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

[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.

 

[jerromyjon]

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

Has anyone proved that (sq root of 2)²=2?

 

[stevendaryl]

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).

I would say that it does. Certain types of questions, of the form "Will state S ever evolve into state S'?" are not solvable, even with perfect knowledge of the initial conditions and perfect knowledge of the laws of motion by which one state evolves into another. Whether chaotic natural systems are an example or not is not clear (to me), but I don't know of any reason to think that they are not subject to Godel's incompleteness.

 

[friend]

I would say that it does. Certain types of questions, of the form "Will state S ever evolve into state S'?" are not solvable, even with perfect knowledge of the initial conditions and perfect knowledge of the laws of motion by which one state evolves into another. Whether chaotic natural systems are an example or not is not clear (to me), but I don't know of any reason to think that they are not subject to Godel's incompleteness.

If Godel's Incompleteness Theorem is the reason of unpredictability in the laws of physics, then there MUST be some underlying system of logic from which those law are derived. What do you suppose that could be? And how does self-reference enter the picture to produce said incompleteness?

 

[stevendaryl]

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.

I think you're mixing things up. As I said, the issue (or an issue---there might be other issues where computability is relevant) is our ability to predict future conditions from current conditions. The way that we know how to do that is through mathematics. We come up with a mathematical description of the state of the system of interest at a particular moment. We come up with a mathematical description of how the state changes with time. That's what a "Theory of Physics" is (possibly we also need some other tools to aid in mapping between observations in the real world to descriptions in our theory). Once you have a theory of physics, you can formulate the question: Is it possible for a system in state $S_1$ to ever evolve into state $S_2$? Once you have a theory of physics, that question is now a completely mathematical question, and such mathematical questions are subject to Godel's incompleteness theorem. There may be questions of that type that cannot be answered.

Now, of course the real world does whatever it is that it does. The real world doesn't care a hoot about Godel's incompleteness theorem. The incompleteness is only relevant to OUR ability to make PREDICTIONS about the world. I think it really does imply that there could be questions (about future conditions based on current conditions) that we have no way of answering, in general.

Of course, Godel's theorem applies to systems of a certain level of complexity. Does the real world have that level of complexity, or not? I guess that's an open question. Maybe our universe is actually finite, and there are only finitely many different possible states it can be in. In that case, the future behavior of the universe would be completely predictable. (But that wouldn't mean that WE could predict it, because maybe the predictions would require a computer larger than our universe.)

 

[friend]

Once you have a theory of physics, that question is now a completely mathematical question, and such mathematical questions are subject to Godel's incompleteness theorem. There may be questions of that type that cannot be answered.

The GIT ONLY applies if you require every mathematical formula provable in math. The laws of physics do not require every possible equation of math provable in math. It only requires a subset of that math. Math is not equivalent to physics.

 

[stevendaryl]

If Godel's Incompleteness Theorem is the reason of unpredictability in the laws of physics, then there MUST be some underlying system of logic from which those law are derived. What do you suppose that could be?

Well, our theories of physics depend at the very least on arithmetic. Here are some arithmetic facts:

1. For any integer $n$, $n+0 = n$
2. For any integer $n$, $n*0 = 0$
3. For any two integers $i$ and $j$, $(i+1) + j = (i+j)+1$
4. For any two integers $i$ and $j$, $(i+1) * j = (i*j)+j$

Any system which is capable of deriving the above general facts is subject to Godelian incompleteness. And if it isn't capable of deriving the above facts, then I would say it was too weak to use for physics.

I think you're confusing two different things: (1) What is the world really doing? (2) What can we predict about the world?

The first is not subject to Godel's theorem--the world does whatever it does, and doesn't care about anybody's theorems. The second is about our ability to reason about the world using mathematics. Godel's theorem certainly applies to that.

So when you talk about the "laws of physics", I'm not sure whether you talking about #1 or #2. I'm not claiming that Godel has anything to do with what the world actually does, but it certainly is relevant to our ability to predict what will happen in the far future.

 

[stevendaryl]

The GIT ONLY applies if you require every mathematical formula provable in math. The laws of physics do not require every possible equation of math provable in math. It only requires a subset of that math. Math is not equivalent to physics.

Math is not equivalent to physics, but predicting the future based on the past certainly involves mathematics. If there are certain mathematical questions that are unsolvable, and those mathematical questions are involved in predicting the future, then those predictions are impossible to make.

Some physical problems have been shown to be unsolvable. Making predictions about chaotic systems may be an example, or it may not be.

