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

[Speakpigeon]

Is Planck length not the accepted minimum measurable length? Anything smaller has no longer any physical attributes and/or relationship to a defined length of 1 meter at all.
It just becomes a mathematical abstraction as suggested by the OP.

Hey, it's your first good point!!! Congratulation!
Still, yes we can't measure below the Planck length but that wouldn't stop an infinity of points from measuring one metre.
EB

 

[Speakpigeon]

QuarkHead, Can you please provide a reference to a peer-reviewed experiment or observation that shows that there are mathematical points in physical space that are in order-preserving bijective correspondence to the mathematical unit interval or real line?

Good point.

And secondly, let me ask you this. If what you say is true, wouldn't the axiom of choice and the continuum hypothesis then be subject to physical experiment? Not to mention simpler axioms such as Powerset or Replacement. Have any such experiments been done? Or even speculatively proposed? Please provide references.

Good point, too.

Where I'm coming from is this:
* Math has this wild crazy symbolic formalization of infinity and infinite sets. If you studied it, and then saw step-by-step how the real numbers are constructed out of the crazy axioms of infinitary set theory, you would cease to believe in the real numbers as physically meaningful.

The set of Rational numbers is apparently quite straightforward to construct and understand and there's already an infinity of rationals between 0 and 1, or indeed between any two rationals you may care to pick, so it's perhaps not very convincing to point a finger at Cantor's "infinitary" dream.

* Rather, you would see them as a highly idealized, abstract model; which we can use to draw inference, and then apply productively to the real world. But nobody really understands why we can do that!

I would have thought evolution is pretty much the best explanation.
And then, presumably our mathematical intuitions do have their limits, anyway. I'm personally highly dubious about the highwire act of mathematicians in some area. Well, in this one area!

This conversation goes back to Euclid. He said a point is without breadth or width, and a line is without breadth, and two lines intersect in a point, and based on this abstract, idealized word game, we put men on the moon.

Before him there was the paradoxes about the divisibility of space and time devised by Greek philosopher Zeno of Elea (c. 490–430 BC).

And yet ... there are no points, there are no lines, there are no circles, there is no perfection to be found in the real world.

Some dude said the world was perfect as it is. I think I would agree with that at least until someone can put a better one on the table.
It's not a question of "perfection" at all. You will have noticed that there's no red colour either in the physical world and I don't think you could claim that red is perfect. Rather, our mathematical intuitions work, at least to some considerable extent, for the same reason that having a red colour to represent things does work to a considerable extent.

So I do believe you are conflating the physical world, on the one hand; with the abstract, idealized mathematical world on the other.

We're all guilty of the same charge in that we tend to take our own perceptions of the world as the world itself. Can't blame us.

It's as if a Greek say, "See, the real world contains points and lines." No, Euclid only said, "Let us play a game with words." And the game is USEFUL. But it is not necessarily ontologically meaningful.

Rest assured that many people agree with you here, at least with the idea. The wording is crap. It's precisely because our intuitions and models are ontologically meaningful to us that we believe they're the real world. So, you must have meant "ontologically true" rather than "ontologically meaningful".

That's my viewpoint. Do you feel that physicists believe differently? That their models are reality? Some do, I'm sure. Others, Newton among them, recognized the distinction.

It takes a village.
EB

 

[arfa brane]

The question: "Can infinity ever be more than a mathematical abstraction?", is also asking "what kind of more than?".
By which I mean, what kind of thing can be more than an abstraction? What does "more than" mean, here?

The Euclidean plane is infinite in extent, but you can have more than one of them in three dimensions, in fact you can have an infinity of planes all intersecting with the same line.

 

[Write4U]

Still, yes we can't measure below the Planck length but that wouldn't stop an infinity of points from measuring one metre.

Can you divide a point into a infinite number of smaller points? A point is a location and has no dimensional or numerical value. Therefore it cannot be used as a divisor or as a measurement of any kind.

The entire concept of division into an infinite number of non-dimensional points is fundamentally wrong. Pick a coordinate location for a point is all you can do.

 

[arfa brane]

Can you divide a point into a infinite number of smaller points?

No. That's something you should know is impossible.

 

[Write4U]

No. That's something you should know is impossible.

Of course I know that. The question was posed to illustrate the conceptual error in using points as a numerical value instead of a location. If you can't divide a point into an infinite number of smaller points, does a point have any dimension. IMO, no.

Can you place an infinite number of points between two points? IMO, no. You can never get to zero distance between two points, which poses a real paradox.

A point is an exact position or location on a plane surface. It is important to understand that a point is not a thing, but a place. We indicate the position of a point by placing a dot with a pencil. This dot may have a diameter of, say, 0.2mm, but a point has no size. No matter how far you zoomed in, it would still have no width. Since a point is a place, not a thing, it has no dimensions.

https://www.mathopenref.com/point.html

Thus it has no mathematical value, other than as a pure abstract 2D coordinate.

Now consider a point location in a moving wavelike spacetime coordinate.....

 

[Speakpigeon]

The question: "Can infinity ever be more than a mathematical abstraction?", is also asking "what kind of more than?".
By which I mean, what kind of thing can be more than an abstraction? What does "more than" mean, here?

The Euclidean plane is infinite in extent, but you can have more than one of them in three dimensions, in fact you can have an infinity of planes all intersecting with the same line.

