Question for the math folk
Let's suppose we have a geometric cube and we call it's surface its limit. Now we cut it up into 2 equal parts and then one of the halves into half too and continue that process, stacking all the parts on top of each other with the larger pieces on the bottom. So we have
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That is not the best diagram but i can't think of another way to do it right now with the limited buttons I have on this tablet. But if you are interested in understanding my question I think you'll grasp it. What I want to know is what is on the top of this structure. Does it go up forever into space or does it stop at a limit? If it stops at a limit, what is right before the limit is hit? If it goes on forever, why do we now have no limit now, when before division we had the clear limit of the surface area?
Thank you
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That is not the best diagram but i can't think of another way to do it right now with the limited buttons I have on this tablet. But if you are interested in understanding my question I think you'll grasp it. What I want to know is what is on the top of this structure. Does it go up forever into space or does it stop at a limit? If it stops at a limit, what is right before the limit is hit? If it goes on forever, why do we now have no limit now, when before division we had the clear limit of the surface area?
Thank you
Comments (24)
Practical purposes would suggest a fairly limited number of divisions before the process becomes pointless.
Suppose the cuts are horizontal, parallel to the surface on which the block rests. Then we can think of all the infinitely many cuts as already there. Whether you see the block as having height 1, or having height 1/2 + 1/4 + 1/8 + 1/16 + ... = 1, it's exactly the same block either way. If you see a block in front of you, you can imagine it having the cuts already there.
There is no "next to last" or "last" cut, for the same reason there's no last element to the infinite series.
ps Wiki has a 2D picture. Not with strictly horizontal cuts but the idea is the same. The area (or volume in the 3D case) doesn't change just because we make a lot of cuts, any more than carving a turkey changes the amount of turkey.
https://en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_%E2%8B%AF
When we do arithmetic, any number can have a half, so 1 plus 1 can really equal 4 in that case, which I believe means that we have to start with numbers which are discrete (numbers so fundamental that they are the landscape of all number theory). Such a state of innocence doesn t seem possible to me with geometry even though I've wanted to find that pure fundamental for space. Every part has a part in order to remain spatial and an infinite gel of uncountable infinities seems to me to put what is blurry at the start of geometry when I wanted it to be clear. I see your point about the series of halves which I presented as already having the limit of the surface area before actually breaking apart the geometrical object. If I can find what is most fundamental despite all this about geometric space I will finally be able to put the question behind me. I found the most basic book I could find on non-Euclidean geometry on Amazon and since I know there is something missing in my understanding about geometric space I am going to go over the basics so carefully that I can find the false assumptions I've made which have thrown a monkey wrench into my understanding of geometry. I hope to become bolder and use my ambition to solve the paradox of Banach and Tarski
It's much simpler than people imagine. It comes up from time to time on this forum. The proof outline on Wiki is very good.
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
At heart it's only a syntactic paradox that's very easy to explain, thought it takes a little work. It's a long sequence of steps, but each step is very understandable. I say it's syntactic because it's based on the fact that the free group on two letters has a paradoxical decomposition. That is, we can do the paradox by only talking about the collection of words that can be made out of two letters and their inverses, without reference to geometry. But perhaps if you have some questions about the paradox I could respond to them without going into the technicalities.
There's nothing to solve, by the way. It's a theorem, not a paradox. It's a veridical paradox, meaning that it's counterintuitive but not actually a true logical paradox. It's entirely Euclidean, it takes place in standard 3-space.
ps -- This is a very good video on Banach-Tarski.
https://www.youtube.com/watch?v=s86-Z-CbaHA
The vsauce video was where I first encountered B\T. His supertask video also showed me that I was not alone in thinking about "Zenonian cubes". I know that mathematicians look at Banach-Tarski with many equations in mind, but I've always looked at it from the angle of Zeno's paradox alone. So my series of questions has been
1) if space is infinitely divisible than it has infinite parts despite the fact that we experience geometric things as finite
2) calculus says that a infinite number can be subsumed by a finite measurement. But in spatial terms how is this possible?
