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

[someguy1]

Check out Max Tegmark.

Max Tegmark explicitly rejects physical infinity.

http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/

I must add that I'm always a little confused when someone says, "Check out X" where X is a prolific scientist with dozens if not hundreds of books and papers and articles on a variety of subjects to their credit. Which of Tegmark's many accomplishments are you referring to?

Max Tegmark is one of the theoretical physicists that I was talking about. He is interesting but ultimately off-base. Actually, I don't think he believes everything he talks about (I could be wrong about that however).

I get that feeling too. He's interesting and provocative, but I don't think he expects his speculative ideas to be taken as actual science. He's probably amused that people take his mathematical universe idea seriously.

Could you explain in a few words what is an "infinite-dimensional complete inner product space" and why it's really necessary in QM?
EB

I'm not a physicist so I'll let this article explain why physicists like Hilbert space. It uses a lot of buzzwords but I can't simplify it. Perhaps one of the physics-minded folks here can do a better job.

https://www.researchgate.net/post/Why_is_the_Hilberts_space_useful_in_quantum_mechanics

From a mathematical point of view, Hilbert space is first, an infinite-dimensional vector space. Just like the reals are a one-dimensional vector space, and the plane is a 2-dimensional vector space, Hilbert space has infinitely many dimensions.

Two, it admits an inner product, which generalizes the dot product from multivariable calculus. And three, it's complete, meaning there are no holes in it. The real numbers are complete, meaning that every sequence that "should" converge does converge. The rational numbers are no complete, because there are holes where the irrationals should be.

It's the completeness that's the most un-physical aspect of Hilbert space and, for that matter, the standard and familiar real numbers. Most real numbers encode an infinite amount of information and can not be computed or described by any algorithm or computer program. The real numbers are extremely UNreal. It's virtually impossible to make a claim that the real numbers represent anything in the physical world. Yet all of physics from Newton onward are based on the real numbers, which require the concept of infinite sets to get off the ground. It's an actual philosophical problem, not always recognized by the physicists themselves.

 

[Seattle]

I get that feeling too. He's interesting and provocative, but I don't think he expects his speculative ideas to be taken as actual science. He's probably amused that people take his mathematical universe idea seriously.

I read his book about the mathematical universe. He says he actually believes it and I think I could find places in there where he says that a physical infinity may exist (I can't remember for sure) but I'm also sure I could find other places where he rejects that.

He does lay out the hypothesis well for the mathematical universe. He does it by bring up Einstein's equivalency principle and by removing "words" thus leaving only the math while giving examples all along the way.

The concept is that something like quarks can be described by only three numbers such as for charge, spin, and type. If we can't see a quark and it can be described by 3 numbers then the quark is essentially just 3 numbers or is equivalent to 3 numbers. If you can't tell the difference between a physical quark and those 3 numbers then they are the same.

The example used was for gravity and acceleration. You can't tell the difference between the two so they are equivalent.

Regarding removing "words" and being left only with mathematics, think of chess. There is a chess board, physical pieces, words to describe them but none of that is the essence of "chess". You can eliminate all that and still have chess.

Computers do this all the time (computerized chess). The algorithm is "chess".

He talks about various types of a multiverse and some require infinity. Black holes and the Big Bang basically require infinity. In Quantum Physics the "Many-Worlds" interpretation requires this as well.

 

[Speakpigeon]

Infinity represents something that is without limit but there is nothing in our physical world that is without limit.

You're claiming to know more than you or anybody actually knows.

Rather than using infinity as a large approximation it is being used as "if anything can happen it will happen and it will happen infinitely many times". It's being used to suggest that there are infinitely many copies of "you" in another Universe. It's being used in Quantum Physics to suggest Superpositions, the Many Worlds interpretation etc.

The bottom line is: We don't know that infinite exists AND we don't know that infinity doesn't exist.
The only rational attitude is to admit you don't know either way and take any assumptions about infinities with a pinch of salt, both assumptions that they exist and assumption that they don't. Your attitude is just dogmatic. Nothing to do with science.
EB

 

[Speakpigeon]

I read his book about the mathematical universe. He says he actually believes it and I think I could find places in there where he says that a physical infinity may exist (I can't remember for sure) but I'm also sure I could find other places where he rejects that.

He does lay out the hypothesis well for the mathematical universe. He does it by bring up Einstein's equivalency principle and by removing "words" thus leaving only the math while giving examples all along the way.

The concept is that something like quarks can be described by only three numbers such as for charge, spin, and type. If we can't see a quark and it can be described by 3 numbers then the quark is essentially just 3 numbers or is equivalent to 3 numbers. If you can't tell the difference between a physical quark and those 3 numbers then they are the same.

The example used was for gravity and acceleration. You can't tell the difference between the two so they are equivalent.

