Can "Infinity" ever be more than a mathematical abstraction?

[someguy1]

Yes, that's mathematically true. The interior of the set of rationals is empty.

However, when you introduce motion, an object can always move a rational distance, or rather, motion can be defined as moving half a distance, then half the remaining distance etc. In which case there are no holes, or motion ignores them. Zeno tried to show that motion over a sum of rational distances is impossible, but obviously it isn't.

Are you trying to convince me that "the rationals are continuous?" I don't get it. You misspoke yourself, I pointed out your factual error. You should just say, "Hey thanks for the clarification," and leave it at that. I can't understand why you're trying to convince me of something you already acknowledged you're wrong about.

Of course it's true that an infinite sum of rationals can converge to an irrational, but that just goes to further falsify your original point. Which you already agreed is false. I don't see this kind of behavior as conducive to rational discussion.

By the way the property of a linearly ordered set that there's always an element strictly between any other two elements is density. The rationals in their usual order are a dense set.

 

[Speakpigeon]

Yes, that's mathematically true. The interior of the set of rationals is empty.
However, when you introduce motion, an object can always move a rational distance

How does that make movement or location continuous?
You could just as well say that an object can always move an integer distance. What about it?

, or rather, motion can be defined as moving half a distance, then half the remaining distance etc.

What distance?
If this distance is a transcendental number, that half of it still is a transcendental number.
Could you clarify?

In which case there are no holes, or motion ignores them.

Well, if motion somehow "ignores them", they're still there.
And your definition based on a series of "half-distances" doesn't "ignores them".

Zeno tried to show that motion over a sum of rational distances is impossible, but obviously it isn't.

Except we totally ignore how exactly motion works and in particular whether there is anything like "a sum of rational distances" involved at all.
EB

 

[Speakpigeon]

By the way the property of a linearly ordered set that there's always an element strictly between any other two elements is density. The rationals in their usual order are a dense set.

Thanks, I wasn't sure about the English word ("compact" in French).
Apparently, there's no other way to talk intuitively of the continuum except to say that unlike the Rationals there's no hole in it.
EB

 

[iceaura]

Thanks, I wasn't sure about the English word ("compact" in French).

In English, "compact" in this context would imply a boundary, so the limits of all internal sequences were also in the set. An open segment of the real line is dense, a closed one compact as well (the endpoints are limits of internal sequences).
(That's not the definition - the normal definition is via "covers" and "subcovers" - but we are looking for physical analogy).

Apparently, there's no other way to talk intuitively of the continuum except to say that unlike the Rationals there's no hole in it.

Imo this use of the metaphor "hole" is muddling the thread topic.
The computable numbers have nothing even abstractly analogous to a physical "hole" in them. A physical universe that included nothing abstractible as a non-computable number could nevertheless include a physically continuous distance and processes of motion etc that abstract as infinite series of real numbers.

 

[Speakpigeon]

In English, "compact" in this context would imply a boundary, so the limits of all internal sequences were also in the set. An open segment of the real line is dense, a closed one compact as well (the endpoints are limits of internal sequences).

Yes, understand the difference. I remember the notion of compactness associated with the limit being "in". Problem is, I can't remember the notion used when the limit is "out". So, I guess, "dense" will do.

Imo this use of the metaphor "hole" is muddling the thread topic.
The computable numbers have nothing even abstractly analogous to a physical "hole" in them.

Yes, I understand that as well. The "hole" can be understood wrongly as somehow a concrete hole in the physical expanse considered, say, space or time, and then this interpretation doesn't make much sense. However, the hole can understood properly as the intuitive equivalent of a sequence of values that is missing its own limit. Where mainly using this notion for an irrational limit of a rational sequence, but we could apply the same idea to an even number hole in a set of uneven numbers.

The analogy is weak in the sense that a physical hole is not normally dimensionless. But I think the point was that the notion of continuum is closely associated with the topological properties of the Reals because the "hole" is meant as something missing in the Rationals relative to the Reals. The physical continuity of movement is irrelevant here.
And arf initially talked about the "rationals" being continuous, not movement.

A physical universe that included nothing abstractible as a non-computable number could nevertheless include a physically continuous distance and processes of motion etc that abstract as infinite series of real numbers.