No. The question is just whether there may be actual, concrete infinities, infinities that would somehow exist for real, for example in the physical world, rather than just as a mathematical abstraction.
EB

 

[Speakpigeon]

Can you divide a point into a infinite number of smaller points? A point is a location and has no dimensional or numerical value. Therefore it cannot be used as a divisor or as a measurement of any kind.

That never was the idea. Please provide actual quotes of mathematicians or physicists talking of dividing "points". That's just nonsense, something again you've failed to understand.
So let me repeat what I already said: the idea of infinity is that in a continuum, you can always find another point in between two points you would have picked first. And you can do that again and again, without any limit. There's always another point between any two points.
This clearly implies that you can divide a continuum, certainly not that you can divide points.

The entire concept of division into an infinite number of non-dimensional points is fundamentally wrong. Pick a coordinate location for a point is all you can do.

This is plain stupid. You should learn first what people mean by infinity before making inane claims.
EB

 

[Write4U]

There's always another point between any two points.

No there is not. If so you would not have a continuum. A continuum is an indivisible Wholeness without dimensional spaces. It cannot be broken, lest it becomes discontinuous.

It is an abstract concept because it does not, cannot exist in a quantum reality, which is by definition discontinuous. You cannot have a quantum event between a quantum event.

 

[arfa brane]

The question is just whether there may be actual, concrete infinities, infinities that would somehow exist for real, for example in the physical world, rather than just as a mathematical abstraction.

Mathematical infinities are concrete. The question should be: "Is there a physical infinity?", perhaps.

But we don't know that the universe is finite, only that we can see just a part of it

 

[arfa brane]

A continuum is an indivisible Wholeness without dimensional spaces. It cannot be broken, lest it becomes discontinuous.

No that's incorrect. A continuum can always be divided, and the idea doesn't make sense unless you also have distances between points.
The rationals are continuous because, given any two rational numbers, you can always find a third rational number between them. Moreover, since there is no largest natural number, there is no smallest rational number.

 

[Write4U]

Thank you all for being so patient. I like to test concepts, especially when they seem to have inherent paradoxes in their conceptual and applied functions.

 

[Speakpigeon]

No there is not. If so you would not have a continuum. A continuum is an indivisible Wholeness without dimensional spaces. It cannot be broken, lest it becomes discontinuous..

Broken?!
Whoa! That's again plain idiotic.
You're just making it up as you go. You come up with irrelevant terminology all your own and then make a fuss about your own invented issues.
Please quote any mathematician, physicist or philosopher claiming a continuum can be "broken".

It is an abstract concept because it does not, cannot exist in a quantum reality, which is by definition discontinuous. You cannot have a quantum event between a quantum event.

Yes, I understand you point. Again, it's the only good point you've ever made in this thread. The real point, though, is that you just don't know that the quantisation of energy implies the quantisation of space and time. You don't know because nobody knows. Come back when you do.
EB

 

[Speakpigeon]

Mathematical infinities are concrete. The question should be: "Is there a physical infinity?", perhaps.

Same thing.

But we don't know that the universe is finite, only that we can see just a part of it

I agree.
EB

 

[Speakpigeon]

No that's incorrect. A continuum can always be divided, and the idea doesn't make sense unless you also have distances between points.
The rationals are continuous because, given any two rational numbers, you can always find a third rational number between them. Moreover, since there is no largest natural number, there is no smallest rational number.

Personally, I don't see the Rationals as continuous since we know of irrational numbers that sort of come in between them, which would seem to me to be the very definition of a discontinuity.
Still, it's true that we can always find another Rational in between any two Rationals, like for the Reals.
However, usually, the idea of the continuum is associated with that of the Reals, I think for the reason I just gave.
EB

 

[Speakpigeon]

Thank you all for being so patient. I like to test concepts, especially when they seem to have inherent paradoxes in their conceptual and applied functions.

And that's perfectly legitimate.
What isn't, though, is to systematically misrepresent like you do the concepts you're discussing.
And debating those things requires a modicum of civility that you lack completely.
I've already explained many times why in this thread, so there's no need to repeat myself. You just have no excuse.
EB

 

[Write4U]

And that's perfectly legitimate.
What isn't, though, is to systematically misrepresent like you do the concepts you're discussing.
And debating those things requires a modicum of civility that you lack completely.
I've already explained many times why in this thread, so there's no need to repeat myself. You just have no excuse.
EB

You should be familiar with this quote

Descartes goes so far as to assert that he will write on this topic "as if no one had written on these matters before".

 

[Speakpigeon]

You should be familiar with this quote

Sure, I myself doubt everything except that I am, and again, it's perfectly legitimate.
So, do I need to repeat myself again and again?
EB

 

[someguy1]

The rationals are continuous because, given any two rational numbers, you can always find a third rational number between them. .

Not true. The rationals are full of holes. Many Cauchy sequences of rationals do not converge in the rationals. The rationals are not complete and the real numbers are. If I had a number line with only rationals on it, the intermediate value theorem would be false because I can draw a line through x = sqrt(2), say, and that does not intersect your rational line. That's not continuous for anyone's intuitive understanding of continuity.

 

[arfa brane]

Not true. The rationals are full of holes.

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.