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

[Speakpigeon]

As I understand it, when an equation yields an infinity the equation is flawed and must be dicarded or modified.
In theoretical mathematics, infinity is used as a control mechanism to test for an equation's potential for expression in finite reality.
Aside from geometry, is Pi not used primarily in "probability theory"?

Most of the time, what you say doesn't make any sense.
You don't support your opinions with any rational argument.
You don't answer questions.
And you don't seem able to learn from experience.
EB

 

[Speakpigeon]

Why should I propose anything? I'm not trying to change math. I'm also not suggesting that all math reflects reality in the physical world. I'm not even suggesting that Tegmark is correct.
However one could argue, as Tegmark has done, that there may be an underlying mathematical basis for reality. I'm not suggesting any more than that. I'm not suggesting that we have a philosophical debate in an attempt to "prove" such a thing.
It's not something that can be tested at the moment lacking such a mathematical equation that appears up to the task.

So, what would be the problem with scientists assuming infinities are physically real?
EB

 

[Speakpigeon]

Tegmark isn't saying the house is just mathematics. He is saying that everything is made of quarks and quarks (according to some) are dimensionless and therefore at its base, everything is mathematical in nature. It's still a stretch but it's not insane.
A computer is really just dealing in 0's and 1's but that's not how we think of the computer world since we are used to dealing with Windows (or whatever OS). Trees, people, buildings are what we are used to dealing with but quarks are more fundamental.

Could someone try to explain how my subjective impression of pain or the deep blue I see could possibly happen at all in a reality that would be "mathematical in nature".
I would be prepared to accept that the physical world is mathematical in nature but then this would logically imply Dualism, with a mathematical physical world on the one side, and the world of our subjective experience on the other, and never the twain shall meet.
EB

 

[Write4U]

Write4U said:
As I understand it, when an equation yields an infinity the equation is flawed and must be discarded or modified.
In theoretical mathematics, infinity is used as a control mechanism to test for an equation's potential for expression in finite reality.

Aside from geometry, is Pi not used primarily in "probability theory"?

Speakpigeon said,
Most of the time, what you say doesn't make any sense.
You don't support your opinions with any rational argument.

Well, I'm just retelling what I heard a theoretical physicist say on tv. Do I need to justify that somehow?

You don't answer questions.

What is the question ?

And you don't seem able to learn from experience.
EB

Lol, are you trying to tell me something?

Re Pi, I asked a question which you seem not prepared to answer. What should I infer from that?

Actually, I got that information from Mario Livio, who demonstrated that Pi could be calculated from dropping a needle on a piece of paper with lines drawn on it and counting how many times the needle fell between the lines and how many time the needle crossed the lines. No circles involved. Cool eh?

 

[Write4U]

So, what would be the problem with scientists assuming infinities are physically real?
EB

How then could there be a beginning? The beginning of infinity...

 

[Seattle]

How then could there be a beginning? The beginning of infinity...

The assumption with the Big Bang/Inflation would be that infinity always existed and the density just started to decrease beginning with the Big Bang.

 

[Write4U]

Could someone try to explain how my subjective impression of pain or the deep blue I see could possibly happen at all in a reality that would be "mathematical in nature".
I would be prepared to accept that the physical world is mathematical in nature but then this would logically imply Dualism, with a mathematical physical world on the one side, and the world of our subjective experience on the other, and never the twain shall meet.
EB

That's where you are wrong, IMO.
In a mathematically physical world, you would also be mathematically physical and every subjective physical experience would be a result of the same mathematical processes which are creating the universe and everything in it.
This is what Tegmark proposes. If everything is of a mathematical nature, then we would not be aware of it, except that we and most all living things would be really good at abstract mathematics, which we and most living organisms are, each in their own area of expertise.

This is supported by Roger Penrose at the universal quantum level and Stuart Hameroff at the microtubular quantum processing in living organisms.
Quantum is a mathematical process, IMO.

 

[Seattle]

Could someone try to explain how my subjective impression of pain or the deep blue I see could possibly happen at all in a reality that would be "mathematical in nature".
I would be prepared to accept that the physical world is mathematical in nature but then this would logically imply Dualism, with a mathematical physical world on the one side, and the world of our subjective experience on the other, and never the twain shall meet.
EB

How can your subjective impression of the deep blue be possible when everything is made of quarks?

 

[someguy1]

Question; does a Hilbert space have to be infinite?