Well, maybe not. A space that would somehow feature only rational distances may have some specific physical property as a result, although my mind comes up with a blank as to what that might possibly be.
Maybe some kind of inflation to fill in all those holes... (/irony)
EB

 

[iceaura]

A space that would somehow feature only rational distances may have some specific physical property as a result,

I can imagine a physical property that required roots, or pi, or e, for abstraction. I cannot imagine one that could involve a noncomputable number - it seems to me that nothing resulting from physical properties could be noncomputable, by definition of "physical property".
If physicists deny the reality of things they cannot measure, what are we to say of things they cannot even compute?

The physical continuity of movement is irrelevant here.

It would, if it existed, answer the thread question - by involving infinities.

 

[Write4U]

If physicists deny the reality of things they cannot measure, what are we to say of things they cannot even compute?

Below Planck scale matter becomes "cloudy". Can we divide a cloud into smaller clouds?
1/2 cloud, 1/4 cloud, 1/64 cloud? Is that computable?

 

[someguy1]

Thanks, I wasn't sure about the English word ("compact" in French).
Apparently, there's no other way to talk intuitively of the continuum except to say that unlike the Rationals there's no hole in it.
EB

Oh my that can't be right or at least I hope it isn't. "Compact" is a technical term that means something entirely different. The closed unit interval [0,1] with its endpoints is compact; and the open unit interval (0,1) without the endpoints is not compact. Compactness has a tricky technical definition so I'll omit it here, but compactness is not the same as being a continuum. For example any finite set is compact, but no finite set could be called a continuum.

Many mathematicians and philosophers have discussed the continuum, and the standard real numbers are not the only model that's been proposed. This SEP article gives some pointers. https://plato.stanford.edu/entries/continuity/

 

[Speakpigeon]

I can imagine a physical property that required roots, or pi, or e, for abstraction. I cannot imagine one that could involve a noncomputable number - it seems to me that nothing resulting from physical properties could be noncomputable, by definition of "physical property".
If physicists deny the reality of things they cannot measure, what are we to say of things they cannot even compute?

Well, we can say that it hurts even though we have zero idea how the quale of pain could be the result of physical processes.

It would, if it existed, answer the thread question - by involving infinities.

No because movement in a Pack-man universe presumably would feel continuous to its inhabitants in the sense that they wouldn't be able to feel there are any holes.
EB

 

[Speakpigeon]

Oh my that can't be right or at least I hope it isn't. "Compact" is a technical term that means something entirely different. The closed unit interval [0,1] with its endpoints is compact; and the open unit interval (0,1) without the endpoints is not compact. Compactness has a tricky technical definition so I'll omit it here, but compactness is not the same as being a continuum. For example any finite set is compact, but no finite set could be called a continuum.

Yes, I understand that. I'm trying to get the vocabulary right.

Many mathematicians and philosophers have discussed the continuum, and the standard real numbers are not the only model that's been proposed. This SEP article gives some pointers. https://plato.stanford.edu/entries/continuity/

Thanks, that should be very interesting.
EB

 

[arfa brane]

Are you trying to convince me that "the rationals are continuous?" I don't get it.

No, and I did post something that was "wrong", and I admit that.

Except, physical distances are continuous no matter how we subdivide them. I can say the distance between two arbitrary points is whatever I want it to be, a rational, an irrational, or a transcendental distance.

Well, if motion somehow "ignores them", they're still there.

So a moving object simply moves or jumps over these "holes" in the rationals?

P.S. If movement is irrelevant, what are we discussing? Or put it this way, how is it that physics can ignore the mathematical objection that the rationals are not continuous, or that some irrational numbers are not computable? In physics, you have real distance, real time and real motion, the real line is continuous with no holes or gaps.

P.P.S. I have, somewhere, an issue of SciAm from the '90s with an article on Zeno's paradox. I'll try to find it because I'm reasonably sure it has some relevant stuff.

 

[someguy1]

I can say the distance between two arbitrary points is whatever I want it to be

I can say the world sits on the back of a giant turtle. It's whatever I want it to be. Science not needed. Is that your argument here? You're certainly not talking physics OR math at this point. You're just making stuff up. And of course there is nothing wrong with making stuff up. Hollywood screenwriters make big bucks for doing exactly that. It's just not science.

 

[arfa brane]

I can say the world sits on the back of a giant turtle. It's whatever I want it to be. Science not needed. Is that your argument here?

I can say mathematics has nothing to do with physics. I can say whatever the hell I like.

