I Godel's ITs & the Physical World: Is a ToE impossible?

[greswd]

Both Hawking and Dyson have said that Godel's incompleteness theorems prove that it is impossible for us to formulate an absolutely fundamental Theory of Everything.

Is that true? Do the theorems apply to the physical world as they apply to the realm of mathematics?

 

[greswd]

http://www.hawking.org.uk/godel-and-the-end-of-physics.html

The original article.

 

[Hornbein]

Both Hawking and Dyson have said that Godel's incompleteness theorems prove that it is impossible for us to formulate an absolutely fundamental Theory of Everything.

Is that true? Do the theorems apply to the physical world as they apply to the realm of mathematics?

I'm not convinced. I don't see why a supertheory couldn't be self-referential. But might be right. Perhaps a more detailed exposition would do it for me.

I'd say that nothing can ever be proved in physics. We have math, phenomena, and a provisional link between them. But I could be wrong. That's really too simple. Maybe there is some way to show that no other theory would work.

You would be interested in Kochen and Conway's Free Will Theorem, which I think deserves much more attention than it has received. It's in the same ballpark but is a thoroughly worked out "proof" instead of an informal lecture.

 

[greswd]

My question isn't really about determinism and freewill.

 

[friend]

Geometry is built on axioms and I'm told it is complete; no theorems exist in the theory that cannot be proved by the existing axioms. And yet geometry can be described with mathematics which is incomplete. So does that make geometry complete or incomplete? I think geometry is complete since the math is limited to that necessary for a description of geometry. Likewise, I think the math used in physics is limited to only that used to describe the theorems of physics (whatever those turn out to be). Physics is not an effort to prove every theorem of math. It's an effort to find the smallest set of physical rule necessary to describe reality.

 

[Demystifier]

Both Hawking and Dyson have said that Godel's incompleteness theorems prove that it is impossible for us to formulate an absolutely fundamental Theory of Everything.

Is that true? Do the theorems apply to the physical world as they apply to the realm of mathematics?

Godel's theorems do not imply many things which people sometimes think they do. In particular, they do not imply that "it is impossible for us to formulate an absolutely fundamental Theory of Everything". I highly recommend to read the book
https://www.amazon.com/dp/1568812388/?tag=pfamazon01-20

 

[pbuk]

Both Hawking and Dyson have said that Godel's incompleteness theorems prove that it is impossible for us to formulate an absolutely fundamental Theory of Everything.

Is that true? Do the theorems apply to the physical world as they apply to the realm of mathematics?

http://www.hawking.org.uk/godel-and-the-end-of-physics.html

The original article.

If you read that lecture you will see that Hawking absolutely does NOT say that GIT proves anything about physics.

 

[greswd]

If you read that lecture you will see that Hawking absolutely does NOT say that GIT proves anything about physics.

oh, you're right. apologies for being such an idiot.

 

[Demystifier]

oh, you're right. apologies for being such an idiot.

Or maybe you are not an idiot. In the last paragraph Hawking says
"Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I'm now glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery." (my bolding)

Hawking is not clear how exactly he arrived at that conclusion, but it seems to be motivated by the Godel's theorem.
Nevertheless, strictly speaking, that conclusion does not follow from the Godel's theorem.

 

[Hornbein]

My question isn't really about determinism and freewill.

True, but Kochen-Conway showed that some things will never be predictable. Would that mean that an "absolutely fundamental TOE" is impossible?

 

[Hornbein]

Geometry is built on axioms and I'm told it is complete; no theorems exist in the theory that cannot be proved by the existing axioms. And yet geometry can be described with mathematics which is incomplete. So does that make geometry complete or incomplete? I think geometry is complete since the math is limited to that necessary for a description of geometry. Likewise, I think the math used in physics is limited to only that used to describe the theorems of physics (whatever those turn out to be). Physics is not an effort to prove every theorem of math. It's an effort to find the smallest set of physical rule necessary to describe reality.

Distinguishing incomplete from complete is quite technical. Diophantine equations are incomplete, but it wasn't easy to prove that. I don't know whether geometry is complete. The first step would be to define geometry, and I wouldn't know how to do that. The old compass and straightedge stuff is almost surely complete.

 

[Demystifier]

True, but Kochen-Conway showed that some things will never be predictable.