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

Does that not indicate that a Hilbertspace needs not necessarily be an infinite mathematical construct only, but is valid in both finite and infinite mathematical dimensions?

Correct, for example the familiar real line and 2D plane are Hilbert spaces. But QM is modeled in an infinite-dimensional Hilbert space.

 

[someguy1]

First, I don't see any logical reason to insist on the idea that the universe should be computable. So, where is the problem exactly?

Well I don't think the universe is computable either. But a lot of people do. That's the CUH, the computable universe hypothesis. You don't have to click around much to find plenty of people these days claiming the world's a computer. That's flat out falsified by any theory that posits that there are a continuum of points.

Tegmark elaborates the MUH into the computable universe hypothesis (CUH), which posits that all computable mathematical structures (in Gödel's sense) exist.

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

Second, maybe the universe doesn't contain non-countable infinities but contains countable ones

Perhaps. But then it can't contain a dense continuum of points, ie a complete mathematical space of points as QM is modeled in. The real numbers are complete and there are uncountably many of them. The rational numbers are countable but they're not complete; they're full of holes.

, and perhaps a countable infinity of countable infinities.

A countable union of countable sets is countable. Theorem of elementary set theory.

So, where is the problem exactly?

If the world is topologically complete it contains uncountably many points hence can't be computable. Yet many QM theorists maintain both contradictory premises. Can't be so.

Third, I'm quite sure a universe that would contain non-countable infinities would still be computable if the computation is itself based on some appropriately non-finite principle, like non-countably infinite memory etc. So, where is the problem exactly?

Of course if you change the meaning of the word computation, you're right. Computation as currently understood refers to finite-length computations with the programs being finite-length strings over an at most countably infinite alphabet. Computer scientists do study infinitary models of computation, but they have no (as yet known) relation to the real world.

I like very much the Banach-Tarski thing but it's not a logical contradiction. It's just a paradox in the sense that it goes against our most basic physical intuitions that from one material thing you can't produce two things identical to the first. So, where is the problem exactly?

If you claim the Banach-Tarski theorem (I agree, not a paradox) is physically realizable, you'll have to justify that reasoning. But if it's not physically realizable, it forces you to drop many mathematical assumptions such as completeness, uncountability, and the axiom of choice as pertaining to the real world. Then you lose the Hilbert space model of QM as being physically meaningful. It's only an idealized model. Which is my point.

 

[someguy1]

Further, you're interpretation of my phrase "Assuming infinity does not exist in any way" is obviously off. As I see it, maths is just an idea, an abstraction we use in our representations and models of the physical world. So, to me, infinity, like any mathematical entity, isn't something that can be said to exist in any way.

You just agreed that infinity DOES exist mathematically; but not physically. Right? Now in what follows you are going to go back and forth contradicting yourself in literally every sentence.

Given this, and considering that Seattle thinks infinity doesn't exist in the physical world,

A point with which you clearly agree ...

my question to Seattle is therefore to explain why infinity is so pervasive in our mathematical representations of the physical world and what alternative method he suggests we should use.

Well that's a good question, and many mathematicians and philosophers regard it as a problem to be addressed. See https://en.wikipedia.org/wiki/Constructivism_(mathematics) or https://en.wikipedia.org/wiki/Intuitionism for example.

There are physicists attempting to rebuild physics on constructivist methods, refusing to admit any mathematical objects that can't be explicitly constructed. They're not getting very far but perhaps someday they might. The point is that it IS in fact a good question as to why infinitary mathematics is so handy in the world, when there is as yet no known actual infinity in the world. That's what this thread is about, yes?

My point is that infinity is at the heart of our most basic mathematical representations of the world.

True. Yet you just agreed that there is no physical infinity. It's very difficult for me to discern what you are trying to say here since you're trying to have it both ways.

It's not just something QM physicists negligently let slip into their equations.

Well in fact it's very arguably exactly that. Infinitary reasoning turns out to be highly useful in physics and we don't know why. And in fact physicists struggle desperately to keep infinities OUT of their equations. https://en.wikipedia.org/wiki/Renormalization

It was already there a very long time ago when human beings started to use symbols to represent integers. As I see it, infinity is a direct consequence of the way the human mind works.

So it's a quality of the human mind and not our external reality. You just veered off in yet another direction. An interesting direction to be sure, but quite at odds with the rest of your post. Now you say that math is in our minds and perhaps not in the world! Now this is something that I happen to believe may well be the case. But it completely blows Tegmark out of the water and it makes physics irrelevant. If our math is only some quality of our mind, what is it that physics is studying when it uses math to understand the world? Are we just ancients looking up at the sky and making up stories about hunters and animals?