I can say that it makes no sense to divide a real distance into rational numbers, because that's mathematical.

You seem to be objecting to my claim that I can say a real distance is an irrational distance. Can you demonstrate why I can't say that? Why am I making it up?

 

[someguy1]

You seem to be objecting to my claim that I can say a real distance is an irrational distance. Can you demonstrate why I can't say that? Why am I making it up?

All scientific measurement is approximate. You can't say any distance is exact at all. Do you agree or disagree with that point?

 

[arfa brane]

All scientific measurement is approximate. You can't say any distance is exact at all. Do you agree or disagree with that point?

I agree that you can't measure a distance exactly, but I disagree that you can't say a distance is irrational or transcendental (equal to π, say). There is the problem of a rational distance having "holes" in it, which doesn't seem to be a problem for physical motion.

 

[someguy1]

I agree that you can't measure a distance exactly, but I disagree that you can't say a distance is irrational or transcendental (equal to π, say). There is the problem of a rational distance having "holes" in it, which doesn't seem to be a problem for physical motion.

I'd like to understand your position better since it's so contrary to science as I understand it.

First, we both agree that no physical measurement or observation is exact. We can say a particle is at position = 1/2, say, on some 0 to 1 scale, but all we mean is that it's at 1/2 plus or minus experimental error.

So we can never measure the exact position of a particle. So now you are saying that even though we can't exactly measure it, it is still possible that the particle is at position pi - 3, say, to pick a well-known irrational between 0 and 1.

But under some interpretations of quantum mechanics, the particle has no definite position at all until it's measured. So it seems that you are making a claim that some particular interpretation of QM is the right one. But this is an issue that physicists and philososophers have argued about since the 1920's. Nobody knows what QM means.

The business with the holes in the rationals is quite secondary to this fundamental problem. You claim a particle has an exact position independent of measurement. That claim is not supported by any actual science. At best you might find some physicists claiming that we can "think about it that way" under some particular interpretation of QM.

Do you agree with what I've written so far? Here'a an overview of what people think about this issue.

https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics

So like I say, forget whether a particle can take a rational or an irrational position. The question is whether it's meaningful to say that it has any exact position at all.

 

[arfa brane]

So like I say, forget whether a particle can take a rational or an irrational position. The question is whether it's meaningful to say that it has any exact position at all.

That may well be true for particles in the frame of QM, but what about Newtonian particles?
And whether or not I can measure a distance exactly, I can still claim that it has a value which is not rational. The thing about real physical distances and the reals themselves is it doesn't matter what mathematics says about topology, measurement is all we have.

Therefore I can claim a distance is a straight line, and it has whatever value I like, such as √2. How does anyone prove I'm wrong?

 

[iceaura]

No because movement in a Pack-man universe presumably would feel continuous to its inhabitants in the sense that they wouldn't be able to feel there are any holes.

But we aren 't talking about feelings - we are talking about calculations in completely sufficient abstract mathematical perceptual structures. Movement in a Pacman universe would disagree with current QED calculations, for example - and those calculations agree fully with experiment.

However, the hole can understood properly as the intuitive equivalent of a sequence of values that is missing its own limit.

That doesn't work for the non-computables. The sequence of values has no more intuitive occupation of "distance" or "space", makes no more of a break in the line, than the limit. There is no hole with any apparent meaning in the abstraction of a physical property.

The question is whether it's meaningful to say that it has any exact position at all.

If it doesn't, the notion of "holes" in the continuity of the computable reals used for abstraction becomes utterly irrelevant instead just misleading.

 

[someguy1]

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If it doesn't, the notion of "holes" in the continuity of the computable reals used for abstraction becomes utterly irrelevant instead just misleading.

Confusing math with physics. In physics, some interpretations of QM say that a particle has no position at all until we measure it. (In other interpretations it has EVERY possible position in some universe. A dubious proposition to investigate scientifically, I'd say). In math, the computable real line is full of holes, since there are uncountably many reals and only countably many computable reals. I can't for the life of me figure out what connection you are making between these two entirely separate things.

 

[iceaura]

Confusing math with physics.

For the fourth time: look at the thread title. It's not a confusion, it's a topic.

In math, the computable real line is full of holes, since there are uncountably many reals and only countably many computable reals.

It is just as accurate to say there are no holes in the computable line, because it contains every limit and real number that can be used as an abstraction of the physical world from which the metaphor "hole" was drawn.