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

[someguy1]

I understand that and there's nothing disingenuous about my question.

And I didn't accuse him of anything therefore I didn't accuse him of denying mathematical infinity.
I asked a question.
If you think you can reply for Seattle, then go for it and answer my question.
EB

The answer to your question is that the axiom of infinity gives us an infinite set of natural numbers 0, 1, 2, 3, 4, ... and then we build on that to formalize the rest of modern math. There are infinitely many decimal digits in pi, one digit (to the right of the decimal point for simplicity) for each natural number. But you already know this, why do you need me to explain it?

Your reply WAS disingenuous, because Seattle did acknowledge that there is infinity in abstract math; and you pretended he said otherwise.

Seattle said: "Can "Infinity" ever be more than a mathematical abstraction?"

And you replied with: "Assuming infinity does not exist in any way ..."

That's disingenuous. Seattle acknowledges that infinity does exist in mathematics; and you pretend he denied that.

 

[QuarkHead]

Hilbert spaces, which are generalizations of the "usual" notion of vector spaces to the complex field, are fundamental in Quantum Physics. In this context they are infinte-dimensional by construction.

 

[someguy1]

Hilbert spaces, which are generalizations of the "usual" notion of vector spaces to the complex field, are fundamental in Quantum Physics. In this context they are infinte-dimensional by construction.

Isn't this a big philosophical problem for QM? Hilbert space is complete. That means that there are no "holes," just as there are no holes in the real numbers. It's a theorem that a complete dense linear continuum must be uncountable. But there are only countably many computable real numbers (because there are only countably many Turing machines). So the Hilbert space formulation of QM immediately contradicts any claim that the universe is a computation or a Turing machine or a computer or an algorithm or any equivalent formulation.

If there is a single noncomputable real number physically instantiated in the world, then it represents an infinite, incompressible amount of information. (For example pi has an infinite decimal expansion but represents only a finite quantity of information because there are numerous finite closed-form expressions that represent pi. So pi is computable. But almost all -- ie all but a set of measure zero -- real numbers are noncomputable).

That's why, absent direct proof of some actually infinite set or quantity in the physical universe, the Hilbert space formulation of QM must be a model and not actual reality.

This is my understanding of the ontological status of Hilbert space. It MUST be a model and not the actual reality, unless you have proof that a physical infinity exists.

If you have facts (not idle speculations or Tegmarkian trolling) to the contrary, I'm most eager to hear them.

Remember, a complete dense linear continuum must be uncountable; and any uncountable set must necessarily contain many noncomputable elements.

ps -- At the risk of making this post too long, let me just remind people what an infinite-dimensional linear space looks like. From calculus you remember what a continuous function is. Now imagine the set of ALL possible continuous functions from the reals to the reals. This is a vector space. Why? Well we can add two functions pointwise to get a third; and we can multiply any function pointwise by a real scalar to get another; and the set of all continuous functions along with pointwise addition and scalar multiplication satisfies the definition of a vector space.

Now, there is a theorem that every vector space has a basis. And this theorem can only be proved with the Axiom of Choice; and furthermore, this theorem is actually equivalent to the Axiom of Choice.

So the set of all continuous functions over the reals is a vector space with a basis. What does such a basis look like? It consists of a set of continuous functions with the property that every conceivable continuous function may be written as a finite linear combination of these functions.

And what is the cardinality of this basis? It must be infinite (proof omitted for the time being). That's what an infinite-dimensional vector space looks like.

So ... tell me this. Is the axiom of choice provable in the physical world? Because if it is, so is the Banach-Tarski paradox.

The argument that Hilbert space is anything BUT an idealized, non-physical model, falls apart the moment you think about it. Physicists don't care because it gives them the right answer. But nobody claims Hilbert space physically exists; or if they do, they haven't really thought about the actual math.

 

[Write4U]

So ... tell me this. Is the axiom of choice provable in the physical world? Because if it is, so is the Banach-Tarski paradox.

Choice by whom?
And why did you decide to dismiss Tegmark's perspective? From wiki;

According to the book Our Mathematical Universe [clarification needed], the shape of the global universe can be explained with three categories:
1. Finite or infinite
2. Flat (no curvature), open (negative curvature), or closed (positive curvature)
3..Connectivity, how the universe is put together, i.e., simply connected space or multiply connected.

There are certain logical connections among these properties. For example, a universe with positive curvature is necessarily finite.[2] Although it is usually assumed in the literature that a flat or negatively curved universe is infinite, this need not be the case if the topology is not the trivial one.

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

This seems perfectly adequate for serious consideration in any discussion of the properties of space and IMO, does not warrant the label of trolling.

Question: Is a torus not a shape that answers to both positive and negative curvature?