They didn't show that.

 

[greswd]

Geometry is built on axioms and I'm told it is complete; no theorems exist in the theory that cannot be proved by the existing axioms.

Who says?

 

[greswd]

True, but Kochen-Conway showed that some things will never be predictable. Would that mean that an "absolutely fundamental TOE" is impossible?

How does things not being predictable lead to the impossibility of a TOE?

 

[greswd]

Hawking is not clear how exactly he arrived at that conclusion, but it seems to be motivated by the Godel's theorem.
Nevertheless, strictly speaking, that conclusion does not follow from the Godel's theorem.

Thanks. I was wondering how a mathematical theorem without experimental basis could lead to physical results.

 

[Hornbein]

They didn't show that.

Such is their claim.

 

[Hornbein]

How does things not being predictable lead to the impossibility of a TOE?

Well, it depends on your definition of a TOE.

 

[greswd]

Well, it depends on your definition of a TOE.

Ok, you give me one and then show how it relates to predictability.

 

[Hornbein]

Ok, you give me one and then show how it relates to predictability.

You are the one who brought it up. I don't care.

 

[greswd]

Such is their claim.

From what I can gather, they did not claim anything regarding whether determinism is true or not, they didn't say that some things will never be predictable.

They just stated a relationship between particles and human free will IF certain conditions are true.

 

[greswd]

You are the one who brought it up. I don't care.

Well, you said "it depends on your definition of a TOE." which implies that you already know something about the relation in question.

I don't know anything about the relation, I don't know which definition of a ToE will work, therefore I'm asking you to tell me what you already know.

 

[Hornbein]

Well, you said "it depends on your definition of a TOE." which implies that you already know something about the relation in question.

I don't know anything about the relation, therefore I'm asking you to tell me what you already know.

I could, but all I would contribute would be a layer of error. I'd recommend you search for the Kochen-Sprecker theorem or paradox, upon which the Conway-Kochen proof is based. I don't know why it hasn't received more notice.

 

[greswd]

I could, but all I would contribute would be a layer of error. I'd recommend you search for the Kochen-Sprecker theorem or paradox, upon which the Conway-Kochen proof is based. I don't know why it hasn't received more notice.

Alright, if you insist. Based on my reading of the KS theorem, I don't see why it prevents us from constructing a ToE.

 

[Hornbein]

Alright, if you insist. Based on my reading of the KS theorem, I don't see why it prevents us from constructing a ToE.

Do whatever you like.

 

[greswd]

Do whatever you like.

Seriously pal, if you have strong convictions, don't be afraid to share it. You may have some errors but that's not an issue as long as you can see them after people have analyzed it.

 

[friend]

Prof. Hawking is concerned about the incompleteness of physics because the laws of physics are described by mathematics which is incomplete by Godel's incompleteness theorem. Godel's proof rests on the ability of creating self-referential statements (This statement is not provable, etc.) And self-referential statements can lead to paradoxes. This brings up the question as to whether the physical world operates by actual principles (that are not just a mathematical description) that constitutes a system which allows self-reference. Some think self-reference is necessary in the physical world in order for consciousness to emerge from nature.

But I've recently read that there are conditions where self-reference in a system does not lead to paradoxes (such as with Godel's incompleteness theorem ?). See for example, in this paper, Thomas Bolander, PhD writes on the top of page 14,

"It can be shown that self-reference can only be vicious (lead to paradoxes) if it involves negation or something equivalent."

And I have to think that the ultimate laws of physics (whatever they end up being) do not involve negation since they should describe only what exists and have nothing to say about what does not exist.

 

[greswd]

Prof. Hawking is concerned about the incompleteness of physics because the laws of physics are described by mathematics which is incomplete by Godel's incompleteness theorem. Godel's proof rests on the ability of creating self-referential statements (This statement is not provable, etc.) And self-referential statements can lead to paradoxes. This brings up the question as to whether the physical world operates by actual principles (that are not just a mathematical description) that constitutes a system which allows self-reference. Some think self-reference is necessary in the physical world in order for consciousness to emerge from nature.

But I've recently read that there are conditions where self-reference in a system does not lead to paradoxes (such as with Godel's incompleteness theorem ?). See for example, in this paper, Thomas Bolander, PhD writes on the top of page 14,

"It can be shown that self-reference can only be vicious (lead to paradoxes) if it involves negation or something equivalent."