3) how can something be spatially finite and infinite is what appears to be "the same respect"?
4) if an object has infinite parts we can take infinite parts out and have a new object, hence Banach Tarski. But isn't this entirely counter intuitive?
5) this is all paradoxical to because of the way I think of objects as finite. What is the way forward?
I wanted to explore the non-Euclidean stuff with more care because it is also counter intuitive and might give me a clue on how to find the fundamental principle of all geometry and space. I'm not trying to prove anything to other people, but trying to find an understanding that satisfies myself. Some are ok with Gabriel 's horn. I don't have peace with it
You should look at it from the paradoxical decomposition of the free group on two generators. There are Wiki pages on the subject. It's simple, it's non-geometric, and it's the heart of the paradox. Regrettably Wiki doesn't have a clear writeup of the phenomenon. I'd like to write it up sometime but I don't know if this site is the right place for expository math of any length. Meanwhile these two pages will have to do.
https://en.wikipedia.org/wiki/Free_group
https://en.wikipedia.org/wiki/Paradoxical_set
Quoting Gregory
Banach-Tarski (B-T from now on) pertains only to mathematical Euclidean space, and most definitely not to the real world. It's a mistake to confuse the two. In particular, B-T involves partitioning 3-spaces into sets that are so jagged and discontinuous that they could not possibly exist in reality. They're non-measurable sets, meaning sets of points that can not possibly be assigned any sensible measure of volume.
Quoting Gregory
I don't know what you mean by subsumed in this context, nor do I relate that to anything in calculus. Can you be more specific so that I can understand your concern?
Quoting Gregory
Can't understand the question, can you please give an example of what you mean? If something is spatially finite it's finite, not infinite. Do you mean how can a finite length, like the unit interval [0,1], contain infinitely many points? That's pretty simple, if nothing else 1/2, 1/4, 1/8, 1/16, ... are infinitely many points contained within the interval. Is that what you mean?
Quoting Gregory
Yes it's a very counterintuitive theorem. A veridical paradox. Counterintuitive but not actually a logical contradiction, on the contrary a provable theorem. But there's much more too it than just removing points from an infinite set. The theorem says that you can partition the unit ball in 3-space into as few as five parts, and move them around rigidly -- that is, preserving all distances -- and end up with two balls. That's the real puzzler, that the motions are rigid.
For example if we take the counting numbers 1, 2, 3, ... we can partition them into two disjoint subsets, the odds and the evens, and each subset is bijectively equivalent to the original set. That's a puzzler in itself. But in B-T we partition the unit ball into five pieces and rigidly rotate the pieces to put together two balls, each the size of the original. That's definitely counterintuitive.
Quoting Gregory
The way forward is the Wikipedia outline of the proof. (1) The free group on two letters has a paradoxical decomposition; (2) The group of rigid transformation of Euclidean 3-space contains a copy of the free group on two letters; (3) Apply the paradoxical decomposition of the free group to the ball in 3-space; (4) Fix up a few dangling anomalies. There are a lot of buzzwords in there but the Wiki proof is pretty decent.
Quoting Gregory
That's pretty ambitious but go for it.
But to be honest, earlier you claimed that 1 + 1 might be 4, and you didn't respond when I asked for clarification. May I suggest nailing that down first. Also, if you seek to understand the true nature of geometry, you need to study Riemann and also Klein, who pointed out that geometry is really group theory. And in fact Banach-Tarski is essentially group-theoretic.
https://en.wikipedia.org/wiki/Bernhard_Riemann
https://en.wikipedia.org/wiki/Felix_Klein
https://en.wikipedia.org/wiki/Erlangen_program
In particular, the modern view of geometry is that a geometry is determined by the collection of transformations that preserve its properties. Euclidean space is defined by Euclidean transformations, etc. Again a little buzzwordy but things to look at.
Quoting Gregory
Do you follow the calculus in Gabriel's horn? The integral of 1/x from 1 to infinity is infinite, and the integral of 1/x^2 is finite. It's just how it is and the proofs are perfectly straightforward.