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

 

[someguy1]

Choice by whom?

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

The rest of your post didn't make sense to me. I said nothing about curvature or anything else.

I said that the Hilbert space formulation of QM must necessarily be an idealization and not an actual reality, absent proof of the existence of physical infinities. And I gave what I believe is a pretty cogent and convincing argument, which you completely ignored.

Shape of the universe? Why are you addressing that remark to me? I said nothing about the shape of the universe. I'm really kind of curious. You and I might not always agree but you seem like a reasonable fellow. Why do you feel that I said something about the shape of the universe? In anything I've ever written on this site, let alone the post you quoted.

Note that the CUH is directly contradicted by the Hilbert space formulation of QM. If you didn't like my calling Tegmark a troll I'm happy to retract that remark. I just want to see an actual scientific reference to someone claiming Hilbert space is an actual thing in the physical universe. Which is cannot be, for the detailed reasons I outlined.

 

[Write4U]

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

The rest of your post didn't make sense to me. I said nothing about curvature or anything else.

I said that the Hilbert space formulation of QM must necessarily be an idealization and not an actual reality, absent proof of the existence of physical infinities. And I gave what I believe is a pretty cogent and convincing argument, which you completely ignored.

I accept that you know what you're talking about. I am just learning and am not qualfied to comment.

Shape of the universe? Why are you addressing that remark to me? I said nothing about the shape of the universe. I'm really kind of curious. You and I might not always agree but you seem like a reasonable fellow. Why do you feel that I said something about the shape of the universe? In anything I've ever written on this site, let alone the post you quoted.

I was not attributing that to you, but in a discussion of infinity it seems to me shape is of primary importance, no?

Note that the CUH is directly contradicted by the Hilbert space formulation of QM. If you didn't like my calling Tegmark a troll I'm happy to retract that remark. I just want to see an actual scientific reference to someone claiming Hilbert space is an actual thing in the physical universe. Which is cannot be, for the detailed reasons I outlined.

Ok, thanks for responding and the link. I'm absorbing.....

 

[Seattle]

Infinity implies a flat or open Universe. If there is no infinity then the shape of it doesn't matter.

All measurements of our Universe indicates a flat Universe but in a very large, but not infinite Universe there may not be enough curvature that we are able to measure. There may also be physical laws that we aren't aware of that would allow for flat but not infinite.

 

[Write4U]

Question; does a Hilbert space have to be infinite?

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions.

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?
I like this apparent aspect of a Hilbert space.

The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space.

 

[Write4U]

Infinity implies a flat or open Universe. If there is no infinity then the shape of it doesn't matter.

All measurements of our Universe indicates a flat Universe but in a very large, but not infinite Universe there may not be enough curvature that we are able to measure. There may also be physical laws that we aren't aware of that would allow for flat but not infinite.

Such as lack of physical matter? The universe is still expanding, no?.....

 

[someguy1]

I accept that you know what you're talking about. I am just learning and am not qualfied to comment. I was not attributing that to you, but in a discussion of infinity it seems to me shape is of primary importance, no?

QuarkHead noted that QM is sometimes formulated in terms of Hilbert space, which is an infinite-dimensional vector space. I pointed out that this must be an abstract mathematical model and not an actual part of the physical universe. That's as far as my post went. Hilbert space is a highly idealized mathematical structure. It's in no way a real thing in the world even if the physicists don't spend any time concerning themselves with that fact.

I was not making any point beyond that. Hilbert space is not an example of a physical infinity. It's an example of a mathematically infinite structure (a complete, infinite-dimensional vector space that admits and inner product) that's useful in physics. Just as the real numbers are useful in physics, without there being any [as yet known] physical representation of the real numbers in the world.

Ok, thanks for responding and the link. I'm absorbing.....

Don't worry about the technical parts esp. the axiom of choice, which is "inside baseball" in set theory. The point is that infinite-dimensional function spaces such as Hilbert spaces are mathematical abstractions with no proven existence in reality. If a mathematical structure is useful for physicists that's all well and good, but that fact doesn't confer any ontological status. This is the same point Tegmark seems to miss. I can measure my house with a ruler but my house is not a ruler.

 

[Write4U]

I can measure my house with a ruler but my house is not a ruler.

I understand what you are saying, but I am just having trouble with that.

If I know the dimensions of my house, why could I not use it as a ruler? A ruler is a mathematical construct which is divided into a set of (symbolic) mathematical measurements.
An eight foot wall is a mathematical construct of 8'. If I divide it in half, I can measure 4', etc.

Why is that against the rules?

 

[Seattle]

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.

 

[Write4U]

Just as the real numbers are useful in physics, without there being any [as yet known] physical representation of the real numbers in the world.