And I have to think that the ultimate laws of physics (whatever they end up being) do not involve negation since they should describe only what exists and have nothing to say about what does not exist.

Dude, you totally sound like a crank. I'm not saying that you are one, but you make a very good impression of one.

You have so many points, and ramble from one to the next. And you're being vague too.

 

[friend]

Dude, you totally sound like a crank. I'm not saying that you are one, but you make a very good impression of one.

You have so many points, and ramble from one to the next. And you're being vague too.

If this is an invitation to say more, I'll have to decline for now.

P.S. A crank is someone who grinds on a subject that has been proven wrong.

 

[Demystifier]

Such is their claim.

I think it is not. But to be sure, can you quote their exact claim with the reference?

 

[stevendaryl]

I don't think that Godel's theorem says anything about the possibility of a Theory of Everything. What it does say is that there might be questions about the ToE that we can't answer.

Let's take a simpler system: A digital computer. I would say that we have a complete theory of how computers work, in the sense that we can with 100% certainty predict future states from past states. But there are questions about a digital computer that we don't know how to answer. For example: If I write a program that, given a number n, searches for the nth Twin Prime, will that program always halt, for all possible n? We don't know the answer to that. But it's not because there are some aspects of how computers work that we don't understand.

If we had a Theory of Everything, there would still be unsolvable problems about physics. But the ToE would allow us to restate such an unsolvable problem about physics into an unsolvable problem about pure mathematics.

 

[jerromyjon]

Prof. Hawking is concerned about the incompleteness of physics because the laws of physics are described by mathematics which is incomplete by Godel's incompleteness theorem. Godel's proof rests on the ability of creating self-referential statements (This statement is not provable, etc.) And self-referential statements can lead to paradoxes. This brings up the question as to whether the physical world operates by actual principles (that are not just a mathematical description) that constitutes a system which allows self-reference. Some think self-reference is necessary in the physical world in order for consciousness to emerge from nature.

But I've recently read that there are conditions where self-reference in a system does not lead to paradoxes (such as with Godel's incompleteness theorem ?). See for example, in this paper, Thomas Bolander, PhD writes on the top of page 14,

"It can be shown that self-reference can only be vicious (lead to paradoxes) if it involves negation or something equivalent."

And I have to think that the ultimate laws of physics (whatever they end up being) do not involve negation since they should describe only what exists and have nothing to say about what does not exist.

Makes total sense to me... agreed. We already have amazing theories to describe as much as can be or need be. To say we have "a complete set of theories" to describe the universe and everything we can get close enough or detailed enough to have observable consequences is true. To say they all don't combine into a complete all-inclusive model somehow or another would NOT be true. It has not been proven either way!

 

[Hornbein]

Prof. Hawking is concerned about the incompleteness of physics because the laws of physics are described by mathematics which is incomplete by Godel's incompleteness theorem.

I think it is quite possible that all of physics could be described by mathematics that is complete. One easy way to do it is to restrict oneself to finite sets. There are other ways as well.

In Hawking's view a Theory of Everything would contain the theory itself. That might lead to a contradiction, but I'll believe that when I see it.

 

[jerromyjon]

In Hawking's view a Theory of Everything would contain the theory itself. That might lead to a contradiction, but I'll believe that when I see it.

I was just thinking about what I think I recall as something dalespam often states that a machine to model a system has to be larger than the system itself. Something to that effect, but where I'm going with this is time. Would it be obviously true that a machine to model atomic physics couldn't "keep up" in real time? I'm mean that is the crux of the problem is "Can physics be condensed and simplified or is it mechanically more complex than it can conceivably be modeled?"

 

[WWGD]

But physics is not developed/modeled axiomatically but instead experimentally, isn't it (except maybe theoretical physics)?

 

[jerromyjon]

But physics is not developed/modeled axiomatically but instead experimentally

I'm not certain but I believe renormalization is basically adjusting the scale of the data where it then fits an axiomatic description. Renormalization group is into theoretic territory but I know very little about it.

 

[Grinkle]

(except maybe theoretical physics)

Per my own lay understanding, except for theoretical physics, the rest is experimental physics.

