can't avoid the pictures.
As for logic, this puts me even further out of my comfort zone but I'm guessing some hybrid would be required: classical logic for the discrete points and fuzzy logic for the continua. This may seem like a patchwork solution to connect the discrete with the continuous but it has parallels to what we do with quantum mechanics to describe our universe (measured objects have definite states and unobserved objects remain in a superposition of potential existence).
I know what you mean here, but reading it makes me uneasy.
I appreciate why it makes you feel uneasy, but consider this claim: Every** graph that you have ever seen was topologically precise and geometrically approximate. For example, if you and I were to independently sketch out a graph of y=x^2-2, our graphs would share few geometric properties. They would be of different sizes and neither would perfectly capture the curvature of the parabola. But when we compare our graphs we still see them being the same because we continuously deform the graphs in our mind to see that they correspond to the same system. And this goes for computer generated plots as well. A computer plotting y=x^2-2 does not place infinite points on your screen but instead calculates the rational coordinates of a finite set of points and imprecisely connects the dots. But once again, we have no problem with it because it is topologically precise. The only difference is that in my plots I continuously deformed the graphs to a state where it was clear that they were not geometrical.
With 'parts-from-whole' constructions, the systems are topological and described perfectly (without approximation). There's no need to imagine a cloud of infinite points, what you see is what you get. It's just that these plots have infinite potential to be cut.
**When I said 'every graph' in the first sentence of this response I was exaggerating a little because some systems cannot be drawn with perfect topological precision. For example, y=sin(1/x) and y=1 cannot be plotted in the same graph because we'd have to draw infinite points where they interest and that's impossible. In such cases I'd just conclude that that pair of functions cannot be plotted simultaneously. I know it's an odd claim, but it's not too far from the complementarity principle in quantum mechanics which holds that objects have certain pairs of complementary properties which cannot all be observed simultaneously.
I see that you're a retired math professor so I'm especially keen to hear your feedback, especially if you see a flaw in my argument. Thanks!
My belief is that we need to go one step further, and apprehend an infinite process, or "irrational process" as actually impossible. But since this process is a potential process, as you describe, this means that it is a possible process which is actually impossible. Therefore the infinite process must be rejected as logically invalid, because it's contradictory.
We can do so much with potentially infinite processes. Not only can we interrupt them to produce rational numbers, but we can work with the underlying algorithms themselves. For example, the following program to outputs the entire list of natural numbers. This program can never be run to completion, but it still is a valid program...I'm talking about it after all and it makes sense even though I've never run it. The same can be said about irrational processes. We need to embrace potential infinity for what it is, not reject it.
I believe that this distinction between the continuum perspective and the points perspective is a very good start, but I don't think it's an either/or question. We need to allow for both. It is the application of both, the two being fundamentally incompatible, which leads to infinity, and the appearance of paradoxes. However, we cannot simply exclude one or the other as unreal, and unnecessary, because there is a very real need for both non-dimensional points, and dimensional lines. You cannot remove the points because this would invalidate all individual units, therefore all number applications would be arbitrary.
In my continuum-based constructions there are still points, it's just that there are only ever finitely many of them and they are not fundamental. Can you expand on a situation where points need to be fundamental?
What I propose is a fundamental division between numerical arithmetic and geometry, which recognizes the incompatibility between these two.
I think we might be on the same page regarding this issue. I mentioned to norm above that I think numbers/equations and graphs are complementary, not equivalent. The problems occur when we think there is a one-to-one correspondence between the two.
I see that you're a retired math professor so I'm especially keen to hear your feedback, especially if you see a flaw in my argument.
What makes me uneasy is using a concept like topological equivalence and then discussing slopes and derivatives, which, as you know, do not carry over in that way. As for your continuum ideas, almost twenty years ago Peter Lynds wrote a paper appearing in Foundations of Physics Letters that postulated time having no instants and instead being composed of intervals. Something like Bergson's notions from a century ago. Here it is.
It's only obvious to me because I only know a handful of real numbers so I assume you're talking about sqrt(2). But it's not a matter of laziness, no finite amount of terms would have allowed me to eliminate any possibility. From this view (when there is no algorithm) it seems like the only important number in a Cauchy sequence is the last one...and there is no last one! Anyway, sorry for putting you in a position having to defend a position you don't support!
This is one of the weird thing about pure math (including computability theory.) We are OK with proving or assuming the existence of a number logically without having to specify that number. The terms of a Cauchy sequence get arbitrarily close, but we don't when that will happen with an arbitrary Cauchy sequence. I'm used to this stuff. Analysis came pretty easy to me, once I entered a logical frame of mind and let go of metaphysics. Intuition is still in play, but it works in tandem with a logical instinct. There's nothing else like it. I also love to program, but programming is different. One builds logical spiderwebs, trapping numbers with inequalities. I found it seductive.
Indeed! But calculus was always haunted by infinity. For a long time, the spirit was 'calculate! faith will come.' The results were good. Reliable technology and predictions emerged from the fast and loose application of a calculus that bothered the philosophers. It's very hard to avoid intuitions of infinity. Ilike the spirit of finitism, but it gets cramped and awkward quickly. Call the largest integer Z. Then Z+1 is larger. Physical arguments against this (like Wildberger's, which I browse) don't convince me, because there's an ideality to math that seems close to its essence. A Turing machine either halts or not on a certain input. I may not know which, but intuitively that's clear to me. Note that a Turing machine is a completely imaginary entity in the first place, at the heart of computability theory. So even your attachment to algorithms is threatened by problems with infinity. Before long we're back to lots of hand waving, no strict definitions. That's fine for practical purposes perhaps, but it's basically an abandonment of math as a distinct discipline.
I think this question is very important. In my view, the topological graphs that I drew actually exist. The geometric graphs that we imagine imagining don't exist, but they are incredibly convenient approximations of what we could do in reality to topological graphs.
The danger here is replacing strict symbolic reasoning with pictures. In some analysis books, there's not a single picture, for epistemological reasons you might say. Pictures often mislead, while obviously being of great use pedagogically for applied math's well-behaved functions.
True, but what if we reinterpret real numbers as real processes which describe continua, not points? Wouldn't we be able to keep the same math? Can't we just say that our algorithms for calculating the 'number' pi can never output the number completely and that pi actually corresponds to those (potentially infinite) algorithms? Why do we need the number pi anyway? We have never precisely used it as a number anyway.
Again, the question is what are we approximating? Why do believe in a single pi in the first place? The circle is already an ideal object. If we decide to think of Turing machines that give decimal expansions as real numbers, then we still need equivalence classes. For each computable real number there are an infinite number of Turing machines that approximate it. Which one will we call pi? Do we put some bound on the number of steps needed to give us digits? Turing machines could be made arbitrarily slow (programmed with wasted motion). (In other words, problems with the real numbers in pure math don't go away when we switch to computability theory --which is itself pure math, awash in idealities. If you want necessary non-empirical truths, I think you are stuck with infinity, unless there's a largest integer. )
Sometimes numbers/equations are needed to describe a system, sometimes graphs are. We can't avoid the pictures.
But pure math has successfully avoided the pictures. It finally triumphed over an uncertain prop. The issue here is systematic reasoning in a strict language which seemingly must be discrete. Perfectly formal proofs of theorems are strings of symbols. The whole 'right or wrong' charm of math is caught up in this. As you may know, some engineers continued using infinitesimals when they were ejected by pure math, simply because they found them convenient and gave good results. Since then, infinitesimals have been rehabilitated via the hyperreals (they were made rigourous, without any metaphysical commitment), and a small minority of calc students learn calculus this way. I have a thin Dover book that presents them.
*Read lots of math books and you may end up an open-minded skeptic who sees the pros and cons of different approaches. You'll like system A for this charming thing and system B for another. They'll all get something right (for your intuition) and something wrong. Meanwhile all of them are correct in the sense of working logically, not appearing to lead to contradictions, and giving the expected applied results that help us build bridges that don't fall down. I'm no expert, but I think Newton's and Euler's calculus (lacking clear foundations) would be enough for most human practical purposes.
Pi only encodes a finite amount of information. [math]\displaystyle \pi = 4 \sum_{k = 0}^\infty \frac{(-1)^k}{2k + 1}[/math]. That's 16 characters if I counted right.
I don't see why this discussion is hung up on such an obvious point. Pi is no more mysterious than 1/3 = ...3333, another number that happens to have an infinite decimal representation. Decimal representation is handy for some applications and not for others. Decimal representation is broken. Some real numbers have infinite representations and others have two distinct representations. You should never confuse a number with any of its representations, so all of the other expressions for pi are just as valid.
Of course I realize that. I'm trying to show Ryan the difficulties with his approach.
I haven't been able to understand Ryan's approach. As in Apocalypse Now, when Kurtz asks, "Are my methods unsound?" And Willard responds: "I don't see any method at all, Sir."
I'm trying to meet him half way, because I do find these issues fascinating. I don't think Ryan has the experience to see math as mathematicians see it. In physics and engineering classes, one can go very far without resolving these issues or even seeing a proof. So a kind of 'outsider's mathematics ' (like outsider art) is a natural result when a person gets mathematically creative. I agree that one has to study some actual proof-driven math to genuinely enter the game, but I also understand the impatience to talk about exciting ideas now. I'm sure that Ryan is learning, and I get to dust off some math training.
Ryan O'ConnorMarch 09, 2021 at 13:43#5081760 likes
I haven't been able to understand Ryan's approach. As in Apocalypse Now, when Kurtz asks, "Are my methods unsound?" And Willard responds: "I don't see any method at all, Sir."
I like your quote and I see where you're coming from, especially given that I'm talking so informally. While I do not have the ability to formally present the idea, I did make a video in which I attempt to describe the intuition behind my method.
Would you please consider watching this 2 minute clip (watch from 1:56-3:48)?
Thanks!
Metaphysician UndercoverMarch 09, 2021 at 14:37#5081970 likes
We can do so much with potentially infinite processes. Not only can we interrupt them to produce rational numbers, but we can work with the underlying algorithms themselves. For example, the following program to outputs the entire list of natural numbers. This program can never be run to completion, but it still is a valid program...I'm talking about it after all and it makes sense even though I've never run it. The same can be said about irrational processes. We need to embrace potential infinity for what it is, not reject it.
I'm in complete agreement that an infinite process is a logically valid process. That's what I said, but it's counterintuitive to think that any process could 'live' forever. This is the same issue which the ancients had with the immortal soul. It's valid logic, but there's something wrong with the premises, which makes the conclusion unsound, despite the fact that the logic is valid. Furthermore, the "immortal soul" served as a very useful moral principle, just like the "infinite process" serves as a very useful mathematical principle, but usefulness does not necessitate truth, if truth is what we are ultimately after..
To say that a process is infinite is to say that it will run forever. That is what you are claiming with "infinite processes". Notice, that to say "X process is infinite", is to say "it will run forever", and that's a statement of necessity, just like to say "the soul is immortal" is to say it will, necessarily live forever. But of course you respect the possibility that the process may for some reason, at some time, cease. Therefore you say that it is "potentially" infinite. It could run forever, if it's allowed to. There is a clear difference between "it will run forever" and "it could run forever".
What I am requesting is that you recognize the constraints of the physical world, to acknowledge that an infinite process is not possible within our universe. Therefore both "it could run forever", and "it will run forever" are excluded as impossible, therefore false premises. From this perspective we see that any problem which requires an infinite process to resolve, is actually not solvable. If a question is not solvable, it is not properly posed, it is not a valid question.
You say: "I have no doubt that math works, it's 'why' that has puzzled me for many years." The simple answer is that it works because it conforms to the constraints of our universe. We can dream up a seemingly infinite number of logically possible axioms which will be completely useless in our universe. It's correspondence with the physical reality which makes them useful. Some people will make a distinction between coherence and correspondence, claiming that coherence is all that is necessary within a system of logic, but coherence itself is fundamentally a correspondence. Fundamental laws of coherence, like non-contradiction, work because they correspond with our universe. So a coherent logical structure has a basic correspondence simply by being coherent.
You might think that this completely contradicts what I said above, "usefulness does not necessitate truth". However, we must maintain the distinction between sufficient and necessary. Proof requires necessity. So despite the fact that correspondence (truth) works, we need to also be aware that there are other reasons why principles, like mathematical axioms, work. And this is dependent on the end, the goal which we have in mind, by which "works" is judged. If the goal is not itself truth, then the axioms are formulated toward that alternative goal, and "work" for that alternative purpose. I find that in the modern sciences, which are the principal users of mathematics, the goal has shifted from truth to prediction, and these two are not the same.
So I think you need to adjust your enquiry from 'why does math work?', to 'what does math do?' The point being that so long as people are applying it, it will work, otherwise they wouldn't be using it. The reason it works is because it's designed, shaped, conformed to the purposes which it is put to, like a finely honed tool. But if that purpose is something other than giving us truth, then sure it "works", but is what it's doing really good?
In my continuum-based constructions there are still points, it's just that there are only ever finitely many of them and they are not fundamental. Can you expand on a situation where points need to be fundamental?
The issue here is with the way that I conceive of the relation between space and time. It is not conventional. With evidence derived from modern observations of phenomena like spatial expansion, it becomes clear to me that we need to give time priority over space in our modeling, to allow that space itself changes with the passing of time. This means that we cannot model spatial existence with a static 3d representation, adding time as a fourth dimension, because we need to allow that the principles for geometrical figures which model 3d space must actually differ being time dependent.
There is a need to produce two dimensions of time, one consistent with our present modeling of space as a static continuity of 3d existence, and the other to allow for the changes which occur to space, they require time as well, but this cannot be the same dimension of time. The finite points you refer to are the points where the two dimensions of time relate, or intersect with each other. So there is a continuum of spatial existence, extended in time, that is the classical 3d modelling. The points within that continuum need to be more fundamental because they represent where the other dimension of time intersects, thus constituting the real possibility for spatial existence. If we propose an infinite continuum of space, and we want to map onto this continuum real finite points of possible existence, then the limited location of those points must be derived from something real. That "real thing" must be more fundamental, because it represents real existence whereas the infinite continuum is the artificial map produced by us. The real points are points of spatial expansion (in this proposed model), which dictate the contortions to classical 3d spatial representations required to account for spatial expansion.
As for your continuum ideas, almost twenty years ago Peter Lynds wrote a paper appearing in Foundations of Physics Letters that postulated time having no instants and instead being composed of intervals.
The problem with this "intervals" of time is that some sort of instants are still require to separate one interval from another. Anything posited to break the apparent continuity of time would require a distinct aspect of time, calling for a second dimension. And if the intervals in any way overlap then there is also a need for multidimensional time.
I like your quote and I see where you're coming from, especially given that I'm talking so informally.
It was wrong of me to poke fun at you without responding to your last two lengthy posts to me. Fact is I wouldn't know where to start so it's better for me to leave it alone. If you can't graph a simple polynomial then there's no conversation to be had. Note that the poly I gave you has a real root at 2 which none of your pictures show. And your claim that there's no such thing as instantaneous velocity is falsified by your car's speedometer. I'm sure you have some interesting ideas but I probably won't engage much going forward, and I'll refrain from indirect remarks even if they are from great movies.
Ryan O'ConnorMarch 10, 2021 at 02:48#5084230 likes
It was wrong of me to poke fun at you without responding to your last two lengthy posts to me. Fact is I wouldn't know where to start so it's better for me to leave it alone. If you can't graph a simple polynomial then there's no conversation to be had. Note that the poly I gave you has a real root at 2 which none of your pictures show. And your claim that there's no such thing as instantaneous velocity is falsified by your car's speedometer.
No problem at all. I appreciate the message! Although I don't expect a response, I do want to say a couple of things. I obviously know how to plot a polynomial in the traditional sense (and I also know how to use plotting programs). If you don't see a polynomial in my graphs it's because you don't understand my view (I'm not blaming you, this may be entirely my fault). Had I chosen to also plot y=0 then you would have seen the points corresponding to the roots. Your speedometer is measuring the average velocity but one measured over quite a short time interval. And I enjoy the quips, even that's all I hear from you.
What makes me uneasy is using a concept like topological equivalence and then discussing slopes and derivatives, which, as you know, do not carry over in that way.
If I didn't use the word topology would you have any other problems with my view? I glimpsed the article and paper. I certainly agree with his postulate: 'there is not a precise static instant in time underlying a dynamical physical process.' It seems obvious, really. But there are statements like 'there is no physical progression or flow of time' and 'a body in relative motion does not have a precisely determined relative position at any time' which I'm not convinced by. Overall, his paper is more like an essay than the type of paper I'd expect to see in a journal (but to be clear, my ideas are no closer to journal standards).
So even your attachment to algorithms is threatened by problems with infinity. Before long we're back to lots of hand waving, no strict definitions. That's fine for practical purposes perhaps, but it's basically an abandonment of math as a distinct discipline.
Maybe, but maybe my lack of strict definitions is simply because my ideas are not mature yet.
The danger here is replacing strict symbolic reasoning with pictures. In some analysis books, there's not a single picture, for epistemological reasons you might say. Pictures often mislead, while obviously being of great use pedagogically for applied math's well-behaved functions.
You have a good point so please allow me to soften my position. Perhaps pictures are only a handy prop in my view but the lack of symbolic reasoning may only reflect that my view is not mature.
There are infinite potential chairs. Must all potential chairs actually exist to give the word chair meaning? The 'chairness' algorithm must be finite otherwise we'd never call anything a chair. Perhaps the same can be said about pi. Perhaps on the deepest level, pi is not the number pi, nor the infinite algorithms used to calculate the number pi, but instead the finite algorithm used to identify which algorithms would generate the number pi.
In other words, problems with the real numbers in pure math don't go away when we switch to computability theory --which is itself pure math, awash in idealities. If you want necessary non-empirical truths, I think you are stuck with infinity, unless there's a largest integer.
I don't think your criticisms of finitism apply to my view. In my view, every system does have a largest number, it's just that there's no universal system containing all possible numbers. For example, in the graph below the largest number is 99498. We could certainly 'cut' the continuum to produce points with coordinates having larger values, but until we actually do that it is meaningless to assign coordinates to those potential points. Could you expand on how I'm stuck with actual infinity?
Since then, infinitesimals have been rehabilitated via the hyperreals (they were made rigourous, without any metaphysical commitment), and a small minority of calc students learn calculus this way.
I've read the Dover book on infinitesimal calculus by Keisler. It must be different from yours because mine isn't so thin. I'm not convinced that there are irrational numbers between the rationals, I'm even less convinced that there are infinitesimals in between the reals. But you're the professional and you've seen the proofs to conclude that the reasoning is rigorous so I don't want to debate about this issue.
I'm trying to meet him half way, because I do find these issues fascinating. I don't think Ryan has the experience to see math as mathematicians see it. In physics and engineering classes, one can go very far without resolving these issues or even seeing a proof. So a kind of 'outsider's mathematics ' (like outsider art) is a natural result when a person gets mathematically creative. I agree that one has to study some actual proof-driven math to genuinely enter the game, but I also understand the impatience to talk about exciting ideas now. I'm sure that Ryan is learning, and I get to dust off some math training.
I am thoroughly enjoying this discussion and I appreciate your pointed questions. So far, my view is that you've clearly demonstrated how far my view is from a formal theory (thanks!) but you haven't identified any flaws yet. You're right, I don't see it as mathematicians see it. And so a mathematician might say that my probability of being right is 0. Thankfully, that means mathematicians still believe I have a chance!
What I am requesting is that you recognize the constraints of the physical world, to acknowledge that an infinite process is not possible within our universe. Therefore both "it could run forever", and "it will run forever" are excluded as impossible, therefore false premises.
A potentially infinite process is one which will not end (unless prematurely terminated). Does this work for you?
From this perspective we see that any problem which requires an infinite process to resolve, is actually not solvable. If a question is not solvable, it is not properly posed, it is not a valid question.
Well, can't the answer to the question simply be the infinite process? For instance, consider the question 'what is the area of a unit circle?' Is this a valid question? In one sense, I think you're right since no rational number will do. But in another sense, I think you're too strict in only accepting rational numbers. I think it's valid to say that the answer is pi, which I believe corresponds to a potentially infinite process. (Well my beliefs are changing a bit as I talk here with norm but I think you get what I'm saying).
I find that in the modern sciences, which are the principal users of mathematics, the goal has shifted from truth to prediction, and these two are not the same.
Only when we understand (truth) can we make a prediction. I think they're connected, but predictions make $$$.
The reason it works is because it's designed, shaped, conformed to the purposes which it is put to, like a finely honed tool.
An engineer may see math as a tool but I imagine a mathematician sees math as something to be understood for the sake of understanding. Like a beautiful painting, it doesn't need any other purpose.
it becomes clear to me that we need to give time priority over space in our modeling, to allow that space itself changes with the passing of time.
What does time mean in the absence of a ticking clock? In other words, if all objects and space are static, has time actually passed? I don't think you have a good reason to believe that time has priority over space. I also don't see why 3D space needs 2 temporal dimensions to change. All of our experiences point to there being only 1 temporal dimension. What evidence do you have to support this claim? I find it a bit hard to follow your later statements, but anyways, until I understand the motivations behind your view, there's no point talking about fundamental points.
Metaphysician UndercoverMarch 10, 2021 at 02:51#5084250 likes
And your claim that there's no such thing as instantaneous velocity is falsified by your car's speedometer.
Velocity is always an average over a duration of time. So-called "instantaneous velocity" is just a derivative from an average. Since velocity is a measure of change, and change without a duration of time is impossible, then also true "instantaneous velocity" is also impossible. It's just a term of convenience, to be able to say that at x point in time, the velocity was such and such. What is really taken is an average over a duration, and from that we can say that the velocity at any particular point in time within that duration was such and such. But you can see from the applicable formula, that "instantaneous velocity" is really just another average. And it's quite obvious that the idea that something has velocity at a point in time, when there is no duration, is nonsensical.
Velocity is always an average over a duration of time. So-called "instantaneous velocity" is just a derivative from an average. Since velocity is a measure of change, and change without a duration of time is impossible, then also true "instantaneous velocity" is also impossible.
Funny you should mention that, since it's so easily disproven.
Consider the speedometer in your car. How do you suppose it works? Is there a tiny little freshman calculus student in there, frantically calculating the limit of the difference quotient moment by moment?
No, actually not. Not even a computer program doing the same. Rather, there is a little induction motor attached to the driveshaft. The faster you travel, the faster the drive shaft spins, the faster the induction motor turns, the more current it outputs. And that current directly drives the needle of your speedometer.
Your speedometer is not a mathematically derived average. It is in fact a direct analog measurement of the instantaneous velocity of your car; subject of course to slight mechanical error common to any physical instrument. The velocity is an actual, physical quantity that can be directly measured -- that IS directly measured -- without recourse to any formal mathematical procedure.
You are confusing the velocity of an object, with the procedure we teach calculus [s]sufferers[/s] students to find the velocity of points moving in the plane or in space.
It's just a term of convenience, to be able to say that at x point in time, the velocity was such and such.
No, as I have just explained, velocity is a directly measurable physical quantity, like the mass or volume of an object, or the wavelength or luminous intensity of a beam of light.
But you can see from the applicable formula, that "instantaneous velocity" is really just another average.
No, because the calculus formalism is not the velocity, it's merely the way we determine velocity given the position function. But we don't need to do that if we have a direct way of measuring the velocity.
But even your remark about the formalism is wrong, because although the value of the difference quotient at any point is not the true velocity, but rather the slope of the secant line; the limit of the difference quotient is exactly the velocity. It is not an approximation.
And it's quite obvious that the idea that something has velocity at a point in time, when there is no duration, is nonsensical.
Yet another instance of the phenomenon whereby something false appears "quite obvious" to you solely by virtue of your lack of knowledge. You wield your ignorance like a weapon. A comic book character, Ignorance Man, whose slogan is "Believe the science!" while knowing none of it. Come to think of it there's rather a lot of that about these days, wouldn't you agree?
I do want to say a couple of things. I obviously know how to plot a polynomial in the traditional sense (and I also know how to use plotting programs). If you don't see a polynomial in my graphs it's because you don't understand my view (I'm not blaming you, this may be entirely my fault).
I agree that I don't understand your viewpoint. I have no trouble with adjoining points at plus/minus infinity to the real line, that's just the two point compactification of the real line. But the rest of it I couldn't follow.
Your speedometer is measuring the average velocity but one measured over quite a short time interval.
Not so, please see my response to @Metaphysician Undercoverhere. Briefly, your speedometer is driven by an induction motor coupled to your driveshaft. It gives a direct analog measurement of instantaneous velocity without any intervening computation.
That's the gist of constructivism. It don't exist unless I can grab it! I was/am strongly attracted to constructivism, but it comes at a cost. Consider some great mathematicians were attracted to Brouwer's ideas, but they found that it was not worth all that had to be sacrificed for it. If you check out constructivist logic, you might like it but also find it disturbing in its own way. Intuitively, some things are true or false even if we can't say which. A Turing machine does or does not halt in some logical sense. There is or there is not a string '7777777' somewhere in the expansion of pi. If I understand correctly, a constructivist would disagree, since a constructivist acknowledges time. There's something like true, false, andundetermined. (Or that's how I remember it. Perhaps someone who knows better will chime in.)
You have a good point so please allow me to soften my position. Perhaps pictures are only a handy prop in my view but the lack of symbolic reasoning may only reflect that my view is not mature.
I see your view as gestating. It's born for a mathematician when there are axioms and a logic. I hope you continue with it as long as you keep enjoying it.
I've read the Dover book on infinitesimal calculus by Keisler. It must be different from yours because mine isn't so thin. I'm not convinced that there are irrational numbers between the rationals, I'm even less convinced that there are infinitesimals in between the reals. But you're the professional and you've seen the proofs to conclude that the reasoning is rigorous so I don't want to debate about this issue.
Consider that the reasoning is dry and formal with no 'metaphysickal' commitment. A person could not even 'believe' in integers and still be great at pure math. (This is what I've seen people not realize, that math is agnostic on pretty much all ambiguous matters.) Do I believe in chess kings? Doesn't matter what I think if I publicly play by the rules. Nothing is hidden, or whatever counts epistemologically is not hidden. The primary obstacle I've seen in the transition to pure math from the calculus sequence is an implicit metaphysics that gets in the way. (I know a graph theory guy who thinks the continuum is a convenient fiction, and so on. The fictions as such are fun to play with. )
Metaphysician UndercoverMarch 10, 2021 at 03:56#5084480 likes
A potentially infinite process is one which will not end (unless prematurely terminated). Does this work for you?
I understand this, but my point is that due to the nature of our universe, any such "potentially infinite process" will be prematurely terminated. So it doesn't make any sense to say that such and such a process could potentially continue forever, because we know that it will be prematurely terminated. Therefore, if we come across a mathematical problem which requires an infinite process to resolve, we need to admit that this problem cannot be resolved, because the necessary infinite process will be terminated prematurely, and the problem will remain unresolved.
Well, can't the answer to the question simply be the infinite process?
I don't think so, because the process is the means by which the answer is produced. If the answer requires an infinite process, and the infinite process will be prematurely terminated, then the answer will not be produced.
For instance, consider the question 'what is the area of a unit circle?' Is this a valid question? In one sense, I think you're right since no rational number will do. But in another sense, I think you're too strict in only accepting rational numbers. I think it's valid to say that the answer is pi, which I believe corresponds to a potentially infinite process. (Well my beliefs are changing a bit as I talk here with norm but I think you get what I'm saying).
The reason I am "too strict", is that I don't believe in coincidence, when it comes to mathematics. Call me superstitious, but I believe that in mathematics, there is a reason for everything being the way that it is. So when it turns out, that a circle cannot have a definite area, then I believe that there is a reason for this. The most likely reason, is that the circle is not a valid object. By "valid" here, I mean true, sound, corresponding with reality.
Here's a sort of anecdote. Aristotle, in his metaphysics posited eternal circular motions for each of the orbits of the planets. Motion in a perfect circle could continue forever because there could be no beginning or ending point on the circumference of the circle, as each point is the same distance from the centre. Of course we've since found out that the orbits are not perfect circles. What we can learn from this, is that despite the fact that the circle is an extremely useful piece of geometry, there is something fundamentally wrong with it, as a mode for representation. It is not real. And, with the irrational nature of pi, the circle actually indicates directly to us, that it is not real. So if we ignore this fact, insisting that we want the circle to be real, or that it must be real because it's so useful, and then we work around the irrational nature, creating patches, and fancy numbering systems to deal with all these seemingly insignificant problems which crop up from employing perfect circles, we are simply deceiving ourselves. We end up believing that the real figures which we are applying the artificial (perfect circles) to are actually the same as the artificial, because all the discrepancies are covered up by the patches.
Metaphysician UndercoverMarch 10, 2021 at 04:03#5084520 likes
No, actually. Not even a computer program doing the same. Rather, there is a little induction motor attached to the driveshaft. The faster you travel, the faster the drive shaft spins, the faster the induction motor turns, the more current it outputs. And that current directly drives the needle of your speedometer.
Your speedometer is not a mathematically derived average. It is in fact a direct analog measurement of the instantaneous velocity of your car; subject of course to slight mechanical error common to any physical instrument. The velocity is an actual, physical quantity that can be directly measured -- that IS directly measured -- without recourse to any formal mathematical procedure.
Oh come on fishfry, you're smarter than this. The current you refer to is just measuring revolutions of the driveshaft. Then the speedometer of the car is scaled to how many revolutions are required to cover a specific distance. It is not measuring the instantaneous velocity of your car. What happens when you use the wrong size tires?
I don't think your criticisms of finitism apply to my view. In my view, every system does have a largest number, it's just that there's no universal system containing all possible numbers. For example, in the graph below the largest number is 99498. We could certainly 'cut' the continuum to produce points with coordinates having larger values, but until we actually do that it is meaningless to assign coordinates to those potential points. Could you expand on how I'm stuck with actual infinity?
In the broader context of my general philosophical views, words don't have exact meanings and context is a dominant factor in what meaning they do have. So I can only grope at what 'actual infinity' means. This is the charm of pure math. There are strict definitions, cold and dry. But I think I've already answered your question. If you have the concept of a rational number, you're already there. Do you believe in a largest rational number? How many rational numbers are there? In general concepts shimmer with infinity.
I am thoroughly enjoying this discussion and I appreciate your pointed questions. So far, my view is that you've clearly demonstrated how far my view is from a formal theory (thanks!) but you haven't identified any flaws yet. You're right, I don't see it as mathematicians see it. And so a mathematician might say that my probability of being right is 0. Thankfully, that means mathematicians still believe I have a chance!
Until a formal system is erected for examination, we're not doing math but only philosophy (but then I love philosophy, so I'm not complaining.) If you remember my first response, I suggested that the issue of fundamentally social. Who are your ideas ultimately for? Mathematicians or metaphysicians?
There are infinite potential chairs. Must all potential chairs actually exist to give the word chair meaning? The 'chairness' algorithm must be finite otherwise we'd never call anything a chair. Perhaps the same can be said about pi. Perhaps on the deepest level, pi is not the number pi, nor the infinite algorithms used to calculate the number pi, but instead the finite algorithm used to identify which algorithms would generate the number pi.
I think you'd probably enjoy looking into computability theory. Do you know about the halting problem? This is some of my favorite math. How do you know that there is a finite algorithim that always halts that can determine if other algorithms generate pi? But then you said that the algo that determines whether another algo generates pi is itself pi? This doesn't make sense. What a person might do is declare a particular Turing machine given a particular input to be a representative of pi and then include all equivalent-in-some-sense Turing machines as different representatives. This would have its own issues, but perhaps you see the charm of equivalence classes? You can start with something relatively concrete and define a notion of equivalence to scoop up the other entities that should be included in the concept. I don't need to know how many such machines there are. I can do some further proofs that show that addition and multiplication are independent of the representatives used. (Actually this technique blew my mind at first. It was when I first started feeling like a mathematician. AFAIK, there's nothing comparable in engineering.)
Oh come on fishfry, you're smarter than this. The current you refer to is just measuring revolutions of the driveshaft. Then the speedometer of the car is scaled to how many revolutions are required to cover a specific distance. It is not measuring the instantaneous velocity of your car. What happens when you use the wrong size tires?
Jeez man it's an analog computer. It gives a direct measurement of a physical quantity. You're saying there's no such thing as velocity. Your scientific nihilism is spreading from math to physics.
I'll take a run at your graphs when I get a chance. You went to some trouble to draw them, you deserve a response.
With you being a crankologist, I'd really benefit from your criticisms and I think you'd enjoy learning my view as I believe I am coming at infinity from a unique angle. As such, I think you'd need a different strategy to take down my ideas (assuming I'm wrong). But your time is short and crankery is infinite so whether you find time or not, it's all good.
your speedometer is driven by an induction motor coupled to your driveshaft. It gives a direct analog measurement of instantaneous velocity without any intervening computation.
You're definition of the instantaneous velocity of a car rests upon a dynamic quantity: the flow of electrons through a wire (i.e. current). So you've only shifted the problem from instantaneous velocity to instantaneous current. Consider this example.
You: Your instantaneous velocity is 10 km/h.
Me: How do you know?
You: Because I'm running right next to you and my instantaneous velocity is 10 km/h.
This begs the question, how do you know your instantaneous velocity? For instantaneous velocity to make sense, it needs to be based only on static quantities that exist at that instant. But just as you can't look at a photograph and determine how fast I'm running, you cannot come up with a meaningful definition of instantaneous velocity.
Now, if you were referring to my GPS-based speedometer then yes, inside my phone is a little freshman calculus student that does the math, not calculus, just a simple delta_s/delta_t.
Edit: If the needle position in your speedometer is indeed only based on instantaneous information it must be doing so like a spring, which deforms as a function of the force applied. One could come up with a correlation between spring deflection and velocity, but this is only approximate. It is not a true measure of instantaneous velocity.
With you being a crankologist, I'd really benefit from your criticisms and I think you'd enjoy learning my view as I believe I am coming at infinity from a unique angle. As such, I think you'd need a different strategy to take down my ideas (assuming I'm wrong). But your time is short and crankery is infinite so whether you find time or not, it's all good.
On the contrary, I have too much time on my hands. I've been over active on this forum lately and I'm feeling the need for a break. I think my reacting negatively to @Wayfarer's helpful link to a Sabine Hossenfelder video was a clue. I'm just crabby lately for the sake of being crabby and when I find myself doing that it's time for a forum break. Sorry @Wayfarer, I apologize.
You're definition of the instantaneous velocity of a car rests upon a dynamic quantity: the flow of electrons through a wire (i.e. current). So you've only shifted the problem from instantaneous velocity to instantaneous current. Consider this example.
Yes but this is true of any physical quantity. How do I measure the mass of a bowling ball? Well I can put it on a scale, but that only measures weight and not mass. The weight would be different on the moon.
So to measure mass, we must observe the bowling ball's acceleration response to force, as described in this fascinating thread.
How can we do that? We can suspend it from a spring with a known spring constant using the formula for a harmonic oscillator. But then you (and @Metaphysician Undercover) will object that all I'm doing is measuring the springiness of the spring.
Or I can measure the centripetal force on a centrifuge, or use a small angle pendulum. But in each case aren't we just measuring something about the apparatus and not the mass itself?
In short, your objection is valid, but overly general. We can't measure any physical quantity at all by your logic. What if I want to measure the wavelength of a beam of light? Well I use a spectrometer, but all that really measures is the prism or the glass or however spectrometers work.
What if I want to measure the temperature of air? I use a thermometer, but that's only measuring the response of mercury or the coil of a metal spring or however thermometers work these days.
So what you and @Meta are saying is that we can't measure ANY physical attributes at all. This is hardly an objection to my point about velocity being directly measurable without recourse to formal calculus. It's a philosophical objection to the idea that we can do any measurements whatsoever, or to the idea that objects even have physical attributes before we measure them by proxy. But that doesn't actually address the different point that I'm making: That moving objects have a velocity, which we can measure directly (by proxy with an induction motor coupled to the driveshaft), without needing formal symbolic methods of calculus.
After all, bowling balls fall to earth with an acceleration of -32 feet/sec^2, and this was true even before Galileo discovered it and Newton modeled it with his law of gravity. You and @Meta can not deny that bowling balls fall down and that they do so with a measurable velocity at any instant of time; without denying the whole of physical science. You don't need calculus to know that falling bowling balls have a velocity. You're both confusing the mathematical model with nature itself. And the fact that all measurements require some intervening apparatus is a red herring.
I think my reacting negatively to Wayfarer's helpful link to a Sabine Hossenfelder video was a clue. I'm just crabby lately for the sake of being crabby and when I find myself doing that it's time for a forum break. Sorry @Wayfarer, I apologize.
Hey no probs, really nothing personal. I thought it might have been relevant, that's all. I find your knowledge of philosophy of maths really interesting.
I was one of the many students who failed terribly at maths. I would feel my grasp of the work slipping away in class, and could never catch up. But later in life, I've come to appreciate maths aesthetically, even though I can't understand it very well. I also think the question of the nature of the reality of number really is important. I was just perusing that article on Hossenfelder's website again, and came across this remark in the combox:
Complex numbers don't exist. For that matter, natural numbers don't exist. They are merely useful fictions.
I say this because I am a mathematical fictionalist. However, there are also many mathematical platonists who would disagree with me.
Honestly, it doesn't make any important difference. Platonists and fictionalists do their mathematics in pretty much the same way. Their philosophical differences don't actually affect the mathematics.
And then there's the Quine - Putnam indispensibility thesis, which argues for platonism to explain why mathematics works so well in physics. However, I happen to think that fictionalism makes more sense of the role of mathematics in physics.
So it is really much ado about nothing. Go with whatever makes most sense to you.
I think that comment is dead wrong. There is a matter of fact about this issue. I think fictionalism really is a white flag in philosophical terms. It might not matter for applying mathematics, but it makes a major difference to your conception of the nature of reality. When you consider the predictive power of maths, the fact that through it you can discover things about reality that you otherwise could never know - how is that reconcilable with the idea that it's a 'useful fiction'?
When you consider the predictive power of maths, the fact that through it you can discover things about reality that you otherwise could never know - how is that reconcilable with the idea that it's a 'useful fiction'?
"The Unreasonable Effectiveness of Mathematics in the Natural Sciences"
My former and late father-in-law's good friend Eugene Wigner raised this question years ago in a famous paper. I agree - I don't perceive it as a useful fiction.
But here
[math]\underset{k=1}{\overset{3}{\mathop R}}\,{{f}_{k}}(z)={{f}_{1}}\circ {{f}_{2}}\circ {{f}_{3}}(z)[/math]
etc. Going the other way involves rules for inverses of compositions.
This sort of thing appears in a general study of infinite compositions, a topic of practically no interest in the larger mathematics community. Abstractions and generalizations are more attractive.
Look up Shota Kojima and infinite compositions. A lot of stuff out there on fractals and simple iteration, but composing different functions endlessly not very much, especially if one considers complex functions that are not holomorphic, which I have done. They are far more interesting IMO.
When teaching almost thirty years I would publish a paper each year just to avoid collapsing intellectually, and all were readily accepted and published, but after I retired in 2000 I decided to do the research for me and anyone else who might be interested, avoiding the formalities of publishing. Hence a collection of informal notes on researchgate. It's amazing the audience one reaches there. Virtually every civilized nation has cropped up with reads of my stuff. But fewer than half read the entire note. Still, that can be 20-30 a week that do. And it's free, unlike most journals - and that irritates me since few journals pay the referees, at least when I was active. (in all fairness, institutions take up the slack with fewer teaching hours, etc.)
I consider the major journals corrupt in their financial practices. But that's just me. Pay no attention.
Ryan O'ConnorMarch 11, 2021 at 01:24#5088170 likes
I'm not denying measurement at all. If an event can be completely described by a photograph then it is instantaneous. If it needs a video to be accurately described then it is a transient event. The only way to report a transient event at an instant is to compress the transient data, e.g. by time averaging it. I have no problem with (at some given instant) reporting a velocity, as long as we recognize that that quantity is not the velocity at that instant, but instead some average velocity over a short interval.
And in the case of a speedometer, I wasn't trying to say that everything is relative to everything else so measurement is meaningless. I was only saying that the problem is not solved by pushing it downstream (from a moving car to moving electrons). Measuring current is a giveaway that you're measuring a transient phenomenon. If you were measuring capacitance on the other hand (which is analogous to a spring) then you could be measuring an instantaneous event.
But it doesn't actually addressing the different point that I'm making: That moving objects have a velocity, which we can approximately measure directly, without needing formal symbolic methods of calculus.
Your original claim was that my rejection of instantaneous velocity is falsified, which I think is false. If you now claim that the speedometer must necessarily be reporting some average or approximate velocity then I have no problem with that.
Your original claim was that my rejection of instantaneous velocity is falsified, which I think is false. If you now claim that the speedometer must necessarily be reporting some average or approximate velocity then I have no problem with that.
What do you think speedometers measure?
I take your point about instantaneous motion, it's related to one of Zeno's paradoxes. If the arrow is not moving at a given instant, how does it know it's moving, or something like that. I don't think I have to resolve that ancient mystery to read my speedometer and know that it's telling my my instantaneous velocity. Even if it's only actually reading a current from an induction motor.
Bonus question. What is the speed of light at any given instant?
Ryan O'ConnorMarch 11, 2021 at 02:27#5088410 likes
Consider some great mathematicians were attracted to Brouwer's ideas, but they found that it was not worth all that had to be sacrificed for it...There's something like true, false, and undetermined.
I do like this idea and I need to find out why it's disturbing.
I see your view as gestating. It's born for a mathematician when there are axioms and a logic. I hope you continue with it as long as you keep enjoying it.
Consider that the reasoning is dry and formal with no 'metaphysickal' commitment. A person could not even 'believe' in integers and still be great at pure math.
It reminds me of the Chinese room argument in AI. Someone locked in a room could be blindly following a set of instructions to take chinese character inputs and generate outputs to convince someone that they speak chinese. It sounds like your view is that math is like this chinese room, a mere set of rules and symbols. I believe that it's more than that. I believe that the messages have content (like the Chinese messages), and if we better understand what it's saying perhaps math will be easier.
My response is 'how many rational numbers are where? Present me with a graph and label all points explicitly and I can tell you what the largest number on your graph is.
Until a formal system is erected for examination, we're not doing math but only philosophy (but then I love philosophy, so I'm not complaining.) If you remember my first response, I suggested that the issue of fundamentally social. Who are your ideas ultimately for? Mathematicians or metaphysicians?
"Without mathematics we cannot penetrate deeply into philosophy. Without philosophy we cannot penetrate deeply into mathematics. Without both we cannot penetrate deeply into anything." Leibniz.
I don't think we have to decide between the two...but you're right, at this point I'm only philosophizing. But hey, I'm on a philosophy forum!
Do you know about the halting problem? This is some of my favorite math. How do you know that there is a finite algorithm that always halts that can determine if other algorithms generate pi?
Yes, I like the halting problem and anything else that deals with incompleteness. I don't know if such an algorithm exists. But as it is we define pi as an equivalence class of particular cauchy sequences. All I'm proposing is that instead of pi being the equivalence class itself that it is the description of that equivalence class. Surely our description is finite, right?
In short, your objection is valid, but overly general. We can't measure any physical quantity at all by your logic. What if I want to measure the wavelength of a beam of light? Well I use a spectrometer, but all that really measures is the prism or the glass or however spectrometers work.
You seem to be missing the point fishfry. Velocity is a measurement of motion, and motion only occurs when time is passing. At an instant zero time passes. Therefore there is no motion at an instant, and no velocity at an instant.
A measurement of velocity requires a determined distance over a determined duration of time. It requires two instants, to determine a duration of time, one to mark the beginning of the period of time, the other to mark the end of the period of time, just like it requires two points to determine a distance. One instant (point in time) is insufficient for a determination of velocity, just like one point is insufficient for a determination of distance.
Velocity is a measurement of motion, and motion only occurs when time is passing. At an instant zero time passes. Therefore there is no motion at an instant, and no velocity at an instant.
Yeah yeah. One of Zeno's complaints. If you look at the arrow at a particular instant it's not moving. How does it know what to do next in terms of direction and speed? Not a bad question actually, one that I won't be able to answer here.
A measurement of velocity requires a determined distance over a determined duration of time. It requires two instants, to determine a duration of time, one to mark the beginning of the period of time, the other to mark the end of the period of time, just like it requires two points to determine a distance. One instant (point in time) is insufficient for a determination of velocity, just like one point is insufficient for a determination of distance.
Yes you already said that. I take the point. If you show me a photo (taken over a sufficiently short time interval, since even a photograph takes time) the arrow appears stationary and you can't determine its velocity.
Yet, it still HAS a velocity, wouldn't you agree? And what does a speedometer measure? A current. And as @Ryan pointed out, even that's a flow of electrons. Since current is a flow, does it exist at an instant? Well I'm sure that a modern physicist would point out that the electromagnetic field exists at every moment. But if you have a magnet in a coil, you have to move the magnet to create a current. I'm really not enough of a physicist or a philosopher to know these things. Good questions though.
Still, would you at least grant me that velocity over a short but nonzero distance exists? And likewise a current flow? Then a moving car has a velocity that can be determined without recourse to formal symbolic manipulations, which was my original point.
Moderator note: When I hit the @ button to search for @Ryan O'Conner his handle doesn't come up, any ideas why?
Metaphysician UndercoverMarch 11, 2021 at 02:58#5088550 likes
Sure, the object is described as moving, it must have a velocity. But it cannot have a velocity at an instant, if no time passes at an instant, just like a point has no spatial extension. That's why points and lines are incompatible, and a line is not composed of points, but points mark off line segments.
So the solution to the issue with velocity, is not to say that it has no velocity, it is to say that there is no such thing as the instant. Time is not composed of instants. So the arrow, or car always has velocity, all the time that it is moving, but that time has no instants. The instant is just an arbitrary point which we insert for the purpose of making a measurement.
Still, would you at least grant me that velocity over a short but nonzero distance exists?
Sure, but the whole point I am arguing in this the thread is that the inclination to reduce the nonzero distance to zero, or even define it as somehow related to zero, produces theoretical absurdities. And this is well demonstrated by these Zeno type paradoxes which speak of time as consisting of instants.
Ryan O'ConnorMarch 11, 2021 at 03:01#5088580 likes
Imagine water flowing uniformly out of the faucet onto a flat plate. Below the flat plate is a spring which compresses. We can determine a relationship between the spring compression and the water flow rate. This is essentially how an analogue speedometer works, but with electrons instead of water.
This description may give the impression that the spring can measure instantaneous velocity but it cannot.
Consider this: the first instant the water hits the plate, the spring is not compressed. It then takes some time for the spring to find the equilibrium position, and only at that time will it report the correct flow rate. During this transient period the spring is not reporting the correct flow rate, but instead some value between zero and the actual flow rate. It's reporting some sort of average.
And as you drive your car continually accelerating and decelerating, the spring behind the needle is continually playing 'catch-up' and thus reporting some sort of average. In fact, it is most meaningful to say that it is always reporting an average.
And as you drive your car continually accelerating and decelerating, the spring behind the needle is continually playing 'catch-up' and thus reporting some sort of average. In fact, it is most meaningful to say that it is always reporting an average.
I'll accept this point. It still has nothing to do with what I originally said, which is that you don't need calculus to determine the instantaneous velocity of a moving object. And I'll concede that by instantaneous I only mean "occurring over a really short time interval."
I have to say I'm not nearly as invested in this point as the number of words written so far, I should probably stop.
Sure, the object is described as moving, it must have a velocity. But it cannot have a velocity at an instant, if no time passes at an instant, just like a point has no spatial extension. That's why points and lines are incompatible, and a line is not composed of points, but points mark off line segments.
So the solution to the issue with velocity, is not to say that it has no velocity, it is to say that there is no such thing as the instant. Time is not composed of instants. So the arrow, or car always has velocity, all the time that it is moving, but that time has no instants. The instant is just an arbitrary point which we insert for the purpose of making a measurement.
Ok. I can live with that. Whether it's a moving arrow or a current driving the speedometer, it's a change occurring over a short interval of time. But my original point was that we don't need calculus to determine the velocity. Actual velocities are not subject to the ancient philosophical mysteries of calculus.
Still, would you at least grant me that velocity over a short but nonzero distance exists?
— fishfry
Sure, but the whole point I am arguing in this the thread is that the inclination to reduce the nonzero distance to zero, or even define it as somehow related to zero, produces theoretical absurdities. And this is well demonstrated by these Zeno type paradoxes which speak of time as consisting of instants.
Ok. Maybe. Let me put to you a hypothetical. An object moves with constant velocity. Does it have a velocity at a given instant?
Likewise does the speed of light have velocity 'c' at a given instant?
I'm kind of done with this topic, the point I'm making isn't worth all this ink. You don't need calculus to do analog measurements. And yes physical measurements depend on time, even if those intervals are tiny. There aren't any actually physical instants as far as we know. Or as far as we don't know. The matter is not answered by current science.
Metaphysician UndercoverMarch 11, 2021 at 03:34#5088700 likes
Ok. Maybe. Let me put to you a hypothetical. An object moves with constant velocity. Does it have a velocity at a given instant?
No, because "a given instant" is not anything real which can be adequately identified. We can attempt to arbitrarily assign an instant to time, to mark a point for the purpose of measurement, but that assignment becomes much more difficult than it appears to be, at first glance. To mark a temporal point in one process or activity, requires a comparison with another process or activity, thus requiring a judgement of simultaneity. According to special relativity such judgements are dependent on the reference frame. Therefore any "given instant" may not be the same instant from one frame to the next, and the question of what a thing's velocity is at a given instant is rather meaningless because it depends on what frame of reference you measure it in relation to.
I'm kind of done with this topic, the point I'm making isn't worth all this ink. You don't need calculus to do analog measurements. And yes physical measurements depend on time, even if those intervals are tiny. There aren't any actually physical instants as far as we know. Or as far as we don't know. The matter is not answered by current science.
I don't think you've adequately considered what is required to produce accuracy in a time related measurement.
The arrow may be momentarily stationary, but it has momentum.
Can you explain this to @Metaphysician Undercover and @Ryan whose handle doesn't show up when you use the @ button?
But actually it's a good question. Suppose there were such a thing as an instant of time, modeled by a real number on the number line. Dimensionless and with zero length. So the arrow is there at a particular instant, frozen in time, motionless. Where does its momentum live? How does it know where to go next?
Does it have, say, "metadata," a data structure attached to it that says, "Go due east at 5mph?" You can see that this is problematic.
Does it have, say, "metadata," a data structure attached to it that says, "Go due east at 5mph?" You can see that this is problematic.
Isn't this just confusing a snapshot of a thing with the thing itself? The movement is not divided into discrete moments until we try to model it using mathematics. Yet mathematics is distinct from the event, as it is merely a way to model the event (albeit, an extremely useful and accurate enough model). We necessarily measure momentum in retrospect, so when we analyse a things momentum at a particular instant it is already a passed instant. However, just because we apply delineation through our act of retrograde analysis, and create a mathematical notion of temporality, that doesn't mean that the thing did not have a momentum at a particular instant. It's only a question of accuracy.
Or pehaps one day scientists will discover some kind of 'instance particles'. Now that would be interesting.
Edit - I guess I lean towards the (Bergsonian) idea of an essential indivisibilty of time/movement.
Metaphysician UndercoverMarch 11, 2021 at 13:10#5089670 likes
Can you explain this to Metaphysician Undercover and @Ryan whose handle doesn't show up when you use the @ button?
The point being, that you cannot take the arrow at a particular moment in time. This is an impossibility because time is always passing, and this would require stopping time at that moment. So, despite the fact that using mathematics to figure hypothetical conditions at particular moments is a very useful thing to do, what it provides us with is a representation which is actually a falsity. Then if people start talking about this situation, with the underlying implication that this mathematics provides us with some sort of truths about these situations, this talk is really a deception or misinformation.
But actually it's a good question. Suppose there were such a thing as an instant of time, modeled by a real number on the number line. Dimensionless and with zero length. So the arrow is there at a particular instant, frozen in time, motionless. Where does its momentum live? How does it know where to go next?
This is the key point to understanding temporal continuity, inertia, Newton's first law, and the overall validity of inductive reasoning in general. We observe that things continue to be as they were, as time passes. Intuition tells us therefore, that they will continue to be as they were, unless something causes them to change, and this intuition is what validates inductive reasoning. However, there is a very real, and very big problem, and this is the reality of change. We see that human beings have the capacity to interfere with, and change the continuity of inert things. Because of the reality of change, we are forced to accept the fact that this continuity is not a necessity. This appears to be the hardest thing for some people, especially those with the determinist mentality, to accept, that the continuity of existence, which we observe, is not necessary. This means that the supposed brute fact, which underlies all inductive principles as supportive to those principles, that things will continue to be as they have been, is itself contingent, not necessary.
When we take a law, like Newton's first law, we view it as something taken for granted. The law tells us the way things are, and it's assumed that it's impossible for things to be otherwise, that's why it's "the law". However, when we apprehend that this is not necessary, then we can grasp the fact that there is a need for a reason why the law upholds. Through principles like the law of sufficient reason, we see that if there is a temporal continuity of existence, described as momentum or inertia, and this feature of existence is not necessary, then there must be a reason for it, a cause of it.
What Aristotle did was posit "matter" as the cause of the temporal continuity of existence. So contrary to the common notion that matter is some sort of physical substance, "matter" by Aristotle's conception is really just a logical principle, adopted to account for the observed temporal continuity of physical existence. It is, in a sense, a placeholder. He didn't know the cause, but logic told him there must be a cause, so he identified it as "matter". In your example of the arrow, we do not know "how does it know where to go next", but we do know that it does. Aristotle attributes this to its "matter", or more precisely he posits "matter" as what causes it to go, where it does go, next. Therefore the theoretical points in time are in reality connected to one another by what is called "matter".
However, just because we apply delineation through our act of retrograde analysis, and create a mathematical notion of temporality, that doesn't mean that the thing did not have a momentum at a particular instant. It's only a question of accuracy.
I'm confused by your post. You make the correct point that the math is just a model of reality, not reality itself. Then you say that the thing DOES have a momentum at a particular instant. Which is what the mathematical model says.
Look up Shota Kojima and infinite compositions. A lot of stuff out there on fractals and simple iteration, but composing different functions endlessly not very much, especially if one considers complex functions that are not holomorphic, which I have done. They are far more interesting IMO.
Kinda off topic, but have you ever seen a generalisation of the iterated composition operator to non-natural indexes? Like... does the following notion make sense in general:
Let's say we have [math]g=f \circ f[/math], does it make sense to think of [math]f[/math] as like half an application of [math]g[/math]?
Are there analogous constructs for an [math]f[/math] which is [math]\frac{1}{\pi}[/math] applications of [math]g[/math]?
Ryan O'ConnorMarch 11, 2021 at 22:36#5091190 likes
I understand this, but my point is that due to the nature of our universe, any such "potentially infinite process" will be prematurely terminated. So it doesn't make any sense to say that such and such a process could potentially continue forever, because we know that it will be prematurely terminated. Therefore, if we come across a mathematical problem which requires an infinite process to resolve, we need to admit that this problem cannot be resolved, because the necessary infinite process will be terminated prematurely, and the problem will remain unresolved.
The beauty of calculus is that in performing a finite number operations (e.g. in manipulating the exponents and coefficients of a polynomial to determine the derivative) we can talk sensibly about a potentially infinite process. If you reject potentially infinite processes as valid mathematical objects then you must reject calculus, and nobody will buy into your philosophy. You've got to stop thinking of the output of a potentially infinite process as the mathematical object. The mathematical object is the process itself. If you're challenging the orthodox view on real numbers then your point is valid, but if you're challenging my view then your point is misdirected.
We end up believing that the real figures which we are applying the artificial (perfect circles) to are actually the same as the artificial, because all the discrepancies are covered up by the patches.
I agree. By treating rationals and irrationals both as the same type of object (i.e. numbers) we blur the line between the output of a finite algorithm and the output of a potentially infinite algorithm.
I agree. By treating rationals and irrationals both as the same type of object (i.e. numbers) we blur the line between the output of a finite algorithm and the output of a potentially infinite algorithm.
What do you make of 1/3 = .333...? Can't you distinguish between a number and one of its representations that you don't happen to like? After all 1/3 is just a shorthand for the grade school division algorithm for 3 divided into 1.
Kinda off topic, but have you ever seen a generalisation of the iterated composition operator to non-natural indexes?
Yes. Goes back a hundred years if my memory serves. Sometimes it's very easy. For example, here is a linear fractional transformation written in terms of fixed points and multiplier.(I'm working on a theorem right now involving this). I think a more general case was dealt with in the discipline of functional equations. Can't recall the work offhand.
If you reject potentially infinite processes as valid mathematical objects then you must reject calculus, and nobody will buy into your philosophy.
I'm not looking for people to buy in, I'm looking for truth. If others are looking for the same thing, they might like to join me. Otherwise I don't really care if people might deceive themselves into thinking that they are engaged in infinite processes. Many think that the soul is eternal, and this doesn't both me either. I consider those two beliefs to be very similar.
There is a fundamental incompatibility between an object and a process, which was demonstrated by Aristotle. If an object changes, it is no longer what it was. We assume a change (process), to account for the object becoming other than it was. So we have object A, then a process, then object B, whereby object A becomes object B. If we represent the intermediary between A and B as another object, C, then object A becomes object C which becomes object B. Now we need to assume a change (process) to account for object A becoming object C, and a process to account for object C becoming object B. We might represent the intermediaries between A and C, and C and B, as objects again, but you can see that we're heading for an infinite regress. So we ought to conclude that "objects" and "processes" are distinct categories.
Pi is a finite number because it's inbetween 3 and 4. But if the length of a circumference is multiplied by pi than you have a length with space corresponding to each number, so the circle has infinite space within a definite finite limit (like being inbetween 3 and 4). Aristotle never understood this stuff
If you have a two foot segment and make it into a circle, suddenly it's pi/r/squared instead of two feet. Hmm. It seems that we must "round to the finite" in everything we do in geometry
Ryan O'ConnorMarch 12, 2021 at 01:49#5091750 likes
What do you make of 1/3 = .333...? Can't you distinguish between a number and one of its representations that you don't happen to like? After all 1/3 is just a shorthand for the grade school division algorithm for 3 divided into 1.
I believe that's a false equality. The correct statement is "the potentially infinite process defined by 0.333... converges to the number 1/3" not "the number 0.333... equals the number 1/3". Decimal notation is flawed in that it cannot be used to precisely represent some rational numbers, like 1/3. If we want a number system which can give a precise notation for any rational number, we should use Stern-Brocot strings, where 1/3 = LL.
Yeah yeah. One of Zeno's complaints. If you look at the arrow at a particular instant it's not moving. How does it know what to do next in terms of direction and speed? Not a bad question actually, one that I won't be able to answer here.
If photographs can't capture motion but videos can, why not conclude that motion happens in the videos? The reason why we are reluctant to come to this conclusion is because we reject the notion of videos being fundamental.
We want points (photographs) to be fundamental and continua (videos) to be composite and as long as we hold this view we will not find a satisfactory resolution to Zeno's paradoxes. If you flip things upside down and see continua as fundamental and points as emergent, then everything makes perfect sense. There's no problem with pausing a video to produce a static image.
Suppose there were such a thing as an instant of time, modeled by a real number on the number line. Dimensionless and with zero length. So the arrow is there at a particular instant, frozen in time, motionless. Where does its momentum live? How does it know where to go next? Does it have, say, "metadata," a data structure attached to it that says, "Go due east at 5mph?" You can see that this is problematic.
It is clear that you appreciate the profoundness of Zeno's Paradox. Zeno presented these paradoxes in response to the criticisms from the 'one from many' camp calling his views ridiculous. Why not consider the 'many from one' view that he supported? He was wayyy ahead of his time so his view did seem to have problems of their own...but in light of modern advancements in physics his view no longer seems crazy.
It still has nothing to do with what I originally said, which is that you don't need calculus to determine the instantaneous velocity of a moving object. And I'll concede that by instantaneous I only mean "occurring over a really short time interval." I have to say I'm not nearly as invested in this point as the number of words written so far, I should probably stop.
Don't stop here, you may just be on your way to becoming a crank! With this admission you have placed yourself on a slippery slope. Instantaneous velocity is no different from the tangent of a function at a point. Do you accept that the derivative corresponds to a limiting process of secants rather than the output of a completed infinite process (i.e. tangent at a point)?
I'm not looking for people to buy in, I'm looking for truth. If others are looking for the same thing, they might like to join me.
Never in the history of mathematics or physics has discovering truth set us backwards. The fact that your philosophy would result in a weaker mathematics is a red flag that you're on the wrong track. Don't get me wrong, I agree that there's a problem, I just don't agree with your resolution.
I made this video on my proposed resolution to Zeno's Paradox. What do you think?
So we ought to conclude that "objects" and "processes" are distinct categories.
I think you're taking my words too literally. Clearly, the process of taking my dog for a walk is not an object with mass and momentum. When I say that processes are valid objects of mathematics, I simply mean that they can be studied in themselves, just as one might write a book entitled 'The Art of Dog Walking'.
Continua based constructions are based on an uncountable infinite amount being manipulate to a finite result. That is new. That's exactly what Newton did.
I don't think you solved Zeno's paradox because you're putting the infinite quantity into philosophically blurry box and focusing just on finite results. Zeno said "the finite" and "the infinite" we're inherently exclusive of each other and so he concluded that idealism was true (a form of idealism wherein there are no 1s and 1s to be added into two)
Ryan O'ConnorMarch 12, 2021 at 02:46#5091940 likes
I don't think you solved Zeno's paradox because you're putting the infinite quantity into philosophically blurry box and focusing just on finite results.
Isn't that what we do with quantum mechanics? We have finite results corresponding to our actual measurements and everything between the measurements is a 'blurry' superposition? Why must everything be 'sharp'?
[in reference me me asking why everything must be sharp] That's just how math is
It's true that in my video I showed a blurry curve in between the measurements, but prior to that I showed the curve being 'topological' [ I use 'topological' only in the sense of the graph having properties which are preserved through continuous deformations]. If instead of saying that I'm 'blurring' the graph, what if we say that I'm reinterpreting it from being a geometric object to a topological object? With this view, your argument can't simply be 'that's just how math is' because topology is certainly acceptable math.
A car traveling 60 mph down the road. Is anyone here going to suggest that at any time, however defined, that car is not moving, or, that there some time, some moment, when it is by no test whatsoever distinguishable from a parked car? The moving car is a reality.
At every time interval we will find the car moving. At every instant in time the car's motion is indistinguishable from that of a parked car. It's Zeno's Arrow Paradox. Did you know that if a quantum system is continuously being observed that it will not evolve? It's called the Quantum Zeno Effect. As confirmed by experiment, motion (i.e. change) only happens when we're not looking. It happens in between the measurements. It happens in between the measured instants in time. And what exists in between the instants? - continua.
Ryan O'ConnorMarch 12, 2021 at 03:14#5092090 likes
Idn. I've been recently working on this question from the angle of non-Euclidean geometry. I'm trying to understand what space even is
When you draw a graph, do you think you actually draw infinite points placed perfectly on the page? Or do you place a few points on the page accurately and then imprecisely connect the dots. What I am proposing is in line with how we've always drawn graphs. And if we're honest with ourselves, it's the only way to draw graphs. It would take an infinite being to draw a graph with perfect precision. Let's not make math dependent on the existence of an infinite being.
Continua is infinitely pointed. So it has instants all over it. If things move at the top level but fundamentally unmoved in subdivisions an object can't move at all. How I see it, we need to say "the infinite" is on one side and "the finite" is on the other and motion is movement between them
Ryan O'ConnorMarch 12, 2021 at 03:28#5092120 likes
Continua is infinitely pointed. So it has instants all over it.
This is the view that we've been indoctrinated with. We can't help from thinking that points (i.e. instants) are fundamental and so we believe that continua are just a collection of points. Zeno's paradoxes reveal that this view has serious problems. Instead, consider the alternative: that continua are fundamental and points are emergent. Points only emerge when a measurement is made. When we draw a graph, we only label a few points where our curves intersect. Those are the only points on the graph. Don't look in between the curves and conclude that there exist uncountably-infinite points there...what lies in between the curves are continua.
Time for you to develop a new axiomatic system, then, that leads to "Truth".
It's not time for me to do that, I'm not a mathematician. There's something called the division of labour. The person who puts one's efforts into pointing at the problems in existing systems need not be the one who produces the repair. Of course the people using the system would probably not like the person pointing and would have the attitude of 'if you think you can produce a better system than ours, then do it'.
