Arguments for discrete time
I have a couple of arguments for time being discrete rather than continuous (actually similar arguments can be used for discrete space too). Thanks in advance for any feedback.
1. A point in space cannot have size=0 because it would only exist in our minds and not reality (no width; insubstantial)
2. Similarly, the point in time ’now’ cannot have length=0 (if it exists for 0 seconds, it does not exist)
3. Or if a ‘now’ had length=0, then a second would contain 1/0=UNDEFINED ‘nows’
4. So ‘now’ has length >0
5. Can’t be length = 1/? because ? does not exist (? + 1 > ? making a nonsense of ?. Or if you define ? + 1 = ?, implies 1 = 0)
6. So a ‘now’ has a finite, non-zero length. Time is composed of a chain of ’nows’ so time must be discrete
Or
a) Imagine a second and a year
b) By the definition of continuous, both time period are graduated identically (to infinite precision).
c) So there must be the same information content in both (same number of time frames: ?)
d) But a year should contain more information than a second
e) Reductio ad absurdum, time must be discrete
1. A point in space cannot have size=0 because it would only exist in our minds and not reality (no width; insubstantial)
2. Similarly, the point in time ’now’ cannot have length=0 (if it exists for 0 seconds, it does not exist)
3. Or if a ‘now’ had length=0, then a second would contain 1/0=UNDEFINED ‘nows’
4. So ‘now’ has length >0
5. Can’t be length = 1/? because ? does not exist (? + 1 > ? making a nonsense of ?. Or if you define ? + 1 = ?, implies 1 = 0)
6. So a ‘now’ has a finite, non-zero length. Time is composed of a chain of ’nows’ so time must be discrete
Or
a) Imagine a second and a year
b) By the definition of continuous, both time period are graduated identically (to infinite precision).
c) So there must be the same information content in both (same number of time frames: ?)
d) But a year should contain more information than a second
e) Reductio ad absurdum, time must be discrete
Comments (282)
Let length of now = L, then:
L/second > L/year
->
L x year > L x second
Equality not satisfied with L=0 or L=1/?
L must be finite and > 0
IE time is discrete
Alternatively, one can realize that mathematics doesn't actually occur per se in the external world, that time is just change/motion, and that change/motion only makes sense relatively (to something changing at a different relative rate/in a different way), so the question of whether it's discrete or continuous is kind of a category error.
If time is just change/motion, why does it run slower when an object travels near the speed of light or is near an intense gravity field?
If time is just change/motion, what mechanism enforces the speed of light speed limit?
The effects on mass (basically a kind of "pulling" on the mass in both cases) is a counteracting force that make changes/motions relatively slower.
Think if you were moving a bunch of helium-filled balloons around. If something were pulling on the balloons, it would be harder to move them around. Near light-speed and intense gravitational fields are how much of a "pulling" difference we need to notice a significant difference in how things are relatively moving/changing.
But the force of gravity is attractive, why should it slow motion?
Anyhow, there are no massive objects involved in the case of an object is travelling near the speed of light. Speed = distance / time so the universe must be time-aware to enforce the speed limit.
Pulling is attractive, right? Pulling is towards something. Imagine you have a simple electron orbiting a proton. If the electron is being pulled on gravitationally, it's more difficult to keep moving in the orbiting motion. Hence a relative slow-down of motion.
Speed increases mass. That's the reason that exceeding light speed isn't possible.
I need to read up on relativity. Thanks.
If mass increases with speed, then the universe must be speed-aware? In some sense the universe must know how fast an object is going to allocate the appropriate mass.
Speed = distance / time so the universe must be time-aware?
What increases is the inertial mass, or in other words, the amount of resistance it has to any change in its motion.
It's just a fact about how relative motion works.
Maybe I'm getting confused, but:
E = mc² ? ? (1 - v² ? c²)
So
m = E × ? (1 - v² ? c²) / c²
So time (in the v term) determines mass. So something in the universe must be aware of time else it could not assign a mass. That suggests time is real?
My view of time isn't the standard physics view of time . . . well, and that's especially the case since physics still leaves time unanalyzed ontologically and just treats it as an instrumental quantity that doesn't need to be pegged otherwise.
Time is definitely real on my view. Only creatures with minds are aware of anything, though, and that's a very small subset of what we find in the universe.
Who says the temporal continuum needs to contain instants? Likewise who says that the spatial continuum needs to be pointy? Perhaps the continuum has no fundamental level at all, no unit with which all other quantities are multiples of.
If that is the case, then the idea of "now" as a snapshot moment in time is mistaken and the passage of time as a succession of said moments (not unlike a succession of strips on a piece of film) is also confused. This is what you seem to implicitly assume in the your argument. Just as objects may be distant from one another without any fundamental length, events simply come and go continuously without any fundamental duration.
I agree. The concept of 'now' presupposes that time exists; and now separates the past from the future - two more meaningless concepts without that of time itself.
Einstein said that space-time is one 'thing' I think and that the 2 cannot be separated. So, since we don't doubt space exists, can we doubt time does too? One thing that fascinates me is that time only flows one way (so far as we know) - whereas movement in space can go back and forth. I considered once whether time could flow just as well backwards, but concluded it couldn't, because the laws of cause and effect wouldn't work in reverse. So surely time must exist, but what it's made of must be similar to what space is made of. And I think space must have some physical form because it is warped by mass so that passing light beams are curved by it. How does the light beam know to travel in a curve?
What structure does time have if it's not a series of instants?
Quoting Mr Bee
But if an event has no duration it would not exist. 'Now' could not exist if it had zero duration. Think about filming someone for zero seconds - you'd have no film right?
How many zero duration 'nows' in a second? 1/0=UNDEFINED. Thats not right.
Quoting Tim3003
Because time has a start (see for example https://thephilosophyforum.com/discussion/4702/argument-from-first-motion), it must be real.
We can also tell the difference between past, present and future so there must be something special about 'now' so the concepts of past and future have meaning. Some sort of positional cursor that regular eternalism/relativity does not incorporate must exist.
Quoting Tim3003
Yes as I mentioned the same arguments work for space too.
One more argument. Consider actual infinity. Aristotle who defined it did not believe in actual infinity. It's a mad concept so I don't believe in it either. Consider a second in the past. The definition of continuous time implies that a second must contain an actual completed infinity of ‘nows’. So all those 'nows' actually existed; thats a funny money number of 'nows' (the cardinality of the set of natural numbers).
Just a series of events without any smallest units. Similarly space without points would simply be space without any fundamental composition. There is no need for time to be a series of instants or fundamental units any more than there is a need for space to be composed of fundamental units or points.
Quoting Devans99
I never said that there are events that are instantaneous nor that time is composed of them. Quite the opposite really.
Problem is that means a second and a year would have the same information content which does not seem right. Clearly more information in a year - the continuum seems paradoxical. Maybe it's one of those concepts that we can conceive of in our minds but never occurs in reality? Reality seems deeply logical and free of paradoxes.
Thing is, you're arguing against the opposite of my position. I never advocated for the point continuum where everything is made up of an infinite number of points (which would lead to your criticism), but instead a gunky continuum that doesn't contain any such fundamental units at all. Personally, I find the point continuum to be a useful tool that doesn't describe reality, but that does not mean I take reality to be discrete, which is a concept I also consider to be problematic in other ways.
I think this again is a circular argument. The past and the future may exist as concepts within that of time in our minds, but they have no concrete reality outside our minds - and that's even if time does exist.
An interesting idea is that of the man with no memory. For him the past does not exist. he simply experiences a continual present. So if none of us had any memories, would the past exist at all? Likewise the future, assuming we had no imagination to picture what it might hold? It seems to me hard to argue that the past exists outside our minds when it is not perceivable in the world or available for us to 'travel' to.
It is a true continuum, such that an instant with no duration, or even a very small finite duration, is a strictly hypothetical discontinuity. Rather than being sharply separated, the past and future meet and overlap in the infinitesimal "now."
The analogy is with a mathematical line, which does not consist of individual dimensionless points or tiny finite segments, but instead can always be divided infinitely into shorter and shorter lines. We arbitrarily mark instants of time and measure the intervals between them, just as we arbitrarily mark points on a line and measure the distances between them.
Because space-time is a true continuum, motion/velocity is a more fundamental reality than either position or duration. Incidentally, recognizing this is the key to dissolving Zeno's famous paradoxes.
Hi. I like this questioning of the 'now.' It seems to be a kind of default metaphysical assumption, and yet I argue that it does not square with experience. I think our very experience of reading complicates this now. Here's an example:
The end of this sentence specifies its beginning.
What we have read is held in suspense and anticipation. The past is determined by the future as the future is determined by the past, at least it seems within the flow of meaning. And given that 'clock time' and its mathematical 'now' exist within this 'meaning time,' we have a fascinating situation.
That's paradoxical. It works ok in the mind but not in reality: If I have a real line length 1 mile, it contains more information than a real line length 1 centimetre. But if they are both continuums then they both contain the same amount of information. Which is impossible. Which is proof by contradiction that continuums do not exist in the real world.
Just curious what definition of "information" you're using.
My own I admit. If you imagine a particle in an interval, the position of the particle relative to the beginning of the interval can be regarded as information. In a continuum, that piece of information (particle position) has infinite precision so infinity many bits of information.
That leads to the paradox of the continuum; all sizes of continuum contain the same amount of information.
Obviously discreteness is the only way out of the paradox.
Why? It begs the question to presuppose discrete units of "information" (i.e., points or finite segments) that comprise a "real line." Again, the "parts" of a true continuum are also true continua.
Quoting Devans99
I repeat: Because space-time is a true continuum, motion/velocity is a more fundamental reality than either position or duration. Marking and measuring the position of a particle at any particular instant imposes an arbitrary discontinuity, like a point on a line.
Your continuum is just magic - how can 1 light year be structurally the same as 1 millimetre? I don't believe in magic so I have to take a different view - no continuums in nature; they are logically unsound and nature does not do unsound.
Also it can't be a continuum because actual infinity does not exist.
An argument from incredulity is not persuasive, and alleging "magic" suggests a lack of interest in engaging in serious philosophical discussion. What exactly do you mean by "structurally the same" in this context?
Quoting Devans99
In what specific sense do you hold that the hypothesis of a real continuum is "logically unsound"? Are you claiming that it is somehow logically impossible, or merely not actual? Either way, why do we nevertheless routinely refer to "the space-time continuum"?
Quoting Devans99
A bare assertion is also not persuasive. Even if I grant the premise, the discontinuity of actuality/existence does not, by itself, rule out the reality of true continua.
I think all this depends on your concept of time. What is time, to you? What do you want to do with it? Before you decide it's quantised, perhaps you should wonder whether it is quantisable (if that's a word)?
Now can indeed be of zero length. Now is a bookmark, not a duration (which would require a start and an end).
How on earth could you construct a continuum? It requires us to construct an actual infinity of possible positions for particles to occupy. Thats impossible. So the fact we could never construct a continuum goes a long way to arguing against its existence.
Quoting aletheist
Actual infinity, if it existed, would be a quantity greater than all other quantities, but:
There is no quantity X such that X > all other quantities because X +1 > X
Now you could define:
? + 1 = ?
But that implies:
1 = 0
Yet again: Because space-time is a true continuum, motion/velocity is a more fundamental reality than either position or duration. We can construct a continuum as the path that a particle traces (or would trace) over an interval of time--which is not a collection (infinite or otherwise) of discrete positions. Besides, how could merely possible positions constitute an actual infinity?
Quoting Devans99
That definition is incorrect, according to the standard mathematics of infinity; mainly because it begs the question by presupposing the discreteness of quantity. In any case, I am advocating the reality of continua, not the actuality of infinity.
How I am 'presupposing the discreteness of quantity'?
X in my proof could be real or natural; I made no assumptions. That is a fair proof that actual infinity is not a quantity IMO.
Quoting aletheist
But a continuum requires actual infinity. If you look at the mathematical models for continua, they use actual infinity (gunky or otherwise). Think about a real-life line segment, say the distance between your eyes and the screen; that HAS to be an actual infinity of points/line segments by the very definition of continuous.
Think about the second that just past; by the definition of continuous; its has to include an actually infinite number of moments/periods of time. They just all happened. All 'cardinality of the set of natural numbers' moments just happened if you can make sense of that.
All I can suggest at this point is looking into the standard mathematics of infinity. I side with Peirce, rather than Cantor, in denying that the real numbers constitute a true continuum.
Quoting Devans99
No, it does not. That is only one way to model it, which I consider incorrect. Another is that the line is a true continuum that does not consist of discrete points or line segments at all. You can arbitrarily mark any such point or segment, but by doing so you introduce a discontinuity into that which is really continuous in itself.
Quoting Devans99
No, they did not. That is only one way to model it, which I consider incorrect. Another is that any finite interval of time is a true continuum that does not consist of discrete instants at all. You can arbitrarily mark any such instant, but by doing so you introduce a discontinuity into that which is really continuous in itself.
Quoting Devans99
I can make sense of that, but it is not the case. A true continuum has a cardinality exceeding that of any infinite set. In Peirce's terminology, a true continuum has a multitude exceeding that of any infinite collection. Between any two points that we mark on a line, there is an inexhaustible continuum of other potential points; between any two instants that we mark in time, there is an inexhaustible continuum of other potential instants.
I've looked at it; its rubbish. They declare in the axiom of infinity that actual infinity exists and prove absolutely nothing. They then move on to patch up all the paradoxes that creates with further illogical axioms.
Quoting aletheist
A point has length 0. How many points on a line segment length 1? 1/0=UNDEFINED.
No doubt they would say the same about your arguments here.
Quoting Devans99
You remain wedded to the mathematics of discrete quantity. Again, there are no points on a continuous line, unless and until we mark them as discontinuities.
My arguments are not shot through with paradoxes. Cantor's, Galileo's, Hilbert's Hotel, Zeno's... what a mess actual infinity has made of maths and science.
Paradoxes indicate an underlying logic error (actual infinity exists).
Contradictions indicate an underlying logic error; paradoxes indicate a need to think more carefully.
How many times must I repeat that I am arguing for real continuity, not actual infinity, and that these are two distinct concepts? that motion (space-time) is more fundamental than position (space) or duration (time) treated separately? that a line does not consist of points, and that a temporal interval does not consist of instants?
I see no model of continuity that does not need actual infinity. If you point me to such a model, I stand corrected, but they all seem to use actual infinity.
Any real continuum can be subdivided infinity so it it exists in the present or the past, it must support an actually infinite number of sub-divisions.
In Peirce's model of a true continuum, the infinity is potential rather than actual. The real is not coextensive with the actual (existence); there are also real possibilities and real (conditional) necessities.
Quoting Devans99
No, any real continuum could potentially be subdivided infinitely; it can never actually be subdivided infinitely. See the difference?
Didn't you know that the mathematics of infinity is a kludge put there to force it into the arithmetic we use on finite numbers? :chin:
Yes I think so. It was all motivated by misplaced belief I think: Cantor an Co thought God was infinite so infinity was shoe-horned into mathematics for that reason.
Nothing wrong with having a finite-sized God IMO.
I would argue that our progress through time is progress through the continuum at the most fundamental level so it requires actual infinity. As does our progress through space when we move. So in both cases our progress through time and space subdivides the continuum to an actual infinity.
I'd say that the mistake you're making here is that you're thinking of spatial extension as a "construction consisting of possible positions for particles to occupy," You're reifying mathematical ideas.
No, that progress itself through the space-time continuum (i.e., motion) is the fundamental reality; any discrete subdivisions of space and time are our arbitrary constructions.
I don't think we will ever agree on this point.
You did not answer my argument that the information content of a larger region of space-time must be larger than a smaller region of space-time? It rules out all forms of continuum with a very reasonable axiom?
Yes, I did.
Quoting aletheist
... or in this case, a "region of space-time."
And yet we can calculate instantaneous velocity.
I do not see how your argument can survive that.
What about the analogy of filming someone for zero seconds? No film would exist. Film is a good analogy for time.
And yet the car is moving at 50km/h.
So, from "A calculation is just purely in our heads" it does not follow that "A velocity is purely in our heads..."
A film zero seconds long is called a Photo.
Photo's require a non-zero length to exist (exposure time).
I think I will maybe adopt 'things need a non-zero length to exist' as an axiom.
As soon as you talk about comparing the "amount" of something, you are quantifying it, and thereby treating it as discrete--i.e., begging the question.
Quoting Devans99
What part of "motion is more fundamental than position" do you still not understand? Giving the position of something to any degree of precision requires measuring its distance from an arbitrary reference point at an arbitrary instant using an arbitrary unit.
Quoting Devans99
Only if you presuppose that time is discrete, like the film in a motion picture.
We can compare analog quantities perfectly well.
Quoting aletheist
You are clearly on a different planet to me. There is just no way a light year has identical granularity and structure to a centimetre; I think I will have to bail out! Thanks for the conversation though. Happy Xmas.
The mistake is assuming that, in itself, any arbitrary portion of space-time has any granularity--i.e., discreteness--at all.
Quoting Devans99
Likewise!
SO if you assume a discrete world, you find that the world is discreet.
(??)
This is pretty directly erroneous.
a) Imagine 2 and 4
b) By the definition of continuous, both numbers are graduated identically (to infinite precision).
c) So there must be the same information content in both
d) But 4 should contain more information than 2...
It's just bad maths.
Imagine an interval of 2 and 4 I mean...
What does that mean?
Only in our minds does the actually infinite exist; its not a real world concept.
No, it doesn't. 4 might be twice two, but what could it mean to say it has twice the information?
What sort of thing is information?
So your mind is not in the real world? Infinity is not a thing like my cat or last Tuesday? What's going on here? Is infinity a thing like my mortgage? Like a unicorn?
What is it?
What do you mean?
But every time his wife told him to take out the trash, he'd reply. "How possibly could I traverse that infinite distance."
The interval 4 is twice the length of 2 so it should contain twice as many real numbers. In our minds both intervals contain an equal and infinite number of real numbers . But I believe that could not be the case for any real life interval - the larger interval would contain more numbers/information.
Quoting Banno
Infinity is an illogical concept. Illogical concepts can exist in our minds but not reality. Faeries, talking trees, infinity we can all imagine but they do not exist in reality IMO.
That is not how real numbers work. By such (il)logic, there should be twice as many integers as even numbers, which is also not the case. A discrete collection of four items obviously does contain twice as many objects as a discrete collection of two items, but a continuum (such as space-time) does not consist of discrete items at all. The unwarranted axiom here is that reality consists entirely of discrete items and collections thereof.