What about the Fibonacci Sequence? Is that not a representation of a real mathematical sequence of values, which can be theoretically applied to infinitely small or large values.

I realize that human numbers are just convenient arbitrary symbols for the real values, but does that matter as long as our symbols can lead us to an understanding of the power of the real values?
This is why I seldom use the word "number" in relation to mathematical values. A number is not a real value in and of itself, it is a symbolic representation of a real value.

This removes the human aspect from the equation. IMO.

 

[someguy1]

If I know the dimensions of my house, why could I not use it as a ruler?

You're right it's a lame metaphor. But if you would humor me and take it a little less literally, you might get a glimpse of the underlying point. That the thing we use to understand another thing, is not necessarily the same thing as the thing we're trying to understand. It's a tool, a model, a conceptual framework. It's not the thing itself. If math describes the world it doesn't necessarily follow that the world is math. Moreover even the premise could be argued. Does math really describe the world? Or is it just the thing human brains use to understand the world? Just as a caterpillar on a leaf on a tree in the forest has a theory. It has a nervous system and it likes to live and it likes to eat. It's not that much different than us. There's simply no reason to feel so certain that "we" are the crown of creation. We're just the next level up. Our physics is just a story, not much different than when the ancients looked up at the sky and told stories about the hunters and animals. I'm not necessarily arguing that point of view but I am pointing out that it is a legitimate point of view. So even the premise "the world is described by math" is arguable. And even if we do grant it, the conclusion doesn't necessarily follow. That's my argument in a nutshell.

But I had our conversation on my mind, something you said, that you're just learning. We all are. I came to my present point of view somewhat recently. I didn't used to know anything about the math of QM. I thought it was hopelessly beyond me. I happened to study some functional analysis a few years ago and it turns out that Hilbert space and the math of QM are all very straightforward from a mathematical point of view. It was a revelation. I don't understand the physics but I understand the nature of the mathematical structure they're using.

Now I can assure you that Hilbert space is the most abstract mathematical object you can imagine. It's shot through and through with infinitary reasoning that can and has been criticized on philosophical grounds. I personally have an extremely difficult time imagining that any of it's physically real. I assume that just as they have done through history, physicists are just using what they need and not worrying about the mathematical fine points. If a theory gives the right answer within enough decimal points we use the theory.

Apparently a lot of people seem to think that Hilbert space is a literally a real thing. I'm wondering if when I say, "Well of course Hilbert space isn't physically real," people think I'm misinformed or saying something contrary to what they believe or have been told.

I wonder if perhaps the public has gotten the wrong idea. I wonder if people think physics is about ultimate truth as opposed to simply crafting the best rational theory we can, always searching and tweaking and waiting for the next revolution. Science is a human activity. The very idea that science is literally about truth is false. It's metaphysics. It's ... to use a word going around ... scientism. Not science.

Well these are a few things on my mind tonight, thanks for reading.

 

[someguy1]

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.

I appreciate your saying "according to some." Nobody knows if a quark is a mathematical point. Maybe it is. Maybe it's a little wiggly thing. Nobody knows. But if someone is claiming that the world is continuous in the sense of Euclidean space, points addressed by mathematical real numbers; well the consequences of that are very strange. The world would be subject to the deepest anomalies and mysteries of set theory. I just don't see it. You'd be making claims about the nature of the continuum. That's philosophy.

And it seems terribly arrogant. To assume we are figuring out how things "really" are. Any such claim by definition is metaphysics and not physics. It's not science.

My take on Tegmark didn't change after I read half his paper and got bored. I re-read the Scott Aaronson review and the Peter Woit review and they're two people whose opinion I respect, and they say, "Interesting but vacuous." So if I were a better person I'd read his book, which I gather is a lot more entertaining than the paper. But I've pretty much made up my mind about this cart-before-horse thing. If the world is math, there are a lot of deep questions nobody's asking.

 

[Write4U]

The world would be subject to the deepest anomalies and mysteries of set theory

If the world is math, there are a lot of deep questions nobody's asking.

I see what you mean, but consider the alternative.
If things in the universe did not work in accordance to some orderly sets of rules, how could we even begin to measure anything or codify its behavior?

IMO that would create a much more complex and undefinable world than a set of inherent behavioral rules which determine how things must work in accordance to their values, as evidenced by the observable regularities (patterns) which are abundant throughout the universe and which lend themselves to analysis with our symbolic mathematical understanding of those regularities.

A daisy employs the same mathematical rule in its petal growth as a spiral galaxy does in its spiral growth. Fibonacci discovered this sequential growth pattern (also known as the Golden Ratio) and codified it. Moreover, many theoretical cosmologists declare they get a sense of "discovery of prior existing conditions", rather than having created a new way of looking at natural phenomena. No hubris there.....