Pi is a finite number because it's inbetween 3 and 4. But if the length of a circumference is multiplied by pi than you have a length with space corresponding to each number, so the circle has infinite space within a definite finite limit (like being inbetween 3 and 4). Aristotle never understood this stuff
At least I'm not alone then, because I haven't got a clue what you're saying.
The fact that your philosophy would result in a weaker mathematics is a red flag that you're on the wrong track.
You demonstrated that you do not grasp the need for the point to be prior to the line, therefore your claim that it would result in a weaker mathematics is based in misunderstanding. What quantum physics demonstrates to us is that points have real existence, and continuities are constructed.
I made this video on my proposed resolution to Zeno's Paradox. What do you think?
I don't see how you get from points to continua. You show measurement points, then you assume that there is some sort of continuum connecting those points. The problem I see, is if certain measurement points are actually possible, then these must be represented as real points which the moving object actually traverses and can be measured at. That's why I give priority to these points, as the real features. The supposed continuum might not have any sort of linear existence at all, in fact we might not have the vaguest idea of how the points are related to each other in the underlying substratum of reality, which produces the appearance of a continuum. For all we know, the object might appear at point A, then completely disappear, and then reappear at point B a moment later, and this is what appears to us as motion.
The reason why I say that priority must be given to the points, is that whatever it is about the underlying substratum which produces the appearance of continuity, this 'power' must be constrained by possible points of appearance. If there wasn't such constraints then we'd have the problem of infinite points where the object could be measured. Furthermore, the nature of spatial expansion demonstrates that there must be points where expansion is centered.
So I find the video mostly acceptable, but what you are really showing is a points based motion, points where the object might be measured to be at, and you are assuming that there is some sort of continuum which underlies the points and connects them. Therefore all you need to do to be consistent with my perspective, is put the points as primary, being the real constraints of real space, and allow that whatever continuum emerges from existence at the points it is a creation produced from the relationships between the points, and this set of relationships comprises the substratum.
When I say that processes are valid objects of mathematics, I simply mean that they can be studied in themselves, just as one might write a book entitled 'The Art of Dog Walking'.
I have doubt in the truth of this. Are processes valid objects of mathematics, or ought they be relegated to physics? Let's start with something simple, assume that a number is an object of quantitative value. So '4' represents such an object, it must be a static and unchanging value to maintain its validity, therefore it cannot be a process. Now let's say that in '2+2', the '+' represents a process. So the inquiry is whether the process represented by '+' is a valid mathematical object to be studied by mathematics. We need to determine what the '+' means. What does it mean to add one unchanging quantitative value signified by '2', to another? Mathematics does not answer this inquiry, it just makes an assumption about how processes like these affect quantitative values. And we can see the same with the other processes, multiplication, division, etc., these process affect quantitative values, but if quantitative values are what are properly referred to as objects, then these processes are something different.
You take a snapshot of a moving car. You look at the photo and ask, "How fast was it going?"
You know, this is more tricky than it looks. Suppose you have a long exposure time. Then you'll see a blurred image, and you can work backwards to determine the velocity. Photographs are not instantaneous. The shutter stays open for a period of time, usually a fraction of a second. During that time the film or digital sensor collects photons. So there's an element of time involved even in a photo. If the object is moving slowly relative to the shutter speed you won't see blur, but in theory the blur is always there. If I can choose the shutter speed I can always tell you how fast the object was moving by analyzing the blur.
@Ryan same point to you. In fact your earlier point is correct, any measurement is taken over time. There's no difference between photo and video. Video after all is just a collection of still images, either analog or digital frames. And a single photo is taken over a period of time, namely the shutter speed.
Metaphysician UndercoverMarch 12, 2021 at 12:08#5093360 likes
In fact your earlier point is correct, any measurement is taken over time.
That's why velocity is always an average, requiring at least two temporal points. Duration is derived, just like distance is. To infer an instantaneous velocity requires a second derivation.
I apologize for calling your statement stupid. I had just had a fight with someone and your comment annoyed me. It seems you are always debating fishfry or someone about numbers and there relation to Kantian synthesis vs an analytic view. To me that's just a discussion about psychology and mathematics does truly take care to define what addition, subtraction, multiplication, and division are. I have not seen where you have a unique insight into the issue. But on eternal motion, Einstein and countless physicists believed in it. Eternal inflation, the "big bounce" , and all these ideas are just noting more than versions of it. I'm very aware of Aristotle's arguments about an accidental series (one that stands on its own) and an essential series (one with supernatural support). I've discussed this with Thomists who have PhD's. There is no consensus on philosophy on this. I think it's a physics mathematica question and that calling on supernatural support is unnecessary since I can describe it in terms of physics. But this isn't the thread to go into that, since I see no connection between it and continua
Why does a segment with a length of finite digits change into a length multiplied by pi (pi×2×r) when the segment is made into a circle? The circumference will have digits going to infinity while as a segment it did not? This must be readily explained in mathematics but I don't remember ever seeing an explanation on it
The person who puts one's efforts into pointing at the problems in existing systems need not be the one who produces the repair........You demonstrated that you do not grasp the need for the point to be prior to the line, therefore your claim that it would result in a weaker mathematics is based in misunderstanding.
I think you are indeed pointing at the problem but when you start talking about your solution involving multiple time dimensions and spatial expansion, I find it hard to follow. It all seems like mumbo jumbo. Give me something concrete to chew on. Does your philosophy produce any graphs or equations? How does your philosophy make sense of the infinities in calculus?
What quantum physics demonstrates to us is that points have real existence, and continuities are constructed.
I don't agree with this claim so I'd like to see your evidence that supports it. What is fundamental in quantum physics is the wave function, a continuum. Definite states (like points) only emerge when a measurement is made.
I'm not going from points to continua. I'm going from continua to points. My graphs and videos aren't seeming to help here so let me expand on this quantum analogy. I'm not a quantum physicist so take this with a grain of salt.
Assume that there exists a wave function of the universe that spans all of time. This is the fundamental object of our universe and it is a continuum. And until the wave function is measured it is meaningless to talk about who lived when and where because a wave function does not describe what is, it describes what could be. It is only when you make a measurement that all of the potential states collapse into a single actual state. When I say that points are emergent, I mean that they only emerge when we make a measurement. We cannot say things like 'there are infinite points on this line' because we have not actually placed infinite points on the paper...what we placed on the paper was a line.
Now let's say that in '2+2', the '+' represents a process.
Put it this way: a computer program that calculates 2+2 is what I mean by 'process' and such a program can be studied (even if the program is never executed).
Ryan O'ConnorMarch 13, 2021 at 00:13#5095670 likes
Insofar as the car is moving and never while it is moving not moving, then any method of description that stops it is simply not reflecting reality, but maybe if anything, something other than reality.
What you don't realize is that it's your description which stops the car from moving. That's Zeno's paradox. Motion is impossible if time is just a collection of instants. My description allows the car to move because I'm allowing for time to be more than just instants. In between the measured instants lies an unmeasured wave function within which motion is possible. Motion happens when we're not looking. It's demonstrated by the Quantum Zeno Effect.
Photographs are not instantaneous. The shutter stays open for a period of time, usually a fraction of a second.......any measurement is taken over time. There's no difference between photo and video. Video after all is just a collection of still images, either analog or digital frames. And a single photo is taken over a period of time, namely the shutter speed.
Imagine a dark room and a quantum sensor (which I'll call a film to stick with the photography analogy). The moment a single photon hits the 'film' we have a 'photograph'. This photograph has no blurriness and captures no motion. It is a photograph, not a video. There is no physical law which states that we cannot know the position of a particle with perfect precision. It's just that if we do know the particle's position precisely, we can't know anything about its momentum (velocity). Static measurement (e.g. position) and dynamic measurements (e.g. velocity) are both valid, important, and distinct from each other.
Videos are not just a collection of still images. They are a collection of still images where each image is displayed for some non-zero duration. If each image was displayed for 0 seconds then you wouldn't get a video, no matter how many images you pile on. It all comes back to 'how can you form a line from a collection of points?' The answer is that you can't. But you can easily go the other way. You can easily cut a line to form a point. You can easily pause a video to produce a still image.
I don't see how QM indeterminacy can be fitted into mathematics at its foundation
It's those damn wave functions! They just seem to be everywhere (at least on this forum). No one mentions path integrals in QT. We can imagine waves, but not functional integrals.
All Godel showed in meta-language is that there are infinite things that cannot be prove in math and infinite things that could be proven (from sure foundations). What space IS cannot be precisely said by mathematics. Affirmation of a double negative is what infinity is, which translates to a double positive. The positives are abstract infinity (a single thing, undivided) and abstract finitude (the finite as idea). They merge to form infinite units, composing a single object that swallows itself. It curves back around like on a sphere. We experience objects in their finitude. In thinking abstractly, you have to go back and forth from the infinite as idea and the thought of "the finite" in dealings with uncountable infinity (in the form of points). I think this is how Leibniz understood it, but anyway it certainly is how Hegel understood Leibniz
Ryan O'ConnorMarch 13, 2021 at 01:46#5096360 likes
Are you just sharing a nice quote from Hawking, or is this supposed to support your argument? If the former, thanks! If the latter, please provide a little explanation.
He said some wanted an ultimate theory that can understand the world in finite ways with finite equations. He said he changed his mind when he reflected on Godel's theorems and realized that the world gives finite results when there is precision from us, who approach the universe as part of it. So there will always be a dialectic between the infinite and the finite in our dealings with the universe.
I am not allowed to quote that article
You're approach is typology but you haven't said anything about the system works. (Topology says how you get results)
An object is bounded by points and a finite surface area. This is how continua is defined. The infinity is in the paths within these bounds, because parts, motions, and paths are uncountably infinite with it
Oh boy my mentions have piled up tonight. I wanted to respond to this before diving back into the maw of the slavering mob I triggered by daring to express a political opinion they don't like. But never mind all that.
The correct statement is "the potentially infinite process defined by 0.333... converges to the number 1/3" not "the number 0.333... equals the number 1/3".
Before going on @Metaphysician Undercover pointed out a distinction we need to nail down. Please respond clearly to this question. Are you talking about
* Mathematics? or
* Some kind of notion of reality that goes beyond math?
If the former, you're just wrong. 1/3 = .3333... because the right hand side is just the sum of the infinite series 3/10 + 3/100 + 3/1000 + ... which equals 1/3 since it's a geometric series as taught in freshman calculus. It does you no good to demonstrate ignorance of the theory of limits. The sum is defined as exactly 1/3, not "approaching 1/3" or "infinitesimally less than 1/3" or any similar confused location that calculus students often have after slogging through the course. Mathematically you're wrong as a matter of fact.
On the other hand if the latter, I have no idea what you mean, because 1/3 = .333... as a matter of formal mathematics. I can prove it directly from the axioms of ZF set theory. There is no question about it. It's as certain as saying that a certain chess position follows from the rules of the game. You'd be better off self-studying real analysis than wasting your time telling me about your mathematical misunderstandings. And if you think it means something reality, you're likewise wasting your breath because I have no interest. I'm doing math, not metaphysics; and frankly I'm not sure 1/3 OR .333... OR the number 47 have any metaphysical meaning at all. That's a completely different subject.
Decimal notation is flawed in that it cannot be used to precisely represent some rational numbers, like 1/3. If we want a number system which can give a precise notation for any rational number, we should use Stern-Brocot strings, where 1/3 = LL.
I don't understand why you like one infinite representation rather than another, but you are riding a hobby horse and making no rhetorical points with me at all. You're wrong on the math and confused on the metaphysics.
If photographs can't capture motion but videos can, why not conclude that motion happens in the videos? The reason why we are reluctant to come to this conclusion is because we reject the notion of videos being fundamental.
No not at all. First, video is nothing more than a series of stills, whether analog or digital frames. Have you ever seen a flip book? We see the illusion of motion from a sequence of still images because our eye-brain system retains a little afterimage for a fraction of a second. That's how movies work, I'm certain you must know that. Back in the old days I believe flip books were a common medium for the pr0n of the day but I found a really nice G-rated contemporary flip book artist. This video is worth checking out for its own sake and makes this point dramatically. Video is nothing more than a rapid sequence of stills. It depends on a quirk of your eye-brain system.
https://www.youtube.com/watch?v=ntD2qiGx-DY
Secondly, how about still photos? If you ever had a DSLR or film SLR you know that one of the important controls is the shutter speed. You hold the shutter open for a period of time, usually a fraction of a second, and during that time interval photons come in and hit the film or digital sensor. If an object is moving side to side in front of you, you can always determine its velocity by setting a long enough shutter speed that you get some motion blur; then you can work backwards from the blur to determine the velocity. You know, the same reason your low-light photos taken with your automatic camera are blurry. The camera compensates for the low light by choosing a slower shutter speed, and your shaky hands cause motion blur.
So still photos actually take time, and videos are just a sequence of stills.
Why are you saying this? I assume you must know it's wrong, no mechanical device is capable of exposing a light-sensitive medium for a true instant. Are you speaking metaphorically? If you set your hypothetical camera to an instantaneous shutter speed no photons could get in and the image would be blank.
to be fundamental and continua (videos) to be composite and as long as we hold this view we will not find a satisfactory resolution to Zeno's paradoxes.
We're not going to solve Zeno's paradoxes here. Better you should try to understand some math, in particular the sum of a geometric series.
If you flip things upside down and see continua as fundamental and points as emergent, then everything makes perfect sense. There's no problem with pausing a video to produce a static image.
Utter nonsense. If you put your webcam on yourself, record at 30 frames per second, and move around, then play it back frame-by-frame, you will most definitely see motion blur in each frame. Unless you are speaking metaphorically about an idealized camera, you are just wrong on the technology. And an idealized camera with infinitesimal shutter speed could not transmit any photons to the light-sensitive medium, so it wouldn't work.
It is clear that you appreciate the profoundness of Zeno's Paradox. Zeno presented these paradoxes in response to the criticisms from the 'one from many' camp calling his views ridiculous. Why not consider the 'many from one' view that he supported? He was wayyy ahead of his time so his view did seem to have problems of their own...but in light of modern advancements in physics his view no longer seems crazy.
We're not going to resolve those ancient issues here.
Don't stop here, you may just be on your way to becoming a crank! With this admission you have placed yourself on a slippery slope. Instantaneous velocity is no different from the tangent of a function at a point. Do you accept that the derivative corresponds to a limiting process of secants rather than the output of a completed infinite process (i.e. tangent at a point)?
What? The tangent line IS defined as the limit of the secant (for a function from the reals to the reals). That's how they motivate it. But in fact that is the DEFINITION of the tangent line. Without the definition, we don't even know what a tangent line is.
The pre-Socratic ontology said that water was pure simplicity and with fire can make perfect movement. It bowls over into the reality of cars, etc (earth and wind) This was seen within the geometry of atomism and Zeno's thesis
So you're no longer talking about cameras but rather quantum sensors? It would be better if you dropped these labored analogies and just made your point some other way. Are you saying photons use quantum tunneling to show up on the sensor without the shutter being open? This is such an unproductive tangent to the discussion. To record an image on a photographic medium you have to expose the medium over a period of time. You're stepping all over your own point, we're not talking about cameras we're talking about something else and this isn't making sense.
Why does a segment with a length of finite digits change into a length multiplied by pi (pi×2×r) when the segment is made into a circle? The circumference will have digits going to infinity while as a segment it did not? This must be readily explained in mathematics but I don't remember ever seeing an explanation on it
Just an artifact of the decimal representation. Doesn't mean anything. Roman numerals had drawbacks, decimal representation has drawbacks. Every notation has some advantages and some drawbacks.
As to why a circle's circumference and radius are incommensurable, I think that's one of God's little jokes. God in Einstein's sense. Not necessarily religious, but a stand-in for all the ineffable mysteries of the world. With a sense of humor thrown in.
Your guess is as good as mine. There's no actual reason. But the diagonal of a square seems like an even simpler example. One unit to the right, one unit up, and the distance between your start and end points is incommensurable with 1. No reason at all, just a shocking surprise.
How can anyone deny the existence of the length of the diagonal? That's the part I don't get.
That's why velocity is always an average, requiring at least two temporal points. Duration is derived, just like distance is. To infer an instantaneous velocity requires a second derivation.
A moving body has an instantaneous velocity, even though our formalism requires two temporally separated measurements. But, I'm willing to concede that you've either made your point or at the very least caused me to doubt mine.
It's FALSE that motion through a point is the same as resting in a fixed point. There is energy that pushes right through. Imagine an arrow being pushed by the air rushing behind as it pushes forward. What's wrong with this picture (of the arrow)? It's that the motion forward is prior to air rushing behind so we can not saw the air pushes the arrow. The arrow moves through any point with forward velocity so it's never ever at rest
I don't agree with this claim so I'd like to see your evidence that supports it. What is fundamental in quantum physics is the wave function, a continuum. Definite states (like points) only emerge when a measurement is made.
What is real and fundamental in quantum physics is the points where particles appear. The wave function is the mathematical apparatus which predicts where particles might appear. Yes, the wave function is fundamental to the model, but what is being modeled is the appearance of particles at specific points. This is why physicists understand light as photons, because the energy appears at, and causes an effect at a point.
You say that points only emerge from a measurement, but a measurement is an interaction between the energy, and the object which is the measuring device. So, such points exist wherever energy is interacting with objects. What this indicates is that energy, though it is modeled as existing in a continuum, (wave function) only interacts with the physical objects which we know, at discrete points. Therefore our only access to observe whatever substratum there is, which is modeled as wave functions, is through an understanding of these points where we can observe interactions.
Sure, you might say that the continuum, or substratum as I call it, is more fundamental, but from the point of view of the model, and this means the mathematics, the points must be fundamental. This is because we only find a route inward, toward understanding the substratum through a mapping of the points where it interacts with the spatial existence we know, observe, and understand. What must be fundamental, and basic to the model, is what we know the best, and this is the points. The substratum is modeled based on the existence of those points where we can observe it The better we know the points, the more reliable our speculation about the substratum will be.
Assume that there exists a wave function of the universe that spans all of time. This is the fundamental object of our universe and it is a continuum. And until the wave function is measured it is meaningless to talk about who lived when and where because a wave function does not describe what is, it describes what could be. It is only when you make a measurement that all of the potential states collapse into a single actual state. When I say that points are emergent, I mean that they only emerge when we make a measurement. We cannot say things like 'there are infinite points on this line' because we have not actually placed infinite points on the paper...what we placed on the paper was a line.
The substratum, which is represented by wave functions, may or may not be a continuum. That a wave function represents it as a continuum doesn't mean it is. Furthermore, a measurement is simply the substratum interacting with a physical object. So if this causes a "collapse", there are collapses occurring all the time, all over the place, as the substratum is interacting with physical objects. And, if measurements are only possible at particular points, then we ought to assume that other interactions between the substratum and physical objects are only possible at particular points, and this is most likely a feature of the substratum itself.
Put it this way: a computer program that calculates 2+2 is what I mean by 'process' and such a program can be studied (even if the program is never executed).
I don't agree. A process which is never executed cannot be studied. It has no existence so it cannot be studied. Let's say that you write out a rule, an algorithm, but the algorithm is never implemented. You can study that rule, but you cannot study the process dictated by that rule, because it does not exist. The rule was never put to work, actualized, it exists only as the potential for the designated process Do you see the difference between the written rule, and the activity which is prescribed or described by that rule? To study one is not the same as studying the other.
[quote="fishfry;509689"]A moving body has an instantaneous velocity,../quote]
Yes, because that is the convention, use some math, and figure out the "instantaneous velocity", just like the convention is to place a zero limit on the example of the op. But what these conventions really represent may not be what one would expect from the terms of usage.
Whether 1+1 creates a new number (Kant) or a set (Frege I think) is a question for phenomenolog psychology, not math. And most of these questions are not useful for modern science.
Yes, because that is the convention, use some math, and figure out the "instantaneous velocity", just like the convention is to place a zero limit on the example of the op. But what these conventions really represent may not be what one would expect from the terms of usage.
A car whose speedometer reads 40mph is going 40mph at that moment even though that's a mechanical approximation. But as I say I'll concede you've moved me off my certainty and I've run out of talking points. After all we don't know the ultimate nature of reality so who's to say if the notion of instantaneous velocity really makes sense. Based on that I'll concede the point.
I've been meaning to ask you: "Descartes great merit here was to have applied geometry to algebra; he was not the first to have applied geometry to geometry... And to Descartes we owe the first systematic classification of curves. After dividing 'geometric curves ' which can be precisely expressed in equations from 'mechanical curves' that cannot, he classified the former into three clases... This new geometry is more than a general theory of quantity: it led to the concept of continuity, from which was developed the theory of function and, in turn, the theory of limits... But he was mistaken in believing that equations of any order could be so resolved." ( Britannica Encyclopedia 1965)
I think applying numbers to geometry is how we apply Godel's numbering to physics
A car going at constant speed passes point A at stopwatch time=0, then passes point B, one mile further at stopwatch time=one minute. You ask, "What was the speed of the car back there at point A?" Your answer, "It was moving at 60 mph at point A".
Silly, but this entire discussion needs termination.
I consider this thread to be about the "geometry of Godel" but nobody uses that phrase and it does spin out of control. Peop!e start talking about space being outside itself and such.
I have two links saved somewhere on Leibniz' relationship to this question. I'll go find them
Marx said Hegel was the best of mathematicians yet Hegel learned what little he knew of the subject from Leibniz, who could easily be the greatest of all mathematicians (so my intuition is saying).
This also relates:
I think applying numbers to geometry is how we apply Godel's numbering to physics
I don't think physics in its present form is amenable to methods of formal logic because there are no axioms for physics. Axiomatizing physics is Hilbert's sixth problem and according to Wiki it's still open. Even so I doubt that the axioms of physics, whatever they may eventually turn out to be, satisfy the premises of Gödel''s incompleteness theorems.
But of course you could Gödel-number anything, Moby Dick or the contents of this post. You assign numbers to the upper and lower-case English letters and punctuation symbols, and you apply Gödel numbering. So if 'abc" is your string, and a = 1, b = 2, c = 3 is your encoding of the symbols, then 'abc' is encoded as [math]2^1 \times 3^2 \times 5^3 = 2 \times 9 \times 125 = 2250[/math].
The idea is to use the unique factorization of integers into prime powers. To go backwards from 2250, we note that it's [math]2^1 \times 3^2 \times 5^3[/math] and read off the exponents, 1, 2, 3 to get 'abc'. You could do this with any piece of text.
As you can see, if you believe that the positive integers exist (whatever existence means for you) then every finite-length document that could ever be written, already exists. It's already encoded as some positive integer. LIkewise if you pixellate a painting finely enough so that the pixellated version is indistinguishable from the original (to the naked eye, say) and encode its color and texture values, you can Gödel-number all possible paintings. Likewise songs, etc.
Or as the Pythagoreans said: All is number!
Metaphysician UndercoverMarch 13, 2021 at 12:04#5097790 likes
After all we don't know the ultimate nature of reality so who's to say if the notion of instantaneous velocity really makes sense.
This is the point. When we use math to figure out things like instantaneous velocity, the volume of a supposed infinitely small tube, etc., it is implied that we know things about reality which we do not. This is a falsely supported certitude.
A car going at constant speed passes point A at stopwatch time=0, then passes point B, one mile further at stopwatch time=one minute. You ask, "What was the speed of the car back there at point A?" Your answer, "It was moving at 60 mph at point A".
There is a flaw with your example jgil. That the car was "going at a constant speed" is just an assumption, so it may not be the truth of the matter. And your answer as to how fast the car was going at point A requires that the assumption be true. So it needs to be proven.
You might lay out a series of such points, at equal distance, and do numerous similar measurements. If your measuring capacity is precise, you'll find that all the measurements will not be exactly the same. The assumption of "constant speed" cannot be validated. That's what we've found out about the nature of reality, motion consists of spurts and starts. So you'd have to establish trends, and the more measurements you took the better your graphing of the trends would be. But you'd be graphing averages which does not tell you the precise amount at any given point.
Yes. Goes back a hundred years if my memory serves. Sometimes it's very easy. For example, here is a linear fractional transformation written in terms of fixed points and multiplier.(I'm working on a theorem right now involving this). I think a more general case was dealt with in the discipline of functional equations. Can't recall the work offhand.
The tangent line IS defined as the limit of the secant (for a function from the reals to the reals).
Are tangent and instantaneous rate of change not the same thing? If you reject the notion of instantaneous rate of change, how can you not reject the notion of tangents?
Are you talking about Mathematics or some kind of notion of reality that goes beyond math?
I'm talking about mathematics. I understand that 0.333... converges to 1/3, but it is only (a useful) convention which states that convergence and equality are the same. If you're saying that it's proved somewhere that the two terms are equivalent then let's leave it at that.
I don't understand why you like one infinite representation rather than another, but you are riding a hobby horse and making no rhetorical points with me at all. You're wrong on the math and confused on the metaphysics.
The Stern-Brocot string for any rational number has finite characters. I don't accept the claim that that LL = LLLRrepeated since LL corresponds to a position in the tree and LLLRrepeated corresponds to a path along the tree. However, let's not waste any more time on 1/3.
Why are you saying this? I assume you must know it's wrong, no mechanical device is capable of exposing a light-sensitive medium for a true instant. Are you speaking metaphorically? If you set your hypothetical camera to an instantaneous shutter speed no photons could get in and the image would be blank.
As with your speedometer argument, your addition of a shutter only complicates the issue without providing any further explanatory power. We add a shutter to cameras only to limit the amount of light that the film is exposed to. Without a shutter we can (at least in principle) still take photographs. Remove the shutter and the instant the first photon hits the film we have a photograph. And if no further photons hit the film we have an image with absolutely no blurriness. This image captures no motion. It is not a video by any definition. I was anticipating you challenging the practicality of such a photograph, which is why I went quantum, but perhaps that's not necessary.
video is nothing more than a series of stills, whether analog or digital frames.
I understand how videos and flipbooks work, and yes we use stills to create them, but the magic ingredient which you are ignoring is time (specifically non-zero intervals of time). We hold each frame for 1/24 seconds before advancing to the next frame.
(I believe) Zeno's paradox was already informally solved by Aristotle. I'm just defending his view. And loosely speaking the view is simply that we start with videos, not stills. With videos we not only can capture motion but we can also pause the video to extract a still. With only stills, motion is not possible, as per Zeno. Zeno's paradox is important and seen as unresolved because the notion of stills being fundamental is deeply rooted in our beliefs.
Metaphysician UndercoverMarch 13, 2021 at 23:11#5099590 likes
Are you gong to argue that the car is not moving at any speed during its traverse of the distance A to B?
Yes, that's about it. Speed is what we assign to the car, it is what we say about it, it has speed. In philosophy we must maintain the distinction between what we say about the thing, and what is really the case, to allow for the real possibility that what we say about the thing might actually be a falsity. If the property which we assign to the thing, "speed", in this example, has faults within its conception (contradictions for example), then despite the fact that it has become acceptable to say this, the concept is defective, and it is really not true to be attributing that property.
You are making a claim about reality (i.e. it's made of events of information). Aristotle slumped into this when he said parts are potential. What exists is the whole composed of all it parts, which are bounded by points (finite) and limit in space (finite). A material body doesn't have math in it. We use imperfect mathematical formulations to understand to described in the field of physics. You can't draw philosophical conclusions from physics is the conclusion. You fell for the Parmedian world view by trying to figure out the logic of his disciple
Are tangent and instantaneous rate of change not the same thing?
No they're not. In functions from the reals to the reals (ignoring the multivariable case where this terminology doesn't make sense) the instantaneous rate of change is the slope of the tangent line.
If you reject the notion of instantaneous rate of change, how can you not reject the notion of tangents?
I don't, you and @Meta do. I'm agreeing that IN REALITY there may not be such a thing. But in math, there most definitely is. Again. you are equivocating math and physics. I'll stipulate to @Meta and your point that instantaneous rates are murky in physics. In math they're perfectly well defined.
I'm talking about mathematics. I understand that 0.333... converges to 1/3, but it is only (a useful) convention which states that convergence and equality are the same. If you're saying that it's proved somewhere that the two terms are equivalent then let's leave it at that.
You can't maintain your credibility while arguing against freshman calculus. here's the proof "somewhere," a somewhere I already linked to earlier.
The Stern-Brocot string for any rational number has finite characters. I don't accept the claim that that LL = LLLRrepeated since LL corresponds to a position in the tree and LLLRrepeated corresponds to a path along the tree. However, let's not waste any more time on 1/3.
On the contrary, 1/3 is the critical case here. If you can't agree that 1/3 = .333... then you fail freshman calculus and high school algebra too. There's nothing more to talk about then.
As with your speedometer argument, your addition of a shutter only complicates the issue without providing any further explanatory power. We add a shutter to cameras only to limit the amount of light that the film is exposed to. Without a shutter we can (at least in principle) still take photographs. Remove the shutter and the instant the first photon hits the film we have a photograph.
You know you keep making claims totally contrary to photographic technology. A single photon is not sufficient to make an impression on any film stock or digital sensor in existence. So you're just flat out wrong here. And if you don't open shutter at all, no photons will come in; and to get sufficient photons in to make an impression on a light-sensitive medium, you need to keep the shutter open for a period of time. Of course I include an electronic shutter that merely activates the sensor for a period of time.
And if no further photons hit the film we have an image with absolutely no blurriness.
There are no single-photon detectors outside of physics labs. But again, I don't know why you're belaboring this point. If you don't think that 1/3 = .333... AND you agree that you are making a mathematical point, then there is no conversation to be had. You're just wrong. Read a calculus book or work through the proof I linked (twice now) on Wiki.
This image captures no motion. It is not a video by any definition. I was anticipating you challenging the practicality of such a photograph, which is why I went quantum, but perhaps that's not necessary.
I see no benefit to this point at all. What of it?
I understand how videos and flipbooks work, and yes we use stills to create them, but the magic ingredient which you are ignoring is time (specifically non-zero intervals of time). We hold each frame for 1/24 seconds before advancing to the next frame.
(I believe) Zeno's paradox was already informally solved by Aristotle. I'm just defending his view. And loosely speaking the view is simply that we start with videos, not stills. With videos we not only can capture motion but we can also pause the video to extract a still. With only stills, motion is not possible, as per Zeno. Zeno's paradox is important and seen as unresolved because the notion of stills being fundamental is deeply rooted in our beliefs.
I don't think we're having the same conversation anymore. But regarding your claim about video, I invite you to wave your arms in front of your webcam while recording, then play it back frame-by-frame. You'll see motion blur. I know this because I've seen it many times. In fact if you are making a video with the intention of capturing still frames you have to make sure to stay motionless for a few moments at a time to avoid motion blur.
To go from A to B you have to go half that distance, since the distance is distance as space. Half of AB is a distance, otherwise you are at the half point instantaneously. So he must go the quarter. But the quarter is spatial it too has a half point mark, ... And a computer can run this activity to an uncountable infinity. There is no paradox as to how motion starts from energy. The question is how it is that the supertask is done in finite bounds (time and space)
Ryan O'ConnorMarch 14, 2021 at 00:35#5100090 likes
What is real and fundamental in quantum physics is the points where particles appear.
My impression is that you're a finitist, so I presume that you believe our universe had a beginning of time. If particles are fundamental, they must have existed at that initial moment, right? Were they concentrated at a point? I take it that you think a measurement involves the interaction of particles, so at the initial instant wouldn't they all be measuring each other? If so, how would they ever move, given the quantum Zeno effect?
Consider this: "QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles." source
You can study that rule, but you cannot study the process dictated by that rule, because it does not exist.
I think you're splitting hairs here. By rule I assume you mean the 'computer program' and by process I assume you mean 'the execution of the computer program'. If so, then we are in agreement, we can study the rule (i.e. the computer program).
The Zeno effect and the anti-Zeno effect refer to how observation changes eternal states. The ancient Arrow paradox is just used to illustrated the effect and the effect does not resolve the Arrow paradox because its not specifically related to it
Ryan O'ConnorMarch 14, 2021 at 01:28#5100340 likes
I'm agreeing that IN REALITY there may not be such a thing. But in math, there most definitely is.
My view is that actual infinity should not be permitted in math any more so that than it is permitted in physics, but that's just my view and it's contrary to contemporary math so I'm willing to leave it at that.
You can't maintain your credibility while arguing against freshman calculus. here's the proof "somewhere," a somewhere I already linked to earlier.
The sum is defined as [math]s=a\frac{1-r^{n+1}}{1-r}[/math]
There is a difference between substituting a finite number in for n and substituting infinity in for n. Obviously the latter is not allowed, so we talk about convergence instead. If we ignore the fact that we're talking about convergence when talking about the sum of an infinite series then we are essentially substituting infinity in for n. It is a matter of convention that we say that geometric series sums to a number. More formally, we should say that it converges to a number. I think 0.333... converges to 1/3 is a valid statement.
You know you keep making claims totally contrary to photographic technology.....There are no single-photon detectors outside of physics labs. But again, I don't know why you're belaboring this point. If you don't think that 1/3 = .333... AND you agree that you are making a mathematical point, then there is no conversation to be had.
Ha! I knew you were going to say this. That's why I had switched to a quantum sensor. With an SLR camera I agree that every photo has some degree of blurriness, but with quantum sensors that's not necessarily the case. There is no law which states that we can't know the position of a particle with perfect precision. Also, I don't think particles are points, but instead excited states of quantum fields.
I don't think we're having the same conversation anymore.
Forget about cameras, sensors, and speedometers - it all boils down to the question of whether a line can be constructed by assembling points. Your earlier post indicated that you agree that this is a mystery (given the orthodox views)? Why not consider alternate views?
Ryan O'ConnorMarch 14, 2021 at 01:51#5100430 likes
You're approach is typology but you haven't said anything about the system works. (Topology says how you get results)
An object is bounded by points and a finite surface area. This is how continua is defined. The infinity is in the paths within these bounds, because parts, motions, and paths are uncountably infinite with it
Perhaps I am misusing terms. Certainly, when I say graphs are topological, I don't use it in the standard sense of the word topology. Instead, I'm using it in a looser sense, simply that the properties (of interest) of the graph are maintained through continuous deformations. The results are largely unchanged in reinterpreting graphs as topological objects (in the sense of topology mentioned above).
As for continua, I simply mean objects with extension.
You are making a claim about reality (i.e. it's made of events of information). Aristotle slumped into this when he said parts are potential. What exists is the whole composed of all it parts, which are bounded by points (finite) and limit in space (finite). A material body doesn't have math in it. We use imperfect mathematical formulations to understand to described in the field of physics. You can't draw philosophical conclusions from physics is the conclusion. You fell for the Parmedian world view by trying to figure out the logic of his disciple
The whole need not be bounded by points. Think of the open interval (-1,1). I think physics can and should inform our philosophy and you're right that I'm influenced by Aristotle and Zeno.
The Zeno effect and the anti-Zeno effect refer to how observation changes eternal states. The ancient Arrow paradox is just used to illustrated the effect and the effect does not resolve the Arrow paradox because its not specifically related to it
I haven't looked for a reference but I assume they named the Quantum-Zeno effect because of the Zeno's Paradox (especially Zeno's Arrow Paradox). If so, I don't see how it's unrelated.
My view is that actual infinity should not be permitted in math any more so that than it is permitted in physics, but that's just my view and it's contrary to contemporary math so I'm willing to leave it at that.
Not all contemporary math. In complex analysis one may move to the Riemann sphere and take the north pole as infinity, but I stick to the complex plane and use expressions like "unbounded" instead. Now, set theory is another animal altogether. Not my cup o' tea, but fishfry is an excellent in-house expert.
Ryan O'ConnorMarch 14, 2021 at 02:00#5100500 likes
We measure the car at 60mph and maybe that's accurate to within a small margin of error.
I said "faults", and I used "contradiction" as an example of a fault. That there is a "margin of error" is another indication of fault. When a small margin of error is ignored or neglected, as if it doesn't exist, one can fall for a paradox like Zeno's, where that small margin of error is infinitely magnified to produce the appearance of contradiction.
My impression is that you're a finitist, so I presume that you believe our universe had a beginning of time. If particles are fundamental, they must have existed at that initial moment, right? Were they concentrated at a point? I take it that you think a measurement involves the interaction of particles, so at the initial instant wouldn't they all be measuring each other? If so, how would they ever move, given the quantum Zeno effect?
I really don't get your question. I was talking about points, not particles, so your question has some underlying presumptions which I don't follow.
Consider this: "QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles." source
I addressed this already. The so-called "underlying quantum fields" are models produced from observations of particles, and are meant to model the interactions of particles. It is implied that there is an underlying substratum which validates this modeling, but the modeling itself, the quantum field theory, does not represent the underlying substratum, it represents the interaction of particles. Until we get accurate and precise modeling of the particles any speculation concerning the substratum is not well informed.
I think you're splitting hairs here. By rule I assume you mean the 'computer program' and by process I assume you mean 'the execution of the computer program'. If so, then we are in agreement, we can study the rule (i.e. the computer program).
OK, so we say that the rule calls for the computer to carry out an endless, or infinite process. We know that the computer cannot succeed in carrying out this request, because it will wear out first, so all the time spent will be wasted, for the computer to be trying to carry out a process it can't. So if we turn to study that rule, should we not put our efforts into avoiding this rule, making it so that the rule never comes up, because it's like a trap which the computer will fall into? Therefore instead of pretending to be having success at carrying out infinite processes, which is self-deception, we should be looking at ways to make sure that such rules are banished.
Ryan O'ConnorMarch 14, 2021 at 02:18#5100580 likes
What is real and fundamental in quantum physics is the points where particles appear.
I believe physicists think particles are only states of quantum fields. If so, you should not be thinking of the point where a particle appears, but instead the continuum where the quantum field exists.
The so-called "underlying quantum fields" are models produced from observations of particles, and are meant to model the interactions of particles.
If you place iron filings over a magnetic field the filings will take a form in line with the field. While it's true that we only see the filings, it is untrue to say that the field is just a model. It's real. The same goes for quantum fields.
So if we turn to study that rule, should we not put our efforts into avoiding this rule, making it so that the rule never comes up, because it's like a trap which the computer will fall into? Therefore instead of pretending to be having success at carrying out infinite processes, which is self-deception, we should be looking at ways to make sure that such rules are banished.
No. If we terminate the potentially infinite process we still get something useful (e.g. the rational approximation of pi on your calculator is a useful button). Also, I would argue that calculus is the study of these 'rules' and calculus is arguably the most useful branch of mathematics. What I agree with you on is that we should not try to carry out the infinite process to completion...that is a fruitless endeavor.
Infinity creates a situation in this question about space that make the question of continua difficult (The reason being that discrete apace is an oxymoron). However in numeral mathematics infinities work ok. Paul Cohen found contradictory proofs in infinite mathematics in the 1960's but the subject simply is not understood properly enough. Maybe a theory of everything which provides the connection between discrete apace (a point) and finite geometry (solids) be found. But nonetheless banishing infinity from mathematics is a move of an ostrich
Ryan O'ConnorMarch 14, 2021 at 03:13#5100770 likes
I had my weekly chat with an old colleague (math prof) living in a retirement home today and brought the subject of instantaneous velocity up. He has his doubts about the existence, as do some of you, and for the same reasons. It doesn't bother me either way. I'll use the term since others will know what I am talking about: a certain mathematical limit.
Ryan O'ConnorMarch 14, 2021 at 03:57#5101060 likes
My view is that actual infinity should not be permitted in math any more so that than it is permitted in physics, but that's just my view and it's contrary to contemporary math so I'm willing to leave it at that.
I think we're done then.
Just out of curiosity though, how do you develop a theory of the real numbers without infinite sets? Even the constructivists allow infinite sets, just not noncomputable ones.
switched to a quantum sensor. With an SLR camera I agree that every photo has some degree of blurriness, but with quantum sensors that's not necessarily the case. There is no law which states that we can't know the position of a particle with perfect precision.
Also, I don't think particles are points, but instead excited states of quantum fields.
Forget about cameras, sensors, and speedometers - it all boils down to the question of whether a line can be constructed by assembling points.
The question doesn't come up in math. We use the phrase "the real line" as an alternate way of saying, "the set of real numbers," but everything can be done without reference to geometry.
Your earlier post indicated that you agree that this is a mystery (given the orthodox views)? Why not consider alternate views?
If I see a coherent one presented I'll engage with it. In the past I've engaged extensively with constructivists on this site and learned a lot about the contemporary incarnations of that viewpoint.
I've also studied the hyperreals of nonstanard analysis. So in fact I'm very open to alternative versions of math, but I don't see that you've presented one. The problem with finitism is that you can't get a decent theory of the real numbers off the ground.
I'm not proposing we ban Infinity altogether. I'm proposing that we restrict ourselves to only use Infinity in a potential sense.
"The philosopher and theologian are conscious of infinity, but from the mathematician's view they do no use it so much as admire it. The mathematician also admits infinity; the great David Hilbert said of it that in all ages this thought has stirred man's imagination most profoundly, and he described the work of G. Cantor as introducing man to the Paradise of the Infinite. But the mathematician also uses infinities..." Leo Zippin
"Every since we first sought number in the object, the series of numbers has begun with 1. Making zero the first of numbers means no longer abstracting them from the object" Jean Piaget
"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."?Hermann Minkowski, 1909
"At a time when Minkowski was giving the geometrical interpretation of special relativity by extending the Euclidean three-space to a quasi-Euclidean four-space that included time, Einstein was already aware that this is not valid, because it excludes the phenomenon of gravitation. He was still far from the study of curvilinear coordinates and Riemannian geometry, and the heavy mathematical apparatus entailed"
https://en.wikipedia.org/wiki/Pseudo-Euclidean_space
https://en.wikipedia.org/wiki/Minkowski_space#cite_ref-14
https://www.quora.com/How-can-I-a-non-mathematician-wrap-my-mind-around-the-Axiom-of-Choice: The reason the Axiom of Choice is (somewhat) controversial is that while it allows us to prove some very useful mathematical statements, it also allows us to prove some less intuitive statements (e.g., the Banach-Tarski paradox).
And finally:
"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." Einstein
(Don't forget Kant's second antimony: (On Atomism)
Thesis:
Every composite substance in the world is made up of simple parts, and nothing anywhere exists save the simple or what is composed of the simple.
Anti-thesis:
No composite thing in the world is made up of simple parts, and there nowhere exists in the world anything simple)
If you place iron filings over a magnetic field the filings will take a form in line with the field. While it's true that we only see the filings, it is untrue to say that the field is just a model. It's real. The same goes for quantum fields.
There is an issue of truth here. There is something there causing the form, and the concept of "field" attempts to account for whatever it is. If the concepts employed are inadequate, then it's not true to say that this is what is there. Here's an example. The ancient Greeks used circles to model the movement of the planets, and Aristotle proposed that the orbits were eternal circular motions. It turned out that these models were wrong, therefore it was not true for them to have been saying that the orbits were circles even though this concept was employed and enabled prediction.
No. If we terminate the potentially infinite process we still get something useful (e.g. the rational approximation of pi on your calculator is a useful button).
Again, there is an issue of truth here. If the process is terminated then it is untrue to say that it is potentially infinite. And if we know that in every instance when such a process is useful, it is actually terminated, then we also know that it is false to say that a potentially infinite process is useful, because it is only by terminating that process, thereby making it other than potentially infinite, that it is made useful. Therefore t is false to say that the potentially infinite process is useful.
But nonetheless banishing infinity from mathematics is a move of an ostrich
No, the opposite is the case. Ignoring the fact that infinities in mathematics is a very real problem, is the type of ignorance which is analogous with the ostrich move.
But he/it doesn't, so the issue of passing particular points is no different from passing any point, and yet all those other points are never mentioned. Why is that, do you suppose? Achilleus - or the Arrow - seems to have no problem whatever passing those. Zeno's then, just an entanglement with words.
What do you think "passing a point" means? Do you mean to say that there are physical points out there, which the arrow can be seen flying by? If so, then you ought to be able to show empirically, the physical existence of such points, and I don't think there will be an infinity of them. If these points are just imaginary, then the arrow doesn't really fly by them and you have created a false scenario, by describing the arrow as flying by points.
I propose that the truth is that the points are imaginary. If this is the case, then any method of measuring motion, velocity and such; which employs points, is really giving us a false measurement. We might be able to find real physical points, which if they exist, would validate such a method, but then these points would not be infinite, so that scenario with infinite would become irrelevant, because we'd have to make a new method of measuring velocity based on empirically verified points. As I explained to you earlier, this is pretty much what relativity theory does, but each empirically verified point turns out to be a different frame of reference, and that the points are at rest relative to each other is very doubtful due to the observed phenomenon of spatial expansion.
Aristotle said infinity is "one". Modern mathematics say it's two (countable, uncountable) with perhaps subdivisions. A Philosophy Overdose audio on youtube said that Paul Cowen proved and then disproved that there is an infinity set between the two of Cantor. This is a challenge for mathematicians and they are not going to listen to a philosopher like you
Daoism at its founding taught that being and non-being create each other. Such dialectical logic is all over philosophy. Yet mathematics is a field that does not use such ideas
"Paul Cowen proved and then disproved that there is an infinite set in between the two of Cantor"
The continuum hypothesis has arguements for and against it too, as can be seen in the WIkipedia article on it. Organizing these ideas into something consistent is going to take work by hundreds of mathematicians and many many years of toil
Philosophy says that a point is a negation of a line, but if the points are ordered they create a double negative, and thus the positive of the line. The line does the same with the solid, and the solid in turn abrogates what comes before and creates dimensions greater than three. Philosophy and mathematics come onto these questions from very different perspectives and we philosophers can't expect our explanations and demands to be accepted by those with high math skills
Sure. (Remember I'm coming at this as a philospher). Pi specifies a a certain precision between the numbers 3 and 4. When it comes to a segment, the precision is at every spot of it. However these points are nothing but precision. They're "not-space" and therefore negative to the segment. But if we turn the segment upside down the finite nature of the points become a positive length of finitude in the segment. The negative can be positive as double negative and therefore we have a segment of a line. This same process happens as we go from dimension to dimension. But remember I think this is probably more philosophy than mathematics. There is not a true supertask in motion because there is not an infinite set of actual lengths being covered. But the division OF the length covered by motion has no end, and this is true of anything spacial
I really hope this was helpful. Space has a strange relationship to itself.
Ryan O'ConnorMarch 14, 2021 at 22:07#5103950 likes
Just out of curiosity though, how do you develop a theory of the real numbers without infinite sets? Even the constructivists allow infinite sets, just not noncomputable ones......The problem with finitism is that you can't get a decent theory of the real numbers off the ground.
I think this is the wrong question to ask. We know what we want (a consistent foundation for calculus) and we think we know how to get it (with real numbers and infinite sets). But real number might not be the answer.
I think a much better question to ask is 'can we build a consistent foundation for calculus without real numbers and infinite sets?' I believe the answer is yes. Now, this doesn't imply that the alternative requires scrapping everything about real numbers. For example, I believe the alternative would have still have the area of a unit circle being pi, it would just mean something different.
If I see a coherent one presented I'll engage with it. In the past I've engaged extensively with constructivists on this site and learned a lot about the contemporary incarnations of that viewpoint. I've also studied the hyperreals of nonstandard analysis. So in fact I'm very open to alternative versions of math, but I don't see that you've presented one.
Like the real numbers, the fundamental objects of the hyperreals are points/numbers. A continuum is constructed by assembling infinite points, each with a corresponding number. And so the hyperreals gets no closer to answering the question 'how can a collection of points be assembled to form a line?' I haven't investigated the constructivists' methods to produce an informed comment but from what I've seen it looks like more of the same. Points/numbers are treated as fundamental.
I don't have a formal theory but I do have an intuition on how the alternative would look. In line with Aristotle's solution to Zeno's Paradox, my view has continua (not points) being fundamental. I think the mathematics of calculus would be almost entirely unaffected in moving to a continuum-based view, but before we could even talk about it, you'd have to first be open to (at least temporarily) shedding your point-based biases and look at graphs in a new light. I would love to hear your criticisms on my earlier post to you where I drew the polynomial in an unorthodox way.
We measure the car at 60mph and maybe that's accurate to within a small margin of error. But at no point during its 60 mph run is its speed zero.
There are two ways to describe the car's motion.
1) Over any non-zero interval of time the car's average velocity is 60mph.
2) At any instant the car's instantaneous velocity is 60mph.
I believe that the latter description does not make sense since velocity is a measure of change over time. But this doesn't imply that I believe that there is some interval for which the car's average velocity is 0mph.
It seems to me that with any line I look at I'm looking at an infinite number of points. Not potential points, but actual points - that is, to the degree that one, or any, point is actual.
This is how we were taught to think, but is it possible that you're just looking at a line? Do you think it's even possible to see a single point? If not, what about 2 points? 3? If something magical happens at infinity when they form a continuum, why not skip the magic and just start with a continuum?
Ryan O'ConnorMarch 15, 2021 at 00:08#5104400 likes
There is an issue of truth here. There is something there causing the form, and the concept of "field" attempts to account for whatever it is. If the concepts employed are inadequate, then it's not true to say that this is what is there.
Sure, physicists could be wrong but that doesn't mean you should stick with outdated information (which could also be wrong).
And if we know that in every instance when such a process is useful, it is actually terminated, then we also know that it is false to say that a potentially infinite process is useful, because it is only by terminating that process, thereby making it other than potentially infinite, that it is made useful.
If you have ever seen ? as the solution to a problem (instead of, say, 3.1415) then the process hasn't been terminated, it hasn't even been initiated. It's incorrect to say that potentially infinite processes are only useful when prematurely terminated.
In line with Aristotle's solution to Zeno's Paradox, my view has continua (not points) being fundamental. I think the mathematics of calculus would be almost entirely unaffected in moving to a continuum-based view,
Why isn't Aristotle's solution just circular because it makes the results of a mathematical construction prior to the construction itself? Plato starts with a line boundless in both directions, then designates bounds to derive segments. Think of it this way, first mark any point on the line as the Origin to divide the line into a half-open dichotomy, then designate any other point as the unit marker to construct a fixed continuous interval. If I give up points as bounds, then how would I have anything but an endless line?
Ryan O'ConnorMarch 15, 2021 at 00:38#5104460 likes
Why isn't Aristotle's solution just circular because it makes the results of a mathematical construction prior to the construction itself?
I'm not a physicist but this is my understanding of how our world works. A wave function of the universe (describing potential/probabilities) exists spanning all of time. When a measurement is made, that wave function locally collapses to a distinct state. So we start with a probabilistic description of the universe which exists prior to the individual moments.
The construction itself certainly must come first. What we disagree on is what that construction is. You see the individual measured points in time being fundamental while I see the continuum wave function being fundamental. Zeno's paradox highlights the weaknesses in your point-based view. In short, we can't build a timeline from points in time.
I think this is the wrong question to ask. We know what we want (a consistent foundation for calculus) and we think we know how to get it (with real numbers and infinite sets). But real number might not be the answer.
I think a much better question to ask is 'can we build a consistent foundation for calculus without real numbers and infinite sets?' I believe the answer is yes. Now, this doesn't imply that the alternative requires scrapping everything about real numbers. For example, I believe the alternative would have still have the area of a unit circle being pi, it would just mean something different.
This is a bit speculative without some sort of indication or hint of specifics. For example as I mentioned, I've looked into constructivism a bit, which is enjoying a modern resurgence due to the influence of computer science and computerized mathematical proof assistants. But you would reject even that.
Like the real numbers, the fundamental objects of the hyperreals are points/numbers. A continuum is constructed by assembling infinite points, each with a corresponding number. And so the hyperreals gets no closer to answering the question 'how can a collection of points be assembled to form a line?' I haven't investigated the constructivists' methods to produce an informed comment but from what I've seen it looks like more of the same. Points/numbers are treated as fundamental.
I don't have a formal theory but I do have an intuition on how the alternative would look. In line with Aristotle's solution to Zeno's Paradox, my view has continua (not points) being fundamental.
Sounds like a Peirceian viewpoint, about which I don't know much. But why am I supposed to be burdened for finding a mathematical viewpoint other than the standard one accepted by almost all the world's mathematicians except for those pesky constructivists and rare finitists and ultrafinitists? Why is this my problem, or math's problem?
I think the mathematics of calculus would be almost entirely unaffected in moving to a continuum-based view, but before we could even talk about it, you'd have to first be open to (at least temporarily) shedding your point-based biases and look at graphs in a new light.
I've looked into constructivism a bit, which is enjoying a modern resurgence due to the influence of computer science and computerized mathematical proof assistants. But you would reject even that.
I certainly like the idea of constructivism in that it is necessary to construct a mathematical object to prove that it exists, and my understanding is that constructivists reject actual infinity. However, I haven't investigated the details of constructivism to know where my views differ. It's on my to do list....and as of now, so too is the Peirceian viewpoint.
But why am I supposed to be burdened for finding a mathematical viewpoint other than the standard one accepted by almost all the world's mathematicians except for those pesky constructivists and rare finitists and ultrafinitists? Why is this my problem, or math's problem?
This is a great question. Nothing is your problem unless you want it to be. But here's my response to your 'all the world's mathematician's' comment: the world's mathematicians are all climbing up the 'point-based' mountain (and certainly doing good work) but it's a competitive space and (I think) most contributions these days are in highly specialized niche areas. I'm here pointing out that there's another place to climb. I can't promise that it's a mountain (but it might be) but I can say that very few people are climbing it. What's the harm in taking a quick look?
I suppose I owe you that. I'm going to find it depressing and frustrating but I'll take a run at it sometime.
...ah okay, so the harm is potential depression and frustration. I'm sure it is frustrating talking math with a non-mathematician, but don't be so certain about it being depressing. There's a chance that you'll like the ideas.
Metaphysician UndercoverMarch 15, 2021 at 02:11#5104720 likes
A program written to spit out the natural numbers one at a time is potentially infinite, regardless of whether it's been executed or interrupted.
We already discussed the difference between the rule ("program" in this case) which sets out, or dictates the process, and the process itself. If the process is interrupted, it ends, and is therefore not infinite. The rule ("program") is never infinite, nor is it potentially infinite, it's a finite, written statement of instruction, like "pi", and "sqrt (2)" are finite statements, even though they may be apprehended as implying a potentially infinite process.
If you have ever seen ? as the solution to a problem (instead of, say, 3.1415) then the process hasn't been terminated, it hasn't even been initiated. It's incorrect to say that potentially infinite processes are only useful when prematurely terminated.
I don't see how you can say this. Pi says that there is a relationship between a circle's circumference and diameter. This information is totally useless if you do not proceed with a truncated version of the seemingly infinite process, such as 3.14. "The solution to the problem is pi" doesn't do anything practical, for anyone, if you cannot put a number to pi.
That leaves the question, when there is no interval of time, is it meaningful to speak of time or anything that requires time.
We can certainly talk about a system at a particular instant of time. For example, we can talk about the time on our clock, and the position and shape of objects at that instant. What we can't talk about is rate of change.
If it makes sense to speak of a time as a "time" when there is no motion, then that same notion applied to a car implies that it's spending most of its time, even while in motion, not in motion.
Let's say that every instant has a corresponding number on my stopwatch. I believe that the numbers on my stopwatch can only ever be computable numbers. Since traditionally we say that the computable numbers are countably infinite, the duration at which the car is not in motion has measure 0. Therefore the car is spending all of it's time in motion. What you must appreciate is that there is 'something' in between the instants and it is during this 'something' when the car moves.
When time is suspended in indeterminacy space is fluid and discrete. When space is seen as knowingly infinite is its parts it is suspended against the discrete nature of time. Motion has an aspect that is spatial and one that is temporal. We do talk of spacetime now, but we must speak of time and space separately in these examples. Saying we have points and instances which are infinite (which must be crossed by motion) is to forget that time is not space, space is not pure mathematics, and that even pi can only be understood as part of a finite number (3)
Ryan O'ConnorMarch 15, 2021 at 03:01#5104790 likes
We already discussed the difference between the rule ("program" in this case) which sets out, or dictates the process, and the process itself. If the process is interrupted, it ends, and is therefore not infinite. The rule ("program") is never infinite, nor is it potentially infinite, it's a finite, written statement of instruction, like "pi", and "sqrt (2)" are finite statements, even though they may be apprehended as implying a potentially infinite process.
Hmm, maybe I wasn't consistent, let me try again.
-The aforementioned program is written with finite characters.
-The execution of the program is potentially infinite.
-The complete output of the program would be actually infinite (if it existed).
-The program spits out numbers as it is being executed, so it doesn't need to be terminated to get something useful from it.
-We can discuss the execution of the program without ever running it (e.g. we can say 'if I executed the program, it would be potentially infinite)
In the end, I think you're splitting hairs here. What's your point?
I don't see how you can say this. Pi says that there is a relationship between a circle's circumference and diameter. This information is totally useless if you do not proceed with a truncated version of the seemingly infinite process, such as 3.14. "The solution to the problem is pi" doesn't do anything practical, for anyone, if you cannot put a number to pi.
? is often written as the solution to a problem - for one it's what they say is the volume of Gabriel's Horn. And there are many cases where ? gets cancelled out, for example cos(2?)=1, so there's no need to evaluate it. Also, who said math had to be practical?
Ryan O'ConnorMarch 15, 2021 at 03:16#5104840 likes
even pi can only be understood as part of a finite number
I'm not sure how to respond to most of your post, but as for pi we typically understand it precisely using some algorithm described with finite characters.
I certainly like the idea of constructivism in that it is necessary to construct a mathematical object to prove that it exists
It is a lot more satisfying to do this than to argue indirectly, IMO. I have told this little story before, but it bears repeating: There was a PhD math student who spent considerable time on his culminating research project creating a mathematical structure about a particular set of functions, until one day he was asked to provide an example of one of these functions. It turned out he was working with an empty set. :sad:
Your ontology is weak. You say a table is one, yet it has 4 legs and a top piece. What number of parts do these have? This process is infinite and it takes a delicate balance to understand all it's intricacies. Motion passes through infinity and the finite, but you want to reduce the question to Aristotle's lame argument,: namely that parts are only potentially there. Bringing in QM isn't going to help your case mr. idealist. The world is real. "Ignore the world and the world will come to you"
Metaphysician UndercoverMarch 15, 2021 at 12:24#5105610 likes
-The program spits out numbers as it is being executed, so it doesn't need to be terminated to get something useful from it.
But spitting out numbers is not something useful. Useful is the application of the numbers towards counting or measuring, or something like that. If the computer is tasked with counting something and does not complete the task it hasn't been useful.
n the end, I think you're splitting hairs here. What's your point?
I can't even remember now, but I believe I said it would be good to rid the system of infinities and you said there is no problem with working with infinities so long as we recognize that they are merely potential.
But what's the point to working with infinities? If an infinity represents an uncompleted tasked, then isn't it better to complete the task before proceeding. After a while the unfinished tasks start to pile up and become a little overwhelming. And if it is a task which is impossible to complete, then to give oneself an infinite task is to set oneself up for failure, so we ought to address the conditions by which this happens so that we can avoid it.
This is probably the crux. "Math does not have to be practical". There's a fundamental element of free choice which lies at the base of all of our understanding of everything. "Has to be" is thereby excluded. And so, we do not have to do anything, nor do we have to figure anything out, or anything like that. One can refuse to move and die if one wants. However, we choose to try and figure things out, we choose to try and understand the nature of reality, and mathematics plays a very big role here. So we need to choose the appropriate mathematics.
Of the mathematicians, the people who dream up axioms, and produce elaborate systems, some might have the attitude that math does not have to be practical, and others might have the attitude that math ought to be practical. But the idea that mathematics does not have to be practical is just an illusion. Each such mathematician will choose a problem, or problems to work on, as that's what mathematics is, working on problems. And problems only exist in relation to practice, as that's what a problem is, a doubtful aspect of practice which needs to be resolved. Without the influence of practice, the need to resolve the issue does not arise, therefore there is no problem. So the reality of the situation is that since mathematicians work on resolving problems, and problems only exist in relation to practice, math is always fundamentally practical, and this fact cannot be avoided. That's why math is classified as an art rather than a science. Despite the huge amount of theory which goes into it, it is theory which is always directed toward solving problems. Therefore, despite the fact that math doesn't have to be practical, it always is practical. If the people who dreamed up axioms and other systems weren't doing something practical (resolving problems), they would have come up with something other than mathematics.
Sounds like a Peirceian viewpoint, about which I don't know much.
I have not had the time, energy, or patience to jump into the substance of this discussion so far, but I can offer a couple of reading suggestions based on these two comments.
The first is John Bell's book, The Continuous and the Infinitesimal in Mathematics and Philosophy. It provides an excellent historical overview followed by chapters specifically on topology, category/topos theory, nonstandard analysis, constructive/intuitionistic mathematics, and smooth infinitesimal analysis/synthetic differential geometry.
The second is my own recent paper, "Peirce's Topical Continuum: A 'Thicker' Theory." Its thesis is that a collection--even an uncountably infinite one, like the real numbers--is bottom-up, such that the parts are real and the whole is an ens rationis; while a true continuum is top-down, such that the whole is real and the parts are entia rationis. Accordingly, five properties are jointly necessary and sufficient for the latter: rationality, divisibility, homogeneity, contiguity, and inexhaustibility.
Reply to jgill
The consensus among Peirce scholars seems to be that SIA/SDG comes the closest among modern mathematical developments to capturing his notion of a true continuum. Bell also has an excellent primer specifically on SIA, and Sergio Fabi wrote something similar for SDG.
Ryan, I only quoted this to mention you. I've noticed that when I hit the @ button and enter Ryan, your handle doesn't come up. Do you know why that is? Moderators, any clues?
Anyway apropos of our convo re video, I thought you might enjoy this. It's a pretty cool video regardless. It's a helicopter taking off with its rotor speed exactly synced to the video frame rate, making the rotors look motionless. So even if things look motionless maybe they're moving. For all we know, still photos are moving like crazy but the lights in our room are blinking. Reality is not what it seems, or something like that. The comments below the video are amusing too.
Thanks for the info. I was not able to get past the JSTOR paywall even though I have a public JSTOR account (the kind they deign to give to us unwashed non-academics). Do you know why? Usually I can read JSTOR articles on the website even if I can't download them.
Ryan O'ConnorMarch 15, 2021 at 21:58#5107430 likes
Your ontology is weak. You say a table is one, yet it has 4 legs and a top piece. What number of parts do these have? This process is infinite and it takes a delicate balance to understand all it's intricacies. Motion passes through infinity and the finite, but you want to reduce the question to Aristotle's lame argument,: namely that parts are only potentially there. Bringing in QM isn't going to help your case mr. idealist. The world is real. "Ignore the world and the world will come to you"
Newton's world was one where the present took the center stage, time marched forward instant by instant, and everything always held a definite state.
Einstein's world is one where simultaneity and the present are relative. This view challenged presentism, making eternalism and the block universe (i.e. the whole universe spanning all time and space) an attractive view.
Bohr's world is one where objects only have definite states when they're measured.
My views does not ignore reality, rather is agrees with the latest developments in physics. Perhaps you're stuck in a classical world, a world plagued by singularities. It at least seems that way given your affinity for infinity. As it is when we grapple with singularities, I doubt that you have a compelling solution to Zeno's Paradox.
Your question about a table seems silly. If you want to talk about 'the whole' you have to talk about the wave function of a whole system, not a table. If you don't like the word 'potential existence', would 'superposition' be more appealing to you?
Consider this: in between any two distinct states of a system lies an unobserved wave function. If we continue to observe the system it will not evolve to a different state (Quantum Zeno Effect). Motion happens when we are not looking. Why are you assuming (as Newton did) that everything always has a definite state?
The greatest error of modernity is saying that the world is information and is not as it appears to us. The world we see transcends any interpretation of QM and psychological studies on mind-matter interaction. What you see is what there is. There is more there, but not less. Any other position is insanity. Zeno's paradox will never have a complete solution, but it is a sign of a healthy position to be comfortable with a paradox
Ryan O'ConnorMarch 16, 2021 at 00:35#5108120 likes
But what good is that? (in reference to talking about a potentially infinite process)
I've said this many times, but I'll say it one more time. We can talk (using finite statements and operations) about potentially infinite processes without ever initiating (let alone completing) the potentially infinite process. That's what calculus is all about (in my view). If you avoid infinity altogether (i.e. actual and potential) then there is no way to get calculus off the ground. Let go of executing the potentially infinite process and embrace the finite statements that talk about the potentially infinite process.
This is probably the crux. "Math does not have to be practical".