Modern mathematicians seem to have forgotten that Aristotle covered up this problem with a sophistry and that Kant presented this problem in one of his antimonies. If I want to know how many parts an oven has or a loaf of bread baked in it, I simply have to ask how many times I can mentally divide it. And it turns out I can do this infinitely, yet the bread and the oven are finite. Mathematicians now longer see this as a problem or as even strange, and I don't know why
Quoting fishfry
Yes, but this is contradicted by infinite divisibility, which all space must have.
Quoting fishfry
Presenting the problem in terms of numbers instead of space obscures the issue
Quoting fishfry
I am trying to comprehend the first few axoims of all geometry, and i'm not sure the specifics of B/T relate. I only was talking about B/T in terms of taking an infinite of points out of another infinity of points.
Quoting fishfry
Sure. If we have two 12 inch rulers, they are equal 1 to 1. However with numbers half of 1 is also a number, so if we apply to this the ruler we have 2 six inches on one side and 2 six inches on the other, hence instead of 1 and 1 being compared, it's 2 and 2. The reason is that in arithmetic you have to have basic numbers that are understood as not divided. In geometry, all space is divisible and its impossible to find the basic unit.
Quoting fishfry
Not precisely. I was good at pre-calculus in high school but in college I only did geometry and that was over ten years ago. I am coming at this from a more basic fundamental level and perhaps I can't avoid highwe mathematical ideas but I had wanted to find the first few axioms of geometry and am confused why it's become to problematic
If I'm understanding you, you're concerned that the unit interval [0,1] contains infinitely many points. Is that correct? If so, how many points do you think it should have?
Quoting Gregory
You are confusing math with physics. And modern physics does not posit infinite divisibility. In fact in physics, space is divisible down to the Planck length, equal to around [math]1.616255... \times10^{-35}[/math] meters. Below this distance, our physics breaks down and we cannot sensibly speak of what goes on or how space is. There's a Planck time as well, a minimum time interval below which our physics breaks down and can't be applied.
Quoting Gregory
Ok. So in the unit interval, consider the spacial intervals [0,1/2), [1/2, 3/4), [3/4, 7/8), and so forth, where the square bracket on the left means that the endpoint is included, and the paren on the right means that the endpoint is excluded. Aren't there infinitely many of those spacial intervals making up the 1-unit long segment? What do you make of that? Why does it trouble you?
Quoting Gregory
Oh darn, I sandbagged myself again. People always bring up Banach-Tarski, and I say, "B-T is at heart a simply syntactic phenomenon that I could describe in a page of exposition if anyone was interested," and they invariably have no interest. One of these days someone's going to say, "I'd like to see that" and I'll do it. But I see once again that you name-checked B-T but don't actually have an interest in it. And I got hopeful, only to be disappointed again. I am telling you that the heart of B-T is simple and surprising and perfectly clear, but nobody wants to hear about it. I pointed you at the references but you had no questions. One of these days ...
Quoting Gregory
Well yes, instead of 1 whole plus 1 whole we have 2 halves plus 2 halves. Surely it's sophistry to claim that this means anything or that it's some kind of paradox or contradiction. If I drive to the store a mile away, I did indeed cover half a mile twice, or a third of a mile three times, or a quarter mile four times. You don't mean for me to take this commonplace fact as some kind of antinomy, do you?
Quoting Gregory
In physics you can take the Planck length as the basic unit. In math, given a line, you can pick any two points, label one point 0 and the other point 1, and that length is your basic unit.
Quoting Gregory
Well, if you take the graph of y = 1/x from 1 to infinity, the area under it is infinite. But the area of 1/x^2 from 1 to infinity is finite. That falls directly out of calculus. And as @andrewk noted in the Gabriel's horn thread, it's analogous to the fact that the infinite series 1/2 + 1/3 + 1/4 + 1/5 ... sums to infinity, yet the series 1/4 + 1/9 + 1/16 + 1/25 + ... has a finite sum. Just a mathematical fact that takes a bit of getting used to, but is undeniably true. The paint business is just a way of making it more confusing.