 

[stevendaryl]

But physics is not developed/modeled axiomatically but instead experimentally, isn't it (except maybe theoretical physics)?

Well, the way that physics can be described is an iteration of:

1. Do experiments.
2. Make up a mathematical model that would allow you to predict the results.
3. Do other experiments to test that model.
4. If it fails, go back to 2 and repeat.

 

[WWGD]

Well, the way that physics can be described is an iteration of:

1. Do experiments.
2. Make up a mathematical model that would allow you to predict the results.
3. Do other experiments to test that model.
4. If it fails, go back to 2 and repeat.

I guess a sort of T.O.T.E https://en.wikipedia.org/wiki/T.O.T.E . ?

 

[jerromyjon]

https://en.wikipedia.org/wiki/T.O.T.E .

The trailing dot was chopped on your link.

 

[Chronos]

I would hazard to guess that an identity is mathematically impossible to prove without becoming self referential and further posit no version of reality [physics] we can construct can entirely escape this same trap. In that sense I would argue any TOE is, by definition, is unprovable.

 

[stevendaryl]

One comment word about Godel's Incompleteness Theorem and how it might come into play in physics.

Incompleteness comes into play when we have an infinite collection of objects, and we try to answer a question about every object in that collection. For example: "Can every even number greater than 2 be written as the sum of two prime numbers?" (Goldbach's conjecture) We can check every instance to see if it's true:

4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 3 + 7
12 = 5 + 7

etc.

But we don't know of any way to prove, once and for all, that it's true for every even number. Godel proved that, no matter how powerful our system of mathematics, as long as it's computable, then there will be questions of that form that we can't answer. We can just guess the answer, and add that guess as a new axiom, but (1) doing so might be inconsistent, and (2) no matter how many new axioms we add, there will still be unanswerable questions of this form.

So how might such questions come into play in physics? Any time a question can be formulated as a search over an infinite set, there is the possibility that Godel's incompleteness theorem will come to bite us. An example might be: Is the solar system stable, or will there come a time in the future where one of the planets crash into the sun, or two planets crash into each other, and one of the planets is ejected from the solar system? We can simulate the future locations of the planets using a computer, so we can use such a simulation to assure ourselves that none of these catastrophic events will happen this year. With a little more computing, we can prove that it won't happen in 2017. We can prove that it won't happen in 2018. But can we prove that it will never happen? There may be no way to prove it. (I'm just using this example to illustrate the flavor of Godel's incompleteness theorem; there very well might be a way to answer this particular question---I'm not an expert on orbital mechanics.)

The point of bringing up the example of orbital mechanics is to give you a flavor of how incompleteness might limit our ability to make definitive statements about physics. Note that, as far as I know, Newtonian mechanics may already be complex enough that we can't answer all questions about it. In quantum mechanics, there might already be questions that we can't answer, for example, see https://www.tum.de/en/about-tum/news/press-releases/short/article/32791/

So Godel's proof is really about the limitations of what we can say about a theory. It's not about our limitations in developing theories. Godel's theorem doesn't imply anything about whether we will eventually develop a quantum theory of gravity, but it suggests that we may not be able to answer all questions about that theory, even after it is developed.

 

[Grinkle]

I may be missing something (I am certainly missing many somethings!) but I don't understand why this question should be a concern. We have many models that describe reality, and the reason they are seen to be incomplete is because of observations about reality. I think that it may be impossible to prove mathematically that 1+1 is 2. That is interesting, but it has no bearing on why F=MA is known to not be accurate as velocities approach c, and why it is known that relativity cannot make predictions when the force of gravity becomes very large.

If we ever formulate a mathematical model of everything that is consistent across all known and hypothesized physical situations, and every prediction that model makes is verified by experiment and observation, I expect few will be concerned that we cannot prove from a finite list of axioms that 1+1 is 2, and so by implication we cannot establish any mathematical proof the model from a purely mathematical perspective. (I have probably not phrased that lat part correctly at all, but I hope that is the gist of it).

 

[stevendaryl]

I may be missing something (I am certainly missing many somethings!) but I don't understand why this question should be a concern. We have many models that describe reality, and the reason they are seen to be incomplete is because of observations about reality. I think that it may be impossible to prove mathematically that 1+1 is 2. That is interesting, but it has no bearing on why F=MA is known to not be accurate as velocities approach c, and why it is known that relativity cannot make predictions when the force of gravity becomes very large.