Maybe we are using 'practical' differently. I mean practical as having an applied benefit, like engineering better gadgets. If you consider the proof of Fermat's Last Theorem 'practical', then by your definition I think all of math is practical.
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I find your philosophy of mathematics to be quite disjoint from the act of doing mathematics. Your view does not explain equations or geometry or axioms, it doesn't provide any explanatory power. It only attempts to delegitimize powerful and proven tools (e.g. calculus).
I am going to talk on the phone tomorrow with my cousin, who is a high profile computer programmer in LA. I will tell him about your claim that infinities play no role in programs and see what HE has to say about that. I feel like you come at these questions from a very limited philosophical perspective and make broad claims about stuff you apparently aren't very familiar with. Expect a reply on this thread tomorrow night
Ryan O'ConnorMarch 16, 2021 at 00:57#5108180 likes
It is a lot more satisfying to do this than to argue indirectly, IMO.
Agreed. It is easy to talk about doing something (e.g. completing an infinite process), it's another thing to actually do it. I think your story is quite fitting because I believe that when we construct objects by assembling points, we never get anywhere. We're always left with a 0D object...an empty set of sorts.
The greatest error of modernity is saying that the world is information and is not as it appears to us. The world we see transcends any interpretation of QM and psychological studies on mind-matter interaction. What you see is what there is. There is more there, but not less. Any other position is insanity. Zeno's paradox will never have a complete solution, but it is a sign of a healthy position to be comfortable with a paradox
I don't think anyone is saying that there only exists information, certainly I'm not saying that. I'm also not denying 'what you see is what there is'. The problem is that you and others are incorrectly filling in the gaps between quantum measurements to assume that a particle is travelling a definite path, and by making this assumption you fall victim to Zeno's Paradoxes. Zeno's Paradox does have a simple solution, Aristotle's solution which I have been promoting here. I don't think anybody has provided a single valid criticism of this view, other than 'it doesn't jive with my classical intuitions'. You need to update your intuitions based on modern physics. Lastly, paradox is a sign that progress can be made, we should not just accept it as being beautiful mystery that will permanently be out of reach. We should be motivated by it.
Aristotle's "solution" was that the whole exists prior to parts. To which I ask:
1) do the parts of the whole exist
2) how many are there of these parts.
The answer to the first question is "yes," (you have an arm for example) and the second answer is "infinite". Even Aristotle agreed that discrete space is an oxymoron. The world exists as a paradox, a round circle right before our eyes. It gives wonder and awe to us who think like Kant and Hegel. You can't fully understand a plant, but saying its just amorphous waves because a quantum physicists says so after getting dizzy from quantum math is not a true, realistic position to hold
Ryan O'ConnorMarch 16, 2021 at 01:35#5108260 likes
I've noticed that when I hit the button and enter Ryan, your handle doesn't come up. Do you know why that is? Moderators, any clues?
I suspect it's the apostrophe...it's always given me computer problems. I've posted a comment in the 'Feedback' section, hopefully a moderator will allow me to change my name.
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That is a cool video, thanks! I've worked for many years in simulating physical systems (in fact I was simulating rotating fan blades today) and when it comes to transient simulations, it's important to select a timestep sufficiently small to adequately resolve the transient fluctuations. Too large and you can incorrectly predict the rotational velocity of a helicopter's blades, you might even predict that they're not moving. As such, it is natural to then want a smaller timestep. But one thing that we never want is to set the timestep = 0 seconds. Not only would the equations blow up, but no matter how many timesteps we run we would never get past the initial state. Motion is impossible when dt=0. So what we do is pick a dt small enough to sufficiently resolve the frequencies of interest. And when we create an animation based on those frames we hold each frame for dt seconds before advancing to the next frame.
And what I haven't been able to get across to you is that a frame held for 0 seconds is totally different than a frame held for dt seconds. The 0 second frame is like a point in time while the dt second frame is like a timeline. We make movies, not by assembling 'points' but by assembling 'lines'. If all we have are points, then we'll never construct anything with more extension than a point. It is because of this sort of reasoning why I earlier called the 'real number line' the 'real number point'.
Now if we go the other way around, we don't face the same problem. If we start with a line we can easily produce points, just as we can produce new endpoints when we cut a string. When we start with continua we get both continua and points. It's a much richer foundation that avoids the paradox.
I'm going to come back to this. The point-based view has reached its limit, it will never be able to solve Zeno's Paradox. But Zeno's Paradox is that loose thread on our sweater that we should pull because shedding ourselves of our inconsistently knitted sweater will reveal that underneath the sweater we're wearing a killer tuxedo.
This is math's problem because mathematicians are the only ones who can 'climb mountains'. I'm just a Joe-schmo who's pointing at some unexplored mound that's hidden behind clouds saying 'that could be a bigger mountain'.
How QM works is a question for quantum physicists. WHY it works is a philosophy question. We can't see with the eye at that level so all the former can do is put philosophical labels on how atoms ect. affect the measuring device. There is no true ontology in that field of research and if you think your body is primarily empty space I'd have to say you have a cognitive distortion
Ryan O'ConnorMarch 16, 2021 at 01:47#5108300 likes
Aristotle's "solution" was that the whole exists prior to parts. To which I ask:
1) do the parts of the whole exist
2) how many are there of these parts.
1) The parts only exist when measured. If nobody is looking, it is meaningless to speak of parts (just as it is meaningless to say that Schrödinger's cat has a definite state).
2) There are only ever finitely many parts. The number depends on how many measurements you've made.
I communicated this to @fryfish a while back on this thread but I don't think you were active at that time so I'll repeat the idea. Your view is that a string is created by assembling a whole bunch of nothing (i.e. 0D points). As long as you have enough nothing then you can eventually produce something. It's crazy.
With Aristotle's view, we start with something, we start with the string. How many endpoints does the string have, let's say 2 for simplicity. Can we create more endpoints? Sure, just cut it more and more times. The more cuts you make, the more endpoints you have. How many endpoints do you have? Well it depends on how many cuts you make, but certainly you will have some finite amount. Now, how many times do you need to cut a string until it disappears - until it's just a whole bunch of 0D points (i.e. nothing)? Does infinity do the trick for you? If you are open minded you have to acknowledge that there's a problem here. And if we can't go from a continuum to infinite points, why would we think that we can go from infinite points to a continuum?
There is no true ontology in that field of research and if you think your body is primarily empty space I'd have to say you have a cognitive distortion
Space is far from empty. Fields permeate all of space.
Ryan O'ConnorMarch 16, 2021 at 01:49#5108340 likes
Is a tomatoe a field? Why not? You're throwing ideas out there that are philosophical and calling them science. This is a philosophy forum. We think critically about what scientists say if it in any way comes into the realm of philosophy. That's what we do! Maybe waves are particles and particles are waves. You can't disprove that. What we know is what our senses say, yet you say our senses and the paradigms they are in are close to having ZERO accuracy. Why trust your use of measuring equipment then? We have no real understanding of what stuff looks like at the quantum level. All we have are vague ideas (which are fine) and a lot of assumptions. But your solution to Zeno's paradox is that the quantum world doesn't exist unless we measure it. Which, well, is more paradoxical than what Zeno proved. But he DID end up being an idealist, so there's victory for you I guess
I believe in the philosophy of Kant, Fitche, Schelling, and Hegel. I interpret them all in terms of materialism (the soul arises from matter) and a combination of process philosophy and traditional ontology. Most philosophers now adays are process philosophers and therefore they don't object to the careless way physicists create philosophical paradigms out of thin air. I read the Tao of Physics last year and it is just ridiculous how a major physicists could have written that! You can't make philosophical claims of that magnitude from physics along. We can point a microscope on a lemon and understand something about the texture of it. But there is a threshold somewhere in there beyond which the microscope can't be proven to be capable of providing evidence of what it really LOOKS LIKE down there. And you are saying that there is nothing there at all unless we measure it. What kind of nonsense
I am not intending to hurt your feelings but simply to attack your world view. It's a common world view these days, and it's kind of ridiculous for people to walk around on sidewalks with their legs while they claim the sidewalk and legs are really evanescent waves simply because scientists interpreted it that way because of how certain phenomena affected their measuring apparati. You can't do physics without philosophy, but a lot of times wrong philosophy comes about because of physics
Metaphysician UndercoverMarch 16, 2021 at 02:55#5108470 likes
I will tell him about your claim that infinities play no role in programs and see what HE has to say about that.
You seem to have missed the gist of the conversation Gregory. I said that when they are rounded off, or "terminated" in Ryan's words, such as the example in the op, or using pi as 3.14, then the so-called infinities are very useful. But a process which is said to be potentially infinite, which will necessarily be terminated at some point, cannot truthfully be said to be potential infinite.
The point-based view has reached its limit, it will never be able to solve Zeno's Paradox.
There is no problem. Mathematical analysis took care of that years ago. Only a few philosophers remain addicted to it. But this is a philosophy forum, so it's OK to quibble. :smile:
Space has always been seen as the hierogamy of infinite and finite powers within some esoteric traditions. Someday I would like to do a study just on the idea of "space" in world cultural and religious traditions
I suspect it's the apostrophe...it's always given me computer problems. I've posted a comment in the 'Feedback' section, hopefully a moderator will allow me to change my name.
Some kind of use-mention problem no doubt.
I deleted the post I wrote. Here's the updated version. First, we must both be frustrated by now. You expressed frustration at trying to tell me something that I thought I had told you three days ago. First you said a still photo stops motion and video shows motion. I pointed out that video is actually a sequence of stills, and that even a still captures motion because it records photons over a nonzero interval of time. Now in your most recent post you are frustrated that you can't explain this to me!
So we're talking past each other. Perhaps we can find agreement at least in that.
Second, it's not the job of math to solve Zeno's paradoxes; and even if it is, it's not an interest of mine. I wish you the best with your efforts in that direction but I can't help. I regard Zeno as a solved problem mathematically via the theory of convergent infinite series; and an unsolved problem physically because our best theories don't allow us to reason below the Planck scale.
And third, dt is a differential form. They don't explain these in calculus so that's why everyone's confused about them. Bottom line is they aren't numbers and they can't be zero OR nonzero. If you use the notation [math]\Delta t[/math] that will be accurate. That's an interval of time, zero or nonzero as the case may be.
Your idea about building points from lines or lines from points just went over my head. I don't understand your intention at all. I'm not trying to solve the nature of the continuum here. Mathematically we construct the real numbers and call them a line, but the geometric visualization is incidental and not essential. As far as the true nature of the world, that's above my pay grade but I would personally be very surprised if it's anything at all like the mathematical real numbers.
And third, dt is a differential form. They don't explain these in calculus so that's why everyone's confused about them. Bottom line is they aren't numbers and they can't be zero OR nonzero.
This may be a little high powered. The differential I-form is [math]f'(x)dx[/math] and it is common practice when teaching elementary calculus or even advanced calculus to simply assign the value
[math]dx=\Delta x[/math] for the independent variable. I used Thomas several times for calculus and Olmsted for advanced calculus. In both the authors explain that dx could be any real number, and then they assume it to be [math]\Delta x[/math]. Of course, dx is originally an infinitesimal. It's not a big deal IMO.
even a still captures motion because it records photons over a nonzero interval of time.
I don't completely agree with this. It's true that macroscopic stills capture some motion but when we look at the microscopic level, it is possible for a still to capture zero motion (e.g. a single photon hitting a quantum sensor). And in a simulation (which is what I'm most familiar with) the stills capture zero motion (note: the velocity field seems to reflect instantaneous velocity but let's not go there).
So we're talking past each other. Perhaps we can find agreement at least in that.
I don't really agree with this. I held the 'point-based' view for most of my life so I think I understand your position, I just don't think it's correct. I think the problem is that you don't understand my position. This is not necessarily your fault, in fact since nobody else here seems to understand my position it's probably my fault.
it's not the job of math to solve Zeno's paradoxes; and even if it is, it's not an interest of mine. I wish you the best with your efforts in that direction but I can't help.
Understood. And to be realistic, since you haven't understood my other arguments I've presented I think it would be a waste of your time.
What we know is what our senses say, yet you say our senses and the paradigms they are in are close to having ZERO accuracy. Why trust your use of measuring equipment then? We have no real understanding of what stuff looks like at the quantum level.
We know much more than what our measuring equipment says. Our measuring equipment gives us data, physics gives us models (that match the data and make predictions). And with these models we can talk sensibly (to some extent) about what reality is like at the quantum level.
But your solution to Zeno's paradox is that the quantum world doesn't exist unless we measure it. Which, well, is more paradoxical than what Zeno proved......And you are saying that there is nothing there at all unless we measure it. What kind of nonsense.
No, unmeasured particles exist in a superposition state (a state of potential). This is the interpretation of QM that physicists are taught in school. It's not paradoxical (there are no contradictions), it's just weird because it violates our classical intuitions. In fact, I believe it's nature's way of cleverly avoiding paradox/contradiction/singularities.
Think about a video game, when Mario is dilly-dallying collecting coins is it reasonable to ask what Bowser is doing at that moment? No, that part of the world just has not been resolved. All we could say is that Bowser is somewhere in his castle waiting for Mario. He has the potential to be in one corner of the room, and the potential to be in the other corner of the room. We can describe his position only as a potential state. Your console is unable to resolve Mario's universe everywhere and at all times so it only resolves what you're looking at. Why would our universe be any different?
I live and breathe physics simulations so I can't help from seeing our universe as a big simulation. And since I don't think any computer can have infinite computing capacity, the computer of our universe has got to cut corners somewhere. And it does it remarkably well. Essentially what you are craving is an infinite computer. You are craving a supernatural world.
And by the way, measurement is not something exclusive to humans.
Metaphysician UndercoverMarch 17, 2021 at 00:11#5112630 likes
But a process which is said to be potentially infinite, which will necessarily be terminated at some point, cannot truthfully be said to be potential infinite.
— Metaphysician Undercover
Care to edit this? I do not understand the last part.
Sorry I left off the suffix, 'ly'. Try this:
But a process which is said to be potentially infinite, which will necessarily be terminated at some point, cannot truthfully be said to be potentially infinite.
That better?
Ryan O'ConnorMarch 17, 2021 at 00:14#5112640 likes
There is no problem. Mathematical analysis took care of that years ago.
Atalanta is walking from x=0 to x=1. What is the first non-zero coordinate that she walks to? I'd like to know how mathematical analysis solved this.
As long as we think that a line is composed of points we cannot answer this question. Motion is only possible when she jumps from one point to another. And what I'm saying is that she doesn't jump over infinite points, she jumps over a continuum.
Atalanta is walking from x=0 to x=1. What is the first non-zero coordinate that she walks to? I'd like to know how mathematical analysis solved this.
There is no first nonzero positive real. The open unit interval (0,1) does not contain its greatest lower bound. Any math major can tell you that. They learned it in real analysis class.
As long as we think that a line is composed of points we cannot answer this question.
If we allow that the real line is made of points (which are just real numbers) then the answer is that there is no first nonzero positive real number. That is the answer, so your claim that "we cannot answer this question" is false.
And what I'm saying is that she doesn't jump over infinite points, she jumps over a continuum
You may not like the real numbers as a model of the continuum (and after all, some mathematicians agree with you); but that doesn't entitle you to mischaracterize the math of the real numbers. There is no smallest positive real number. Why is that a problem. In your model of a continuum, whatever it is, is there a smallest real number?
Ok. "Is said" seems gratuitous. ?, I'm told, in decimal expansion never ends. To use it as a number, it's usually truncated at some point. That is just a number, nothing infinite about it at all, potential or otherwise. But why confuse the two? One stands in for the other to get an approximation. What is the issue about "potential" anything?
This was Ryan's term. You'd have to go back to see what Ryan was talking about. Essentially Ryan suggested replacing infinities with infinite processes. Since the supposed infinite processes could never be completed they are assumed to be potentially infinite. I argued that every supposed potentially infinite process will for some reason or another, at some point be terminated. If this is the case then it is incorrect to even call them potentially infinite.
"But a process which is said to be potentially infinite, cannot truthfully be said to be potentially infinite." Eh? Sure it can. Or do you mean that the never ending decimal expression of ? actually ends?
You left out the important phrase: "which will necessarily be terminated". The infinite process would continue forever, by definition. Since forever never arrives Ryan says we ought to call it potentially infinite. Ryan suggested that we could put an end to a potentially infinite process, by rounding of pi for example, yet still say that it is potentially infinite. Obviously though, if someone puts an end to a process, it is not potentially infinite. So Ryan proceeded to distinguish between the rule which produces the supposed potentially infinite process, and the process itself, trying to place the potential for infinity within the rule rather than the process.
Ryan O'ConnorMarch 17, 2021 at 01:48#5112800 likes
@fishfry I'm trying to let you off the hook on talking to me about the nature of continua but you keep coming back for more. I'm not complaining!
If we allow that the real line is made of points (which are just real numbers) then the answer is that there is no first nonzero positive real number. That is the answer, so your claim that "we cannot answer this question" is false.
Exactly. And since there is no first nonzero positive real then she cannot take her first step. Her journey doesn't begin. Motion is impossible. When I said that 'we cannot answer this question' I should have said 'we cannot answer this question satisfactorily.'
Now who's claiming a line is made of points? You are the one doing that!
My view doesn't exclude points. Let me try this again but with a much more digestible image.
I suspect you would say that you see infinite points there. I don't. In this picture I see 5 points, each connected by lines. That's it. WYSIWYG. And if we relate this back to Atalanta's story, I would say that we have still photos of her at 0, 1/3, 1/2, 3/4, and 1. That's all we've got. We are not justified to say that at some time she was at 1/10 or at 7/8 because we don't have the photographs. At one moment we see her at 0, we blink, and when we open our eyes she's at 1/3. Motion happens when we blink, when we are not looking. And I've said it before but perhaps it might sink in this time that this is consistent with QM. If we are continuously observing a quantum system it will not evolve. It will only evolve when it is not being observed. Change happens when we blink. Not surprisingly, it's called the Quantum Zeno Effect.
that doesn't entitle you to mischaracterize the math of the real numbers. There is no smallest positive real number.
Perhaps my point didn't come across. All I was trying to say was that 'there is no smallest positive real number' means that Atalanta cannot decide what point to go to first.
In your model of a continuum, whatever it is, is there a smallest real number?
In my view, we cannot speak of the 'completely measured continuum' which is what I would call 'the real number line'. We cannot do so because it would require infinite measurements and that's impossible. Instead, we must only speak of systems which we can actually look at (at least in principle), like the one above. And in that case, the smallest positive number is precisely 1/3.
As I've shown, its demonstratable that space is infinitely packed. Our eyes can even see it in matter. Whether waves will be revealed to be made of particles or visa versa is irrelevant because there is always another level. Quantum physicists descriptions of reality are not necessarily accurate. They have to fit true philosophy. Why so you get your philosophy from scientists i wonder
Exactly. And since there is no first nonzero positive real then she cannot take her first step.
Why not? Explain to me exactly why someone can't put one foot in front of the other and take a step. How does the mathematical theory of the real numbers preclude anyone from doing that? And -- a question that I keep asking you and that you never answer -- why does any mathematical theory have anything to do with physics?
Her journey doesn't begin. Motion is impossible. When I said that 'we cannot answer this question' I should have said 'we cannot answer this question satisfactorily.'
It's perfectly obvious to anyone who ever took a step -- that's most two year olds -- how the process works. Why do you think the modern theory of the real numbers prevents anyone from walking? "Oh no I can't walk, they're teaching Dedekind cuts to the math majors at the university!"
I genuinely can not understand your point. How does the mathematical theory of the real numbers prevent anyone from taking a step?
I suspect you would say that you see infinite points there.
I would say that when you draw the real number line, that's a visual depiction of the mathematical real line, which itself is an abstract object that cannot be depicted. Like drawing an old guy in a robe with a white beard to symbolize God. Of course the picture itself on my laptop screen is made of pixels, which are discrete, and there are only finitely many of them. Why do you either (a) fail to appreciate both of these points, or (b) think I don't?
And if we relate this back to Atalanta's story, I would say that we have still photos of her at 0, 1/3, 1/2, 3/4, and 1. That's all we've got. We are not justified to say that at some time she was at 1/10 or at 7/8 because we don't have the photographs. At one moment we see her at 0, we blink, and when we open our eyes she's at 1/3.
Yes, motion in the real world is pretty much like that.
Motion happens when we blink, when we are not looking.
According to the Copenhagen interpretation, we have no idea what happens when we're not looking. According to Many Worlds, everything happens. How is this relevant to the conversation?
Has it occurred to you that perhaps you are not personally possessed of the ultimate truth about how the universe works?
And I've said it before but perhaps it might sink in this time that this is consistent with QM.
It's consistent with the Copenhagen interpretation of QM but not with the Many Worlds interpretation. And don't forget that Feynman said we can model reality by assuming that you get from point A to point B by integrating over every possible path, no matter how circuitous. You are just getting yourself tangled up here. You're confusing interpretations of QM, which itself is just a mathematical model that agrees with experiment up to the limit of Congressional funding for particle accelerators; with reality itself. You confuse math with physics, and you confuse physics with ultimate reality. Two series category errors.
That's because you seem to think that the mathematical theory of the real number precludes my getting off my ass and taking a walk. I'll try that one out on my doctor when he tells me to get more exercise. "They're teaching Zermelo-Fraenkel set theory, which leads to Dedekind cuts, so it's all I can do to just change the channel, let alone stand up from the couch."
This is your thesis. Don't you see how silly it is?
All I was trying to say was that 'there is no smallest positive real number' means that Atalanta cannot decide what point to go to first.
How does the mathematical theory of the real numbers prevent someone from taking a walk? Before Dedekind had his clever idea, were people able to walk? And then the day he published, they couldn't? Isn't what you are saying patent nonsense?
In my view, we cannot speak of the 'completely measured continuum' which is what I would call 'the real number line'. We cannot do so because it would require infinite measurements and that's impossible. Instead, we must only speak of systems which we can actually look at (at least in principle), like the one above. And in that case, the smallest positive number is precisely 1/3.
Now you're just being silly, since if you claim 1/3 is the smallest positive real number I'll just divide it by 2 (using the field axioms) and note that 0 < 1/6 < 1/3. Your claim stands refuted.
Atalanta is walking from x=0 to x=1. What is the first non-zero coordinate that she walks to? I'd like to know how mathematical analysis solved this.
Zeno's paradox when seen from the perspective of pure mathematics is easily dealt with using the limit concept, but giving it an anthropomorphic twist makes it absurd. And the idea of a first non-zero coordinate shows a very limited knowledge of mathematics.
Would you agree a wise intellectual warrior should first know his enemy before striking?
The Numberphile video on Zeno's paradox expresses concern about what is at the end of an infinite series with no final term. The mathematician said he wanted a *physicists* to explain it to him
The Numberphile video on Zeno's paradox expresses concern about what is at the end of an infinite series with no final term. The mathematician said he wanted a *physicists* to explain it to him
He asks whether actual time and actual space can be divided according to the math of ZP.
Oh you watched it, good. He said it "melts the brain" to think of an infinite series going to a destination without a final term. Math, I think you're saying, explains it in its own ways, but there are aspects which still trouble philosophers
Oh you watched it, good. He said it "melts the brain" to think of an infinite series going to a destination without a final term.
I found the vid at https://www.youtube.com/watch?v=u7Z9UnWOJNY. I didn't want to watch the whole thing so I skipped to 12:00 and he said that "in the physical world" it's not the same, but I couldn't find the exact quote about brain melting. As I mentioned to @Ryan, Zeno is solved mathematically by virtue of the theory of infinite convergent series, as the Numberphile guy says; and it's unsolved physically, as he also seems to agree.
In most discussions online people seem to believe the problem is solved by calculus, but that's not true. Only the mathematical problem is solved. The physical problem remains, and even moreso today, because we know about the Planck scale. Below a certain point, our physical theories are not applicable to space or time. The tortoise or the arrow or whatever only have to iterate through 35 or so steps before reaching the point where nothing sensible can be said.
At some point, you can't sensibly divide a distance in half, nor an interval of time. Physics doesn't allow us to get a sensible answer. We don't know whether space and time are arbitrarily divisible; but we DO know that our current understanding of physics doesn't let us sensibly speculate.
Metaphysician UndercoverMarch 17, 2021 at 11:30#5113570 likes
I left it out because it is a nonrestrictive clause. Further, any necessary termination is for a reason external to the process itself, usually to make an approximation.
I don't know what you're talking about tim. To say "X will necessarily be terminated" seems very restrictive to me. Obviously, the cause of termination of the process is external, that's Newton's first law. But how's that relevant?
If A to B is seen as infinite fractions (half, quarter, eighth, ect) and as one travels it the mover changes color every fraction, he is not a definite color by the time he reaches B. He is kind a blur. But he can reach B because, on the flip side, A to B is a finite segment and must be finite in actuality. So there is the infinite approach and the finite way to see it. For some of us the tension between the infinite and finite here is strained and we feel there is and can only be something off about this. But as you say, it's telling us that there are aspects of reality that transcend us, and oddball paradoxes like this are just like jokes put into reality by God
Ryan O'ConnorMarch 17, 2021 at 21:34#5115640 likes
its demonstratable that space is infinitely packed.
It is? Can you point me to the demonstration? This may be going off on a tangent, but the Bekenstein bound poses an upper limit on the information that can be contained within a given volume. Or are you talking about mathematical space being infinitely packed?
Quantum physicists descriptions of reality are not necessarily accurate. They have to fit true philosophy.
I believe that physics, math, and philosophy all inform each other. Physicists don't have it all figured out, but philosophers need to update their views based on the latest advancements in physics (and math).
Modern physics only says how stuff work. Do this, that happens. That sort of thing. The electrons might be doing what the protons are said to do or the other way around. For every wave there could be a hidden particle. We don't need pilot theory to know this, but if that's works for ye that' fine. My point is much of what QM says is false because they are clothing their findings in the language of an unprovable philosophy. They can predict things, but that is all QM can do.
As for matter, it's spatial? Yes. Is all space infinitely divisible? We can't conceive it as discrete, so our natural lights say yes. So your computer is infinitely packed. This is intuitive for me at least. That objects are in a sense very finite is also true, but General Relativity goes well with infinitely parted objects in the sense I mean that. GR is a great theory, and not just because it has my initials :) Lastly, there are different kinds of density and ZP is about the relationship between volume and mass. Until you grasp this you'll be in the dark of QM topology
Ryan O'ConnorMarch 18, 2021 at 00:06#5116230 likes
Zeno's paradox when seen from the perspective of pure mathematics is easily dealt with using the limit concept, but giving it an anthropomorphic twist makes it absurd. And the idea of a first non-zero coordinate shows a very limited knowledge of mathematics.
Would you agree a wise intellectual warrior should first know his enemy before striking?
Since both you and @fishfry reacted the same way, I've learned that I shouldn't try to make a point in the form of a question. I know that there is no first non-zero coordinate on the real number line, that's exactly what I was trying to highlight. Let me try again. Before she arrives at x=1, do you believe that she must first cross all points between 0 and 1? And before she arrives at x=0.5, do you believe that she must first cross all points between 0 and 0.5? If so, then we can take this reasoning to its limit and say that to move she must first reach the first non-zero coordinate. And if there is no first non-zero coordinate, then she cannot move. This is Zeno's Argument which is what trying to highlight, but apparently didn't convey very well.
In the Numberphile on Zeno's Paradox, James Grime says the following:
"I want to give you the mathematician's point of view for this, because, well, some say that the mathematicians have sorted this out........So something like this-- an infinite sum-- behaves well when, if you take the sum and then you keep adding one term at a time, so you've got lots of different sums getting closer and closer to your answer. If that's the case, if your partial sums--that's what they're called-- are getting closer and closer to a value, then we say that's a well-behaved sum, and at infinity, it is equal to it exactly. And it's not just getting closer and closer but not quite reaching. It is actually the whole thing properly."
Is your view that the problem (of Atalanta travelling from x=0 to x=1) is resolved by completing an infinite process?
I think the most compelling solution to Zeno's Paradox (of Achilles and the tortoise) that is often presented is by looking at the situation holistically. If you ask what are the velocities of Achilles and the tortoise you can work backwards to calculate the instant when Achilles passes the tortoise. It seems so simple when you think of it this way. This is the type of thinking that I'm promoting: starting with the whole and working backwards to determine instants.
Why not? Explain to me exactly why someone can't put one foot in front of the other and take a step. How does the mathematical theory of the real numbers preclude anyone from doing that?
Are you suggesting that with each step someone sweeps over infinite points? In other words, are you suggesting that motion involves the completion of a supertask?
Why do you think the modern theory of the real numbers prevents anyone from walking?
My problem is that the theory of real numbers implies that continua are made up solely of points. To me, this means that to move from x=0 to x=1, Atalanta must sweep over infinite points (she must complete a supertask). Instead of saying that there are infinite points, I'm saying that she only needs to sweep over a finite number of things. For example, in the image below, Atalanta needs to sweep over only 4 line segments. It’s a task that does not require any supernatural powers.
And -- a question that I keep asking you and that you never answer -- why does any mathematical theory have anything to do with physics?
Math and physics are not necessarily related, but as you point out, we move all the time. I take that to mean that nature has found a way to avoid the traps of Zeno's Paradoxes in the physical universe. As such, nature may provide some clues on how to solve Zeno's Paradoxes in the mathematical universe. The cheap solution is to say that continua are discrete. I think nature has found a far more elegant solution.
I genuinely can not understand your point. How does the mathematical theory of the real numbers prevent anyone from taking a step?
Let me try from a different angle. Let us say Atalanta lives in a perfect resolution simulation. As such, as she traverses from x=0 to x=1, she must at some instant in time be at each of the intermediate points. If we take out all the fancy graphics of the simulation, the computer is essentially just outputting a list of coordinates from 0 to 1. But we know that the real numbers are not listable (countable). There must be something wrong with the story. Do you agree?
I would say that when you draw the real number line, that's a visual depiction of the mathematical real line, which itself is an abstract object that cannot be depicted.
I'm not trying to trick you by using pixels on your screen or atoms on my paper. Take my drawing and 'idealize it'. I know that a truly 1D object cannot be depicted (it takes up no area after all). We should move to 2D to make it depictable but you didn't like my earlier graphs so to keep things simple, let's just assume that we're looking at it with our mind's eye. This is a representation of how I see the idealized image:
I see 4 line segments with the numbers corresponding to the gaps in between the line segments. Points vanish upon idealization.
Fine. What of it. How would anyone's interpretation of that picture prevent them from walking or allow them to walk?......This is your thesis. Don't you see how silly it is?.......Before Dedekind had his clever idea, were people able to walk? And then the day he published, they couldn't? Isn't what you are saying patent nonsense?
One can hold incorrect views on the laws of nature, but they will nevertheless follow the true laws. Similarly, one can hold incorrect views on the nature of continua which imply that motion is impossible, but they will still be able to walk. What I'm saying is that your views imply that motion is impossible. I'm not saying anything about your ability to walk.
According to the Copenhagen interpretation, we have no idea what happens when we're not looking. According to Many Worlds, everything happens. How is this relevant to the conversation?
Many worlds provides a different interpretation on wavefunction collapse, but it still holds that in between quantum measurements what exists is a superposition, not a definite state. QM is only relevant if it informs our solution to Zeno's Paradox. I'm bringing it up because without QM my view sounds absurd. But you're right, I don't know the correct interpretation of QM. I also don't know what exactly entails a quantum measurement.
Has it occurred to you that perhaps you are not personally possessed of the ultimate truth about how the universe works?
Of course. Truth comes out with formalized theories. I'm merely sharing an atypical intuition, which has neither been proven right or wrong. Ideas are cheap, including mine.
Now you're just being silly, since if you claim 1/3 is the smallest positive real number I'll just divide it by 2 (using the field axioms) and note that 0 < 1/6 < 1/3. Your claim stands refuted.
No, I'm not being silly. 1/3 is the smallest non-zero number in this system.
In this system 1/6 is the smallest non-zero number.
Once we place another point on that line, we have a distinctly different system. This system is composed of 6 points and 5 line segments. In this system 1/6 is the smallest non-zero number. You have refuted nothing.
As I mentioned to Ryan, Zeno is solved mathematically by virtue of the theory of infinite convergent series
As I mentioned to jgill, James Grime in the Numberphile video suggests that the solution kind of involves the completion of infinite tasks. Is this the view you hold?
My point is much of what QM says is false because they are clothing their findings in the language of an unprovable philosophy. They can predict things, but that is all QM can do.
No doubt, QM is in dire need of philosophical progress. But we can't downplay the value of physics. Experiments and corresponding models ground our reasoning.
Is all space infinitely divisible? We can't conceive it as discrete, so our natural lights say yes. So your computer is infinitely packed.
Infinitely divisible does not imply infinitely packed. We can cut a 'mathematical' string indefinitely but never will the string be in infinite pieces.
I've been arguing that to be infinitely divisible means that it has infinite parts. These seem identical to me
These are not identical. A 'mathematical' string is infinitely divisible because I can cut it to no end. I end up with infinite parts only when done cutting it, but am I ever done cutting it?
Since both you and fishfry reacted the same way, I've learned that I shouldn't try to make a point in the form of a question.
@keystone, I'll reply to this one before the other one. I had no trouble @-ing you so at least that worked. I'll continue to quote you as Ryan here but I assume that if you quote me back you'll be on your @keystone account and everything will be clear.
I know that there is no first non-zero coordinate on the real number line, that's exactly what I was trying to highlight. Let me try again. Before she arrives at x=1, do you believe that she must first cross all points between 0 and 1? And before she arrives at x=0.5, do you believe that she must first cross all points between 0 and 0.5?
As I have asked you several times, is this a mathematical or a physical scenario? If mathematical, the answer is yes. And @jgillI made the same point. By the intermediate value theorem, a continuous function passes through all intervening points between one value and another. And I assume continuity is one of your assumptions here. Is it? Essentially you're asking about the nature of continuity.
If this is a physical scenario, the answer is that one, nobody knows for sure; and two, the question is meaningless with respect to contemporary physics because we cannot reason sensibly below the Planck length. Does she hop from quark to quark? No physicist would regard that as a meaningful question.
By the way I looked up [url=https://www.thoughtco.com/greek-mythology-alanta-1525976[/url], and having been a student of Greek mythology way back in the day I was pleasantly informed. She's the goddess of running. The link I gave, which is better than her Wiki link, is full of good stories.
So: I say again, as I have said several times: Mathematically there is no question that she passes through every point indexed by a real number. Physically, the question is open in general, and meaningless in current theory. Even a goddess can't run between quarks.
If so, then we can take this reasoning to its limit and say that to move she must first reach the first non-zero coordinate. And if there is no first non-zero coordinate, then she cannot move. This is Zeno's Argument which is what trying to highlight, but apparently didn't convey very well.
You're perfectly well conveying your refusal to read what I'm writing. If this is a mathematical thought experiment, she does pass through every point in the closed unit interval [0,1]. If this is a physical thought experiment, the question is metaphysically open and physically meaningless. So which question are you asking?
In the Numberphile on Zeno's Paradox, James Grime says the following:
"I want to give you the mathematician's point of view for this, because, well, some say that the mathematicians have sorted this out........So something like this-- an infinite sum-- behaves well when, if you take the sum and then you keep adding one term at a time, so you've got lots of different sums getting closer and closer to your answer. If that's the case, if your partial sums--that's what they're called-- are getting closer and closer to a value, then we say that's a well-behaved sum, and at infinity, it is equal to it exactly. And it's not just getting closer and closer but not quite reaching. It is actually the whole thing properly."
That's right. Mathematically the sum of an infinite series is defined as the limit of the sequence of partial sums; and the limit of a sequence is defined as the number the sequence gets (and stays) arbitrarily close to. And the Numberphile guy says exactly that.
Is your view that the problem (of Atalanta travelling from x=0 to x=1) is resolved by completing an infinite process?
Depends on whether this is a mathematical or physical thought experiment. The physical question is certainly not resolved by the math, but I've stated that many times already.
I think the most compelling solution to Zeno's Paradox (of Achilles and the tortoise) that is often presented is by looking at the situation holistically. If you ask what are the velocities of Achilles and the tortoise you can work backwards to calculate the instant when Achilles passes the tortoise. It seems so simple when you think of it this way. This is the type of thinking that I'm promoting: starting with the whole and working backwards to determine instants.
If you would stop conflating math and physics, and avoiding answering whether you are asking a mathematical or physical question, all would be clear.
On the other hand, it's perfectly obvious that motion does occur in the real world (unless we live in a block universe or we're all programs running in the great computer in the sky or brains in vats or I'm just dreaming all this.
Aren't you the one saying you agree with Aristotle? He believed that the reason bowling balls fall down is that they're made of "stuff" and so is the earth, and like attracts like. Aristotle needs an update too.
Are you suggesting that with each step someone sweeps over infinite points? In other words, are you suggesting that motion involves the completion of a supertask?
No, I'm suggesting that the mathematical concept of the real line doesn't apply to the true nature of physical space; or that if it does, this fact lies far beyond contemporary physics. Even physics doesn't claim it does. Can't we stop here and nail down this point?Why are you deliberately ignoring this point, which I have made to you over and over? Nobody knows if space is infinitely divisible; and everyone knows that we don't know this.
Human beings and even goddesses can not traverse the mathematical real line because the latter is a mathematical abstraction. They can only traverse physical space, and nobody knows if physical space is like the mathematical real line or not. And we DO know that contemporary physics can not address the question.
There is a dynamism in the resting of matter, something that Heraclitus glimpsed and could only call "fire". Perhaps he knew of Zeno type paradoxes and referred to it specifically in his union of opposites (SEP article is great on him). One thing we know for sure is that infinity and " the finite " are opposites
No, this is a confusion of "infinitely divisible" with "infinitely divided." The former means potentially having infinitely many parts, while the latter means actually having infinitely many parts. A true continuum is infinitely divisible, but this does not entail that it is infinitely divided. It is a whole such that in itself it has no actual parts, only potential parts. These are indefinite unless and until someone marks off distinct parts for a particular purpose, such as measurement, even if this is done using countably infinite rational numbers or uncountably infinite real numbers. A continuous line does not consist of such discrete points at all, but we could (theoretically) mark it with points exceeding all multitude.
Mathematically there is no question that she passes through every point indexed by a real number.
This is one particular mathematical model of the interval--the dominant modern one, to be sure, but not the only one. Again, it is not mathematically necessary to treat a spatial interval as somehow consisting of unextended points. We can understand them instead as denoting locations in space, not constituents of space. As such, she does not really "pass through" them, we just just use them to track her progress.
Although the parts exist together, they exist even though they are not separated. A continuum is an infinite substance. Nothing potentially has parts. It HAS the parts whether they are separated or not
Nothing potentially has parts. It HAS the parts whether they are separated or not
I obviously disagree. Again, a true continuum has no definite parts except those that we deliberately mark off within it for a particular purpose. It is infinitely divisible, but not actually divided.
One foot does not potentially have two united six inches. The 12 inches are there
Neither the foot nor the inches are "there" unless and until we mark them. They are arbitrary units of length for measuring things, not intrinsic to space itself.
Reply to Gregory
A one-dimensional line does not "contain" anything. We can mark it with as many zero-dimensional points as we can imagine, but those points are not parts of the line itself.
Metaphysician UndercoverMarch 18, 2021 at 17:04#5119200 likes
No, this is a confusion of "infinitely divisible" with "infinitely divided." The former means potentially having infinitely many parts, while the latter means actually having infinitely many parts. A true continuum is infinitely divisible, but this does not entail that it is infinitely divided. It is a whole such that in itself it has no actual parts, only potential parts. These are indefinite unless and until someone marks off distinct parts for a particular purpose, such as measurement, even if this is done using countably infinite rational numbers or uncountably infinite real numbers. A continuous line does not consist of such discrete points at all, but we could (theoretically) mark it with points exceeding all multitude.
If this is true, what you describe here, then it is impossible that "a true continuum is infinitely divisible. If marking points on a continuous line does not constitute dividing it, then there is nothing to indicate that the continuous line is divisible at all. And if dividing it once would break it's continuity, then a continuum cannot be infinitely divisible because dividing it once would prove it to be discontinuous.
Therefore it is a contradiction to say "a true continuum is infinitely divisible". We ought to say instead, "if it were divisible it would not be a true continuum". A true continuum is indivisible.
In itself, yes; but we can still "divide" it at will to suit our purposes. That is what I mean when I say that the whole is real and the parts are entia rationis, creations of thought. For example, we can conceive space itself as continuous and indivisible, but we can nevertheless mark it off using arbitrary and discrete units for the sake of locating and measuring things that exist within space.
This is one particular mathematical model of the interval--the dominant modern one, to be sure, but not the only one. Again, it is not mathematically necessary to treat a spatial interval as somehow consisting of unextended points. We can understand them instead as denoting locations in space, not constituents of space. .
If you view real numbers as locations where points might live, then she passes through each location. I don't see how this changes anything. Also I've previously referenced the constructive, hyperreal, and Peircian concepts of the real line, so I don't think I can be accused of narrow mindedness on the topic. Unless you reject the intermediate value theorem, my point stands. And even the constructivists have their own version of IVT, which they make work by restricting functions to being computable ones. Some version of the IVT is always valid regardless of one's model of the real line.
As such, she does not really "pass through" them, we just just use them to track her progress
Distinction without a difference. IMO. She passes through each point or she passes through each location that is the address of a point. Not sure what you are getting at.
Points are the limit that infinite divisions approach. Space does lead to a pure contradiction. Our minds are not precise enough to understand it. Every part can be divided by the mind endlessly.
She passes through each point or she passes through each location that is the address of a point. Not sure what you are getting at.
Put another way, there are no real points or locations that she passes through, only the ones that we invent to describe her motion. Again, space does not consist of such discrete points or locations, we introduce them for our own purposes. Not sure I can convey what I mean any more clearly than that.
Not sure I can convey what I mean any more clearly than that.
How many times must I say the same thing? I don't get it. To be fair you may not be reading my replies to @keystone aka @Ryan, but rest assured that I have made the point you are making many times over.
There is no true space that cannot be divided by the mind. And this applies to the spaces that are divided from each other. And again and again. Understanding the contradiction is a type of wisdom on the subject. It's not scary. "A healthy mind can accept a paradox" (Chesterton)
The "Internet Encyclopedia of Philosophy" has a superb article on "Inconsistent Mathematics", which is a field of study also called paraconsistent mathematics. It mentions Zeno's paradox
(I believe all of Zeno's arguments are reduced to one single insight: spatial objects are finite with infinite parts. If this does not give you a headache one day of your life then you never understood it. The argument that motion through a point is really "rest" I think is false. Zeno had one great insight and it was an insight into mathematics, not physics)
Unless you reject the intermediate value theorem, my point stands. ... Some version of the IVT is always valid regardless of one's model of the real line.
Well, the IVT is not valid in smooth infinitesimal analysis. As Bell states in his book that I suggested a while back, "the classical intermediate value theorem, often taken as expressing an 'intuitively obvious' property of continuous functions, is false in smooth worlds." The Wikipedia article on SIA explains:
Intuitively, smooth infinitesimal analysis can be interpreted as describing a world in which lines are made out of infinitesimally small segments, not out of points. These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved. The construction of discontinuous functions fails because a function is identified with a curve, and the curve cannot be constructed pointwise. We can imagine the intermediate value theorem's failure as resulting from the ability of an infinitesimal segment to straddle a line.
It is presumably less surprising that the Banach-Tarski paradox also does not arise in SIA.
In other words space is not described by the mathematical real line. As I've written in this thread at least ten times now.
So we agree, then? The mathematical real line is an extremely useful model of a continuous line, but like all representations, it does not capture every aspect of its object--in this case, a true continuum.
So we agree, then? The mathematical real line is an extremely useful model of a continuous line, but like all representations, it does not capture every aspect of its object--in this case, a true continuum.
I have never claimed otherwise. If I've somehow failed to communicate that, I'll try to do better. But I would disagree with one aspect of what you said. The mathematical real line fails to describe what's physically provable about space, in terms of people walking and "passing through every point." But why should space be a continuum at all? That's an open question. How would anyone know what a "true continuum" even is? The idea of a continuum, let a lone a "true" one, is a conceptual abstraction. It's like arguing about which mathematical model of the tooth fairy is correct.
I would ask you: What is a "true continuum," and how would you know one if you saw it? Are you claiming there is such a thing in the world? Or if not, and if it's only a conceptual abstraction, how can anything be an accurate model of it?
ps -- SIA denies excluded middle and so falls into the general category of constructivism. No wonder many common standard theorems are false in such a framework. I've already mentioned that IVT is false in constructivist math but they patch that up by demanding that all functions are computable. In SIA, all functions are continuous.
But why should space be a continuum at all? That's an open question.
I agree, it is a hypothesis--one that I happen to find much more plausible than space consisting of discrete parts. I would say the same about time, which Peirce considered to be "the continuum par excellence, through the spectacles of which we envisage every other continuum."
How would anyone know what a "true continuum" even is?
I guess it comes down to definitions. Modern mathematicians stipulate that the real numbers constitute the (analytical) continuum, but (at least arguably) that approach is not entirely consistent with the common-sense notion of what it means to be continuous.
I agree, it is a hypothesis--one that I happen to find much more plausible than space consisting of discrete parts. I would say the same about time, which Peirce considered to be "the continuum par excellence, through the spectacles of which we envisage every other continuum."
Your intuition is seriously at odds with modern physics. Do you think physics is wrong? How do you square this.
Secondly I wanted to repeat in case you missed the ps to my last post, that SIA denies excluded middle. So it's a flavor of constructivism. No wonder that IVT is false, it's false in constructive math. And no wonder Banach-Tarski is false, constructivism denies the axiom of choice.
I guess it comes down to definitions. Modern mathematicians stipulate that the real numbers constitute the (analytical) continuum, but (at least arguably) that approach is not entirely consistent with the common-sense notion of what it means to be continuous.
Your common sense notion is incompatible with modern physics. And you are failing to distinguish between physical space, which we think is "really there," and the idea of the continuum, which is just a philosophical concept with no actual referent in the real world.
Your intuition is seriously at odds with modern physics. Do you think physics is wrong?
No, I think that anyone who interprets the Planck length as a discrete constituent part of space is wrong. I interpret it instead as a limitation on the precision of measurement, or as Wikipedia puts it, "the minimum distance that can be explored. ... The Planck length is sometimes misconceived as the minimum length of space-time, but this is not accepted by conventional physics, as this would require violation or modification of Lorentz symmetry."
Secondly I wanted to repeat in case you missed the ps to my last post, that SIA denies excluded middle.
I did not see your PS until now, but I am well aware that the logic of SIA is what has come to be known as constructive or intuitionistic. Peirce was skeptical of excluded middle, but for very different philosophical reasons than Brouwer and Heyting--he believed that reality itself does not conform to it, because it is fundamentally continuous and general, rather than discrete and individual. He stated this in slightly different ways in alternate drafts of the same text.
Peirce (NEM 3:758, 1893):To speak of the actual state of things implies a great assumption, namely that there is a perfectly definite body of propositions which, if we could only find them out, are the truth, and that everything is really either true or in positive conflict with the truth. This assumption, called the principle of excluded middle, I consider utterly unwarranted, and do not believe it.
Peirce (NEM 3:759-760, 1893):No doubt there is an assumption involved in speaking of the actual state of things ... namely, the assumption that reality is so determinate as to verify or falsify every possible proposition. This is called the principle of excluded middle. ... I do not believe it is strictly true.
Metaphysician UndercoverMarch 19, 2021 at 00:33#5120670 likes
For example, we can conceive space itself as continuous and indivisible, but we can nevertheless mark it off using arbitrary and discrete units for the sake of locating and measuring things that exist within space.
The proposed units would be arbitrary, but I do not think you could call them discrete. And, according to the issues brought forward by special relativity, the supposed "same unit" would be different depending on the frame of reference, or more precisely, we could not determine the "same unit" from different frames of reference. This makes the whole idea of measuring the continuum through the means of units rather difficult. I suggest that if the application of units works well for measuring space and time, they are probably not actually continuous.
The speculation which is the reverse of yours is that continuity is what is artificial. The continuum is something created by human minds, and physical existence contains no such continuity. Problems such as Zeno's paradoxes arise because we apply principles of continuity to a physical world which is discontinuous.
"Pierre Gassendi in the early 17th century mentioned Zeno’s paradoxes as the reason to claim that the world’s atoms must not be infinitely divisible. Pierre Bayle’s 1696 article on Zeno drew the skeptical conclusion that, for the reasons given by Zeno, the concept of space is contradictory." Internet Encyclopedia of Philosophy
I think Bayle's conclusion is the only one that is eventually reached by logic on this. The great mathematician Alfred Whitehead write on the subject of "wholes and parts" and yet Wikipedia said some of his results appear to be wrong. This subject is a tangle which however allows you to view it with intuition instead of logic and get something out of it instead of eternal frustration. I have no doubt no one will find "THE answer" to this riddle. Yet it can be accepted as a trick played on us by the fathomless
As I have asked you several times, is this a mathematical or a physical scenario?
I've mentioned QM and physics in an attempt to support my view, but the focus of my argument has always been on the mathematical scenario. From now on, unless explicitly stated otherwise, I'm talking about the mathematical scenario. I'm assuming continuity. The space that I'm talking about is infinitely divisible. There's no planck length/time to suggest any possible notion of discreteness.
In other words space is not described by the mathematical real line. As I've written in this thread at least ten times now.
Are you referring to mathematical space or physical space? I assume in this context you're referring to physical space, right? Is it fair to assume that you believe that mathematical space can be modelled with the real numbers?
Mathematically there is no question that she passes through every point indexed by a real number....If this is a mathematical thought experiment, she does pass through every point in the closed unit interval [0,1].
A thought experiment is exactly what I'd like us to do. To perform an experiment, we cannot just say 'she passes through every point', we must actually conduct the experiment. The best way to do this is to envision that Atalanta lives in a simulation and we must understand how that simulation works. I will grant you a computer with no restrictions (e.g. infinite memory and speed). The only constraint is that the simulated universe must be consistent. How does the simulation allow her to pass through every point in the closed interval [0,1]?
One might say that this simulation can be easily performed in 1 second (e.g. the time interval [0,1]). There's a 1-to-1 correspondence between positions and instants in time, so for each instant the simulation outputs the matching coordinate (e.g. at 0.4 seconds, Atalanta is at x=0.4). But hold on. That would mean that the computer is effectively outputting a complete list of the real numbers between 0 and 1 and Cantor showed that such a list is impossible. For her to move, the simulation must be a lot more clever than that. Any ideas?
Aren't you the one saying you agree with Aristotle? He believed that the reason bowling balls fall down is that they're made of "stuff" and so is the earth, and like attracts like. Aristotle needs an update too.
Aristotle was wrong on many things. We should not resuscitate his bad ideas without good reason...I just think that his ideas on potential infinity deserve another look now that our intuitions have been altered due to modern physics. (note: I'm talking about intuitions from physics providing further insight into longstanding paradoxes in the philosophy of mathematics, such as the mathematical Zeno's paradox.)
No, I think that anyone who interprets the Planck length as a discrete constituent part of space is wrong. I interpret it instead as a limitation on the precision of measurement
If we take this reasoning to its limit then we end up with the whole being an assembly of a bunch of 0-Dimensional objects (i.e. nothing). Is that what you believe? Everything is made up of nothing?
If we take this reasoning to its limit then we end up with the whole being an assembly of a bunch of 0-Dimensional objects (i.e. nothing). Is that what you believe? Everything is made up of nothing?
No. The zeros (points) is approached by infinite space that is also finite. That is the only way to put this question honestly. "In the early 19th century, Hegel suggested that Zeno’s paradoxes supported his
view that reality is inherently contradictory." Internet Encyclopedia of Philosophy
It's said everywhere that modern math found the answer to this paradox, to which I use words of Heraclitus "Hearing they do not understand, like the deaf. Of them does the saying bear witness: 'present, they are absent'" I say this only because most people don't understand what the paradox means so there "explanations" completely miss the mark
Do you expect to be in the news as the guy who solved this ancient problem? As the mathematician said in video linked earlier in this discussion, greater minds than yours have wrestled with this problem and failed. Not so long ago some writer came out with his "solution" and got a lot of praise until everyone realized that it was Henri Bergson's answer that had plagiarized and it was not a true solution to the problem to begin with
I've mentioned QM and physics in an attempt to support my view, but the focus of my argument has always been on the mathematical scenario. From now on, unless explicitly stated otherwise, I'm talking about the mathematical scenario. I'm assuming continuity. The space that I'm talking about is infinitely divisible. There's no planck length/time to suggest any possible notion of discreteness.
Ok. Then for clarity let me add another important condition. The rational numbers are infinitely divisible, but they are not complete. In addition to infinite divisibility, we need to require that every nonempty set of real numbers has a least upper bound. Otherwise the resulting system fails to be an adequate model of continuity. Just noting this for accuracy.
Are you referring to mathematical space or physical space?
By space I mean physical space. And by physical space I sometimes mean "true" physical space as in ultimate reality, if there even is such a thing; and other times I mean the theories of contemporary physics.
A thought experiment is exactly what I'd like us to do. To perform an experiment, we cannot just say 'she passes through every point', we must actually conduct the experiment. The best way to do this is to envision that Atalanta lives in a simulation and we must understand how that simulation works. I will grant you a computer with no restrictions (e.g. infinite memory and speed). The only constraint is that the simulated universe must be consistent. How does the simulation allow her to pass through every point in the closed interval [0,1]?
There are no computers involved in this. Computers are far too limited, since Turing machines are restreicte to finite sequences of instructions and finitely many steps. A Turing machine already has an unbounded tape (memory) and speed of execution is not a factor at all. So this is a bad model of mathematics. For example, no computer program can approximate or generate the decimal digits of a noncomputable number. So I reject this idea totally. The set of computable numbers is a countably infinite proper subset of all real numbers, so they are missing a lot.
A function that passes through the point a at one time and b at a later time must necessarily pass through every intervening point. If one is a constructivist this becomes false, but the constructivists patch the problem by restricting attention to computable functions to make it true again.
One might say that this simulation can be easily performed in 1 second (e.g. the time interval [0,1]). There's a 1-to-1 correspondence between positions and instants in time, so for each instant the simulation outputs the matching coordinate (e.g. at 0.4 seconds, Atalanta is at x=0.4). But hold on. That would mean that the computer is effectively outputting a complete list of the real numbers between 0 and 1 and Cantor showed that such a list is impossible. For her to move, the simulation must be a lot more clever than that. Any ideas?
Your idea is totally invalid. Computers, even theoretical abstract ones, are not relevant. What can be computed is a tiny subset of what can be mathematically proved to exist. As I noted, Turing machines can only generate the countably infinite set of computable real numbers. The noncomputable reals, of which there are far far more, are entirely beyond the range of computation.
The concept of computable real numbers was first elucidated by Turing himself. His great 1936 paper is called, "On Computable Numbers, with an Application to the Entscheidungsproblem." Never mind what the Entscheidungsproblem is. What's important is that it was Turing himself who first pointed out this profound limitation in the ability of computers to represent or characterize real numbers. The exact limitation your simulation or computation idea is bumping into. If we restrict ourselves to only numbers that are computable, then you CAN drive a truck through the real number line, because the computable real line is full of holes. [A skinny truck of course, the width of a point].
In fact the computable numbers, just like the rationals, are infinitely divisible but not complete.
Aristotle was wrong on many things. We should not resuscitate his bad ideas without good reason...I just think that his ideas on potential infinity deserve another look now that our intuitions have been altered due to modern physics. (note: I'm talking about intuitions from physics providing further insight into longstanding paradoxes in the philosophy of mathematics, such as the mathematical Zeno's paradox.)
Just pointing out that you want to refer back to Aristotle as authoritative in some things but "in need of updating" in others. Cherry-picking Aristotle as it were.
I interpret it instead as a limitation on the precision of measurement, or as Wikipedia puts it, "the minimum distance that can be explored. ...
It is not exactly a measurement problem. The fact that I can't measure exactly one meter with a meter stick is the problem of the approximation of measurement.
The Planck scale represents the point at which contemporary physics breaks down and is not applicable; so that we can not rationally discuss or compute what happens below those lengths, durations, and energies. That's subtly different than just the approximateness of measurement.
but this is not accepted by conventional physics, as this would require violation or modification of Lorentz symmetry."
We're perfectly in agreement on this point except that my physics is not strong enough to catch the reference to Lorentz symmetry. I found a physics.SE thread linking the terms but didn't read it. Should I?
I did not see your PS until now, but I am well aware that the logic of SIA is what has come to be known as constructive or intuitionistic.
Ok. Then it's fair for me to note that I already mentioned constructivism in replies to @keystone and can't personally do anything about the legion of neo-intuitionists running about these days, though I would if I could :-)
Peirce was skeptical of excluded middle, but for very different philosophical reasons than Brouwer and Heyting--he believed that reality itself does not conform to it, because it is fundamentally continuous and general, rather than discrete and individual. He stated this in slightly different ways in alternate drafts of the same text.
That's interesting. The differences among the LEM-deniers are too subtle for me to appreciate. But then again I don't have much interest in reality when it comes to math. I don't feel a need to justify math in the name of reality. And I can always fall back on history: complex numbers, non-Euclidean geometry etc., to show that no matter how weird math gets, people often find a use for it. I am not defending math as any kind of model of reality any more than I would defend the game of chess on that basis. I'm immune to criticisms based on anyone's idea of what's real because I don't think math is particularly real. On my formalist days at least. The rest of the time I take a Platonist view. I have no hard convictions in this regard.
To speak of the actual state of things implies a great assumption, namely that there is a perfectly definite body of propositions which, if we could only find them out, are the truth, and that everything is really either true or in positive conflict with the truth. This assumption, called the principle of excluded middle, I consider utterly unwarranted, and do not believe it.
— Peirce (NEM 3:758, 1893)
Yes but does he deny the way the knight moves in chess? LEM can be taken as a rule in a formal game. I would never defend LEM by saying the world works that way. How could I pretend to know such a thing? Peirce could be 100% right yet LEM-based math is still an entertaining pastime. I never speak of the absolute state of things. You see you are trying to get me to take the other side of a question I reject entirely. I don't say, "LEM-based math is right because the world works that way." I would never say such a thing. I'm not making any claims about the world, nor about LEM-based math's applicability to the world. I say, "LEM-based math is interesting in its own right, and if the arc of history is turning against it via the neo-intuitionists, so be it." Sort of like how Cubs fans must have felt about night games [Chicago's Wrigley Field only got lights as recently as 1988].
You say that standard math doesn't apply to the "true continuum." I say to you, "Yes I certainly agree. And by the way, what do you mean by true continuum." That's the conversation I believe I'm trying to have. I'm not claiming knowledge of the world or staking out any metaphysical position.
No doubt there is an assumption involved in speaking of the actual state of things ... namely, the assumption that reality is so determinate as to verify or falsify every possible proposition. This is called the principle of excluded middle. ... I do not believe it is strictly true.
— Peirce (NEM 3:759-760, 1893)
I have no argument or complaint. He may be right for all I know. That doesn't bear on any point I'm making. You are arguing against multiple strawmen positions I'm not taking.
I don't feel a need to justify math in the name of reality.
Oh, I completely agree. Again, mathematics is the science of drawing necessary conclusions about strictly hypothetical states of things. Whether those premisses match up with reality is a matter of metaphysics, not mathematics. They can be just about anything imaginable, although some of the most interesting cases come about when we remove a previously taken-for-granted axiom like the parallel postulate in geometry or excluded middle in logic, but still manage to come up with a consistent and useful system.
You say that standard math doesn't apply to the "true continuum." I say to you, "Yes I certainly agree. And by the way, what do you mean by true continuum." That's the conversation I believe I'm trying to have.
I can only cover so much ground in this format. My long answer is the paper that I provided.
The rational numbers are infinitely divisible, but they are not complete. In addition to infinite divisibility, we need to require that every nonempty set of real numbers has a least upper bound. Otherwise the resulting system fails to be an adequate model of continuity. Just noting this for accuracy.
I have concerns with infinite sets (including the set of all rational numbers) and real numbers, but those concerns are not essential to my argument. As such, I accept these two conditions.
No, I'd be very surprised if this turns out to be true. The mathematical real numbers are far too strange to be real in the sense of physical reality.
Your comment was in response to me asking "Is it fair to assume that you believe that mathematical space can be modelled with the real numbers?" By mathematical space, I just mean continua and given that above you mention that real numbers can adequately model continuity, I will assume that you misread my question.
There are no computers involved in this. Computers are far too limited...So I reject this idea totally.
I wasn't granting you a Turing machine, I was granting you an infinite computer with no restrictions. But fine, let's take algorithms out of the picture, and I'll grant you God and the Axiom of Choice. My only requirement is that everything God does must be consistent. Now, going back to my original question, how God move Atalanta from x=0 to x=1? As I mentioned in my last response, he can't advance her point by point because that would be equivalent to listing the real numbers, which is impossible. So how would he do it?
A function that passes through the point a at one time and b at a later time must necessarily pass through every intervening point.
Your statement/position implies that all that exist between a and b are points. The way I see it is that the function must pass through the intervening spaces. So in the image below, to get from 0 to 1, the function must pass through the 4 continua represented by the following open intervals: [math](0,\frac{1}{3}), (\frac{1}{3},\frac{1}{2}), (\frac{1}{2},\frac{3}{4}), \text{and} (\frac{3}{4},1)[/math]. We both believe that the function cannot skip the intervening objects, I just believe that there are finite intervening objects and you believe that there are infinite intervening objects. I think this difference is what makes Zeno's Paradoxes a problem for the point-based view. Also, my view is not restricted to computable functions.
Just pointing out that you want to refer back to Aristotle as authoritative in some things but "in need of updating" in others. Cherry-picking Aristotle as it were.
Oh come on, all I'm saying is that the idea didn't originate from me. I'm not claiming to be right on anyone's authority. You talk as if I must either agree with everything he taught or disagree with everything he taught. Sometimes things come back into fashion...like the mullet, right? That's due for a resurgence soon.
Do you expect to be in the news as the guy who solved this ancient problem? As the mathematician said in video linked earlier in this discussion, greater minds than yours have wrestled with this problem and failed.
I am probably wrong, but your statement isn't proof of that. Like I said to fryfish, my view has neither been formalized or proven correct. If I provided a formal theory and proved it correct then that would certainly be newsworthy, but as it stands I'm just looking for flaws in my ideas.
I agree, this is a better concise summary of what it means.
Ok. My point, which I sadly forgot to make, is that even with the clarified fact that the Planck scale is the scale below which contemporary physics can't be applied, your position -- that the physical world embodies or instantiates or contains or is a "true continuum" -- is not supported by contemporary physics. And I asked you what a true continuum is, and how you'd know one if you saw it. And how exactly would you see it? Good questions all.
Heh, we are in the same boat on that, I was just quoting Wikipedia.
Yes but even with the clarified definition of Planck scale my point still applies. That your belief that there exists a "true continuum" is incompatible with contemporary physics. If I said inconsistent that was too strong. I should have said, "not supported by." That would be more accurate.
Oh, I completely agree. Again, mathematics is the science of drawing necessary conclusions about strictly hypothetical states of things. Whether those premisses match up with reality is a matter of metaphysics, not mathematics. They can be just about anything imaginable, although some of the most interesting cases come about when we remove a previously taken-for-granted axiom like the parallel postulate in geometry or excluded middle in logic, but still manage to come up with a consistent and useful system.
Ok. I still want to know, and think there could be an interesting discussion around, your claim that there exists a "true continuum." You said math doesn't model it, as if there even is any such thing to be modeled.
No, that was not my intention, and I am sorry that I came across that way. I was just trying to provide more background about my own position.
Ok, but I still don't understand your position. You claimed that math doesn't model the "true continuum." I have two questions. What is a true continuum, and what makes you think any such thing exists in the physical world?
I can only cover so much ground in this format. My long answer is the paper that I provided.
Can I get a short answer? What is a true continuum and what makes you think such a thing exists in the physical world, such that the question of whether it's accurately modeled by math is a meaningful question? After all, we can't ask if math models the tooth fairy. And I claim the tooth fair is in the same category of things as the "true continuum" -- a fairytale. Except that I can't trade the continuum a tooth for a quarter
I have concerns with infinite sets (including the set of all rational numbers) and real numbers, but those concerns are not essential to my argument. As such, I accept these two conditions.
I'm just clarifying that we often take "infinitely divisible" as a mathematical continuum, but this is very loose speech. The actual condition required is completeness in the metric sense.