We can't do without it I don't think.

Are you sure? There are finitists and even ultra-finitists who argue otherwise, even in mathematics.

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

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

So, assuming infinity does not exist in any way in the physical world, how do you do basic arithmetic like 1 = 3 x 1/3 = 3 x 0.333... = 0.999... so that it reflects the finiteness of the world?

First, the theory of limits goes quite a bit beyond "basic arithmetic." It took 200 years after Newton to get the logic right.

But the way we do math is just to do math, without regard for its applicability to the world. If the physicists come along and say, "Hey this bit over here is useful," more power to them. But physics isn't math. Plenty of math has no known applicability to the world.

Clearly, 1 isn't equal to any finite decimal part, like, say, 0.99999999999. We normally understand "0.999..." to mean an infinity of 9's so that it makes sense to see 0.999... as equal to 1.

Yes. Mathematical infinity is assumed in standard modern math. It was not always so.

If you think infinities don't exist, the conventional way of interpreting "0.999..." has to be discarded for good.

But infinities DO exist in mathematics. Why? Because WE ASSUME IT. We make up, out of thin air, an axiom that says in effect that an infinite set exists.

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

Same for very many other arithmetic operations, like 1/7, 1/11, 1/17, 1/29 etc. So, what do you propose instead?

Rational numbers aren't very good examples of infinity in math, because their decimal expressions are the output of perfectly finite processes such as the long division algorithm taught in grade school. But what I propose is what pretty much everyone does:

* If we're mathematicians, we use math on its own terms.

* If we're physicists (or economists, or biologists, or any other specialist who uses math) we use the parts of math that we find handy and applicable and that helps us to get results. And we don't worry too much about the deeper meaning of the math.

* If we're philosophers, we ask, "How can it be that this highly abstract math stuff seems to apply so well to the world?" Eugene Wigner wrote a famous essay about this: "The Unreasonable Effectiveness of Mathematics in the Natural Sciences."

https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

And pi? The number pi is understood as having an infinity of decimal digits, without any repeating sequence ever.

Pi is also not a very good example, since it's the output of a perfectly finite algorithm. There are many well-known closed-form expressions and computer algorithms that generate the digits of pi. Even though pi is irrational, it's computable.

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

There are in fact non-computable numbers (lots of them in fact) and they do present many philosophical problems.

What do you propose to do instead?

Instead of what? I'm perfectly comfortable with infinitary mathematics. The physicists are happy using infinitary mathematics and they don't worry that there are no actual infinities in the world. The philosophers study the mystery of how abstract math can so accurately help us understand the world when math itself is built on such un-worldly concepts. Every professor has something to do and everyone's happy.

 

[Write4U]

If the world is topologically complete it contains uncountably many points hence can't be computable. Yet many QM theorists maintain both contradictory premises. Can't be so.

If I understand prof. Tong (Cambridge) the fundamental spacetime fabric exhibits both continuous wavelike (fluid) fields and quantized point particle properties and behaviors.

Is that not why we have particle duality and the uncertainty effect where we can measure the continuous wave behavior patterns and quantized value point position, but just not at the same time?

 

[someguy1]

If I understand prof. Tong (Cambridge) the fundamental spacetime fabric exhibits both continuous wavelike (fluid) fields and quantized point particle properties and behaviors.

Is that not why we have particle duality and the uncertainty effect where we can measure the continuous wave behavior patterns and quantized value point position, but just not at the same time?

I have no idea. Can you supply a link?

There's a difference between a "pointlike" particle in a MODEL of physics, and a solid, incontrovertible proof of the existence of such a thing in the real world. The former is commonplace; the second, nonexistent as far as I know.

Of course the world **EXHIBITS** pointlike and wavelike behaviors. Everybody knows this. I don't follow what thesis this supports. That our approximate models are literally true about the world? That's false. It's always been false throughout history. It's not likely to become true via pop science from celebrity Ted talkers and the like.

 

[Write4U]

I have no idea. Can you supply a link?

There's a difference between a "pointlike" particle in a MODEL of physics, and a solid, incontrovertible proof of the existence of such a thing in the real world. The former is commonplace; the second, nonexistent as far as I know.