Regarding removing "words" and being left only with mathematics, think of chess. There is a chess board, physical pieces, words to describe them but none of that is the essence of "chess". You can eliminate all that and still have chess.

Computers do this all the time (computerized chess). The algorithm is "chess".

He talks about various types of a multiverse and some require infinity. Black holes and the Big Bang basically require infinity. In Quantum Physics the "Many-Worlds" interpretation requires this as well.

I'll assume that your formulation here may not be exactly that of Tegmark.
So,where in my view you, or possibly Tegmark, go wrong in there is that you don't seem to see that there is a fundamental difference between being "the same" and being "equivalent".
I would agree that if you can't tell the difference between a physical quark and three numbers then the quark is equivalent to those three numbers, i.e. logically equivalent, which really goes without saying. It's clear, though, that you couldn't possibly prove that, beyond logical equivalence, they are also "the same", i.e. the same thing, i.e. ontologically the same. I hope you understand the difference.
I don't see any problem assuming that the physical world is logically equivalent to our mathematical description of it. Claiming that the universe is a mathematical universe is merely using a fuzzy expression. Not helpful.
You, and Tegmark, should also apply the same line of reasoning to the concept of infinity, namely that if you can't see the difference between a model assuming infinities and the physical world then the two are equivalent.
And then, you can relax. That they are equivalent logically doesn't mean they are the same, i.e. it doesn't mean that using infinities in your model implies the existence of actual infinities in the physical world.
So, I would say, the problem is entirely in the fallacy of equating logical equivalence with ontological identity. Just sloppy thinking.
Possibly on both sides I wouldn't be surprised, though.
EB

 

[Write4U]

Which of Tegmark's many accomplishments are you referring to?

I like his proposition that the universe does not have "some mathematical properties", but has "only mathematical properties", and that the mathematical properties are surprisingly simple and fundamentally consist of some 33 numerical values (and combinations thereof) and a handful of mathematical equations. He does admit that there are still some missing pieces of the puzzle.
See the NOVA clip of "The Great Math Mystery" above.

 

[Speakpigeon]

I'm not a physicist so I'll let this article explain why physicists like Hilbert space. It uses a lot of buzzwords but I can't simplify it. Perhaps one of the physics-minded folks here can do a better job.
https://www.researchgate.net/post/Why_is_the_Hilberts_space_useful_in_quantum_mechanics
From a mathematical point of view, Hilbert space is first, an infinite-dimensional vector space. Just like the reals are a one-dimensional vector space, and the plane is a 2-dimensional vector space, Hilbert space has infinitely many dimensions.
Two, it admits an inner product, which generalizes the dot product from multivariable calculus. And three, it's complete, meaning there are no holes in it. The real numbers are complete, meaning that every sequence that "should" converge does converge. The rational numbers are no complete, because there are holes where the irrationals should be.

OK, thanks, I can connect that to what mathematics I learnt a very long time ago. Just with a different vocabulary, in French.

It's the completeness that's the most un-physical aspect of Hilbert space and, for that matter, the standard and familiar real numbers. Most real numbers encode an infinite amount of information and can not be computed or described by any algorithm or computer program. The real numbers are extremely UNreal. It's virtually impossible to make a claim that the real numbers represent anything in the physical world. Yet all of physics from Newton onward are based on the real numbers, which require the concept of infinite sets to get off the ground. It's an actual philosophical problem, not always recognized by the physicists themselves.

Yes, I would agree that it's a mistake to make a claim as to the existence of infinities. But, equally, it's a mistake to make any claim as to the inexistence of them. We should just admit that our epistemology is limited and that we just don't know either way.
Now if someone could prove that there's an actual problem with using infinities in our mathematical models then they should come forward and if what they say is true I'm sure all specialists will be able to recognise this.
So, for now, I will believe that nobody has done that yet.
EB

 

[Speakpigeon]

I like his proposition that the universe does not have "some mathematical properties", but has "only mathematical properties", and that the mathematical properties are surprisingly simple and fundamentally consist of some 33 numerical values (and combinations thereof) and a handful of mathematical equations. He does admit that there are still some missing pieces of the puzzle.
See the NOVA clip of "The Great Math Mystery" above.

This reminds me of the consensus among scientists that mostly physics was done just before Planck and Einstein smashed their belief to pieces.
EB

 

[Write4U]

The concept is that something like quarks can be described by only three numbers such as for charge, spin, and type. If we can't see a quark and it can be described by 3 numbers then the quark is essentially just 3 numbers or is equivalent to 3 numbers. If you can't tell the difference between a physical quark and those 3 numbers then they are the same.

I think I am in agreement with your interpretation, however I believe that Tegmark with his use of symbolic numbers, he means to indicate just symbolically relatable values and equatable symbolic mathematical functions. Perhaps somewhat similar to the concept of relativity which presents an infinite number of value relationships, with a few constants mixed in.