Your comment was in response to me asking "Is it fair to assume that you believe that mathematical space can be modelled with the real numbers?"
Depends on the space. In math there are metric spaces, topological spaces, measure spaces, probability spaces, Sobolov spaces, function spaces, and many many other kinds of things called spaces. So the answer is no, without further qualification or clarification.
By mathematical space, I just mean continua and given that above you mention that real numbers can adequately model continuity, I will assume that you misread my question.
What is a continuum? You ask if the real numbers can model a continuum and I don't know what the question means. The real numbers are commonly identified with "the continuum" but one can challenge that on philosophical grounds, hence the history of intuitionism etc.
I was granting you an infinite computer with no restrictions.
I wish you'd take the trouble to read what you yourself wrote. You said a computer (or simulation) with infinite memory. A Turing machine already has unbounded memory so you haven't added anything without supplying further qualification. Which you didn't supply.
But fine, let's take algorithms out of the picture, and I'll grant you God and the Axiom of Choice. My only requirement is that everything God does must be consistent. Now, going back to my original question, how God move Atalanta from x=0 to x=1?
In the physical world? I have no idea and neither does anyone else. In math? There's a function f(t) = t that's 0 at time 0, 1 at time 1, and that passes through every intervening point. Or that passes through every intervening location where there could potentially be a point as @aletheist noted.
As I mentioned in my last response, he can't advance her point by point because that would be equivalent to listing the real numbers, which is impossible. So how would he do it?
f(t) = t. Or any of infinitely many other functions that have f(0) = a and f(1) = b. I don't follow why you're making a mountain of a mathematical molehill. Or what God has to do with any of this.
Your statement/position implies that all that exist between a and b are points.
On the standard mathematical real line? Yes that's true. You think otherwise? But I don't need to use the philosophically loaded word points. I can say that between any two real numbers all that exists are other real numbers. You disagree in some sense? Be specific.
The way I see it is that the function must pass through the intervening spaces. So in the image below, to get from 0 to 1, the function must pass through the 4 continua represented by the following open intervals: (0,13),(13,12),(12,34),and(34,1)(0,13),(13,12),(12,34),and(34,1). We both believe that the function cannot skip the intervening objects, I just believe that there are finite intervening objects and you believe that there are infinite intervening objects. I think this difference is what makes Zeno's Paradoxes a problem for the point-based view. Also, my view is not restricted to computable functions.
If I go out to the nearby highway and remove all the mile markers and road signs does that make them disappear? You're confusing labeling with existence. If I remove the label from a can of soup the can still contains soup. You're making a very disingenuous point.
I find your claim silly and not at all a serious argument or position.
Oh come on, all I'm saying is that the idea didn't originate from me. I'm not claiming to be right on anyone's authority. You talk as if I must either agree with everything he taught or disagree with everything he taught. Sometimes things come back into fashion...like the mullet, right? That's due for a resurgence soon.
... your position -- that the physical world embodies or instantiates or contains or is a "true continuum" -- is not supported by contemporary physics.
That is not an accurate statement of my position. I hold that space is a true continuum, but not that it is something physical; rather, it is the real medium within which everything physical exists. Ditto for time, albeit in a different respect (obviously).
And I asked you what a true continuum is, and how you'd know one if you saw it. And how exactly would you see it?
We all perceive space and time, and some of us formulate the hypothesis that each is truly continuous in a way that no collection of numbers, even an uncountably infinite one, could ever fully capture because of their intrinsic discreteness. Nevertheless, this does not preclude the real numbers (for example) from serving as an extremely useful model of continuity for the vast majority of practical purposes.
You said math doesn't model it, as if there even is any such thing to be modeled.
I have said before, and I just said again, that the real numbers do very successfully model a continuum. They just do not constitute a true continuum. That requires a different mathematical conceptualization, and smooth infinitesimal analysis turns out to be a promising candidate.
That is not an accurate statement of my position. I hold that space is a true continuum, but not that it is something physical; rather, it is the real medium within which everything physical exists. Ditto for time, albeit in a different respect (obviously).
Hmmmm. The way I'm hearing this, and correct me if I'm misunderstanding you, is that there's an absolute space against which everything else happens. A fixed, universal frame of reference. I hope you can see the problem there. And if that's not what you mean, then I don't understand what you're saying.
After all in general relativity spacetime is a manifold. It's like a twisted space without the ambient space. Like a globe, the earth say, embedded in Euclidean 3-space ... except there is no ambient space. There's only the globe, with its intrinsic curvature.
And again you have used the phrase true continuum, which you haven't defined.
I do appreciate that you've put something on the table that I can at least begin to ask questions about.
We all perceive space and time, and some of us formulate the hypothesis that each is truly continuous in a way that no collection of numbers, even an uncountably infinite one, could ever fully capture because of their intrinsic discreteness.
I understand that you mean that the real numbers are composed of individuals, namely the real numbers. But the real numbers are not a discrete set. The integers are a discrete set, because around each integer you can draw a little circle that doesn't contain any other integers. You can't do that with the real numbers. So I get that Peircians don't like the fact that the reals are the union of their singleton points. But I don't see how you can label some alternative model "true" without evidence.
Nevertheless, this does not preclude the real numbers (for example) from serving as an extremely useful model of continuity for the vast majority of practical purposes.
Yes of course, we agree on that. And for my part I go farther. The reals are fun to study and if the physicists find them useful that's great because it means the universities will continue to fund the math department. Else they'd fire the whole lot. But the reals are far too weird to represent anything real. If nothing else, the reals as we understand them depend on the vagaries of which set-theoretic axioms we choose. If anyone thought the real numbers were "really real" then physics postdocs would get grants to count the number of points in a meter in order to determine whether the continuum hypothesis is true. Since nobody has applied for any such grant, I take that as evidence that nobody takes the real numbers seriously as an accurate model of anything physical. They're only an abstract mathematical model, one with deeply strange properties.
Yes, you did give me specific things to be unclear about and to push back on. Is my end of the conversation hopeless unless I read Bell and Peirce?
But I do think you need to explain yourself about this mythical background space in your first paragraph, which sounds suspiciously like the luminiferous aether, whose existence was disproved by the famous Michelson-Morly experiment, leading Einstein to special and then general relativity. There is no preferred frame of reference in the universe according to modern physics. There is no fixed background against which all other physical things can be measured.
pd -- Reading your paper, with much eye-glazing I'm ashamed to say. I did come across this:
"...he restates the second and third properties of Time as a continuum: any lapse can be made up of two lapses that have a common instant between them, although it need not "have a final and an initial instant." For example, the present is "assignable" as the "limiting instant" between the past, which has no initial instant, and the future, which has no final instant."
This sounds suspiciously like the idea of a Dedekind cut. Except that there's a point missing, as in the union of the open intervals (0,1) and (1,2). Am I understanding that right?
The way I'm hearing this, and correct me if I'm misunderstanding you, is that there's an absolute space against which everything else happens. A fixed, universal frame of reference.
No, that is not what I am saying. I am not really talking about physics at all, just a hypothetical/mathematical conceptualization that might have phenomenological and metaphysical applications.
Brouwer's revenge. The intuitionists are back with a vengeance. I don't doubt the historical momentum. It's the influence of the computers.
Peirce came before Brouwer, and my interest in SIA/SDG has nothing to do with intuitionism or computers. If Peirce had followed through on his skepticism of excluded middle and omitted what we now (ironically) call "Peirce's Law" from his 1885 axiomatization of classical logic, then he would have effectively invented what we now (unfortunately) call "intuitionistic logic" and it might be known instead as "synechistic logic"; i.e., the logic of continuity.
Is my end of the conversation hopeless unless I read Bell and Peirce?
Maybe not hopeless, but I suspect that there is a "curse of knowledge" aspect here on my part, given my immersion over the last few years in Peirce's writings and the secondary literature that they have prompted.
This sounds suspiciously like the idea of a Dedekind cut. Except that there's a point missing, as in the union of the open intervals (0,1) and (1,2). Am I understanding that right?
Peirce would say that there is no point missing, because there are no points at all until we deliberately mark one as the limit that two adjacent portions of the line have in common. If we make a cut there, then the one point becomes two points, since each interval has one at its newly created "loose end."
Depends on the space. In math there are metric spaces, topological spaces, measure spaces, probability spaces, Sobolov spaces, function spaces, and many many other kinds of things called spaces. So the answer is no, without further qualification or clarification.
What is a continuum? You ask if the real numbers can model a continuum and I don't know what the question means. The real numbers are commonly identified with "the continuum" but one can challenge that on philosophical grounds, hence the history of intuitionism etc.
I mean continuum in the context of the geometrical objects of extension studied in elementary calculus, the objects that we typically describe using the cartesian coordinate system.
In the physical world? I have no idea and neither does anyone else. In math? There's a function f(t) = t that's 0 at time 0, 1 at time 1, and that passes through every intervening point. Or that passes through every intervening location where there could potentially be a point as aletheist noted.
I'm talking about the mathematical world. The two sentences in this quote are quite different. The first sentence essentially states that it passes through infinite intervening points. The second sentence states that it passes through all intervening locations where there could be points. I actually agree with the second sentence.
f(t) = t. Or any of infinitely many other functions that have f(0) = a and f(1) = b. I don't follow why you're making a mountain of a mathematical molehill. Or what God has to do with any of this.
What I'm trying to convey is that no matter where Atalanta's mathematical universe lives (whether in an infinite computer or the mind of God) it is impossible to construct Atalanta's journey from points because that would amount to listing the real numbers. The only way to build her universe is to deconstruct it from a continuum, working your way down from the big picture to specific instants.
When an engineer tries to solve Zeno's Paradox (of Achilles and the Tortoise) they ask questions about the system as a whole, specifically 'What are the speed functions of Achilles and the Tortoise from the beginning to the end of time?' With that information we don't have to advance forward in time, instant by instant. We just find where their two position functions intersect and conclude that Achilles passes the tortoise at that instant. And if this mathematical universe lives in that engineer's mind, that's the only actual instant that exists. Sure, the engineer could calculate their positions at other instants in time, but the engineer isn't going to calculate their positions at all times. That would be unnecessary...and impossible.
I'm sure you agree with the above paragraph (and perhaps are a little offended that I'm positioning it as the engineer's solution...hehe) but my point is that knowing a function doesn't imply that we can describe it completely using points. Any attempt to do so would be akin to listing the real numbers.
On the standard mathematical real line? Yes that's true. You think otherwise? But I don't need to use the philosophically loaded word points. I can say that between any two real numbers all that exists are other real numbers. You disagree in some sense? Be specific.
I like when you earlier said 'every intervening location where there could potentially be a point'. It is worth creating a distinction between actual points and potential points. If we make that distinction, then I agree with you that there are only (actual and potential) points between a and b. What I would disagree with is the claim that there are only actual points between a and b. Actual points are discrete while potential points form a continuum. So instead of saying that there are finite actual points and infinite potential points between a and b, I think it is much better to say that there are finite actual points and finite continua between a and b. For example, in the image below, there are 3 actual points and 4 continua between 0 and 1.
You're confusing labeling with existence....I find your claim silly and not at all a serious argument or position.
If we start with continua, the actual points only exist when we make a measurement. It seems like you agreed with aletheist on this. With a continuum-based view, when we make a measurement, we are not labeling points that existed all along, we are bringing them into existence (i.e. actualizing them). Until then they are potential points and can only be described as a part of a collection (i.e. a continuum), which I described using an interval. I am totally serious about this argument. My view is only silly when seen from a point-based view because you assume that all we can talk about are actual objects...an infinite number of them.
Peirce would say that there is no point missing, because there are no points at all until we deliberately mark one as the limit that two adjacent portions of the line have in common. If we make a cut there, then the one point becomes two points, since each interval has one at its newly created "loose end."
We don't have two adjacent portions before the marker. The marker is a pointer that demarcates but does not split the line thereby making it discontinuous. The marker is not part of the line. A materialist overlay is unnecessary in math. It's cheaper to think of it as an abstract pointer anywhere to an abstract endless line in one dimensional space.
No, that is not what I am saying. I am not really talking about physics at all, just a hypothetical/mathematical conceptualization that might have phenomenological and metaphysical applications.
Ok not physics. Thanks for the clarification, I'm sure you can see that this idea would not hold up as physics.
But if it's a "hypothetical conceptualization," how can you claim with a straight face that the standard real line doesn't model the "true continuum?" I do understand Peirce's point that the real line isn't a continuum because it's made up of individual points. But I am objecting to your claim that it's meaningful to say whether something is or isn't a good model of an abstract idea. What if my true continuum isn't the same as yours? Just as you can't say whether set theory is a good model of the tooth fairy. One conceptual fiction's like another. You have a vague idea (ok perhaps not vague to you) of a true continuum, and you're saying the real line isn't it. How can I agree or disagree with that statement, without sharing your inner visions on the nature of the true continuum?
If Peirce had followed through on his skepticism of excluded middle and omitted what we now (ironically) call "Peirce's Law" from his 1885 axiomatization of classical logic, then he would have effectively invented what we now (unfortunately) call "intuitionistic logic" and it might be known instead as "synechistic logic"; i.e., the logic of continuity.
I agree with you that Peirce should be more famous. Didn't realize he pioneered LEM rejection.
Maybe not hopeless, but I suspect that there is a "curse of knowledge" aspect here on my part, given my immersion over the last few years in Peirce's writings and the secondary literature that they have prompted.
I wish I could dispatch a clone to read up on Peirce. It's on my to-do list, none of which ever gets done.
Thanks for the attempt, sorry for the resulting effect.
Most philosophical papers glaze my eyes. Some I find very clear, but many confuse me. Not just yours. I'll take another run at it in light of this interesting conversation. But I must admit that given a finite block of time, I'd be more likely to spend it trying to learn some new standard math rather than philosophy. I find philosophy very hard to grab on to.
Peirce would say that there is no point missing, because there are no points at all until we deliberately mark one as the limit that two adjacent portions of the line have in common.
Is this @keystone's point about points not existing on the number line till we mark them with labels? Sounds similar. Surely Peirce must have been familiar with Dedekind. Dedekind cuts are a clever idea, because we can logically construct the reals given only the rationals. With Peirce's description, I don't know what we've got. Philosophy versus math again. Perhaps I shouldn't even try to talk about philosophy.
If we make a cut there, then the one point becomes two points, since each interval has one at its newly created "loose end."
Grrrr. That makes no sense. If we divide the rationals into two classes, those whose square is less than 2 and those whose square is greater than 2, we can define sqrt(2) as that exact pair of classes of rationals. That's Dedekind's idea. Are you saying Peirce would make sqrt(2) both the largest of the smaller set and the smallest of the larger set? Having trouble with this. How can one point become two points? That's a more mysterious trick than Banach-Tarski.
In this thread I find jgill and @fishfry, who I believe are or were professional mathematicians,
@jgill is a professional mathematician. I don't say was, because it's not the kind of status one loses by virtue of being retired.
I am a failed math grad student. My own zeal is that of the fallen priest (I'm thinking Richard Burton in Night of the Iguana and probably some other roles too). Nobody has missionary zeal as strong as one who has been cast out of the order they strove to join.
I mean continuum in the context of the geometrical objects of extension studied in elementary calculus, the objects that we typically describe using the cartesian coordinate system.
I've taken and taught calculus and have no idea what the "geometrical objects of extension" are. The objects of the Cartesian coordinate system are ordered pairs of real numbers; or in the general case, ordered n-tuples. The pictures are for intuition and visualization. The actual objects of study are analytic. Just real numbers and ordered lists (in computer parlance) of same.
I'm talking about the mathematical world. The two sentences in this quote are quite different. The first sentence essentially states that it passes through infinite intervening points. The second sentence states that it passes through all intervening locations where there could be points. I actually agree with the second sentence.
Distinction without a difference. You can think of the real line as a set of points or as a set of locations or addresses. Just like a street is a collection of houses or it's a collection of addresses where there might be houses or there might be empty lots. Except that by the completeness of the real numbers, there are no empty lots. But if you want to think of real numbers as locations on a line, that's perfectly ok.
When an engineer tries to solve Zeno's Paradox (of Achilles and the Tortoise) they ask questions about the system as a whole, specifically 'What are the speed functions of Achilles and the Tortoise from the beginning to the end of time?' With that information we don't have to advance forward in time, instant by instant. We just find where their two position functions intersect and conclude that Achilles passes the tortoise at that instant.
See, you DO believe in the intermediate value theorem.
And if this mathematical universe lives in that engineer's mind, that's the only actual instant that exists. Sure, the engineer could calculate their positions at other instants in time, but the engineer isn't going to calculate their positions at all times. That would be unnecessary...and impossible.
but my point is that knowing a function doesn't imply that we can describe it completely using points.
On the contrary. A function is a collection of individual ordered pairs. That's the set-theoretic definition of a function. It's not the same as the path of a moving point as it was for Newton. But he was doing physics when he had that viewpoint.
I like when you earlier said 'every intervening location where there could potentially be a point'. It is worth creating a distinction between actual points and potential points.
You can think of real numbers as locations on the real line if you like. Locations or addresses. But the completeness of the real numbers means there is a point at every location.
But more to the point, "point" is just another name for a real number. The set of real numbers is the collection of all the real numbers and vice versa. You are thinking "points" are things separate from real numbers, but mathematically they are not. Or in n-space, points are n-tuples of real numbers.
If we make that distinction, then I agree with you that there are only (actual and potential) points between a and b. What I would disagree with is the claim that there are only actual points between a and b. Actual points are discrete while potential points form a continuum.
There are no such things as an actual or potential point. There are only real numbers and n-tuples of real numbers.
So instead of saying that there are finite actual points and infinite potential points between a and b, I think it is much better to say that there are finite actual points and finite continua between a and b. For example, in the image below, there are 3 actual points and 4 continua between 0 and 1.]/quote]
Nonsense. You keep repeating this and I keep calling it nonsense (last time I called it silly) but I'll soon run out of adjectives and also of patience. This isn't going anywhere. I disagree with your view and don't find there to be any meaningful content in it.
[quote="keystone;512652"]
If we start with continua, the actual points only exist when we make a measurement. It seems like you agreed with aletheist on this.
No no no no no I did not. @aletheist said that this was Peirce's point of view and I noted that this seemed similar to yours. I admit that when Peirce says it, I say, "Peirce, interesting guy. I wish I knew more about him." And when you say it, I say, "Nonsense." But in either case I do think the idea is mathematical nonsense. Philosophically I have no idea what you or Peirce could possibly mean by this and I emphatically deny ever giving the slightest consideration to the idea. It's wrong.
I do know that the intuitionists try to tie existence to human cognition, and this point I also disagree with despite there being many smart people on board with the idea.
With a continuum-based view, when we make a measurement, we are not labeling points that existed all along, we are bringing them into existence (i.e. actualizing them).
Ok. I can't keep disagreeing with the thing you keep repeating over and over. I disagree strongly with what you said here.
Until then they are potential points and can only be described as a part of a collection (i.e. a continuum), which I described using an interval. I am totally serious about this argument.
I understand that. And I am totally serious when I say this idea is nonsense utterly devoid of content. Although when @aletheist tells me Peirce said it, I scratch my chin and go, "Hmmm, that Peirce sure was an interesting guy." But what I think to myself is, "Nonsense."
I don't see that either of us has said anything new in a long time.
I agree with this. I need to study more to either accept that it's nonsense or find a way to better communicate it. Until then, we're just wasting our time. Let's not waste any more time. I really appreciate your patience sticking this out with me on this up until now. Thanks!
I agree with this. I need to study more to either accept that it's nonsense or find a way to better communicate it. Until then, we're just wasting our time. Let's not waste any more time. I really appreciate your patience sticking this out with me on this up until now. Thanks!
No prob, likewise. After all when Peirce says the same thing I go, "Hmmmm I need to learn more." So maybe there's something to it. Who's to say.
The solution to Zeno that has been proposed here is that parts appear on the geometric item only when noticed. Did not Parmenides say thought is being? Is his philosophy not early Greek idealism? Are not these new QM ideas just modern idealism? I have accepted that unbounded space is identical to bounded space not just because its a bigger idea than these other opinions, but because it's true
How can I agree or disagree with that statement, without sharing your inner visions on the nature of the true continuum?
I guess it comes down to the meaning of the concept of continuity. Someone immersed in modern mathematics, where the real numbers are routinely called a continuum, is understandably satisfied with that definition. Someone like Peirce who objects to finding any discrete parts whatsoever in something that is supposed to be continuous can never accept it. He was motivated primarily by logical considerations rather than mathematical ones.
According to a paper by Conor Mayo-Wilson, "Peirce and Brouwer seemed to have no knowledge of each other's work." However, they were indirectly linked through Lady Victoria Welby, with whom Peirce exchanged a fair amount of correspondence including some of his most important writings about semeiotic, and whose ideas about significs were later adopted by a group of Dutch thinkers that eventually included Brouwer.
Surely Peirce must have been familiar with Dedekind.
Yes, Dedekind's name appears in a bunch of his writings, and his most fundamental disagreement with him was about the relationship between mathematics and logic. For Peirce, logic (generalized as semeiotic) depends on mathematics, as does every other positive science; while for Dedekind, mathematics is a branch of logic.
Are you saying Peirce would make sqrt(2) both the largest of the smaller set and the smallest of the larger set?
Peirce would not talk about "sets"--or "collections," his usual term--when referring to a continuum at all. By definition, a collection consists of discrete parts, which are ontologically prior to the whole ("bottom-up"); while in a continuum, the whole is ontologically prior to the parts ("top-down").
Because the only points at all are the ones that we create by marking them. When we mark a line without separating it, we create one point. When we separate the line, we create two points, one at the discontinuous end of each resulting portion. When we put them back together, we have only one point again. The points are never parts of the line itself, because they are of lower dimensionality. Every part of a line is one-dimensional, but a point is dimensionless. SIA seeks to capture this with its infinitesimal segments that are long enough to have "direction" but too short to be curved.
What Peirce questions is not the LEM, but instead the applicability of it as referenced. To be sure, he calls it the "principle of the excluded middle," and in my opinion the substitution of "principle" for "law" makes all the difference.
Indeed, I believe that his use of "principle" rather than "law" for excluded middle is very deliberate. As he wrote elsewhere, "Logic requires us, with reference to each question we have in hand, to hope some definite answer to it may be true. That hope with reference to each case as it comes up is, by a saltus [leap], stated by logicians as a law concerning all cases, namely, the law of excluded middle. This law amounts to saying that the universe has a perfect reality."
Consequently, classical logic is strictly applicable only where "a recognized universe [of discourse] is definite (so that no assertion can be both true and false of it), individual (so that any assertion is either true or false of it), and real (so that what is true or false of it is independent of any judgment of man or men, unless it be that of the creator of the universe, in case this is fictive)."
In philosophical terms, since all objects are spatial and subject to one or another type of geometry, we have to say that objects are finite in form and infinite in content. To be perfectly honest when contemplating a ball or a cup will lead to this conclusion. There are, I admit, many types of geometry, and if someone finds a way to explain "the spatial" in a way that is comprehensive and avoids paradox, I am all ears. (I like how non-Euclidean geometry is on an infinite curve that revolves back into compactness. The weirdness of it gives me a faint hope that Zeno's paradox could be solved, but the final result might be way over my head)
If you can forever take a part from a solid and there always be a part remaining, it is not fully finite but instead has an aspect of infinity. Common sense says something should be either finite or infinite but not both
The ancient Chinese knew about this: "One of the few surviving lines from the school, 'a one-foot stick, every day take away half of it, in a myriad ages it will not be exhausted,' resembles Zeno's paradoxes."
https://en.wikipedia.org/wiki/School_of_Names
Zeno added motion to the question of infinite divisibility to make the question even more difficult. I think mathematics does apply to objects in that two haves will have the same volume when united. I see no reason why these processes of division can't go on forever in real objects, although I recognize that objects are finite. Hence the paradox: "the infinite" WITHIN "the finite". Infinite content
I recognize that our thoughts are imperfect in this regard. But I think they are interesting because they lead to either Parmenides's speculations or to the ever-changing "fire" of Heraclitus. All three thinkers spoke of very ancient ideas, and Zeno leads to one of the other two
If matter does not perfectly conform to geometry, then this alone is the answer to infinite divisibility problem. I am more likely to accept a contradiction than accept that objects only exist when perceived, as others have said on this thread. Anyway, maybe I should read more Wittgenstein
Reply to aletheist I've been scanning and reading Bell's book that you kindly linked. Although I've been a math person for many years I've been concerned only with certain areas of classical complex analysis and a few abstract vector spaces. So, when I read that in smooth infinitesimal analysis (SIA) all functions from the reals to the reals are continuous I immediately think, what of a step function, like Heaviside function, how can that be interpreted as continuous? However, I gather that in accordance with the axioms of SIA such functions may not be considered in the first place, as they are not defined on the (augmented) reals. In other words, all functions defined in the system are continuous by definition. Am I correct? If so, then the claim that all functions from the reals to the reals are continuous is misleading. But perhaps I'm not interpreting things properly. :chin:
I like the infinitesimal line segment approach to continuity. It's how I program many of the functions I graph on the computer, although I'm using short real line segments.
Perhaps you and/or fishfry would comment on these ideas and elaborate on them to make SIA clearer. Sometimes in math very arcane subjects arise from elementary observations. Like reading a research paper in which simple motivation is left out, replaced by a series of odd looking lemmas leading to the proof of a theorem. :cool:
Consequently, classical logic is strictly applicable only where "a recognized universe [of discourse] is ... real (so that what is true or false of it is independent of any judgment of man or men, unless it be that of the creator of the universe, in case this is fictive)."
This is strange. Do you understand it? Because I do not. Try reading it closely and see if it doesn't begin to seem to you that the writer is confused about his subject.
It makes perfect sense to me. Again, the basic definition of real is being such as it is regardless of what anyone thinks about it. If we are talking about a fictional universe, then what is true or false of it depends entirely on what its creator decides about it, but not on what anyone else thinks about it. In Shakespeare's "Hamlet," the title character is the prince of Denmark and kills Claudius because Shakespeare says so; but no one can now truthfully claim that within the universe of that play, Hamlet is the king of Spain and spares Claudius. That is why there are objectively right and wrong answers on tests that students of English literature have to take after reading it.
And I see "logic requires us." Logic does no such thing, nor can. Hmm.
This strikes me as merely shorthand for your own characterization of logic as a game with certain rules. In the case of classical logic, one of those rules is excluded middle--every constituent of the universe of discourse must be treated as "individual (so that any assertion is either true or false of it)," which "amounts to saying that the universe [of discourse] has a perfect reality."
In other words, all functions defined in the system are continuous by definition. Am I correct?
I believe so, as this seems to be simply what it means for a world to be smooth. As Bell says on p. 276, "If we think of a smooth world as a model of the natural world, then the Principle of Microstraightness guarantees not just the Principle of Continuity--that natural processes occur continuously, but also the Principle of Microuniformity, namely, the assertion that any such process may be considered as taking place at a constant rate over any sufficiently small period of time." For me this is reminiscent of the following passage.
C. S. Peirce:Accepting the common-sense notion [of time], then, I say that it conflicts with that to suppose that there is ever any discontinuity in change. That is to say, between any two instantaneous states there must be a lapse of time during which the change is continuous, not merely in that false continuity which the calculus recognizes but in a much stricter sense. Not only must any given instantaneous value, s, implied in the change be itself either absolutely unchanging or else always changing continuously, but also, denoting an instant of time by t, so likewise must, in the language of the calculus, ds/dt, d^2s/dt^2, d^3s/dt^3, and so on endlessly, be, each and all of them, either absolutely unchanging or always changing continuously.
A step function obviously has a discontinuity that violates this requirement. Of course, the "false continuity which the calculus recognizes" is that of the real numbers, which Peirce elsewhere calls a "pseudo-continuum."
How can this be when the "true or false of it" is exactly dependent on people and not otherwise?
I do not understand the question. What is true or false of a fictional universe is only dependent on what its creator thinks about it. It is such as it is regardless of what anyone else thinks about it. Quoting tim wood
How do you characterize classical logic?
It codifies how we can properly draw necessary conclusions about states of things that are definite, thus conforming to non-contradiction; individual, thus conforming to excluded middle; and real, in the sense that they are such as they are regardless of what we think about them.
Indeed, if it's a fictional universe, then no proposition about it is true - except the proposition that it is a fictional universe.
No proposition about it is true of the real universe, but certain propositions about that fictional universe are true within that fictional universe. The proposition "Hamlet was the prince of Denmark" is false in the real universe, but true in the fictional universe of Shakespeare's play.
But you have evaded the point. Truth and falsity are assigned to propositions. If no propositions, or, if no one to assign truth or falsity, then no truth or falsity. You need the assigner.
I did not realize that this was the point, and in any case I disagree. Truth and falsity are not assigned to propositions, they are properties of propositions. The proposition "the earth revolves around the sun" was true before there were any humans around to express that proposition or to assign truth to it.
Nobody has mentioned the Stadium paradox. If the two moving columns are made of 3 discrete parts each, and pass another other column from opposite sides in a discrete second, is the relative time between the moving columns even a thing? Wouldn't the relative time be less than a pure instant then? How does modern physics theory on time and motion deal with this? Maybe I should start a new thread. I've read that "A History of Greek Mathematics" by Heath discussrd this neglected paradox of Zeno in Chapter VIII. I personally haven't thought of it in awhile
The proposition that the sun - and everything else - revolved around the earth was a proposition propounded by folks who believed and had proofs of it. For them it was a true proposition. But just one not altogether in accord with the facts.
Truth or falsity is not a matter of human belief. A proposition that is "not altogether in accord with the facts" is not a true proposition, no matter how many people hold it to be true. A proposition about the world is true if and only if it denotes a reality as its object and signifies a fact as its interpretant.
Reply to tim wood
You seem to be conflating knowledge with truth. The standard philosophical definition of knowledge is justified true belief. I can believe a proposition, and even be justified in believing it, but that is not what makes it true.
Whether I know that something is real is irrelevant. Again, the real is that which is such as it is regardless of what anyone (including me) thinks about it.
It doesn't. Yours just a convenient fiction - a non-truth.
So, you believe that the earth does not really revolve around the sun? Like all beliefs, that one is also fallible. Since they are contradictory, one of us affirms a true proposition and the other affirms a false proposition; but which is true and which is false does not depend on what either of us (or anyone else) thinks about the matter.
Let's start with this: does the earth revolve around the sun?
So many posts on this thread. Here's a question for those who are entranced with the notion math should begin with continua: From the OP is there a continuum of discussion relating Gabriel's Horn with the Earth revolving around the sun? :chin:
Comments (496)
Quite. I must have. Paint however is not mathematics, it consists of molecules of a finite and measurable size, or else it is no longer paint. Though perhaps "paint" is implied to be simply the smallest unit of measure available, ie. an atom or the smallest known subatomic particle being a quark (which is still measurable). Or does it transcend even this? Basically, how can something be infinitely long if it's slowly decreasing from a measurable state or point? Writing off the world we live in for numbers that could in theory go on forever.. it's still kinda confusing lol. Almost seems kinda like arguing over a glass being half full or half empty. Which of course is actually simple. If it was an empty glass filled up, it is half full. If it was a full glass, with half removed, it's half empty. Unfortunately, this does seem to boggle the mind. Hopefully not by design.
I never liked math. Perhaps someone who does can shine some light on this for the both of us? Basically, what is the largest number that can be reached, and is it any closer to infinity than 1? What is the smallest unit of measurement to reach infinity? What is the smallest fraction that can be reached without there being nothing left, etc. The universe, at least the world we live in, seems to disprove the existence of 'infinity' beyond "oh look a solid figure".
There are other examples, like the Cantor set, which is dispersed throughout the interval (0,1) - like being completely dissolved in a fluid - but nevertheless has no volume whatsoever.
Physical volume and mathematical volume don't have to correspond!
No that's not true. Divide the infinite surface area into sections 1 unit long, as for example the positive real number line is partitioned by the integers 1, 2, 3, 4, ...
Between 1 and 2 you use 1/2 gallon of paint. Between 2 and 3 you use 1/4 gallon. Between 3 and 4 you use 1/8 gallon, and so forth. You will then cover the entire infinite surface of the cone with only one gallon of paint. Of course this is a-physical, since the thickness of the paint would soon be far less than the width of an atom. But mathematically you can indeed cover an infinite surface area with a finite amount of paint.
I didn't see the video, but it was referenced on Reddit this morning with the same misconception, so I wonder if Numberphile perhaps confused people on this point.
ps -- It's perhaps easier to see this in two dimensions. Imagine the graph of y = 1/x from 1 to infinity. We know from calculus that the area under the curve is infinite. But for any positive integer n, the area under the curve on the interval [n, n+1] is finite. So you just cover each such interval with a finite amount of paint according to some convergent infinite series such as 1/2 + 1/4 + 1/8 + ... = 1.
You painted the internal surface of the horn, but not the external one, which is bigger, even if just infinitesimally. How is this accounted for?
The solution lies in the meaning that we silently assume for the word 'paint', which is to cover an area with a layer of liquid paint with a constant, nonzero thickness. We cannot 'paint' the horn in that sense because the volume required would be the area (infinite) multiplied by the thickness (nonzero), which means an infinite volume.
In the filled horn, the internal surface is indeed covered by paint. The thickness of the covering layer at a particular place equals the radius of the horn at that place. Since the radius decreases towards zero, the thickness of the layer just keeps getting thinner. There is no nonzero thickness of layer with which we can say that filling the horn has 'painted' it in the above-defined sense.
This solution does not rely on any physical limits such as Planck lengths or diameters of paint molecules. It simply lies in the definition of 'paint' requiring a constant, nonzero thickness of layer. If we remove that requirement then we can 'paint' any infinite two-dimensional surface (including painting the horn externally, by the way*) with any nonzero volume of paint, however small. We might have to make some technical restrictions such as only allowing surfaces that can be smoothly embedded in 3D Euclidean space. But those restrictions would only interest mathematicians as they would not exclude any surfaces non-mathematicians might imagine.
The mathematical principle behind this is that the function f(x) = 1/x has no convergent integral from 0 to infinity whereas the function g(x) = 1/x^2 does.
Or, avoiding calculus, the sum 1/1 + 1/2 + 1/3 + ... diverges to infinity whereas the sum 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/ 36 + ....converges to a value around 1.645.
* As fishfry points out, the horn has no distinction between its 'internal' and 'external' surface. But we can meaningfully define 'painting the horn internally (externally)' as follows. First define the inside I of the horn as the set of points x such that the shortest line segment connecting x to the horn's axis does not intersect the horn's surface S. The outside O of the horn is all of 3-space excluding the inside and the surface.
Then an internal (external) painting of S is a subset A of Euclidean 3-space (which we think of as being 'full of paint'), such that, for every point P on S, there exists an open neighbourhood N(P) whose intersection with I (O) lies entirely within A.
Under this definition, which we note specifies no minimum thickness of paint, we can make both an internal and an external 'painting' of S using a finite volume of paint.
Reminds me of this: https://youtu.be/5LWfXhggC70
It's just semantic trickery of the mind in the end that creates these apparent paradoxes.
They are not very interesting (in my opinion).
Correct, but the paint can't be uniform since it must be spread over an infinite surface area. The thickness of the paint has to decrease as some convergent infinite series: 1/2 gallon for the area between x = 1 and x = 2, 1/2 gallon for the next chunk, and so forth. So nothing you said is in conflict with the story, which is simply that 1/x has an infinite area but its solid of revolution has finite volume.
Because if the thickness is constant its volume must be infinite. Think of the infinitely long real line and a rectangle above it with height 1/10. Width x height is infinite. Say the height is 1/million. Same thing. 1/zillion. Same thing. As long as the height is nonzero and uniform, the total area across an infinite surface must be infinite.
Quoting tim wood
Nothing to do with the previous sentence. And the whole idea of painting your living room is a red herring, since this is a mathematical and not a physical situation. It's like people complaining about Hilbert's hotel that there can't be any such hotel, or how would the maid make up all those beds, and so forth. The attempt to put the puzzle in "practical" terms ends up confusing the issue. But the point still stands that if the surface to be painted is infinite and the paint layer is uniform and of positive thickness, no matter how small, the volume of paint must be infinite.
Quoting tim wood
Since that's true, that's a hint that a cardinality argument will be of no use here.
Quoting tim wood
There's a big problem, which is that there is no such thing as the "inner or outer surface" of a line. For example, what is the "upper surface" of the real line? There clearly isn't one because if the surface is any positive height above or below the line, it's not the "surface," and if it's directly on the line, it's the line.
Same argument with a circle. There is no inner and outer surface of a circle. Do you follow this point? It's crucial, since the "inner and outer surface" of a two-dimensional surface in 3-space is being bandied about in this thread and there is no such thing.
What are the inner and outer surfaces of a sphere? Can you name a point on the inner surface? No. Because if the point is zero distance from the sphere, it's on the sphere. And if it's a positive distance inside the sphere, it's not the "inner surface" because there are many points strictly between that point and the sphere.
In fact the "surface of a sphere" can only be the sphere itself. Like FDR said! The only surface of the sphere is sphere itself. Ok nevermind.
Quoting tim wood
Bearing in mind that there is no such thing as the inner or outer surfaces of the horn, which is a tw- dimensional object living in 3-space, I'll play along. But remember the premise is false already.
Quoting tim wood
Ok. But think of a circle in the plane. Can you paint it's "inner" or "outer" surfaces? No, because there's no such thing. If you paint the interior of a circle you either include the points on the circle or you don't. If you do, you've painted the circle itself. But if you don't, then there's always a space between your paint and the circle into which you could have put more paint; that is, gotten a little closer to the circile.
I hope you see this, if not please say so, because this point is essential. There are no inner or outer surfaces to a 1-D object living in 2-D, or a 2-D object living in 3-D.
Quoting tim wood
Why? The cardinality of any circle concentric to a given circle is the same. Ancient proof, just draw rays from the center, there's your bijection.
Quoting tim wood
This doesn't make sense (because there are no inner and outer surfaces) and doesn't follow logically even if I granted you that there are such things. It's your own cardinality argument. Any continguous chunk of 3-space has the same cardinality as the unit interval on the line.
Quoting tim wood
There are no inner or outer surfaces. But you haven't got an argument here even if there were. After all the interval []0,1] has the same cardinality as [0,2] which is twice as long. You already made that point earlier. I don't see that you've made an argument.
Quoting tim wood
At least half a dozen so far. But more to the point you haven't made an argument, only claims that you yourself refuted at the beginning by noting that every contiguous subset of 3-space has the same cardinality as the real line or as a finite segment of the real line.
Quoting tim wood
Which of the many errors I pointed out do you disagree with? Starting from the false concept of inner and outer surfaces of 2-D shapes living in 3-space? Just think a sphere in 3-space or (easier) a circle in the plane. Name a point on the "inner" or "outer" surface? You can't. If you pick a point on the circle that's on the circle. But if the point is any positive distance whatsoever off the circle, it's not the inner or outer surface because there are points strictly between your point and the circle.
Quoting tim wood
This I didn't understand. But imagine filling the inside of a circle in the plane. If you tell me you're including the circle in your painted region, then the points on the circle are painted. But if not, then there's some positive distance between painted region and the circle, so that's not the "inner or outer" surface after all.
In the end perhaps casting this problem in terms of paint causes more problems than it solves. But just saying, "1/x is square-integrable but not integrable" (a point made earlier by @andrewk) doesn't have quite the same ring to it.
Yes. It's a finite volume with infinite surface area. It's a veridical paradox: "A veridical paradox produces a result that appears absurd, but is demonstrated to be true nonetheless." In other words it's a true fact (provable in freshman calculus) that's so counterintuitive it seems impossible. But there is no actual paradox. There's no statement that's both true and false. Math is filled with these kinds of things.
https://en.wikipedia.org/wiki/Paradox
Quoting tim wood
Whatever that means. It's a shape with finite volume and infinite surface area. That's all you can say. Yes it's "fillable with paint" if you want to view it that way. But if you push too hard on that visualization it gets confusing.
Quoting tim wood
I agree with that, about actual paint cans. But I don't agree with that about Gabriel's horn, because we're not filling anything with paint. And there is no inner surface. I went at this in detail in my previous post. You understand that for example a circle in the plane has no "inner surface." The horn is a two-dimensional surface in 3-space. It's not like a paint can. That's another source of bad intuition. A paint can has thickness, with an inner and outer surface. But the horn has no thickness and no inner or outer surfaces.
Quoting tim wood
Well, to the extent that I accept that, and I sort of do, it's a veridical paradox. We have a finite volume that has infinite surface area, and it's a seeming common-sense paradox without being an actual logical contradiction.
Quoting tim wood
I didn't actually watch this particular video, I've seen this example in the past. But I disagree with that statement because we could paint the outside of the horn by using pi/2 gallons of paint on the segment between 1 and 2; and pi/4 gallons between 2 and 3; and pi/8 gallons between 3 and 4; and so forth. We'd use pi gallons of paint to cover the entire infinite surface area. Of course we can't paint it uniformly, unless we imagine paint of zero thickness, but then we can't sensibly have any meaningful volume of paint at all.
Quoting tim wood
I agree with that. I saw this example years ago and it hasn't lost its force. It's a real puzzler. The best you can say is that it's a veridical paradox. There is no actual logical contradiction; only a violation of common sense. But the math is clear. The horn has finite volume and infinite surface area.
Quoting tim wood
I don't know the answer. It's a veridical paradox. How can an infinite surface area enclose a finite volume, or a finite volume have an infinite surface area? I don't know. It just does. I don't suppose that's satisfactory. That's why Numberphile got everyone talking about it. For what it's worth, Wiki has my solution and nothing better:
https://en.wikipedia.org/wiki/Gabriel%27s_Horn
If you didn't find my explanation compelling you won't be satisfied by Wiki's identical explanation (except by virtue of Wiki's authority). And as I say, nobody on Wikipedia could come up with a better explanation. It's just one of those things in math that, as von Neumann said, you don't understand; you just get used to it.
ps -- Here's a tl;dr:
* It's just one of those things. It's counterintuitive but not a true paradox.
* The horn is two-dimensional whereas a paint can has thickness. That's throwing off our intuition. Paint cans have an inside and outside surface. The horn doesn't.
Volume V of a cylinder = pi * r^2 * h
Surface area A of a cylinder = 2 * pi * r * h
for a cylinder with radius r and height h
Ratio of A to V = (2 * pi * r * h)/(pi * r^2 * h) = 2/r
As r approaches 0, V too approaches 0 but, oddly, A doesn't.
If you would have listened to me in those other threads, where I explained the deficiencies in mathematical axioms, especially those which make what is mathematically indefinite into some thing definite, like the closure at the end of the infinitely long horn, you would have no problem with this issue.
It's quite obvious that the contradiction is in "infinitely long horn". If you pour paint in the top, and it is infinitely long, the paint never reaches the bottom. But if you assume the horn has a closure, a bottom, just like you might assume that .999... is 1, then the horn gets filled.
The film clip just demonstrates the inconsistencies in how "infinite" is dealt with by mathematical axioms. In one group of axioms, "infinite" is allowed to be a real mathematical object, so the infinite sutface requires an infinite amount of paint. But in cases like .999... and the volume of the horn, the infinite is rounded off and given closure.
Bullpucky. There is no "closure at the bottom of the horn." You just make stuff up and claim mathematicians said it when they didn't. That's called a strawman argument.
Quoting Metaphysician Undercover
Bullpucky bullpucky bullpucky. YOU said that, not any mathematician, ever. As always you take your own mathematical ignorance and project it onto mathematics itself. You wield your ignorance like a weapon.
ps -- Let me not be so ill mannered. Perhaps you could explain to me your version of calculus in which the area of 1/x from 1 to infinity isn't infinite, and the area under 1/x^2 from 1 to infinity isn't finite. If you could elucidate your version of calculus then I'd be enlightened. I seek your wisdom. Or perhaps you reject calculus entirely. I'd like to hear your perspective on this.
Do you happen to understand that there is no "closure at the bottom" of the cone? That this is NOT anything that any mathematician says or thinks? That is is entirely something you made up and then claim to mock? A strawman, as they say. Do you apprehend this point?
One divided by infinity is not zero, it is indefinite. If you assume that one divided by infinity equals zero, you assume that the value for y reaches zero, therefore closure. It's very clearly stated in the YouTube video, he says we're taking the value of y to zero. However, this clearly contradicts the premise that the horn continues infinitely. The real issue is that integrals are approximations, and infinity has no place in an approximation. So that method of integration is simply not applicable to an infinitely long cylinder.
Absolutely correct. Also absolutely irrelevant, since nothing in this problem involves dividing one by infinity. I'm afraid this looks like yet another case of your misunderstanding the math, making up a story about what the math is saying, then arguing with your own story. A classic strawman argument.
But you know, you did say something mathematically correct, that one divided by infinity is not defined. I give you credit for that.
Quoting Metaphysician Undercover
Which it doesn't. So if we assume a falsehood we can prove anything by Ex falso quodlibet". If 2 + 2 = 5 then I am the Pope.
Quoting Metaphysician Undercover
and I am the Pope, since your antecedent is false. Go in peace and sin no more. You can cos as much as you like.
Quoting Metaphysician Undercover
Likewise.
The cone or horn is NEVER closed. Pick any point on the real line and the cone is still open at the bottom. There is no "point at infinity" in this problem nor is there one in the real numbers. Again you misunderstand the math, make up a story, then argue against your own story. Your strawman has by now had all the straw beaten right out of it.
Quoting Metaphysician Undercover
Ok to be fair, as I've mentioned, I've seen Gabriel's horn before but didn't watch the video, so I don't know if perhaps he said something misleading. But it's true that as x gets arbitrarily large, y gets arbitrarily close to zero. That's what's meant by "going to zero." It's a technical phrase meaning that as x increases without bound, y gets as close as you like to zero. But x never "becomes infinite" nor does y ever become zero. The mathematical phrasing is a clever and subtle way of talking about these things WITHOUT saying that x becomes infinite or that y becomes zero. It's your mathematical ignorance of this terminology that's leading you into error. And since you repeatedly do this, when Wikipedia and other online sources could easily explain these things to you, I must assume at some point you choose not to learn the math, but rather to flail at strawmen of your own creation. I don't mean to sound uncharitable but if you have a better explanation I'm open to hearing it.
The tl;dr here is that as x increases without bound, but remains at all times finite; y gets as close as you like -- "arbitrarily close" as they say -- to zero. x does NOT "become infinite" nor does y ever become zero. That's the math. It's well-known. They teach it to college freshman. These days they even teach it to high school students. Not that it's taught well at that level or that anyone understands it. I'll stipulate that nobody comes out of calculus class with a clear understanding of the logical fine points. But it's all on Wiki.
Quoting Metaphysician Undercover
Since your beliefs are false, you haven't got a contradiction; only confusion caused by your lack of mathematical knowledge. But now that I've explained it to you, you can no longer claim to be ignorant.
Quoting Metaphysician Undercover
Integrals are exact. The Riemann sums that define them are approximations, but the integrals are the limits of the Riemann sums and they are perfectly exact. The volume of the horn is exactly pi and the surface area is "infinite," which has a technical meaning: It's greater than any real number you could name. It's not the metaphysical infinite. I think that's a point of confusion in these conversations.
Integrals are perfectly exact, that's the takeaway.
Quoting Metaphysician Undercover
Well they're not approximations. And secondly, "infinity" is not a magic thing that you should allow to cloud your mind. Infinity in this context is nothing more than a shorthand for "increases without bound." We say "x goes to infinity" meaning that x increases without bound, but is at every point finite. Mathematical infinity is not the metaphysical infinity. Perhaps that needs to be said more often. It's more of a technical gadget or shorthand for a quantity that increases without bound, or is larger than any finite number.
Quoting Metaphysician Undercover
Of course it is. It's freshman calculus. It's been working well since the days of Archimedes, formalized into a systematic scientific tool in the late 17th century, and put on a logically rigorous footing since the late 19th century.
It's not a matter of what is "meant by 'going to zero'", it's a question of what value is given in the calculation. Look at 6:25 in the video where he's doing the calculation, plugging the values into the formula. He says "well one over infinity that's zero, so you get nothing from that".
Quoting fishfry
Clearly you are using a different calculation than the one in the video then. If you know of a method to figure out the volume of that horn, which avoids rounding off the infinitely small dimeter to zero, then maybe you should present it for us.
Nonsense :roll:
Quoting tim wood
Quoting Outlander
Probably best, then, to avoid topics like this one.
Try Wiki.
https://en.wikipedia.org/wiki/Gabriel%27s_Horn#Mathematical_definition
I can't speak for Numberphile, which I generally don't watch because the guy annoys me. Perhaps they were trying to simplify the fact that [math]\displaystyle \lim_{x \to \infty} \frac{1}{x} = 0[/math] and you were confused by their simplification. In which case it's on them and not on you. But now that I've explained it to you, it's up to you. Truly if they said that on Numberphile they shouldn't have exactly for this reason. It confuses people.
When we see the expression 1/infinity it's a shorthand for the limit and people generally know that. But if the video is intended (as it apparently is) for people who don't know that, they shouldn't have done it.
ps -- Ok I watched that section of the original video. That's a shorthand they do in integration problems, formally you'd replace it with a limit. I guess now I'd be in a position of trying to defend calculus pedagogy, that's hopeless. In general 1/infinity isn't defined but when it comes up in problems like this you can take it to be zero. I can't argue with you that there's a bit of flimflam to the whole enterprise. I can only say that the procedure can be made rigorous, but at the level of calculus problems, it rarely is. I don't expect that to be a satisfying answer, it's not to me either.
pps -- For integration problems we can consider ourselves working in the extended real numbers. These are the standard reals with special symbols [math]\pm \infty[/math].
One of the rules for these symbols is that if [math]a[/math] is a positive real number, the symbol [math]\frac{a}{\infty} = 0[/math].
That's actually the formally correct answer to your question and I probably should have thought of it earlier. The extended reals serve as a shorthand so that we don't have to use cumbersome limits to talk about expressions involving infinity.
I think the real issue is that it's cumbersome to talk about limits when the subject is infinite, because it's contradictory. Is the horn closed (limited), or is it infinite (unlimited). Clearly the premise is that it is unlimited, infinite, and any mathematical axioms which deal with it by imposing a limit, are not truthfully adhering to the premise. And that's why the appearance of a paradox arises.
No it's not. We have a logically rigorous theory of the calculus used in the example
Quoting Metaphysician Undercover
It's infinite. But let me ask you this. Are you familiar with the graph of the function [math]y = \frac{1}{x}[/math]? Isn't it infinite on the right? You can go out as far as you like, right? Did you need me to copy a picture from the web? No matter how far you go, like x = a zillion, there's a point on the graph at y = 1/zillion. Right? It's the cross-section of Gabriel's horn. We can do the integration to see that the area is infinite. And if you don't like calculus, there's a simple visual demonstration that I could provide.
Is that what you're objecting to? That the area under 1/x from 1 to infinity is infinite? Or what mathematical fact are you objecting to?
Quoting Metaphysician Undercover
You're equivocating the word limit, as in "limited," versus the mathematical theory of limits to infinity. A cheap rhetorical trick. How can you accept confusing yourself like this? Surely you know better. Or you could just ask.
Quoting Metaphysician Undercover
The apparent paradox arises because 1/x has an infinite integral and 1/x^2 has a finite one.
I'm objecting to the method employed by the person in the YouTube clip, which replaces the stated infinite limit, (approaches zero) with zero, as I referenced above. By that method, the equation for the volume of the horn resolves to pi, as stated in the op. If you have a method to figure out the volume of the horn without that substitution, then you might present it. If not, then we probably don't disagree, and we're just wasting our time talking past each other.
I suggest that the proper representation is that the volume is necessarily indefinite, rather than finite, and there is no paradox. This means that the amount of paint required to fill the horn cannot be determined. Therefore no act of pouring a determined amount of paint into the horn will fill it
Check my reference above, 6:25 in the video, where one over infinity is taken to be zero. Otherwise you do not get pi.
If I'm not mistaken, the part you object to is when he's proving that the cross-sectional area is infinite; that is, the area under 1/x from 1 to infinity. I could be wrong, I only looked at a couple of seconds of the vid. Either way though, I already explained that he's implicitly using the extended real number system as a shorthand for the cumbersome use of limits.
Quoting Metaphysician Undercover
That was already clear two years ago. But I'm surprised to see you still complaining about something I already explained to you -- that we're working in the extended reals -- and that you're rejecting freshman calculus. I must admit you're making progress ... from being confused about 2 + 2 = 4 to being confused about integrating 1/x.
Quoting Metaphysician Undercover
Nonsense.
Quoting Metaphysician Undercover
There is already no paradox, only a veridical paradox.
Quoting Metaphysician Undercover
So you don't agree with the determination of the volumes of solids of revolution. Ok. Whatever. An intellectual nihilist. Throw out all the engineering, all the physics, all the physical science simply because you aren't willing to take the time to understand it.
Quoting Metaphysician Undercover
Yeah yeah.
ps -- You know, if you said, "Modern calculus does get the right answers and it's useful for physics and engineering; but the mystery of the ultimate nature of infinitesimal quantities, whether mathematical or physical, is not satisfactorily addressed by the formalism," that would be an intelligent criticism.
But to deny the mathematical result of computing the volume of the solid of revolution ... that's just ignorance for its own sake.
:lol: Why?
The part I am bringing your attention to (and I'm not objecting to it, I'm bringing your attention to it, as relevant to the so-called paradox), is where he determines the limits to the radius of the horn. The radius is given as represented by y, which is equal to 1/x. Then when he plugs the values into the equation, 1/x becomes 1/infinity, which he says is zero.
Sorry timwood because right now I feel like a drunk driver being asked to conduct a sobriety test on himself.
As a civil libertarian I am always conflicted. On the one hand, sobriety checkpoints are unconstitutional as they are a search without probable cause. On the other hand as a driver I'm perfectly happy to get some drunks off the road.
[math]\frac{1}{\infty} = 0[/math] in the extended real number system, which is always the implicit domain of integration problems. The Numberphile guy didn't mention it since it's taken for granted: either glossed over, in the case of freshman calculus; or explicitly formalized, in the case of a more rigorous class in real analysis. Either way it's perfectly rigorous. We could live without it by substituting the phrase, "increases without bound" instead of infinity, and plugging in limits as needed. But that's far more messy and confusing than simply defining the extended real numbers as a notational shorthand.
https://en.wikipedia.org/wiki/Extended_real_number_line
And to be fair, and to hold you to your own words, you ARE objecting, because you have denied that the volume of the horn is pi, when in fact it is exactly pi.
Get to the point if you don't mind me saying. Either tell me where I'm wrong or stop wasting your time. :smile:
We now return you to your normal programming.
:ok: :up: By the way, what's "normal programming"? Do you have one yourself? And also, you haven't gotten round to pointing out the error, if there's one, in my argument. Please focus on the issue.
Whatever my vatkeepers have scheduled for me today.
Quoting TheMadFool
I scrolled back and did not find the argument you're referring to, can you please repeat it?
So, as I said this implies a zero radius and therefore closure of the horn. That's the reason for the appearance of a paradox. To figure the volume of the horn requires that zero is taken as the limit, rather than the unlimited "infinity".
Quoting fishfry
The volume of the horn is figured to be pi only when the infinitely small radius, as stipulated by the premise, is taken to be zero, as required for calculating the volume.
Do you have a cat? Because someone in control of your account typed:
Quoting TheMadFool
Not so? If it wasn't you, perhaps your cat used your keyboard while you were otherwise occupied.
No, the horn is not closed. There are many online calculus tutorials and classes available that explain the theory of limits.
Quoting Metaphysician Undercover
That's now how limits are defined. What's literally true here is that as [math]a[/math] gets arbitrarily large, the volume of the solid between 1 and [math]a[/math] gets arbitrarily close to [math]\pi[/math]. Since that is the case, we say that the limit is [math]\pi[/math]. That's the definition of a limit.
https://en.wikipedia.org/wiki/Limit_(mathematics)
You mentioned sobriety checkpoints, and I went off-topic for a moment to express my ambivalence about them in my dual societal role as both a civil libertarian and a driver. Like the scene in Full Metal Jacket where Joker is confronted by a Colonel for wearing a peace sign and having "Born to Kill" written on his helmet, and he responds, "The duality of man. The Jungian thing Sir!" That was another little off-topic excursion.
https://rolandscivilwar.wordpress.com/2017/06/24/full-metal-jacket-pvt-jokers-born-to-killpeace-and-the-jungian-duality-of-man/
And now that you mention it ... perhaps that's what Gabriel's horn is ultimately about. A shape that reflects aspects of both the finite and the infinite. I think Jung would approve.
No, the reason for the appearance of a paradox is that the shape has finite volume and infinite area; that those are two completely different kinds of things; and that our intuitions about paint "connect" the two. We just "know" that if we buy a gallon of paint and start painting the walls, we'll run out at some point and cover a certain area of the walls. But the reason that is, as already mentioned by @andrewk early on in this thread (and apparently severely underappreciated), is that painting areas with paint requires some thickness of paint.
You make it sound like the problem is that Gabriel's horn stretches out for an infinite while, but that's actually a red herring. You get the same exact problem with any shape of finite inner volume/infinite area, such as extruding a Koch snowflake and giving it a bottom. The only reason Gabriel's horn is noteworthy is that it's a curiosity with a built in toy homework exercise for people taking a calculus class.
It's quite simple. The radius is y, which is represented at the one limit as y=1/x=0. From someone who adamantly argues that = means "the same as", I don't see that you have anything to argue. The diameter is taken to be 0 at that point in the solution. This means that the horn is closed at that point, represented within that method of figuring out the volume. There's nothing to discuss, it's clear and obvious.
The nature of "a limit" is irrelevant and just a ruse. What matters is the values that are plugged into the equation, the values which bring about the apparent paradoxical conclusion. If y=0 then the radius is zero at that point, which is what is employed in that method. If the premise is that the radius never reaches zero, then the solution does not truthfully adhere to the premise.
Quoting InPitzotl
The shape is only said to have finite volume because of the method employed to determine the volume. As explained above, this method assumes a point where the radius of the shape is zero. Therefore this method contradicts the premise of the problem, which states that the horn continues infinitely without reaching a zero radius.
If we were to properly consider that infinite extension of the horn, then despite the fact that it is an infinitely small circumference, being an infinitely long extension of that circumference gives us an infinitely large volume. That volume could not ever be filled.
Quoting InPitzotl
Andrewk does not provide a solution. The inside of the horn has a non-zero diameter with infinite extension. This means that there is an infinite surface area on the inside of that horn, to be covered with paint, just like there is on the outside.
If the argument is that the thickness of the paint prevents it from going into that tiny channel, then we're just arguing physical properties, which has already been dismissed, as not what is to be discussed. Therefore if we are assuming that the paint can go around the tiny surface on the outside, it can also go around the tiny surface on the inside, because we are not discussing the physical properties which would prevent the thought experiment. If that were the case, we'd just reject the whole idea of an infinitely long horn as ridiculous in the first place.
The questioning of the shape's volume is only said to be problematic because the questioner thinks that the questioner knows what he is talking about.
Quoting Metaphysician Undercover
The method makes no such assumption.
[math]\lim_\limits{x\rightarrow \infty}{f(x)}=L \Rightarrow \forall \epsilon >0, \exists M:x > M \Rightarrow L-\epsilon \leq f(x)\leq L+\epsilon[/math]
In other words:
[math]\lim_\limits{x\rightarrow \infty}{f(x)}=0 \Rightarrow \forall \epsilon >0, \exists M:x > M \Rightarrow -\epsilon \leq f(x)\leq \epsilon[/math]
Limits don't need points at infinity to work.
Quoting Metaphysician Undercover
So? Areas have no volume. The paint analogy tricks you into thinking they at least relate when, in fact, they don't. A cubic foot of paint is roughly 7.5 gallons. A gallon of paint can paint about 400 square feet of wall; thus our cubic foot can paint about 3000 square feet of wall. Such paint has a specific layer width of 1/3000 feet. The intuitive equating of a particular area of paint to a particular volume requires such a specific nonzero layer; in the case of actual paint, 1/3000 feet thick layer.
But that inside Gabriel's horn you refer to is less than 1/3000 units across beyond 3000; so if you fill it, you're filling it with less than a layer's worth. It's less than a thousandths of a layer's worth beyond 3,000,000; less than a billionth beyond 3,000,000,000,000.
Quoting Metaphysician Undercover
The physical properties are implied by the intuitions, so they're imported by a back door; the paradox is always phrased about paint filling versus painting the horn... that's tricking you to use your intuitions of paint to compare a volume ("filling") to an area ("painting"). You don't need a shape extending into the infinite to make this paradox "work"; you just need a finite volume/infinite area, and to be tricked into thinking areas relate to volume. The actual thickness of the surface of Gabriel's horn is 0 units, so that infinite surface area actually doesn't consume any meaningful volume at all. Comparing the infinite surface area to the finite volume is simply a false comparison.
It depends on what you mean by paint.
Quoting tim wood
If filling the inside means the inside is painted, then there's no positive minimal thickness of paint required to paint things; i.e., paint can be 0 thick (quick proof; assume there is such a thickness t... then if you go out 1/t on the horn, and toss the finite part, you're left with a horn whose insides are too thin to paint). At that point you're basically just mapping points to points, and there are plenty enough points in a tiny droplet of paint to map to the infinite surface area of the horn.
If you just think of this as whether or not you can spread a drop of mathematical paint indefinitely thin, then I don't quite see any intuitive conflict to build a paradox from.
I don't see how this is relevant. Are you forgetting that it's infinitely long? It doesn't really matter how many time less than a layers worth of paint you're putting in there, it's infinitely long, so whatever the layer is, it will be multiplied by infinity, and therefore enough paint to make an infinite number of those "less than layers worth" layers on the inside of that horn..
You've got this backwards. I have 400 square feet of wall coated with 1/3000th of a foot of paint. How much paint is that? Well, given that the entire area is covered with a 1/3000th foot layer, we can just multiply 400 by 1/3000 and we get about 1/7.5. Now what's this you were saying about the horn?
Quoting Metaphysician Undercover
...ah, yes. Tiny bit of a problem you have there, though. It doesn't matter how many times less than a layer's worth of paint you've got inside, the horn can only have an inner layer that thick if it's thick enough on the inside. And only a finite portion of the horn so qualifies. The rest of the infinite horn (the infinite portion) is too small to have an inner layer that thick. You can't just multiply your paint's thickness by infinity if it isn't covering the infinite area. So where are you getting this crazy notion that you can multiply your layer's thickness by infinity?
What about the ratio between A (surface area of the cylinder) and V (the volume of the cylinder)?
A = 2 * pi * r * h
V = pi * r^2 * h
A / V = 2 / r ??!!
One way of making sense of this is as below:
2 * pi * r = C = Circumference of a circle with radius r [1 dimensional object]
pi * r^2 = = E = Area of a circle with radius r [2 dimensional object]
A = C * h
V = E * h
As r -> 0, the 2 dimensional object (E) collapses to a point and subsequently, all objects based on E, like the 3 dimensional object V, vanishes.
But as r -> 0, there's still h to deal with; the cylinder becomes a 1 dimensional object, a line with length h to be specific.
The physical properties of the paint being incompatible with an infinite horn, was already rejected as not the subject of this discussion. If we were discussing whether the molecules of paint could fit down inside an infinitely small tube, we might just as well reject the infinitely small tube as a nonsensical proposition in the first place.
See? You don't even know what you're discussing!
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
That is what we're discussing. You made a claim that there's some number, you call it "whatever the layer is", that you multiply by infinity. The problem is, there's only a finite portion of the horn that can fit "whatever the layer is", and there's an infinite portion of the horn too thin to have "whatever the layer is" be the thickness of "whatever the layer is". Because of this, you cannot multiply "whatever the layer is" by infinity and get anything meaningful.
You're the one who keeps dragging physical properties of paint into this. I'm just talking about your wrongness; your "whatever the layer is" that you think you get to multiply by infinity.
We could be having the same argument about your ability to fit a y by y by x rectangular prism in the center of infinite volume (V=y*y*x, with x being "infinitely long"). There's no such prism you can fit in the horn; no matter what your y is, it will only go in so far. The only way you get to have an infinitely long prism in there is if y is 0; and 0*0*infinity is indeterminate.