Quoting Gregory
Well Euclid's axioms are a fine set of basic axioms. And if you drop the parallel postulate and replace it with either zero or many parallels through a point parallel to a given line, you get various flavors of non-Euclidean geometry. In modern times, Tarski's axiomitization of Euclidean geometry is of interest.
https://en.wikipedia.org/wiki/Tarski%27s_axioms
Something discrete. Yet "discrete space" is impossible if it is to remain space.
Quoting fishfry
This is a commonly held sophistry. As i demonstrated on this thread, everything in this world is made of infinite parts and I BELIEVE the conclusion is that everything is finite and infinite in the exact same respect. That last part is what I was trying to explore
Quoting fishfry
I don't see how anyone with a brain wouldn't want to know how to get two objects out of one without referring to infinities. Such a theorem is incredible and I hope you do codify it into a thesis that others will read and appreciate. I for one am having trouble with it because it's of such a nature which I do not think I will understand it by READING it, as opposed to having it explained in person where I can cross examine every step. Reading it is just to much for me
Quoting fishfry
That is arbitrary, as is the Plank length
Quoting fishfry
Something I need to consider more, thanks.
Quoting fishfry
"Sir Henry Savile remarked in 1621, there were only two blemishes in Euclid, the theory of parallels and the theory of proportion. It is now known that these are almost the only points in which Euclid is free from blemish. Countless errors are involved in his first eight propositions. That is to say, not only is it doubtful whether his axioms are true, which is a comparatively trivial matter, but it is certain that his propositions do not follow from the axioms which he enunciates."
Bertrand Russell
It becomes very confusing, which is why I was trying to find something basic about space that I could use as "first principles" in a Cartesian fashion
The integers are discrete. And if you have any space whatsoever that's made of points, you can define the distance between two points to be 1 if the points are different, and 0 if they're the same. This is called the discrete metric. It makes any space into a discrete space; even an uncountably infinite space.
https://en.wikipedia.org/wiki/Discrete_space
But you didn't really answer the question. You said you're troubled that the unit interval [0,1] can be partitioned into infinitely many intervals of the form [0,1/2), [1/2, 3/4), [3/4, 7/8), and so forth. I asked you what is your objection and I am still unclear as to why you are troubled by this example. In fact it seems to relate to your original question about dividing a block. This is the one-dimensional version of the same question.
Quoting Gregory
The Planck length is a fundamental aspect of modern physics. And by modern I mean since 1899, when Planck came up with the idea. He noted that it's defined only in terms of the speed of light, Newton's gravitational constant, and Planck's constant. His idea was that the Planck length was universal, in the sense that aliens would come up with it.
Here's Sabine Hossenfelder discussing the Planck length.
http://backreaction.blogspot.com/2020/02/does-nature-have-minimal-length.html
Quoting Gregory
You claimed it. You haven't demonstrated it. Such a demonstration, that the world is infinitely divisible, would be a breakthrough on the order of Newton or Einstein.
Quoting Gregory
The Banach-Tarski theorem does depend on the axiom of infinity. You do have to be able to treat infinite collections as sets. No getting around that. And at one point in the proof you have to invoke the axiom of choice to pick a set consisting of one point from each of an uncountably infinite collection of sets. So there's no question that infinity is involved.
Quoting Gregory
One of these days ...
Quoting Gregory
Should I come over with a blackboard and chalk?
Quoting Gregory
The Planck length is not arbitrary, Planck noted that an alien would discover it. A mathematical unit of length, formed by choosing two points on a line and calling one 0 and the other 1, is indeed arbitrary. But any choice at all will do, since any choice of points can be transformed into any other by a translation and a linear scale factor.
Quoting Gregory
Are you familiar with the fact that the harmonic series diverges? This is the basis of the proof that the area under 1/x from 1 to infinity diverges. That is, 1 + 1/2 + 13 + 1/4 + 1/5 + 1/6 + ... diverges to infinity. Very cool proof if you haven't seen it.
https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)
"Sir Henry Savile remarked in 1621, there were only two blemishes in Euclid, the theory of parallels and the theory of proportion. It is now known that these are almost the only points in which Euclid is free from blemish. Countless errors are involved in his first eight propositions. That is to say, not only is it doubtful whether his axioms are true, which is a comparatively trivial matter, but it is certain that his propositions do not follow from the axioms which he enunciates."