I believe a great example can be found in the mathematical relationships between notes (waves) and chords (harmonics) in music.

Pythagoras (6th century BC) observed that when the blacksmith struck his anvil, different notes were produced according to the weight of the hammer.
Number (in this case amount of weight) seemed to govern musical tone...

With string instruments it is the string length.

http://www.aboutscotland.com/harmony/prop.html

 

[Speakpigeon]

It's the completeness that's the most un-physical aspect of Hilbert space and, for that matter, the standard and familiar real numbers. Most real numbers encode an infinite amount of information and can not be computed or described by any algorithm or computer program.

Yes, except that an infinite number of computers or even an infinitely long algorithm running on an infinitely fast computer would do it.
The point is that you cannot encode infinity using a finite model and that we are apparently ourselves finite information systems. So, nothing is demonstrated by this, either way.
EB

 

[Sarkus]

Ok, but then should we not always use the qualified term "observable universe" as being the expanding physical universe within a larger infinite metaphysical Universe?

Usually (I.e. within most contexts) the term "universe" implies the observable universe. However, given the subject matter in this case one should probably be more explicit with regard precisely which universe you are describing.
If you're talking about the universe being infinite, for example, then this most certainly is not the observable universe. The observable universe has a defined limit at a given point in time, and gradually even though it gets bigger, more and more objects will move beyond observational capability, due to the expansion of space.

And a little aside; does a 3D infinity not have a logical (mathematical) center from which each dimension extends infinitely?

If each dimension extends infinitely then there is no centre (or alternatively every point is a centre), as any point you take is the same with regard how far along each dimension you are. You can then only talk about relative positions and not absolute, and relative to an arbitrary centre point. If you are at point (a,b,c) in your 3D space, then each axis extends infinitely far in each direction, irrespective of what a, b, and c actually are. The origin simply becomes an arbitrary point to aid with relative referencing.

 

[Speakpigeon]

Yes , I saw him first briefly on a NOVA program;

and was initially impressed, but after reading and watching some of his lectures I was becoming a little skeptical as well.
However I still believe that the fundamental workings of the universe are based on a form of mathematical processing of inherent discreet physical values and functions, which we have been able to symbolize with a mathematical language, mathematics.

The Fibonacci Sequence is just too pervasive to be ignored as a fundamental mathematical property or potential of spacetime. The Exponential Function is another.

Yes, well, one fallacy is at 10:11 and then 10:15. There Tegmark assumes fast and loose that the little character in the video game could be made to be "conscious" and "feel" the video game world like we do our own physical world.
I'd say, first do it and then you can talk about it.
So, it's just fuzzy analogising and all the thing is bound to be only that.
Some people seem to just gobble the stuff and ask for more.
EB

 

[Write4U]

This reminds me of the consensus among scientists that mostly physics was done just before Planck and Einstein smashed their belief to pieces.
EB

Why so? If one can represent a universal enfolded potential or unfolded expression of reality by using symbolic mathematical language, the does it not follow that the universe in fact has mathematical properties. The symbolic numerical language is arbitrary as long as the relationships between the symbols of values and functions is an accurate representation of relative values and functions in reality.

We can mathematically choreograph tap dancing using symbolic representations of the moves.
We can mathematically represent dress ties (Half Windsor = Li Ro Ci Lo Ri Co T).
We can mathematically represent the word MATHEMATICS using binary code, alphabetical letters, morse code (sound).
We can create curves by using simple straight lines.
We can even create shapes from numerical relationships, such as found in the number 4/3.

https://www.ted.com/talks/roger_ant...en_secret_to_understanding_the_world#t-196686

 

[Seattle]

You're claiming to know more than you or anybody actually knows.

The bottom line is: We don't know that infinite exists AND we don't know that infinity doesn't exist.
The only rational attitude is to admit you don't know either way and take any assumptions about infinities with a pinch of salt, both assumptions that they exist and assumption that they don't. Your attitude is just dogmatic. Nothing to do with science.
EB

My approach has everything to do with science. Of course no one knows the answers to these questions. The point is that since no infinities have been observed there is less reason to give that concept credence. We don't usually give equal credence to unicorns existing and to unicorns not existing even though we don't know for sure that they don't exist.

 

[Write4U]

Yes, well, one fallacy is at 10:11 and then 10:15. There Tegmark assumes fast and loose that the little character in the video game could be made to be "conscious" and "feel" the video game world like we do our own physical world.
I'd say, first do it and then you can talk about it.
So, it's just fuzzy analogising and all the thing is bound to be only that.
Some people seem to just gobble the stuff and ask for more.
EB

Practically speaking I agree with you and already indicated my problems with some of his assertions. Nevertheless, a weakness in a presentation does not necessarily invalidate the entire proposition.
Moreover, it can be a simple matter of communication. That often is the weakest link in a proposition. How does one verbally present an abstract idea clearly and unambiguously without the mathematics.......
Mathematics can exquisitely describe universal functions. IMO, that indicates a mathematical nature and reality to the universe. I find it elegant and eminently suitable to survive Occam's Razor.