Obviously, if we can assume that the horn is infinitely long, with an infinitely small diameter, we can also assume that the paint can go infinitely thin. That this is not an issue of the real physical properties of a real horn, or the real physical properties of real paint was determined in the first couple of replies in the thread. Do you agree that if the horn is allowed to go infinitely thin, then the paint must play by the same rules, and be allowed to go infinitely thin as well?
You're kind of mixing two things in here. Imagine a mathematical bag; inside the bag, we'll put all finite numbers. All of them, mind you, but only the finite ones. There are an infinite number of numbers in this bag, but infinity isn't in the bag. So if we admit that the horn is infinitely long, we're not necessarily admitting that there's an infinity-point on the horn (a point where there's an infinitely small diameter).
But if you want to talk about an infinite point on the horn, you're free to do so (it's just not implied by admitting the horn is infinitely long). You can even say the horn is infinitely thin at that point. But that doesn't help you here:
Quoting Metaphysician Undercover
...because now you're talking about an infinitely tiny quantity multiplied by infinity. And that's still not meaningful.
Quoting Metaphysician Undercover
I already discussed that here.
Quoting InPitzotl
In other words you want the paint to follow different rules for the inside of the horn, than for the outside of the horn, allowing a finite volume of paint to cover an infinite surface area on the inside, but not the outside. What's the point, if you can just make up whatever rules you want?
Why can't we just do the same thing with the infinite surface area on the outside of the horn as well? We can just say that any tiny amount of paint can be spread out over the infinite outside surface area, just like you say for the inside.
Wrong. You're only confusing yourself here. I haven't specified any rules for paint at all, much less different rules for the inside and outside. Rather, I've talked about three things:
(A) Intuitive paint, which is based on real life paint (not necessarily physics; a painter can get by just fine if he believes atomic theory is a conspiracy, so long as he follows the rules of intuitive paint whereby he calculates he needs 8 gallon cans to paint 3000 square feet).
(B) tim wood's paint, which he explicitly said wasn't intuitive paint and must meet particular conditions
(C) MU's paint, a strange kind of thing that supposedly lets us multiply some number by infinity and get some meaningful conclusion out of it
(A) is where all of the confusion sets in; it's why we think there's a paradox when there is in fact nothing here. When I use my gallon of paint to paint a 20 foot by 20 foot square, and I think that the thing I painted was an area, I've just tricked myself into thinking volumes (gallons) relate to areas (square feet); namely, that 1 gallon=1/7.5 cubic feet relates to 400 square feet. But that intuitive relation is illusory; the 20x20 swatch is actually 1/3000 foot thick, making that paint layer a volume not an area. Gabriel's horn has infinite surface area, but holds a finite volume. But as I've said repetitively, areas have no volume. "An infinite number of square feet" conveys no meaningful amount of cubic feet.
(B) is some new kind of paint tim wood was proposing; whatever that was, it's something that by filling the horn counts as painting its inside. That's the discussion I linked you to. But that sort of paint necessarily must count a width 0 layer of paint (or if you prefer, infinitely thin) as painting the inside surface. There's no separate rule for the outside; (B) kind of paint is just as good for the outside as the inside, and you don't even need pi paint to paint either... any tiny droplet would do. In fact, any volume of paint would paint any area, even if the volume is small and the area infinite.
Then we have C paint, whereby MU is trying to justify that multiplying some number by infinity means that volumes really do relate to areas, and/or demonstrate that he should not have to understand a conversation before he pretends he's contributing to it.
The volume of the horn is only determined as finite when the infinite radius is rounded off to zero at some determinable length, as is demonstrated by the YouTube video. Otherwise the infinite length of the horn ensures that the volume is infinite.
Wrong. The video used limits (and integrals, which are built off of limits). Limits don't round off to zero.
Here's how a limit works:
[math]\lim_\limits{x \rightarrow \infty}{\frac{1}{x}}=0[/math]
...and it's equal to 0 exactly... not rounded off, because L=0 meets the conditions:
[math]\lim_\limits{x \rightarrow \infty}{\frac{1}{x}}=L \Rightarrow \forall \epsilon > 0, \exists M : x > M \Rightarrow L - \epsilon < \frac{1}{x} < L + \epsilon[/math]
...and that can be shown generically for any ?. For such ?, simply choose [math]M=\frac{1}{\epsilon}[/math].
No other number works for that limit; a billionth doesn't work for example; ?=two billionths betrays it, because for all x's greater than two billion, [math]\frac{1}{x} < \frac{1}{2,000,000,000}[/math] which is more than two billionths away from one billionth.
Likewise, every close to 0 integer doesn't work; only 0 exactly works.
Incidentally, integration and limits were used in that video, so this:
Quoting Metaphysician Undercover
...is just uninformed non-sense.
Sure chief, one over infinity is zero, and that's not a matter of rounding off. Tell me another.
Quoting InPitzotl
Quoting Metaphysician Undercover
At this point, it's just denial, and you're unqualified to continue this discussion with. Hardly surprising, given this is the same exact thing you failed to grasp in the other thread.
LOL! Look who's in denial!
Saying that 1/infinity equals zero is obviously an instance of rounding off. There's nothing wrong with rounding off. We do it all the time with pi, square roots, etc.. That's how we get the job done by rounding off. If we couldn't round off, we couldn't get the job done in many instances. So you shouldn't be embarrassed by it. You should be embarrassed by insisting that it's not an instance of rounding off, when it clearly is, though.
It's right there under your nose and you can't see it. You read:
[math]\lim_\limits{x \rightarrow \infty}{\frac{1}{x}} = 0[/math]
...as "saying 1/infinity equals 0". But that's not what it says, and it's not what it means. I told you what it means, and showed you a link.
Quoting Metaphysician Undercover
Okay, so there's nothing wrong with rounding off. And okay, we do it all of the time. So what? It's cute and all that you're trying to "counsel" me so that I can "cope" with rounding off, but your projection of some imagined psychological trauma is a red herring. Limits still aren't rounding off.
Quoting Metaphysician Undercover
Clearly it's not rounding off, though. Clearly, you just don't understand what a limit is. And that's okay, MU. Not understanding something isn't the end of the world. It's nothing to be embarrassed about; there's a lot of knowledge in the world and not everyone knows everything. There's nothing to be embarrassed about by admitting that you don't know something. But you should be embarrassed by insisting that you understand when, clearly, you don't.
Rounding off implies that there's a stated answer a, and a real answer b, and that a is not b but is "close enough" to it.
But there's no such thing as an [math]L \neq 0[/math] such that:
[math]\forall \epsilon > 0, \exists M : x > M \Rightarrow L - \epsilon < \frac{1}{x} < L + \epsilon[/math]
There is, however, such a thing as an L such that this condition is met; namely, L=0.
So the limit here is met by the value 0 exactly. This is a binary thing; either something works, aka meets the definitive criteria, or it doesn't work, aka it doesn't meet the criteria. 0 falls in the "meets the criteria" camp; it's the real answer b. "0 approximately" falls in the doesn't meet the criteria camp. There's genuinely no rounding off here. There is only an infinite amount of MU confusion.
Quoting InPitzotl
I saw it, and heard it distinctly stated, on the Youtube video at the referenced point, he distinctly says "well one over infinity that's zero, so you get nothing from that". And, you said: your method is the same as the one on the video. You can insist, as fishfry stated, that this is the convention in such procedures, to take one divided by infinity as zero. But that is to round off, so we need to respect the fact that the solution to the question of the volume of the horn, is a rounded off solution.
Quoting InPitzotl
"Infinite" means unlimited. When you apply limits to the unlimited, you are either contradicting or rounding off. Which charge do you prefer? If you will not accept the fact that you are rounding off, then the paradox arises due to the contradiction.
Quoting InPitzotl
Yes, and that describes exactly what is the case with the volume of Gabriel's horn. The "real answer" is that the horn is, stipulated by the stated premise, as infinitely long. And, the volume of an infinitely long container cannot be determined. The "close enough" answer is produced by applying the limits of integration, which works with approximations. The appearance of a paradox arises, because "close enough" fails the task intended by the proposition "infinite". In other words, there is no such thing as "close enough" when we're talking about the infinite.
Quoting InPitzotl
To impose a limit on the infinite is to contradict. So it is not the determined value, 0, which is wrong. That value is correct according to the terms of application, but the judgement that the situation is suited to the application, is wrong. The example stipulates that the horn is of infinite length. Therefore to impose a limit on its length is a mistaken procedure.
In common practice, this is not a problem. We don't often encounter infinitely long things which we want to figure the volume of. But when we encounter an infinitely long thing proposed in a thought experiment, it is of paramount importance that we stay true to the premises, and have one consistent way of interpreting "infinite", or else someone will claim that the results are paradoxical.
The mathematics others have presented ...still complex and fascinating and obviously I haven't watched the YouTube clip...
But you didn't understand it.
Quoting Metaphysician Undercover
Yes, but the method is integration.
Quoting Metaphysician Undercover
But that is not the method; that is just a shortcut. The method is to apply a limit.
Quoting Metaphysician Undercover
Sure. But remember the bag I was talking about earlier? That bag has all of the finite numbers in it, but no infinite numbers. This is the situation here. The bag has real numbers in it; there's no such thing as an infinite real number though. Nevertheless, "unlimited" describes the extent of such numbers on the number line in the positive direction. And that ? symbol when used in the limit is used to represent just that... that's not a number, it's just shorthand for representing the infinite extent of the numbers in my bag.
For example, a lower limit of 1 and an upper limit of 2 refers to "all of the numbers in the bag that are greater than or equal to 1, and less than or equal to 2". By contrast, a lower limit of 1 and an upper limit of ? simply means: "all of the numbers in the bag that are greater than or equal to 1".
Quoting Metaphysician Undercover
Nonsense. You just made that up.
Quoting Metaphysician Undercover
Those things don't follow, there isn't a paradox here in the first place, and anyone trying to play the if-you're-wrong-that-means-I'm-right card should have their philosophy license revoked.
Quoting Metaphysician Undercover
If you don't understand the method, you're unqualified to critique it.
Quoting Metaphysician Undercover
Wrong. The real issue is very simple... areas aren't volumes. "Paint" tricks you into thinking they are. It is very interesting to note that you never actually tried to address this explanation, just as you never commented on the extruded Koch snowflake with a bottom having the same "issue". It's no wonder you're trying to play the if-you're-wrong-that-means-I'm-right card.
Quoting Metaphysician Undercover
That sounds like a confused equivocation. A 1x1x1 cube by definition is 1 cubic units of volume, but it has an infinite number of points. It's not a contradiction to say that it has an unlimited number of points but a limited volume. Gabriel's horn has an unlimited extent into the x axis in the positive direction; that means it is unlimited... in extent... along the x axis. And that's it. It doesn't mean that the horn surrounds an infinite volume, as your equivocation is apparently meant to imply, any more than the fact that the 1x1x1 cube contains an infinite number of points suggests it should have an infinite volume.
Yes, apply a limit to what is stipulated by the premise, as without limit. That is the mistake. Can't you see that it is stipulated that there is no limit to the length of the horn, therefore to apply a limit is to contradict the premise?
Quoting InPitzotl
You cannot put all the finite numbers in a bag, because there is an infinite quantity of them. This example provides nothing of relevance.
Quoting InPitzotl
This is not relevant either. It is stipulated in the Gabriel's horn example, that there is no lower limit. It is stipulated that the horn continues infinitely. That means no limit. It is therefore not a case of having an upper limit and a lower limit, and to represent it as such is a mistake. To impose a lower limit (such as zero) is to contradict the premise of the example. We are not talking about how many numbers there are between two numbers here, we are talking about an unlimited length. To impose a limit on that length, a point where the diameter of the horn reaches zero, is to contradiction the premise of the example.
Quoting InPitzotl
I did addressed this. If the horn can go infinitely thin, then so can the paint. They must play by the same rules.
Quoting InPitzotl
Again, we're not talking about an infinite number of points within a confined space, so that is irrelevant. The horn is infinitely long therefore there is no confined space. If someone were to say to you that there is an infinite extension of the universe, would you think that this implies a confined space? If someone says to you, take this line which continues infinitely, would you think that they were talking about a line segment which extends between two points?
This seems to be where your misunderstanding lies. You want to make this into an issue of a confined, "limited" space, but it is clearly stipulated that the horn is infinitely long, therefore there is no such confined space.
Volume of a cylinder V = pi* r^2 * h where r is the radius and h is height
r approaches zero and h approaches infinity
A = 2 * pi * r * h = 2 * pi * (r approaching zero) * (h approaching infinity)
V = pi * (r approaching zero) * (r approaching zero) * (h approaching infinity)
(r approaching zero) * (h approaching infinity) = 1
So,
A = 2 * pi * 1 = 2 * pi
V = pi * (r approaching zero) * 1 = pi * (r approaching zero) = 0
As you can see, A plateaus to 2 * pi but V becomes 0.
Suppose r=1/n and h=n^2. Then V -> pi. You are not describing Gabriel's Horn.
Quoting TheMadFool
This is mysterious. One should make pronouncements about topics familiar to one.
(Not being a philosopher, this makes me wonder if some of the "sophisticated" philosophical arguments on the forum are any better) :roll:
Take it all with a grain of salt.
Quoting jgill
We can't suppose anything we want. Gabriel's horn begins with the assumption that r = 1 and h extends from 0 to infinity.
I can see that you're equivocating. You're confusing "limit" as a method with "limit" as a point beyond which you don't go.
Quoting Metaphysician Undercover
The phrase "all the finite numbers" is itself such a bag.
Quoting Metaphysician Undercover
Of course it's relevant. The 1/? that you're whining about is the 1/? in the video at 6:20. Right? That 1/? is 1/x with ? substituted in it. That 1/x is the 1/x from [math]\pi \frac{-1}{x}[/math]. And that is from [math]\int_{1}^{\infty}{\pi \frac{1}{x^2}dx}[/math]. And that is a Riemann integral; it's applied over the range where we want to take this volume. For Gabriel's horn, the lower limit here is 1, and there is no upper limit. To simply plug in infinity here as if it's a number is to say something different than what you yourself said in the prior post... that "infinite" simply means unlimited.
According to the rules of the method being applied, this is an improper integral. For improper integrals of this type, the calculation for the infinite portion is performed using a limit, whose definition I gave earlier.
Quoting Metaphysician Undercover
If there's no lower limit, then what's up with this big giant yellow arrow pointing to the lower limit during the segment where Gabriel's horn is defined?:
https://youtu.be/yZOi9HH5ueU?t=56
...not to mention Tom saying straight up, "and what we do first of all, is we're going to chop it here at 1"?
Quoting Metaphysician Undercover
No, you didn't address this, because if this were indeed the case... if you could paint infinitely thin, then you can paint an infinite area with a finite amount of paint. And if you can do that, then there's no paradox. And you can do that, and there is no real paradox. But such infinitely thin paint simply becomes an empty metaphor... it's equivalent to what I was saying here. But you replied to that post saying that it wasn't the "real answer", and as recently as here you were peddling this one:
Quoting Metaphysician Undercover
But your proposed "real answer" doesn't address the extruded Koch snowflake, because that isn't infinitely long. The thing you're proposing isn't the real answer does address the Koch snowflake, because the infinite area on its perimeter is not a volume.
Quoting Metaphysician Undercover
You're reasoning by equivocation; "the extent is infinite, therefore the volume is infinite" simply doesn't follow.
A phrase is a bag? Come on Pitzotl, you're reaching for straws. Get back to the subject.
Quoting InPitzotl
That's not true at all. Your interpretation of "infinite" is dreadful. You cannot paint an infinite area regardless of how much paint you have, because no matter how much painting you do there is always more to be painted. That's the issue with Gabriel's horn. It's infinitely long, so no matter how much paint you pour in the top, it never reaches the bottom.
It doesn't matter what you propose as the volume of the horn, you still cannot fill it with paint . Suppose you conclude it's 3.1 gallons,. You pour that in, but you haven't filled the horn because it hasn't reached the bottom.
Quoting InPitzotl
We're discussing Gabriel's horn not snowflakes. How my answer relates to a snowflake is irrelevant. I don't see why you feel the need to bring up so many irrelevant issues.
Quoting InPitzotl
It seems you do not know the meaning of "equivocation".
Yes.
Quoting Metaphysician Undercover
But the subject is the paradox of Gabriel's horn; it's literally the title of this thread. Gabriel's horn is an object defined using algebraic geometry. Algebraic geometry defines points in a space using coordinates using number lines. Number lines are defined with real numbers.
And that's what this bag is; it's the real numbers.
Ironically, your accusation and your imperative to me to get back to the subject is itself an avoidance of discussing this subject.
Quoting Metaphysician Undercover
These are non-issues. In the paradox that is the subject of this thread, we're concerned primarily with the quantities not the length of the tasks. These are mathematical spaces; in the span of 30 seconds we define the entire infinite horn in an algebraic geometry space... doing infinite things is certainly not a problem here. Your imagined alleged problem has a lot more problems than you're letting on... the 3.1 gallons not going to the bottom is child's play. How are you upturning the horn in a gravitational field, and where do you put the planet? How does that planet manage to exert a field on the top of the horn anyway? But the biggest and most relevant question of all here is... do these really sound like mathematical questions?
Quoting Metaphysician Undercover
We're discussing the presumed paradox of Gabriel's horn. The presumptions that appear to introduce the conflicts is the point of the thread. Those presumptions are based on the fact that the quantity of the surface are "on the outside" is infinite while the quantity of volume "on the inside" is finite... the "outside-ness" of the surface of which is really an irrelevant detail that is part of the intuitive distraction of using paint to compare areas to volume. You are presuming that the paradox is solved by questioning the volume, but you haven't even shown a good reason to doubt the volume much less the flaw in the presumptions leading to the paradox.
Nonsense. This has nothing to do with algebraic geometry. G's Horn is elementary calculus. :roll:
You guys should just let this go and get back to epistemological metaphysics where accuracy is optional.
What do you mean this has nothing to do with algebraic geometry?:
Gabriel's horn is the algebraic variety defined by the polynomial z^2+y^2=(1/x)^2 starting at x=1.
Quoting jgill
Sure, you use calculus to analyze the surface area of and volume surrounded by this object, as they did in the video. But that doesn't preclude the fact that you're studying geometric properties of an algebraic variety.
This is where your mistaken. The paradox assumes a spatial form, created from mathematical principles, and names this form Gabriel's horn. Therefore the subject is this form which is created by mathematics, not the quantities which are used to create it. You need to divorce yourself from the numbers, from the mathematics, look at the form described, directly, and ask how is it to be measured.
This might be why you seem to be having so much difficulty understanding the problem. You want to use the same numbers which create the form, to measure the form. But when infinity is induced, this represents a failing in the capacity of the numbering system, so it is impossible to measure the form produced with the same numbers which produce it. That's what happens with the square root of two, and pi. There is a failure in the numbering system's capacity to measure the spatial form produced because of an inherent incommensurability. So we have a simple solution, we round off. Or, if necessary we can move toward employing more complex numbering systems, real numbers for example.
Quoting InPitzotl
This is your false presumption, which is misleading you, that "the quantity of volume 'on the inside' is finite". If you would approach the problem with an open mind, rather than with what I see as a false presumption, we could probably make better progress in this discussion.
Here's what I propose. I'll justify my presumption, and you justify your presumption. I see that the spatial form which we are talking about, Gabriel's horn, is infinitely long. Therefore it is impossible, in theory, to precisely figure its volume. The volume therefore, in theory, is indefinite, being "infinite", unbounded, just like the extent of the natural numbers is "infinite", indefinite, or unbounded. We can however figure the volume of such forms, for practical purposes, by rounding off.
Will you justify your presumption that the volume is finite?
Quoting InPitzotl
I told you already, Gabriel's horn is a spatial form with an infinite length. That's very good reason to doubt the volume. All you said was "You're reasoning by equivocation; 'the extent is infinite, therefore the volume is infinite'. Clearly there is no equivocation. A spatial form which has an unlimited (infinite) extension in one of its dimensions, will have an unlimited (infinite) volume accordingly
Nothing you described justifies a concern about the length of tasks.
Quoting Metaphysician Undercover
The spatial form is given by Cartesian coordinates with three axes at right angles; that defines a space where the set of all points are (x, y, z) coordinates with x, y, z being reals.
Formatting for brevity.
"You want to use the same numbers which create the form, to measure the form." If it works.
"But when infinity is induced, this represents a failing in the capacity of the numbering system, so it is impossible to measure the form produced with the same numbers which produce it." That does not follow.
"That's what happens with the square root of two, and pi." The square root of two and pi are real numbers. See above. Also, square root of 2 is not infinite; it's between 1.4 and 1.5. Pi is not infinite; it's between 3.1 and 3.2.
Quoting Metaphysician Undercover
It's not a presumption; it's the result of a calculation. The fundamental issue here is that you're critiquing the methods without understanding what they are or why they are employed. Take your critique of Tom at 6:20 in the video for example. There, Tom is calculating the results of an integral. Tom has an improper integral, and this describes the method for its evaluation:
When Tom says: "Well one over infinity, that's zero", that's okay, so long as we know it's a shortcut for:
[math]\hspace{2 em}\lim_\limits{x \rightarrow \infty}{\frac{1}{x} }[/math]
...which is exactly 0, as shown previously by definition of that limit. You have confused this with saying that 1/infinity=0. That's baseless; infinity is not a real number; the domain of the integral is the same domain as the x axis, and infinity isn't even in that domain. The method isn't "plug in infinity", and there's a reason it isn't.
Quoting Metaphysician Undercover
That does not follow.
Quoting Metaphysician Undercover
That was already shown, and you're mischaracterizing the problem. The most fundamental problem here is that you're objecting to the efficacy of these methods without understanding what the methods are being employed or why they are employed. The other big problem is the obvious bias portrayed in objecting to the efficacy of the method before understanding these things.
Quoting Metaphysician Undercover
That does not follow.
Well you distinctly said it was a presumption. "Those presumptions are based on the fact that the quantity of the surface are "on the outside" is infinite while the quantity of volume "on the inside" is finite..."
We've been through all this. Your so-called calculation, "limx??1xlimx??1x...which is exactly zero by definition..." is nothing but a rounding off. See, it's zero by definition, not by calculation.
Quoting InPitzotl
Whereas one can describe the collection of points in 3-space comprising GH with the zeros of
[math]P\left( x,y,z \right)={{z}^{2}}{{x}^{2}}+{{y}^{2}}{{x}^{2}}-1[/math], the paradox of GH does not emanate from that perspective, but from elementary calculus. Why even bring varieties up since it is irrelevant to the issue being discussed, and participants of the thread might well be familiar with the rudiments of calculus, but have little acquaintance with algebraic geometry?
I see that we have a fundamental difference of opinion concerning the logic of spatial areas. I think that it is illogical to believe that a 3d spatial form with an infinite extension in one dimension could have a finite volume. You disagree.
Well ok. What would you say is the volume of the solid of revolution of [math]y = \frac{1}{x}[/math] between 1 and [math]\infty[/math] when the curve is revolved around the x-axis? Here's the theory and formula, if you forgot.
https://en.wikipedia.org/wiki/Solid_of_revolution
If the extension is infinite, the volume cannot be figured. You can only figure the volume by assuming that there is an end, a limit, and this is rounding off. But then you are not figuring the volume of an infinite extension.
LOL. Well then how do you know the area under the curve is infinite then?
Wow, someone who actually understands.
Quoting Ryan O'Connor
Of course it's a fictitious object.
Quoting fishfry
Because it cannot be measured. That's what infinite means.
And the lazier definition of it typically given, such as the one in the video, is that we start with the curve 1/x starting at x=1 and rotate it about the x axis. Then the resulting object we call GH (with the same "starting at x=1" specification). But that is precisely this object. And we're talking about this object, so it's relevant for that reason.
So what are you on about jgill? And I mean that question literally; I have no idea what you're actually objecting to. Incidentally, no, calculus doesn't give us the GH paradox... broken intuitions do. I also find it a bit strange to claim that calculus is used to define the object; rather, it's used to analyze the object (surface area/volume in this case).
soph·ist
/?säf?st/
* a paid teacher of philosophy and rhetoric in ancient Greece, associated in popular thought with moral skepticism and specious reasoning.
* a person who reasons with clever but fallacious arguments.
The surface area of the horn has no limit - as we consider larger and larger horns that area continues to increase, so we can loosely say that it has 'infinite' area. But as MU notes, I think it's more appropriate to say that we cannot measure the area (or I would argue that the horn simply doesn't exist). What's paradoxical is that the volume does have a limit. But that's not to say that the horn has a definite volume. We cannot complete the potentially infinite process associated with the limit any more than we can explicitly list all decimal digits of pi, so we cannot claim that it has a definite volume. The best anyone can do to offer a measure of the volume is to prematurely terminate the potentially infinite process and output a rational approximation of the volume.
...is there? Back to "intuitive paint" based on real paint, 1 cubic foot of paint can paint 3000 square feet of wall. With that in mind, we can construct a Gabriel's horn with units of feet, chopped off at a finite length, such that it holds less than 3.15 cubic feet of paint, but takes over 1000 cubic feet to paint the outside.
Let me clarify this...there's something better than giving a numerical measure....the best we can do is to stick with an algorithm which defines a potentially infinite process. And that's how we define pi.
I don't see the paradox - as long as we're allowed to play with the thickness we can come up any combination of numbers for the finite horns. What's important is that the numbers in your example are all finite.
I don't see the original paradox; a square foot of area has no meaningful volume.
But the same exact questions arise in the finite scenarios. "If it's holding about 3.15 cubic feet of paint, isn't that already painting the inside? If so how come it takes 1000 cubic feet then to paint the outside?"
For example, why compare an imaginary horn with real paint?
If it's a real horn, then there must be a certain limit beyond which the internal space cannot be diminished while still maintaining the coherence of molecular and atomic structure of what constitutes the physicality of anything (a horn).
So, why compare a metaphysical horn with actual paint?
If we postulate paint made up of a 'quantized super fluid with infinite tension/compression (whichever fits) capability' (e.g. the fundamental matter of cosmic space, just an idea), then we can paint infinity.
=> If 'infinite' is compared to 'finite' then obviously, ultimately, there would be incongruence - call it paradox, chaos, whatever...
Math 631 (Algebraic Geometry) (U of Mich):
"Intended Level: Graduate students past the alpha algebra (593/594) courses. Students should either already know or be concurrently taking commutative algebra (Math 614). Students should also know the basic definitions of topology — we won't be using any deep theorems, but we will use topological language all the time. Basic familiarity with smooth manifolds will be very helpful, as much as what we do is the hard version of things that are done more easily in a first course on manifolds. Undergraduate students intending to take this course should speak to me about your background during the first week of classes."
This description speaks for itself. Correct me if I am mistaken, but it appears you have tossed in AG to impress the readers of this thread. If you are indeed a mathematics professor and feel AG is necessary, then I would understand. Are you? I was one for many years and we never had an undergraduate course in AG, although some schools do. GH always came up in a standard calculus course. Tell me where you are coming from and why you found it essential to define GH this way.
By the way, you should now go to the Wikipedia article on GH and inject your considered opinion. It's a nice piece and never mentions AG. You apparently think it should. Again, if you are or were a professional math person and have strong feelings about this I will understand.
Jeez I'm with @jgill here. This is a standard example from freshman calculus. The integral of 1/x from 1 to infinity is infinite and the integral of 1/x^2 is finite. Or 1/x is square integrable but not integrable if you prefer. Algebraic geometry is a little high-powered in this context, it's not needed. Your judgment is off from letting yourself be trolled by @Metaphysician Undercover.
Quoting fishfry
I'm open to suggestions, but all I'm after here is a description of the space and the object. This is related to the conversation. This came up several times:
Quoting Metaphysician Undercover
...referring to 6:20 in the video. So MU thinks the process is to substitute infinity in, as one would do in a proper integral. The point being made here is that in contrast to the proper integral limit, the ? in the improper integral with infinite limit isn't even a coordinate in the space.
This is not really what I'm saying. What I mean is that if the value on the x axis, or y for that matter, is proposed as infinite, then it is wrong to assign a limit, such as zero, to the perpendicular axis. By saying that the one is infinite you say that there is no limit to how small the value of the other can be, and zero is an incorrect representation. In other words it is incorrect to speak of a limit here, and to say that infinity is a limit is nonsense.
What I think is that there is a fundamental incommensurability between two distinct dimensions of space, demonstrated by the irrationality of pi and the square root of two, which indicates that representing spatial forms with perpendicular axis is a sort of misrepresentation. Spatial forms, as we know them, cannot be accurately represented this way.
So if we look at the difference between a straight line and a curved line, we see that one requires two dimensions, fundamentally, while the other defines (or establishes the limits of) one dimension. The curved line, even at an infinitesimally small length requires two dimensions to be mapped by a straight-lined coordinate system. This would mean that even the most simple thing in a spatial reality (a point particle for example), would require a multi-dimensional representation. And, the multi-dimensional representation would get it wrong because of the incommensurability between two distinct dimensions.
Therefore the curved (real) line will never be commensurable with the straight (artificial) line. And, when we map the curved line with a straight-lined coordinate system, the incommensurability shows as infinity. Gabriel's horn is a curved line being mapped by a straight-lined coordinate system, and the incommensurability is evident.
The further question, which comes to my mind, is which is the proper way to represent space. Which way represents how space really is? Is the proposed "real" curvature just an illusion created by deficiencies of observation, and there is no such thing as a true arch, or perfect circle. Or does all of space consist only in multi-dimensional, non-straight relations, and our artificial dimensional straight-lined coordinate systems are incapable of giving precise representations of multidimensional existence? Or is the apparent incompatibility due to something else, which we have not yet grasped? Perhaps we ought not be representing multi-dimensional lines with perfect curves, archs, and circles. Maybe we need to banish this type of object from geometry as not properly representing reality.
But that is really what's going on at 6:20; the ? there is the ? symbol of the upper limit of the integral, and it is a sentinel; a placeholder meaning unlimited. There's a notation here that requires filling in a spot for the lower limit and a spot for the upper limit. "Usually" you would fill that in with something like 1 and 2. To show you're doing this same kind of thing, but there is no upper limit, you put ? there. The 1/x comes from that integral.
Quoting Metaphysician Undercover
You told me that infinite just means unlimited. Try taking that seriously for a moment. Don't say it's infinite, just say it's unlimited.
Quoting Metaphysician Undercover
But there is a limit to how small the value can be; for any real number > 1, 1/x cannot be less than 0, whereas it can be less than any other positive real number. The limit is saying something similar... that the farther out you go, the closer you get to 0 (and that you can get arbitrarily close). The limit in its definitive form can be used to show that this is only true for 0; it is not true to say that the farther out you go, the closer you get to 1 billionth. It is only true to say that the farther out you go, the closer you get to 0 (arbitrarily so).
Quoting Metaphysician Undercover
Again, pi and square root of 2 are real numbers. So the spaces involved have those as coordinates.
Quoting Metaphysician Undercover
From here down you're pontificating about physical space, which is not the space being used here.
Actually I referred to 6:26, when he says one over infinity, that's zero.
Quoting InPitzotl
Let's remove this necessity of a "real number", maybe that's what's misleading you. Is there any limit to how small the value can be? No, that's what's meant by infinitely small, we can conceive that there is always a further value, smaller than any value which we put a number to. That's the same as what's meant when we say that the natural numbers are infinite, only in the inverse direction. We can conceive that there is a further value, larger than any value which we put any number to.
Therefore, what is meant is that there is no limit to how small the value can be, just like in the natural numbers there is no limit to how big the value can be. This, I think, is what's misleading you. You keep thinking that there must be a limit to how small the value can be, but what is indicated by "infinite" is that there is no such limit. If a limit was intended, we'd employ "infinitesimal", which indicates that there is a smallest possible. But "infinite" indicates that there is no limit to how small we can go.
Quoting InPitzotl
What is arbitrary is the choice of "0" here, as the representation of some non-existent limit. There is no limit, the value can keep getting smaller and smaller, always beyond any numerical representation which you might give it, that's what's indicated by "infinite". So there is no point in representing this value as getting close to some imaginary limit, "0". The value keeps changing without ever reaching that proposed limit, so in reality it never gets any closer to that limit. There is always infinite more values to cover before it gets there. The value really never gets any closer to the proposed limit, it's always infinitely far away, so the limit is completely irrelevant. Therefore "the farther out you go, the closer you get" is not an accurate representation at all. because the whole point in saying "infinite" is to say that there is no end, so it's impossible that the end is getting any closer.
You are essentially saying that it takes <3.15 ft3 to paint the finite horn AND it takes >3.15 ft3 to paint the finite horn. If film thickness doesn't explain the apparent contradiction, then I would conclude that your problem definition is invalid. Perhaps you cannot claim that 1 ft3 can only paint 3000 ft2 of wall. I think focusing on physical paint is a distraction.
Quoting InPitzotl
I agree that an area has no volume, so a single drop of mathematical paint could paint any surface of arbitrarily large size. However, we are not justified to claim that it can paint a surface of actually infinite area. That requires a leap of though which we are not in a position to make.
Quoting InPitzotl
Consider the Stern-Brocot tree. If L=1/2, LL=1/3, LLL=1/4, and so on, then L_repeated is a "real number" which it cannot be less than. I suspect you'll argue that L_repeated = 0, which brings us to the classical debate of whether 0.9_repeated=1. I would argue that 0.9_repeated describes a potentially infinite process and is not a number in the same sense that 1 is (dare I say that 0.9_repeated is not a rational number), but that's a debate for another time. In any case, consider this: if a number at the 'bottom' of the Stern-Brocot tree is equivalent to both of its neighbors, are any of the real numbers actually distinct from each other?
Quoting Metaphysician Undercover
Given that y continually approaches 0 as x increases, the limit is 0. We don't need a point at (?,0) for the limit to be equal to 0. I think the problem is that your definition of limit doesn't match the standard definition. With that said, I sympathize with your view and I think your argument would be stronger if you focused on the limit used to calculate the volume of GH. Such a limit is a whole different beast since it converges to pi - an irrational 'number'. Just because the interval corresponding to V can get arbitrarily small as a approaches infinity, it doesn't necessarily mean that V has a definite value. Just as it is impossible to explicitly compute all decimal digits of pi, the best anyone can do here is either (1) provide a small interval for V or better yet (2) leave V in algorithmic form (i.e. pick your favorite formula for pi and don't bother to compute it).
You seem to be imagining a hypothetical number "so big that" 1/x dips below 0. This sounds like speculative fantasy to me. Your reasoning that it dips below 0 could equally be applied to an imagined consequence that it levels off at and stays at 0, or that it rises again. There's basically no meaning to this.
Quoting Ryan O'Connor
Those are loosely based on real numbers. A gallon of paint can paint approximately 400 square feet . A gallon is about 1/7.5 square feet.
But thinking like an engineer, our really long Gabriel's horn is "mostly" "essentially" a negligible sized needle... the first portion that's much bigger is essentially a fixed amount of error. If our paint is 1/3000 feet thick (as the volume to area-covered ratio implies), then most of our paint on the outside is a cylinder with radius 1/3000. We could imagine this as if we're squeezing paint out of a tiny hole like a roll of toothpaste; in that sense, there's no limit to the amount of "paste" we can squeeze out of the tube.
Quoting Ryan O'Connor
Already had that thread with MU.
Most of the area is on the "needle" portion but most of the volume is enclosed by the "horn" portion. I find it odd that you're focusing on the needle portion. The typical area and volume calculations for GH are based on limits as a approaches infinity (not the other way around) - in other words, as the horn gets longer and longer. But sure...
Quoting InPitzotl
You lost me here. Are you saying that the paint on the outside is a cylinder of radius 1/3000 and infinite length and asking why that infinite amount of paint doesn't agree with the finite amount of paint needed to fill needle?
Quoting InPitzotl
Fair enough.
Almost; I'm talking about arbitrarily large but finite amounts of paint. And the only point here is that it's not really surprising this "outside" can be indefinitely long with a layer while the inside is limited.
It seems like you're implying that finite math is equally paradoxical by comparing the infinitely long needle with an infinitely long horn. If you want to say something about finite math you need to talk about a horn of finite length.
I didn't say anything about dipping below zero. Where did you get that idea from? I said it doesn't ever get any closer to zero. There is always an infinitude of values between it and zero, so it's really not ever getting any closer to zero. Zero is off the scale, it's literally not part of the scale, as it is excluded by virtue of being impossible. So there is always an infinity of values between the value of y and zero. Since there is always that infinity of values between any given value of y, and zero, it makes no sense to say that it is getting closer to zero.
Quoting Ryan O'Connor
As I explained y does not in any way approach zero. It is always infinitely far away from zero, no matter what value it has, therefore it never gets any closer to zero, and cannot be said to approach zero. Zero is imposed as an arbitrary limit, on something which, by definition, has no limit. If the line came to an end at a particular value, we could say that value is the limit. But it doesn't, the line continues onward without limit. It doesn't reach zero, yet continues infinitely, so zero is right off the scale, irrelevant as unobtainable.
Consider this example.
First proposition: The natural numbers are infinite, therefore there is no highest number.
Second proposition: 20 is closer to the highest number than 10 is.
Do you see how the second proposition contradicts the first? We have the very same type of contradiction when we say that x can increase infinitely without y ever reaching zero, yet we also say that it is getting closer to zero. Zero has been excluded from the scale as an impossibility, just like the highest natural number. So we can't say that one point is closer to zero than another. How does that make any sense? It's just like saying that one number is closer to the highest number than another when there is no highest number. Here it's the lowest number, we're talking about and that's not zero because there's an infinitude of numbers to go through, making zero impossible, just like the highest number is impossible.
That's not the proper way to speak, to say that one number is closer than another to the highest number. One number is higher than another, but it is not closer to the highest number, because there is no higest number. Likewise, with the value of y, one value is lower, and another higher, but we can't say that one value is closer to the lowest number because there is no lowest number, just like there is no highest natural number. There is just more and more numbers, and zero cannot be posited as the lowest of those numbers, because it is not one of them, as unobtainable, impossible, outside the bounds.
From this:
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
That does not follow. 1 is closer to 0 than 2 is, despite the infinite number of points between 0 and 1.
Quoting Metaphysician Undercover
I assume by off the scale you mean 0 will never be reached. But that's not required for 0 to be the limit.Quoting Metaphysician Undercover
Sure it does; but it's a bit more precise than this. The limit specifies that it's possible to get arbitrarily close to 0. "Arbitrarily" here is used in a strong sense that includes all positive distances at once.
Quoting InPitzotl
I told you, you need to stop thinking about the numbers which make up the coordinate system which produces the shape. They are outside the shape, irrelevant, and insufficient for measuring the shape. There is no zero point to start from in our measurement, nor is there a zero ending point. The value of one dimension is allowed to increase indefinitely such that there is no highest value, and correspondingly, the value of the other dimension is allowed to decrease indefinitely such that there is no lowest value. Zero does not enter the picture. It is excluded, (just like "highest value" is excluded, so is "lowest value", or zero) and therefore cannot be a part of the measurement scheme.
Quoting InPitzotl
See, zero is irrelevant. At any point on the shape, the y value is "arbitrarily" close to zero. So it is false to say that it gets any closer, at any point, because it's always the same, "arbitrarily close". This arbitrariness indicates that zero is completely irrelevant to any valid measurement.
It appears to me, like you want to allow the x value to increase indefinitely, without limit, assuming no highest number, but you will not allow the y value to decrease in the corresponding way. You want to limit the y value's decrease with an imposed zero. This removes the symmetry from the shape, and is a false representation of it.
Your entire response is misguided. The limit is not describing a point in the shape. If it were, it would be an empty concept; the limit would just be a function evaluation. Take:
[math]\hspace{2 em}f(x)= \left\{ \begin{array}{ll} \frac{x^2-1}{x-1} & x \neq 1 \\ 8 & x=1\end{array} \right.[/math]
The value of this piecewise function at f(1) is 8, as explicitly denoted. But:
[math]\hspace{2 em}\lim_\limits{x \rightarrow 1}{f(x)}=2[/math]
Despite explicitly talking about 1, which is explicitly defined by the bottom piece, the limit is about that top piece, which doesn't even have a value at x=1, nor is there an f(x)=2.
Your notion of 'close' that is based on the number of points between A and B can only have value in a number system which is not dense in the reals, such as the integers. For example, since there are 3 integers between 0 and 4 but 6 integers between 0 and 7, we can conclude that so 4 is closer to 0 than 7. If you want to restrict your mathematics to the integers then your notion of 'close' is suitable. However, such a mathematics is far less powerful than orthodox math so I suspect that you'd have a very tough time convincing anyone to adopt your view.
But as I said before, I sympathize with your position. In one breath we say that we use limits to avoid actual infinity but then in the next we say that the limit can be a number inseparably tied to actual infinity (e.g. pi). But I would argue that the resolution to such contradictory thinking is simple: just don't say that the limit is a number with actually infinite digits. Instead, keep the limit in algorithmic form - say that the limit is a potentially infinite process which is described by [pick your favorite] algorithm for pi, for example, 4(1-1/3+1/5-1/7+1/9-...)
You don't seem to quite grasp why I reject "closer". The line is known to never reach the limit, that's the point with "infinite". So it makes no sense to say that it is getting closer to the limit. It is impossible for the line to reach the limit. so it is impossible that one point is closer to the limit than another. The proposed limit is outside the parameters by which the line can be measured.
The issue is how to best measure the shape which is Gabriel's horn. Orthodox mathematics, while it is very good at other things, fails here, as is evident from the appearance of the paradox. The reason it fails, I believe, is because it describes a feature which very clearly cannot be described in terms of limits, as getting closer to a limit, and that's nonsensical.
So I don't agree with your resolution, because you still want to apply limits, where the shape denies the application of limits. If measurement necessarily involves the application of limits, then we regard this as having created a shape which cannot be precisely measured, just like a circle.
Perhaps I don't grasp it, or perhaps I just don't agree with it. Let's assume it's the former. Please tell me whether the following points aligns with your view:
1) One can travel along y=1/x in the positive-x direction, without bound.
2) The limit of the journey corresponds to the final destination, which if anything would be (?,0).
3) The point (?,0) does not exist (since ? is not a number) therefore there is no limit.
I surely disagree. There is no "final destination." That's @MU's error, why are you amplifying it?
I'm trying to condense his argument into a few points in hopes that it brings to focus where the misunderstanding lies.
Quoting fishfry
Yes that's what I'm saying, there is no final destination, so to even produce any representation (such as ?,0), as if it is a final destination, is a misrepresentation amounting to contradiction.
There is a particular line we are talking about, and #1 ought to state that this line is extended "without bound", which means "there is no limit". So #3 ought to read "the point (?,0) does not exist because there is no limit". Now we could add #4: "?" is a description of the entirety of the line (not a point on the line), and 0 is completely unrelated to the line, therefore irrelevant.
Here's another way to look at the position of zero. It is an ideal, like infinite is an ideal. The two are opposing ideals, like hot and cold are opposing ideals. The line takes the characteristic of the one ideal, the infinite, therefore the opposing ideal, zero, is excluded. It's just like if we were talking about the absolute, ideal hot, cold would be completely excluded.
This is where the misunderstanding is. Nobody is saying that there is a final destination or that (?,0) exists. The standard definition of a limit does not require a final destination, it only requires one to approach a number as they advance along the journey. A limit is the process of approaching not the act of arriving. And you must admit that any workable definition of 'approach' will have one approach y=0 as they travel along y=1/x. [As mentioned before, your definition of approach involving looking at the number of intermediate numbers is not workable for number systems which are dense in the reals, e.g. the rational numbers].
In short, I believe our disagreement is simply the result of us having a different definition of limit.
Did you read the rest of my post? What I'm saying is that 0 is not even relevant. That's what I've been trying to explain, that to describe the value of y as approaching 0 is a false representation. Y is always infinitely far away from zero, because zero is impossible on that line. The value for y never "approaches 0". It is correct to say that the value gets lower and lower, but it is incorrect to say that it approaches 0, because no matter how low it gets it never approaches 0. 0 is not at all relevant to this line.
To say that y approaches zero is an inaccurate simplification, nothing but a rounding off in your description. The real description is that the value of y gets lower and lower without ever approaching zero. Of course the true description is "not workable", that's the nature of any infinity. The appearance of infinity is the result of something being not workable. To make an infinity into something workable is to provide a false representation.
Quoting Ryan O'Connor
I think where we disagree is in the role that zero can play in this measurement. You think 0 can play a role, as the value which y approaches. I think that since the line has no start nor end, 0 is not an applicable number.
Quoting Metaphysician Undercover
Our disagreement is also due to us having different definitions of approach. We both agree that y gets lower and lower (and perhaps you would even agree that the greatest value which y never reaches is 0) but I call that approach and you call that not approaching. Let us agree to disagree on definitions!
There is no such value. Might it be 100? Or 1000?. y=1/x, x>1 has a greatest lower bound, 0, which it never reaches. Your attempts at the philosophy of mathematics may never bear fruit if you consider this a cogent statement.
You're right. You said what I was intending to say. Thanks for the correction!
As I explained, by way of example, to assume such a "greatest value", or "lowest value" is contradiction. When we say that the natural numbers are infinite, and therefore have no highest value, it's contradiction to say that 20 is closer to the highest value than 10. Likewise, when there is no lowest value, it's contradiction to say that .01 is closer to the lowest value than .02.
What is misleading in the example of Gabriel's horn, is that the x and y axes are set to converge at 0, at a right angle in relation to each other. This proposed point of convergence creates the illusion that 0 is a valid value, where x and y are 'the same". However, as I described earlier, the spatial representation of two dimensions at right angles to each other is actually a false representation, making the two dimensions incommensurable, as demonstrated by the irrationality of the square root of two. This incommensurability indicates that the two proposed lines, x and y, cannot actually be modeled as intersecting, and sharing a common point at 0.
So this false idea that x and y actually meet each other at that point, 0, is what misleads you into thinking that 0 is a valid measurement.
No it's not. When you get in your car and start driving to the store, do you experience infinite acceleration? What's that feel like, exactly? According to special relativity, you should be pressed against the back of your seat with infinite force. You'd be crushed before you drove a foot. What do you say?
ps -- Let's do the math. Say I'm at rest and start moving at 1 unit/second or whatever. In physics we need to give the units but in math we'll just say the velocity is 1. So at 1 second we've gone 1 unit, at 2 seconds we've gone two units, etc.
So our position function is p(t) = t; and our velocity is always 1, which is consistent with the first derivative of position being the derivative of t with respect to t, or 1.
Now this is tricky and this is where you got yourself confused. What was our instantaneous acceleration at 0? After all we weren't moving and then a tiny moment later we were. Well, the graph of our position look like this:
This function is not differentiable at zero. There is no instantaneous velocity at zero and no definite acceleration either. I agree that this is counterintuitive, and your intuition is not uncommon. But it's wrong. Clearly it's wrong. If you experienced infinite acceleration even for a moment, every atom in your body would be flattened like so many pancakes.
I actually agree with you about the intuition. If we're not moving, how do we start moving? It's a bit of a mystery actually, I'm not sure what physicists say about this. Well I guess I do know. If we're a steel ball in Newton's cradle, or we're a ball on a pool table, we start moving when we get smacked by another ball that transfers its momentum to us. But how does our velocity go instantaneously from zero to nonzero? The Newtonian physics works out, but not the intuition.
Quoting Metaphysician Undercover
A little woo-woo-y there @MU. By the way, how do the zero-dimensional, zero-length points in the unit interval make up the one-dimensional, length 1 unit interval? That's actually another mystery, despite the fact that we have a mathematical formalism that says [math]\int_0^1 dx = 1[/math]. The math works but we have no metaphysical explanation that I know of.
Is the area infinitely large or merely infinite in the sense of 'unbounded'?
I addressed the issue in my post. There is only a need to conclude infinite acceleration if we assume absolute rest, zero velocity, but relativity denies absolute rest. If something had an absolute zero velocity, and changed from that zero velocity to having a positive velocity, this would imply a point in time (not a short duration) when the thing goes from zero velocity to a positive velocity. At that point in time, since there is no duration, but there is acceleration, the acceleration would be infinite. We avoid this problem with relativity theory which denies the reality of rest, and makes any supposed zero point in time into an extended duration.
Quoting Metaphysician Undercover
You are equating 'approaching' with 'arriving at'. If I could only tell you one piece of information about my trip I would tell you my destination. But if my trip never ends there are some situations where I could still give you some useful information since in some situations I could still tell you which direction I'm pointing (e.g. what I'm approaching). But I think we are both firm with our incompatible definitions and we've said what could be said. I think our words would be much better spent on the new topics of this thread.
Quoting fishfry
This reminds me of Diogenes the Cynic's rebuttal to Zeno's paradoxes. According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions. I'm not convinced that Diogenes appreciated that profundity of Zeno's paradoxes.
Quoting fishfry
Again, Zeno's Paradox. The issue is that we're looking at things with a 'whole-from-parts' view. We want to advance forward one point at a time and can't seem to get anywhere...our intuition is saying that it doesn't make sense how a line can be formed from points. But with a 'parts-from-whole' view it's easy. We start with the whole (the unobserved wave function of the universe spanning all of time) and then we make (quantum) measurements. At one measurement we're here and then at the next measurement we're there. Change doesn't happen at points (or instants in time). Instantaneous velocity makes no sense. Change it happens in between the points. And if we draw a graph using the 'parts-from-whole' view as I mentioned in the Have we really proved the existence of irrational numbers? thread, the change of a function happens in between the points...across the unmeasured curves.
If I point out to @Metaphysician Undercover that he can get in his car and drive to the store without being crushed before he drives the first inch; am I failing to appreciate the profundity of his beliefs? I think not!
I addressed this in my post. The position and velocity functions are not differentiable at time zero. So there's no well-defined acceleration. Nor as others pointed out does relativity bail us out. Relative to your own frame of reference, you are at zero velocity at time zero and nonzero velocity a short time afterward. You have to come to terms with that.
Collision is a notoriously messy scenario - both physically and mathematically. Better to think of a ball in Newton's cradle at its highest point: at that point it is instantaneously at rest, then it starts moving again. Voila, motion from rest. Or easier still, just pick up that ball, gently release your grip and let it fall to the ground. Same deal, and we even know pretty exactly what its acceleration is when it starts moving. This doesn't seem so unintuitive to me.
I remember struggling with the concept of acceleration when it was first introduced - in middle school, I guess. It started making sense after a while. But some people just can't come to grips with such abstract concepts. Most of them have the good sense to leave it alone and apply themselves to something they are better at. Those who can't leave it alone become lifetime cranks, like MU. Or philosophers :)
FWIW, and because no one has mentioned it yet, 'infinite limits' are taught in calculus as usefully specific ways to indicate divergence.
You can even write [math] f(x) = x [/math] and then [math]f(\infty) = \infty[/math], but only as a cute abbreviation for something more technical. Later there is the extended real line, represented as [math] [-\infty, \infty] [/math], but there's nothing magical about this, no more than there's anything magical about [math] 0! = 1 [/math]. It's all philosophically agnostic. Indeed, I know one mathematician who thinks the world is discrete and that continuity is a fiction, and then I know another who believes the reverse. Another dislikes philosophy altogether, and still another more has read Kant's CPR in German.
It's "absolute rest" which I said is a problem, because this makes a point in time into a real situation rather than a perspective (reference frame) dependent designation. That's a point when no time passes relative to the thing at absolute rest.
Quoting tim wood
This "at rest" which you refer to isn't real, because the earth is moving. Your hand is never at rest. So the moving of your hand is just a change in the existing motion of your hand, it is not an act of acceleration from rest. Physicists might represent it as an acceleration from rest, but the point I am arguing is that this is really an incorrect representation, which serves the purpose, just like representing Gabriel's horn as approaching 0 is an incorrect representation, which serves a purpose.
Quoting Ryan O'Connor
No I'm not equating these two. If there is no such thing as the lowest point, then it is impossible to be "approaching" the lowest point. In the case of the natural numbers, do you see that there is no such thing as "approaching" the highest number? We recognize that there is no such thing as "the highest number", so it doesn't make sense to say that if a person is counting higher and higher, they are "approaching" the highest number. You can never approach the highest number. If you can apprehend this, then why can't you turn it around, and see that when infinity is at the low end, there is no such thing as "the lowest number", and it doesn't make any sense to say that someone counting lower and lower is "approaching" the lowest number?
Quoting Ryan O'Connor
OK, this is a good point. The question here is what grounds or substantiates "direction". You imply that direction must be grounded by going toward something, but you forget that it might equally be substantiated by going away from something. In Gabriel's horn we have both, moving away from one axis, and moving toward the other. The axes are artificial confines, imposed as standards of measurement, and through the descriptive term of "infinite", the line of the form is stipulated as going beyond the capacity of the measuring scale. Therefore to understand that line we can no longer employ those measurement axes.
This is the problem we have here. Generally we assign infinite capacity to the measuring tool, and this allows us the capability to measure anything with that tool. The natural numbers are infinite for example, and this allows that we might count absolutely any multitude of objects. In Gabriel's horn, we have turned the table. We propose an infinite shape to be measured. Of course we cannot measure it, because it is defined as infinite, meaning that we cannot measure it, the thing is stipulated as going beyond the capacity of the measuring tool. So there is a trick hidden in the proposal, it's asking us to measure what cannot be measured by the tool. Then when we look at the shape, we see it getting further and further from the one axis, and we conclude, 'that's impossible to measure'. But we also see it getting closer and closer to the other axis, and intuition tells us, 'that's a finite distance which can be measured'. However, we must adhere to the principles of the construction, which state that the shape will appear to approach the axis, to a point beyond our capacity to measure the distance between them. Therefore we must resist our intuition and inclination to say that this distance is measurable.
So we must remove the axes as incapable of giving us the scale required for the measurement. The axes are what produced the form, which is by that construction, infinite and therefore immeasurable. Therefore the axes cannot be used to measure that form, because it has been constructed by them, as immeasurable. Now we have no basis for the terms of "farther from" or "closer to", because these values have been stipulated as going beyond our capacity to measure. What we are left with now, is just theoretical values, to be assumes as spatial distances, values which we acknowledge cannot actually be measured as spatial distances. Now we are really not talking about "farther from" or "closer to" any more, even though the numbering system employed was originally derived from that. We have explicitly gone beyond our capacity to determine farther from or closer to, and all we are talking about now is a higher value and a lower value. If we do not divorce the value from the spatial distance, we are just left with a spatial distance which is impossible to measure.
The point now, is that since we have done what the example requires, and taken the values beyond our capacity for making spatial measurements, we cannot use spatial references to ground or substantiate "direction". All we have now is a higher value and a lower value, and the stipulation that each of these may continue infinitely. The two directions (values) are actually defined in relation to each other. As the one gets a lot larger, the other gets a tiny bit tinier. And so long as we allow that the one can continue to get a lot larger, we must allow that the other can get a tiny bit tinier. But these "directions" must be thought of solely as numerical values, because we have gone beyond the relevance of spatial distances as dictated by the proposed example. So we cannot look at them as spatial directions of "farther" or "closer" or else we just fall back into the stipulated impossible to measure..
Quoting fishfry
So the point I'm making, is that zero is completely arbitrary, and represents nothing real, just like in the case of Gabriel's horn. That's why we must decline this idea of "approaching zero". It is extremely useful in practice, yes, for sure it serves the purpose. But this is an exercise in theory, and we need to be able to go beyond what works in practice to be able to see that the principles which we employ in practice mislead us in our metaphysical efforts to understand the true nature of reality. The existence of paradoxes such as Zeno's demonstrate an incompatibility between theory and practice, and these incompatibilities expose where we misunderstand the true nature of reality.
Quoting norm
This is a different, but related issue, the difference between discrete and continuous. The issue is not whether the world is discrete or continuous, it is to find compatibility between the two. In practice the world is continuous (time passes continuously), but in theory the world is discrete (represented by distinct units, numbers). Simply modeling the world as discrete, or modeling the world as continuous, is fine either way, until someone approaches you with an example of the other, and makes a paradox jump out at you.
This has been mentioned to @Metaphysician Undercover repeatedly. For years.
Because the wind blows.
I think he's touching on something important. If time can be broken down into a collection of instants and if at one instant we're stationary and the next instant we're not then in one sense it does appear that we have undergone infinite acceleration. Pointing to our physical reality and suggesting that it proves he's wrong is besides the point. The real issue is with our assumption: that time can be broken down into a collection of instants. Or more generally, that a line is composed of infinite points. Anyway, you've already mentioned that you are puzzled by this so the analogy to Diogenes does not exactly fit so don't put much weight on my 'throwaway' comment.
Quoting Metaphysician Undercover
I don't have a problem with saying 'x approaches infinity' in the context of a potentially infinite process. I interpret it as 'the value of x is continuously growing'. 'x approaches infinity provides some information about the journey, even if we never can arrive at some final destination.
Consider the graph linked here: https://imgur.com/FZANGZ8
This is not a typical graph in that it spans all possible values of x and y. Think of it topologically in that it is a single system which maintains its topological properties when undergoing continuous deformations. In this plot, there is a point at (1,1) and a pseudo-point at (?,0). In this context, it makes sense to say that we're starting at (1,1), travelling along y=1/x and heading towards the pseudo-point at (?,0). By plotting Gabriel's Horn like this, (?,0) is no longer out of sight, it's right there in front of us. And because of that we have the ability to use it in some contexts without requiring infinite measuring capacity.
Quoting Metaphysician Undercover
I would argue that objects are continuous but measurements are discrete. This allows us to use the richness of mathematics that calculus offers while avoiding the paradoxes of actual infinity.
Second clause does not follow from the first. A mathematical line is composed of points. But there is no "next" point after any given point. You are confused on this ... point.
Quoting Ryan O'Connor
You're thinking of Andrew Cuomo.
How can we travel from one point to the another without traversing through the intermediate points in sequence? This is essentially Zeno's paradox and if you cannot offer a resolution to it then you are not justified to claim that a line is composed of points.
Quoting fishfry
And you even admit that your view is shrouded in mystery. Why not consider the alternative...that a line is not composed of points, but instead points emerge from lines? Why won't you consider my...line...of thought?
What does that mean?
ps -- I'm not making a geometric statement. The real number line is composed of real numbers. How can you disagree with that?
One theory that I've toyed with is that we have intuitions of both the continuous and the discrete that don't play nice together. Measurements are clearly discrete, as you say, but we also can draw the unit square and its diagonal and try to measure it 'perfectly' or 'ideally' and discover irrational numbers. The arithmetization of analysis was maybe driven by epistemological concerns. We want proofs in a universal language, and pictures aren't computer-checkable. (?)
Imagine that lines are fundamental, not composite objects. Take a string and mentally label the two endpoints -? and ?. In my world, this string only has two points - the endpoints (I'd actually call them pseudo-points but that's not important here). Cut the string somewhere and mentally label the cut (i.e. the gap) 42. This string now has an additional 'point'. It doesn't matter where this cut was along the string as long as you follow this rule: if you cut the string somewhere between 42 and ? then that cut will be mentally labelled with a number greater than 42 (and if you cut the string somewhere between 42 and -? then that cut will be mentally labelled with a number less than 42).
The only object in my example was a string, which I'm using to describe a line. There are only a finite number of points on the string and they correspond to the cuts...the absence of string. The points only emerge when you make cuts and the numbers and their proper ordering are contingent on you (a 'computer') actively thinking about them. If you forget about the mental labels, then the numbers no longer exist. You are left with a string...a line...a continuum....and nothing more.
This way of building the parts from the whole is not new. It was (loosely) how Aristotle resolved Zeno's Paradox. I believe that this construction is paradox-free, but I'd love to hear your criticisms.
Quoting fishfry
What if instead of 'the real number line', I defined the object that's constructed from all of the real numbers 'the real number point'? Does my definition make it so? It may seem like I'm giving a silly example but I truly think you'd have a tough time explaining why your definition is better than mine, and I think this because I believe that the real numbers are singular. But since you are not convinced by Wildberger's arguments, what are the chances that an untrained engineer could convince you with informal reasoning. Perhaps your time is better spent telling me why my 'parts-from-whole' view is wrong rather than hearing me informally complain about why I think your 'whole-from-parts' view is wrong...
Quoting norm
We can perhaps use Newton's method or some other algorithm to produce better and better approximations of sqrt(2) but trying to measure a 'perfect' value doesn't imply that you've discovered it. Perhaps all that you've discovered is an algorithm...and not an irrational number.
I said your hand is not at absolute rest because the earth is moving. And you asked me how do I know this. So I answered that. How am I supposed to know that you were asking me something other than what you were asking me?
Quoting tim wood
It isn't absurd, it just shows that the idea of an "instant", a zero point in time is absurd. The conclusion of an infinite acceleration is only produced from the idea that there is a point in time when a thing goes from resting to moving. Obviously then, what is absurd is the idea of a point in time, not the idea of rest. And the falsity of this idea (of a point in time) is borne out by special relativity which describes simultaneity (being how we determine a point in time) as frame dependent. There are no real points in time, they are arbitrarily assigned according to a frame of reference.
Quoting Ryan O'Connor
I don't like this phrase "approaches infinity" because it reifies infinity as a thing which is approached. Either the process is designated as infinite, or it is not. Suppose we don't know whether it is or it is not infinite, then we might say that it is potentially infinite, because we don't know. Perhaps, for some reason we are not ready to commit to infinity, like if someone worked out pi to hundreds of decimal points, and was still not convinced that it would go forever. That person would say that it appears to approach infinity, and it is potentially infinite, but I think it might still reach an end at some point, so I won't admit that it's actually infinite.
Don't you think that we have enough evidence to make the judgement, it is infinite? Suppose we say that it appears to be infinite, but it's still possible that it is not. How is this compatible with "approaches infinity"? Then you'd be claiming that there is some actual thing called "infinity", and the values appear to be heading in that direction. But how does this make any sense? What sort of thing could that be, which is called "infinity"?
Quoting Ryan O'Connor
But what is the meaning of that point which you label as (?,0)? How can ? represent a point? You say it's a "pseudo-point". I assume that this means that it's not a valid point. What's the point in having a non-valid point? I can see how it's useful in practice, but this is an exercise in theory.
I am obviously wrong in this opinion.
When I say potential infinite, I don't mean 'something that might be infinite'. I mean a process that certainly goes on to no end. We are certain that if you begin to write out the digits of pi that that process would never end.
Quoting Metaphysician Undercover
I call it a pseudo-point because it's not an actual point on the graph but instead a potential point. It's a point that you can approach but never arrive at. By 'never arrive at' I mean you can't plug one of its coordinates into a calculator to compute the other.
The north pole of the Riemann sphere. But carry on.
Well, yes! I'm certainly open to that view. Irrational numbers are fictions, constructions. The Cauchy sequence construction sorta-kinda says that a real number just is a streaming approximation.
For instance: 1, 1.4, 1.41, 1.414, 1.4142, .... just 'is' the square root of 2, if only we could ignore the extra complexity of equivalence classes. We can't, because any subsequence of the one above also represents root(2), and that's just the beginning of equivalent sequences.
Have you looked into Dedekind cuts? Consider the set of all rational numbers q such that q < 0 or q^2 < 2 for q >= 0. That set of rational numbers just is root(2), and we don't have to worry about equivalence classes. There are something like 15 constructions of the real numbers that I've heard of and looked into (some quite briefly, because some are quite complex and strange.)
If you want to use algorithms (an idea I like), it seems you need to either use mainstream computability theory or rebuild that too. But the computable numbers have measure 0, so you'll have to rebuild measure theory or stick with early analysis.
But to your original point: I'm happy with the word 'invented.' My point is that people wanted to connect symbol math to pictorial math and discovered that our two basic forms of intuition (discrete and continuous) don't play well together. (Consider Zeno's paradoxes in this light.)
Amazing, the things which mathematicians will come up with, in an attempt to solve their problems, instead of simply recognizing that the dimensional representation of space is wrong.
Quoting Metaphysician Undercover
Nothing more need be said. But it will be said anyway.
Incidentally, GR and SR use dimensional representations of space. This includes with GR the use of Penrose diagrams, which have multiple infinities.
Why call this "potential infinite" then? If you are certain that the process goes without end, then you are certain that it is actually infinite.
We might however use terms like potential and actual to distinguish between things which have real existence in the world, and things which are completely imaginary. In this case, there is no such infinite process actually going on, so we say that if someone endeavoured to carry out that process they would find it unending, therefore infinite,. But since they would never arrive at "the infinite", we'd say that such an infinity only exists potentially. One could never prove it to be infinite by reaching infinity.