Bertrand Russell[/quote]
That's why Tarski and others have cleaned up Euclidean geometry in the 20th century.
Quoting Gregory
Not sure what you're after. To me, Cartesian means two copies of the real numbers at right angles to each other, with the point of intersection labeled (0,0), and all the analytic geometry following from that by labeling points in the plane as pairs (x,y) of real numbers. You can do quite a lot with that, especially if you put in a third axis so you can do 3-D, and even a 4th axis with a funny relationship between it and the other 3 so you can do special relativity. You can go pretty far using nothing more than mutually perpendicular copies of the real number line. That's Cartesian. To me, anyway.
I'd like to see that!
(But you may want to make a new thread for that. I suspect that few folks are looking at Gregory threads. I only looked because I saw your response.)
Quoting fishfry
This isn't really relevant to the topic, but just to get one common misconception out of the way: Sabine Hossenfelder talks about the smallest structure that can be resolved in space. This doesn't mean that space is discrete, made up of Planck-length cells or anything like that. Space and time in standard fundamental theories of physics are continua, just as in classical physics. This you can readily see from any dynamical equation, such as Schrodinger equation.
Since space is continuous, it has infinitely many potential parts, but its only actual parts are those that we create by marking them off.
Quoting Gregory
That is because there is no basic unit intrinsic to space itself, only arbitrary constructs that we impose in order to measure length/area/volume.
Not exactly, Einstein himself discovered that light could be quantised with the photo electric effect, and depending on the type of number used, the results are technically discreet since they are approximations. The equations are ultimately calculated to yield discreet results where irrational numbers and fundamental constants are concerned, but the solution to the problems are mostly described with continua, yes. Fundamental constants, for example are improved year on year so that they are effectively approximations.
If you have a proof that space is continuous that would be a discovery on a par with the revolutions of Newton and Einstein.
Quoting aletheist
I just pointed out that Max Planck introduced the Planck length as something aliens would discover. It's intrinsic to space itself as I understand it. I'm no expert but this is my understanding. It's not like inches or centimeters. It's a fundamental length that a physicist anywhere in the universe would discover.
Sure, but space itself is not an object in that sense. It has no actual parts, only potential parts, and it is obviously not spatial in the same sense as a physical body. It is the continuous medium in which discrete objects exist.
Quoting fishfry
Likewise if you have a proof that space is discrete. They are two different mathematical assumptions.
Quoting fishfry
Yes, but I understand it to be a limitation on measurement, not a discrete unit of space itself.
I've made no such claim. You did. And they're physical, not mathematical assumptions.
Thanks, maybe I'll get started on it.
Yes indeed. Me too. :chin:
When speaking of something that exists, the phrase "potential parts" is an oxymoron. Aristotle was in a bind over what Zeno had said years before him and so tricked his readers by saying objects are infinitely divisible only potentially. Yet he was wrong. Objects have parts and the only way "potential parts" can mean anything as an expression is to have it refer to conscious awareness of those parts, something people are unwilling to do. They would rather deny that material objects are truly spatial, deny they know what space means, and host of other tricks to avoid what I've been addressing in this thread. Pyrhonnians throughout history have used the "infinitely divided finite" as proof that we can't know anything truthfully. I for one want to make as much sense of it as I can and it doesn't bother in itself. It does bother me when some says "sure" and then repeats the "only potentially has those parts" thing when I thought we had just settled that. I guess I finally have gotten it into my head that people are truly truly bothered by the question "how many parts does a table actually have" so maybe I just won't bring this stuff up anymore on this forum. But thanks especially to fishfry for sharing his wide knowledge
I disagree, everything that is physical occupies space, but space itself is not physical.
Quoting Gregory
I agree, but again, space is not an object that exists--something that reacts with other like things in the environment. Instead, it is a reality--something that is as it is regardless of what anyone thinks about it.