After all, we would have to use mathematics to disprove the mathematics.....

 

[Seattle]

Yes, well, one fallacy is at 10:11 and then 10:15. There Tegmark assumes fast and loose that the little character in the video game could be made to be "conscious" and "feel" the video game world like we do our own physical world.
I'd say, first do it and then you can talk about it.
So, it's just fuzzy analogising and all the thing is bound to be only that.
Some people seem to just gobble the stuff and ask for more.
EB

So now the standard is that one has to do something before they can talk about it or it's fuzzy? That's pretty fuzzy logic in and of itself.

 

[iceaura]

The number 2 is the limit of the infinite sum 1 + 1/2 + 1/4 + - - -

Does the number 2 exist, as more than a mathematical abstraction?
Is it possible to have 2 things, or does the impossibility of adding one thing and a half thing and a quarter thing and so forth in an infinite series mean we can never actually have two things?

 

[Seattle]

The number 2 is the limit of the infinite sum 1 + 1/2 + 1/4 + - - -

Does the number 2 exist, as more than a mathematical abstraction?
Is it possible to have 2 things, or does the impossibility of adding one thing and a half thing and a quarter thing and so forth in an infinite series mean we can never actually have two things?

You've proved my point with your comment about the impossibility of having an infinite series.

You can't have everything that can happen, happen and do so infinitely many times, unless there is actually an infinity. You can find many uses for infinity in math. We haven't found infinite space, energy, density, etc. in the real world.

Anything you could contrive in the real world would (so far) be artificial due to some framework that was imposed such of speaking of a singularity at the Earth's poles due to all the lines of longitude converging there (for instance).

 

[Speakpigeon]

Why so? If one can represent a universal enfolded potential or unfolded expression of reality by using symbolic mathematical language, the does it not follow that the universe in fact has mathematical properties. The symbolic numerical language is arbitrary as long as the relationships between the symbols of values and functions is an accurate representation of relative values and functions in reality.

We can mathematically choreograph tap dancing using symbolic representations of the moves.
We can mathematically represent dress ties (Half Windsor = Li Ro Ci Lo Ri Co T).
We can mathematically represent the word MATHEMATICS using binary code, alphabetical letters, morse code (sound).
We can create curves by using simple straight lines.
We can even create shapes from numerical relationships, such as found in the number 4/3.

https://www.ted.com/talks/roger_ant...en_secret_to_understanding_the_world#t-196686

The point is that given one mathematical model that's good with what we've observed so far, there is an infinite number of potential realities that would comply with the model, even if we limit ourselves to assuming realities that would only have quantifiable properties, which is a huge assumption given subjective experience.
So, again, the real physical world is only shown at best to be logically equivalent to our models, not ontologically equivalent.
EB

 

[Speakpigeon]

My approach has everything to do with science. Of course no one knows the answers to these questions. The point is that since no infinities have been observed there is less reason to give that concept credence. We don't usually give equal credence to unicorns existing and to unicorns not existing even though we don't know for sure that they don't exist.

We don't give credence to unicorns only because they're crap at predicting anything about the physical world. In fact people did give credence to unicorns and still do and none of them was able to use them to do any good science. The situation is very different with infinity. People apparently do good science assuming it. Maybe it's ultimately wrong and so maybe at some point in the future that will come out clearly but meanwhile each scientist doing good predictive science should be given a break. Although, I guess the topic should be discussed. But whatever consensus there would be, I think it would be better to let each scientist decides for themselves what more promising. Better theories will prove themselves.
And then maybe this discussion is just a faint echo of the fight for a budget among scientists.
EB

 

[Speakpigeon]

Mathematics can exquisitely describe universal functions. IMO, that indicates a mathematical nature and reality to the universe.

Sure, I'm myself minded to conceive of reality as just somehow coming out of logical possibilities. That's definitely a very seductive idea, certainly since QM, and more recently with more scientists sort of toying with this idea or heading that way.
Yet, there's a fundamental problem with this idea. We have a limited epistemology. The only things we actually know are our qualia. The rest we get to have a glimpse of it only through the kind of massive quantification that we're using when we do science. That's all we can do. It's even an ideological bias in scientists. If you can't measure it, it doesn't exist! So, yes, if you start with the assumption that If you can't measure it, it doesn't exist, then obviously, what you get in the end will be a mathematical universe.
Well, no. All you get is a mathematical model of the universe. Inevitably and irrespective of the nature of reality.
EB