This is why I didn't like your use of "infinity". You used it as if it signified something with actual existence, which one could be approaching. It seemed as if you thought that if you carried out such an activity, then after a designated point you could be said to be "approaching infinity", when in reality you know that you would never be approaching infinity, because you clearly recognize that the process would never end without ever reaching infinity. Therefore that is a deceptive us of words, when you claim to be approaching something which you clearly acknowledge you can never reach.
Exactly, look at the problems which envelope modern cosmology and quantum physics, due to the use of inadequate spatial representations, and a massively complex and fractured numbering system with imaginary numbers etc., required to cope with these problems.
The problem with Platonism (Pythagorean Idealism), is that once we assign to mathematics the status of objects, the objects obtain the status of unchangeable, and therefore eternal truth. Then it appears impossible that these mathematical constructs could be wrong. So when a problem emerges, something which cannot be figured out under the current mathematical system, a fix must be created, axiomatized, and added to the system. These fixes are nothing other than exceptions to the fundamental rules, causing the whole mathematical structure to get extremely top heavy with fix after fix. The fundamental rules which ought to be seen as faulty if they require such exceptions, cannot be apprehended as such, because they are already axiomatized as objects, eternal truths of Platonism.
Some time ago, at this forum, there was a discussion concerning the validity of Euclid's fifth postulate, the parallel postulate. It was argued that the geometry required by GR rendered the parallel postulate invalid. To any rational human being, this would indicate that the fundamental postulates of dimensional space need to be revisited and recreated. But human laziness inclines us to avoid this task, and simply make exceptions to the rules in an attempt to establish compatibility between what science now tells us about spatial existence, and what the rules created thousands of years ago say about spatial existence. Then we rationalize our laziness by saying that mathematical principles are eternal Platonic objects, and couldn't be changed even if we wanted to change them.
...and what is that problem?
Quoting Metaphysician Undercover
...and how are you fixing that? And why should we trust a guy whining about lack of reality when it's the same guy who claims it takes an infinite amount of acceleration to move an object at rest?
I don't understand your idea at all. Suppose the position of a particle at time [math]t[/math] is given by [math]f(t) = t^3 - 5 t^2 + 9t - 6[/math]. Find the acceleration of the particle at t = 47.
How do you do that problem after you've thrown out 350 years of calculus and our understanding of the real numbers? What happens to the whole of physics and physical science? Statistics and economics? Are you prepared to reformulate all of it according to your new principles? And what principles are those, exactly? That there aren't real numbers on the real number line?
I find this example unsatisfying given that everything important is contained in the ellipses. You are no better just writing "For instance: ... just 'is' the square root of 2". And so that equivalence class could just as well correspond to 42. The only way to give it meaning is to state the algorithm used for generating the sequence, which is why I think non-computable numbers are questionable since there is no algorithm behind them.
Quoting norm
Here's a dumb question for you: how can the rational numbers (of which there are only aleph-0) can be cut in c unique ways? For example, if there are 2 numbers, then there's only 1 unique cut. If there are 3 numbers, then there are only 2 unique cuts. If we approach the limit, how do we end up with more cuts than numbers?
Quoting norm
I think our problem is that we're using numbers to model a continuum. As I'm discussing with fishfry in this thread, I think we should do the opposite and instead use a continuum to model numbers. I think flipping this on its head avoids the paradoxes, allows objects to have non-zero measure, and does not require us to decide between the discrete and continuous because they actually do play well together.
Quoting Metaphysician Undercover
This is standard terminology. Check out this wikipedia page
Quoting Metaphysician Undercover
Who says only actual things be approached? I can certainly approach a mirage.
Quoting TaySan
One day they may reveal the pokémon stored in that ball and the answer will be clear.
What I said is that I know your hand is not at absolute rest. And, I also said that if there is something at absolute rest, that thing would have to go through infinite acceleration to start moving. You, and some others here maybe, are the ones claiming infinite acceleration is absurd. I suggest to you, that instead of thinking about the absurdity of infinite acceleration, you simply apprehend absolute rest as absurd.
Then there is no need to think about infinite acceleration because it is only the idea of absolute rest which produces that idea. Make sense?
Quoting InPitzotl
That problem is, the deficiency in our capacity to measure. Obviously, if the thing appears to be infinite, this means that we do not have adequate capacity to measure it.
Quoting InPitzotl
I told you already. We need to revisit our spatial representations, from the beginning. Go back to the fundamental principles, armed with what we now know about space and time, derived from modern science, and rework them all, from the very beginning, starting with the most basic relationship between the non-dimensional point, and the dimensional line. The fundamental (Euclidean) axioms of geometry provide us with inadequate modelling principles which incapacitates our attempt to understand a large portion of spatial existence. Ryan gets it:
Quoting Ryan O'Connor
Quoting Ryan O'Connor
I have a slightly different way of looking at this issue. I give priority to non-dimensional existence, represented as points. Points can be related to each other through lines, but these are still non-dimensional, meaning that these lines do not correspond to real physical, dimensional existence, they are ideals, by which we relate other ideals, points. To establish a correspondence between the non-dimensional, ideal, the point, and the dimensional, real physical existence, we need to bring time in as the first "dimension". Time must be the most fundamental mathematical representation as simply order. Order can be expressed without spatial reference. An understanding of the passing of time will determine the relationships which points can have with each other in real dimensional existence, and geometrical figures must be produced in accordance with the principles derived.
Because I'm lazy, I'm going find the velocity of the particle instead of acceleration. First, I want to show you three valid graphs of y=t3-5t2+9t-6 (valid in my construction, that is).
You have to remember that these plots are topological so even though it looks linear in the first image I could have just as well drawn it with squiggles. The key thing is that I've only made a finite number of 'cuts' (exactly like cutting the string in my last post to you, but this time I'm cutting a 2D continuum). All of the information contained in these three graphs is correct. It's just that the graphs have the potential to be cut infinitely more times. The more cuts I make, the more points will emerge and I (a 'computer') will make sure that the points have coordinate values consistent with the functions.
First, I need to point out that (in my view) velocity at an instant is meaningless. It's equivalent to 0/0. Velocity applies, not to points, but to curves. In the third graph I've highlighted a section of the function between x=47 and x=48. As normal, I can calculate the average velocity across this interval using the coordinates as follows (99498-93195)/(48-47)=6303.
But obviously we're not satisfied with that. We want to shrink this interval as much as possible. And we can do so by making cuts closer and closer to x=47 and finding the average velocity across those shrinking intervals. This is what the limit describes (in my construction), it is a potentially infinite process.
What calculus does is describe the potential of that process. And I believe that when calculus was made rigorous by going from numbers (infinitesimals) to processes (limits) some 'infinite-like' numbers (irrational numbers) were left behind. I believe that to complete the job, we need to reinterpret irrational numbers as irrational processes. Calculus is the study of potentially infinite processes. In my view, the math is the same, dy/dt=3t2-10t+9. It's just that the philosophy is different.
This may seem like a trivial difference, but I believe that with this continuum-based view (as opposed to the standard points-based view) many paradoxes are no longer paradoxical. In fact, I can't even think of a paradox with this view (especially given our refined intuitions developed through quantum-mechanics).
I hope I was clear in my explanation. I'd love to hear your feedback, especially if you have criticisms. Thanks!
Consider though that ellipsis are just shorthand that lazy mathematicians use for one another. In this case, it should be obvious how the sequence proceeds. Lots of different algorithms can give the exact same sequence, and that's why equivalence classes are necessary.
Personally I think non-computable numbers are questionable, but root(2) is of course not one of them and is ultimately a finite object (since there's a finite Turing [actually many!] machine that gives arbitrarily good approximations. )
Just to be clear, I think metaphysics in general is a bust. To talk about numbers is (for me) to talk about the talk about numbers. So I'm just quoting some mainstream math, pointing out some of the issues.
Quoting Ryan O'Connor
Don't forget the jump between finite and infinite sets! A typical limit operation doesn't make sense here. I do like your question, because it highlights the strangeness of math. (The diagonal proofs, when I first saw them, were lots of what seduced me away from engineering toward pure math and proofs. I've come up with some of my own to prove smaller subsets of R uncountable, where the diagonal is not perfect...fun stuff, engineering with infinity, in pure thought.)
Quoting Ryan O'Connor
I'm all for bold ideas. I don't know of any paradoxes. I think 'discomforts' is better. AFAIK, mainstream math works, is correct (even if we can't prove it.) The only problem is that it offends lots of peoples intuition here and there.
I know what you mean here, but reading it makes me uneasy.
I like this string. It's maybe how we first think of the line. This idea would require a radical change in the foundations, sounds like even set theory is jettisoned. If you could rewrite a calculus textbook so that calculations come out the same (so as not to clash with mainsteam math in applications), it could be presented as a pedagogical alternative. The vibe I'm getting is pre-proofs applied mathematics, because there's no mention of axioms. You are offering a nice intuitive foundation. If you want a 'normal discourse' (an objective discipline where disputes can be resolved somehow) you'll have to come up with rules, something like axioms. It could be a calculus in the old sense of the world, with rules like arithmetic except in a calculus context. (I think non-math people experience calculus this way already.)
Epistemologically speaking, why should people trust the results? If you get mainstream results, all is well. You could just present Newton's calculus with different metaphors. What would you do with limits? Infinite sums?
To sum up, nice basic metaphor, but you'd probably have your hands full fleshing it out (which is not meant to be discouraging)
If you look at the definition of a limit, it's actually timeless. For all epsilon > 0 , there exists a delta > 0 such that ETC. So there is a leap from the intuition of the potentially infinite approximation process. The fundamental question is something like: what are we approximating? A limit is a real number, a point, and not the process (in the mainstream view). Different processes can converge to the same point. (Subsequences make this obvious, but it's not only subsequences.)
We can draw the symbol root(2) confidently because we can prove that it exists from the axioms, (IVT) entirely without pictures. The desire to free math from pictures should perhaps be addressed here. Can your system free itself from pictures? A theory of continua would presumably have to be symbolically established. Would classical logic work? Would you still find the system charming if the pictures were secondary and only props for the intuition? (Just trying to ask productive questions. Hope they inspire you!)
This is very good insight. My belief is that we need to go one step further, and apprehend an infinite process, or "irrational process" as actually impossible. But since this process is a potential process, as you describe, this means that it is a possible process which is actually impossible. Therefore the infinite process must be rejected as logically invalid, because it's contradictory.
Rejection of the impossible needs to be recognized as the greatest epistemological tool which human beings possess. It is the basis for certainty in knowledge. In our quest for an understanding of "what is" we narrow the field of possibilities by rejecting any proposed possibility which can be determined as impossible.
In the case of the infinite process, we have a very difficult judgement. It appears to be a reasonable and logically valid possibility. But for some reason it's extremely repugnant to intuition, and the reason why it is repugnant cannot be properly identified, so as to prove logically that it is impossible. It appears as absolutely impossible to show, or demonstrate the impossibility of infinite possibility, because it cannot be done with an empirical demonstration. There is however a logical demonstration commonly known as the cosmological argument, in Aristotle's Metaphysics, and I assume that other forms may be produced.
Quoting Ryan O'Connor
I believe that this distinction between the continuum perspective and the points perspective is a very good start, but I don't think it's an either/or question. We need to allow for both. It is the application of both, the two being fundamentally incompatible, which leads to infinity, and the appearance of paradoxes. However, we cannot simply exclude one or the other as unreal, and unnecessary, because there is a very real need for both non-dimensional points, and dimensional lines. You cannot remove the points because this would invalidate all individual units, therefore all number applications would be arbitrary.
What I propose is a fundamental division between numerical arithmetic and geometry, which recognizes the incompatibility between these two. Numbers refer to discrete units, and geometrical constructs refer to a continuum. We ought to recognize that reality consists of both these aspects, and the metaphysical question which we are faced with is to determine in which circumstances each system is applicable. Of course the need to establish a system of correlation between the discrete and continuous will never go away, this is mathematics, but we need to get a firm handle on which aspects of reality are discrete, and which are continuous, before we can axiomatize that correlation in a way which might serve us adequately.
Quoting norm
The problem I see here, is that a process is fundamentally continuous. If it were not continuous we would identify it as a number of different processes. We apply a point, to limit or individualize a process, marking a beginning or ending. But if these points are arbitrarily applied, i.e. not in reference to real points in the apparently continuous world, the numbering system will not provide us with truth (in the sense of correspondence).
When we look at the physical world empirically, it appears like we find true limits in spatial extensions, the boundaries of objects. The reality of these spatial limits have always justified individuals, units, numbers, and quantity, with distinct spatial objects forming discrete entities. However, now that we've come to look closer at these entities, the boundaries have become vague, and when we take the physical object right down to the micro level, those boundaries are not at all valid. So we find that the empirical evidence has really been misleading us, we think that there are spatial boundaries, and distinct entities when there really is not, spatial existence is continuous.
Likewise, if we look at time rationally, (because we cannot look at it empirically), we find the exact opposite situation. Our intuition tells us that time is continuous, that's how it seems to be, time is continually passing. However, we know that there is a big difference between past and future. Since there is such a difference, it is necessary to assume a boundary between past and future. If such a boundary exists, (and it is logically necessary that it does, to maintain a difference between past and future), then this boundary will provide the necessary principles for discrete units of time.
Quoting tim wood
Do you understand the principle, that if you have two premises which you do not know whether they are true or not, and they lead to a logical conclusion which is obviously false, then one or both of the premises must be false? The two premises are, rest may be absolute, and there is motion. These two premises lead to the conclusion of infinite acceleration. The conclusion is obviously false. The second premise, "there is motion" looks true, so the first must be false.
I give up. I've tried numerous times to answer your question, and it appears I have no idea what you're asking. It seems like you're having the same difficulty with the word "absolute", which you had the last time we discussed the use of that word. You just carry on as if the word isn't there, and insist that it has no meaning. So, onward with your diatribe if it makes you feel good, tim.
Actually, it seems you're having problems with the word "acceleration". If an object at rest remains at rest, it is ipso facto not accelerating. Likewise, if an object at rest accelerates, it ipso facto starts moving. But somehow in MU land, absolute means something can stay at rest and still accelerate, so long as it's not accelerating infinitely. Whatever that means.
It's only obvious to me because I only know a handful of real numbers so I assume you're talking about sqrt(2). But it's not a matter of laziness, no finite amount of terms would have allowed me to eliminate any possibility. From this view (when there is no algorithm) it seems like the only important number in a Cauchy sequence is the last one...and there is no last one! Anyway, sorry for putting you in a position having to defend a position you don't support!
Quoting norm
Yes, something magical happens at infinity...
Quoting norm
There are a lot of infinity-related paradoxes which offend students' intuitions. Given that it drew you in, perhaps the strangeness is a strength (not a weakness) of infinity. I have no doubt that math works, it's 'why' that has puzzled me for many years.
Quoting norm
I haven't worked it out, but IF this is a valid perspective I don't think the math would change much. I think that (in a way) this is consistent with how we've been thinking all along (as I described to jgill below). We'd write the same equations and draw the same pictures, we'd just think about it differently. I think it's a matter of philosophy, not math. But you're totally correct, this is only an intuition...an idea...a formal theory is a whole different thing. And you're right...I'd have my hands full fleshing it out. And to be honest, I don't have the necessary skills to flesh it out.
Quoting norm
I believe that my view is in agreement with limits and infinite sums as long as you think of them as descriptions of unending processes. For example, 1+1/2+1/4+1/8+... corresponds to the unending process of computing larger and larger partial sums which approach but never arrive at 2.
Quoting norm
I think this question is very important. In my view, the topological graphs that I drew actually exist. The geometric graphs that we imagine imagining don't exist, but they are incredibly convenient approximations of what we could do in reality to topological graphs.
Quoting norm
True, but what if we reinterpret real numbers as real processes which describe continua, not points? Wouldn't we be able to keep the same math? Can't we just say that our algorithms for calculating the 'number' pi can never output the number completely and that pi actually corresponds to those (potentially infinite) algorithms? Why do we need the number pi anyway? We have never precisely used it as a number anyway.
Quoting norm
These are good questions indeed, thanks!! I don't know the answers to them, but here are my thoughts. There's a convenience to the completeness of the real numbers. We get a one-to-one correspondence between our numbers/equations and our graphs. If in our graphs, lines cross at a point, then we can always find the coordinates of that point (and most often they'll be irrational numbers). If we calculate the derivative of a function at a point, then we can always draw a tangent at that point. With my view, we cannot always do these as they would amount to completing an infinite process. We need to be clever to avoid actual infinity and I think the way to do it is to use both numbers/equations and graphs. Sometimes numbers/equations are needed to describe a system, sometimes graphs are. We can't avoid the pictures.
As for logic, this puts me even further out of my comfort zone but I'm guessing some hybrid would be required: classical logic for the discrete points and fuzzy logic for the continua. This may seem like a patchwork solution to connect the discrete with the continuous but it has parallels to what we do with quantum mechanics to describe our universe (measured objects have definite states and unobserved objects remain in a superposition of potential existence).
Quoting jgill
I appreciate why it makes you feel uneasy, but consider this claim: Every** graph that you have ever seen was topologically precise and geometrically approximate. For example, if you and I were to independently sketch out a graph of y=x^2-2, our graphs would share few geometric properties. They would be of different sizes and neither would perfectly capture the curvature of the parabola. But when we compare our graphs we still see them being the same because we continuously deform the graphs in our mind to see that they correspond to the same system. And this goes for computer generated plots as well. A computer plotting y=x^2-2 does not place infinite points on your screen but instead calculates the rational coordinates of a finite set of points and imprecisely connects the dots. But once again, we have no problem with it because it is topologically precise. The only difference is that in my plots I continuously deformed the graphs to a state where it was clear that they were not geometrical.
With 'parts-from-whole' constructions, the systems are topological and described perfectly (without approximation). There's no need to imagine a cloud of infinite points, what you see is what you get. It's just that these plots have infinite potential to be cut.
**When I said 'every graph' in the first sentence of this response I was exaggerating a little because some systems cannot be drawn with perfect topological precision. For example, y=sin(1/x) and y=1 cannot be plotted in the same graph because we'd have to draw infinite points where they interest and that's impossible. In such cases I'd just conclude that that pair of functions cannot be plotted simultaneously. I know it's an odd claim, but it's not too far from the complementarity principle in quantum mechanics which holds that objects have certain pairs of complementary properties which cannot all be observed simultaneously.
I see that you're a retired math professor so I'm especially keen to hear your feedback, especially if you see a flaw in my argument. Thanks!
Quoting Metaphysician Undercover
We can do so much with potentially infinite processes. Not only can we interrupt them to produce rational numbers, but we can work with the underlying algorithms themselves. For example, the following program to outputs the entire list of natural numbers. This program can never be run to completion, but it still is a valid program...I'm talking about it after all and it makes sense even though I've never run it. The same can be said about irrational processes. We need to embrace potential infinity for what it is, not reject it.
[b]while ( i > 0):
print (i)
i = i + 1[/b]
Quoting Metaphysician Undercover
In my continuum-based constructions there are still points, it's just that there are only ever finitely many of them and they are not fundamental. Can you expand on a situation where points need to be fundamental?
Quoting Metaphysician Undercover
I think we might be on the same page regarding this issue. I mentioned to norm above that I think numbers/equations and graphs are complementary, not equivalent. The problems occur when we think there is a one-to-one correspondence between the two.
What makes me uneasy is using a concept like topological equivalence and then discussing slopes and derivatives, which, as you know, do not carry over in that way. As for your continuum ideas, almost twenty years ago Peter Lynds wrote a paper appearing in Foundations of Physics Letters that postulated time having no instants and instead being composed of intervals. Something like Bergson's notions from a century ago. Here it is.
This is one of the weird thing about pure math (including computability theory.) We are OK with proving or assuming the existence of a number logically without having to specify that number. The terms of a Cauchy sequence get arbitrarily close, but we don't when that will happen with an arbitrary Cauchy sequence. I'm used to this stuff. Analysis came pretty easy to me, once I entered a logical frame of mind and let go of metaphysics. Intuition is still in play, but it works in tandem with a logical instinct. There's nothing else like it. I also love to program, but programming is different. One builds logical spiderwebs, trapping numbers with inequalities. I found it seductive.
Quoting Ryan O'Connor
Indeed! But calculus was always haunted by infinity. For a long time, the spirit was 'calculate! faith will come.' The results were good. Reliable technology and predictions emerged from the fast and loose application of a calculus that bothered the philosophers. It's very hard to avoid intuitions of infinity. Ilike the spirit of finitism, but it gets cramped and awkward quickly. Call the largest integer Z. Then Z+1 is larger. Physical arguments against this (like Wildberger's, which I browse) don't convince me, because there's an ideality to math that seems close to its essence. A Turing machine either halts or not on a certain input. I may not know which, but intuitively that's clear to me. Note that a Turing machine is a completely imaginary entity in the first place, at the heart of computability theory. So even your attachment to algorithms is threatened by problems with infinity. Before long we're back to lots of hand waving, no strict definitions. That's fine for practical purposes perhaps, but it's basically an abandonment of math as a distinct discipline.
Quoting Ryan O'Connor
The danger here is replacing strict symbolic reasoning with pictures. In some analysis books, there's not a single picture, for epistemological reasons you might say. Pictures often mislead, while obviously being of great use pedagogically for applied math's well-behaved functions.
Quoting Ryan O'Connor
Again, the question is what are we approximating? Why do believe in a single pi in the first place? The circle is already an ideal object. If we decide to think of Turing machines that give decimal expansions as real numbers, then we still need equivalence classes. For each computable real number there are an infinite number of Turing machines that approximate it. Which one will we call pi? Do we put some bound on the number of steps needed to give us digits? Turing machines could be made arbitrarily slow (programmed with wasted motion). (In other words, problems with the real numbers in pure math don't go away when we switch to computability theory --which is itself pure math, awash in idealities. If you want necessary non-empirical truths, I think you are stuck with infinity, unless there's a largest integer. )
But pure math has successfully avoided the pictures. It finally triumphed over an uncertain prop. The issue here is systematic reasoning in a strict language which seemingly must be discrete. Perfectly formal proofs of theorems are strings of symbols. The whole 'right or wrong' charm of math is caught up in this. As you may know, some engineers continued using infinitesimals when they were ejected by pure math, simply because they found them convenient and gave good results. Since then, infinitesimals have been rehabilitated via the hyperreals (they were made rigourous, without any metaphysical commitment), and a small minority of calc students learn calculus this way. I have a thin Dover book that presents them.
*Read lots of math books and you may end up an open-minded skeptic who sees the pros and cons of different approaches. You'll like system A for this charming thing and system B for another. They'll all get something right (for your intuition) and something wrong. Meanwhile all of them are correct in the sense of working logically, not appearing to lead to contradictions, and giving the expected applied results that help us build bridges that don't fall down. I'm no expert, but I think Newton's and Euler's calculus (lacking clear foundations) would be enough for most human practical purposes.
Pi only encodes a finite amount of information. [math]\displaystyle \pi = 4 \sum_{k = 0}^\infty \frac{(-1)^k}{2k + 1}[/math]. That's 16 characters if I counted right.
I don't see why this discussion is hung up on such an obvious point. Pi is no more mysterious than 1/3 = ...3333, another number that happens to have an infinite decimal representation. Decimal representation is handy for some applications and not for others. Decimal representation is broken. Some real numbers have infinite representations and others have two distinct representations. You should never confuse a number with any of its representations, so all of the other expressions for pi are just as valid.
https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80
I agree completely. I'm trying to show Ryan the difficulties with his approach.
I haven't been able to understand Ryan's approach. As in Apocalypse Now, when Kurtz asks, "Are my methods unsound?" And Willard responds: "I don't see any method at all, Sir."
I love that movie.
I'm trying to meet him half way, because I do find these issues fascinating. I don't think Ryan has the experience to see math as mathematicians see it. In physics and engineering classes, one can go very far without resolving these issues or even seeing a proof. So a kind of 'outsider's mathematics ' (like outsider art) is a natural result when a person gets mathematically creative. I agree that one has to study some actual proof-driven math to genuinely enter the game, but I also understand the impatience to talk about exciting ideas now. I'm sure that Ryan is learning, and I get to dust off some math training.
I like your quote and I see where you're coming from, especially given that I'm talking so informally. While I do not have the ability to formally present the idea, I did make a video in which I attempt to describe the intuition behind my method.
Would you please consider watching this 2 minute clip (watch from 1:56-3:48)?
Thanks!
I'm in complete agreement that an infinite process is a logically valid process. That's what I said, but it's counterintuitive to think that any process could 'live' forever. This is the same issue which the ancients had with the immortal soul. It's valid logic, but there's something wrong with the premises, which makes the conclusion unsound, despite the fact that the logic is valid. Furthermore, the "immortal soul" served as a very useful moral principle, just like the "infinite process" serves as a very useful mathematical principle, but usefulness does not necessitate truth, if truth is what we are ultimately after..
To say that a process is infinite is to say that it will run forever. That is what you are claiming with "infinite processes". Notice, that to say "X process is infinite", is to say "it will run forever", and that's a statement of necessity, just like to say "the soul is immortal" is to say it will, necessarily live forever. But of course you respect the possibility that the process may for some reason, at some time, cease. Therefore you say that it is "potentially" infinite. It could run forever, if it's allowed to. There is a clear difference between "it will run forever" and "it could run forever".
What I am requesting is that you recognize the constraints of the physical world, to acknowledge that an infinite process is not possible within our universe. Therefore both "it could run forever", and "it will run forever" are excluded as impossible, therefore false premises. From this perspective we see that any problem which requires an infinite process to resolve, is actually not solvable. If a question is not solvable, it is not properly posed, it is not a valid question.
You say: "I have no doubt that math works, it's 'why' that has puzzled me for many years." The simple answer is that it works because it conforms to the constraints of our universe. We can dream up a seemingly infinite number of logically possible axioms which will be completely useless in our universe. It's correspondence with the physical reality which makes them useful. Some people will make a distinction between coherence and correspondence, claiming that coherence is all that is necessary within a system of logic, but coherence itself is fundamentally a correspondence. Fundamental laws of coherence, like non-contradiction, work because they correspond with our universe. So a coherent logical structure has a basic correspondence simply by being coherent.
You might think that this completely contradicts what I said above, "usefulness does not necessitate truth". However, we must maintain the distinction between sufficient and necessary. Proof requires necessity. So despite the fact that correspondence (truth) works, we need to also be aware that there are other reasons why principles, like mathematical axioms, work. And this is dependent on the end, the goal which we have in mind, by which "works" is judged. If the goal is not itself truth, then the axioms are formulated toward that alternative goal, and "work" for that alternative purpose. I find that in the modern sciences, which are the principal users of mathematics, the goal has shifted from truth to prediction, and these two are not the same.
So I think you need to adjust your enquiry from 'why does math work?', to 'what does math do?' The point being that so long as people are applying it, it will work, otherwise they wouldn't be using it. The reason it works is because it's designed, shaped, conformed to the purposes which it is put to, like a finely honed tool. But if that purpose is something other than giving us truth, then sure it "works", but is what it's doing really good?
Quoting Ryan O'Connor
The issue here is with the way that I conceive of the relation between space and time. It is not conventional. With evidence derived from modern observations of phenomena like spatial expansion, it becomes clear to me that we need to give time priority over space in our modeling, to allow that space itself changes with the passing of time. This means that we cannot model spatial existence with a static 3d representation, adding time as a fourth dimension, because we need to allow that the principles for geometrical figures which model 3d space must actually differ being time dependent.
There is a need to produce two dimensions of time, one consistent with our present modeling of space as a static continuity of 3d existence, and the other to allow for the changes which occur to space, they require time as well, but this cannot be the same dimension of time. The finite points you refer to are the points where the two dimensions of time relate, or intersect with each other. So there is a continuum of spatial existence, extended in time, that is the classical 3d modelling. The points within that continuum need to be more fundamental because they represent where the other dimension of time intersects, thus constituting the real possibility for spatial existence. If we propose an infinite continuum of space, and we want to map onto this continuum real finite points of possible existence, then the limited location of those points must be derived from something real. That "real thing" must be more fundamental, because it represents real existence whereas the infinite continuum is the artificial map produced by us. The real points are points of spatial expansion (in this proposed model), which dictate the contortions to classical 3d spatial representations required to account for spatial expansion.
Quoting jgill
The problem with this "intervals" of time is that some sort of instants are still require to separate one interval from another. Anything posited to break the apparent continuity of time would require a distinct aspect of time, calling for a second dimension. And if the intervals in any way overlap then there is also a need for multidimensional time.
It was wrong of me to poke fun at you without responding to your last two lengthy posts to me. Fact is I wouldn't know where to start so it's better for me to leave it alone. If you can't graph a simple polynomial then there's no conversation to be had. Note that the poly I gave you has a real root at 2 which none of your pictures show. And your claim that there's no such thing as instantaneous velocity is falsified by your car's speedometer. I'm sure you have some interesting ideas but I probably won't engage much going forward, and I'll refrain from indirect remarks even if they are from great movies.
No problem at all. I appreciate the message! Although I don't expect a response, I do want to say a couple of things. I obviously know how to plot a polynomial in the traditional sense (and I also know how to use plotting programs). If you don't see a polynomial in my graphs it's because you don't understand my view (I'm not blaming you, this may be entirely my fault). Had I chosen to also plot y=0 then you would have seen the points corresponding to the roots. Your speedometer is measuring the average velocity but one measured over quite a short time interval. And I enjoy the quips, even that's all I hear from you.
Quoting jgill
If I didn't use the word topology would you have any other problems with my view? I glimpsed the article and paper. I certainly agree with his postulate: 'there is not a precise static instant in time underlying a dynamical physical process.' It seems obvious, really. But there are statements like 'there is no physical progression or flow of time' and 'a body in relative motion does not have a precisely determined relative position at any time' which I'm not convinced by. Overall, his paper is more like an essay than the type of paper I'd expect to see in a journal (but to be clear, my ideas are no closer to journal standards).
Quoting norm
I don't think it's a trivial assumption.
Quoting norm
Or as some quantum physicists say 'shut up and calculate'. I'm an engineer, I get that. But the armchair philosopher in me is not satisfied.
Quoting norm
Maybe, but maybe my lack of strict definitions is simply because my ideas are not mature yet.
Quoting norm
You have a good point so please allow me to soften my position. Perhaps pictures are only a handy prop in my view but the lack of symbolic reasoning may only reflect that my view is not mature.
Quoting norm
There are infinite potential chairs. Must all potential chairs actually exist to give the word chair meaning? The 'chairness' algorithm must be finite otherwise we'd never call anything a chair. Perhaps the same can be said about pi. Perhaps on the deepest level, pi is not the number pi, nor the infinite algorithms used to calculate the number pi, but instead the finite algorithm used to identify which algorithms would generate the number pi.
Quoting norm
I don't think your criticisms of finitism apply to my view. In my view, every system does have a largest number, it's just that there's no universal system containing all possible numbers. For example, in the graph below the largest number is 99498. We could certainly 'cut' the continuum to produce points with coordinates having larger values, but until we actually do that it is meaningless to assign coordinates to those potential points. Could you expand on how I'm stuck with actual infinity?
Quoting norm
I've read the Dover book on infinitesimal calculus by Keisler. It must be different from yours because mine isn't so thin. I'm not convinced that there are irrational numbers between the rationals, I'm even less convinced that there are infinitesimals in between the reals. But you're the professional and you've seen the proofs to conclude that the reasoning is rigorous so I don't want to debate about this issue.
Quoting norm
I am thoroughly enjoying this discussion and I appreciate your pointed questions. So far, my view is that you've clearly demonstrated how far my view is from a formal theory (thanks!) but you haven't identified any flaws yet. You're right, I don't see it as mathematicians see it. And so a mathematician might say that my probability of being right is 0. Thankfully, that means mathematicians still believe I have a chance!
Quoting Metaphysician Undercover
A potentially infinite process is one which will not end (unless prematurely terminated). Does this work for you?
Quoting Metaphysician Undercover
Well, can't the answer to the question simply be the infinite process? For instance, consider the question 'what is the area of a unit circle?' Is this a valid question? In one sense, I think you're right since no rational number will do. But in another sense, I think you're too strict in only accepting rational numbers. I think it's valid to say that the answer is pi, which I believe corresponds to a potentially infinite process. (Well my beliefs are changing a bit as I talk here with norm but I think you get what I'm saying).
Quoting Metaphysician Undercover
I don't think math is subordinate to physics. Both offer a path to truth.
Quoting Metaphysician Undercover
Only when we understand (truth) can we make a prediction. I think they're connected, but predictions make $$$.
Quoting Metaphysician Undercover
An engineer may see math as a tool but I imagine a mathematician sees math as something to be understood for the sake of understanding. Like a beautiful painting, it doesn't need any other purpose.
Quoting Metaphysician Undercover
What does time mean in the absence of a ticking clock? In other words, if all objects and space are static, has time actually passed? I don't think you have a good reason to believe that time has priority over space. I also don't see why 3D space needs 2 temporal dimensions to change. All of our experiences point to there being only 1 temporal dimension. What evidence do you have to support this claim? I find it a bit hard to follow your later statements, but anyways, until I understand the motivations behind your view, there's no point talking about fundamental points.
Velocity is always an average over a duration of time. So-called "instantaneous velocity" is just a derivative from an average. Since velocity is a measure of change, and change without a duration of time is impossible, then also true "instantaneous velocity" is also impossible. It's just a term of convenience, to be able to say that at x point in time, the velocity was such and such. What is really taken is an average over a duration, and from that we can say that the velocity at any particular point in time within that duration was such and such. But you can see from the applicable formula, that "instantaneous velocity" is really just another average. And it's quite obvious that the idea that something has velocity at a point in time, when there is no duration, is nonsensical.
Funny you should mention that, since it's so easily disproven.
Consider the speedometer in your car. How do you suppose it works? Is there a tiny little freshman calculus student in there, frantically calculating the limit of the difference quotient moment by moment?
No, actually not. Not even a computer program doing the same. Rather, there is a little induction motor attached to the driveshaft. The faster you travel, the faster the drive shaft spins, the faster the induction motor turns, the more current it outputs. And that current directly drives the needle of your speedometer.
Your speedometer is not a mathematically derived average. It is in fact a direct analog measurement of the instantaneous velocity of your car; subject of course to slight mechanical error common to any physical instrument. The velocity is an actual, physical quantity that can be directly measured -- that IS directly measured -- without recourse to any formal mathematical procedure.
You are confusing the velocity of an object, with the procedure we teach calculus [s]sufferers[/s] students to find the velocity of points moving in the plane or in space.
Quoting Metaphysician Undercover
No, as I have just explained, velocity is a directly measurable physical quantity, like the mass or volume of an object, or the wavelength or luminous intensity of a beam of light.
Quoting Metaphysician Undercover
No, as I've just explained.
Quoting Metaphysician Undercover
You're confusing freshman calculus with the actual, directly measurable instantaneous velocity of a moving body.
Quoting Metaphysician Undercover
No, because the calculus formalism is not the velocity, it's merely the way we determine velocity given the position function. But we don't need to do that if we have a direct way of measuring the velocity.
But even your remark about the formalism is wrong, because although the value of the difference quotient at any point is not the true velocity, but rather the slope of the secant line; the limit of the difference quotient is exactly the velocity. It is not an approximation.
Quoting Metaphysician Undercover
Yet another instance of the phenomenon whereby something false appears "quite obvious" to you solely by virtue of your lack of knowledge. You wield your ignorance like a weapon. A comic book character, Ignorance Man, whose slogan is "Believe the science!" while knowing none of it. Come to think of it there's rather a lot of that about these days, wouldn't you agree?
I'll take a run at your graphs when I get a chance. You went to some trouble to draw them, you deserve a response.
Quoting Ryan O'Connor
I agree that I don't understand your viewpoint. I have no trouble with adjoining points at plus/minus infinity to the real line, that's just the two point compactification of the real line. But the rest of it I couldn't follow.
Quoting Ryan O'Connor
I'll take a more detailed look at what you wrote and try to frame some specific questions.
Quoting Ryan O'Connor
Not so, please see my response to @Metaphysician Undercover here. Briefly, your speedometer is driven by an induction motor coupled to your driveshaft. It gives a direct analog measurement of instantaneous velocity without any intervening computation.
Quoting Ryan O'Connor
Often that's all I can manage. And I find many subjects in philosophy are best responded to with an old pop tune or a line from a film.
That's the gist of constructivism. It don't exist unless I can grab it! I was/am strongly attracted to constructivism, but it comes at a cost. Consider some great mathematicians were attracted to Brouwer's ideas, but they found that it was not worth all that had to be sacrificed for it. If you check out constructivist logic, you might like it but also find it disturbing in its own way. Intuitively, some things are true or false even if we can't say which. A Turing machine does or does not halt in some logical sense. There is or there is not a string '7777777' somewhere in the expansion of pi. If I understand correctly, a constructivist would disagree, since a constructivist acknowledges time. There's something like true, false, and undetermined. (Or that's how I remember it. Perhaps someone who knows better will chime in.)
Quoting Ryan O'Connor
I see your view as gestating. It's born for a mathematician when there are axioms and a logic. I hope you continue with it as long as you keep enjoying it.
Quoting Ryan O'Connor
Consider that the reasoning is dry and formal with no 'metaphysickal' commitment. A person could not even 'believe' in integers and still be great at pure math. (This is what I've seen people not realize, that math is agnostic on pretty much all ambiguous matters.) Do I believe in chess kings? Doesn't matter what I think if I publicly play by the rules. Nothing is hidden, or whatever counts epistemologically is not hidden. The primary obstacle I've seen in the transition to pure math from the calculus sequence is an implicit metaphysics that gets in the way. (I know a graph theory guy who thinks the continuum is a convenient fiction, and so on. The fictions as such are fun to play with. )
I understand this, but my point is that due to the nature of our universe, any such "potentially infinite process" will be prematurely terminated. So it doesn't make any sense to say that such and such a process could potentially continue forever, because we know that it will be prematurely terminated. Therefore, if we come across a mathematical problem which requires an infinite process to resolve, we need to admit that this problem cannot be resolved, because the necessary infinite process will be terminated prematurely, and the problem will remain unresolved.
Quoting Ryan O'Connor
I don't think so, because the process is the means by which the answer is produced. If the answer requires an infinite process, and the infinite process will be prematurely terminated, then the answer will not be produced.
Quoting Ryan O'Connor
The reason I am "too strict", is that I don't believe in coincidence, when it comes to mathematics. Call me superstitious, but I believe that in mathematics, there is a reason for everything being the way that it is. So when it turns out, that a circle cannot have a definite area, then I believe that there is a reason for this. The most likely reason, is that the circle is not a valid object. By "valid" here, I mean true, sound, corresponding with reality.
Here's a sort of anecdote. Aristotle, in his metaphysics posited eternal circular motions for each of the orbits of the planets. Motion in a perfect circle could continue forever because there could be no beginning or ending point on the circumference of the circle, as each point is the same distance from the centre. Of course we've since found out that the orbits are not perfect circles. What we can learn from this, is that despite the fact that the circle is an extremely useful piece of geometry, there is something fundamentally wrong with it, as a mode for representation. It is not real. And, with the irrational nature of pi, the circle actually indicates directly to us, that it is not real. So if we ignore this fact, insisting that we want the circle to be real, or that it must be real because it's so useful, and then we work around the irrational nature, creating patches, and fancy numbering systems to deal with all these seemingly insignificant problems which crop up from employing perfect circles, we are simply deceiving ourselves. We end up believing that the real figures which we are applying the artificial (perfect circles) to are actually the same as the artificial, because all the discrepancies are covered up by the patches.
Oh come on fishfry, you're smarter than this. The current you refer to is just measuring revolutions of the driveshaft. Then the speedometer of the car is scaled to how many revolutions are required to cover a specific distance. It is not measuring the instantaneous velocity of your car. What happens when you use the wrong size tires?
In the broader context of my general philosophical views, words don't have exact meanings and context is a dominant factor in what meaning they do have. So I can only grope at what 'actual infinity' means. This is the charm of pure math. There are strict definitions, cold and dry. But I think I've already answered your question. If you have the concept of a rational number, you're already there. Do you believe in a largest rational number? How many rational numbers are there? In general concepts shimmer with infinity.
Quoting Ryan O'Connor
Until a formal system is erected for examination, we're not doing math but only philosophy (but then I love philosophy, so I'm not complaining.) If you remember my first response, I suggested that the issue of fundamentally social. Who are your ideas ultimately for? Mathematicians or metaphysicians?
Quoting Ryan O'Connor
I think you'd probably enjoy looking into computability theory. Do you know about the halting problem? This is some of my favorite math. How do you know that there is a finite algorithim that always halts that can determine if other algorithms generate pi? But then you said that the algo that determines whether another algo generates pi is itself pi? This doesn't make sense. What a person might do is declare a particular Turing machine given a particular input to be a representative of pi and then include all equivalent-in-some-sense Turing machines as different representatives. This would have its own issues, but perhaps you see the charm of equivalence classes? You can start with something relatively concrete and define a notion of equivalence to scoop up the other entities that should be included in the concept. I don't need to know how many such machines there are. I can do some further proofs that show that addition and multiplication are independent of the representatives used. (Actually this technique blew my mind at first. It was when I first started feeling like a mathematician. AFAIK, there's nothing comparable in engineering.)
Bet you haven't seen this:
[math]\begin{align}& ArcTan(z)=\underset{k=1}{\overset{\infty }{\mathop L}}\,\frac{2z}{1+\sqrt{1+\tfrac{1}{{{4}^{k}}}{{z}^{2}}}},\text{ }\underset{k=1}{\overset{n}{\mathop L}}\,{{g}_{k}}(z)={{g}_{n}}\circ {{g}_{n-1}}\circ \cdots \circ {{g}_{1}}(z),\text{ } \\ & \pi =4ArcTan\left( 1 \right) \\ \end{align}[/math]
That's pretty rad!
One of my inventions (probably! You can't tell in mathematics.) :smile:
I'm going to guess that the proof is nontrivial. (Well, I'd be shocked if it was easy!)
I hadn't seen that symbol for composition before. I can actually use that in something that I need to get around to writing up.
Jeez man it's an analog computer. It gives a direct measurement of a physical quantity. You're saying there's no such thing as velocity. Your scientific nihilism is spreading from math to physics.
Arctan(1) is the proof of the Leibniz formula. What's the meaning of [math]L[/math]?
With you being a crankologist, I'd really benefit from your criticisms and I think you'd enjoy learning my view as I believe I am coming at infinity from a unique angle. As such, I think you'd need a different strategy to take down my ideas (assuming I'm wrong). But your time is short and crankery is infinite so whether you find time or not, it's all good.
Quoting fishfry
You're definition of the instantaneous velocity of a car rests upon a dynamic quantity: the flow of electrons through a wire (i.e. current). So you've only shifted the problem from instantaneous velocity to instantaneous current. Consider this example.
You: Your instantaneous velocity is 10 km/h.
Me: How do you know?
You: Because I'm running right next to you and my instantaneous velocity is 10 km/h.
This begs the question, how do you know your instantaneous velocity? For instantaneous velocity to make sense, it needs to be based only on static quantities that exist at that instant. But just as you can't look at a photograph and determine how fast I'm running, you cannot come up with a meaningful definition of instantaneous velocity.
Now, if you were referring to my GPS-based speedometer then yes, inside my phone is a little freshman calculus student that does the math, not calculus, just a simple delta_s/delta_t.
Edit: If the needle position in your speedometer is indeed only based on instantaneous information it must be doing so like a spring, which deforms as a function of the force applied. One could come up with a correlation between spring deflection and velocity, but this is only approximate. It is not a true measure of instantaneous velocity.
On the contrary, I have too much time on my hands. I've been over active on this forum lately and I'm feeling the need for a break. I think my reacting negatively to @Wayfarer's helpful link to a Sabine Hossenfelder video was a clue. I'm just crabby lately for the sake of being crabby and when I find myself doing that it's time for a forum break. Sorry @Wayfarer, I apologize.
Quoting Ryan O'Connor
Yes but this is true of any physical quantity. How do I measure the mass of a bowling ball? Well I can put it on a scale, but that only measures weight and not mass. The weight would be different on the moon.
So to measure mass, we must observe the bowling ball's acceleration response to force, as described in this fascinating thread.
https://physics.stackexchange.com/questions/179269/how-do-we-measure-mass
How can we do that? We can suspend it from a spring with a known spring constant using the formula for a harmonic oscillator. But then you (and @Metaphysician Undercover) will object that all I'm doing is measuring the springiness of the spring.
Or I can measure the centripetal force on a centrifuge, or use a small angle pendulum. But in each case aren't we just measuring something about the apparatus and not the mass itself?
In short, your objection is valid, but overly general. We can't measure any physical quantity at all by your logic. What if I want to measure the wavelength of a beam of light? Well I use a spectrometer, but all that really measures is the prism or the glass or however spectrometers work.
What if I want to measure the temperature of air? I use a thermometer, but that's only measuring the response of mercury or the coil of a metal spring or however thermometers work these days.
So what you and @Meta are saying is that we can't measure ANY physical attributes at all. This is hardly an objection to my point about velocity being directly measurable without recourse to formal calculus. It's a philosophical objection to the idea that we can do any measurements whatsoever, or to the idea that objects even have physical attributes before we measure them by proxy. But that doesn't actually address the different point that I'm making: That moving objects have a velocity, which we can measure directly (by proxy with an induction motor coupled to the driveshaft), without needing formal symbolic methods of calculus.
After all, bowling balls fall to earth with an acceleration of -32 feet/sec^2, and this was true even before Galileo discovered it and Newton modeled it with his law of gravity. You and @Meta can not deny that bowling balls fall down and that they do so with a measurable velocity at any instant of time; without denying the whole of physical science. You don't need calculus to know that falling bowling balls have a velocity. You're both confusing the mathematical model with nature itself. And the fact that all measurements require some intervening apparatus is a red herring.
Hey no probs, really nothing personal. I thought it might have been relevant, that's all. I find your knowledge of philosophy of maths really interesting.
I was one of the many students who failed terribly at maths. I would feel my grasp of the work slipping away in class, and could never catch up. But later in life, I've come to appreciate maths aesthetically, even though I can't understand it very well. I also think the question of the nature of the reality of number really is important. I was just perusing that article on Hossenfelder's website again, and came across this remark in the combox:
I think that comment is dead wrong. There is a matter of fact about this issue. I think fictionalism really is a white flag in philosophical terms. It might not matter for applying mathematics, but it makes a major difference to your conception of the nature of reality. When you consider the predictive power of maths, the fact that through it you can discover things about reality that you otherwise could never know - how is that reconcilable with the idea that it's a 'useful fiction'?
"The Unreasonable Effectiveness of Mathematics in the Natural Sciences"
My former and late father-in-law's good friend Eugene Wigner raised this question years ago in a famous paper. I agree - I don't perceive it as a useful fiction.
Quoting fishfry
Here's what's going on, with a simple example:
[math]\begin{align}& {{g}_{2}}\circ {{g}_{1}}(z)={{g}_{2}}\left( {{g}_{1}}(z) \right) \\ & {{g}_{1}}(z)=2z+1,\text{ }{{g}_{2}}(z)={{z}^{2}}-3\text{ }\Rightarrow \text{ }{{g}_{2}}\circ {{g}_{1}}(z)={{\left( 2z+1 \right)}^{2}}-3=4{{z}^{2}}+4z-2 \\ & \underset{k=1}{\overset{2}{\mathop{L}}}\,{{g}_{k}}(z)={{g}_{2}}\circ {{g}_{1}}(z)={{g}_{2}}\left( {{g}_{1}}(z) \right) \\ \end{align}[/math]
[math]\underset{k=1}{\overset{\infty }{\mathop{L}}}\,{{g}_{k}}(z)=\underset{n\to \infty }{\mathop{\lim }}\,\underset{k=1}{\overset{n}{\mathop{L}}}\,{{g}_{k}}(z)[/math]
The process arises using techniques from functional equations. For example,
[math]\begin{align}& f(z)={{e}^{z}}-1\text{ }\Rightarrow \text{ }f(2z)=f(z)\left( f(z)+2 \right) \\ & f(z)=z(z+2)\circ f(z/2)=\left( \frac{{{z}^{2}}}{4}+z \right)\circ 2z\circ f(z/2)=\left( \frac{{{z}^{2}}}{4}+z \right)\circ 2f(z/2) \\ & \text{ =}\left( \frac{{{z}^{2}}}{4}+z \right)\circ \left( 2{{z}^{2}}+4z \right)\circ f(z/4)=\left( \frac{{{z}^{2}}}{4}+z \right)\circ \left( \frac{{{z}^{2}}}{8}+z \right)\circ 4z\circ f(z/4)=\cdots \\ & \text{ =}\underset{k=1}{\overset{n}{\mathop R}}\,\left( \frac{{{z}^{2}}}{{{2}^{k+1}}}+z \right)\circ {{z}_{n}},\text{ }{{z}_{n}}\to z \\ \end{align}[/math]
But here
[math]\underset{k=1}{\overset{3}{\mathop R}}\,{{f}_{k}}(z)={{f}_{1}}\circ {{f}_{2}}\circ {{f}_{3}}(z)[/math]
etc. Going the other way involves rules for inverses of compositions.
This sort of thing appears in a general study of infinite compositions, a topic of practically no interest in the larger mathematics community. Abstractions and generalizations are more attractive.
Iterated composition. Hadn't seen that notation before. Thanks.
A cursory search shows that I can't find it used anywhere. "Today I learned!"
Oh -- L and R. Nice. I thought iterated functions were getting a lot of attention these days. Fractals and such.
When teaching almost thirty years I would publish a paper each year just to avoid collapsing intellectually, and all were readily accepted and published, but after I retired in 2000 I decided to do the research for me and anyone else who might be interested, avoiding the formalities of publishing. Hence a collection of informal notes on researchgate. It's amazing the audience one reaches there. Virtually every civilized nation has cropped up with reads of my stuff. But fewer than half read the entire note. Still, that can be 20-30 a week that do. And it's free, unlike most journals - and that irritates me since few journals pay the referees, at least when I was active. (in all fairness, institutions take up the slack with fewer teaching hours, etc.)
I consider the major journals corrupt in their financial practices. But that's just me. Pay no attention.
I'm not denying measurement at all. If an event can be completely described by a photograph then it is instantaneous. If it needs a video to be accurately described then it is a transient event. The only way to report a transient event at an instant is to compress the transient data, e.g. by time averaging it. I have no problem with (at some given instant) reporting a velocity, as long as we recognize that that quantity is not the velocity at that instant, but instead some average velocity over a short interval.
And in the case of a speedometer, I wasn't trying to say that everything is relative to everything else so measurement is meaningless. I was only saying that the problem is not solved by pushing it downstream (from a moving car to moving electrons). Measuring current is a giveaway that you're measuring a transient phenomenon. If you were measuring capacitance on the other hand (which is analogous to a spring) then you could be measuring an instantaneous event.
Quoting fishfry
Your original claim was that my rejection of instantaneous velocity is falsified, which I think is false. If you now claim that the speedometer must necessarily be reporting some average or approximate velocity then I have no problem with that.
What do you think speedometers measure?
I take your point about instantaneous motion, it's related to one of Zeno's paradoxes. If the arrow is not moving at a given instant, how does it know it's moving, or something like that. I don't think I have to resolve that ancient mystery to read my speedometer and know that it's telling my my instantaneous velocity. Even if it's only actually reading a current from an induction motor.
Bonus question. What is the speed of light at any given instant?
I do like this idea and I need to find out why it's disturbing.
Quoting norm
That's a reasonable statement.
Quoting norm
It reminds me of the Chinese room argument in AI. Someone locked in a room could be blindly following a set of instructions to take chinese character inputs and generate outputs to convince someone that they speak chinese. It sounds like your view is that math is like this chinese room, a mere set of rules and symbols. I believe that it's more than that. I believe that the messages have content (like the Chinese messages), and if we better understand what it's saying perhaps math will be easier.
Quoting norm
My response is 'how many rational numbers are where? Present me with a graph and label all points explicitly and I can tell you what the largest number on your graph is.
Quoting norm
"Without mathematics we cannot penetrate deeply into philosophy. Without philosophy we cannot penetrate deeply into mathematics. Without both we cannot penetrate deeply into anything." Leibniz.
I don't think we have to decide between the two...but you're right, at this point I'm only philosophizing. But hey, I'm on a philosophy forum!
Quoting norm
Yes, I like the halting problem and anything else that deals with incompleteness. I don't know if such an algorithm exists. But as it is we define pi as an equivalence class of particular cauchy sequences. All I'm proposing is that instead of pi being the equivalence class itself that it is the description of that equivalence class. Surely our description is finite, right?
@jgill
Incredible!
You seem to be missing the point fishfry. Velocity is a measurement of motion, and motion only occurs when time is passing. At an instant zero time passes. Therefore there is no motion at an instant, and no velocity at an instant.
A measurement of velocity requires a determined distance over a determined duration of time. It requires two instants, to determine a duration of time, one to mark the beginning of the period of time, the other to mark the end of the period of time, just like it requires two points to determine a distance. One instant (point in time) is insufficient for a determination of velocity, just like one point is insufficient for a determination of distance.
Not for the first time I'm sure.
Quoting Metaphysician Undercover
Yeah yeah. One of Zeno's complaints. If you look at the arrow at a particular instant it's not moving. How does it know what to do next in terms of direction and speed? Not a bad question actually, one that I won't be able to answer here.
Quoting Metaphysician Undercover
Yes you already said that. I take the point. If you show me a photo (taken over a sufficiently short time interval, since even a photograph takes time) the arrow appears stationary and you can't determine its velocity.
Yet, it still HAS a velocity, wouldn't you agree? And what does a speedometer measure? A current. And as @Ryan pointed out, even that's a flow of electrons. Since current is a flow, does it exist at an instant? Well I'm sure that a modern physicist would point out that the electromagnetic field exists at every moment. But if you have a magnet in a coil, you have to move the magnet to create a current. I'm really not enough of a physicist or a philosopher to know these things. Good questions though.
Still, would you at least grant me that velocity over a short but nonzero distance exists? And likewise a current flow? Then a moving car has a velocity that can be determined without recourse to formal symbolic manipulations, which was my original point.
Moderator note: When I hit the @ button to search for @Ryan O'Conner his handle doesn't come up, any ideas why?
Sure, the object is described as moving, it must have a velocity. But it cannot have a velocity at an instant, if no time passes at an instant, just like a point has no spatial extension. That's why points and lines are incompatible, and a line is not composed of points, but points mark off line segments.
So the solution to the issue with velocity, is not to say that it has no velocity, it is to say that there is no such thing as the instant. Time is not composed of instants. So the arrow, or car always has velocity, all the time that it is moving, but that time has no instants. The instant is just an arbitrary point which we insert for the purpose of making a measurement.
Quoting fishfry
Sure, but the whole point I am arguing in this the thread is that the inclination to reduce the nonzero distance to zero, or even define it as somehow related to zero, produces theoretical absurdities. And this is well demonstrated by these Zeno type paradoxes which speak of time as consisting of instants.
Imagine water flowing uniformly out of the faucet onto a flat plate. Below the flat plate is a spring which compresses. We can determine a relationship between the spring compression and the water flow rate. This is essentially how an analogue speedometer works, but with electrons instead of water.
This description may give the impression that the spring can measure instantaneous velocity but it cannot.
Consider this: the first instant the water hits the plate, the spring is not compressed. It then takes some time for the spring to find the equilibrium position, and only at that time will it report the correct flow rate. During this transient period the spring is not reporting the correct flow rate, but instead some value between zero and the actual flow rate. It's reporting some sort of average.
And as you drive your car continually accelerating and decelerating, the spring behind the needle is continually playing 'catch-up' and thus reporting some sort of average. In fact, it is most meaningful to say that it is always reporting an average.
I'll accept this point. It still has nothing to do with what I originally said, which is that you don't need calculus to determine the instantaneous velocity of a moving object. And I'll concede that by instantaneous I only mean "occurring over a really short time interval."
I have to say I'm not nearly as invested in this point as the number of words written so far, I should probably stop.
Ok. I can live with that. Whether it's a moving arrow or a current driving the speedometer, it's a change occurring over a short interval of time. But my original point was that we don't need calculus to determine the velocity. Actual velocities are not subject to the ancient philosophical mysteries of calculus.
Still, would you at least grant me that velocity over a short but nonzero distance exists?
— fishfry
Quoting Metaphysician Undercover
Ok. Maybe. Let me put to you a hypothetical. An object moves with constant velocity. Does it have a velocity at a given instant?
Likewise does the speed of light have velocity 'c' at a given instant?
I'm kind of done with this topic, the point I'm making isn't worth all this ink. You don't need calculus to do analog measurements. And yes physical measurements depend on time, even if those intervals are tiny. There aren't any actually physical instants as far as we know. Or as far as we don't know. The matter is not answered by current science.
No, because "a given instant" is not anything real which can be adequately identified. We can attempt to arbitrarily assign an instant to time, to mark a point for the purpose of measurement, but that assignment becomes much more difficult than it appears to be, at first glance. To mark a temporal point in one process or activity, requires a comparison with another process or activity, thus requiring a judgement of simultaneity. According to special relativity such judgements are dependent on the reference frame. Therefore any "given instant" may not be the same instant from one frame to the next, and the question of what a thing's velocity is at a given instant is rather meaningless because it depends on what frame of reference you measure it in relation to.
Quoting fishfry
I don't think you've adequately considered what is required to produce accuracy in a time related measurement.
The arrow may be momentarily stationary, but it has momentum.
Can you explain this to @Metaphysician Undercover and @Ryan whose handle doesn't show up when you use the @ button?
But actually it's a good question. Suppose there were such a thing as an instant of time, modeled by a real number on the number line. Dimensionless and with zero length. So the arrow is there at a particular instant, frozen in time, motionless. Where does its momentum live? How does it know where to go next?
Does it have, say, "metadata," a data structure attached to it that says, "Go due east at 5mph?" You can see that this is problematic.
Isn't this just confusing a snapshot of a thing with the thing itself? The movement is not divided into discrete moments until we try to model it using mathematics. Yet mathematics is distinct from the event, as it is merely a way to model the event (albeit, an extremely useful and accurate enough model). We necessarily measure momentum in retrospect, so when we analyse a things momentum at a particular instant it is already a passed instant. However, just because we apply delineation through our act of retrograde analysis, and create a mathematical notion of temporality, that doesn't mean that the thing did not have a momentum at a particular instant. It's only a question of accuracy.
Or pehaps one day scientists will discover some kind of 'instance particles'. Now that would be interesting.
Edit - I guess I lean towards the (Bergsonian) idea of an essential indivisibilty of time/movement.
The point being, that you cannot take the arrow at a particular moment in time. This is an impossibility because time is always passing, and this would require stopping time at that moment. So, despite the fact that using mathematics to figure hypothetical conditions at particular moments is a very useful thing to do, what it provides us with is a representation which is actually a falsity. Then if people start talking about this situation, with the underlying implication that this mathematics provides us with some sort of truths about these situations, this talk is really a deception or misinformation.
Quoting fishfry
This is the key point to understanding temporal continuity, inertia, Newton's first law, and the overall validity of inductive reasoning in general. We observe that things continue to be as they were, as time passes. Intuition tells us therefore, that they will continue to be as they were, unless something causes them to change, and this intuition is what validates inductive reasoning. However, there is a very real, and very big problem, and this is the reality of change. We see that human beings have the capacity to interfere with, and change the continuity of inert things. Because of the reality of change, we are forced to accept the fact that this continuity is not a necessity. This appears to be the hardest thing for some people, especially those with the determinist mentality, to accept, that the continuity of existence, which we observe, is not necessary. This means that the supposed brute fact, which underlies all inductive principles as supportive to those principles, that things will continue to be as they have been, is itself contingent, not necessary.
When we take a law, like Newton's first law, we view it as something taken for granted. The law tells us the way things are, and it's assumed that it's impossible for things to be otherwise, that's why it's "the law". However, when we apprehend that this is not necessary, then we can grasp the fact that there is a need for a reason why the law upholds. Through principles like the law of sufficient reason, we see that if there is a temporal continuity of existence, described as momentum or inertia, and this feature of existence is not necessary, then there must be a reason for it, a cause of it.
What Aristotle did was posit "matter" as the cause of the temporal continuity of existence. So contrary to the common notion that matter is some sort of physical substance, "matter" by Aristotle's conception is really just a logical principle, adopted to account for the observed temporal continuity of physical existence. It is, in a sense, a placeholder. He didn't know the cause, but logic told him there must be a cause, so he identified it as "matter". In your example of the arrow, we do not know "how does it know where to go next", but we do know that it does. Aristotle attributes this to its "matter", or more precisely he posits "matter" as what causes it to go, where it does go, next. Therefore the theoretical points in time are in reality connected to one another by what is called "matter".
I'm confused by your post. You make the correct point that the math is just a model of reality, not reality itself. Then you say that the thing DOES have a momentum at a particular instant. Which is what the mathematical model says.
I'm mostly in agreement with you on this point, as reading the rest of my post would have indicated.
Yes, I read the rest of your post, as reading the rest of my post should have indicated to you.
Kinda off topic, but have you ever seen a generalisation of the iterated composition operator to non-natural indexes? Like... does the following notion make sense in general:
Let's say we have [math]g=f \circ f[/math], does it make sense to think of [math]f[/math] as like half an application of [math]g[/math]?
Are there analogous constructs for an [math]f[/math] which is [math]\frac{1}{\pi}[/math] applications of [math]g[/math]?
The beauty of calculus is that in performing a finite number operations (e.g. in manipulating the exponents and coefficients of a polynomial to determine the derivative) we can talk sensibly about a potentially infinite process. If you reject potentially infinite processes as valid mathematical objects then you must reject calculus, and nobody will buy into your philosophy. You've got to stop thinking of the output of a potentially infinite process as the mathematical object. The mathematical object is the process itself. If you're challenging the orthodox view on real numbers then your point is valid, but if you're challenging my view then your point is misdirected.
Quoting Metaphysician Undercover
I agree. By treating rationals and irrationals both as the same type of object (i.e. numbers) we blur the line between the output of a finite algorithm and the output of a potentially infinite algorithm.
What do you make of 1/3 = .333...? Can't you distinguish between a number and one of its representations that you don't happen to like? After all 1/3 is just a shorthand for the grade school division algorithm for 3 divided into 1.
Yes. Goes back a hundred years if my memory serves. Sometimes it's very easy. For example, here is a linear fractional transformation written in terms of fixed points and multiplier.(I'm working on a theorem right now involving this). I think a more general case was dealt with in the discipline of functional equations. Can't recall the work offhand.
[math]\begin{align}& f(z)=\frac{(\alpha -K \beta )z+\alpha \beta (K -1)}{(1-K )z+(\alpha K -\beta )},\text{ }{{F}_{1}}(z)=f(z),\text{ }{{F}_{n}}(z)=f\left( {{F}_{n-1}}(z) \right) \\ & \Rightarrow {{F}_{n}}(z)=\frac{(\alpha -{{K }^{n}}\beta )z+\alpha \beta ({{K }^{n}}-1)}{(1-{{K }^{n}})z+(\alpha {{K }^{n}}-\beta )} \\ & Define\text{ }{{F}_{t}}(z):=\frac{(\alpha- {{K }^{t}}\beta )z+\alpha \beta ({{K }^{t}}-1)}{(1-{{K }^{t}})z+(\alpha {{K}^{t}}-\beta )},\text{ }t\in R \\ \end{align}[/math]
I'm not looking for people to buy in, I'm looking for truth. If others are looking for the same thing, they might like to join me. Otherwise I don't really care if people might deceive themselves into thinking that they are engaged in infinite processes. Many think that the soul is eternal, and this doesn't both me either. I consider those two beliefs to be very similar.
Quoting Ryan O'Connor
There is a fundamental incompatibility between an object and a process, which was demonstrated by Aristotle. If an object changes, it is no longer what it was. We assume a change (process), to account for the object becoming other than it was. So we have object A, then a process, then object B, whereby object A becomes object B. If we represent the intermediary between A and B as another object, C, then object A becomes object C which becomes object B. Now we need to assume a change (process) to account for object A becoming object C, and a process to account for object C becoming object B. We might represent the intermediaries between A and C, and C and B, as objects again, but you can see that we're heading for an infinite regress. So we ought to conclude that "objects" and "processes" are distinct categories.
Time for you to develop a new axiomatic system, then, that leads to "Truth".
Quoting Metaphysician Undercover
Agreed.