IMO, a notable fact is that our mathematics allow us to make predictions at those metaphysical levels, such as the existence of the Higgs field, which was purely speculative until our applied mathematics actually produced the Higgs boson, a necessary potential for the accretion of mass in fundamental particles.

As I understand it, the Cern collider has been improved since then, and it is expected to yield several more fundamental particles which are projected to form additional potential fields, which produce the fundamental building blocks of matter itself.
The conversion engine of pure energy into matter.

IMO, if it is not magical, it must be a form of hierarchical mathematical orders, which exist as "enfolded" potentials, and ultimately become "unfolded" (expressed) in reality as we experience it. (Bohmian Mechanics)

 

[Speakpigeon]

The answer to your question is that the axiom of infinity gives us an infinite set of natural numbers 0, 1, 2, 3, 4, ... and then we build on that to formalize the rest of modern math. There are infinitely many decimal digits in pi, one digit (to the right of the decimal point for simplicity) for each natural number. But you already know this, why do you need me to explain it?
Your reply WAS disingenuous, because Seattle did acknowledge that there is infinity in abstract math; and you pretended he said otherwise.
Seattle said: "Can "Infinity" ever be more than a mathematical abstraction?"
And you replied with: "Assuming infinity does not exist in any way ..."
That's disingenuous. Seattle acknowledges that infinity does exist in mathematics; and you pretend he denied that.

No, I didn't. I repeat my post here so you can check again:

Assuming infinity does not exist in any way, how do you do basic arithmetic like 1 = 3 x 1/3 = 3 x 0.333... = 0.999...?
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. If you think infinities don't exist, the conventional way of interpreting "0.999..." has to be discarded for good. Same for very many other arithmetic operations, like 1/7, 1/11, 1/17, 1/29 etc. So, what do you propose instead?
And pi? The number pi is understood as having an infinity of decimal digits, without any repeating sequence ever. What do you propose to do instead?
Where do I "pretend he denied"?​

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.
Given this, and considering that Seattle thinks infinity doesn't exist in the physical world, 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.
My point is that infinity is at the heart of our most basic mathematical representations of the world. It's not just something QM physicists negligently let slip into their equations. 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. We can't do without it I don't think.
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?
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. If you think infinities don't exist, the conventional way of interpreting "0.999..." has to be discarded for good. Same for very many other arithmetic operations, like 1/7, 1/11, 1/17, 1/29 etc. So, what do you propose instead?
And pi? The number pi is understood as having an infinity of decimal digits, without any repeating sequence ever. What do you propose to do instead?
EB

 

[Write4U]

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

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"?

 

[Seattle]

No, I didn't. I repeat my post here so you can check again:

Assuming infinity does not exist in any way, how do you do basic arithmetic like 1 = 3 x 1/3 = 3 x 0.333... = 0.999...?
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. If you think infinities don't exist, the conventional way of interpreting "0.999..." has to be discarded for good. Same for very many other arithmetic operations, like 1/7, 1/11, 1/17, 1/29 etc. So, what do you propose instead?
And pi? The number pi is understood as having an infinity of decimal digits, without any repeating sequence ever. What do you propose to do instead?
Where do I "pretend he denied"?​

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.
Given this, and considering that Seattle thinks infinity doesn't exist in the physical world, 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.
My point is that infinity is at the heart of our most basic mathematical representations of the world. It's not just something QM physicists negligently let slip into their equations. 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. We can't do without it I don't think.
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?
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. If you think infinities don't exist, the conventional way of interpreting "0.999..." has to be discarded for good. Same for very many other arithmetic operations, like 1/7, 1/11, 1/17, 1/29 etc. So, what do you propose instead?
And pi? The number pi is understood as having an infinity of decimal digits, without any repeating sequence ever. What do you propose to do instead?
EB

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.

 

[Speakpigeon]

Isn't this a big philosophical problem for QM? Hilbert space is complete. That means that there are no "holes," just as there are no holes in the real numbers. It's a theorem that a complete dense linear continuum must be uncountable. But there are only countably many computable real numbers (because there are only countably many Turing machines). So the Hilbert space formulation of QM immediately contradicts any claim that the universe is a computation or a Turing machine or a computer or an algorithm or any equivalent formulation.

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?
Second, maybe the universe doesn't contain non-countable infinities but contains countable ones, and perhaps a countable infinity of countable infinities. So, where is the problem exactly?
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?

So ... tell me this. Is the axiom of choice provable in the physical world? Because if it is, so is the Banach-Tarski paradox.

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?
EB