Of course the world **EXHIBITS** pointlike and wavelike behaviors. Everybody knows this. I don't follow what thesis this supports. That our approximate models are literally true about the world? That's false. It's always been false throughout history. It's not likely to become true via pop science from celebrity Ted talkers and the like.

Yes, I may have given a very poor review of that presentation. One must see the lecture to appreciate the full scope of it and the latest areas of investigation. The lecture is designed for the layman and starts simply but leads one very gently into more exotic and deeper properties of spacetime and the formation of the fundamental building blocks (quanta) of energy and matter.

Apparently these lectures have been a Cambridge tradition for 200 years, starting during Faraday's tenure. I found it worth the time and will view it several more times to get the full scope and proper understanding of the state of theoretical science in cosmology.

 

[Speakpigeon]

How can your subjective impression of the deep blue be possible when everything is made of quarks?

Yes. Where could the blue and the pain come out of if the world was only dimensionless quarks?
EB

 

[Speakpigeon]

How then could there be a beginning? The beginning of infinity...

Please explain what's the logical problem if there's no beginning.
EB

 

[Seattle]

Yes. Where could the blue and the pain come out of if the world was only dimensionless quarks?
EB

How can my computer screen display text when everything is fundamentally just a binary set of zeros and ones?

 

[Speakpigeon]

Well I don't think the universe is computable either. But a lot of people do. That's the CUH, the computable universe hypothesis. You don't have to click around much to find plenty of people these days claiming the world's a computer. That's flat out falsified by any theory that posits that there are a continuum of points.
Tegmark elaborates the MUH into the computable universe hypothesis (CUH), which posits that all computable mathematical structures (in Gödel's sense) exist.
https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis
Perhaps. But then it can't contain a dense continuum of points, ie a complete mathematical space of points as QM is modeled in. The real numbers are complete and there are uncountably many of them. The rational numbers are countable but they're not complete; they're full of holes.
A countable union of countable sets is countable. Theorem of elementary set theory.
If the world is topologically complete it contains uncountably many points hence can't be computable. Yet many QM theorists maintain both contradictory premises. Can't be so.
Of course if you change the meaning of the word computation, you're right. Computation as currently understood refers to finite-length computations with the programs being finite-length strings over an at most countably infinite alphabet. Computer scientists do study infinitary models of computation, but they have no (as yet known) relation to the real world.

So, we could have a computable universe with actual countable infinities.
We could have a computable universe with actual uncountable infinities if the computation is itself based on some appropriately non-finite principle.
We don't know because we don't yet understand how computation could be understood as based on non-finite principle.
So, basically, we don't yet know enough to conclude either way.
So, where would be the problem with scientists assuming infinities, including uncountable ones?
EB

 

[Speakpigeon]

If you claim the Banach-Tarski theorem (I agree, not a paradox) is physically realizable, you'll have to justify that reasoning. But if it's not physically realizable, it forces you to drop many mathematical assumptions such as completeness, uncountability, and the axiom of choice as pertaining to the real world. Then you lose the Hilbert space model of QM as being physically meaningful. It's only an idealized model. Which is my point.

I think Banach-Tarski reflects our sense of the "materiality" of the physical world, i.e. we cannot multiply bread, money, energy, atoms etc. Whether that's really true that we can't I don't know, and I don't think anybody does. We essentially assume it's true based on our long but limited experience of the physical world. Time will tell.
However, I can think of at least one properly physical, although not material, thing that maybe does the Banach-Tarski trick. Space. This would be exemplified by the period of cosmological inflation early on in the history of the universe and by the notion of the curvature of space, where we can conceive of areas of space expanding to forme bubbles. I don't think we know whether such things really exist but it seems science is seriously considering it. Combine this with the failure of science so far to prove that space is not continuous and you get something like the physical possibility of Banach-Tarski. So no material Banach-Tarski, in accordance with our intuition, but maybe physical Banach-Tarski, in contradiction with what some people seem to say.
EB

 

[Speakpigeon]

You just agreed that infinity DOES exist mathematically; but not physically. Right? Now in what follows you are going to go back and forth contradicting yourself in literally every sentence.

No.
I'm saying that the mathematical concept of infinity isn't the kind of thing that could be said to exist. It's just an idea we have.
I'm agnostic as to whether there are actual infinities in the physical world.

Still, I don't see what would be the problem in having actual infinities.
In particular, there's no logical problem that I can see.
So, we have to get back to what science knows and there we have to accept science doesn't. So, let scientists assume whatever they feel is best. At least until such a time as you can prove they're wrong.
EB