Pi is a finite number because it's inbetween 3 and 4. But if the length of a circumference is multiplied by pi than you have a length with space corresponding to each number, so the circle has infinite space within a definite finite limit (like being inbetween 3 and 4). Aristotle never understood this stuff
I'm thankful for that. :roll:
I believe that's a false equality. The correct statement is "the potentially infinite process defined by 0.333... converges to the number 1/3" not "the number 0.333... equals the number 1/3". Decimal notation is flawed in that it cannot be used to precisely represent some rational numbers, like 1/3. If we want a number system which can give a precise notation for any rational number, we should use Stern-Brocot strings, where 1/3 = LL.
Quoting fishfry
If photographs can't capture motion but videos can, why not conclude that motion happens in the videos? The reason why we are reluctant to come to this conclusion is because we reject the notion of videos being fundamental.
We want points (photographs) to be fundamental and continua (videos) to be composite and as long as we hold this view we will not find a satisfactory resolution to Zeno's paradoxes. If you flip things upside down and see continua as fundamental and points as emergent, then everything makes perfect sense. There's no problem with pausing a video to produce a static image.
Quoting fishfry
It is clear that you appreciate the profoundness of Zeno's Paradox. Zeno presented these paradoxes in response to the criticisms from the 'one from many' camp calling his views ridiculous. Why not consider the 'many from one' view that he supported? He was wayyy ahead of his time so his view did seem to have problems of their own...but in light of modern advancements in physics his view no longer seems crazy.
Quoting fishfry
Don't stop here, you may just be on your way to becoming a crank! With this admission you have placed yourself on a slippery slope. Instantaneous velocity is no different from the tangent of a function at a point. Do you accept that the derivative corresponds to a limiting process of secants rather than the output of a completed infinite process (i.e. tangent at a point)?
Quoting fishfry
Only if you consider 0/0 a valid velocity.
Never in the history of mathematics or physics has discovering truth set us backwards. The fact that your philosophy would result in a weaker mathematics is a red flag that you're on the wrong track. Don't get me wrong, I agree that there's a problem, I just don't agree with your resolution.
I made this video on my proposed resolution to Zeno's Paradox. What do you think?
Quoting Metaphysician Undercover
I think you're taking my words too literally. Clearly, the process of taking my dog for a walk is not an object with mass and momentum. When I say that processes are valid objects of mathematics, I simply mean that they can be studied in themselves, just as one might write a book entitled 'The Art of Dog Walking'.
Continua based constructions are based on an uncountable infinite amount being manipulate to a finite result. That is new. That's exactly what Newton did.
I don't think you solved Zeno's paradox because you're putting the infinite quantity into philosophically blurry box and focusing just on finite results. Zeno said "the finite" and "the infinite" we're inherently exclusive of each other and so he concluded that idealism was true (a form of idealism wherein there are no 1s and 1s to be added into two)
Isn't that what we do with quantum mechanics? We have finite results corresponding to our actual measurements and everything between the measurements is a 'blurry' superposition? Why must everything be 'sharp'?
That's just how math is
It's true that in my video I showed a blurry curve in between the measurements, but prior to that I showed the curve being 'topological' [ I use 'topological' only in the sense of the graph having properties which are preserved through continuous deformations]. If instead of saying that I'm 'blurring' the graph, what if we say that I'm reinterpreting it from being a geometric object to a topological object? With this view, your argument can't simply be 'that's just how math is' because topology is certainly acceptable math.
Idn. I've been recently working on this question from the angle of non-Euclidean geometry. I'm trying to understand what space even is
At every time interval we will find the car moving. At every instant in time the car's motion is indistinguishable from that of a parked car. It's Zeno's Arrow Paradox. Did you know that if a quantum system is continuously being observed that it will not evolve? It's called the Quantum Zeno Effect. As confirmed by experiment, motion (i.e. change) only happens when we're not looking. It happens in between the measurements. It happens in between the measured instants in time. And what exists in between the instants? - continua.
When you draw a graph, do you think you actually draw infinite points placed perfectly on the page? Or do you place a few points on the page accurately and then imprecisely connect the dots. What I am proposing is in line with how we've always drawn graphs. And if we're honest with ourselves, it's the only way to draw graphs. It would take an infinite being to draw a graph with perfect precision. Let's not make math dependent on the existence of an infinite being.
Continua is infinitely pointed. So it has instants all over it. If things move at the top level but fundamentally unmoved in subdivisions an object can't move at all. How I see it, we need to say "the infinite" is on one side and "the finite" is on the other and motion is movement between them
This is the view that we've been indoctrinated with. We can't help from thinking that points (i.e. instants) are fundamental and so we believe that continua are just a collection of points. Zeno's paradoxes reveal that this view has serious problems. Instead, consider the alternative: that continua are fundamental and points are emergent. Points only emerge when a measurement is made. When we draw a graph, we only label a few points where our curves intersect. Those are the only points on the graph. Don't look in between the curves and conclude that there exist uncountably-infinite points there...what lies in between the curves are continua.
Quoting Gregory
I think your intuition is close. But I would say that motion happens in the chasm of space (filled with infinite potential) in between "the finite".
It's not time for me to do that, I'm not a mathematician. There's something called the division of labour. The person who puts one's efforts into pointing at the problems in existing systems need not be the one who produces the repair. Of course the people using the system would probably not like the person pointing and would have the attitude of 'if you think you can produce a better system than ours, then do it'.
Quoting Gregory
At least I'm not alone then, because I haven't got a clue what you're saying.
Quoting Ryan O'Connor
You demonstrated that you do not grasp the need for the point to be prior to the line, therefore your claim that it would result in a weaker mathematics is based in misunderstanding. What quantum physics demonstrates to us is that points have real existence, and continuities are constructed.
Quoting Ryan O'Connor
I don't see how you get from points to continua. You show measurement points, then you assume that there is some sort of continuum connecting those points. The problem I see, is if certain measurement points are actually possible, then these must be represented as real points which the moving object actually traverses and can be measured at. That's why I give priority to these points, as the real features. The supposed continuum might not have any sort of linear existence at all, in fact we might not have the vaguest idea of how the points are related to each other in the underlying substratum of reality, which produces the appearance of a continuum. For all we know, the object might appear at point A, then completely disappear, and then reappear at point B a moment later, and this is what appears to us as motion.
The reason why I say that priority must be given to the points, is that whatever it is about the underlying substratum which produces the appearance of continuity, this 'power' must be constrained by possible points of appearance. If there wasn't such constraints then we'd have the problem of infinite points where the object could be measured. Furthermore, the nature of spatial expansion demonstrates that there must be points where expansion is centered.
So I find the video mostly acceptable, but what you are really showing is a points based motion, points where the object might be measured to be at, and you are assuming that there is some sort of continuum which underlies the points and connects them. Therefore all you need to do to be consistent with my perspective, is put the points as primary, being the real constraints of real space, and allow that whatever continuum emerges from existence at the points it is a creation produced from the relationships between the points, and this set of relationships comprises the substratum.
Quoting Ryan O'Connor
I have doubt in the truth of this. Are processes valid objects of mathematics, or ought they be relegated to physics? Let's start with something simple, assume that a number is an object of quantitative value. So '4' represents such an object, it must be a static and unchanging value to maintain its validity, therefore it cannot be a process. Now let's say that in '2+2', the '+' represents a process. So the inquiry is whether the process represented by '+' is a valid mathematical object to be studied by mathematics. We need to determine what the '+' means. What does it mean to add one unchanging quantitative value signified by '2', to another? Mathematics does not answer this inquiry, it just makes an assumption about how processes like these affect quantitative values. And we can see the same with the other processes, multiplication, division, etc., these process affect quantitative values, but if quantitative values are what are properly referred to as objects, then these processes are something different.
Discrete curves?
Uhhh
Quoting Metaphysician Undercover
I thought you were Aristotilean. You must be aware that Aristotle rejected points (infinitesimals) and instants
Aristotle also posited eternal circular motion, which is nonsense.
Eternal circular motion is fine. What is stupid is what you said about math not defining what addition means
You know, this is more tricky than it looks. Suppose you have a long exposure time. Then you'll see a blurred image, and you can work backwards to determine the velocity. Photographs are not instantaneous. The shutter stays open for a period of time, usually a fraction of a second. During that time the film or digital sensor collects photons. So there's an element of time involved even in a photo. If the object is moving slowly relative to the shutter speed you won't see blur, but in theory the blur is always there. If I can choose the shutter speed I can always tell you how fast the object was moving by analyzing the blur.
@Ryan same point to you. In fact your earlier point is correct, any measurement is taken over time. There's no difference between photo and video. Video after all is just a collection of still images, either analog or digital frames. And a single photo is taken over a period of time, namely the shutter speed.
OK then, show me this perpetual motion which you know about.
Quoting fishfry
That's why velocity is always an average, requiring at least two temporal points. Duration is derived, just like distance is. To infer an instantaneous velocity requires a second derivation.
I apologize for calling your statement stupid. I had just had a fight with someone and your comment annoyed me. It seems you are always debating fishfry or someone about numbers and there relation to Kantian synthesis vs an analytic view. To me that's just a discussion about psychology and mathematics does truly take care to define what addition, subtraction, multiplication, and division are. I have not seen where you have a unique insight into the issue. But on eternal motion, Einstein and countless physicists believed in it. Eternal inflation, the "big bounce" , and all these ideas are just noting more than versions of it. I'm very aware of Aristotle's arguments about an accidental series (one that stands on its own) and an essential series (one with supernatural support). I've discussed this with Thomists who have PhD's. There is no consensus on philosophy on this. I think it's a physics mathematica question and that calling on supernatural support is unnecessary since I can describe it in terms of physics. But this isn't the thread to go into that, since I see no connection between it and continua
Why does a segment with a length of finite digits change into a length multiplied by pi (pi×2×r) when the segment is made into a circle? The circumference will have digits going to infinity while as a segment it did not? This must be readily explained in mathematics but I don't remember ever seeing an explanation on it
I just brought it up as a topic. fishfry is of course correct. :smile:
Quoting Gregory
This is indeed a puzzle.
I think you are indeed pointing at the problem but when you start talking about your solution involving multiple time dimensions and spatial expansion, I find it hard to follow. It all seems like mumbo jumbo. Give me something concrete to chew on. Does your philosophy produce any graphs or equations? How does your philosophy make sense of the infinities in calculus?
Quoting Metaphysician Undercover
I don't agree with this claim so I'd like to see your evidence that supports it. What is fundamental in quantum physics is the wave function, a continuum. Definite states (like points) only emerge when a measurement is made.
Quoting Metaphysician Undercover
I'm not going from points to continua. I'm going from continua to points. My graphs and videos aren't seeming to help here so let me expand on this quantum analogy. I'm not a quantum physicist so take this with a grain of salt.
Assume that there exists a wave function of the universe that spans all of time. This is the fundamental object of our universe and it is a continuum. And until the wave function is measured it is meaningless to talk about who lived when and where because a wave function does not describe what is, it describes what could be. It is only when you make a measurement that all of the potential states collapse into a single actual state. When I say that points are emergent, I mean that they only emerge when we make a measurement. We cannot say things like 'there are infinite points on this line' because we have not actually placed infinite points on the paper...what we placed on the paper was a line.
Quoting Metaphysician Undercover
Put it this way: a computer program that calculates 2+2 is what I mean by 'process' and such a program can be studied (even if the program is never executed).
No, continuous curves.
Quoting tim wood
What you don't realize is that it's your description which stops the car from moving. That's Zeno's paradox. Motion is impossible if time is just a collection of instants. My description allows the car to move because I'm allowing for time to be more than just instants. In between the measured instants lies an unmeasured wave function within which motion is possible. Motion happens when we're not looking. It's demonstrated by the Quantum Zeno Effect.
Quoting fishfry
Imagine a dark room and a quantum sensor (which I'll call a film to stick with the photography analogy). The moment a single photon hits the 'film' we have a 'photograph'. This photograph has no blurriness and captures no motion. It is a photograph, not a video. There is no physical law which states that we cannot know the position of a particle with perfect precision. It's just that if we do know the particle's position precisely, we can't know anything about its momentum (velocity). Static measurement (e.g. position) and dynamic measurements (e.g. velocity) are both valid, important, and distinct from each other.
Videos are not just a collection of still images. They are a collection of still images where each image is displayed for some non-zero duration. If each image was displayed for 0 seconds then you wouldn't get a video, no matter how many images you pile on. It all comes back to 'how can you form a line from a collection of points?' The answer is that you can't. But you can easily go the other way. You can easily cut a line to form a point. You can easily pause a video to produce a still image.
I don't see how QM indeterminacy can be fitted into mathematics at its foundation
"Quantum indeterminacy is the apparent necessary incompleteness in the description of a physical system" [ source ]
Doesn't that sound a little like Gödel's incompleteness theorems?
It's those damn wave functions! They just seem to be everywhere (at least on this forum). No one mentions path integrals in QT. We can imagine waves, but not functional integrals.
Let's let the master speak on that:
http://yclept.ucdavis.edu/course/215c.S17/TEX/GodelAndEndOfPhysics.pdf
All Godel showed in meta-language is that there are infinite things that cannot be prove in math and infinite things that could be proven (from sure foundations). What space IS cannot be precisely said by mathematics. Affirmation of a double negative is what infinity is, which translates to a double positive. The positives are abstract infinity (a single thing, undivided) and abstract finitude (the finite as idea). They merge to form infinite units, composing a single object that swallows itself. It curves back around like on a sphere. We experience objects in their finitude. In thinking abstractly, you have to go back and forth from the infinite as idea and the thought of "the finite" in dealings with uncountable infinity (in the form of points). I think this is how Leibniz understood it, but anyway it certainly is how Hegel understood Leibniz
Are you just sharing a nice quote from Hawking, or is this supposed to support your argument? If the former, thanks! If the latter, please provide a little explanation.
He said some wanted an ultimate theory that can understand the world in finite ways with finite equations. He said he changed his mind when he reflected on Godel's theorems and realized that the world gives finite results when there is precision from us, who approach the universe as part of it. So there will always be a dialectic between the infinite and the finite in our dealings with the universe.
I am not allowed to quote that article
You're approach is typology but you haven't said anything about the system works. (Topology says how you get results)
An object is bounded by points and a finite surface area. This is how continua is defined. The infinity is in the paths within these bounds, because parts, motions, and paths are uncountably infinite with it
Oh boy my mentions have piled up tonight. I wanted to respond to this before diving back into the maw of the slavering mob I triggered by daring to express a political opinion they don't like. But never mind all that.
Quoting Ryan O'Connor
Before going on @Metaphysician Undercover pointed out a distinction we need to nail down. Please respond clearly to this question. Are you talking about
* Mathematics? or
* Some kind of notion of reality that goes beyond math?
If the former, you're just wrong. 1/3 = .3333... because the right hand side is just the sum of the infinite series 3/10 + 3/100 + 3/1000 + ... which equals 1/3 since it's a geometric series as taught in freshman calculus. It does you no good to demonstrate ignorance of the theory of limits. The sum is defined as exactly 1/3, not "approaching 1/3" or "infinitesimally less than 1/3" or any similar confused location that calculus students often have after slogging through the course. Mathematically you're wrong as a matter of fact.
On the other hand if the latter, I have no idea what you mean, because 1/3 = .333... as a matter of formal mathematics. I can prove it directly from the axioms of ZF set theory. There is no question about it. It's as certain as saying that a certain chess position follows from the rules of the game. You'd be better off self-studying real analysis than wasting your time telling me about your mathematical misunderstandings. And if you think it means something reality, you're likewise wasting your breath because I have no interest. I'm doing math, not metaphysics; and frankly I'm not sure 1/3 OR .333... OR the number 47 have any metaphysical meaning at all. That's a completely different subject.
Quoting Ryan O'Connor
I don't understand why you like one infinite representation rather than another, but you are riding a hobby horse and making no rhetorical points with me at all. You're wrong on the math and confused on the metaphysics.
Quoting Ryan O'Connor
No not at all. First, video is nothing more than a series of stills, whether analog or digital frames. Have you ever seen a flip book? We see the illusion of motion from a sequence of still images because our eye-brain system retains a little afterimage for a fraction of a second. That's how movies work, I'm certain you must know that. Back in the old days I believe flip books were a common medium for the pr0n of the day but I found a really nice G-rated contemporary flip book artist. This video is worth checking out for its own sake and makes this point dramatically. Video is nothing more than a rapid sequence of stills. It depends on a quirk of your eye-brain system.
https://www.youtube.com/watch?v=ntD2qiGx-DY
Secondly, how about still photos? If you ever had a DSLR or film SLR you know that one of the important controls is the shutter speed. You hold the shutter open for a period of time, usually a fraction of a second, and during that time interval photons come in and hit the film or digital sensor. If an object is moving side to side in front of you, you can always determine its velocity by setting a long enough shutter speed that you get some motion blur; then you can work backwards from the blur to determine the velocity. You know, the same reason your low-light photos taken with your automatic camera are blurry. The camera compensates for the low light by choosing a slower shutter speed, and your shaky hands cause motion blur.
So still photos actually take time, and videos are just a sequence of stills.
Quoting Ryan O'Connor
Why are you saying this? I assume you must know it's wrong, no mechanical device is capable of exposing a light-sensitive medium for a true instant. Are you speaking metaphorically? If you set your hypothetical camera to an instantaneous shutter speed no photons could get in and the image would be blank.
Quoting Ryan O'Connor
We're not going to solve Zeno's paradoxes here. Better you should try to understand some math, in particular the sum of a geometric series.
https://en.wikipedia.org/wiki/Geometric_series
Quoting Ryan O'Connor
Utter nonsense. If you put your webcam on yourself, record at 30 frames per second, and move around, then play it back frame-by-frame, you will most definitely see motion blur in each frame. Unless you are speaking metaphorically about an idealized camera, you are just wrong on the technology. And an idealized camera with infinitesimal shutter speed could not transmit any photons to the light-sensitive medium, so it wouldn't work.
Quoting Ryan O'Connor
We're not going to resolve those ancient issues here.
Quoting Ryan O'Connor
What? The tangent line IS defined as the limit of the secant (for a function from the reals to the reals). That's how they motivate it. But in fact that is the DEFINITION of the tangent line. Without the definition, we don't even know what a tangent line is.
Quoting Ryan O'Connor
You're confusing the formalism with reality.
So you're no longer talking about cameras but rather quantum sensors? It would be better if you dropped these labored analogies and just made your point some other way. Are you saying photons use quantum tunneling to show up on the sensor without the shutter being open? This is such an unproductive tangent to the discussion. To record an image on a photographic medium you have to expose the medium over a period of time. You're stepping all over your own point, we're not talking about cameras we're talking about something else and this isn't making sense.
Just an artifact of the decimal representation. Doesn't mean anything. Roman numerals had drawbacks, decimal representation has drawbacks. Every notation has some advantages and some drawbacks.
As to why a circle's circumference and radius are incommensurable, I think that's one of God's little jokes. God in Einstein's sense. Not necessarily religious, but a stand-in for all the ineffable mysteries of the world. With a sense of humor thrown in.
Your guess is as good as mine. There's no actual reason. But the diagonal of a square seems like an even simpler example. One unit to the right, one unit up, and the distance between your start and end points is incommensurable with 1. No reason at all, just a shocking surprise.
How can anyone deny the existence of the length of the diagonal? That's the part I don't get.
A moving body has an instantaneous velocity, even though our formalism requires two temporally separated measurements. But, I'm willing to concede that you've either made your point or at the very least caused me to doubt mine.
It's FALSE that motion through a point is the same as resting in a fixed point. There is energy that pushes right through. Imagine an arrow being pushed by the air rushing behind as it pushes forward. What's wrong with this picture (of the arrow)? It's that the motion forward is prior to air rushing behind so we can not saw the air pushes the arrow. The arrow moves through any point with forward velocity so it's never ever at rest
:cool:
What is real and fundamental in quantum physics is the points where particles appear. The wave function is the mathematical apparatus which predicts where particles might appear. Yes, the wave function is fundamental to the model, but what is being modeled is the appearance of particles at specific points. This is why physicists understand light as photons, because the energy appears at, and causes an effect at a point.
You say that points only emerge from a measurement, but a measurement is an interaction between the energy, and the object which is the measuring device. So, such points exist wherever energy is interacting with objects. What this indicates is that energy, though it is modeled as existing in a continuum, (wave function) only interacts with the physical objects which we know, at discrete points. Therefore our only access to observe whatever substratum there is, which is modeled as wave functions, is through an understanding of these points where we can observe interactions.
Sure, you might say that the continuum, or substratum as I call it, is more fundamental, but from the point of view of the model, and this means the mathematics, the points must be fundamental. This is because we only find a route inward, toward understanding the substratum through a mapping of the points where it interacts with the spatial existence we know, observe, and understand. What must be fundamental, and basic to the model, is what we know the best, and this is the points. The substratum is modeled based on the existence of those points where we can observe it The better we know the points, the more reliable our speculation about the substratum will be.
Quoting Ryan O'Connor
The substratum, which is represented by wave functions, may or may not be a continuum. That a wave function represents it as a continuum doesn't mean it is. Furthermore, a measurement is simply the substratum interacting with a physical object. So if this causes a "collapse", there are collapses occurring all the time, all over the place, as the substratum is interacting with physical objects. And, if measurements are only possible at particular points, then we ought to assume that other interactions between the substratum and physical objects are only possible at particular points, and this is most likely a feature of the substratum itself.
Quoting Ryan O'Connor
I don't agree. A process which is never executed cannot be studied. It has no existence so it cannot be studied. Let's say that you write out a rule, an algorithm, but the algorithm is never implemented. You can study that rule, but you cannot study the process dictated by that rule, because it does not exist. The rule was never put to work, actualized, it exists only as the potential for the designated process Do you see the difference between the written rule, and the activity which is prescribed or described by that rule? To study one is not the same as studying the other.
[quote="fishfry;509689"]A moving body has an instantaneous velocity,../quote]
Yes, because that is the convention, use some math, and figure out the "instantaneous velocity", just like the convention is to place a zero limit on the example of the op. But what these conventions really represent may not be what one would expect from the terms of usage.
A process that is active is mental use of the four functions. This can be applied to reality but not perfectly
A car whose speedometer reads 40mph is going 40mph at that moment even though that's a mechanical approximation. But as I say I'll concede you've moved me off my certainty and I've run out of talking points. After all we don't know the ultimate nature of reality so who's to say if the notion of instantaneous velocity really makes sense. Based on that I'll concede the point.
I've been meaning to ask you: "Descartes great merit here was to have applied geometry to algebra; he was not the first to have applied geometry to geometry... And to Descartes we owe the first systematic classification of curves. After dividing 'geometric curves ' which can be precisely expressed in equations from 'mechanical curves' that cannot, he classified the former into three clases... This new geometry is more than a general theory of quantity: it led to the concept of continuity, from which was developed the theory of function and, in turn, the theory of limits... But he was mistaken in believing that equations of any order could be so resolved." ( Britannica Encyclopedia 1965)
I think applying numbers to geometry is how we apply Godel's numbering to physics
Silly, but this entire discussion needs termination.
I have two links saved somewhere on Leibniz' relationship to this question. I'll go find them
https://www.humanities.mcmaster.ca/~rarthur/articles/lsi-final.pdf?fbclid=IwAR34tqTaNU4OY-YBbMA-ZpsGIxLFev1ZV8QRvHQF7UprFiXtdx9RgYoLHGc
Marx said Hegel was the best of mathematicians yet Hegel learned what little he knew of the subject from Leibniz, who could easily be the greatest of all mathematicians (so my intuition is saying).
This also relates:
https://en.wikipedia.org/wiki/Benacerraf%27s_identification_problem
Thanks. Wasn't aware of that. Kind of anti-set theory regarding natural numbers.
I don't think physics in its present form is amenable to methods of formal logic because there are no axioms for physics. Axiomatizing physics is Hilbert's sixth problem and according to Wiki it's still open. Even so I doubt that the axioms of physics, whatever they may eventually turn out to be, satisfy the premises of Gödel''s incompleteness theorems.
But of course you could Gödel-number anything, Moby Dick or the contents of this post. You assign numbers to the upper and lower-case English letters and punctuation symbols, and you apply Gödel numbering. So if 'abc" is your string, and a = 1, b = 2, c = 3 is your encoding of the symbols, then 'abc' is encoded as [math]2^1 \times 3^2 \times 5^3 = 2 \times 9 \times 125 = 2250[/math].
The idea is to use the unique factorization of integers into prime powers. To go backwards from 2250, we note that it's [math]2^1 \times 3^2 \times 5^3[/math] and read off the exponents, 1, 2, 3 to get 'abc'. You could do this with any piece of text.
As you can see, if you believe that the positive integers exist (whatever existence means for you) then every finite-length document that could ever be written, already exists. It's already encoded as some positive integer. LIkewise if you pixellate a painting finely enough so that the pixellated version is indistinguishable from the original (to the naked eye, say) and encode its color and texture values, you can Gödel-number all possible paintings. Likewise songs, etc.
Or as the Pythagoreans said: All is number!
This is the point. When we use math to figure out things like instantaneous velocity, the volume of a supposed infinitely small tube, etc., it is implied that we know things about reality which we do not. This is a falsely supported certitude.
Quoting jgill
There is a flaw with your example jgil. That the car was "going at a constant speed" is just an assumption, so it may not be the truth of the matter. And your answer as to how fast the car was going at point A requires that the assumption be true. So it needs to be proven.
You might lay out a series of such points, at equal distance, and do numerous similar measurements. If your measuring capacity is precise, you'll find that all the measurements will not be exactly the same. The assumption of "constant speed" cannot be validated. That's what we've found out about the nature of reality, motion consists of spurts and starts. So you'd have to establish trends, and the more measurements you took the better your graphing of the trends would be. But you'd be graphing averages which does not tell you the precise amount at any given point.
Thanks!
Are tangent and instantaneous rate of change not the same thing? If you reject the notion of instantaneous rate of change, how can you not reject the notion of tangents?
Quoting fishfry
I'm talking about mathematics. I understand that 0.333... converges to 1/3, but it is only (a useful) convention which states that convergence and equality are the same. If you're saying that it's proved somewhere that the two terms are equivalent then let's leave it at that.
Quoting fishfry
The Stern-Brocot string for any rational number has finite characters. I don't accept the claim that that LL = LLLRrepeated since LL corresponds to a position in the tree and LLLRrepeated corresponds to a path along the tree. However, let's not waste any more time on 1/3.
Quoting fishfry
As with your speedometer argument, your addition of a shutter only complicates the issue without providing any further explanatory power. We add a shutter to cameras only to limit the amount of light that the film is exposed to. Without a shutter we can (at least in principle) still take photographs. Remove the shutter and the instant the first photon hits the film we have a photograph. And if no further photons hit the film we have an image with absolutely no blurriness. This image captures no motion. It is not a video by any definition. I was anticipating you challenging the practicality of such a photograph, which is why I went quantum, but perhaps that's not necessary.
Quoting fishfry
I understand how videos and flipbooks work, and yes we use stills to create them, but the magic ingredient which you are ignoring is time (specifically non-zero intervals of time). We hold each frame for 1/24 seconds before advancing to the next frame.
Quoting fishfry
(I believe) Zeno's paradox was already informally solved by Aristotle. I'm just defending his view. And loosely speaking the view is simply that we start with videos, not stills. With videos we not only can capture motion but we can also pause the video to extract a still. With only stills, motion is not possible, as per Zeno. Zeno's paradox is important and seen as unresolved because the notion of stills being fundamental is deeply rooted in our beliefs.
Sorry tim, but if I already knew the truth, then I wouldn't be looking for it, would I?
Quoting tim wood
Yes, that's about it. Speed is what we assign to the car, it is what we say about it, it has speed. In philosophy we must maintain the distinction between what we say about the thing, and what is really the case, to allow for the real possibility that what we say about the thing might actually be a falsity. If the property which we assign to the thing, "speed", in this example, has faults within its conception (contradictions for example), then despite the fact that it has become acceptable to say this, the concept is defective, and it is really not true to be attributing that property.
This is exactly where you go off the rails. There is no "infinitely small tube." This is your ignorance speaking again.
Quoting Metaphysician Undercover
Nobody is making any claims about reality. Gabriel's horn is a strictly mathematical example.
You are making a claim about reality (i.e. it's made of events of information). Aristotle slumped into this when he said parts are potential. What exists is the whole composed of all it parts, which are bounded by points (finite) and limit in space (finite). A material body doesn't have math in it. We use imperfect mathematical formulations to understand to described in the field of physics. You can't draw philosophical conclusions from physics is the conclusion. You fell for the Parmedian world view by trying to figure out the logic of his disciple
No they're not. In functions from the reals to the reals (ignoring the multivariable case where this terminology doesn't make sense) the instantaneous rate of change is the slope of the tangent line.
Quoting Ryan O'Connor
I don't, you and @Meta do. I'm agreeing that IN REALITY there may not be such a thing. But in math, there most definitely is. Again. you are equivocating math and physics. I'll stipulate to @Meta and your point that instantaneous rates are murky in physics. In math they're perfectly well defined.
Quoting Ryan O'Connor
You can't maintain your credibility while arguing against freshman calculus. here's the proof "somewhere," a somewhere I already linked to earlier.
https://en.wikipedia.org/wiki/Geometric_series
Quoting Ryan O'Connor
On the contrary, 1/3 is the critical case here. If you can't agree that 1/3 = .333... then you fail freshman calculus and high school algebra too. There's nothing more to talk about then.
Quoting Ryan O'Connor
You know you keep making claims totally contrary to photographic technology. A single photon is not sufficient to make an impression on any film stock or digital sensor in existence. So you're just flat out wrong here. And if you don't open shutter at all, no photons will come in; and to get sufficient photons in to make an impression on a light-sensitive medium, you need to keep the shutter open for a period of time. Of course I include an electronic shutter that merely activates the sensor for a period of time.
Quoting Ryan O'Connor
There are no single-photon detectors outside of physics labs. But again, I don't know why you're belaboring this point. If you don't think that 1/3 = .333... AND you agree that you are making a mathematical point, then there is no conversation to be had. You're just wrong. Read a calculus book or work through the proof I linked (twice now) on Wiki.
Quoting Ryan O'Connor
I see no benefit to this point at all. What of it?
Quoting Ryan O'Connor
Then why did you give the impression you didn't?
Quoting Ryan O'Connor
I don't think we're having the same conversation anymore. But regarding your claim about video, I invite you to wave your arms in front of your webcam while recording, then play it back frame-by-frame. You'll see motion blur. I know this because I've seen it many times. In fact if you are making a video with the intention of capturing still frames you have to make sure to stay motionless for a few moments at a time to avoid motion blur.
To go from A to B you have to go half that distance, since the distance is distance as space. Half of AB is a distance, otherwise you are at the half point instantaneously. So he must go the quarter. But the quarter is spatial it too has a half point mark, ... And a computer can run this activity to an uncountable infinity. There is no paradox as to how motion starts from energy. The question is how it is that the supertask is done in finite bounds (time and space)
My impression is that you're a finitist, so I presume that you believe our universe had a beginning of time. If particles are fundamental, they must have existed at that initial moment, right? Were they concentrated at a point? I take it that you think a measurement involves the interaction of particles, so at the initial instant wouldn't they all be measuring each other? If so, how would they ever move, given the quantum Zeno effect?
Consider this: "QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles." source
Quoting Metaphysician Undercover
I think you're splitting hairs here. By rule I assume you mean the 'computer program' and by process I assume you mean 'the execution of the computer program'. If so, then we are in agreement, we can study the rule (i.e. the computer program).
A particle is not the system. It passes through the system
The Zeno effect and the anti-Zeno effect refer to how observation changes eternal states. The ancient Arrow paradox is just used to illustrated the effect and the effect does not resolve the Arrow paradox because its not specifically related to it
My view is that actual infinity should not be permitted in math any more so that than it is permitted in physics, but that's just my view and it's contrary to contemporary math so I'm willing to leave it at that.
Quoting fishfry
The sum is defined as [math]s=a\frac{1-r^{n+1}}{1-r}[/math]
There is a difference between substituting a finite number in for n and substituting infinity in for n. Obviously the latter is not allowed, so we talk about convergence instead. If we ignore the fact that we're talking about convergence when talking about the sum of an infinite series then we are essentially substituting infinity in for n. It is a matter of convention that we say that geometric series sums to a number. More formally, we should say that it converges to a number. I think 0.333... converges to 1/3 is a valid statement.
Quoting fishfry
Ha! I knew you were going to say this. That's why I had switched to a quantum sensor. With an SLR camera I agree that every photo has some degree of blurriness, but with quantum sensors that's not necessarily the case. There is no law which states that we can't know the position of a particle with perfect precision. Also, I don't think particles are points, but instead excited states of quantum fields.
Quoting fishfry
Forget about cameras, sensors, and speedometers - it all boils down to the question of whether a line can be constructed by assembling points. Your earlier post indicated that you agree that this is a mystery (given the orthodox views)? Why not consider alternate views?
Perhaps I am misusing terms. Certainly, when I say graphs are topological, I don't use it in the standard sense of the word topology. Instead, I'm using it in a looser sense, simply that the properties (of interest) of the graph are maintained through continuous deformations. The results are largely unchanged in reinterpreting graphs as topological objects (in the sense of topology mentioned above).
As for continua, I simply mean objects with extension.
Quoting Gregory
This is challenged by Zeno's Arrow Paradox.
Quoting Gregory
The whole need not be bounded by points. Think of the open interval (-1,1). I think physics can and should inform our philosophy and you're right that I'm influenced by Aristotle and Zeno.
Quoting Gregory
I haven't looked for a reference but I assume they named the Quantum-Zeno effect because of the Zeno's Paradox (especially Zeno's Arrow Paradox). If so, I don't see how it's unrelated.
Not all contemporary math. In complex analysis one may move to the Riemann sphere and take the north pole as infinity, but I stick to the complex plane and use expressions like "unbounded" instead. Now, set theory is another animal altogether. Not my cup o' tea, but fishfry is an excellent in-house expert.
You're right, I shouldn't have generalized.
I said "faults", and I used "contradiction" as an example of a fault. That there is a "margin of error" is another indication of fault. When a small margin of error is ignored or neglected, as if it doesn't exist, one can fall for a paradox like Zeno's, where that small margin of error is infinitely magnified to produce the appearance of contradiction.
Quoting Ryan O'Connor
I really don't get your question. I was talking about points, not particles, so your question has some underlying presumptions which I don't follow.
Quoting Ryan O'Connor
I addressed this already. The so-called "underlying quantum fields" are models produced from observations of particles, and are meant to model the interactions of particles. It is implied that there is an underlying substratum which validates this modeling, but the modeling itself, the quantum field theory, does not represent the underlying substratum, it represents the interaction of particles. Until we get accurate and precise modeling of the particles any speculation concerning the substratum is not well informed.
Quoting Ryan O'Connor
OK, so we say that the rule calls for the computer to carry out an endless, or infinite process. We know that the computer cannot succeed in carrying out this request, because it will wear out first, so all the time spent will be wasted, for the computer to be trying to carry out a process it can't. So if we turn to study that rule, should we not put our efforts into avoiding this rule, making it so that the rule never comes up, because it's like a trap which the computer will fall into? Therefore instead of pretending to be having success at carrying out infinite processes, which is self-deception, we should be looking at ways to make sure that such rules are banished.
I believe physicists think particles are only states of quantum fields. If so, you should not be thinking of the point where a particle appears, but instead the continuum where the quantum field exists.
Quoting Metaphysician Undercover
If you place iron filings over a magnetic field the filings will take a form in line with the field. While it's true that we only see the filings, it is untrue to say that the field is just a model. It's real. The same goes for quantum fields.
Quoting Metaphysician Undercover
No. If we terminate the potentially infinite process we still get something useful (e.g. the rational approximation of pi on your calculator is a useful button). Also, I would argue that calculus is the study of these 'rules' and calculus is arguably the most useful branch of mathematics. What I agree with you on is that we should not try to carry out the infinite process to completion...that is a fruitless endeavor.
Infinity creates a situation in this question about space that make the question of continua difficult (The reason being that discrete apace is an oxymoron). However in numeral mathematics infinities work ok. Paul Cohen found contradictory proofs in infinite mathematics in the 1960's but the subject simply is not understood properly enough. Maybe a theory of everything which provides the connection between discrete apace (a point) and finite geometry (solids) be found. But nonetheless banishing infinity from mathematics is a move of an ostrich
I'm not proposing we ban Infinity altogether. I'm proposing that we restrict ourselves to only use Infinity in a potential sense.
It's a useful concept, especially in an applied sense like engineering.
I think we're done then.
Just out of curiosity though, how do you develop a theory of the real numbers without infinite sets? Even the constructivists allow infinite sets, just not noncomputable ones.
Quoting Ryan O'Connor[/quote]
We're just not having the same conversation.
Quoting Ryan O'Connor
The question doesn't come up in math. We use the phrase "the real line" as an alternate way of saying, "the set of real numbers," but everything can be done without reference to geometry.
Quoting Ryan O'Connor
If I see a coherent one presented I'll engage with it. In the past I've engaged extensively with constructivists on this site and learned a lot about the contemporary incarnations of that viewpoint.
I've also studied the hyperreals of nonstanard analysis. So in fact I'm very open to alternative versions of math, but I don't see that you've presented one. The problem with finitism is that you can't get a decent theory of the real numbers off the ground.
"The philosopher and theologian are conscious of infinity, but from the mathematician's view they do no use it so much as admire it. The mathematician also admits infinity; the great David Hilbert said of it that in all ages this thought has stirred man's imagination most profoundly, and he described the work of G. Cantor as introducing man to the Paradise of the Infinite. But the mathematician also uses infinities..." Leo Zippin
"Every since we first sought number in the object, the series of numbers has begun with 1. Making zero the first of numbers means no longer abstracting them from the object" Jean Piaget
"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."?Hermann Minkowski, 1909
"At a time when Minkowski was giving the geometrical interpretation of special relativity by extending the Euclidean three-space to a quasi-Euclidean four-space that included time, Einstein was already aware that this is not valid, because it excludes the phenomenon of gravitation. He was still far from the study of curvilinear coordinates and Riemannian geometry, and the heavy mathematical apparatus entailed"
https://en.wikipedia.org/wiki/Pseudo-Euclidean_space
https://en.wikipedia.org/wiki/Minkowski_space#cite_ref-14
https://www.quora.com/How-can-I-a-non-mathematician-wrap-my-mind-around-the-Axiom-of-Choice: The reason the Axiom of Choice is (somewhat) controversial is that while it allows us to prove some very useful mathematical statements, it also allows us to prove some less intuitive statements (e.g., the Banach-Tarski paradox).
And finally:
"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." Einstein
(Don't forget Kant's second antimony: (On Atomism)
Thesis:
Every composite substance in the world is made up of simple parts, and nothing anywhere exists save the simple or what is composed of the simple.
Anti-thesis:
No composite thing in the world is made up of simple parts, and there nowhere exists in the world anything simple)
And we can paint an infinite space with finite paint,
Then it's clear humans struggle with the union of mass and volume
There is an issue of truth here. There is something there causing the form, and the concept of "field" attempts to account for whatever it is. If the concepts employed are inadequate, then it's not true to say that this is what is there. Here's an example. The ancient Greeks used circles to model the movement of the planets, and Aristotle proposed that the orbits were eternal circular motions. It turned out that these models were wrong, therefore it was not true for them to have been saying that the orbits were circles even though this concept was employed and enabled prediction.
Quoting Ryan O'Connor
Again, there is an issue of truth here. If the process is terminated then it is untrue to say that it is potentially infinite. And if we know that in every instance when such a process is useful, it is actually terminated, then we also know that it is false to say that a potentially infinite process is useful, because it is only by terminating that process, thereby making it other than potentially infinite, that it is made useful. Therefore t is false to say that the potentially infinite process is useful.
Quoting Gregory
No, the opposite is the case. Ignoring the fact that infinities in mathematics is a very real problem, is the type of ignorance which is analogous with the ostrich move.
Quoting tim wood
What do you think "passing a point" means? Do you mean to say that there are physical points out there, which the arrow can be seen flying by? If so, then you ought to be able to show empirically, the physical existence of such points, and I don't think there will be an infinity of them. If these points are just imaginary, then the arrow doesn't really fly by them and you have created a false scenario, by describing the arrow as flying by points.
I propose that the truth is that the points are imaginary. If this is the case, then any method of measuring motion, velocity and such; which employs points, is really giving us a false measurement. We might be able to find real physical points, which if they exist, would validate such a method, but then these points would not be infinite, so that scenario with infinite would become irrelevant, because we'd have to make a new method of measuring velocity based on empirically verified points. As I explained to you earlier, this is pretty much what relativity theory does, but each empirically verified point turns out to be a different frame of reference, and that the points are at rest relative to each other is very doubtful due to the observed phenomenon of spatial expansion.
Aristotle said infinity is "one". Modern mathematics say it's two (countable, uncountable) with perhaps subdivisions. A Philosophy Overdose audio on youtube said that Paul Cowen proved and then disproved that there is an infinity set between the two of Cantor. This is a challenge for mathematicians and they are not going to listen to a philosopher like you
Daoism at its founding taught that being and non-being create each other. Such dialectical logic is all over philosophy. Yet mathematics is a field that does not use such ideas
"Paul Cowen proved and then disproved that there is an infinite set in between the two of Cantor"
The continuum hypothesis has arguements for and against it too, as can be seen in the WIkipedia article on it. Organizing these ideas into something consistent is going to take work by hundreds of mathematicians and many many years of toil
Philosophy says that a point is a negation of a line, but if the points are ordered they create a double negative, and thus the positive of the line. The line does the same with the solid, and the solid in turn abrogates what comes before and creates dimensions greater than three. Philosophy and mathematics come onto these questions from very different perspectives and we philosophers can't expect our explanations and demands to be accepted by those with high math skills
Please clarify this.
Sure. (Remember I'm coming at this as a philospher). Pi specifies a a certain precision between the numbers 3 and 4. When it comes to a segment, the precision is at every spot of it. However these points are nothing but precision. They're "not-space" and therefore negative to the segment. But if we turn the segment upside down the finite nature of the points become a positive length of finitude in the segment. The negative can be positive as double negative and therefore we have a segment of a line. This same process happens as we go from dimension to dimension. But remember I think this is probably more philosophy than mathematics. There is not a true supertask in motion because there is not an infinite set of actual lengths being covered. But the division OF the length covered by motion has no end, and this is true of anything spacial
I really hope this was helpful. Space has a strange relationship to itself.
I think this is the wrong question to ask. We know what we want (a consistent foundation for calculus) and we think we know how to get it (with real numbers and infinite sets). But real number might not be the answer.
I think a much better question to ask is 'can we build a consistent foundation for calculus without real numbers and infinite sets?' I believe the answer is yes. Now, this doesn't imply that the alternative requires scrapping everything about real numbers. For example, I believe the alternative would have still have the area of a unit circle being pi, it would just mean something different.
Quoting fishfry
Like the real numbers, the fundamental objects of the hyperreals are points/numbers. A continuum is constructed by assembling infinite points, each with a corresponding number. And so the hyperreals gets no closer to answering the question 'how can a collection of points be assembled to form a line?' I haven't investigated the constructivists' methods to produce an informed comment but from what I've seen it looks like more of the same. Points/numbers are treated as fundamental.
I don't have a formal theory but I do have an intuition on how the alternative would look. In line with Aristotle's solution to Zeno's Paradox, my view has continua (not points) being fundamental. I think the mathematics of calculus would be almost entirely unaffected in moving to a continuum-based view, but before we could even talk about it, you'd have to first be open to (at least temporarily) shedding your point-based biases and look at graphs in a new light. I would love to hear your criticisms on my earlier post to you where I drew the polynomial in an unorthodox way.
Thanks. I would need a greater knowledge of philosophical theory to really understand. :smile:
There are two ways to describe the car's motion.
1) Over any non-zero interval of time the car's average velocity is 60mph.
2) At any instant the car's instantaneous velocity is 60mph.
I believe that the latter description does not make sense since velocity is a measure of change over time. But this doesn't imply that I believe that there is some interval for which the car's average velocity is 0mph.
Quoting tim wood
This is how we were taught to think, but is it possible that you're just looking at a line? Do you think it's even possible to see a single point? If not, what about 2 points? 3? If something magical happens at infinity when they form a continuum, why not skip the magic and just start with a continuum?
Sure, physicists could be wrong but that doesn't mean you should stick with outdated information (which could also be wrong).
Quoting Metaphysician Undercover
A program written to spit out the natural numbers one at a time is potentially infinite, regardless of whether it's been executed or interrupted.
Quoting Metaphysician Undercover
If you have ever seen ? as the solution to a problem (instead of, say, 3.1415) then the process hasn't been terminated, it hasn't even been initiated. It's incorrect to say that potentially infinite processes are only useful when prematurely terminated.
Why isn't Aristotle's solution just circular because it makes the results of a mathematical construction prior to the construction itself? Plato starts with a line boundless in both directions, then designates bounds to derive segments. Think of it this way, first mark any point on the line as the Origin to divide the line into a half-open dichotomy, then designate any other point as the unit marker to construct a fixed continuous interval. If I give up points as bounds, then how would I have anything but an endless line?
I'm not a physicist but this is my understanding of how our world works. A wave function of the universe (describing potential/probabilities) exists spanning all of time. When a measurement is made, that wave function locally collapses to a distinct state. So we start with a probabilistic description of the universe which exists prior to the individual moments.
The construction itself certainly must come first. What we disagree on is what that construction is. You see the individual measured points in time being fundamental while I see the continuum wave function being fundamental. Zeno's paradox highlights the weaknesses in your point-based view. In short, we can't build a timeline from points in time.
Quoting magritte
We don't need points to bound the open interval (-?,+?).
In the transitions of time, motion covers space as if it is fixed ( "discrete") and as if it is undifferentiated continuity. It's two sides of a coin
This is a bit speculative without some sort of indication or hint of specifics. For example as I mentioned, I've looked into constructivism a bit, which is enjoying a modern resurgence due to the influence of computer science and computerized mathematical proof assistants. But you would reject even that.
Quoting Ryan O'Connor
Sounds like a Peirceian viewpoint, about which I don't know much. But why am I supposed to be burdened for finding a mathematical viewpoint other than the standard one accepted by almost all the world's mathematicians except for those pesky constructivists and rare finitists and ultrafinitists? Why is this my problem, or math's problem?
Quoting Ryan O'Connor
You're just waving your hands.
Quoting Ryan O'Connor
I suppose I owe you that. I'm going to find it depressing and frustrating but I'll take a run at it sometime.
I certainly like the idea of constructivism in that it is necessary to construct a mathematical object to prove that it exists, and my understanding is that constructivists reject actual infinity. However, I haven't investigated the details of constructivism to know where my views differ. It's on my to do list....and as of now, so too is the Peirceian viewpoint.
Quoting fishfry
This is a great question. Nothing is your problem unless you want it to be. But here's my response to your 'all the world's mathematician's' comment: the world's mathematicians are all climbing up the 'point-based' mountain (and certainly doing good work) but it's a competitive space and (I think) most contributions these days are in highly specialized niche areas. I'm here pointing out that there's another place to climb. I can't promise that it's a mountain (but it might be) but I can say that very few people are climbing it. What's the harm in taking a quick look?
Quoting fishfry
...ah okay, so the harm is potential depression and frustration. I'm sure it is frustrating talking math with a non-mathematician, but don't be so certain about it being depressing. There's a chance that you'll like the ideas.
We already discussed the difference between the rule ("program" in this case) which sets out, or dictates the process, and the process itself. If the process is interrupted, it ends, and is therefore not infinite. The rule ("program") is never infinite, nor is it potentially infinite, it's a finite, written statement of instruction, like "pi", and "sqrt (2)" are finite statements, even though they may be apprehended as implying a potentially infinite process.
Quoting Ryan O'Connor
I don't see how you can say this. Pi says that there is a relationship between a circle's circumference and diameter. This information is totally useless if you do not proceed with a truncated version of the seemingly infinite process, such as 3.14. "The solution to the problem is pi" doesn't do anything practical, for anyone, if you cannot put a number to pi.
What about the halting problem?
We can certainly talk about a system at a particular instant of time. For example, we can talk about the time on our clock, and the position and shape of objects at that instant. What we can't talk about is rate of change.
Quoting tim wood
Let's say that every instant has a corresponding number on my stopwatch. I believe that the numbers on my stopwatch can only ever be computable numbers. Since traditionally we say that the computable numbers are countably infinite, the duration at which the car is not in motion has measure 0. Therefore the car is spending all of it's time in motion. What you must appreciate is that there is 'something' in between the instants and it is during this 'something' when the car moves.
When time is suspended in indeterminacy space is fluid and discrete. When space is seen as knowingly infinite is its parts it is suspended against the discrete nature of time. Motion has an aspect that is spatial and one that is temporal. We do talk of spacetime now, but we must speak of time and space separately in these examples. Saying we have points and instances which are infinite (which must be crossed by motion) is to forget that time is not space, space is not pure mathematics, and that even pi can only be understood as part of a finite number (3)
Hmm, maybe I wasn't consistent, let me try again.
-The aforementioned program is written with finite characters.
-The execution of the program is potentially infinite.
-The complete output of the program would be actually infinite (if it existed).
-The program spits out numbers as it is being executed, so it doesn't need to be terminated to get something useful from it.
-We can discuss the execution of the program without ever running it (e.g. we can say 'if I executed the program, it would be potentially infinite)
In the end, I think you're splitting hairs here. What's your point?
Quoting Metaphysician Undercover
? is often written as the solution to a problem - for one it's what they say is the volume of Gabriel's Horn. And there are many cases where ? gets cancelled out, for example cos(2?)=1, so there's no need to evaluate it. Also, who said math had to be practical?
I'm not sure how to respond to most of your post, but as for pi we typically understand it precisely using some algorithm described with finite characters.
It is a lot more satisfying to do this than to argue indirectly, IMO. I have told this little story before, but it bears repeating: There was a PhD math student who spent considerable time on his culminating research project creating a mathematical structure about a particular set of functions, until one day he was asked to provide an example of one of these functions. It turned out he was working with an empty set. :sad:
Your ontology is weak. You say a table is one, yet it has 4 legs and a top piece. What number of parts do these have? This process is infinite and it takes a delicate balance to understand all it's intricacies. Motion passes through infinity and the finite, but you want to reduce the question to Aristotle's lame argument,: namely that parts are only potentially there. Bringing in QM isn't going to help your case mr. idealist. The world is real. "Ignore the world and the world will come to you"
But spitting out numbers is not something useful. Useful is the application of the numbers towards counting or measuring, or something like that. If the computer is tasked with counting something and does not complete the task it hasn't been useful.
Quoting Ryan O'Connor
But what good is that?
Quoting Ryan O'Connor
I can't even remember now, but I believe I said it would be good to rid the system of infinities and you said there is no problem with working with infinities so long as we recognize that they are merely potential.
But what's the point to working with infinities? If an infinity represents an uncompleted tasked, then isn't it better to complete the task before proceeding. After a while the unfinished tasks start to pile up and become a little overwhelming. And if it is a task which is impossible to complete, then to give oneself an infinite task is to set oneself up for failure, so we ought to address the conditions by which this happens so that we can avoid it.
Quoting Ryan O'Connor
Obviously, that's not a real solution.
Quoting Ryan O'Connor
This is probably the crux. "Math does not have to be practical". There's a fundamental element of free choice which lies at the base of all of our understanding of everything. "Has to be" is thereby excluded. And so, we do not have to do anything, nor do we have to figure anything out, or anything like that. One can refuse to move and die if one wants. However, we choose to try and figure things out, we choose to try and understand the nature of reality, and mathematics plays a very big role here. So we need to choose the appropriate mathematics.
Of the mathematicians, the people who dream up axioms, and produce elaborate systems, some might have the attitude that math does not have to be practical, and others might have the attitude that math ought to be practical. But the idea that mathematics does not have to be practical is just an illusion. Each such mathematician will choose a problem, or problems to work on, as that's what mathematics is, working on problems. And problems only exist in relation to practice, as that's what a problem is, a doubtful aspect of practice which needs to be resolved. Without the influence of practice, the need to resolve the issue does not arise, therefore there is no problem. So the reality of the situation is that since mathematicians work on resolving problems, and problems only exist in relation to practice, math is always fundamentally practical, and this fact cannot be avoided. That's why math is classified as an art rather than a science. Despite the huge amount of theory which goes into it, it is theory which is always directed toward solving problems. Therefore, despite the fact that math doesn't have to be practical, it always is practical. If the people who dreamed up axioms and other systems weren't doing something practical (resolving problems), they would have come up with something other than mathematics.
Quoting fishfry
I have not had the time, energy, or patience to jump into the substance of this discussion so far, but I can offer a couple of reading suggestions based on these two comments.
The first is John Bell's book, The Continuous and the Infinitesimal in Mathematics and Philosophy. It provides an excellent historical overview followed by chapters specifically on topology, category/topos theory, nonstandard analysis, constructive/intuitionistic mathematics, and smooth infinitesimal analysis/synthetic differential geometry.
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.307.7578&rep=rep1&type=pdf
The second is my own recent paper, "Peirce's Topical Continuum: A 'Thicker' Theory." Its thesis is that a collection--even an uncountably infinite one, like the real numbers--is bottom-up, such that the parts are real and the whole is an ens rationis; while a true continuum is top-down, such that the whole is real and the parts are entia rationis. Accordingly, five properties are jointly necessary and sufficient for the latter: rationality, divisibility, homogeneity, contiguity, and inexhaustibility.
https://www.jstor.org/stable/10.2979/trancharpeirsoc.56.1.04
The consensus among Peirce scholars seems to be that SIA/SDG comes the closest among modern mathematical developments to capturing his notion of a true continuum. Bell also has an excellent primer specifically on SIA, and Sergio Fabi wrote something similar for SDG.
https://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf
https://pages.physics.ua.edu/staff/fabi/InvitationSDG.pdf
Ryan, I only quoted this to mention you. I've noticed that when I hit the @ button and enter Ryan, your handle doesn't come up. Do you know why that is? Moderators, any clues?
Anyway apropos of our convo re video, I thought you might enjoy this. It's a pretty cool video regardless. It's a helicopter taking off with its rotor speed exactly synced to the video frame rate, making the rotors look motionless. So even if things look motionless maybe they're moving. For all we know, still photos are moving like crazy but the lights in our room are blinking. Reality is not what it seems, or something like that. The comments below the video are amusing too.
https://www.youtube.com/watch?v=yr3ngmRuGUc
Thanks for the info. I was not able to get past the JSTOR paywall even though I have a public JSTOR account (the kind they deign to give to us unwashed non-academics). Do you know why? Usually I can read JSTOR articles on the website even if I can't download them.
Newton's world was one where the present took the center stage, time marched forward instant by instant, and everything always held a definite state.
Einstein's world is one where simultaneity and the present are relative. This view challenged presentism, making eternalism and the block universe (i.e. the whole universe spanning all time and space) an attractive view.
Bohr's world is one where objects only have definite states when they're measured.
My views does not ignore reality, rather is agrees with the latest developments in physics. Perhaps you're stuck in a classical world, a world plagued by singularities. It at least seems that way given your affinity for infinity. As it is when we grapple with singularities, I doubt that you have a compelling solution to Zeno's Paradox.
Your question about a table seems silly. If you want to talk about 'the whole' you have to talk about the wave function of a whole system, not a table. If you don't like the word 'potential existence', would 'superposition' be more appealing to you?
Consider this: in between any two distinct states of a system lies an unobserved wave function. If we continue to observe the system it will not evolve to a different state (Quantum Zeno Effect). Motion happens when we are not looking. Why are you assuming (as Newton did) that everything always has a definite state?
The greatest error of modernity is saying that the world is information and is not as it appears to us. The world we see transcends any interpretation of QM and psychological studies on mind-matter interaction. What you see is what there is. There is more there, but not less. Any other position is insanity. Zeno's paradox will never have a complete solution, but it is a sign of a healthy position to be comfortable with a paradox
I've said this many times, but I'll say it one more time. We can talk (using finite statements and operations) about potentially infinite processes without ever initiating (let alone completing) the potentially infinite process. That's what calculus is all about (in my view). If you avoid infinity altogether (i.e. actual and potential) then there is no way to get calculus off the ground. Let go of executing the potentially infinite process and embrace the finite statements that talk about the potentially infinite process.
Quoting Metaphysician Undercover
Maybe we are using 'practical' differently. I mean practical as having an applied benefit, like engineering better gadgets. If you consider the proof of Fermat's Last Theorem 'practical', then by your definition I think all of math is practical.
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I find your philosophy of mathematics to be quite disjoint from the act of doing mathematics. Your view does not explain equations or geometry or axioms, it doesn't provide any explanatory power. It only attempts to delegitimize powerful and proven tools (e.g. calculus).
I am going to talk on the phone tomorrow with my cousin, who is a high profile computer programmer in LA. I will tell him about your claim that infinities play no role in programs and see what HE has to say about that. I feel like you come at these questions from a very limited philosophical perspective and make broad claims about stuff you apparently aren't very familiar with. Expect a reply on this thread tomorrow night
Agreed. It is easy to talk about doing something (e.g. completing an infinite process), it's another thing to actually do it. I think your story is quite fitting because I believe that when we construct objects by assembling points, we never get anywhere. We're always left with a 0D object...an empty set of sorts.
Quoting aletheist
Thanks for the recommendations, both seem like interesting reads and your paper seems especially relevant.
Quoting Gregory
I don't think anyone is saying that there only exists information, certainly I'm not saying that. I'm also not denying 'what you see is what there is'. The problem is that you and others are incorrectly filling in the gaps between quantum measurements to assume that a particle is travelling a definite path, and by making this assumption you fall victim to Zeno's Paradoxes. Zeno's Paradox does have a simple solution, Aristotle's solution which I have been promoting here. I don't think anybody has provided a single valid criticism of this view, other than 'it doesn't jive with my classical intuitions'. You need to update your intuitions based on modern physics. Lastly, paradox is a sign that progress can be made, we should not just accept it as being beautiful mystery that will permanently be out of reach. We should be motivated by it.
Aristotle's "solution" was that the whole exists prior to parts. To which I ask:
1) do the parts of the whole exist
2) how many are there of these parts.
The answer to the first question is "yes," (you have an arm for example) and the second answer is "infinite". Even Aristotle agreed that discrete space is an oxymoron. The world exists as a paradox, a round circle right before our eyes. It gives wonder and awe to us who think like Kant and Hegel. You can't fully understand a plant, but saying its just amorphous waves because a quantum physicists says so after getting dizzy from quantum math is not a true, realistic position to hold
I suspect it's the apostrophe...it's always given me computer problems. I've posted a comment in the 'Feedback' section, hopefully a moderator will allow me to change my name.
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That is a cool video, thanks! I've worked for many years in simulating physical systems (in fact I was simulating rotating fan blades today) and when it comes to transient simulations, it's important to select a timestep sufficiently small to adequately resolve the transient fluctuations. Too large and you can incorrectly predict the rotational velocity of a helicopter's blades, you might even predict that they're not moving. As such, it is natural to then want a smaller timestep. But one thing that we never want is to set the timestep = 0 seconds. Not only would the equations blow up, but no matter how many timesteps we run we would never get past the initial state. Motion is impossible when dt=0. So what we do is pick a dt small enough to sufficiently resolve the frequencies of interest. And when we create an animation based on those frames we hold each frame for dt seconds before advancing to the next frame.
And what I haven't been able to get across to you is that a frame held for 0 seconds is totally different than a frame held for dt seconds. The 0 second frame is like a point in time while the dt second frame is like a timeline. We make movies, not by assembling 'points' but by assembling 'lines'. If all we have are points, then we'll never construct anything with more extension than a point. It is because of this sort of reasoning why I earlier called the 'real number line' the 'real number point'.
Now if we go the other way around, we don't face the same problem. If we start with a line we can easily produce points, just as we can produce new endpoints when we cut a string. When we start with continua we get both continua and points. It's a much richer foundation that avoids the paradox.
Quoting fishfry
I'm going to come back to this. The point-based view has reached its limit, it will never be able to solve Zeno's Paradox. But Zeno's Paradox is that loose thread on our sweater that we should pull because shedding ourselves of our inconsistently knitted sweater will reveal that underneath the sweater we're wearing a killer tuxedo.
This is math's problem because mathematicians are the only ones who can 'climb mountains'. I'm just a Joe-schmo who's pointing at some unexplored mound that's hidden behind clouds saying 'that could be a bigger mountain'.
1) The parts only exist when measured. If nobody is looking, it is meaningless to speak of parts (just as it is meaningless to say that Schrödinger's cat has a definite state).
2) There are only ever finitely many parts. The number depends on how many measurements you've made.
I communicated this to @fryfish a while back on this thread but I don't think you were active at that time so I'll repeat the idea. Your view is that a string is created by assembling a whole bunch of nothing (i.e. 0D points). As long as you have enough nothing then you can eventually produce something. It's crazy.
With Aristotle's view, we start with something, we start with the string. How many endpoints does the string have, let's say 2 for simplicity. Can we create more endpoints? Sure, just cut it more and more times. The more cuts you make, the more endpoints you have. How many endpoints do you have? Well it depends on how many cuts you make, but certainly you will have some finite amount. Now, how many times do you need to cut a string until it disappears - until it's just a whole bunch of 0D points (i.e. nothing)? Does infinity do the trick for you? If you are open minded you have to acknowledge that there's a problem here. And if we can't go from a continuum to infinite points, why would we think that we can go from infinite points to a continuum?
This is why I called you an idealist.
Space is far from empty. Fields permeate all of space.
Name calling doesn't win debates.
Is a tomatoe a field? Why not? You're throwing ideas out there that are philosophical and calling them science. This is a philosophy forum. We think critically about what scientists say if it in any way comes into the realm of philosophy. That's what we do! Maybe waves are particles and particles are waves. You can't disprove that. What we know is what our senses say, yet you say our senses and the paradigms they are in are close to having ZERO accuracy. Why trust your use of measuring equipment then? We have no real understanding of what stuff looks like at the quantum level. All we have are vague ideas (which are fine) and a lot of assumptions. But your solution to Zeno's paradox is that the quantum world doesn't exist unless we measure it. Which, well, is more paradoxical than what Zeno proved. But he DID end up being an idealist, so there's victory for you I guess
I believe in the philosophy of Kant, Fitche, Schelling, and Hegel. I interpret them all in terms of materialism (the soul arises from matter) and a combination of process philosophy and traditional ontology. Most philosophers now adays are process philosophers and therefore they don't object to the careless way physicists create philosophical paradigms out of thin air. I read the Tao of Physics last year and it is just ridiculous how a major physicists could have written that! You can't make philosophical claims of that magnitude from physics along. We can point a microscope on a lemon and understand something about the texture of it. But there is a threshold somewhere in there beyond which the microscope can't be proven to be capable of providing evidence of what it really LOOKS LIKE down there. And you are saying that there is nothing there at all unless we measure it. What kind of nonsense
Quoting Ryan O'Connor
I am not intending to hurt your feelings but simply to attack your world view. It's a common world view these days, and it's kind of ridiculous for people to walk around on sidewalks with their legs while they claim the sidewalk and legs are really evanescent waves simply because scientists interpreted it that way because of how certain phenomena affected their measuring apparati. You can't do physics without philosophy, but a lot of times wrong philosophy comes about because of physics
You seem to have missed the gist of the conversation Gregory. I said that when they are rounded off, or "terminated" in Ryan's words, such as the example in the op, or using pi as 3.14, then the so-called infinities are very useful. But a process which is said to be potentially infinite, which will necessarily be terminated at some point, cannot truthfully be said to be potential infinite.
There is no problem. Mathematical analysis took care of that years ago. Only a few philosophers remain addicted to it. But this is a philosophy forum, so it's OK to quibble. :smile:
I think you are wrong in assuming a computer can't, in its way of knowing, understand infinity
You should try posting an unpopular opinion in the politics-related threads around here. Namecalling is all they've got.
Some kind of use-mention problem no doubt.
I deleted the post I wrote. Here's the updated version. First, we must both be frustrated by now. You expressed frustration at trying to tell me something that I thought I had told you three days ago. First you said a still photo stops motion and video shows motion. I pointed out that video is actually a sequence of stills, and that even a still captures motion because it records photons over a nonzero interval of time. Now in your most recent post you are frustrated that you can't explain this to me!
So we're talking past each other. Perhaps we can find agreement at least in that.