If we ever formulate a mathematical model of everything that is consistent across all known and hypothesized physical situations, and every prediction that model makes is verified by experiment and observation, I expect few will be concerned that we cannot prove from a finite list of axioms that 1+1 is 2, and so by implication we cannot establish any mathematical proof the model from a purely mathematical perspective. (I have probably not phrased that lat part correctly at all, but I hope that is the gist of it).

Well, it is certainly not true that "it is impossible to prove that 1+1=2". That's about the easiest proof there is.

As I pointed out, mathematical incompleteness certainly can come into play in physics. Certain questions of pure physics, such as "Is this configuration of planets stable?" or certain questions about spectral gaps in semiconductors (https://www.tum.de/en/about-tum/news/press-releases/short/article/32791/) might unsolvable. Even if we have the complete theory of everything, there would still be questions that we couldn't answer about that theory.

 

[Grinkle]

Well, it is certainly not true that "it is impossible to prove that 1+1=2". That's about the easiest proof there is.

Sorry, I saw that on a pop-math documentary years ago and I am certainly mis-quoting and simultaneously taking whatever I do remember out of context.

If one is ok to unbound the life of the solar system ignoring that the sun will explode sometime, then what you argue makes sense to me. I can see how even with a solid theory of orbital mechanics it might be impossible to use that theory to prove that orbits are stable for an infinite time. Is that truly a consequence of Godel's theorem? (not implying rhetorically that I doubt it, I am really asking)

 

[stevendaryl]

If one is ok to unbound the life of the solar system ignoring that the sun will explode sometime, then what you argue makes sense to me. I can see how even with a solid theory of orbital mechanics it might be impossible to use that theory to prove that orbits are stable for an infinite time. Is that truly a consequence of Godel's theorem? (not implying rhetorically that I doubt it, I am really asking)

I don't actually know if that problem is unsolvable, or not. I suspect that it is, but I don't know of a proof.

But in any case, that's the sort of problem that Godel's theorem says might be unsolvable, proving that something is true for all integers, or all times.

 

[Grinkle]

Thanks, Steven, I was troubled by something and you helped me bring it into focus. I opine that we needn't be any more concerned over whether a theory of everything is impossible to prove that we are concerned about whether a theory of anything (eg orbital mechanics) is impossible to prove. There is nothing special or limiting about a theory of everything that will suddenly invoke the consequences of Godel's theorem, those limitations such as they are have been with us all along.

 

[friend]

In order for Godel's incompleteness theorem to apply to physics, you'd have to prove that the mathematical laws of physics were derived from some system of logic. For those theorems only apply to systems of logic. But presently, we don't know if the laws of physics can be derived from logic. Presently it is all just guess-work that is confirmed or rejected by experiments. Perhaps in the future we can find some principles of logic that dictate where the laws of physics come from. Then we can think about whether those principles allow self-reference of a kind that results in inconsistency.

 

[friend]

Even if we have the complete theory of everything, there would still be questions that we couldn't answer about that theory.

The "laws of physics" only predict what kinds of events will happen, not that any particular event will happen. That's why they are called general laws. I don't think GIT has anything to do with the accuracy of predictions based on initial conditions (such as whether the solar system is stable or not). Just because we don't have enough information to be completely accurate does not mean that the rules we use are inconsistent or incomplete.

 

[XilOnGlennSt]

Math is the handmaiden of Science, not the queen of Science.

Put another way: Math is analytical. Science is empirical. Math deals with proofs. Science deals with tests. If a scientific theory is not open to refutation by tests, then it is not science.

By some definitions, a TOE would be a theory that unifies quantum gravity with GUT theories. By this definition, there may be convincing "TOE"s. But even if we somehow stumble on the actual underpinnings of reality, we can never prove it, but only verify and re-verify it, until someday we ultimately exhaust our bandwidth.

Others like to assert that Science can, someday, explain everything, by stating axioms and inference rules. I side with Hawking and Godel on this, that such assertions can never be confirmed.

 

[Demystifier]

Well, it is certainly not true that "it is impossible to prove that 1+1=2". That's about the easiest proof there is.

Sure, Whitehead and Russell needed only a few hundred pages to prove that.