Now, I do not dispute that actuality consists entirely of discrete items and collections thereof; but I deny that reality--that which is as it is, regardless of what anyone thinks about it--is limited to actuality. Specifically, there are real continua of potential items that are not even remotely exhausted by their discrete instantiations. Being and existence are not coextensive.
Twice as many integers as even numbers makes sense. We can count them:
1, 2, 3, 4, 5, 6, ...
Vs
2, 4, 6, ...
Appears to be twice as many integers by counting up-to 6. We can us our knowledge of extrapolation to conclude that there are twice as many integers than even numbers.
Not so much.
There's a lot of cute maths around infinity. But your OP depends on continuity. Check out
Some of them. They do nothing to resolve the paradox of the arrow, so far as I can tell.
You are claiming:
- Twice as many integers as even numbers within a finite interval
- An equal amount of integers and even numbers in an infinite interval
This is nonsense.
I said nothing whatsoever about "equal amount" or "infinite interval," concepts that mistakenly treat infinity as if it were extremely large, but still finite. How many integers are there? Infinitely many. How many even numbers are there? Infinitely many. If we paired up each integer with an even number, when would we run out of even numbers, but still have integers left? Never.
Note that whether an actual infinity is possible or impossible is completely irrelevant here. This is mathematics, which is the science of drawing necessary conclusions from formal hypotheses. The definitions of "integer" and "even number" constrain us to recognize these somewhat counterintuitive relations between them.
As summarized by Wikipedia, the arrow paradox states, "If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible." If motion is a more fundamental reality than position, and space-time is a true continuum, then both premises here are false--nothing is ever completely motionless, and time is not composed of discrete instants.
Ah. I just saw this.
You may be right.
He is right. We talk about the magic of infinity and thats just it; its magic not maths. Galileo's paradox I've already mentioned but its pretty central to why infinity is contradictory:
https://en.wikipedia.org/wiki/Galileo%27s_paradox
So there are clearly more numbers than squares in all intervals. Except an infinite interval when there are the same amount. So we have assumed actual infinity exists and derived a contradiction - Galileo's paradox is proof by contradiction that actual infinity does not exist.
One more time: Mathematical infinity is not an actual infinity, but it is a real infinity. If we paired up every number with its square, when would we run out of one or the other? Never. How is this a contradiction?
There are less squares than numbers in the following intervals:
0-10
0-1000
0-1000000
So via extrapolation, there must be less squares than numbers in ALL intervals.
The fact that actual infinity contradicts the above means we can induce that actual infinity does not exist.
Mathematical infinity is not actual infinity. Which part of this do you still not understand?
Mathematical infinity in set theory is actual infinity.
Mathematical infinity in calculus is potential infinity.
?
No, it is not. Like all mathematical theories, set theory--especially as applied to infinite sets--is based on certain hypothetical formalizations that may or may not correspond to anything actual.
The problem being that we cannot pair them up because there is an infinite number of either one of them. So your conditional "if we paired up each integer with an even number", is a statement of an impossibility, and therefore must be dismissed as a false premise.
Quoting aletheist
Again, an impossible conditional. "If we did X" when X is impossible.
Infinite means in-finite, not finite or not discrete. It is only finite things which have a size. If God is infinite that does not mean God is infinitely large. In a sense even finite objects are infinite because they are not composed of discrete points, whether those points are of zero dimension or some dimension.
As others have said you seem to be confusing yourself by reifying mathematical concepts.
It's called a bijection.
(Fairly basic high school mathematics, if memory serves.)
Kind of odd to just deny something without really knowing about it. :brow:
I already know about it. The fact remains that we cannot actually pair them because there is an infinite number of them. So the proposition states something which is, by definition, impossible (i.e.it is contradictory). Therefore we ought to reject it as a falsity, as is customary for propositions which are recognized as self-contradictory. Whether or not it's basic high school mathematics is irrelevant. If it's a falsity it ought to be rejected.
A proposition is not contradictory merely by virtue of stating something that is actually impossible, only if it states something that is logically impossible--which is certainly not the case here. Mathematics has to do with the hypothetical, not the actual.
As I said, it is impossible "by definition". This means that it is logically impossible, contradictory. "Infinite" is commonly defined in such a way that it is impossible, by definition, to pair up infinite things, because the task would never be complete. It is only by changing the definition of "infinite" to something else, that this might become possible. But then "infinite" loses it's meaning, so what's the point? You simply change the definition of "infinite" to create the illusion that the logically impossible is actually possible.
The numbers are infinite. The bijection is necessarily incomplete. Therefore, in this case the bijection "does not", and that contradict "does".
Bijection gives non-sensical answers: it says there are the same amount of even integers as there are integers. But any finite interval; there are twice as many integers as even integers. And the infinite interval is composed of infinitely many finite intervals. Thats a contradiction that proves bijection is plain wrong.
If your first statement were true, then your second statement would also be true. But your first statement is false, so your second statement is also false.
That is actual impossibility, not logical impossibility. It is completely irrelevant to pure mathematics--the science of drawing necessary conclusions about formal hypotheses--whether anyone could ever actually pair up the members of infinite sets.
Why is my first statement false?
? = ? * 100
So an infinite interval is composed of infinity many finite intervals (of 100 length in this case).
Interesting. So consider length, the human concept of length. It has no length, being a non-physical thing. But it exists. And remember that axioms are guesses, that we call axioms partly to make them sound more credible and scientific, but mainly because we cannot prove them, so we assert them instead, without a shred of evidence or justification. If we could prove them, we would. When we can't we pretend: axioms.
Because an infinite interval is not composed of infinitely many finite intervals. Until you understand that, we will continue going in circles.
...always remembering that it is mathematically invalid to divide both sides of the equation by infinity. You did remember that, right? :chin:
I would argue that length is a concept and concepts can exist in our minds only. Maybe I should have said: 'things need a non-zero length to exist in reality'. Anything can exist in our minds. Talking trees and Santa. Reality is material and you cannot have a material thing with length=0.
An axiom should be more than a guess IMO. The original definition of axiom was 'self evident truth'.
Quoting Pattern-chaser
Sorry I have given up on the maths of infinity. What is the point of having a quantity (infinity) that you can do nothing with mathematically; you cannot add/subtract/multiply/divide without hitting a contradiction... sort of my point... every way we turn, infinity leads to contradictions. It's too illogical to be a real world concept.
Quite right. But mathematicians needed infinity, so they shoe-horned it into their arithmetic and algebra, even though it is obvious that it doesn't and can't fit there, as you describe. :up: But them's the rules (of how infinity is handled), so your attempt to ridicule it is pointless. The whole idea is ridiculous but necessary, so that's that.
And a self-evident-truth is ... a guess. For if we could manage anything more - ideally, proof - then we would. And when we can't, we guess. Self-evidently, of course. :wink:
You can do all kinds of things with infinity mathematically, but what you cannot do is treat it as if it were just another quantity. Infinity is a different kind of thing from any discrete number, no matter how large (or small).
How about cannot treat infinity as a quantity because it is not a quantity?
When definitions deny the possibility of something due to contradiction, this is a logical impossibility, like a square circle is a logical impossibility. That one could make a bijection of infinite numbers is logically impossible because the definition of "infinite" (what it means to be infinite), contradicts the definition of "bijection" (what "bijection" means). Therefore "infinite bijection" is excluded as a possibility because it is contradictory, i.e. logically impossible.
I don't know what you would be referring to with a distinction between "actual impossibility" and "logical impossibility", because all impossibilities are logical impossibilities. The only way that we have of demonstrating, or knowing that, something is impossible is through logic. Therefore actual impossibilities are logical impossibilities, because all impossibilities are logical impossibilities. An actual impossibility might be one based in sound logic, while a not-actual impossibility might be one based in unsound logic.
As usual, equating the logical with the actual leads to absurdity. Logical possibility is much broader than actual possibility.
If pigs had large and powerful wings, then pigs could fly. The truth of this hypothetical proposition is not affected by the fact that pigs do not actually have large and powerful wings. If one were to pair all of the integers with the even numbers, then one would never run out of even numbers while still having integers left. Again, the truth of this hypothetical proposition is not affected by the fact that one cannot actually pair all of the integers with even numbers.
A square circle is logically impossible because the definition of a square and the definition of a circle are mutually exclusive. There is no such incompatibility between the definition of an integer and the definition of an even number; in fact, the alleged paradox is rooted in those very definitions, which place no finite limitation on either set.
You might finally be on to something there, depending on exactly what you mean by it.
How is this relevant?
Quoting aletheist
Right, let's hold that thought.
Quoting aletheist
The incompatibility is not between "integer" and "even number", it is between "pairing" and "infinite". "Pairing" is a task which requires completion. If the task is incomplete, they are not actually paired, and the "pairing" attempt is a failure. "Infinite" denies the possibility of completion, therefore the "pairing", is of logical necessity, in the case of the infinite, a failure. Therefore either the integers are infinite in which case pairing is impossible, or the integers are not infinite, in which case pairing is possible.
It illustrates that actual impossibility does not entail logical impossibility.
Quoting Metaphysician Undercover
No; the whole point here is that pairing the members of infinite sets cannot actually be completed, yet it is still logically possible.
You two don't know MU and Devan. They will just insist there is a contradiction. When you ask then to formally show the contradiction, they will just say it's weird, or that it's not possible to actually map two infinite sets or something like that. They won't actually address the point because they've misunderstood some fundamental things, from confusing distinct modalities to the reasons why mathematicians were rationally forced to accept infinity. MU in particular is so off base that he rejects the mathematical definition of a set ("Sets must be constructed by literally putting things together BY DEFINITION").
But that's irrelevant because I was only arguing logical impossibility all along, which as I explained is the only real form impossibility.
Quoting aletheist
It is the definition of "infinite" which necessitates that pairing infinite sets is impossible. How on earth do you assert that it is still logically possible without changing the definition of "infinite"? That's MindForged's tactic, to produce a different definition of "infinite", but that leaves "infinite" as utter nonsense. As Devan's99 has demonstrated over and over again, the definition of "infinite" employed by set theory is illogical.
What a short memory you have. I actually demonstrated the contradiction to you in numerous different ways, because each time I demonstrated it you would change the goal posts in an effort to avoid my demonstration. That's why I had to demonstrate it to you in so many different ways, you kept trying to wiggle out from under the crushing force of blatant contradiction.
As explained in my last several posts, pairing infinite numbers is contradictory due to the definitions of "pairing" and "infinite". If you want to get back under the crushing force of contradiction, and try to wiggle out again, then be my guest, and try to demonstrate to me how this is not contradictory. But since your memory seems to be very short, let me remind you that you were not able to wiggle out last time.
why?
Actually, I take that back. Mapping an infinity of one sort against anther is a common mathematical practice. So you are wrong, or talking about something else.
A: "one..........two...........three"
Now does it make sense to dispute this, by arguing that I could have counted the same interval twice as fast?
For mustn't any supposedly 'counterfactual' argument refer to a newly constructed interval, B, and not to the past interval A that no longer exists and therefore cannot itself be re-measured?
B: "one,two,three,four,five,six"
If one accepts the counterfactual argument that A might have been counted differently, then one is led to ask how fast the same interval could have been counted, which leads to the further question as to whether there is a limit. In which case the above statement of A isn't a definition of A but merely one of many possible descriptions of A, namely that it just so happened to begin and end when I was counting.
On the other hand, if one rejects the counterfactual argument then A has an exact length of 3 by definition, and there is nothing more to be said about it.
Pairing, and mapping, are all activities just like counting is an activity. You cannot count an infinite number because this is contradictory to the definition of "infinite". You cannot pair an infinite number for the very same reason. You cannot measure an infinite number, nor can you map an infinite number, for the very same reason that these are activities which require completion to be successful.
You might assert that you have mapped an infinite number, and even show me your map. But since I cannot show you the infinite number (because this is contradictory), I cannot show you that your map does not correspond with the infinite number.
All I can do is demonstrate logically that it is impossible to map an infinite number because this is contradictory. Do you recognize the truth of "it is contradictory to claim that you could measure an infinity"? Do you recognize the truth of "to map something requires that it be measured in some way"? What makes you think that mathematicians have done what is logically impossible, mapped infinity? How naïve are you? Suppose I told you, that if you keep going in this direction, counting, you will eventually reach infinity. Would you believe that I have mapped infinity?
Tell me the exact formal definition of a mathematical mapping and infinity within the context of form mathematics and prove the contradiction. Don't do this BS where you talk big but repeatedly leave crucial terms undefined by implicitly assuming colloquial vagueness of the terms when you know full well that's not how definitions work in formal disciplines. You don't have an argument, this is pure bluster on your part.its been known for about a century that the Axiom of Infinity does not add any contradictions to ZF set theory, which on all accounts appears to be consistent. Formally derive the contradiction from the actual definitions used in mathematics or just admit you're straw Manning mathematics.
Not only is there no contradiction entailed, if there were your proof that there was would ensure that you received the Fields Medal. But curious that it will forever be beyond your grasp, almost like you're making fundamental missteps.
One more time: The fact that no one can actually pair all of the integers with corresponding even numbers has no bearing whatsoever on its logical possibility.
Quoting Metaphysician Undercover
Actually impossible, but not logically impossible. Just ask a mathematician.
And yet we can; and yet we do, map series of infinite numbers, one against the other.
So this comes down to Meta vs. mathematics.
@MindForged is right. What you have shown is that you refuse to understand mathematics.
If you have a problem with my terms (they are English), then address my posts and tell me where the problems are. If my terms are not related to mathematics, then don't worry about them, they pose no threat to this field which you hold sacred.
Quoting aletheist
And here's my "one more time". It is only a "fact" by definition, therefore the impossibility is logical. The only reason why no one can actually pair the integers is because they are stated to be infinite, and by this definition, it is impossible to do such. Therefore it is logically impossible to do such.
Quoting Banno
What I have shown is that I cannot understand mathematics because the language of mathematics contradicts my native language, English. This renders mathematics as incoherent and unintelligible to me. I know that you don't care about this. So be it.
In case you've never noticed this, claiming that you've dome something, and actually doing it, are two different things.
Sorry, that is not how logical impossibility is defined. It would have to be something that is impossible for anyone even to conceive (like a square circle), not something that is merely impossible for anyone to do. Again, the latter is actual impossibility, which has absolutely no relevance whatsoever to pure mathematics.
Quoting Metaphysician Undercover
My native language is also English, and I see no contradiction whatsoever with the language of mathematics that we have been discussing here. The same is true of any and every English-speaking mathematician in the world. In any case, it frankly seems rather foolish to keep making definitive (and incorrect) pronouncements about a subject that, by your own admission, you cannot even understand.
This is the exactly what I was talking about . The issue is you're misrepresenting what is being said. It should be patently obvious mathematicians do not define mapping (pairing) and infinity so as to make them jointly inapplicable. Just saying "I'm speaking English" isn't even beginning to honestly address this obvious fact. If your terms are not related to mathematics then you have absolutely no argument against the mathematical results relating to infinity. You're simply talking about something else.
As I said, it's a simple and clear case refusing to simply read how the terms are defined and then pretending to have discovered a problem because some colloquial definitions of some words conflict with the colloquial definitions of other words. Mathematics is formal, our definitions need to be stated up front/known beforehand and remain consistent throughout the calculation. If there is a contradiction, show the formal contradiction. Prove the system to be trivial.
That's exactly my argument, it's logically impossible, impossible to conceive of, just like a square circle is logically impossible. The point is that people claim to be able to conceive of square circles, just like they claim to be able to conceive of pairing infinite numbers. People claim all sorts of weird things, like a polygon with infinite sides. They do this by violating, or changing the definitions of the terms.
Quoting MindForged
That's the point. In English we know that pairing infinite numbers is impossible, just like we know that counting infinite numbers is impossible. The way that we use and define "infinite" and the way that we use and define "pairing", ensures that this is impossible. if mathematicians want to define these two terms in a different way, so that it is possible to pair an infinite number, that's their prerogative. I am not here to police mathematicians. However, we ought to be clear that this "mathematical" language is inconsistent with common English, and also inconsistent with how "infinite" is represented in philosophy.
Quoting MindForged
You may have noticed that I have no arguments against the mathematical results relating to infinity, although others like Devans99 do. I really don't care about the mathematical results relating to infinity, because what "infinity" means to a mathematician is something completely different from what "infinity" means to me, a philosopher. And, I think it's quite obvious that the mathematicians have it wrong, (they've created an illusory "infinity"), so I'm really not interested in the conclusions which they might derive from their false premises.
You have offered no argument for this claim, you have merely asserted it over and over; and now you have completely undermined your own position by freely acknowledging that you are not using the relevant terms in accordance with how they are carefully defined within mathematics, such that there is no logical impossibility whatsoever.
Can you not read? Or do you have an extremely short memory? Let me reiterate. "Pairing", like "counting" is a human activity which is not successful unless it is completed. "Infinite" is defined in such a way that human activities such as counting and pairing cannot be completed an infinite number of times. Therefore it is logically impossible to count, or pair, an infinite number.
Of course I've spelled this out for you numerous times already, and you've simply ignored it, claiming some unreasonable distinction between logically impossible and actually impossible. So I expect you to continue with this unreasonable ploy.
Quoting aletheist
This in no way undermines my position. I've acknowledged this from the beginning, defining "infinite" in a different way allows for infinite pairing. My position is that this mathematical definition of "infinite" is mistaken because it doesn't properly represent what "infinite" refers to in common usage, and in philosophy. The mathematical notion of "infinite" is illusory.
In mathematics, pairing and counting are not activities at all; they are concepts, and there is no requirement that they ever actually be completed, or even be capable of actually being completed. One more time: Mathematics has to do with the hypothetical, not the actual.
Quoting MindForged
Exactly. No one is disputing that some mathematical definitions of terms are inconsistent with their colloquial (or even philosophical) meanings, but that has no bearing whatsoever on whether the associated concepts are logically possible.
Right, so in mathematics you can count the infinite numbers without counting the infinite numbers.
That's exactly why I say it's a false premise. If you want to ignore that contradiction, and accept this hypothetical as a true premise that's your prerogative. I think this hypothetical is clearly and obviously false though, so I reject it as false, and I would not employ it as a premise, like mathematicians do.
What you refuse to acknowledge, is that I reject it on the basis that it is logically impossible by way of contradiction, because "infinite" means cannot be counted. You might claim that it's not logically impossible because "infinite" means something other than this in mathematics, but I think that's wrong, as an illusory definition of 'infinite".
That's my opinion and I will continue to defend it until someone demonstrates to me that I am wrong, that as something capable of being be paired or being counted is a better representation of what "infinite" truly represents. Simply pointing out that my opinion is inconsistent with the opinions of many mathematicians, does not demonstrate that my opinion is wrong.
You make a big deal out of an unimportant point. No one ought care about how these are defined in natural language because the meaning is often in flux, is context sensitive and still has multiple definitions. Sometimes we say "infinite" and mean Aristotle's potential infinity , sometimes we mean a completed infinity (as in the cardinality of an infinite set) and other times we just mean some arbitrarily large number that we leave unspecified. Philosophy always makes recourse to.mathematics in understanding infinity, I don't know why you think otherwise.
Quoting Metaphysician Undercover
How do you not see the contradiction between "I have no arguments against the mathematical results of infinity" and "I think it's quite obvious that the mathematicians have it wrong"? It's one or the other, either you're not arguing against it and thus you cannot say it's wrong, or else you're saying it's wrong and thus have some argument against it.
The actual understanding of infinity is the one that came from mathematics courtesy of Cantor and Dedekind. Philosophers almost uniformly appeal to this rigorous understanding they gave us because it let's us come to grips with understanding this intuitive mathematical concept that we only vaguely understand in natural language. It caused many paradoxes in philosophical areas (e.g. Zeno's paradoxes) that were banished once mathematicians (not philosophers) gave a real regimentation of the concept. So of course we should privilege the mathematical understanding, which philosophers do. It's applicable to many areas of philosophy, mathematics, science, you name it. And it still accords with many intuitions about infinity, though not all of them (which doesn't matter since the intuitive understanding of infinity creates paradoxes).
I would add Peirce here, although as in the case of philosophy, unfortunately his contributions are widely overlooked.
Indeed, Peirce independently invented quantification; and he disagreed with Cantor and Dedekind about the real numbers comprising a continuum, because he viewed numbers of any kind as intrinsically discrete. He was primarily driven by a philosophical interest in true continuity, rather than a mathematical interest in infinity.
The fact that philosophy has a different definition of infinite which is inconsistent with your mathematical definition of "completed infinity" is clear evidence that philosophy does not make recourse to mathematics for its understanding of reality.
Quoting MindForged
All this demonstrates is that you are very selective in the philosophy which you read. Cantor's representation of "infinite" was confronted by Russell, and hence replaced by Zermelo-Fraenkel. But any thorough reading on the subject will reveal that the issue is far from settled.
[quote=Internet Encyclopedia of Philosophy]Nevertheless there was, and still is, serious philosophical opposition to actually infinite sets and to ZF's treatment of the continuum, and this has spawned the programs of constructivism, intuitionism, finitism and ultrafinitism, all of whose advocates have philosophical objections to actual infinities. Even though there is much to be said in favor of replacing a murky concept with a clearer, technical concept, there is always the worry that the replacement is a change of subject that has not really solved the problems it was designed for. [/quote]
Notice specifically, "..there is always the worry that the replacement is a change of subject that has not really solved the problems it was designed for". This is my argument. By redefining "infinite" mathematics is not even dealing with what we generally refer to as "infinite'. It has created a completely new concept of "infinite". It has put aside the true concept of "infinite" which derives its meaning from continuity, in favour of an illusory one, a completed one, in order to create the illusion that it has resolved the problems of infinity. In reality the new concept of "infinite" has just distracted us from the true infinite. "Completed" is not a word which one could use to describe "infinite" in the way that we commonly use the word. Of course, your claim is that the "completed infinity" is the true concept of infinity. I disagree.
Quoting MindForged
It's as I said, I have no arguments against the conclusions drawn by mathematicians from their concept of "infinite", what you call the "results". I do not even know these conclusions, or results, and I have no interest in them. I am arguing against their premise, their concept of "infinite". This is not contradictory, just a simple statement of fact, I am not arguing against the results (conclusions), I am arguing against the premise (their concept of "infinite"). And, I have no interest in these results.
Quoting aletheist
Right, tell that to MindForged, who seems to think that mathematicians have resolved the philosophical problem of "infinity". In reality, mathematicians have redefined "infinity" to suit their own purposes, neglecting the real problem of infinity, which is associated with continuity. And this might lead some naïve philosophers to think that mathematicians have resolved the problem of infinity. All they've really done is created a new problem, a divided concept of "infinity".
Philosophy does not have a different definition of infinity outside the colloquial ones which are inconsistent. A potential infinity is just that: potential, as in not an infinity. Check out any relevant excerpts from mereological and ontological work that relate to infinity and in virtually all of them infinity is understood based on the long-standing mathematical definition of it. The reasons for this should be obvious.
Quoting Metaphysician Undercover
There was opposition to Cantor early on, to be sure. But that was gone in relatively short order. The Cantorian understanding of infinity is the understanding of infinity in modern mathematics, any claim that ZFC is markedly different from Cantor is just false. It's completely settled in current mathematics. In fact, check the paragraph immediately before the one you quoted from the IEP:
Quoting Metaphysician Undercover
All you're proving is as I said, that some colloquial definitions conflict with others. Who cares? If those definitions lead to insoluble paradoxes and cannot be applied where they ought to (in mathematics) then they need replacing. With the proper understanding of infinity and a developed calculus, we solves Zeno's paradoxes when philosophers could not because they did not have a workable definition of infinity outside the vaguely defined one. Mathematics does not wholecloth redefine infinity, it still has most of the properties it intuitively ought to have (continuously extendable, for instance), but has the unique benefit of being perfectly and probably consistent.
And it's funny that you mention constructivism and such from the IEP. The problem is - ignoring that constructivism does not eliminate all infinities - is that those are in the extreme minority, even in philosophy. Worse, the quote you mentioned is talking about the continuum, not infinity in general. Thats the size of the real numbers, not of the set of natural numbers which is still completed in intuitionistic mathematics (constructivism). Ultrafinitism is widely regarded as just above crankery, funnily enough because they argue similarly to you that we should reject infinity and essentially pretend that the counter-intuitive properties of infinity should be treated as if they are contradictions even though it's not even arguable because we have formal proofs that infinity does not introduce any inconsistencies in standard mathematics.
Quoting Metaphysician Undercover
You haven't argued against it save to say that it's different than the colloquial one in *some* respects. So if the definitions cannot be agreed upon, we need only look at the results. Your view of infinity neuters mathematics because then calculus goes out the window as that requires several sizes of infinity (you're dealing with the real numbers, for one) and science since it is predicated on ZFC and uses calculus everywhere (not to mention all current spacetime theories that aren't mostly speculative (e.g. Lopp-quantum gravity) explicitly assume space and time are a continuum). That's just a useless definition at that point, especially as it then runs counter to other colloquial views on infinity.
Quoting Metaphysician Undercover
Peirce was writing in the exact time that Cantor and Dedekind's work on infinity was contentious. It's just dishonest to pretend that has any bearing in the status of that work among philosophers and mathematicians today.
I am not talking about potential vs. actual. I am talking about "infinity" as boundless (philosophical conception), and "infinity" as completed (mathematical conception). The two are incompatible.
Quoting MindForged
Choosing one conception and rejecting the other does not resolve the incompatibility. Nor does it resolve the paradoxes involved with the one conception, by choosing the other conception. That's simply an act of ignorance.
if you could demonstrate that "infinity" (the philosophical concept) as boundless, and incomplete, is an incoherent, unintelligible conception, then we'd have reason to reject it in favour of the other, mathematical conception. Until then it remains a valid concept which is incompatible with the mathematical concept of "infinitely".
On the other hand, I reject the mathematical conception because I believe it was created solely for the purpose of giving the illusion that the issues involved with the philosophical concept of "infinite", as boundless and incomplete, could be resolved in this way, by replacing the conception. Despite your claims about how calculus and science rely on this conception of "infinite", I believe it serves no purpose other than to create the illusion that the problems involved with the philosophical concept of "infinity" have been resolved. In reality, mathematics could get along fine without this conception of "infinity". It would just be different, having different axioms. And, since this conception of infinity is just a distraction for mathematics, mathematics would probably be better without it.
Not sure if this was directed at me, but to clarify--I did not mean to imply that Peirce's objections to Cantor and Dedekind carry much weight among philosophers and mathematicians today. They do not, which I happen to think is unfortunate, but only from the standpoint of understanding true continuity as utterly incompatible with discrete mathematics. I readily acknowledge that treating the real numbers as if they constituted a continuum works just fine for most practical purposes within mathematics.
That's not a philosophical conception, that's as much the colloquial conception as anything else. The two aren't incompatible either since in a sense even standard math has infinity as boundless. After all, take some arbitrary infinite set and new members can be added to it.
Quoting Metaphysician Undercover
Yes it does. These are competing theories of what infinity is and this it is not a debate that it fundamentally any different than any other disagreement in philosophy regarding how to define or conceptualize something. Zeno's paradoxes are not resolvable under the colloquial understanding of infinity, but they are resolved by appeal to modern mathematics (calculus) which requires the hierarchy of infinities. It's not an act of ignorance, it's then use of a better theory of infinity because it's both usable in mathematics and it resolves issues that existed previously. Under your view absolutely nothing can ever replace a previous misconception because to change ones accepted theory of a concept entails just changing the subject.
Quoting Metaphysician Undercover
This is like rejecting nominalism because one thinks it was created to give the illusory victory of overcoming issues with platonism or something. In reality, mathematics cannot get by without the conception of infinity it uses. Otherwise you're doing something like constructivist mathematics which is more limited, using different tpea of analysis (e.g. smooth infinitesimal analysis and the like) and is significantly more limited in the proofs that can be made since Excluded Middle cannot be placed inside the universal quantifier. Virtually all science uses the ZFC set theory which includes the axiom of infinity. If the math all worked out without that axiom it would not be asserted as an axiom. You're essentially supposing all mathematicians are idiots who don't realize they have an unneeded or useless axiom despite the many criticisms of the formalism (including Cantor's work on infinity) of a century ago.
So, yes, in a way, time is discrete.
Another way to look at it is to consider nerve thresholds. There's a limit to our senses. If something moves fast all we see is a blur. Fast enough and it disappears. Think of a bullet.
So, there is a lower limit of time we can experience and "make sense of".
But you're talking about a more fundamental property of time; that it doesn't flow but actually hops from moment to moment.
One problem with your argument is how you use NOW.
NOW is a moment in time. If it is 7:00 AM NOW then it means SEVEN hours have passed since midnight. NOW is just an arbitrarily chosen DISTANCE from a given starting point. It doesn't have a size I agree but it isn't meant to.
So, yes, NOW=0.
Suppose you're given a length of time 2 seconds.
How many NOWs are there in it?
We do the division x=2 seconds ÷ NOW. As NOW approaches zero (the real NOW=0) x approaches infinity. That is the correct answer because between any two points in a continuous model of time there should be infinite points or NOWS.
Are you familiar with Platonic dialectics? We determine the meaning of a word by referring to how it is used in our society. This mean that the colloquial conception is the correct one. If mathematics is using a conception of "infinite" which is inconsistent with the colloquial conception, then this is an indication that they have not properly represented "infinite"?
Quoting MindForged
Zeno's paradoxes were adequately resolved by Aristotle's distinction between actual and potential. The colloquial conception of "inifinite" is consistent with this distinction, though the colloquial understanding does not all the time include an understanding of this distinction, so Zeno's paradoxes may appear to one who holds the colloquial conception but does not understand Aristotle's resolution. The modern world of scientific discovery has long ago rejected Aristotelian physics, and with it the Aristotelian distinction between actual and potential. The concept of energy is clear evidence that this distinction has been lost to modern science. Because these principles, which resolve Zeno's paradoxes, were lost to modern science, Zeno's paradoxes reappeared as valid paradoxes.
The principles of modern mathematics do not resolve Zeno's paradoxes because the philosophers of mathematics have simply produced an illusory conception of "infinite", which is inconsistent with what we are referring to in colloquial use of the term. That's sophistry, and Platonic dialectics was developed as a means to root out and expose such sophistry. The sophists would define a word like "virtue" in a way which suited their purposes, and then profess to be teachers of this. However, Socrates exposed that what they were teaching as "virtue" was just their own little conception, which was completely inconsistent with the colloquial meaning of "virtue" (what the members of society in general regarded as virtue). Philosophers of mathematics have engaged in the same form of sophistry. They teach their own private conception of "infinite" which is completely inconsistent with what we generally mean by "infinite" (the colloquial meaning of the word), creating the illusion that this resolves Zeno's paradoxes.
Quoting MindForged
This is not true. What I am arguing is that if we change the defining features of a thing, then we are not talking about the same thing any more. Therefore we ought to give it a different name so as to avoid confusion. This is not a case of correcting a misconception, it is a case of introducing a new conception. We cannot say that one is a correction of a misconception, because they are distinct conceptions, having distinct defining features. The new conception ought to be named by a word which will not cause confusion with the old conception, or any sort of equivocation. For example, if the defining feature of parallel lines is that they will never meet, and someone says that they've come up with a new geometry in which parallel lines meet, then we ought not call these lines parallel, but use a term other than "parallel" in order to avoid confusion and the appearance of contradiction. They are distinct conceptions, not a correction of a misconception. Likewise, the new conception in mathematics, which is called "infinite" ought to bear another name like "transfinite" so as not to confuse the conception with what we commonly call "infinite".
Quoting MindForged
That's ridiculous. I am saying no such thing, and I resent that because I have great respect for mathematicians, they are as far from "idiot" as you can get. But the mathematicians which I know do not create the axioms, as this is more of an activity of philosophical speculation. And I do believe that much of the philosophical speculations which provide the foundation for modern science and mathematics is misguided. And I would not call a misguided philosopher an idiot, because much philosophy is hit and miss, trial and error.
Alternatively, as Peirce argued, there are no instants (NOWs) in any continuous interval of time, and there are no points in any continuous segment of a line. Time does not consist of instants and space does not consist of points; instead, those are arbitrary discontinuities that we mark within continuous space-time for our own purposes, such as measurement. The real numbers thus serve as an adequate model of a continuum for almost all uses within mathematics, but they do not constitute a true continuum.
I agree with this representation, but the problem which TheMadFool points to is that there appears to be real points of discontinuity within time, which are represented by the concept of Planck time. So if time is represented as a true continuity, as you say Peirce suggests, how do we account for these fundamental units which cannot be further divided. What type of point would mark the beginning and end of these units of time?
I think the question is whether the Planck time is properly described as a discrete "unit of time" or as a limitation on our ability to mark and measure time, which in itself is truly continuous. Needless to say, I lean toward the latter.
So, what can model a true continuum?
I'm a bit confused here. I consider the real numbers to be an adequate model of any continuum for we can measure an arbitrarily small or large quantity without any problem.
I agree that the real numbers are adequate for modeling, marking, and measuring discrete quantities, no matter how small. My understanding is that conceptualizing a true continuum requires the acceptance of infinitesimals. Again, for Peirce this was primarily a philosophical matter--related to his self-described "extreme scholastic realism" regarding generals--rather than a mathematical one.
I agree. And infinitesimals are just 1/?. And actual infinity is not a number:
1. If actual infinity is a number, there must be a number larger than any given number.
2. But that’s contradictory.
3. Can’t be a number AND larger than any number.
4. So actual infinity is not a number
5. Invention of magic numbers runs contrary to the spirit of science.
If actual infinity is not a number, a mathematical continuum does not exist IMO. Likely neither in reality...
No one is arguing for actual infinity, or that infinity is a number, or (for that matter) that infinitesimals are numbers. Again, numbers are intrinsically discrete, and thus useful for marking and measuring. On the other hand, switching from mathematics to philosophy, a real general is a true continuum--between any two actual instantiations, there are potential instantiations beyond all multitude.
Quoting aletheist
So those instantiations are not multitudes. So the actual instantiations are not multitudes. So you can't compare the two.
Mathematical modeling is representation for a particular purpose. I already acknowledged that the real numbers serve as an adequate model of a continuum for almost all uses within mathematics, even though they do not themselves constitute a true continuum. Likewise, the mathematical models that I routinely create as part of my job are extremely useful within my professional field of structural engineering, even though they obviously are not actual structures being subjected to gravity, wind, earthquakes, etc.
And again, no one is arguing that true continua exist or are actual, but that they are real--they are as they are, regardless of what anyone thinks about them. Thus defined, reality is not coextensive with existence/actuality--there are also real qualities/possibilities and real habits/laws (conditional necessities); but that obviously takes us far beyond the thread topic.
But that is a contradictory statement--"real" means "as it is regardless of what anyone thinks about it," so anything that is "in our minds only" cannot be real.
No I mean some things can exist in our mind and not in reality or mathematics. Illogical things like inanimate objects that talk. Or infinity. Or a true continua.
But by definition, if it exists in our minds only, it does not exist anywhere else. We conventionally class the contents of our minds as not part of reality. But you could take the opposite definition I guess.
Right, we call such things fictions. Unicorns are not real because they are as they are only because people think of them that way.
Quoting Devans99
There is nothing illogical about infinity or true continua, and the fact that they cannot exist (be actual) does not entail that they cannot be real.
The problem is the word is not used in one single way. The way you prefer is inconsistent at times and at others only vaguely understood and has limits in both application to mathematics and in just straightforward analysis (e.g. understanding the divisibility of time and space). The notion of a definition being "incorrect" because it diverges from colloquial usage is absurd. It captures most or all of the features of that usage but without any contradictions at all.
Quoting Metaphysician Undercover
This is a claim there is no reason to accept. Aristotle believed actual infinities were impossible but they are not. So the entire justification for his distinction between which infinities were possible is empty this side of Cantor. You cannot study a variable which does not exist within a fixed domain. For there to be a potential infinity, there has to be a predefined set of values that can be occupied otherwise the domain is precluded from study as it can change arbitrarily by an arbitrary amount. That domain is actually infinite. And so to understand that domain you need to understand and apply the mathematical regimentation of the concept of infinity.
Quoting Metaphysician Undercover
This is just an outright misrepresentation. Mathematicians did not simply redefine infinity to mean something contrary to its colloquial usage to disingenuously prove things about it. And hold this thought, I'll come back to it later when you say something inconsistent with the above.
Quoting Metaphysician Undercover
There is so much wrong here that it shows a deep lack of understanding of mathematics. "Defining features" are, ironically by definition, established by the definition in use. Otherwise we would never had words whose meaning varies across context and circumstance due to the resemblance in those varying contexts. Contrary to your geometry misunderstanding, the reason parallel lines can meet is that the reason they cannot meet in Euclidean Geometry is because of how space is understood there (as planar). In Riemannian geometry, Euclidean space is understood simply as a space with a curvature of 0. But if space is curved then the provably such lines do intersect, such as on the surface of a sphere (i.e. lines of longitude). The actual Euclidean definition of what a parallel line is does not say the lines will not intersect. It's that if you have some infinite Line J, and a point P not on that line, no lines passing through P intersect with L. This does not hold if the space is different. It's only when one misstates the Parallel Postulate that it sounds contradictory to have intersecting parallel lines. You are forgetting that these notions are defined by the geometry, not separate from them.
Quoting Metaphysician Undercover
So you say this, but remember what you said earlier?:
Quoting Metaphysician Undercover
So you're not calling them idiots, you have respect for them, but you're calling them (or at least certainly comparing them to) sophists? Ok man. Alternative views of math are fine but you're muddying the waters by suggesting the idea is "illusory" simply because you don't think it completely preserves the intuitive idea of infinity (which is contradictory and fails to solve paradoxes). If a paradox arises in some domain of discourse then something has to change to resolve the paradox. Aristotle unwittingly required assuming the existence of the actual infinite in order to deploy the idea of a potential infinity (because the possibility space must be predefined to have any meaning, and indefinite segments are only possible in an infinitely divisible or extensive space), and the intuitive notion of an actual infinity is inconsistent.
So we looked to the mathematicians (Cantor, Bolzano, Dedekind and co.) who gave a concrete, comprehensible theory of the infinite that removed all contradiction while retaining most of the natural ideas about infinity. That's called success, not illusion and sophistry.
Maths describes reality to a high degree... no infinity in maths suggest no infinity in nature.
They are so!
Actual infinity is not a number:
1. If actual infinity is a number, there must be a number larger than any given number.
2. But that’s contradictory.
3. Can’t be a number AND larger than any number.
4. So actual infinity is not a number
5. Invention of magic numbers runs contrary to the spirit of science.
Do you understand the point of Hilbert's Hotel? David Hilbert was a mathematician, not someone who rejected the notion of infinity as contradictory. The point of the "paradox" (not an actual paradox, just a strange thought experiment) is to point out that infinity is weird and does not work the way most people could naturally understand without learning some of the mathematical logic underlying our theories of infinity.
Infinity is defined to be bigger than anything else. That means there can only be one infinity by definition.
No it isn't. Infinity in mathematics simply means some set's members can be put into a one-to-one correspondence with a proper subset of the parent set. What falls out of this is there can be multiple sizes of infinity. Infinity in math is not "The biggest number" or whatever. Aleph-null is larger than any natural number certainly, but aleph-null is smaller than the size of the continuum. Cantor proved this fact with a proof by contradiction (the Diagonal Argument).
I suspect that I have encountered them, but it was a while back and I never got very far. I am aware of a few different approaches that seek to capture a true continuum mathematically, such as nonstandard analysis and smooth infinitesimal analysis. Again, I concede that the real numbers are an adequate model for almost all mathematical purposes.
Quoting MindForged
Like Peirce, I prefer to say that it is really infinite, but not actually infinite. I also join Peirce in denying that numbers exist--i.e., I am not a mathematical Platonist--even while affirming that they are real.
Quoting MindForged
Indeed, and that is the precisely point that you and I have been trying to make throughout this thread, albeit from somewhat different perspectives. Trying to treat infinity/continuity no differently from finite/discrete quantities is what leads to severe misunderstandings.
Quoting MindForged
Right, and Peirce proved that the power of the set of all subsets of a given set is always greater than the power of the original set itself--which entails that there is no largest multitude (his term for aleph). What he called a true continuum is "supermultitudinous," larger than any multitude, and thus impossible to construct from (or divide into) discrete elements. You might find this introductory article about "Peirce's Place in Mathematics" interesting.
Oh I wasn't asking you to concede anything, I just started googling for some stuff and came across it. :) it looks like an attempt to recapture infinitesimals so it caught my eye.
Quoting aletheist
Oh that's fine, I'm just using the terminology I saw others using. I'm not sure I'm a math platonist either (undecided). I just meant it's a real one (in the sense that it's not just some continuously iterated task that still comes out to a finite number at every step).
Quoting aletheist
I'll take a look at it! From your description it sounds like what Cantor referred to as the Absolute Infinite though it doesn't exist in ZF.
...which is what makes them real! They are not real in the same way that rocks and crocodiles are real. They're real like Harry Potter and mathematics are. For unicorns, Harry and mathematics are all human inventions that have no existence in the space-time universe that science so ably describes. They were invented for quite different reasons, admittedly, but none of them are "real" as you use the term. :up:
As I understand the issue, the physical evidence indicates that there is likely a real limit the spatial-temporal existence at this level. If this is the case then there is a separation, a lack of correspondence, between the conceptualization of a continuous space and time, and real physical existence. I believe that Peirce suggested replacing "infinite" with "infinitesimal", in our conceptions, as a way to deal with this problem.
Quoting TheMadFool
It is impossible that any numbers can model a true continuum because all conceptions of numbers are based in a conception of units, such that a number signifies either a unit or a multitude of units, and this is incompatible with a true continuum. This issue is very similar to the issue with true infinity. The "infinity" is defined as something which numbers cannot count, it is impossible by definition, that numbers can count infinity. Likewise, it is impossible by definition that numbers can represent a true continuum.
Quoting aletheist
Infinitesimals does not resolve the problem because infinitesimals are units. So to model a continuum as infinitesimals is to model it as composed of discrete units.
Quoting MindForged
Obviously, this is what I disagree with. The mathematical conception of "infinite" clearly contradicts the colloquial definition of "infinite", I've demonstrated this over and over again, so you know what I mean and I will not demonstrate it here again. You simply assert that it does not contradict, while the evidence is clear, that it does.
Neither of the two distinct conceptions have contradiction inherent within. There is no contradiction inherent within the colloquial concept. Where the problem lies is in applying the concepts to what we consider as the real world. Each conception, the colloquial and the mathematical, has its own set of problems involved with application.
Quoting MindForged
That's not a proper representation of what Aristotle argued. He used the argument to separate "eternal" from "infinite" because Ideas, Forms, were described as eternal, and "infinite" was an idea. So he proceeded to demonstrate that "eternal" and "infinite" were incompatible. What he demonstrated is that anything eternal is necessarily actual, while anything infinite has the nature of potential. The latter, that the infinite belongs in the class of potential, must be read as a definition, a description, derived from observation. All instances of "infinite" are conceptual, ideas, and ideas are classed in the category of potential. From this premise, along with other premises, the conclusion that anything that is eternal is necessarily actual is derived.
This idea which you state, that actual infinities are possible, is produced from the conflation of Aristotle's two distinct aspects of reality, actual and potential, and consequentially time and space, in the modern conception of "energy". So the claim "actual infinities are possible" (which would be contradictory if we adhered to Aristotle's distinction between actual and possible), only demonstrates a failure to maintain Aristotle's principles. it doesn't mean that "actual infinities are possible" has any coherent meaning.
Quoting MindForged
The whole point of "potential", under Aristotle's philosophy is that it cannot be studied as such. What we know, study, and understand, are all forms and forms are by definition actualities. "Matter" being classed as "potential", just like "ideas", is that part of reality which is impossible for us to understand. Potential is defined that way, it defies the law of excluded middle. There is an aspect of reality which is impossible for us human beings to understand because it violates the laws of logic, and this is "potential". Therefore, by its very definition, it is precluded from the study which you refer to. To assign a set of values, in order to study that domain is simple contradiction.
Quoting MindForged
See, you have taken the category which is defined by "that which cannot be studied", "potential", which consists of matter, ideas, and the infinite, and you've applied some values (which is contradictory), and now you claim that this thing "infinite" is no longer in that category, it's in the category of actual. All you have done is changed the subject.
Quoting MindForged
Yes it does, read the above.
Quoting MindForged
This is clearly false, and indicates a misunderstanding of how logic and understanding proceeds. We identify a thing (law of identity), this thing as identified, becomes our subject, and we proceed to understand it through predication. The "defining features", how the subject is defined, ensures that the subject represents the object. This is known as correspondence, truth. It is evident therefore, that "defining features" is determined by correspondence between the logical subject and the object which is said to correspond to that subject, and not by "the definition in use". If it were the "definition in use" which defined the subject everything would be random with no correspondence to reality It is clear that "the definition in use" must be consistent with the known correspondence, truth. When the definition in use is not consistent to provide a correspondence with the identified object, we can correct the definition, saying that you are using an incorrect definition.
Quoting MindForged
So this is very wrong because you have reified space, as if "space" were the subject, and there is a corresponding object which has been identified as "space". There is no such object being described here in geometry. The objects are all mathematical, conceptual, such as a "line". My point was that if there are two distinct concepts of "line", then there are two distinct objects referred to by that name "line" corresponding to the defining features which constitute the two distinct subjects under that name, "line". Therefore "line" ought not be used for identification of both of these objects.
Reality and existence are not synonyms. Harry Potter and unicorns are not real, because their properties depend upon the thinking of an individual mind (e.g., J. K. Rowling) or a finite group of minds. Rocks and crocodiles are real, because their properties do not have any such dependence.
How to characterize mathematics is not as straightforward. I endorse Charles Sanders Peirce's definition of it, which he adapted from that of his father Benjamin, one of the foremost mathematicians of the 19th century: the science which draws necessary conclusions about hypothetical states of things. This clearly rules out mathematical Platonism, which claims that objects such as numbers exist; they are (logical) possibilities, rather than actualities. Nevertheless, it may still be the case that those objects are real, in the sense that we discover their properties, rather than inventing them.
There are two fundamental mistakes here: first, infinitesimals are not units; second, a continuum is not composed of infinitesimals.
Quoting Metaphysician Undercover
Forms are only actualities when and where they are instantiated in concrete particulars. In themselves, as distinct from Matter, they are real possibilities.
Quoting Metaphysician Undercover
It violates the laws of classical (bivalent) logic, but that is not the only kind of logic available to us. For example, we can reason without the law of excluded middle using intuitionist logic. In fact, the law of excluded middle does not apply to infinitesimals; rather than discrete points, they are analogous to indefinite "neighborhoods" with an inexhaustible supply of potential points.
Yes, infinitesimals are units, they are bounded necessarily in order to give them the status of distinct things "infinitesimals". without separation between them, there could not be the plurality indicated by "infinitesimals". Under Peirce's philosophy though, the boundaries are vague. However, it is necessary that either the boundaries are real in order that the infinitesimals are real, or else the boundaries are not real, in which case neither are the infinitesimals.
Quoting aletheist
Right, but doing this only gives "potential" a different definition, just like mathematics gives "infinite" a different definition. Producing a different definition only means that the corresponding object is not the same object. So we are no longer talking about what Aristotle identified as "potential" we are talking about something different.
No, infinitesimals are not units, and they are not "distinct things."
Quoting Metaphysician Undercover
False, infinitesimals are real but indefinite--i.e., potential not actual.
Quoting Metaphysician Undercover
How so? By "potential" I simply mean real possibility, rather than actuality. As Peirce put it, "the word 'potential' means indeterminate yet capable of determination in any special case" (CP 6.185; 1898, emphasis in original).
You seem to have little understanding of what "infinitesimals" refers to. An infinitesimal is defined by Leibniz as an entity, a unity. It was used as an approach to mathematizing space, time, and matter which were previously conceived of as continuities. There was an inconsistency between mathematics which deals with distinct units, and these concepts, space, time, and matter, which were based in an assumption of continuity. If these continuities, space, time, and matter, could be conceived of as composed of infinitesimals (units, like monads) we could establish compatibility between a continuum and the numbers..
Peirce did nothing to change the fundamental nature of infinitesimals as units. The problem of course, is that if the things which we knew of as continuous, space, time, and matter, are really composed of these units, infinitesimals, then they are not truly continuous. So the question is whether these basic, primitive intuitions which hold space, time, and matter as continuous are correct, or are these things which appear to be continuous, more appropriately represented by the discrete units, infinitesimals. If the latter is the truth then we ought to be able to distinguish real boundaries between one infinitesimal and another. It does not suffice to say that the boundaries are there, but they are vague and cannot be determined. To give reality to the infinitesimals we need to determine those boundaries.
Right back at you.
Quoting Metaphysician Undercover
As I already stated, a continuum is not composed of infinitesimals. Moreover, there is no ultimate compatibility between a continuum and the numbers--or anything else discrete.
Quoting Metaphysician Undercover
The only reason for positing infinitesimals (in contrast to points) is to preserve those basic, primitive intuitions of continuity, rather than resorting to (wrongly) treating a continuum as if it were composed of discrete units.
That's the point, they are incompatible. And, the introduction of infinitesimals does not make them compatible. It's only an illusion.
Quoting aletheist
The problem with points is that a point is by definition, dimensionless. It takes up no space on a line, only divides a line. Therefore the line cannot be made up of points. Even an infinity of dimensionless points cannot account for the existence of the continuous, one dimensional line, which is supposed to exist between two points. There is a fundamental incompatibility between the point which is dimensionless, and the line, which is dimensional. Infinitesimal points were introduced to allow that a multitude of infinitesimals may have dimensionality, therefore a continuous line could be conceived of as being composed of infinitesimal points. This is an attempt to establish compatibility between the non-dimensional point, and the dimensional line.
Continuum to me implies a smooth transition between two points. I agree that, numerically, we need to define a unit before we can make sense of a given quantity.
However, we can choose arbitrarily small units and do any measurement. What I mean is we can make any quanitification arbitrarily smooth depending on our needs for accuracy or whatever.
Also, the real numbers includes ALL points possible. In other words no point on the number line is left out. Isn't that sufficient for a continuum model?
You misread what you quoted. I said the mathematical conception has no contradictions, I didn't say it was identical to the colloquial definition. The colloquial understanding of infinity includes, for example, a notion of unboundedness. And yet we know infinities are in some sense bounded, and people will readily admit that the real between 0 and 1 are infinite despite that clearly being a bounded array of values. That's just an obvious case of a colloquial, folk conception being contradictory and hence the need for a formal understanding which we got from mathematics.
Quoting Metaphysician Undercover
Oh my Lord, you aren't saying anything different. This is essentially just that an actual infinite isn't possible, but now because of our observations instead of an inherent contradiction. We don't derive from observation that the infinite is relegated to ideas. All current theories of spacetime that are more grounded than speculative (e.g. LQG isnt mainstream right now) require space and time to be infinitely divisible and no observation contradicts this at all. In fact, attempting to make those finite will result in inconsistencies more than likely. So no, observation does not require one to class the infinite as merely potential, a mere idea that cannot be found in the world.
And in any case, the way you're talking about potentiality sounds contradictory. It's being spoken of as if it's ineffable. And yet you're telling me about it and what makes it ineffable... Which means youre talking about it, so it's not ineffable.
Quoting Metaphysician Undercover
This sounds incredibly wacky. For one, even if there is some metaphysical violation of Excluded Middle, that doesn't preclude it from human understanding nor does that make reality "violate the laws of logic" because there are not "the" laws of logic. There are many such sets of laws, and some drop Excluded Middle. It would certainly be a surprise to the Intuitionists that they don't understand constructive mathematics or their own logic because it doesn't assume EM as an axiom.
Quoting Metaphysician Undercover
Ok so now you admit what you earlier rejected? Previously I said that under your view, a previous misconception of disagreement about a concept cannot changed because to do so is to change the subject. In which case progress is impossible because people cannot have different theories about the same concept. There goes all of philosophy.
If the category cannot be studied how do you know anything about it? If you don't know anything about it, how can it be studied? If you do know something about it you must be studying it by some means. In which case the distinctions you're making don't seem motivated by anything other than philosophical prejudice.
Quoting Metaphysician Undercover
What a highly idealized and incoherent procedure you're suggesting. The definition in use communicates how we believe the object to be. Predication is how we understand the object to have the attributes it has, it's not how we understand the object itself (that's an intuitive project, one done reflexively most of the time). This doesn't leave it random. If I say the term "Beep" refers to my dog, so long as my use is consistent absolutely everyone understands what I'm talking about (if they speak English). That's why people can make up words and have them often times be understood by people not perceiving what we are referring to.
And this is all besides the point anyway. The nonsense you tried to pass off earlier was the idea that there are "defining features" of things like parallel lines despite now knowing that these terms are defined by the user's (implicitly or explicitly). They don't have inherent definitions, they're defined within a certain domain. So the idea that you tried to push that parallel lines don't intersect is purely based off the underlying assumptions of the geometry in which you made an assumption of, it is not true writ large in geometry. The Parallel Postulate is only true inasmuch as it's assumed to be so in a geometry and anyone saying otherwise is just misinformed about how mathematical formalisms work.
Quoting Metaphysician Undercover
I haven't reified anything, I didn't treat these as anything other than abstract mathematical constructs. Are you truly unable to talk about the basic properties of a geometry? There are different kinds of space indifferent geometries. Some are curved, some aren't as they are planes. Terms like "line" are usually left as undefined primitives in geometry so there is no confusion here because they are essentially take to be the same object in a different background (a different space) or else as a similar objects in different spaces so there's no benefit to calling one a line and calling nother "line-ish".
Continuum is uninterrupted. "Two points" implies two distinct places and therefore a boundary which separates them.
Quoting TheMadFool
So I assume that the "smooth transition" you refer to is arbitrary and not real?
Quoting MindForged
I agree that the mathematical conception has no inherent contradiction. But the only sense in which "an infinity" is bounded is by the terms of its definition. All infinites which we speak of are bounded by the context in which the word is used. If someone mentions an infinity of a particular item, then the infinity is bounded, defined as consisting of only this item. Likewise if we are talking about an infinity of real numbers between 0 and 1, the infinity is bounded, limited by those terms. However, we are not discussing particular infinities here, which may be understood as particular (though imaginary) objects, we are discussing the concept of "infinite". We are not discussing conceptual entities which are said to be infinite, we are discussing what it means to be infinite. The reals between 0 and 1 is a conceptual entity which is said to be infinite. We are asking what does it mean when we say that this is infinite.
Quoting MindForged
This is false. Anytime "infinite" is used to refer to something boundless, or endless, it refers to something made up by the mind, something imaginary or conceptual. We do not ever observe with our senses anything which is boundless or endless, because the capacities of our senses are limited and could not observe such a thing. Since the capacities of our senses are finite we know that anything which is said to be infinite is a creation of our minds, it is conceptual, ideal.
Quoting MindForged
Spacetime is conceptual. This is the problem I had with your last post, you reified "space", making it into some sort of an object to justify your position. In reality, "space" is purely conceptual. We do not sense space at all, anywhere, it is a constructed concept which helps us to understand the world we live in. Furthermore, "infinitely divisible" is an imaginary activity, purely conceptual. We never observe anything being infinitely divided, we simply assume, in our minds, that something has the potential to be thus divided.
Quoting MindForged
I never defined "potentiality" as ineffable. It may appear to you that potentiality is contradictory ifyou do not understand the concept, but Aristotle was very specific and explicit in his description of what the term refers to, to ensure that his conception does not defy the law of non-contradiction. As that which may or may not be, he allowed potentiality to defy the law of excluded middle. You will find that some modern philosophies though, such as dialectical materialism, and dialetheism allow for violation of the law of non-contradiction. They may conceive of potentiality, or matter as contradictory.
Quoting MindForged
it's not "incredibly whacky" it's the central point of our discussion. There are always things which escape human understanding. They are things which human beings do not understand. I would place "infinite" in that category, but you seem to think that "infinite" is understood, so let's go back four or five hundred years and say that at that time, "infinite" was not understood. The reason why it could not be understood was that it appeared to defy the law of non-contradiction, in the form of paradoxes. When some premises lead to conclusions which contradict what we know as basic fact, then there is a problem in our understanding. These are things which escape human understanding.
When this problem occurs, there are two principal possibilities of the cause of the problem. One is that the fundamental premises, how the terms are defined, are faulty, the other is that the fault is within the logical process. So our subject is "infinite", a few hundred years ago,when it escaped human understanding. The term produced problems which caused the appearance that it could not be understood. The logicians at the time decided that the best way to proceed was to change the premises, the defining terms of "infinite". What I am arguing is that misunderstanding is not due to faulty premises, but to faulty logical process. Zeno's paradoxes deceive the logician through means such as ambiguity or equivocation, by failing to properly differentiate between whether the aspects of reality referred to by the words, have actual, or potential existence. That's what Aristotle argued. So the logician gets confused by a conflation of actual problems and potential problems, which require different types of logic to resolve, and are resolved in different ways. Instead of disentangling the potential from the actual, the logicians took the easy route, which was to redefine the premises. All this does is to bury the problem deeper in a mass of confusion.
Quoting MindForged
You haven't addressed the issue here. You only support these claims with a reified "space", assuming that space is a physical object to be studied, and not a conceptual object. It's beginning to appear like this is the crux of the differences which we have. Do you distinguish between objects which exist solely in the mind, imaginary things, concepts and ideas, and things which have physical existence in the world, like rocks and trees, and planets? If so, then when we use words to refer to things, we ought not confuse whether the thing referred to is ideal, a concept, or a physical object. An infinity, just like infinite, is always an ideal, a concept, never something in the physical world.
Quoting MindForged
What's this then?
Quoting MindForged
See, you are treating "space" as if it is something described by geometry. In reality, since we can use various different geometries to describe the various types of objects we sense, there is no such thing as "space". We might be able produce a concept of "space" from this geometry, and another concept of "space" from this other geometry, but it really makes no sense to talk about "how space is", or "if space is curved...", because there is no such thing as "space", not even as a concept.
This is why your geometrical examples are irrelevant, and way off the mark. You are talking about geometry as if it is created to describe some sort of "space". Then you need to bring in some principles to account for the intuition that space is in some sense infinite. However, this is totally uncalled for. We produce principles of geometry to measure the objects which we encounter, and we do not encounter any infinite objects. It is only when we approach these concepts of geometry attempting to synthesize a concept of "space", that we find "infinite" as inherent within the principles themselves. Then there is a problem because we want to produce a concept of space which allows for the infinite because we have been confronted with the infinite as inherent within the concepts used to describe objects.. But this is only a pseudo problem. The infinite is not part of the objects which we measure and therefore it ought not be part of any concept of space. The infinite is only a part of the mathematical tools which we use to measure with. So any attempt to bring the infinite into our conception of "space" as a the thing being measured is a mistake.
"Infinitesimal point" is self-contradictory--points are, by definition, dimensionless and indivisible; infinitesimals are, by definition, dimensional and potentially divisible without limit. As I have stated repeatedly, in my view an infinitesimal is not a discrete unit of any kind, and a continuum is not composed of infintesimals.
Quoting TheMadFool
Yes, as I have acknowledged, the real numbers serve as a sufficient model of a continuum for almost all purposes within mathematics; but it is still a model, not a true continuum itself.
That's exactly the problem with infinitesimals, their dimensionality is ambiguous. If it is not a discrete unit in any way, then it has no form and therefore no specific dimensionality. However, you say that it is not dimensionless. Therefore it has dimension but its dimensionality is completely undefined and ambiguous. But dimensionality is purely conceptual, and must be defined. Now we have an infinitesimal which is defined as having no specific dimensions yet it is not non-dimensional. Therefore it is either completely irrelevant to our conceptualization of dimensionality, or it occupies some vague, ambiguous, undefined position within our conceptualization of dimensionality which could only serve to mislead us.
Why is that necessarily a problem? An infinitesimal indeed has no specific or definite or measurable dimensionality, yet it does have real dimensionality.
The main idea that I keep trying to emphasize is that a true continuum is not a composition of discrete units of any kind; it is a top-down concept, not bottom-up. Instead of division, perhaps a more perspicuous approach is to think in terms of magnification. No matter how much we were to "zoom in" on any portion of a truly continuous line, what we would always "see" is a continuous line, rather than a point or other discrete unit. Likewise, no matter how much we were to "zoom in" on an infinitesimal, what we would always "see" is a continuous line, rather than a point or other discrete unit.
You don't see this as a problem? Imagine if I told you about something which has no specific, definite, or measurable colour, yet it does have real colour. What could this possibly mean, other than something contradictory? It doesn't have any measurable colour but it has real colour.
Quoting aletheist
But a line is specifically one dimensional, and an infinitesimal is not. So if you zoomed in on an infinitesimal why would you see it as a one dimensional line rather than as three dimensional, four dimensional, or even an infinity of dimensions for that matter? If it might be an infinity of dimensions, then the purpose of the infinitesimal is self-defeating.
Again, where is the problem if that "something" is mathematical--i.e., hypothetical--rather than actual? Are you claiming that reality is limited to that which is specific, definite, and measurable? If so, on what grounds?
A color is a quality, so its mode of being is that of possibility. Between any two "measurable" shades of red (for example)--e.g., identified by RGB hexidecimal code or electromagnetic wavelength to an arbitrary degree of precision--there are intermediate shades beyond all multitude. All of them are real, regardless of whether they ever exist by being instantiated in actual concrete particulars.
Quoting Metaphysician Undercover
I thought it was obvious in context that I was talking about a one-dimensional infinitesimal for the sake of conceptual simplicity. Its "length" is non-zero, yet smaller than any assignable value. As such, how could we measure it, even in principle?
You mean, like saying that there is a number which has no definite value, but it is nevertheless a number? What nonsense is that? Mathematical objects exist as specific definite things. That's what gives mathematical objects their actual existence, the definition. To say that there is a mathematical object which is indefinite is nonsense. That's why the attempt by speculative logicians and mathematicians to bring "infinite" into the realm of mathematical objects is doomed to failure as inherently contradictory.
Quoting aletheist
Each of those shades of colour is measurable though. You have defined the infinitesimal as having an immeasurable dimensionality, yet still having a dimensionality. Since dimensionality constitutes being measurable, this is like saying that infinitesimals have something measurable (dimensionality), which cannot be measured. That's blatant contradiction.
Quoting aletheist
The point being that you defined infinitesimals as having no specific, or definite, or measurable dimensionality, so it is contradictory to talk about a "one-dimensional infinitesimal".
An infinitesimal is not a number.
Quoting Metaphysician Undercover
What is nonsense is claiming that mathematical objects have actual existence at all. In themselves, numbers (for example) are aspatial and atemporal, and do not react to or interact with anything else.
Quoting Metaphysician Undercover
False. Again, between any two measurable shades, there are intermediate potential shades beyond all multitude that cannot be measured, even in principle. That is what it means to be a true continuum.
Quoting Metaphysician Undercover
It straightforwardly begs the question to define dimensionality as "being measurable," when what is at issue is the logical (not actual) possibility of dimensionality that is not measurable. Measurement entails discreteness, but we are talking about true continuity.
Quoting Metaphysician Undercover
By definition, a one-dimensional infinitesimal has dimensionality, even though it cannot be measured along that one dimension. Its "length" relative to any finite/discrete unit is less than any assignable value, but nevertheless not zero.
How is it that mathematics is necessary for building things, yet numbers do not interact with anything? That's the nonsensical claim, that numbers do not interact with anything. I suppose engineering could be done without numbers? And if numbers are necessary, how so if they don't interact with anything? Obviously numbers interact with things or else they could not be necessary for building things.
Quoting aletheist
Of course they're not measurable shades if they're not actual shades, only potential shades. They are simply imaginary, so of course they cannot be measured. Is this how you conceive of the continuum as well? Is it simply imaginary as your example seems to indicate? I think it's purely imaginary, don't you?
Quoting aletheist
This is what is nonsense. Your one-dimensional infinitesimal is just a short line. You arbitrarily claim that its length is less than any assignable value, but there is no such limit to our capacity to assign a length value because numbers are infinite. So all you are doing is attempting to limit, arbitrarily, our capacity to measure a length, by saying that this length, the infinitesimal length, is such a limit.
Wow, do you really think that mathematics is necessary for building things? That would be news to the ancients, or to any young child even today who builds things while playing. Mathematics is certainly useful for analyzing, designing, and building things--especially large, complex things--but it is by no means necessary.
Quoting Metaphysician Undercover
I suppose it depends on how you define "engineering." At this stage of my own career as a structural engineer, I spend most of my time making high-level decisions that involve the exercise of practical judgment obtained through experience, rather than crunching numbers.
Quoting Metaphysician Undercover
Really? Where can I find a number so that I may interact with it?
Quoting Metaphysician Undercover
I have consistently characterized a continuum and an infinitesimal as real--that which is as it is, regardless of what any individual mind or finite group of minds thinks about it--but not actual.
Quoting Metaphysician Undercover
This is exactly backwards--what is arbitrary is the insistence that anything must be measurable in order to be real.
Infinite in your head only, not mathematically: width of a number is 0. How many in an interval sized 1? 1 / 0 = UNDEFINED.
Infinity is greater than any assignable quantity; which implies is not a quantity (can't be a quantity and greater than any assignable quantity).
When you add one to it, nothing changes; clearly not a quantity. So it should not be present in mathematics. Which means no mathematical continua.
If its not a quantity, which it is not by definition, we should not assign it to physical quantities like, time, size, mass etc...
Actually you seem to have misunderstood what I meant. I didn't mean mathematics is necessary for building all things, but for some things. So my argument remains the same. Of these things which mathematics is necessary to build, the mathematics must somehow interact with things in order that these things get built.
Quoting aletheist
I don't see how my computer could have been built without mathematics. Regardless, let's just say that mathematics is useful for building things, as you say. How could mathematics be useful in building things unless it somehow interacted with things? You might say that the human being is a medium between the thing built and the mathematics, but the human being is also a thing, and the mathematics must interact with that thing in order for it to build the things which the mathematics is useful for. So the mathematics still interacts with things, even though the human being, as a thing is a medium between the mathematics and the thing built..
Quoting aletheist
I never made any such claim so instead of addressing my concerns you are just changing the subject.
Again, where can I find such mathematics so that I may interact with it? We can only interact with that which is actual, which is why both words have the same root; but mathematics deals entirely with the hypothetical. We use mathematics to model the actual, but that is not interacting with mathematics as if it were something that exists.
This is mistaken. My point is simple. Infinity is often intuitively understood as unbounded quantitatively. In other words, given any arbitrary number N there is some number N+1 that can be accessed from N. No set end point, basically. However, it's clearly the case that in the interval of reals between 0 and 1 that 1 is an end point, yet people will when asked refer to that as infinite despite having a set, determinable end. So clearly the colloquial understanding of this infinite is not consistent.
Quoting Metaphysician Undercover
I didn't say we perceive infinity, I said our observations do not demonstrate that infinity is merely an idea. In fact, take the set of all observations ever made and assume they are of finite things. So what? All that tells us is that those observations are finite and so the next ones made will likely be finite. It doesn't entail that they are necessarily the case, you (and Aristotle) arbitrarily define them to be such. Worse, if you accept standard mathematics at all you have to agree that time and space are infinitely divisible. We have the math to make perfectly logical sense of this and all current physics assumes this is the case (even if you wanted to suggest loop quantum gravity, I could suggest the equally speculative string theory where space can be infinitely divided again).
Quoting Metaphysician Undercover
I also never observe my own brain activity, that doesn't entail my brain doesn't exist as an object. I don't observe exoplanets, that doesn't mean their existence is purely conceptual. Sensing a thing is not identical to that thing not existing. Furthermore, space is a thing. It is not even in question that space has properties, such as our being curved for instance. We can actually see curved space (gravitational lensing), so even then your criteria has been satisfied. And bearing properties is pretty much a fundamental requirement and sufficient condition for being an object.
Quoting Metaphysician Undercover
Whatever Aristotle may have said, refer to what you said before:
You said it's impossible for humans to understand and yet clearly you think you can explain what about it makes it impossible for humans to understand. So unless you can explain something about things you can't possibly understand it sounds like you're contradicting yourself.
Quoting Metaphysician Undercover
You don't realize the game you're playing. Aristotle is doing the exact same thing. By your own admission it's Aristotle who is partitioning infinite into the category of ideas and away from reality, thereby changing the definitions of potential and actual. After all, in plain English "potential" is understood as a modal term, as a synonym for "possible". But for something to possibly be the case there must be some state of affairs where it obtains. Colloquially and philosophically, a potential can be actualized otherwise it's an impossibility. So no, you're just ignoring it when you do it because it's presumed to be acceptable for you to do so and only because it's you doing it. It's a convenient standard for you to have.
Quoting Metaphysician Undercover
I never said space was physical. An object, sure. It has properties after all and we have studied these properties. I'll come to this in a moment.
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
I don't really see how you are saying anything here because you're moving between unrelated points. When I referred to "how space is" I was talking about the actual structure of spacetime, not the more vague, general concept of space. Some abstract geometric spaces are curved and some are not. Whether or not the structure of the actual space of the universe falls into one or the other is a physics question, and current physicsand observational evidence says space is curved. If you cannot even accept this there's no point in this.
Quoting Metaphysician Undercover
Er, yes and no. The canonical application of geometry is to understand the spatial structure of the actual world. But I never said that's what geometry itself is about, it's about the study of abstract spatial structures (if you object that geometry isn't about I'm sorry you are so wrong I don't know how anything short of a mathematics textbook being regurgitated would correct you).
Quoting Metaphysician Undercover
I lost interest the moment I realized you treated measurement of objects as a fundamental concern of a geometry axioms. I don't encounter any perfect spheres, so surely it must be totally uncalled for to apply geometrical principles to reality where some object is arbitrarily similar to a perfect spheres since there cannot be any such thing in reality. Do you see why your view of geometry makes no sense to me?
You do not understand the concept of cardinality, do you? The size of the interval is the size of the continuum, aleph-1.
Quoting Devans99
Why do you keep asserting this as fact? That is not how "infinity" is defined in mathematics because it's too hazy and informal a definition. An infinite set is, e.g. the transfinite Cardinals, a set whose members can be put into a one-on-one correspondence with a proper subset of themselves. There is absolutely no mention of being "greater than any assignable quantity, you're just wrong. Find one mathematics textbook that formally defines and describes infinity that way. Go on, I'm sure you can do it... (Obviously I'm being sarcastic here).
Quoting Devans99
That's not an argument, that's a statement that you cannot justify. As it happens, it's perfectly understandable why the CARDINALITY doesn't change. The set does change if you add a new unique element, but the size cannot change by mere finite additions because we can still world new numbers to put into a function with the new element that was added.
I was just helping a friend out with a website. The RGB color values could be changed by the file in question as needed, but there seemed to be an oddity. Because color values are cappped at 255 per channel, adding any number to that channel resulted in no change to the value of that channel. I suppose 255 must not be a quantity then since adding to it did not cause the value to change. Saturation mathematics must be incoherent despite it's use in computer science then!
It's a little funny that mathematics by unjustified dogmatic assertion went out of vogue for everyone besides the ultrafinitists.
You can find this mathematics right in your mind. It's really there, and actual. An hypothesis has actual existence whether or not you believe it to be true.
Quoting MindForged
Actually, that is what is mistaken. In the case of the reals between 0 and 1, it is quite obvious that 1, like 0, is a defining point, and therefore a beginning point rather than an end point. What is claimed is that there is an endless number of points between 0 and 1. It is impossible that there could be a direction of procedure, because if we started at zero, and tried to progress toward 1 as an ending point, it is impossible to name the first real number after zero. Any named number after 0 would have more numbers between it and 0, and we'd have to turn around and go back, heading away from 1. Therefore, in this instance it is false to represent 1 as an "end point". There are two defined start points, 0 and 1, with an infinity of points between them. No end point.
Quoting MindForged
I guess you've never heard of inductive reasoning. Inductive reasoning is how we draw logical conclusions called generalizations, from observations. The bigger issue though which you didn't seem to grasp, is that all observations themselves, are necessarily finite.
Quoting MindForged
I know, I believe I already described to you how definitions are arbitrary. But those definitions which demonstrate a true correspondence are considered to be true definitions. So if we observe that the clear sky is always a similar colour, and we name this colour as "blue", so that everyone calls this colour blue, then we can define "blue" as "the colour of the sky". There is true correspondence because that is how people use the word "blue". But if everyone is referring to the "infinite" as endless, and we decide to define "infinite" in some other way, then we do not have true correspondence.
Quoting MindForged
Why is this "worse"? Time and space are purely conceptual, just like numbers. Numbers are conceptual and infinite. If time and space are concepts produced from mathematics, why wouldn't they be infinite as well? A conclusion reflects its premises. The premise is that numbers are infinite. If time and space are concepts created from numbers they will reflect this infinity. Unless we allow that time and space are concepts created by something other than mathematics, they will necessarily be infinite. If time and space are other than mathematical, what would be the basis of these concepts, observation? Observations are necessarily finite. Therefore we have an incompatibility between the concepts of space and time which are consistent with mathematics, and the concepts of space and time which are consistent with observations. This has manifested as Zeno's paradoxes.
Quoting MindForged
The problem is that things unobserved do not enter into conceptions produced from observations So, even if there is a real thing out there, like space or time, which is truly infinite, the limitations of our senses deny us the capacity to observe the infinity of this thing. That's the classical, or colloquial understanding of "infinite", that it's impossible for the human being to observe. Let's assume that space and time are infinite, as the mathematical conceptions tell us, but our observations are incapable of corroborating this due to the limitations of our senses. Now we have the platform for Zeno-type paradoxes between the mathematical concepts of space and time, and the observational concepts. What do you think is the appropriate procedure to resolve the incompatibility? Do we face the fact that our observations are limited, and therefore fail us in this realm, and maintain a pure infinite in our concepts of space and time, or do we denigrate the pure infinite concept, and produce a new concept of "infinite" which is more consistent with our faulty observations? The latter is what the logicians have done, and what you seem to insist was the right thing.
Quoting MindForged
What you're not respecting, is that for Aristotle ideas are part of reality. He was a student of Plato and was well trained in an ontology that holds ideas as real. To place infinity into the category of ideal, would only remove it from reality, if you proceed like aletheist above, on the preconceived notion that ideas are not real. Infinity, as well as mathematical ideas are very real for both Plato and Aristotle, so placing "infinity" into the category of ideal is not partitioning it away from reality.
Quoting MindForged
You seem to be missing the point. I agree with what you have described here, a possibility is defined by actuality, what actually is. This is a specific possibility, it is only correctly "a possibility" if the actuality permits, otherwise it's impossible. Now let's move to the more general, "potential" what it means to be possible. What is it about reality which makes tings "possible"? What is the nature of contingency? We know that actuality defines a particular possibility as possible instead of impossible, but possibilities are not confined to one, they are by nature numerous. What do they have in common by which they are all possible? What actuality can we refer to in order to define what it means to have numerous things under the same name, as possible?
Quoting MindForged
Yes, I see why my view of geometry makes no sense to you. You're speaking nonsense, and if this represents how you apprehend "geometry", your apprehension must be nonsensical as well. Did you just claim, that just because you haven't ever encountered a perfect sphere, you may conclude that geometry wasn't created for the purpose of measuring objects? What kind of nonsense is that?
As usual, this reflects conflation of the real with the actual.
Quoting Metaphysician Undercover
If numbers are infinite, and mathematics is actual, then I guess there is such a thing as an actual infinity after all. Right?
Quoting Metaphysician Undercover
Recognize that continuous motion through space-time is a more fundamental reality than discrete positions in space and/or discrete instants in time. We arbitrarily impose the latter for the sake of measurement and calculation.
Quoting Metaphysician Undercover
Where on earth have I ever suggested that ideas are not real? Perhaps this reflects yet another conflation, this time between two definitions of "idea"--the content of an actual thought vs. anything whose mode of being is its mere possibility of representation. The latter is real even if it never actually gets represented, which means ...
Quoting Metaphysician Undercover
... this is incorrect. Possibility is a distinct mode of being from actuality--and from (conditional) necessity, as well; none of them is dependent on either of the others. That is precisely why we must carefully distinguish logical possibility from actual possibility. Mathematics deals with that which is logically possible, regardless of whether it is actually possible.
"Infinite" is a description of the numbers, as such it is qualitative, not quantitative. Mathematics cannot deal with the concept of "infinite" which is a description of mathematical objects made from outside the principles of mathematics, because it is not a mathematical principle. That's the problem here. The point being that whatever category you put the mathematical objects into, the descriptive term "infinite" is of another category, as the difference between the territory and the map, one being the object, the other a description of the object. The problem occurs when we attempt to make "infinite" a mathematical object.
Quoting aletheist
You insistently claim that numbers have no actual existence. The #1 definition of "real" in my OED is actually existing. So I concluded that you do not believe numbers to be real. You insist that numbers cannot interact with things in the world. Now you claim that ideas are real. I guess you use "real" in another way, to allow for something which is real, but cannot interact with our world. What sense is there in this, to allow for something real, which cannot interact with anything else in the world? So I haven't any idea what you might mean by "real" now because you seem to be claiming that there are real things which cannot in any way interact with the world we sense.
Quoting aletheist
I don't see how "possibility" is at all relevant in your reality. It cannot interact with the actual world, as a distinct mode of being, so how could it be relevant? And "actual possibility" implies that the possibility is interacting with the world, but this contradicts what you've already claimed.
I am afraid that I cannot make heads or tails of your first paragraph because of the confusion it exhibits regarding the meaning of terms--infinite, qualitative, quantitative, mathematics/mathematical, category.
Quoting Metaphysician Undercover
Yes, but I claim just as insistently that numbers are nevertheless real, because ...
Quoting Metaphysician Undercover
I have stated this explicitly and repeatedly--I deny that reality and actuality/existence are synonymous. Reality consists of that which is as it is regardless of what any individual mind or finite group of minds thinks about it. Actuality/existence is that which reacts with other like things in the environment. Reality includes some possibilities and some conditional necessities that may or may not ever be actualized.
Quoting Metaphysician Undercover
There is no contradiction. Something is logically possible if it is merely capable of representation; something is actually possible only if it is also capable of actualization.
Nonsense. The whole argument you're making assumes there needs to be counting - or as you called it, an "order of procedure" - in order for there to be an end point. And this is just false. The only way your argument could work would be by the hilarious assumption that the quantity of real numbers between any arbitrary interval were finite. Defining the start and end of something does not mean that end is not an end point. For goodness sake, a "race" has a defined start point and end point and no one would object "But sir, if you define the starting point and end point at once it's a defined point, not an end". The end point of an interval is not defined as the end of where you stop counting, come on. It's just the set of numbers you're quantifying over.
Quoting Metaphysician Undercover
And I guess you've never heard that induction does not yield necessary conclusions like deduction does. The set of all observations simply, as I said, makes it more likely that the next observation will be of something finite. You claiming that they are necessarily finite is either begging the question (because you're presuming we can't observe some object that has some property which is infinite) or you're conflating induction with deduction. There are no necessary conclusions for inductive reasoning.
Quoting Metaphysician Undercover
The problem is that is not the exclusive colloquial definition of infinite for reasons I've already mentioned.
Quoting Metaphysician Undercover
What you seem to be missing is that the math is used to model the world and so far no model of finite space or time has any particular empirical or theoretical backing. You'd have to pin your hopes on something like Loop Quantum Gravity, but that's highly speculative and has about as going for it currently as String Theory does (not to say it won't change), whereas time and space are still modelled as continuums in both quantum mechanics and relativity. If the current models are accepted, it's just a performative contradiction to give them credence but to arbitrarily say some of their fundamental assumptions are to be presumptively excluded from reality.
Quoting Metaphysician Undercover
This is just false. Mathematics already resolved Zeno's paradoxes so clearly adopting the mathematical models of spacetime does not create paradoxes given we know how to resolve the apparent issues Zeno saw with having them be infinitely divisible. Zeno made fundamentally mistaken assumptions about the consequences of trying to cross an infinite series, they turned out to be negated.
Quoting Metaphysician Undercover
Word game. By "reality" I meant being actual. You've already said you don't think this is possiblefor infinity, I was showing you how you were holding a hypocritical standard that only applies when other people use definitions you don't like, but you're perfectly find doing it yourself even if it's not the colloquial, "true" definition.
Quoting Metaphysician Undercover
Contingency and possibility are not the same thing. Necessary truths are also possible truths (because possibility just means truth in at least one possible world). Contingency means some modal proposition is true in some worlds and false in others. That aside, I don't see the relevancy in your questions about modality. In nearly all cases, potential is just a synonym for the term "possible". E.g. Every English speaker can readily understand "I have the potential to be a doctor", but sentences like "I have the potential to be a doctor but I cannot be a doctor" have to be disambiguated since it switches between two different types of modality (logical possibility and physical possibility), otherwise it's a flat contradiction. You can consistently say "I have the potential to be a doctor but in actuality I functionally cannot".
If you say X is a potential infinity but it cannot be actualized you are either contradicting yourself or you're switching between 2 different modalities. If it's the former, well that's not workable on pain of absurdity. If X cannot be actualized it's not a potential anything, the label doesn't fit. If it's the latter, then you're playing a shell game. You have to argue that Infinity isn't metaphysically possible, it's not contradictory so it's not inherently off the table for a consistency issue.
Quoting Metaphysician Undercover
Dude, just prior to the part you quoted I said that the canonical application of geometry was for measurement:
Quoting MindForged
My point is you are confusing the canonical use of the thing with the thing itself, and that's just an obvious mistake. The most canonical use of arithmetic is for counting things. That doesn't mean arithmetic is just about counting. the canonical application of geometry is to measure things, but measurement isn't a geometric operation, it doesn't appear in the mathematical formalism of geometry. Geometry itself is about study certain types of mathematical structures with certain types of mathematical objects (points, lines, planes and so on). Theory and application are not the same thing.
Right, you think that there could be an "end point" without an order. You really like to argue by way of contradiction, don't you?
Quoting MindForged
Yes, a race has a definite order of procedure, doesn't it? There could be no start point or end point without an order of procedure. Sorry, but contradiction just doesn't cut it. I produced a whole argument, and instead of addressing it, you dismiss it as "nonsense" by asserting a contradiction. As if you could prove someone's argument as nonsense by making a contradictory assertion.
Quoting MindForged
Of course it's begging the question, it's the definition. I suppose if I assumed that a square is an equilateral rectangle you'd accuse me of begging the question.
So, what kind of infinite thing (infinity) do you think you could observe?
Quoting MindForged
That's what I meant, "actual". If you saw my discussion with aletheist, you'd see that. For Aristotle, ideas, concepts, have actual existence, actualized by the human mind. This is the argument he uses against Platonic idealism. Ideas cannot be eternal, because only actual things can be eternal, and ideas are only given actual existence by the human mind, so they have a beginning and are therefore not eternal.
Quoting MindForged
More of the same, nonsense. The issue was whether or not we "produce principles of geometry to measure the objects which we encounter". You're just avoiding the issue by turning to a division between theory and application, as a diversion. Face the reality, even theoretical geometry is produced with the intent of measuring the objects which we encounter.
I didn't assert a contradiction. Your claim was that you have to be able to count the series in order to declare an end point, which is false. A race can be run backwards, it can be run from the middle out to either end, etc. The order is irrelevant. A race doesn't end at a random point, the end is defined when the beginning is defined.
Quoting Metaphysician Undercover
Disingenuous. The point is you cannot define it as necessarily impossible and then claim to have proven it to be the case. Induction only yields probable conclusions, you claimed the conclusion that infinity was impossible to actualiz was necessarily false, and you brought up inductive generalizations to prove that. You made a non sequitur, induction cannot give you necessary conclusions.
Quoting Metaphysician Undercover
Space and time. I observe and experience them, and our best models of them require the assumption that they are infinitely divisible.
Quoting Metaphysician Undercover
You say things like I can't just quote what was said before. The question was whether or not geometry was about measuring things. I said that was the canonical use of the discipline, but that's not what the discipline itself was about. That you're trying to claim the division being pointed out avoiding the issue is ridiculous. I brought it up because you said this:
Quoting Metaphysician Undercover
But that's absurd. We don't produce axioms in geometry to measure things, that's just a very useful feature of geometry. The common assumption was that reality could not be any other way than as a model of a Euclidean Space until Non-Euckidean geometry came along and Relativity gave credence to thinking our actual space was best modelled as a pseudo Riemannian space. Funnily, Euclid made as a base assumption in his geometry that space was an infinite plane but I'm sure you'll object to that without question begging and ignoring that Euclid's assumption contradicts your claim that geometrical axioms are about measurement. It's about studying abstract math structures of a certain kind.
Anyway, theory and application aren't the same and the idea that geometry is fundamentally about measurement is wrong. If you think otherwise, show where measurement appears in the formalism of common geometries.
Do the assumptions underlying our best mathematical models of something qualify as observations and experiences of the real object itself? Our best mathematical models of buildings for structural analysis consist of finite elements, but no one would seriously claim that we observe and experience real buildings as collections of finite elements.
In any case I would suggest that space-time is an example of observable continuity, rather than observable infinity; and since the concept of infinite divisibility has proven problematic in past discussions, I would suggest infinite magnification as an alternative. No matter how much you were to "zoom in" on space-time, you would always "see" a four-dimensional continuum, all the way down to the infinitesimal level; never a discrete point at a discrete instant.
Quoting MindForged
Exactly, and the same is true of mathematics in general. We generate formal hypotheses and work out their necessary consequences, only some of which turn out to be useful for measuring or otherwise analyzing actual phenomena. That is precisely why we make a distinction between "pure" and "applied" mathematics.
I should clear this up. I posit that those models, well evidenced as they are, are the best explanation of why we make the kinds of observations we make (e.g. not reaching any sort of discrete unit of space no matter the magnification). And since both QM and Relativity have some type of continuity to spacetime, we ought to accept this until such time as we have reason not to. I don't mean I completely see the infinite totality if a thing, but that whatever credence we give to observations in establishingnor refuting the actuality of infinities, our current observations of space and time don't seem to contradict this possibility at all.
That is fair. Can you elaborate on how QM supports the continuity of space-time? What is your interpretation of the Planck length and Planck time?
All that said, I'll give it a go. From the name, many people think QM must say space and time are quantized (discrete/, but many values in QM are continuous (position for instance), and space and time are among those values. In and of itself that doesn't mean too much, since you probably could modify the theory to use discrete values for these instead (although in practice there's no benefit to doing so).
If there's a minimum distance you start messing up a lot of other things in physics, especially as it relates to Relativity currently, such as people in different reference frames measuring different Planck length due to relativistic effects. S you'd probably have to drop Lorentz Invariance, but somewhat recent experimental observations (2011) haven't borne out high enough violations of it that would be expected if spacetime were discrete at some scale:
https://www.nature.com/articles/nature08574
Planck length and time are just measurement limitations at best. It's plausible that Planck lengths are the smallest measurable lengths, but there's no current reason to thinks it's a fundamental chunk of reality. Like these are somewhat taken arbitrarily. Inagine I was measuring things based on, I don't know, the radius of the Earth. That wouldn't mean everything else should be measured as if they were a multiple or divisor of the planet's radius. Natural units are all well and good. Some make a lot of signifance out of the Planck length but all we can really say that isn't speculation is that it sets a limit on the non-negligible effects of Quantum gravity. Our models of quantum gravity would only work down to that scale, so we'd need something more to model anything that exists at a smaller scale.
Thanks--all of that is consistent with my understanding, as well. :up:
This is what you said:
Quoting MindForged
You are saying that there can be an end point without an order of procedure. That's contradictory, "end" implies order, by definition.. Your dismissal of my argument as "nonsense" relies on the truth of this contradiction. Since it is impossible that a contradiction is true, you need to go back and address my argument properly.
Quoting MindForged
Let me see if I can understand what you're saying. You're saying that we do not produce axioms in geometry for the purpose of measuring things, we do it for some other purpose, maybe just for fun, or some arbitrary, random purpose. Then, voila, it just so happens by some random chance, that the principles of geometry prove to be very useful for measuring objects. Come on, get real.
Quoting MindForged
Are you serious? Measurement is everywhere in the formalism of geometry. 360 degrees in a circle is a measurement. Pythagorean theorem is a principle of measurement. I can't understand how you can appear to be so intelligent MindForged, but then fill your posts with such silly and even ridiculous statements.
It doesn't make sense to carry on this discussion because you just defend your position with contradictions and statements of random nonsense. Then you pretend that what I am saying doesn't make any sense in relation to your statements of contradiction and random nonsense.
.
Metaphysician Undercover tend to wander off in some direction of own makings, yet imposing own ideas on other things. :)
As far as I can tell, @Devans99 just doesn't have much familiarity with the mathematics.
You do not have much familiarity with basic logic. A number cannot be larger than any number and be a number at the same time.
None of you will address this point directly.
Cool. It's a a good thing "A number larger than any number" is not the definition of infinity in mathematics. The closest correct description of infinity that resembles what you're saying would be to say every infinite number is larger than any finite number. There's no logical error that results from infinity.
...and the difference between "real" and "actual" is...? :chin:
The real is that which is as it is regardless of what any individual mind or finite group of minds thinks about it. The actual (or existent) is that which reacts with other like things in the environment. Hence reality and actuality are not coextensive--besides real actualities (e.g., individual events), there are also real possibilities (e.g., qualities) and real conditional necessities (e.g., laws of nature) that cannot be reduced to collections of their actual instantiations.
Time is discrete.
From this follows that space must be discrete, where one unit of space equals one unit of time multiplied with the speed of light.
Assumption 2: Every physical process can be expressed mathematically.
Then it follows:
The logical framework that underpins a theory of everything must be based on natural numbers. This means, by the incompleteness theorem, that this system cannot be complete and consistent at the same time. Meaning, there are two options:
There exists phenomena in this universe that cannot be described by a theory of everything,
or
the theory of everything must produce false predictions.
Or, in other words, a theory of everything for such a universe cannot exist.
Quoting Karl
Is incorrect. Logical theories are not based on numbers, number systems are reasoned about by logical systems. Further, scientific theories use real numbers (decimal numbers) just as much as of not enormously more than natural numbers. Calculus is all about the real numbers, for example. And even further than that, your mention of Gödel's Incompleteness Theorems is probably false. Even in physics, theories aren't fully mathematical and are more like quasi-empirical. Gödel's Incompleteness Theorems apply purely to formal systems capable of expressing arithmetic, a theory of everything is usually thought of as a theory sufficient to explain all of fundamental physics (say by a theory of quantum gravity to unify GR and QM).
If we consider the length of ‘now’, it cannot be zero seconds because that gives a divide by zero error when we work out the number of ‘nows’ in any finite interval (eg 1 second gives 1/0=undefined). Also if now had zero length, it would not exist.
Could the length of now be 1/? ? Problem with that is no matter how many ‘nows’ elapse, the total time period elapsed is still infinitesimal (1/? + 1/? = 2/? etc…). So with now length 1/?, we’d be stuck forever at a single point in time no matter how many ‘nows’ elapse.
So the length of now must be greater than 1/?, IE a finite number (else time would not ‘flow’).
That makes time discrete.
Or "now"--like any other durationless instant--is simply an arbitrary human construct that marks continuous space-time, rather than a real constituent of time itself.
I don't see how you can have a 'durationless instant' surely a contradiction in terms? To say someone eats for zero seconds is to say they don't eat. A durationless instant indicates non-existence.
No, time is not an independent "thing" that changes, it is the (fourth) dimension of space-time that corresponds to spatial change. As I keep pointing out, motion through continuous space-time is a more fundamental reality than discrete positions in space or moments in time, which we arbitrarily mark for the sake of measurement and analysis.
Quoting Devans99
Exactly--time is not composed of durationless instants, and space is not composed of dimensionless points. Those are human constructs, which are very useful for certain purposes, but not real.
Right, change occurs at the present, now. And since change requires that time passes, it is very reasonable to discuss the duration of now.
Who can deny that the natural numbers are infinite. If you do then you'll have to furnish us the greatest possible natural number and I just add 1 to that and add 1 to that and so on.
The corollaries of Cantorian infinity are hard to digest. How can even numbers equal the natural numbers? It's ''obvious'' that the even numbers are half of the natural numbers, the other half being odd.
Cantor's treatment may be to blame. He used an old trick in the book viz. 1 to 1 correspondence. This works well for finite numbers (pls note this). If I pick up a stone each for every sheep I have then 3 stones = 3 sheep. 1 to 1 correspondence is a basic mathematical tool in our bag.
Cantor used the above principle of matching two infinities.
Even numbers = {0, 2, 4,...}
Natural numbers {1, 2, 3,...}
As you can see we match with the formula 2n-2
Natural number ( n )...Even number (2n-2)
1...2(1)-2...0
2...2(2)-2...2
3...2(3)-2...4
.
.
.
n...2n-2
As you can see, or as Cantor would like you to see, every even number has a matching natural number without any unmatched numbers. So, if 1 to 1 correspondence is applicable to infinities then we must conclude that the infinity of even numbers is equivalent to the infinity of natural numbers.
I understand your argument that 4 cm is twice as long as 2cm and it seems quite mad to think they both contain the ''same'' infinite number of points. However, 1 to 1 correspondence says this is true.
Can you explain your point in the context I've provided above? Thanks.
How do we describe time then? The only models of a continua I've seen have used points or line segments to model it. In both cases its valid to discuss the length of the point or line segment representing 'now'.
Quoting TheMadFool
The natural numbers are in our minds only so sure they can be infinite there. Trees can dance in our minds too. But try writing out an infinite list of natural numbers... its impossible.
I doubt infinity can exist in the real world. It's not a quantity so why should real world quantities ever take the value of infinity or 1/infinity? That leads to the suspicion that time must be discrete.
A couple of other points:
- It sounded perfectly reasonable that matter was infinitely divisible but then we found out different. Maybe the same will happen for space and time?
- A continuum has the property that its parts are each equal to its whole. That is a somewhat mad conception. Should not be part of math IMO - too illogical.
You can say time is a human construct but it represents something that does/did exist in reality. There was ‘then’ and there is ‘now’ and there is a non-zero distance between them measured in units of what we call time. They both have a length. It can’t be zero because:
- We would not be able to perceive something of zero seconds long
- Time does not flow if ‘now’ is zero length: now + now = now.
If the length of now was 1/?, time would still not flow.
Here is a different argument that time exists:
If time does not exist it has no properties.
- Time has the property that it flows from one moment to the next enabling cause and effect
- Time has the property that it slows close to the speed of light
- Time has the property that it slows in intense gravity
So time exists
We describe time as continuous--it is not composed of discrete instants or very short durations. Likewise, we describe a line as continuous--it is not composed of discrete dimensionless points or very short line segments. "Now" is an arbitrary human construct that separates what we call "the past" from what we call "the future," but time itself does not really include any such discontinuity.
Quoting Devans99
But that is not what I am saying. Time is real and continuous; a durationless instant is an arbitrary human construct.
Quoting Devans99
Measurement is a human construct. We indeed mark two instants as "then" and "now," and measure the non-zero interval of time between them by comparing it to an arbitrary unit--e.g., one second as "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom."
And what is your definition of a continuum? All the mathematical definitions I've seen use instances or short durations of some form. What do you mean by continuous?
How on earth would you ever construct a continuum? By what magic processes do you construct something with the property 'each part is equal to its whole'? If it is a challenge for us to even conceive of a workable continuum then surely that suggests that nature would incorporate such an illogical concept.
Quoting aletheist
But when we measure time intervals, we are measuring something real. The measurement is not real but the measured is.
It’s hard to imagine or explain my point sorry.
Over trillions of years of density in space, tiny particles called atoms start forming.
Atoms start grouping and knocking into each other, causing friction (heat)
Random nebula clouds from clustered atoms.
Atoms compact further into solid forms and gas forms (asteroids and stars)
One part in space is unusually warmer and bigger than the rest, but is dying out from its mass.
Heat cools down, star collapses on its self under its own gravity so quickly to the point atoms bind.
Causing a sudden shock outwards! To this day that explosion is still expanding, and it’s gravity is still strong, any moment our universe can retract so quickly that we wouldn’t even know what happened.
This one point is a singularity. Literally meaning 1. There will always be 1 something. Except in maths
From the perspective of the foton, no time has passed, does that mean fotons (and thus light) doesn't exist? time is relative, so sure it can exist for 0 seconds. to light itself there is no time, hence light is timeless, There is no time, there is only spacetime. separating the two is creating a false dichotomy that is what is causing your troubles understanding reality.
Exactly what I said before--not composed of discrete parts. If we were to "zoom in" on a continuous line, we would never "see" anything other than a continuous line.
Quoting Devans99
Who said anything about "constructing" a continuum? It is the more fundamental reality.
Quoting Devans99
Who said anything about such an alleged property?
Geometry reflects reality. If we can't construct it geometrically, its probably does not exist.
Quoting aletheist
If you sub-divide a continuum you get two continua identical to the one you started with - the parts are equal to the whole. Thats a unique and illogical property of continua.
"Construct" implies building something up from discrete constituents, which cannot be done in the case of a true continuum. I have never claimed that true continua exist, only that they are real. I have already given a specific example of a true continuum in geometry--a line.
Quoting Devans99
Who (besides you) has attributed any such property to a continuum? What I said was that if we were to "zoom in" on a continuous line, we would never "see" anything other than a continuous line.
Is this analogy apt to the issue?
Mathematical infinity is an abstract concept. It doesn't claim any physical representation, does it?
What is time anyway?
Is it real (then your argument works) or is it too an abstract concept (your argument is faulty)?
Do you know of planck time? It is, supposedly, the smallest unit (ergo discrete) of time in physics.
If we think along the lines of planck time then, since we are limited by our biology to a smallest discrete unit of time that is sensible to us, time can be thought of as discrete.
Can any human ever experience a millisecond? If you get shot in the head you die without experiencing the pain of the bullet because our nerves don't conduct information fast enough. We're dead before we know it.
That's as far as I'm willing to go on discrete time.
But nature is logical so maths can explain it because it is logical also. Actual infinity is not a logical concept so does not fits in maths or nature.
Quoting TheMadFool
Thats the question. We can measure time, does that make it real? I can't think of anything we can measure that is not real.
Quoting aletheist
If we were to try that with a real line, we'd see discrete atoms.
If we start with the common sense notion that there must be more points/intervals in a large line compared to a small line then a continua immediately violates this with ? = ?. Continua are illogical, reality is logical, hence continua don't exist in reality.
Infinity is counterintuitive, is the comment I usually encounter. That it's illogical may not be true. How many natural numbers are there?
Quoting Devans99
Well, I think time is the exception to the rule that measurement entails something is real and the converse may not be true as well.
Time, as far as I know, is ''measured'' in terms of oscillatory phenomena - pendulum swings, vibrations of the Caesium atom, etc.
What are we actually ''measuring''? Looks to me like we're just counting the rhythmic beats of the pendulum or the Caesium atom. There's nothing like an object to which we put a measuring scale and say it's x cm/inches long.
You mean a physical line, which is not the same thing. If you were to "zoom in" on space-time itself--not any physical object within space-time--you would never "see" anything other than continuous space-time.
Quoting Devans99
Continua are perfectly logical, just not in strict accordance with the logic of discrete quantity. It straightforwardly begs the question to insist that the latter is the only version of logic that corresponds to reality.
Whats logical about ? + 1 = ? (implies 1 = 0)? In fact infinity is invariant under all arithmetic operations; what's logical about something that when you change it, it does not change?
In the the mind there are an infinite number of natural numbers, but infinity itself is an illogical concept, so that conception is flawed and cannot exist in reality. The natural numbers do not exist in reality so talking about how many there are in reality makes no sense. How many exist in our mind? A potential rather than actual infinity exist as thats all we can ever visualise.
Quoting TheMadFool
We use rhythmic beats to measure time but time could exist independently of the beating mechanism and indeed enable the beating mechanism. So time enables motion rather than motion defines time.
So think of a clock and next door empty space, I'm saying I think time flows equally for both even though there is movement only in one.
If the world only had 3 dimensions, everything would be static. Time is a degree of freedom even if you don't class its as a first class dimension. So in that sense, time is as real as space.
Nature isn't 'logical', logic is just useful in understanding nature when used appropriately. Mathematics is used to give us models of nature but no model ever really captures things perfectly because nature is just too complicated. And an actual infinity is perfectly consistent no matter how many times you just claim it not to be, whether or not it can be physically instantiated.
Quoting Devans99
A real line is not composed of atoms or anything else so this is nonsense. A line is an abstract object, you have to investigate it's properties mathematically. And in basically any geometry you like a line is not finitely divisible.
Quoting Devans99
Lines are not composed of points. In a real-valued n-dimensional space, points are defined by distance from the original. But crucially, *points don't have a width* so an infinite sum of points would never give you a line. Points can be used to mark the beginning and end of a line but they cannot define them.
Further, all you're really saying is "If we assume my position that everything is discrete, then opposing views are incoherent", which, well, who cares? You're position is less credible because you're not telling anyone why they should accept your assumptions and your criticism of continuum and infinities has yet to start with accurstely representing infinity.
Quoting Devans99
That does not imply 1=0. Finite additions to an infinite number *by definition* cannot change the sum. It's the definition of infinity that it does not change by finite modifications. And saying infinity is invariant under all arithmetic operations is patently false. Take the smallest infinite number Aleph-null. Now take the power set of Aleph-null. The Cardinality has increased, it's size is now that of the continuum, Aleph-One.
That's because "space-time" is purely conceptual. If you "zoom in" on a concept defined as continuous you'll never see anything other than continuity, otherwise the concept would be contradictory.
Can you give an example of something illogical from nature/reality?
Quoting MindForged
Yes but you cannot actually infinitely divide a line - it would take forever. So thats a potential infinity rather than an actual infinity you can describe at best geometrically. It's impossible to describe actual infinity geometrically, mathematically or otherwise so/as it does not exist.
The point was logic is about abstract objects, it has nothing to do with nature. I didn't say nature was illogical, it's non-logical. Some models work better than others at explaining it, but that's the best one can do.
Quoting Devans99
This is your fundamental confusion. No one is saying you can actually do a calculation an infinite number of times in a finite period because that's a process that is defined in terms of an activity done in small periods of time. But we know mathematically that the cardinality of the continuum is such that it can be put into a one-on-one correspondence with a proper subset of itself. That's the definition of infinity, hence a line (which is defined in terms of real numbers, i.e. the continuum) is infinite.
This categorization has absolutely nothing to do with some mechanical process. Your argument would be like saying the natural numbers are finite because I cannot count to a number called infinity. This is just a misunderstanding on your part on what these words mean.
But the one-on-one correspondence procedure yields nonsense like Galileo's paradox. And the continuum does not have a cardinality... Cantor should never have made such numbers up. It's down to a deficiency in the core of set theory; the polymorphic definition of set supports two different object types: finite sets and descriptions of set. The first have a cardinality, the 2nd do not. They are different kinds of objects with different properties and need to be treated differently. Cantor tried to shoe-horn both objects into a common facade and ended up making up magic numbers for cardinality - definitely not the right approach.
Begging the question. You're saying it's nonsense because it results in infinities having the same size when you think it shouldn't, no argument given on your part. That's the very thing under debate, you cannot point to it as if it supported your point at all.
Quoting Devans99
This is ridiculous and incoherent. The semantic method of defining a set is easily proven to be coherent. I definine a set P as the set of all red objects in my room. As it happens, that set has three members, so its cardinality is 3. This is just as coherent as defining a set P explicitly with the members {My phone case, my pen, an apple}. In fact, defining sets semantically is far more efficient and is used by people every day all the time. You give no argument that semantically defined sets don't have a cardinality and by any reasonable means they do have cardinality.
There is nothing "shoehorned" here, you object to it because it's patently obvious that semantically defining sets let's one define infinite sets as easily as one defines finite sets, e.g. the set N of natural numbers is defined as having every whole number 0 and greater as a member. No one is confused by what members populate that set, and the cardinality is infinite per Dedekind and Cantor.
Space-time is part of the model in physics. What is modeled is the way things behave, the way events occur. There's nothing about the model which says that space-time is something real. In fact, to describe space as curved, is to separate space from time, which is not an aspect of the model. So saying "space is curved" is just speaking metaphorically.
No, space-time is real--it is as it is regardless of what any individual mind or finite group of minds thinks about it.
Quoting Metaphysician Undercover
On the contrary, space-time is the continuous medium (reality) within which discrete things react and discrete events occur (existence).
That's your opinion. Got any support for that opinion?
Quoting aletheist
Where does this idea of a continuous medium come from? Things themselves are the medium of separation between you and I. The medium consists only of discrete things. The continuum is purely conceptual, it's our tool for measuring the discrete things which form the medium.
Quoting MindForged
Right, it's part of the model, not what is modeled, that was my point. It's theoretical like a perfect circle is theoretical. So we could take a model of a perfect circle, and map real things against it like the orbits of the planets, and see how they vary from the perfect circle. The circle is conceptual, the orbits are real
Quoting MindForged
Actually, the model is deficient in its capacity to account for things like gravity and acceleration, so principles are added to allow for the model to be flexible. This gives the appearance that an aspect of the model, space-time is fluid, behaving. In reality the model just changes itself in an attempt to account for the things which it can't properly model. So if you happen to believe that space-time is a real entity, you'll believe that it changes according to those principles which have been added to allow for flexibility of the model.
Take my analogy of the circle for example. Suppose that when it was found out that the orbits of the planets were not real circles, we adjusted the concept of "circle" such that each planet would have a circle for its orbit, and we just dropped the idea of a perfect circle. Then we could claim that a circle is a real thing. That's what you're doing with space-time. The concept has been adjusted to allow for things like gravity, dark matter, dark energy, spatial expansion, etc., and instead of recognizing that this indicates that reality is other than the concept, you claim that this is an indication that the concept is of something real.
The issue is more contentious than that as perusing the physics stack exchange on the subject would show.
https://physics.stackexchange.com/questions/33273/is-spacetime-discrete-or-continuous
Can you (or anyone else) establish or change the properties of space-time just by thinking differently about them? Or is space-time something that we must investigate in order to ascertain what its properties are, regardless of what we think about it?
Quoting Metaphysician Undercover
Sorry, that is not what it means to be a medium.
Quoting Metaphysician Undercover
That's your opinion. Got any support for that opinion?
Yes, it's an evolving concept. It came into existence and changed when necessary. Einstein realized the concept of special relativity by thinking differently about space and time. And he realized general relativity by thinking differently about special relativity and gravity.
Quoting aletheist
A medium is what exists in the middle, between two places. What exists between you and I, as the medium, is discrete things.
What did you have in mind as a "medium"?
Quoting aletheist
Empirical evidence demonstrates to us that all which exists in the world is discrete things. These are the things we sense. We conceive of continuities and continuums, but we never ever encounter such in the empirical world. So the evidence indicates that continuities and continuums are conceptual whereas the physical world consists of discrete things.
Of course it is a concept, but the issue is whether it is "purely conceptual," as you claim. Why did it have to evolve? Because our understanding changed. Einstein had to think differently in order to resolve observed anomalies that were inconsistent with the thinking of his predecessors. Space-time always was and always will be as it is, regardless of how we think about it; our ultimate goal in studying it is to think about it correctly.
Quoting Metaphysician Undercover
A medium cannot consist of discrete things or discrete events, because it is the environment in which those things react and events occur.
Quoting Metaphysician Undercover
I agree--but continuous space-time is the real environment in which those discrete things exist.
Quoting Metaphysician Undercover
What is the warrant for holding that whatever does not exist is necessarily conceptual, rather than real but in a different mode of being? If the discrete things and events that we can and do observe behave in ways that are consistent with continuity, why would we rule out its reality?
But isn't this the mistaken belief that infinity is like other numbers, which it isn't.
If that's the way you approach the problem then zero is also nonsense because we can do 4÷2 but not 4÷0.
When we're faced with the situation 4÷0 we don't say 0 is nonsense or illogical. Rather we tell ourselves that 0 is a ''special'' number that needs, well, ''special'' treatment. We then say 4÷0 is undefined.
Similarly, when we see ? + 1 = ?, it doesn't mean 0=1. Infinity is a special number and normal arithmetic doesn't apply to it.
Quoting Devans99
I'm still not convinced. There was a theory once called the aether theory, the aether being a medium in which light travels. Given a region of space, say that in a 3cm × 3cm × 3cm box, we could ''measure'' the volume of aether as 27 cubic cm.
Only after Michelson-Morley did their experiment the aether theory was cast into the wastebasket. See? We can ''messure'' things that don't exist.
Could time be like that? Measurable but not real.
OK, let's assume that a "medium" is the environment that things exist in. Where do you get the idea that this environment does not consist of discrete things? It seems quite evident that all there is around discrete things is other discrete things. That's what "react" means, one discrete thing is interacting with another. And discrete things overlap each other in their spatial existence, one molecule overlaps another for example, and they interact in this way. There is no need to assume that there is a medium between, or around the discrete things, that's just imaginary, conceptual.
Quoting aletheist
But this is false. Relativity theory depends on the assumption that there is no such medium in which things exist. If there was such a medium, it would exist as an absolute, against which all the motions of things could be mapped, in an absolute way. But this contradicts the very premise of relativity, that there is no such absolute, that all motions are relative. So it is clearly impossible that space-time is "the real environment", or "medium", within which discrete things exist, because that assumption would blatantly contradict the premise upon which the concept of "space-time" is constructed.
Quoting aletheist
They do not though, that is the point. No discrete things, or events, behave in a way which is consistent with continuity, that's a big problem. We map those things with a conceptual structure which assumes continuity. But the models, which allow for infinity, as a feature of continuity, are unable to account for the beginnings and endings of discrete existence. That's a fundamental problem.
4÷0 does not make sense. How can you split 4 loafs into 0 parts? So we have to exclude it from arithmetic for logical reasons. Apart from that, zero behaves fine under the arithmetic operators. Zero has a small quirk but infinity not working with any arithmetic operators is in a completely different league - there is no valid logical reason to exclude infinity from arithmetic and it gives nonsense results with all of them. All other 'numbers' in maths work with the arithmetic operators or analogies of them. Infinity is unlike any other number so we can deduce it is not a number.
If you look at how infinity defined: the cardinality of the set of natural numbers, it must be a positive integer. But there is no integer X wit the property is is greater than all other integers because X+1>X.
Quoting TheMadFool
Space seems real though; its has vacuum/dark energy associated with it and fields so I'd argue they were measuring something real.
If we take a spacial dimension way, maybe we can see what it would be like without time: A 3D cube would become a 2D square. A square is a 2D object with no depth so it does not exist in 3D space (depth=0 so no volume so does not exist).
Imagine then a 3D+time object being changed to 3D-only object by taking time away. A hypercube becomes a cube. By analogy it would cease to exist in 3D+time as it has length zero in time. The object would exist for zero seconds thus not exist in our reality.
So I think time is required for things to exist. We are not living in a pure 3D world, it is a 3D world plus something that enables motion (time).
Time is like that. Clock-like objects exist, and we can measure the passing of time using them, but time itself is not an observable - there is no time operator in quantum mechanics. Time may not be real, and may be no more than a correlation phenomenon. There are certainly ideas that future quantum theories will not contain c-number parameters on which observables depend.
In some quantum gravity models, time is absent, notably in the Wheeler-DeWitt model. If such models are on the right track, the arguments for discrete and continuous time seems moot.
It seems quite evident to me that there must be a real context within which discrete things exist and react. For example, we say that they have extension in space-time.
Quoting Metaphysician Undercover
First, I am arguing for the reality of space-time, not its existence; as I have stated repeatedly, these terms are not synonymous. Second, there is no necessity for something real to be absolute--the whole point of relativity is that space-time is really relative; as I have also stated repeatedly, continuous motion through space-time is a more fundamental reality than discrete positions in space or discrete moments in time.
Quoting Metaphysician Undercover
All discrete things and events behave in a way which is consistent with the continuity of space-time. Since you deny this, further discussion would likely be a waste of time.
The "context" is our own measurement of them. We measure their existence, and say that they have "extension" but extension is just how they are represented in the model which provides the basis for measurement. How could they have a measurement without being measured? What is measured is the thing itself, and so it is said to have "extension" as extension is assigned to it through measurement.
Quoting aletheist
You said, "continuous space-time is the real environment in which those discrete things exist." I simply pointed out that this is contradictory because "space-time" is a concept that is derived from relativity theory, which has a premise that denies the possibility that discrete thing exist in such a medium.
Therefore, if you want to assign "real" or "reality" to space-time, you need to describe this reality in a way other than as the medium or environment, within which discrete things exist. I suggest we describe "space-time" as real, in the sense of being conceptual, so we provide the necessary separation between the things measured and the means of measurement. This way we avoid the contradiction involved in saying that it is the real environment, or medium in which discrete things exist. Discrete things do not exist in any medium. There is nothing to warrant that assumption.
Quoting aletheist
That's clearly false, and disproven by quantum mechanics. Fundamental particles do not behave in a way consistent with the continuity of space-time.
I see it the other way around--measurement is arbitrary; we impose it by comparing something to a discrete unit, but the underlying reality itself is continuous.
Quoting Metaphysician Undercover
I am not aware of any such premise. Relativity theory is the basis for the current scientific understanding of the space-time continuum.
Quoting Metaphysician Undercover
There is nothing to warrant the assumption that discrete things can exist and interact without a continuous medium within which to do so.
Quoting Metaphysician Undercover
I am not aware of any such evidence. I view the Planck length and Planck time as limitations on observation and measurement, not real discrete units of space-time.
I think I see what you're saying, but I believe the situation is more complex than you make it out to be. I agree that we measure in discrete units, numbers, and that there is arbitrariness in the measurement.. But I think that the idea that the thing measured is continuous is an assumption that we make which is made to support arbitrary the measurement. It is not based in evidence. The evidence, such as Zeno's paradoxes indicates that the underlying reality is discrete. Nevertheless, if the thing measured is continuous, then the discrete units of measurement may be arbitrary without any negative effect to the validity of the measurement. So we assume continuity of the underlying reality because this validates arbitrary units of measure. However, the assumption of continuity creates problems like Zeno's paradoxes which demonstrate that the underlying reality is likely not continuous. So I conclude that in reality things are discrete, but since our units of measurement are arbitrary, based on the assumption that the underlying reality of things is continuous, our arbitrary units are incommensurable with the underlying real units. Therefore we have problems with some measurements which need to be precise.
Quoting aletheist
Right, but the space-time continuum is understood by physicists as conceptual. It is not understood as a medium within which things exist. This would contradict the fundamental principle of relativity, that things do not exist within such a medium. The "space-time continuum" is the fundamental principles upon which a coordinate system can be constructed, just like "Euclidean space" is. They are each, "space-time continuum", and "Euclidean space", fundamental concepts upon which models are build. We don't say that things exist in the medium of Euclidean space, because we recognize that Euclidean space is completely conceptual. Likewise with "space-time continuum". Physicists don't say that things exist in the medium of the space-time continuum, because they recognize that this is just a conceptual structure.
Quoting aletheist
I already explained this to you. Discrete things overlap in their existence, just like molecules as discrete things, overlap one another. There is no place in the physical realm for a continuous medium, or a need to assume one. The only reason for assuming a continuous medium is to justify the easy choice of arbitrary units of measurement. If we recognized that the underlying reality consists of discrete units rather than assuming a continuity, then we could not justify the use of arbitrary units of measurement because our units would have to be commensurate with the real units. Instead, we take the easy way, assume continuity and make arbitrary units of measurement. And this produces measurement problems.
Quoting aletheist
Are you unaware of the uncertainty principle, the measurement problem, and quantum entanglement? These are evidence that fundamental particles do not behave in a way which is consistent with the continuity of space-time.
Huh? The assumption of discreteness is what creates problems like Zeno's paradoxes. As I have said before, recognizing that continuous motion through space-time is a more fundamental reality than discrete positions in space or discrete moments in time dissolves Zeno's paradoxes.
Quoting Metaphysician Undercover
I suspect that would be news to many physicists.
Quoting Metaphysician Undercover
Who said anything about the physical realm? This is conflating reality and existence again.
Quoting Metaphysician Undercover
I am not aware of any reason to interpret them as inconsistent with the continuity of space-time.
" Einstein's theory of relativity established the opinion
that traditional philosophical doctrine concerning time has been shaken
to the core through the theory of physics. However, this widely held
opinion is fundamentally wrong. The theory of relativity in physics does
not deal with what time is but deals only with how time, in the sense of
a now-sequence, can be measured. [It asks] whether there is an absolute
measurement of time, or whether all measurement is necessarily relative,
that is, conditioned.* The question of the theory of relativity could not
be discussed at all unless the supposition of time as the succession of
a sequence of nows were presupposed beforehand. If the doctrine of
time, held since Aristotle, were to become untenable, then the very
possibility of physics would be ruled out. [The fact that] physics, with
its horizon of measuring time, deals not only with irreversible events,
but also with reversible ones and that the direction of time is reversible
attests specifically to the fact that in physics time is nothing else than the
succession of a sequence of nows. "
Heidegger's point is that whether one posits a continuum or discrete units, time as a counting of nows misses the irreversible, creative basis of temporaity. From biology we know that you cannot un-fry and egg, but in physics temporal phenomena are presumed to work just as well in reverse as forward.
The discreteness in Zeno's paradoxes is not an assumption, it's a fact of the measurement system, the numbers represent discrete units. The moving items must cover the discrete units of distance, in the discrete units of time, provided by the measurement system. The measurement system gives us discrete units. However, continuity in the actual distance and time is assumed under the claim of infinite divisibility. The paradoxes are created by that assumption of continuity.
Quoting aletheist
Are you telling me that the observed behaviour of quantum particles which cannot be explained by the laws of physics, does not indicate to you that the behaviour of these discrete units is inconsistent with the continuity of space-time?
Quoting Joshs
Heidegger was quite advanced in his understanding of time. Do you see the fact revealed in this quote? How the physicists tend to deal with this problem, Einstein included, is to deny that time is anything other than a tool for measuring. By denying that time is even something which is measured, the question of "what time is", is left as inapplicable.
But 4-4=0. 0 isn't nonsense. Likewise the infinity of natural numbers isn't nonsense.
Infinity + 1 = Infinity but just as you're not allowed to divide by zero, you're not allowed to do ''normal'' arithmetic with infinity.
Do you know that our forefathers didn't know how to count beyond 2. There was 1, 2, and many (meaning an indefinite amount or infinity). I hear that some Amazonian and African tribes still count so.
In a group there were either 1 or 2 or many (infinite) people.
In their case 2+1= 3 was infinity.
So, 3+1=4 for us at present but for our ancestors it was infinity/many(3)+1=infinity/many(4)
Do you suppose we're in that stage, like our ancestors, as regards the notion of infinity?
In this context, Georg Cantor made a huge contribution by showing, through the simple logic of 1 to 1 correspondence, that some infinities were bigger than others. In fact he showed there are infinite infinities.
Cantor's achievement could be likened to us, at present, discovering that 2+1= 3 and 3+1=4 which to our forefathers was 2+1=infinity or infinity+1=infinity.
As for time, I'm still in doubt whether it's real or not.
Can you explain? Thanks.
Spacetime is modelled. Like what are you talking about? When I say it's part of the model I mean we have a set of propositions in a theory based on observable evidence which is closed under logical consequence. It's not just this background thing that is immaterial to the meat and potatoes, Spacetime is a real thing.
Quoting Metaphysician Undercover
What? General relativity gives us an incredibly accurate understanding of gravity and acceleration. Spacetime is deformed by massive objects, the model isn't changing. That's a prediction of the model and one which is true. We have to account for the deformation of space by the planet Earth in order for our satellites to orbit properly. We literally observe this warping in distance pictures of galaxies because dark matter causes gravitational lensing, the distortion of Spacetime. Distortion is behavior, spacetime isn't somehow unaffected in the way you are insisting, no argument is given, nor evidence presented. Like come on, you're not giving anything serious to overturn the overwhelmingly minority position you hold as compared to physicists on the issue.
Whats most powerful about scientific development isnt its accuracy so much as its capacity, along with the development of philosophy, to enact qualitative shifts in its perspective over time.
Lee Smolen is one of those natural scientists (Ilya Prigogine is another) who would argue that a revolution of philosophical worldview within physics is necessary to keep pace with where philosophy has already gone after Darwin . with respect to temporality. This shift in thinking would not necessitate the invalidation of any of the prior empirical results , but rather a re-envisioning of the significance of those results within a metatheoretical framework that would open up new horizons of discovery.
No, I can't explain how models like the Wheeler-DeWitt equation are come by, but I can point our a rather beautiful feature of such models.
In the Wheeler-DeWitt model (for example) time is absent, because the universe as a whole is at rest. This is because the universal wavefunction is in an eigenstate of its Hamiltonian. This in turn is necessitated because otherwise physical quantities would depend on the unphysical c-number parameter t.
Now, because the universe is in an eigenstate of its hamiltonian, it is NOT in an eigenstate of the position of the hands on the face of a clock, or the state of any clock-like object. Rather it is in a superposition of those states. Time is thus a correlation phenomenon.
Now for the bad news. It seems to me, that this model is irresistibly yields a Many-Worlds interpretation of Reality, in which different times are just special cases of different worlds, that are connected by the laws of physics.
Right, spacetime a real concept, just like unicorn is. The fact that it's extremely useful separates it from the concept of a unicorn, which is not so useful. However, this just places it more like the concept of Santa Clause, or the perfect circle, a very useful concept.
Quoting MindForged
No it doesn't it just gives us the means for modelling the effects of gravity. General relativity gives us no understanding of gravity itself, none at all. If it did, it could point us to the graviton.
Quoting MindForged
I've talked to many physicists, and your claims, that space-time is more than just a conceptual tool, is just not consistent with what these physicists tell me. You're just taking an extremely speculative metaphysical proposition, and claiming that physicists believe this proposition. Maybe some do.
Acceleration is something which physics has never been able to properly model. The basic problem is that a thing at rest, must go from zero velocity to a positive velocity at some particular time, so its acceleration would be infinite at that particular time. Relativity, in a way, sidesteps this problem by denying absolute rest, and different frames of reference can be employed. However, this just creates a convoluted relationship between potential energy and kinetic energy, and so, the fact that acceleration is not actually being properly modeled is hidden under this complexity.
We have been over all of this before. Infinite divisibility is a red herring. Continuous motion through space-time is the fundamental reality. An interval of space does not consist of infinitely many discrete positions, and an interval of time does not consist of infinitely many discrete instants.
Quoting Metaphysician Undercover
I am telling you that I am not aware of any reason to interpret them as inconsistent with the continuity of space-time.
False, spacetime is real as in it's part of the model of physical reality as understood by both QM and Relativity. It's not merely extremely useful, the fact that it can be distorted by matter is a fundamental finding in modern physics. Ideologically-motivated rejection of this, a century later, is ridiculous.
Quoting Metaphysician Undercover
Whether or not a graviton exists isn't even understood. There are numerous problems in trying to even add the graviton to the Standard Model of physics (namely the current inability to renormalize certain results at certain levels of energy). Quantum field theory can explain gravity in terms of particle exchange but this doesn't work out at Planck scales so it's obviously not workable currently. GR's understanding of gravity is that it's not a force in the usual sense (just a feature of space in the presence of matter), so no gravitons are needed to explain what we observe at large scales. Is there more to learn? Sure, but that doesn't make the current models inaccurate, just incomplete. That's a non sequitur.
Quoting Metaphysician Undercover
"I've talked to many" (likely just a few) is the textbook example of an anecdote. What I'm taking is the standard model and understanding of gravity and spacetime.
Like I suspected but could be mistaken, change is essential for the concept of time. So, the universe at rest isn't changing and ergo, no time.
What of this wavefunction? I guess it's a theoretical wavefunction and having no physical counterpart, fails to provide the tick-tock of a clock.
How does one define a wavefunction of such kind? A wave must have, if I remember correctly, at least 2 dimensions. What are they? Thanks.
The universe is completely stationary, just like the block-universe of general relativity, except this one (probably best to give it a more appropriate name - the multiverse) is much bigger.
Quoting TheMadFool
Maybe I should have just used the word "state" or "quantum state", but in realist theories, the wavefunction is a mathematical object that is in one-to-one correspondence with reality, and evolves in accordance with reality. Fortunately in this particular case it doesn't evolve.
What does "continuous motion" mean other than that the intervals of time and distances are infinitely divisible?
Quoting MindForged
Yeah sure, it's part of the model, we've been through this already. The model is conceptual, so it's real in the same sense that models are real, and concepts are real. Are you familiar with the analogy of the map and the terrain? Spacetime is part of the map.
Quoting MindForged
Right, see why I say that your claim that general relativity provides us with an understanding of gravity is false? If general relativity provided us with an understanding of gravity we wouldn't have to question whether or not a graviton exists.
Quoting MindForged
Right, so space-time is different where there is matter from where there is no matter. That's what I meant, instead of the concept accounting for the existence of gravity, the concept changes to account for gravity. That's like saying that I have a concept of the boiling of water, "water boils at 100 degrees Celsius". But for some reason water on top of a mountain boils at a different temperature. Instead of producing a new concept of the boiling of water, employing pressure as well temperature,, I just adjust the concept, to say that water boils at 100 degrees Celsius, with some exceptions for elevation. There are exceptions to the concept of space-time to allow for differences in the presence of matter, but space-time, as a concept, does not incorporate within itself, an understanding of gravity.
Infinite divisibility is an insufficient criterion for continuity. After all, the rational numbers are infinitely divisible--thus serving as the basis for Zeno's paradoxes--but no one takes them to be truly continuous. I now find magnification to be a more perspicuous illustration--no matter how much we were to "zoom in" on continuous space-time, we would only ever "see" continuous space-time--never discrete point-instants.
For various purposes, we arbitrarily mark points and instants, and then measure the distances and intervals of time between them by comparing them with arbitrarily established standard units. There is no unit of distance or time embedded in the universe itself, only velocity--the speed of light--consistent with my contention that continuous motion is the more fundamental reality.
Quoting Metaphysician Undercover
No, space-time itself is the terrain, and mathematical models of it are the map. The latter can be incorrect precisely because they purport to represent something real--that which is as it is regardless of what any individual mind or finite group of minds thinks about it.
Isn't zooming in basically the same thing as dividing? Anyway, space-time is completely conceptual, there is no such thing as zooming in on it. If you were zooming in on something, it would be an object, like a molecule or an atom or something like that. You couldn't even produce an absolute vacuum, and try to zoom in on it, whatever that would mean, because the absolute vacuum is purely conceptual. The idea of zooming in on an absolute vacuum is nonsensical, and the idea of zooming in on space-time is even more nonsensical (if that makes any sense) than the idea of zooming in on an absolute vacuum.
Quoting aletheist
Space-time is a mathematical model, just like a triangle is a mathematical model. I don't see where you think you might find this thing called space-time other than in the minds of physicists.
Conflating reality and existence, as usual. Cheers.
I have long argued that Harry Potter and Sherlock Holmes are as real as Donald Trump and Barack Obama, but not in the same sense/way, of course. I think you're saying the same here? :chin:
What J. K. Rowling and Arthur Conan Doyle have written about Harry Potter and Sherlock Holmes, respectively, is real--their books (and all derivative literature, movies, etc.) are as they are regardless of what anyone thinks about them. However, Harry Potter and Sherlock Holmes are not themselves real, they are fictional--they only are as they are because, and to the extent that, their creators (and others) have thought about them.
As I see it, Potter and Holmes have not actually influenced anyone as real agents; those effects are more properly attributed to Rowling and Doyle, along with the creators of subsequent derivative works.