Second, it's not the job of math to solve Zeno's paradoxes; and even if it is, it's not an interest of mine. I wish you the best with your efforts in that direction but I can't help. I regard Zeno as a solved problem mathematically via the theory of convergent infinite series; and an unsolved problem physically because our best theories don't allow us to reason below the Planck scale.
And third, dt is a differential form. They don't explain these in calculus so that's why everyone's confused about them. Bottom line is they aren't numbers and they can't be zero OR nonzero. If you use the notation [math]\Delta t[/math] that will be accurate. That's an interval of time, zero or nonzero as the case may be.
Your idea about building points from lines or lines from points just went over my head. I don't understand your intention at all. I'm not trying to solve the nature of the continuum here. Mathematically we construct the real numbers and call them a line, but the geometric visualization is incidental and not essential. As far as the true nature of the world, that's above my pay grade but I would personally be very surprised if it's anything at all like the mathematical real numbers.
I haven't much to add beyond this.
This may be a little high powered. The differential I-form is [math]f'(x)dx[/math] and it is common practice when teaching elementary calculus or even advanced calculus to simply assign the value
[math]dx=\Delta x[/math] for the independent variable. I used Thomas several times for calculus and Olmsted for advanced calculus. In both the authors explain that dx could be any real number, and then they assume it to be [math]\Delta x[/math]. Of course, dx is originally an infinitesimal. It's not a big deal IMO.
I wouldn't dare, I've already had my fair share of backlash in the math group.
Quoting fishfry
I don't agree with this. You're missing the key ingredient: time. A video is a sequence of stills, each held for some interval of time.
Quoting fishfry
I don't completely agree with this. It's true that macroscopic stills capture some motion but when we look at the microscopic level, it is possible for a still to capture zero motion (e.g. a single photon hitting a quantum sensor). And in a simulation (which is what I'm most familiar with) the stills capture zero motion (note: the velocity field seems to reflect instantaneous velocity but let's not go there).
Quoting fishfry
I don't really agree with this. I held the 'point-based' view for most of my life so I think I understand your position, I just don't think it's correct. I think the problem is that you don't understand my position. This is not necessarily your fault, in fact since nobody else here seems to understand my position it's probably my fault.
Quoting fishfry
Understood. And to be realistic, since you haven't understood my other arguments I've presented I think it would be a waste of your time.
Quoting fishfry
You're right. This was purely a laziness thing because I didn't want to find the symbol for delta. Thanks for the correction.
Quoting fishfry
Okay, given that our discussions have been focused on the nature of a continuum, I think this is a pretty good reason to wrap them up.
No, I'm saying that my philosophy is in agreement with modern physics.
Quoting Gregory
We know much more than what our measuring equipment says. Our measuring equipment gives us data, physics gives us models (that match the data and make predictions). And with these models we can talk sensibly (to some extent) about what reality is like at the quantum level.
Quoting Gregory
No, unmeasured particles exist in a superposition state (a state of potential). This is the interpretation of QM that physicists are taught in school. It's not paradoxical (there are no contradictions), it's just weird because it violates our classical intuitions. In fact, I believe it's nature's way of cleverly avoiding paradox/contradiction/singularities.
Think about a video game, when Mario is dilly-dallying collecting coins is it reasonable to ask what Bowser is doing at that moment? No, that part of the world just has not been resolved. All we could say is that Bowser is somewhere in his castle waiting for Mario. He has the potential to be in one corner of the room, and the potential to be in the other corner of the room. We can describe his position only as a potential state. Your console is unable to resolve Mario's universe everywhere and at all times so it only resolves what you're looking at. Why would our universe be any different?
I live and breathe physics simulations so I can't help from seeing our universe as a big simulation. And since I don't think any computer can have infinite computing capacity, the computer of our universe has got to cut corners somewhere. And it does it remarkably well. Essentially what you are craving is an infinite computer. You are craving a supernatural world.
And by the way, measurement is not something exclusive to humans.
Sorry I left off the suffix, 'ly'. Try this:
But a process which is said to be potentially infinite, which will necessarily be terminated at some point, cannot truthfully be said to be potentially infinite.
That better?
Atalanta is walking from x=0 to x=1. What is the first non-zero coordinate that she walks to? I'd like to know how mathematical analysis solved this.
As long as we think that a line is composed of points we cannot answer this question. Motion is only possible when she jumps from one point to another. And what I'm saying is that she doesn't jump over infinite points, she jumps over a continuum.
There is no first nonzero positive real. The open unit interval (0,1) does not contain its greatest lower bound. Any math major can tell you that. They learned it in real analysis class.
Quoting Ryan O'Connor
If we allow that the real line is made of points (which are just real numbers) then the answer is that there is no first nonzero positive real number. That is the answer, so your claim that "we cannot answer this question" is false.
Quoting Ryan O'Connor
Now who's claiming a line is made of points? You are the one doing that!
Quoting Ryan O'Connor
You may not like the real numbers as a model of the continuum (and after all, some mathematicians agree with you); but that doesn't entitle you to mischaracterize the math of the real numbers. There is no smallest positive real number. Why is that a problem. In your model of a continuum, whatever it is, is there a smallest real number?
This was Ryan's term. You'd have to go back to see what Ryan was talking about. Essentially Ryan suggested replacing infinities with infinite processes. Since the supposed infinite processes could never be completed they are assumed to be potentially infinite. I argued that every supposed potentially infinite process will for some reason or another, at some point be terminated. If this is the case then it is incorrect to even call them potentially infinite.
Quoting tim wood
You left out the important phrase: "which will necessarily be terminated". The infinite process would continue forever, by definition. Since forever never arrives Ryan says we ought to call it potentially infinite. Ryan suggested that we could put an end to a potentially infinite process, by rounding of pi for example, yet still say that it is potentially infinite. Obviously though, if someone puts an end to a process, it is not potentially infinite. So Ryan proceeded to distinguish between the rule which produces the supposed potentially infinite process, and the process itself, trying to place the potential for infinity within the rule rather than the process.
Quoting fishfry
Exactly. And since there is no first nonzero positive real then she cannot take her first step. Her journey doesn't begin. Motion is impossible. When I said that 'we cannot answer this question' I should have said 'we cannot answer this question satisfactorily.'
Quoting fishfry
My view doesn't exclude points. Let me try this again but with a much more digestible image.
I suspect you would say that you see infinite points there. I don't. In this picture I see 5 points, each connected by lines. That's it. WYSIWYG. And if we relate this back to Atalanta's story, I would say that we have still photos of her at 0, 1/3, 1/2, 3/4, and 1. That's all we've got. We are not justified to say that at some time she was at 1/10 or at 7/8 because we don't have the photographs. At one moment we see her at 0, we blink, and when we open our eyes she's at 1/3. Motion happens when we blink, when we are not looking. And I've said it before but perhaps it might sink in this time that this is consistent with QM. If we are continuously observing a quantum system it will not evolve. It will only evolve when it is not being observed. Change happens when we blink. Not surprisingly, it's called the Quantum Zeno Effect.
Quoting fishfry
Perhaps my point didn't come across. All I was trying to say was that 'there is no smallest positive real number' means that Atalanta cannot decide what point to go to first.
Quoting fishfry
In my view, we cannot speak of the 'completely measured continuum' which is what I would call 'the real number line'. We cannot do so because it would require infinite measurements and that's impossible. Instead, we must only speak of systems which we can actually look at (at least in principle), like the one above. And in that case, the smallest positive number is precisely 1/3.
As I've shown, its demonstratable that space is infinitely packed. Our eyes can even see it in matter. Whether waves will be revealed to be made of particles or visa versa is irrelevant because there is always another level. Quantum physicists descriptions of reality are not necessarily accurate. They have to fit true philosophy. Why so you get your philosophy from scientists i wonder
You keep baiting me by posting bad math.
Quoting Ryan O'Connor
I like talking to you too even though I don't understand the points you're trying to make
Quoting Ryan O'Connor
Why not? Explain to me exactly why someone can't put one foot in front of the other and take a step. How does the mathematical theory of the real numbers preclude anyone from doing that? And -- a question that I keep asking you and that you never answer -- why does any mathematical theory have anything to do with physics?
Quoting Ryan O'Connor
It's perfectly obvious to anyone who ever took a step -- that's most two year olds -- how the process works. Why do you think the modern theory of the real numbers prevents anyone from walking? "Oh no I can't walk, they're teaching Dedekind cuts to the math majors at the university!"
I genuinely can not understand your point. How does the mathematical theory of the real numbers prevent anyone from taking a step?
Quoting Ryan O'Connor
I would say that when you draw the real number line, that's a visual depiction of the mathematical real line, which itself is an abstract object that cannot be depicted. Like drawing an old guy in a robe with a white beard to symbolize God. Of course the picture itself on my laptop screen is made of pixels, which are discrete, and there are only finitely many of them. Why do you either (a) fail to appreciate both of these points, or (b) think I don't?
Quoting Ryan O'Connor
Fine. What of it. How would anyone's interpretation of that picture prevent them from walking or allow them to walk?
Quoting Ryan O'Connor
Yes, motion in the real world is pretty much like that.
Quoting Ryan O'Connor
According to the Copenhagen interpretation, we have no idea what happens when we're not looking. According to Many Worlds, everything happens. How is this relevant to the conversation?
Has it occurred to you that perhaps you are not personally possessed of the ultimate truth about how the universe works?
Quoting Ryan O'Connor
It's consistent with the Copenhagen interpretation of QM but not with the Many Worlds interpretation. And don't forget that Feynman said we can model reality by assuming that you get from point A to point B by integrating over every possible path, no matter how circuitous. You are just getting yourself tangled up here. You're confusing interpretations of QM, which itself is just a mathematical model that agrees with experiment up to the limit of Congressional funding for particle accelerators; with reality itself. You confuse math with physics, and you confuse physics with ultimate reality. Two series category errors.
Quoting Ryan O'Connor
Only in some interpretations. You are making statements about things nobody knows.
Quoting Ryan O'Connor
According to some interpretations, not others. What if a cat blinks? A microbe? A cameral lens? What exactly is a measurement? Nobody knows.
https://en.wikipedia.org/wiki/Measurement_problem
How does the mathematical theory of the real numbers prevent me from walking? You're not thinking through your own argument.
Quoting Ryan O'Connor
That's because you seem to think that the mathematical theory of the real number precludes my getting off my ass and taking a walk. I'll try that one out on my doctor when he tells me to get more exercise. "They're teaching Zermelo-Fraenkel set theory, which leads to Dedekind cuts, so it's all I can do to just change the channel, let alone stand up from the couch."
This is your thesis. Don't you see how silly it is?
Quoting Ryan O'Connor
How does the mathematical theory of the real numbers prevent someone from taking a walk? Before Dedekind had his clever idea, were people able to walk? And then the day he published, they couldn't? Isn't what you are saying patent nonsense?
Quoting Ryan O'Connor
Now you're just being silly, since if you claim 1/3 is the smallest positive real number I'll just divide it by 2 (using the field axioms) and note that 0 < 1/6 < 1/3. Your claim stands refuted.
Zeno's paradox when seen from the perspective of pure mathematics is easily dealt with using the limit concept, but giving it an anthropomorphic twist makes it absurd. And the idea of a first non-zero coordinate shows a very limited knowledge of mathematics.
Would you agree a wise intellectual warrior should first know his enemy before striking?
He asks whether actual time and actual space can be divided according to the math of ZP.
Oh you watched it, good. He said it "melts the brain" to think of an infinite series going to a destination without a final term. Math, I think you're saying, explains it in its own ways, but there are aspects which still trouble philosophers
I found the vid at https://www.youtube.com/watch?v=u7Z9UnWOJNY. I didn't want to watch the whole thing so I skipped to 12:00 and he said that "in the physical world" it's not the same, but I couldn't find the exact quote about brain melting. As I mentioned to @Ryan, Zeno is solved mathematically by virtue of the theory of infinite convergent series, as the Numberphile guy says; and it's unsolved physically, as he also seems to agree.
In most discussions online people seem to believe the problem is solved by calculus, but that's not true. Only the mathematical problem is solved. The physical problem remains, and even moreso today, because we know about the Planck scale. Below a certain point, our physical theories are not applicable to space or time. The tortoise or the arrow or whatever only have to iterate through 35 or so steps before reaching the point where nothing sensible can be said.
At some point, you can't sensibly divide a distance in half, nor an interval of time. Physics doesn't allow us to get a sensible answer. We don't know whether space and time are arbitrarily divisible; but we DO know that our current understanding of physics doesn't let us sensibly speculate.
I don't know what you're talking about tim. To say "X will necessarily be terminated" seems very restrictive to me. Obviously, the cause of termination of the process is external, that's Newton's first law. But how's that relevant?
If A to B is seen as infinite fractions (half, quarter, eighth, ect) and as one travels it the mover changes color every fraction, he is not a definite color by the time he reaches B. He is kind a blur. But he can reach B because, on the flip side, A to B is a finite segment and must be finite in actuality. So there is the infinite approach and the finite way to see it. For some of us the tension between the infinite and finite here is strained and we feel there is and can only be something off about this. But as you say, it's telling us that there are aspects of reality that transcend us, and oddball paradoxes like this are just like jokes put into reality by God
It is? Can you point me to the demonstration? This may be going off on a tangent, but the Bekenstein bound poses an upper limit on the information that can be contained within a given volume. Or are you talking about mathematical space being infinitely packed?
Quoting Gregory
I believe that physics, math, and philosophy all inform each other. Physicists don't have it all figured out, but philosophers need to update their views based on the latest advancements in physics (and math).
Modern physics only says how stuff work. Do this, that happens. That sort of thing. The electrons might be doing what the protons are said to do or the other way around. For every wave there could be a hidden particle. We don't need pilot theory to know this, but if that's works for ye that' fine. My point is much of what QM says is false because they are clothing their findings in the language of an unprovable philosophy. They can predict things, but that is all QM can do.
As for matter, it's spatial? Yes. Is all space infinitely divisible? We can't conceive it as discrete, so our natural lights say yes. So your computer is infinitely packed. This is intuitive for me at least. That objects are in a sense very finite is also true, but General Relativity goes well with infinitely parted objects in the sense I mean that. GR is a great theory, and not just because it has my initials :) Lastly, there are different kinds of density and ZP is about the relationship between volume and mass. Until you grasp this you'll be in the dark of QM topology
Since both you and @fishfry reacted the same way, I've learned that I shouldn't try to make a point in the form of a question. I know that there is no first non-zero coordinate on the real number line, that's exactly what I was trying to highlight. Let me try again. Before she arrives at x=1, do you believe that she must first cross all points between 0 and 1? And before she arrives at x=0.5, do you believe that she must first cross all points between 0 and 0.5? If so, then we can take this reasoning to its limit and say that to move she must first reach the first non-zero coordinate. And if there is no first non-zero coordinate, then she cannot move. This is Zeno's Argument which is what trying to highlight, but apparently didn't convey very well.
In the Numberphile on Zeno's Paradox, James Grime says the following:
"I want to give you the mathematician's point of view for this, because, well, some say that the mathematicians have sorted this out........So something like this-- an infinite sum-- behaves well when, if you take the sum and then you keep adding one term at a time, so you've got lots of different sums getting closer and closer to your answer. If that's the case, if your partial sums--that's what they're called-- are getting closer and closer to a value, then we say that's a well-behaved sum, and at infinity, it is equal to it exactly. And it's not just getting closer and closer but not quite reaching. It is actually the whole thing properly."
Is your view that the problem (of Atalanta travelling from x=0 to x=1) is resolved by completing an infinite process?
I think the most compelling solution to Zeno's Paradox (of Achilles and the tortoise) that is often presented is by looking at the situation holistically. If you ask what are the velocities of Achilles and the tortoise you can work backwards to calculate the instant when Achilles passes the tortoise. It seems so simple when you think of it this way. This is the type of thinking that I'm promoting: starting with the whole and working backwards to determine instants.
Quoting fishfry
Are you suggesting that with each step someone sweeps over infinite points? In other words, are you suggesting that motion involves the completion of a supertask?
Quoting fishfry
My problem is that the theory of real numbers implies that continua are made up solely of points. To me, this means that to move from x=0 to x=1, Atalanta must sweep over infinite points (she must complete a supertask). Instead of saying that there are infinite points, I'm saying that she only needs to sweep over a finite number of things. For example, in the image below, Atalanta needs to sweep over only 4 line segments. It’s a task that does not require any supernatural powers.
Quoting fishfry
Math and physics are not necessarily related, but as you point out, we move all the time. I take that to mean that nature has found a way to avoid the traps of Zeno's Paradoxes in the physical universe. As such, nature may provide some clues on how to solve Zeno's Paradoxes in the mathematical universe. The cheap solution is to say that continua are discrete. I think nature has found a far more elegant solution.
Quoting fishfry
Let me try from a different angle. Let us say Atalanta lives in a perfect resolution simulation. As such, as she traverses from x=0 to x=1, she must at some instant in time be at each of the intermediate points. If we take out all the fancy graphics of the simulation, the computer is essentially just outputting a list of coordinates from 0 to 1. But we know that the real numbers are not listable (countable). There must be something wrong with the story. Do you agree?
Quoting fishfry
I'm not trying to trick you by using pixels on your screen or atoms on my paper. Take my drawing and 'idealize it'. I know that a truly 1D object cannot be depicted (it takes up no area after all). We should move to 2D to make it depictable but you didn't like my earlier graphs so to keep things simple, let's just assume that we're looking at it with our mind's eye. This is a representation of how I see the idealized image:
I see 4 line segments with the numbers corresponding to the gaps in between the line segments. Points vanish upon idealization.
Quoting fishfry
One can hold incorrect views on the laws of nature, but they will nevertheless follow the true laws. Similarly, one can hold incorrect views on the nature of continua which imply that motion is impossible, but they will still be able to walk. What I'm saying is that your views imply that motion is impossible. I'm not saying anything about your ability to walk.
Quoting fishfry
Many worlds provides a different interpretation on wavefunction collapse, but it still holds that in between quantum measurements what exists is a superposition, not a definite state. QM is only relevant if it informs our solution to Zeno's Paradox. I'm bringing it up because without QM my view sounds absurd. But you're right, I don't know the correct interpretation of QM. I also don't know what exactly entails a quantum measurement.
Quoting fishfry
Of course. Truth comes out with formalized theories. I'm merely sharing an atypical intuition, which has neither been proven right or wrong. Ideas are cheap, including mine.
Quoting fishfry
No, I'm not being silly. 1/3 is the smallest non-zero number in this system.
In this system 1/6 is the smallest non-zero number.
Once we place another point on that line, we have a distinctly different system. This system is composed of 6 points and 5 line segments. In this system 1/6 is the smallest non-zero number. You have refuted nothing.
Quoting fishfry
As I mentioned to jgill, James Grime in the Numberphile video suggests that the solution kind of involves the completion of infinite tasks. Is this the view you hold?
Quoting Gregory
No doubt, QM is in dire need of philosophical progress. But we can't downplay the value of physics. Experiments and corresponding models ground our reasoning.
Quoting Gregory
Infinitely divisible does not imply infinitely packed. We can cut a 'mathematical' string indefinitely but never will the string be in infinite pieces.
I've been arguing that to be infinitely divisible means that it has infinite parts. These seem identical to me
Quoting Gregory
These are not identical. A 'mathematical' string is infinitely divisible because I can cut it to no end. I end up with infinite parts only when done cutting it, but am I ever done cutting it?
The parts are always there!
@keystone, I'll reply to this one before the other one. I had no trouble @-ing you so at least that worked. I'll continue to quote you as Ryan here but I assume that if you quote me back you'll be on your @keystone account and everything will be clear.
Quoting Ryan O'Connor
As I have asked you several times, is this a mathematical or a physical scenario? If mathematical, the answer is yes. And @jgillI made the same point. By the intermediate value theorem, a continuous function passes through all intervening points between one value and another. And I assume continuity is one of your assumptions here. Is it? Essentially you're asking about the nature of continuity.
If this is a physical scenario, the answer is that one, nobody knows for sure; and two, the question is meaningless with respect to contemporary physics because we cannot reason sensibly below the Planck length. Does she hop from quark to quark? No physicist would regard that as a meaningful question.
By the way I looked up [url=https://www.thoughtco.com/greek-mythology-alanta-1525976[/url], and having been a student of Greek mythology way back in the day I was pleasantly informed. She's the goddess of running. The link I gave, which is better than her Wiki link, is full of good stories.
So: I say again, as I have said several times: Mathematically there is no question that she passes through every point indexed by a real number. Physically, the question is open in general, and meaningless in current theory. Even a goddess can't run between quarks.
Quoting Ryan O'Connor
You're perfectly well conveying your refusal to read what I'm writing. If this is a mathematical thought experiment, she does pass through every point in the closed unit interval [0,1]. If this is a physical thought experiment, the question is metaphysically open and physically meaningless. So which question are you asking?
Quoting Ryan O'Connor
That's right. Mathematically the sum of an infinite series is defined as the limit of the sequence of partial sums; and the limit of a sequence is defined as the number the sequence gets (and stays) arbitrarily close to. And the Numberphile guy says exactly that.
Quoting Ryan O'Connor
Depends on whether this is a mathematical or physical thought experiment. The physical question is certainly not resolved by the math, but I've stated that many times already.
Quoting Ryan O'Connor
If you would stop conflating math and physics, and avoiding answering whether you are asking a mathematical or physical question, all would be clear.
On the other hand, it's perfectly obvious that motion does occur in the real world (unless we live in a block universe or we're all programs running in the great computer in the sky or brains in vats or I'm just dreaming all this.
Quoting Ryan O'Connor
Aren't you the one saying you agree with Aristotle? He believed that the reason bowling balls fall down is that they're made of "stuff" and so is the earth, and like attracts like. Aristotle needs an update too.
No, I'm suggesting that the mathematical concept of the real line doesn't apply to the true nature of physical space; or that if it does, this fact lies far beyond contemporary physics. Even physics doesn't claim it does. Can't we stop here and nail down this point?Why are you deliberately ignoring this point, which I have made to you over and over? Nobody knows if space is infinitely divisible; and everyone knows that we don't know this.
Human beings and even goddesses can not traverse the mathematical real line because the latter is a mathematical abstraction. They can only traverse physical space, and nobody knows if physical space is like the mathematical real line or not. And we DO know that contemporary physics can not address the question.
Quoting Gregory
No, this is a confusion of "infinitely divisible" with "infinitely divided." The former means potentially having infinitely many parts, while the latter means actually having infinitely many parts. A true continuum is infinitely divisible, but this does not entail that it is infinitely divided. It is a whole such that in itself it has no actual parts, only potential parts. These are indefinite unless and until someone marks off distinct parts for a particular purpose, such as measurement, even if this is done using countably infinite rational numbers or uncountably infinite real numbers. A continuous line does not consist of such discrete points at all, but we could (theoretically) mark it with points exceeding all multitude.
Quoting fishfry
This is one particular mathematical model of the interval--the dominant modern one, to be sure, but not the only one. Again, it is not mathematically necessary to treat a spatial interval as somehow consisting of unextended points. We can understand them instead as denoting locations in space, not constituents of space. As such, she does not really "pass through" them, we just just use them to track her progress.
Although the parts exist together, they exist even though they are not separated. A continuum is an infinite substance. Nothing potentially has parts. It HAS the parts whether they are separated or not
I obviously disagree. Again, a true continuum has no definite parts except those that we deliberately mark off within it for a particular purpose. It is infinitely divisible, but not actually divided.
Quoting Gregory
Neither the foot nor the inches are "there" unless and until we mark them. They are arbitrary units of length for measuring things, not intrinsic to space itself.
The continuum could easily be a dog instead of a segment. He has legs regardless of whether you point them out or cut them off
Responses like this are why I rarely bother jumping into these discussions.
There seems to be no reason the segment has to be parted into infinity in order to contain infinity. That seems irrelevant to me
A one-dimensional line does not "contain" anything. We can mark it with as many zero-dimensional points as we can imagine, but those points are not parts of the line itself.
If this is true, what you describe here, then it is impossible that "a true continuum is infinitely divisible. If marking points on a continuous line does not constitute dividing it, then there is nothing to indicate that the continuous line is divisible at all. And if dividing it once would break it's continuity, then a continuum cannot be infinitely divisible because dividing it once would prove it to be discontinuous.
Therefore it is a contradiction to say "a true continuum is infinitely divisible". We ought to say instead, "if it were divisible it would not be a true continuum". A true continuum is indivisible.
Quoting aletheist
Again, this is contradiction. If it cannot actually be divided, then it is false to claim that it is divisible, in any sense.
In itself, yes; but we can still "divide" it at will to suit our purposes. That is what I mean when I say that the whole is real and the parts are entia rationis, creations of thought. For example, we can conceive space itself as continuous and indivisible, but we can nevertheless mark it off using arbitrary and discrete units for the sake of locating and measuring things that exist within space.
If you view real numbers as locations where points might live, then she passes through each location. I don't see how this changes anything. Also I've previously referenced the constructive, hyperreal, and Peircian concepts of the real line, so I don't think I can be accused of narrow mindedness on the topic. Unless you reject the intermediate value theorem, my point stands. And even the constructivists have their own version of IVT, which they make work by restricting functions to being computable ones. Some version of the IVT is always valid regardless of one's model of the real line.
Quoting aletheist
Distinction without a difference. IMO. She passes through each point or she passes through each location that is the address of a point. Not sure what you are getting at.
Put another way, there are no real points or locations that she passes through, only the ones that we invent to describe her motion. Again, space does not consist of such discrete points or locations, we introduce them for our own purposes. Not sure I can convey what I mean any more clearly than that.
In other words space is not described by the mathematical real line. As I've written in this thread at least ten times now.
Quoting aletheist
How many times must I say the same thing? I don't get it. To be fair you may not be reading my replies to @keystone aka @Ryan, but rest assured that I have made the point you are making many times over.
(I believe all of Zeno's arguments are reduced to one single insight: spatial objects are finite with infinite parts. If this does not give you a headache one day of your life then you never understood it. The argument that motion through a point is really "rest" I think is false. Zeno had one great insight and it was an insight into mathematics, not physics)
Well, the IVT is not valid in smooth infinitesimal analysis. As Bell states in his book that I suggested a while back, "the classical intermediate value theorem, often taken as expressing an 'intuitively obvious' property of continuous functions, is false in smooth worlds." The Wikipedia article on SIA explains:
It is presumably less surprising that the Banach-Tarski paradox also does not arise in SIA.
Quoting fishfry
So we agree, then? The mathematical real line is an extremely useful model of a continuous line, but like all representations, it does not capture every aspect of its object--in this case, a true continuum.
I have never claimed otherwise. If I've somehow failed to communicate that, I'll try to do better. But I would disagree with one aspect of what you said. The mathematical real line fails to describe what's physically provable about space, in terms of people walking and "passing through every point." But why should space be a continuum at all? That's an open question. How would anyone know what a "true continuum" even is? The idea of a continuum, let a lone a "true" one, is a conceptual abstraction. It's like arguing about which mathematical model of the tooth fairy is correct.
I would ask you: What is a "true continuum," and how would you know one if you saw it? Are you claiming there is such a thing in the world? Or if not, and if it's only a conceptual abstraction, how can anything be an accurate model of it?
ps -- SIA denies excluded middle and so falls into the general category of constructivism. No wonder many common standard theorems are false in such a framework. I've already mentioned that IVT is false in constructivist math but they patch that up by demanding that all functions are computable. In SIA, all functions are continuous.
I agree, it is a hypothesis--one that I happen to find much more plausible than space consisting of discrete parts. I would say the same about time, which Peirce considered to be "the continuum par excellence, through the spectacles of which we envisage every other continuum."
Quoting fishfry
I guess it comes down to definitions. Modern mathematicians stipulate that the real numbers constitute the (analytical) continuum, but (at least arguably) that approach is not entirely consistent with the common-sense notion of what it means to be continuous.
Your intuition is seriously at odds with modern physics. Do you think physics is wrong? How do you square this.
Secondly I wanted to repeat in case you missed the ps to my last post, that SIA denies excluded middle. So it's a flavor of constructivism. No wonder that IVT is false, it's false in constructive math. And no wonder Banach-Tarski is false, constructivism denies the axiom of choice.
Quoting aletheist
Your common sense notion is incompatible with modern physics. And you are failing to distinguish between physical space, which we think is "really there," and the idea of the continuum, which is just a philosophical concept with no actual referent in the real world.
No, I think that anyone who interprets the Planck length as a discrete constituent part of space is wrong. I interpret it instead as a limitation on the precision of measurement, or as Wikipedia puts it, "the minimum distance that can be explored. ... The Planck length is sometimes misconceived as the minimum length of space-time, but this is not accepted by conventional physics, as this would require violation or modification of Lorentz symmetry."
Quoting fishfry
I did not see your PS until now, but I am well aware that the logic of SIA is what has come to be known as constructive or intuitionistic. Peirce was skeptical of excluded middle, but for very different philosophical reasons than Brouwer and Heyting--he believed that reality itself does not conform to it, because it is fundamentally continuous and general, rather than discrete and individual. He stated this in slightly different ways in alternate drafts of the same text.
To identify a point on a line is not to divide that line. So it's not really a matter of dividing which you are talking about.
Quoting aletheist
The proposed units would be arbitrary, but I do not think you could call them discrete. And, according to the issues brought forward by special relativity, the supposed "same unit" would be different depending on the frame of reference, or more precisely, we could not determine the "same unit" from different frames of reference. This makes the whole idea of measuring the continuum through the means of units rather difficult. I suggest that if the application of units works well for measuring space and time, they are probably not actually continuous.
The speculation which is the reverse of yours is that continuity is what is artificial. The continuum is something created by human minds, and physical existence contains no such continuity. Problems such as Zeno's paradoxes arise because we apply principles of continuity to a physical world which is discontinuous.
I think Bayle's conclusion is the only one that is eventually reached by logic on this. The great mathematician Alfred Whitehead write on the subject of "wholes and parts" and yet Wikipedia said some of his results appear to be wrong. This subject is a tangle which however allows you to view it with intuition instead of logic and get something out of it instead of eternal frustration. I have no doubt no one will find "THE answer" to this riddle. Yet it can be accepted as a trick played on us by the fathomless
I've mentioned QM and physics in an attempt to support my view, but the focus of my argument has always been on the mathematical scenario. From now on, unless explicitly stated otherwise, I'm talking about the mathematical scenario. I'm assuming continuity. The space that I'm talking about is infinitely divisible. There's no planck length/time to suggest any possible notion of discreteness.
Quoting fishfry
Are you referring to mathematical space or physical space? I assume in this context you're referring to physical space, right? Is it fair to assume that you believe that mathematical space can be modelled with the real numbers?
Quoting fishfry
A thought experiment is exactly what I'd like us to do. To perform an experiment, we cannot just say 'she passes through every point', we must actually conduct the experiment. The best way to do this is to envision that Atalanta lives in a simulation and we must understand how that simulation works. I will grant you a computer with no restrictions (e.g. infinite memory and speed). The only constraint is that the simulated universe must be consistent. How does the simulation allow her to pass through every point in the closed interval [0,1]?
One might say that this simulation can be easily performed in 1 second (e.g. the time interval [0,1]). There's a 1-to-1 correspondence between positions and instants in time, so for each instant the simulation outputs the matching coordinate (e.g. at 0.4 seconds, Atalanta is at x=0.4). But hold on. That would mean that the computer is effectively outputting a complete list of the real numbers between 0 and 1 and Cantor showed that such a list is impossible. For her to move, the simulation must be a lot more clever than that. Any ideas?
Quoting fishfry
Aristotle was wrong on many things. We should not resuscitate his bad ideas without good reason...I just think that his ideas on potential infinity deserve another look now that our intuitions have been altered due to modern physics. (note: I'm talking about intuitions from physics providing further insight into longstanding paradoxes in the philosophy of mathematics, such as the mathematical Zeno's paradox.)
Quoting aletheist
Yay!
If we take this reasoning to its limit then we end up with the whole being an assembly of a bunch of 0-Dimensional objects (i.e. nothing). Is that what you believe? Everything is made up of nothing?
No. The zeros (points) is approached by infinite space that is also finite. That is the only way to put this question honestly. "In the early 19th century, Hegel suggested that Zeno’s paradoxes supported his
view that reality is inherently contradictory." Internet Encyclopedia of Philosophy
It's said everywhere that modern math found the answer to this paradox, to which I use words of Heraclitus "Hearing they do not understand, like the deaf. Of them does the saying bear witness: 'present, they are absent'" I say this only because most people don't understand what the paradox means so there "explanations" completely miss the mark
Do you expect to be in the news as the guy who solved this ancient problem? As the mathematician said in video linked earlier in this discussion, greater minds than yours have wrestled with this problem and failed. Not so long ago some writer came out with his "solution" and got a lot of praise until everyone realized that it was Henri Bergson's answer that had plagiarized and it was not a true solution to the problem to begin with
I mean no offense
Ok. Then for clarity let me add another important condition. The rational numbers are infinitely divisible, but they are not complete. In addition to infinite divisibility, we need to require that every nonempty set of real numbers has a least upper bound. Otherwise the resulting system fails to be an adequate model of continuity. Just noting this for accuracy.
Quoting keystone
By space I mean physical space. And by physical space I sometimes mean "true" physical space as in ultimate reality, if there even is such a thing; and other times I mean the theories of contemporary physics.
Quoting keystone
Yes.
Quoting keystone
No, I'd be very surprised if this turns out to be true. The mathematical real numbers are far too strange to be real in the sense of physical reality.
Quoting keystone
There are no computers involved in this. Computers are far too limited, since Turing machines are restreicte to finite sequences of instructions and finitely many steps. A Turing machine already has an unbounded tape (memory) and speed of execution is not a factor at all. So this is a bad model of mathematics. For example, no computer program can approximate or generate the decimal digits of a noncomputable number. So I reject this idea totally. The set of computable numbers is a countably infinite proper subset of all real numbers, so they are missing a lot.
A function that passes through the point a at one time and b at a later time must necessarily pass through every intervening point. If one is a constructivist this becomes false, but the constructivists patch the problem by restricting attention to computable functions to make it true again.
Quoting keystone
Your idea is totally invalid. Computers, even theoretical abstract ones, are not relevant. What can be computed is a tiny subset of what can be mathematically proved to exist. As I noted, Turing machines can only generate the countably infinite set of computable real numbers. The noncomputable reals, of which there are far far more, are entirely beyond the range of computation.
The concept of computable real numbers was first elucidated by Turing himself. His great 1936 paper is called, "On Computable Numbers, with an Application to the Entscheidungsproblem." Never mind what the Entscheidungsproblem is. What's important is that it was Turing himself who first pointed out this profound limitation in the ability of computers to represent or characterize real numbers. The exact limitation your simulation or computation idea is bumping into. If we restrict ourselves to only numbers that are computable, then you CAN drive a truck through the real number line, because the computable real line is full of holes. [A skinny truck of course, the width of a point].
In fact the computable numbers, just like the rationals, are infinitely divisible but not complete.
Quoting keystone
Just pointing out that you want to refer back to Aristotle as authoritative in some things but "in need of updating" in others. Cherry-picking Aristotle as it were.
I have never done that, so I'm happy to see that I am therefore not the target of that particular criticism.
Quoting aletheist
It is not exactly a measurement problem. The fact that I can't measure exactly one meter with a meter stick is the problem of the approximation of measurement.
The Planck scale represents the point at which contemporary physics breaks down and is not applicable; so that we can not rationally discuss or compute what happens below those lengths, durations, and energies. That's subtly different than just the approximateness of measurement.
Quoting aletheist
Not by me and frankly never by me.
Quoting aletheist
We're perfectly in agreement on this point except that my physics is not strong enough to catch the reference to Lorentz symmetry. I found a physics.SE thread linking the terms but didn't read it. Should I?
Quoting aletheist
Ok. Then it's fair for me to note that I already mentioned constructivism in replies to @keystone and can't personally do anything about the legion of neo-intuitionists running about these days, though I would if I could :-)
Quoting aletheist
That's interesting. The differences among the LEM-deniers are too subtle for me to appreciate. But then again I don't have much interest in reality when it comes to math. I don't feel a need to justify math in the name of reality. And I can always fall back on history: complex numbers, non-Euclidean geometry etc., to show that no matter how weird math gets, people often find a use for it. I am not defending math as any kind of model of reality any more than I would defend the game of chess on that basis. I'm immune to criticisms based on anyone's idea of what's real because I don't think math is particularly real. On my formalist days at least. The rest of the time I take a Platonist view. I have no hard convictions in this regard.
Quoting aletheist
Yes but does he deny the way the knight moves in chess? LEM can be taken as a rule in a formal game. I would never defend LEM by saying the world works that way. How could I pretend to know such a thing? Peirce could be 100% right yet LEM-based math is still an entertaining pastime. I never speak of the absolute state of things. You see you are trying to get me to take the other side of a question I reject entirely. I don't say, "LEM-based math is right because the world works that way." I would never say such a thing. I'm not making any claims about the world, nor about LEM-based math's applicability to the world. I say, "LEM-based math is interesting in its own right, and if the arc of history is turning against it via the neo-intuitionists, so be it." Sort of like how Cubs fans must have felt about night games [Chicago's Wrigley Field only got lights as recently as 1988].
You say that standard math doesn't apply to the "true continuum." I say to you, "Yes I certainly agree. And by the way, what do you mean by true continuum." That's the conversation I believe I'm trying to have. I'm not claiming knowledge of the world or staking out any metaphysical position.
Quoting aletheist
I have no argument or complaint. He may be right for all I know. That doesn't bear on any point I'm making. You are arguing against multiple strawmen positions I'm not taking.
I agree, this is a better concise summary of what it means.
Quoting fishfry
Heh, we are in the same boat on that, I was just quoting Wikipedia. :cool:
Quoting fishfry
Oh, I completely agree. Again, mathematics is the science of drawing necessary conclusions about strictly hypothetical states of things. Whether those premisses match up with reality is a matter of metaphysics, not mathematics. They can be just about anything imaginable, although some of the most interesting cases come about when we remove a previously taken-for-granted axiom like the parallel postulate in geometry or excluded middle in logic, but still manage to come up with a consistent and useful system.
Quoting fishfry
No, that was not my intention, and I am sorry that I came across that way. I was just trying to provide more background about my own position.
Quoting fishfry
I can only cover so much ground in this format. My long answer is the paper that I provided.
Quoting fishfry
Again, I apologize for giving you that impression.
infinite = 1/infinity
Try to cross multiply
There is only "can increase without limit" but the limit is never reached.
I have concerns with infinite sets (including the set of all rational numbers) and real numbers, but those concerns are not essential to my argument. As such, I accept these two conditions.
Quoting fishfry
Your comment was in response to me asking "Is it fair to assume that you believe that mathematical space can be modelled with the real numbers?" By mathematical space, I just mean continua and given that above you mention that real numbers can adequately model continuity, I will assume that you misread my question.
Quoting fishfry
I wasn't granting you a Turing machine, I was granting you an infinite computer with no restrictions. But fine, let's take algorithms out of the picture, and I'll grant you God and the Axiom of Choice. My only requirement is that everything God does must be consistent. Now, going back to my original question, how God move Atalanta from x=0 to x=1? As I mentioned in my last response, he can't advance her point by point because that would be equivalent to listing the real numbers, which is impossible. So how would he do it?
Quoting fishfry
Your statement/position implies that all that exist between a and b are points. The way I see it is that the function must pass through the intervening spaces. So in the image below, to get from 0 to 1, the function must pass through the 4 continua represented by the following open intervals: [math](0,\frac{1}{3}), (\frac{1}{3},\frac{1}{2}), (\frac{1}{2},\frac{3}{4}), \text{and} (\frac{3}{4},1)[/math]. We both believe that the function cannot skip the intervening objects, I just believe that there are finite intervening objects and you believe that there are infinite intervening objects. I think this difference is what makes Zeno's Paradoxes a problem for the point-based view. Also, my view is not restricted to computable functions.
Quoting fishfry
Oh come on, all I'm saying is that the idea didn't originate from me. I'm not claiming to be right on anyone's authority. You talk as if I must either agree with everything he taught or disagree with everything he taught. Sometimes things come back into fashion...like the mullet, right? That's due for a resurgence soon.
I don't understand this statement.
Quoting Gregory
I am probably wrong, but your statement isn't proof of that. Like I said to fryfish, my view has neither been formalized or proven correct. If I provided a formal theory and proved it correct then that would certainly be newsworthy, but as it stands I'm just looking for flaws in my ideas.
Ok. My point, which I sadly forgot to make, is that even with the clarified fact that the Planck scale is the scale below which contemporary physics can't be applied, your position -- that the physical world embodies or instantiates or contains or is a "true continuum" -- is not supported by contemporary physics. And I asked you what a true continuum is, and how you'd know one if you saw it. And how exactly would you see it? Good questions all.
Quoting aletheist
Yes but even with the clarified definition of Planck scale my point still applies. That your belief that there exists a "true continuum" is incompatible with contemporary physics. If I said inconsistent that was too strong. I should have said, "not supported by." That would be more accurate.
Quoting aletheist
Ok. I still want to know, and think there could be an interesting discussion around, your claim that there exists a "true continuum." You said math doesn't model it, as if there even is any such thing to be modeled.
Quoting aletheist
Ok, but I still don't understand your position. You claimed that math doesn't model the "true continuum." I have two questions. What is a true continuum, and what makes you think any such thing exists in the physical world?
Quoting aletheist
Can I get a short answer? What is a true continuum and what makes you think such a thing exists in the physical world, such that the question of whether it's accurately modeled by math is a meaningful question? After all, we can't ask if math models the tooth fairy. And I claim the tooth fair is in the same category of things as the "true continuum" -- a fairytale. Except that I can't trade the continuum a tooth for a quarter
Quoting aletheist
No prob. I'm still curious about what's a true continuum and why you think there is such a thing.
I'm just clarifying that we often take "infinitely divisible" as a mathematical continuum, but this is very loose speech. The actual condition required is completeness in the metric sense.
Quoting keystone
Depends on the space. In math there are metric spaces, topological spaces, measure spaces, probability spaces, Sobolov spaces, function spaces, and many many other kinds of things called spaces. So the answer is no, without further qualification or clarification.
Quoting keystone
What is a continuum? You ask if the real numbers can model a continuum and I don't know what the question means. The real numbers are commonly identified with "the continuum" but one can challenge that on philosophical grounds, hence the history of intuitionism etc.
Quoting keystone
If you use the word computation it's a Turing machine by default unless you explicitly say otherwise.
Quoting keystone
I wish you'd take the trouble to read what you yourself wrote. You said a computer (or simulation) with infinite memory. A Turing machine already has unbounded memory so you haven't added anything without supplying further qualification. Which you didn't supply.
Quoting keystone
In the physical world? I have no idea and neither does anyone else. In math? There's a function f(t) = t that's 0 at time 0, 1 at time 1, and that passes through every intervening point. Or that passes through every intervening location where there could potentially be a point as @aletheist noted.
Quoting keystone
f(t) = t. Or any of infinitely many other functions that have f(0) = a and f(1) = b. I don't follow why you're making a mountain of a mathematical molehill. Or what God has to do with any of this.
Quoting keystone
On the standard mathematical real line? Yes that's true. You think otherwise? But I don't need to use the philosophically loaded word points. I can say that between any two real numbers all that exists are other real numbers. You disagree in some sense? Be specific.
Quoting keystone
If I go out to the nearby highway and remove all the mile markers and road signs does that make them disappear? You're confusing labeling with existence. If I remove the label from a can of soup the can still contains soup. You're making a very disingenuous point.
I find your claim silly and not at all a serious argument or position.
Quoting keystone
Argumentum ad mulletus.
That is not an accurate statement of my position. I hold that space is a true continuum, but not that it is something physical; rather, it is the real medium within which everything physical exists. Ditto for time, albeit in a different respect (obviously).
Quoting fishfry
We all perceive space and time, and some of us formulate the hypothesis that each is truly continuous in a way that no collection of numbers, even an uncountably infinite one, could ever fully capture because of their intrinsic discreteness. Nevertheless, this does not preclude the real numbers (for example) from serving as an extremely useful model of continuity for the vast majority of practical purposes.
Quoting fishfry
I have said before, and I just said again, that the real numbers do very successfully model a continuum. They just do not constitute a true continuum. That requires a different mathematical conceptualization, and smooth infinitesimal analysis turns out to be a promising candidate.
Quoting fishfry
I gave it a shot, hope it helps.
Hmmmm. The way I'm hearing this, and correct me if I'm misunderstanding you, is that there's an absolute space against which everything else happens. A fixed, universal frame of reference. I hope you can see the problem there. And if that's not what you mean, then I don't understand what you're saying.
After all in general relativity spacetime is a manifold. It's like a twisted space without the ambient space. Like a globe, the earth say, embedded in Euclidean 3-space ... except there is no ambient space. There's only the globe, with its intrinsic curvature.
And again you have used the phrase true continuum, which you haven't defined.
I do appreciate that you've put something on the table that I can at least begin to ask questions about.
Quoting aletheist
I understand that you mean that the real numbers are composed of individuals, namely the real numbers. But the real numbers are not a discrete set. The integers are a discrete set, because around each integer you can draw a little circle that doesn't contain any other integers. You can't do that with the real numbers. So I get that Peircians don't like the fact that the reals are the union of their singleton points. But I don't see how you can label some alternative model "true" without evidence.
Quoting aletheist
Yes of course, we agree on that. And for my part I go farther. The reals are fun to study and if the physicists find them useful that's great because it means the universities will continue to fund the math department. Else they'd fire the whole lot. But the reals are far too weird to represent anything real. If nothing else, the reals as we understand them depend on the vagaries of which set-theoretic axioms we choose. If anyone thought the real numbers were "really real" then physics postdocs would get grants to count the number of points in a meter in order to determine whether the continuum hypothesis is true. Since nobody has applied for any such grant, I take that as evidence that nobody takes the real numbers seriously as an accurate model of anything physical. They're only an abstract mathematical model, one with deeply strange properties.
Quoting aletheist
Ok.
Quoting aletheist
LOL. As Reagan said to Jimmy Carter, "There you go again." I wish you would define a true continuum.
Quoting aletheist
Brouwer's revenge. The intuitionists are back with a vengeance. I don't doubt the historical momentum. It's the influence of the computers.
Quoting aletheist
Yes, you did give me specific things to be unclear about and to push back on. Is my end of the conversation hopeless unless I read Bell and Peirce?
But I do think you need to explain yourself about this mythical background space in your first paragraph, which sounds suspiciously like the luminiferous aether, whose existence was disproved by the famous Michelson-Morly experiment, leading Einstein to special and then general relativity. There is no preferred frame of reference in the universe according to modern physics. There is no fixed background against which all other physical things can be measured.
pd -- Reading your paper, with much eye-glazing I'm ashamed to say. I did come across this:
"...he restates the second and third properties of Time as a continuum: any lapse can be made up of two lapses that have a common instant between them, although it need not "have a final and an initial instant." For example, the present is "assignable" as the "limiting instant" between the past, which has no initial instant, and the future, which has no final instant."
This sounds suspiciously like the idea of a Dedekind cut. Except that there's a point missing, as in the union of the open intervals (0,1) and (1,2). Am I understanding that right?
No, that is not what I am saying. I am not really talking about physics at all, just a hypothetical/mathematical conceptualization that might have phenomenological and metaphysical applications.
Quoting fishfry
Peirce came before Brouwer, and my interest in SIA/SDG has nothing to do with intuitionism or computers. If Peirce had followed through on his skepticism of excluded middle and omitted what we now (ironically) call "Peirce's Law" from his 1885 axiomatization of classical logic, then he would have effectively invented what we now (unfortunately) call "intuitionistic logic" and it might be known instead as "synechistic logic"; i.e., the logic of continuity.
Quoting fishfry
Maybe not hopeless, but I suspect that there is a "curse of knowledge" aspect here on my part, given my immersion over the last few years in Peirce's writings and the secondary literature that they have prompted.
Quoting fishfry
Thanks for the attempt, sorry for the resulting effect.
Quoting fishfry
Peirce would say that there is no point missing, because there are no points at all until we deliberately mark one as the limit that two adjacent portions of the line have in common. If we make a cut there, then the one point becomes two points, since each interval has one at its newly created "loose end."
Okay, that makes sense.
Quoting fishfry
I mean continuum in the context of the geometrical objects of extension studied in elementary calculus, the objects that we typically describe using the cartesian coordinate system.
Quoting fishfry
Okay, my mistake.
Quoting fishfry
I'm talking about the mathematical world. The two sentences in this quote are quite different. The first sentence essentially states that it passes through infinite intervening points. The second sentence states that it passes through all intervening locations where there could be points. I actually agree with the second sentence.
Quoting fishfry
What I'm trying to convey is that no matter where Atalanta's mathematical universe lives (whether in an infinite computer or the mind of God) it is impossible to construct Atalanta's journey from points because that would amount to listing the real numbers. The only way to build her universe is to deconstruct it from a continuum, working your way down from the big picture to specific instants.
When an engineer tries to solve Zeno's Paradox (of Achilles and the Tortoise) they ask questions about the system as a whole, specifically 'What are the speed functions of Achilles and the Tortoise from the beginning to the end of time?' With that information we don't have to advance forward in time, instant by instant. We just find where their two position functions intersect and conclude that Achilles passes the tortoise at that instant. And if this mathematical universe lives in that engineer's mind, that's the only actual instant that exists. Sure, the engineer could calculate their positions at other instants in time, but the engineer isn't going to calculate their positions at all times. That would be unnecessary...and impossible.
I'm sure you agree with the above paragraph (and perhaps are a little offended that I'm positioning it as the engineer's solution...hehe) but my point is that knowing a function doesn't imply that we can describe it completely using points. Any attempt to do so would be akin to listing the real numbers.
Quoting fishfry
I like when you earlier said 'every intervening location where there could potentially be a point'. It is worth creating a distinction between actual points and potential points. If we make that distinction, then I agree with you that there are only (actual and potential) points between a and b. What I would disagree with is the claim that there are only actual points between a and b. Actual points are discrete while potential points form a continuum. So instead of saying that there are finite actual points and infinite potential points between a and b, I think it is much better to say that there are finite actual points and finite continua between a and b. For example, in the image below, there are 3 actual points and 4 continua between 0 and 1.
Quoting fishfry
If we start with continua, the actual points only exist when we make a measurement. It seems like you agreed with aletheist on this. With a continuum-based view, when we make a measurement, we are not labeling points that existed all along, we are bringing them into existence (i.e. actualizing them). Until then they are potential points and can only be described as a part of a collection (i.e. a continuum), which I described using an interval. I am totally serious about this argument. My view is only silly when seen from a point-based view because you assume that all we can talk about are actual objects...an infinite number of them.
We don't have two adjacent portions before the marker. The marker is a pointer that demarcates but does not split the line thereby making it discontinuous. The marker is not part of the line. A materialist overlay is unnecessary in math. It's cheaper to think of it as an abstract pointer anywhere to an abstract endless line in one dimensional space.
Ok not physics. Thanks for the clarification, I'm sure you can see that this idea would not hold up as physics.
But if it's a "hypothetical conceptualization," how can you claim with a straight face that the standard real line doesn't model the "true continuum?" I do understand Peirce's point that the real line isn't a continuum because it's made up of individual points. But I am objecting to your claim that it's meaningful to say whether something is or isn't a good model of an abstract idea. What if my true continuum isn't the same as yours? Just as you can't say whether set theory is a good model of the tooth fairy. One conceptual fiction's like another. You have a vague idea (ok perhaps not vague to you) of a true continuum, and you're saying the real line isn't it. How can I agree or disagree with that statement, without sharing your inner visions on the nature of the true continuum?
Quoting aletheist
Ok, yes you pointed that out to me earlier. Was Brouwer familiar with Peirce? It's interesting that these ideas were already floating around.
Quoting aletheist
I agree with you that Peirce should be more famous. Didn't realize he pioneered LEM rejection.
Quoting aletheist
I wish I could dispatch a clone to read up on Peirce. It's on my to-do list, none of which ever gets done.
Quoting aletheist
Most philosophical papers glaze my eyes. Some I find very clear, but many confuse me. Not just yours. I'll take another run at it in light of this interesting conversation. But I must admit that given a finite block of time, I'd be more likely to spend it trying to learn some new standard math rather than philosophy. I find philosophy very hard to grab on to.
Quoting aletheist
Is this @keystone's point about points not existing on the number line till we mark them with labels? Sounds similar. Surely Peirce must have been familiar with Dedekind. Dedekind cuts are a clever idea, because we can logically construct the reals given only the rationals. With Peirce's description, I don't know what we've got. Philosophy versus math again. Perhaps I shouldn't even try to talk about philosophy.
Quoting aletheist
Grrrr. That makes no sense. If we divide the rationals into two classes, those whose square is less than 2 and those whose square is greater than 2, we can define sqrt(2) as that exact pair of classes of rationals. That's Dedekind's idea. Are you saying Peirce would make sqrt(2) both the largest of the smaller set and the smallest of the larger set? Having trouble with this. How can one point become two points? That's a more mysterious trick than Banach-Tarski.
@jgill is a professional mathematician. I don't say was, because it's not the kind of status one loses by virtue of being retired.
I am a failed math grad student. My own zeal is that of the fallen priest (I'm thinking Richard Burton in Night of the Iguana and probably some other roles too). Nobody has missionary zeal as strong as one who has been cast out of the order they strove to join.
I've taken and taught calculus and have no idea what the "geometrical objects of extension" are. The objects of the Cartesian coordinate system are ordered pairs of real numbers; or in the general case, ordered n-tuples. The pictures are for intuition and visualization. The actual objects of study are analytic. Just real numbers and ordered lists (in computer parlance) of same.
Quoting keystone
Distinction without a difference. You can think of the real line as a set of points or as a set of locations or addresses. Just like a street is a collection of houses or it's a collection of addresses where there might be houses or there might be empty lots. Except that by the completeness of the real numbers, there are no empty lots. But if you want to think of real numbers as locations on a line, that's perfectly ok.
Quoting keystone
Computers are too limited and the mind of God is too expansive. The mathematical universe lives in the world of symbolic math.
Quoting keystone
I don't have to name all the people in China to know there are a billion of them.
Quoting keystone
That's just not true mathematically.
Quoting keystone
See, you DO believe in the intermediate value theorem.
Quoting keystone
Right. It's unnecessary.
Quoting keystone
Yes I do, but it's trivial and doesn't support your point.
Quoting keystone
Why? Believe me when I get offended around here I let the offender know about it.
Quoting keystone
On the contrary. A function is a collection of individual ordered pairs. That's the set-theoretic definition of a function. It's not the same as the path of a moving point as it was for Newton. But he was doing physics when he had that viewpoint.
Quoting keystone
No. Not true. To say that a function is a collection of ordered pairs does not mean we are required to explicitly list them.
Quoting keystone
You can think of real numbers as locations on the real line if you like. Locations or addresses. But the completeness of the real numbers means there is a point at every location.
But more to the point, "point" is just another name for a real number. The set of real numbers is the collection of all the real numbers and vice versa. You are thinking "points" are things separate from real numbers, but mathematically they are not. Or in n-space, points are n-tuples of real numbers.
Quoting keystone
There are no such things as an actual or potential point. There are only real numbers and n-tuples of real numbers.
Quoting keystone
No no no no no I did not. @aletheist said that this was Peirce's point of view and I noted that this seemed similar to yours. I admit that when Peirce says it, I say, "Peirce, interesting guy. I wish I knew more about him." And when you say it, I say, "Nonsense." But in either case I do think the idea is mathematical nonsense. Philosophically I have no idea what you or Peirce could possibly mean by this and I emphatically deny ever giving the slightest consideration to the idea. It's wrong.
I do know that the intuitionists try to tie existence to human cognition, and this point I also disagree with despite there being many smart people on board with the idea.
Quoting keystone
Ok. I can't keep disagreeing with the thing you keep repeating over and over. I disagree strongly with what you said here.
Quoting keystone
I understand that. And I am totally serious when I say this idea is nonsense utterly devoid of content. Although when @aletheist tells me Peirce said it, I scratch my chin and go, "Hmmm, that Peirce sure was an interesting guy." But what I think to myself is, "Nonsense."
Quoting keystone
I don't see that either of us has said anything new in a long time.
I agree with this. I need to study more to either accept that it's nonsense or find a way to better communicate it. Until then, we're just wasting our time. Let's not waste any more time. I really appreciate your patience sticking this out with me on this up until now. Thanks!
No prob, likewise. After all when Peirce says the same thing I go, "Hmmmm I need to learn more." So maybe there's something to it. Who's to say.
I appreciate this and would be content to leave it at that.
Quoting fishfry
I guess it comes down to the meaning of the concept of continuity. Someone immersed in modern mathematics, where the real numbers are routinely called a continuum, is understandably satisfied with that definition. Someone like Peirce who objects to finding any discrete parts whatsoever in something that is supposed to be continuous can never accept it. He was motivated primarily by logical considerations rather than mathematical ones.
Quoting fishfry
According to a paper by Conor Mayo-Wilson, "Peirce and Brouwer seemed to have no knowledge of each other's work." However, they were indirectly linked through Lady Victoria Welby, with whom Peirce exchanged a fair amount of correspondence including some of his most important writings about semeiotic, and whose ideas about significs were later adopted by a group of Dutch thinkers that eventually included Brouwer.
Quoting fishfry
Yes, Dedekind's name appears in a bunch of his writings, and his most fundamental disagreement with him was about the relationship between mathematics and logic. For Peirce, logic (generalized as semeiotic) depends on mathematics, as does every other positive science; while for Dedekind, mathematics is a branch of logic.
Quoting fishfry
Peirce would not talk about "sets"--or "collections," his usual term--when referring to a continuum at all. By definition, a collection consists of discrete parts, which are ontologically prior to the whole ("bottom-up"); while in a continuum, the whole is ontologically prior to the parts ("top-down").
Quoting fishfry
Because the only points at all are the ones that we create by marking them. When we mark a line without separating it, we create one point. When we separate the line, we create two points, one at the discontinuous end of each resulting portion. When we put them back together, we have only one point again. The points are never parts of the line itself, because they are of lower dimensionality. Every part of a line is one-dimensional, but a point is dimensionless. SIA seeks to capture this with its infinitesimal segments that are long enough to have "direction" but too short to be curved.
Indeed, I believe that his use of "principle" rather than "law" for excluded middle is very deliberate. As he wrote elsewhere, "Logic requires us, with reference to each question we have in hand, to hope some definite answer to it may be true. That hope with reference to each case as it comes up is, by a saltus [leap], stated by logicians as a law concerning all cases, namely, the law of excluded middle. This law amounts to saying that the universe has a perfect reality."
Consequently, classical logic is strictly applicable only where "a recognized universe [of discourse] is definite (so that no assertion can be both true and false of it), individual (so that any assertion is either true or false of it), and real (so that what is true or false of it is independent of any judgment of man or men, unless it be that of the creator of the universe, in case this is fictive)."
Infinite subdivisions imply an infinite within the finite. How these opposites coincide is the issue
If you can forever take a part from a solid and there always be a part remaining, it is not fully finite but instead has an aspect of infinity. Common sense says something should be either finite or infinite but not both
The ancient Chinese knew about this: "One of the few surviving lines from the school, 'a one-foot stick, every day take away half of it, in a myriad ages it will not be exhausted,' resembles Zeno's paradoxes."
https://en.wikipedia.org/wiki/School_of_Names
Zeno added motion to the question of infinite divisibility to make the question even more difficult. I think mathematics does apply to objects in that two haves will have the same volume when united. I see no reason why these processes of division can't go on forever in real objects, although I recognize that objects are finite. Hence the paradox: "the infinite" WITHIN "the finite". Infinite content
Can God divide a soccer ball into infinite parts?
I recognize that our thoughts are imperfect in this regard. But I think they are interesting because they lead to either Parmenides's speculations or to the ever-changing "fire" of Heraclitus. All three thinkers spoke of very ancient ideas, and Zeno leads to one of the other two
If matter does not perfectly conform to geometry, then this alone is the answer to infinite divisibility problem. I am more likely to accept a contradiction than accept that objects only exist when perceived, as others have said on this thread. Anyway, maybe I should read more Wittgenstein
I like the infinitesimal line segment approach to continuity. It's how I program many of the functions I graph on the computer, although I'm using short real line segments.
Perhaps you and/or fishfry would comment on these ideas and elaborate on them to make SIA clearer. Sometimes in math very arcane subjects arise from elementary observations. Like reading a research paper in which simple motivation is left out, replaced by a series of odd looking lemmas leading to the proof of a theorem. :cool:
Quoting tim wood
It makes perfect sense to me. Again, the basic definition of real is being such as it is regardless of what anyone thinks about it. If we are talking about a fictional universe, then what is true or false of it depends entirely on what its creator decides about it, but not on what anyone else thinks about it. In Shakespeare's "Hamlet," the title character is the prince of Denmark and kills Claudius because Shakespeare says so; but no one can now truthfully claim that within the universe of that play, Hamlet is the king of Spain and spares Claudius. That is why there are objectively right and wrong answers on tests that students of English literature have to take after reading it.
Quoting tim wood
This strikes me as merely shorthand for your own characterization of logic as a game with certain rules. In the case of classical logic, one of those rules is excluded middle--every constituent of the universe of discourse must be treated as "individual (so that any assertion is either true or false of it)," which "amounts to saying that the universe [of discourse] has a perfect reality."
Too bad they don't have this is book form for sale anymore
Full disclosure, I am not a mathematician, so my ability to address the details of SIA is admittedly limited.
Quoting jgill
I believe so, as this seems to be simply what it means for a world to be smooth. As Bell says on p. 276, "If we think of a smooth world as a model of the natural world, then the Principle of Microstraightness guarantees not just the Principle of Continuity--that natural processes occur continuously, but also the Principle of Microuniformity, namely, the assertion that any such process may be considered as taking place at a constant rate over any sufficiently small period of time." For me this is reminiscent of the following passage.
A step function obviously has a discontinuity that violates this requirement. Of course, the "false continuity which the calculus recognizes" is that of the real numbers, which Peirce elsewhere calls a "pseudo-continuum."
I do not understand the question. What is true or false of a fictional universe is only dependent on what its creator thinks about it. It is such as it is regardless of what anyone else thinks about it.
Quoting tim wood
It codifies how we can properly draw necessary conclusions about states of things that are definite, thus conforming to non-contradiction; individual, thus conforming to excluded middle; and real, in the sense that they are such as they are regardless of what we think about them.
Quoting tim wood
Fair enough, but the point is that it does not depend on what anyone else thinks (or says) about it.
Quoting tim wood
No proposition about it is true of the real universe, but certain propositions about that fictional universe are true within that fictional universe. The proposition "Hamlet was the prince of Denmark" is false in the real universe, but true in the fictional universe of Shakespeare's play.
Quoting tim wood
I did not realize that this was the point, and in any case I disagree. Truth and falsity are not assigned to propositions, they are properties of propositions. The proposition "the earth revolves around the sun" was true before there were any humans around to express that proposition or to assign truth to it.
Truth or falsity is not a matter of human belief. A proposition that is "not altogether in accord with the facts" is not a true proposition, no matter how many people hold it to be true. A proposition about the world is true if and only if it denotes a reality as its object and signifies a fact as its interpretant.
Whatever is such as it is regardless of what anyone thinks about it.
Quoting tim wood
The earth revolves around the sun--provided that it really does, as I believe to be the case.
You seem to be conflating knowledge with truth. The standard philosophical definition of knowledge is justified true belief. I can believe a proposition, and even be justified in believing it, but that is not what makes it true.
On the contrary, it is quite clear to me that a fact is the real state of things that a true proposition signifies.
Whether I know that something is real is irrelevant. Again, the real is that which is such as it is regardless of what anyone (including me) thinks about it.
Quoting tim wood
I explicitly stipulated that this is a fact if and only if the earth really revolves around the sun. Like all our beliefs, this one is fallible.
Quoting tim wood
So, you believe that the earth does not really revolve around the sun? Like all beliefs, that one is also fallible. Since they are contradictory, one of us affirms a true proposition and the other affirms a false proposition; but which is true and which is false does not depend on what either of us (or anyone else) thinks about the matter.
Quoting tim wood
How do you know this? I am not asking how you justify that belief, but on what basis it is (purportedly) true.
Quoting tim wood
I would prefer that you just come out and make whatever point is on your mind.
So many posts on this thread. Here's a question for those who are entranced with the notion math should begin with continua: From the OP is there a continuum of discussion relating Gabriel's Horn with the Earth revolving around the sun? :chin: