Can Gödel's Theorem Inform Our Understanding of Physical Reality?

[Giulio Prisco]

TL;DR Summary

What are the implications of Gödel theorem for fundamental science and metaphysics?

What does Gödel’s theorem say about physical reality? Does Gödel’s theorem imply that no finite mathematical model can capture physical reality? Does the nondeterminism found in quantum and chaos physics - it’s impossible to predict (prove) the future from the present and the laws of physics - have something to do with Gödel’s incompleteness?
Many scientists e.g. Stanley Jaki, Freeman Dyson, and recently Stephen Hawking ("According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted…") have formulated this intuition.
But I'm not aware of any proof (or very strong semi-rigorous argument) that causal openness in physical reality follows from Gödel's theorems (or the related results of Turing, Chaitin etc.).
Can anyone give me links/ideas?

 

[PeroK]

I imagine you have already read a lot on the subject. As far as I can see, there isn't a direct connection from Gödel to physical theories. It's entirely possible, for example, that ultimately the universe has a finite number of possible states, which would make Gödel's theorems irrelevant.

It's not a question of predictability, as such. E.g. a physical system that probabilitistically changes (e.g. radioactive decay) may be perfectly simple mathematically. Its unpredictability is not related to the foundations of mathematics, but is simply an axiom of the mathematical model.

With all these questions, I feel you need to be much more specific about what you are saying or asking. What particular aspect of physical theories are affected by the fundamentals of mathematics? E.g. if we abandon the axiom of choice and instead assume that every subset of $\mathbb R$ is Borel-measurable, how would that change in the foundations of mathematics affect physics? If at all?

 

[Giulio Prisco]

I imagine you have already read a lot on the subject. As far as I can see, there isn't a direct connection from Gödel to physical theories. It's entirely possible, for example, that ultimately the universe has a finite number of possible states, which would make Gödel's theorems irrelevant.

It's not a question of predictability, as such. E.g. a physical system that probabilitistically changes (e.g. radioactive decay) may be perfectly simple mathematically. Its unpredictability is not related to the foundations of mathematics, but is simply an axiom of the mathematical model.

With all these questions, I feel you need to be much more specific about what you are saying or asking. What particular aspect of physical theories are affected by the fundamentals of mathematics? E.g. if we abandon the axiom of choice and instead assume that every subset of $\mathbb R$ is Borel-measurable, how would that change in the foundations of mathematics affect physics? If at all?

Thanks! Let me think about that...

 

[Giulio Prisco]

I imagine you have already read a lot on the subject. As far as I can see, there isn't a direct connection from Gödel to physical theories. It's entirely possible, for example, that ultimately the universe has a finite number of possible states, which would make Gödel's theorems irrelevant.

Good point.

It's not a question of predictability, as such. E.g. a physical system that probabilitistically changes (e.g. radioactive decay) may be perfectly simple mathematically. Its unpredictability is not related to the foundations of mathematics, but is simply an axiom of the mathematical model.

With all these questions, I feel you need to be much more specific about what you are saying or asking. What particular aspect of physical theories are affected by the fundamentals of mathematics? E.g. if we abandon the axiom of choice and instead assume that every subset of $\mathbb R$ is Borel-measurable, how would that change in the foundations of mathematics affect physics? If at all?

As far as I understand, that wouldn't invalidate Gödel theorem, which applies to systems based on weaker sets of axioms than the full ZFC with choice. Please correct me if I am wrong.

In general, I don't think changes in the (or better "a") foundations of mathematics can affect physics.
I tend to think of mathematics as the physics of little things that can be counted - a construction motivated by physical experience and the need to stay alive long enough to reproduce. So thinking beings very different from us might develop entirely different forms of mathematics, and it would be the physics of the real world that discriminates between one or another forms of mathematics. But perhaps Gödel theorem is sufficiently general to apply to all.

 

[PeroK]

As far as I understand, that wouldn't invalidate Gödel theorem, which applies to systems based on weaker sets of axioms than the full ZFC with choice. Please correct me if I am wrong.

You can't invalidate Gödel's theorem. I gave an example of the practical effect that Gödel's theorem has on mathematics (it was recommended to me nearly 40 years ago as a potential PhD topic!). To produce a non-measurable set, you must invoke the axiom of choice. And, certainly, measure-theory is central to mathematical physics. Another example is that every Hilbert Space has an orthonormal basis - which in general relies on Zorn's lemma.

The question is whether these undecidable propositions are relevant to physics. In both these cases, the mathematics is probably not relevant - in the sense that any subset of $\mathbb R$ that is used in physics may be assumed to be measurable and every separable Hilbert space has a basis without invoking the axiom of choice.

From that point of view it appears that the subset of mathematics that is used in mathematical physics is unaffected by undecidable propositions - although I'm not claiming that categorically!

 

[PeroK]

@Giulio Prisco by contrast, Gödel's theorem and the axiom choice are highly relevant for modern pure mathematics. It's a part of everyday life!

For example, take a look at the first problem from our monthly maths challenge from last year:

https://www.physicsforums.com/threads/math-challenge-june-2020.989862/

I had a go at this problem and began to suspect that it was undecidable and depended on the axiom of choice. And, indeed, that turned out to be true, via the Baire Category Theorem:

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

Does this sort of thing happen in mathematical physics?

 

[Demystifier]

A must read: https://www.amazon.com/dp/1568812388/?tag=pfamazon01-20

 

[Demystifier]

It's entirely possible, for example, that ultimately the universe has a finite number of possible states, which would make Gödel's theorems irrelevant.

Yes. A lot of trouble in pure math arises from the axiom that an infinite set exists.

 

[Demystifier]

although I'm not claiming that categorically!

I'm glad you don't. I can follow arguments based on logic and set theory, but when someone uses category theory I get lost.

 

[MathematicalPhysicist]

Yes. A lot of trouble in pure math arises from the axiom that an infinite set exists.

Is it really an axiom?
Just start counting: $1,2,3,\ldots$ it never stops.

 

[Demystifier]

Just start counting: $1,2,3,\ldots$ it never stops.

You just stopped, at 3 followed by the dots.

 

[MathematicalPhysicist]

You just stopped, at 3 followed by the dots.

the dots mean that the counting never stops.
We all know there's no last number in the naturals.

 

[Demystifier]

We all know there's no last number in the naturals.

Yes, but the full set of natural numbers is our mental invention. In nature we only see finite subsets of it. Observed physics could, in principle, be formulated with a set in which the largest number is, say, $100^{100^{100}}$. Or if it's not big enough, then the busy beaver of that would certainly suffice. https://en.wikipedia.org/wiki/Busy_beaver

 

[MathematicalPhysicist]

Yes, but the full set of natural numbers is our mental invention. In nature we only see finite subsets of it. Observed physics could, in principle, be formulated with a set in which the largest number is, say, $100^{100^{100}}$. Or if it's not big enough, then the busy beaver of that would certainly suffice. https://en.wikipedia.org/wiki/Busy_beaver

You can only grasp a finite portion of the universe. Beyond the horizon nothing is known.

 

[Demystifier]

You can only grasp a finite portion of the universe. Beyond the horizon nothing is known.

Even if there was no horizon, we could only observe a finite amount of data. The size of the universe has nothing to do with that.

 

[Fra]

Yes, but the full set of natural numbers is our mental invention. In nature we only see finite subsets of it. Observed physics could, in principle, be formulated with a set in which the largest number is, say, $100^{100^{100}}$. Or if it's not big enough, then the busy beaver of that would certainly suffice. https://en.wikipedia.org/wiki/Busy_beaver

One question one could ask from an agent interpretation is; how far can an agent count? or, what observational resolution does it have? Does this have any relations to the agents internal coding? mass? And if this is so, would this natural regulator, have any impact on the agent-agent interactions?

All these interesting questions are LOST at the starting point, as you pull the real number out of the sleeve. It is also a problem already in Jayens and others reconstruction of probability theory. For this reason even starting from pure probilistic reasoing is not innocent.

Edit: Note that this objection is not suggesting that the universe has to be finite, just that any intrinsic perspective should be limited. We have to be able to measure this. And if the possibilities are always infinte, we face situations where infinity fights infinity, and a post-construction of regularisation of formally non-sensiable expressions is required from us. I think we can do better.

/Fredrik

 

[PeroK]

One question one could ask from an agent interpretation is; how far can an agent count? or, what observational resolution does it have? Does this have any relations to the agents internal coding? mass? And if this is so, would this natural regulator, have any impact on the agent-agent interactions?

All these interesting questions are LOST at the starting point, as you pull the real number out of the sleeve. It is also a problem already in Jayens and others reconstruction of probability theory. For this reason even starting from pure probilistic reasoing is not innocent.

Edit: Note that this objection is not suggesting that the universe has to be finite, just that any intrinsic perspective should be limited. We have to be able to measure this. And if the possibilities are always infinte, we face situations where infinity fights infinity, and a post-construction of regularisation of formally non-sensiable expressions is required from us. I think we can do better.

/Fredrik

I must confess I have no idea what any of that means - especially the last paragraph.

 

[Fra]

I must confess I have no idea what any of that means - especially the last paragraph.

I see that I brought in too much thinking at once without giving it the time needed to formulate it. For clarity I might as well have deleted that last paragraph. The two first ones, was the main points anyway.

(What I associated to in that paragraph was relations between the cardinality or the observable event spaces and renormalisation problems in general, as we may miss out "natural regulators" which has a physical interpretation. The result can be formal expressions that are divergent, but which shouldn't had to be, if only the physical complexions was counted to start with)

/Fredrik

 

[PeroK]

I see that I brought in too much thinking at once without giving it the time needed to formulate it. For clarity I might as well have deleted that last paragraph. The two first ones, was the main points anyway.

(What I associated to in that paragraph was relations between the cardinality or the observable event spaces and renormalisation problems in general, as we may miss out "natural regulators" which has a physical interpretation. The result can be formal expressions that are divergent, but which shouldn't had to be, if only the physical complexions was counted to start with)

/Fredrik

That's not any better!

 

[Fractal matter]

Summary:: What are the implications of Gödel theorem for fundamental science and metaphysics?

What does Gödel’s theorem say about physical reality? Does Gödel’s theorem imply that no finite mathematical model can capture physical reality? Does the nondeterminism found in quantum and chaos physics - it’s impossible to predict (prove) the future from the present and the laws of physics - have something to do with Gödel’s incompleteness?
Many scientists e.g. Stanley Jaki, Freeman Dyson, and recently Stephen Hawking ("According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted…") have formulated this intuition.
But I'm not aware of any proof (or very strong semi-rigorous argument) that causal openness in physical reality follows from Gödel's theorems (or the related results of Turing, Chaitin etc.).
Can anyone give me links/ideas?

"The incompleteness theorems apply only to formal systems which are able to prove a sufficient collection of facts about the natural numbers."
"For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system."

Basically, the 1st Gödel’s theorem is about natural numbers. The set on axioms should contain natural numbers in it for the Gödel’s theorem to apply. In the opposite case the Gödel’s theorem doesn't apply. Gödel’s theorem states that if you take a set on axioms that contain natural numbers there are statements, that are undecidable(neither true not false). Those can not be proved to be true or false, considering that set of axoims, and need to more powerful set of axioms to be proven. The cause is the infinity as i understand it.

Yes. A lot of trouble in pure math arises from the axiom that an infinite set exists.