 

[friend]

Well, our theories of physics depend at the very least on arithmetic. Here are some arithmetic facts:

1. For any integer $n$, $n+0 = n$
2. For any integer $n$, $n*0 = 0$
3. For any two integers $i$ and $j$, $(i+1) + j = (i+j)+1$
4. For any two integers $i$ and $j$, $(i+1) * j = (i*j)+j$

Any system which is capable of deriving the above general facts is subject to Godelian incompleteness. And if it isn't capable of deriving the above facts, then I would say it was too weak to use for physics.

I think you're confusing two different things: (1) What is the world really doing? (2) What can we predict about the world?

The first is not subject to Godel's theorem--the world does whatever it does, and doesn't care about anybody's theorems. The second is about our ability to reason about the world using mathematics. Godel's theorem certainly applies to that.

So when you talk about the "laws of physics", I'm not sure whether you talking about #1 or #2. I'm not claiming that Godel has anything to do with what the world actually does, but it certainly is relevant to our ability to predict what will happen in the far future.

What are you saying, that 1+1=2 is incomplete? Of course I'm talking about epistemology and not ontology. By laws of physics one is inherently talking about the language used to describe reality. In this case the language used is math.

 

[stevendaryl]

What are you saying, that 1+1=2 is incomplete?

No, I said that any system that is strong enough to derive the following general statements is incomplete:

1. x+0 = x
2. x+(y+1) = (x+y)+1
3. x*0 = 0
4. x*(y+1) = x*y + x

Of course I'm talking about epistemology and not ontology. By laws of physics one is inherently talking about the language used to describe reality. In this case the language used is math.

And that's almost certainly subject to Godel's incompleteness theorem.

 

[friend]

Math is not equivalent to physics, but predicting the future based on the past certainly involves mathematics. If there are certain mathematical questions that are unsolvable, and those mathematical questions are involved in predicting the future, then those predictions are impossible to make.

Some physical problems have been shown to be unsolvable. Making predictions about chaotic systems may be an example, or it may not be.

Then what is the physical mechanism that implements the self-reference used in GIT? Godel was able to construct a metalanguage about mathematics using mathematics itself. This enabled him to construct the self-referential statement "This statement is not provable" - all within math itself. Now, if the laws of physics are incomplete by GIT, then where does that self-reference occur in the subset of the math used for physics? How is the metalanguage constructed within the math for physics? The math used in physics is used to refer to objects of physics, and it does not refer to other mathematical objects.

 

[stevendaryl]

Then what is the physical mechanism that implements the self-reference used in GIT?

I think you're confused about Godel's incompleteness theorem. The PROOF involves self-reference, but the conclusion has nothing to do with self-reference.

For example, we don't know how to answer the following question:

Can every even integer greater than 2 be written as the sum of two prime numbers?

That question doesn't involve self-reference. But we don't know how to solve it. (For that particular question, there is no proof one way or the other as to whether it is solvable using standard mathematics, or not.)

A general class of problems is this: Given a polynomial equation with integer coefficients, can we decide whether it has integer solutions? That general problem is known to be unsolvable.

It doesn't have anything to do with self-reference, other than the fact that self-reference is used to prove incompleteness.

 

[martinbn]

Can every even integer greater than 2 be written as the sum of two prime numbers?

That question doesn't involve self-reference. But we don't know how to solve it. (For that particular question, there is no proof one way or the other as to whether it is solvable using standard mathematics, or not.)

Is it undecideable? Just because we cannot solve the question yet, it doesn't mean it cannot be solved. Take Fermat's theorem, there was no proof for a long time. It turned out to be true. So is your example an example that shows incompleteness or is it irrelevent?

 

[stevendaryl]

Is it undecideable? Just because we cannot solve the question yet, it doesn't mean it cannot be solved. Take Fermat's theorem, there was no proof for a long time. It turned out to be true. So is your example an example that shows incompleteness or is it irrelevent?

It's an example of the type of problem that is unsolvable: Do all elements of this infinite set have property P?

Every specific yes/no question is potentially solvable someday. But there are sets of yes/no questions such that it is provable that it is impossible to solve every problem in the set.

I've already given one example: Does such and such polynomial equation have an integer solution? For any method you choose to answer such questions, there will be questions that your method gets wrong.

Mathematically, there is no function which given an equation (written in ascii, for definiteness) will return True or False depending on whether the equation has an integral solution.

 

[stevendaryl]

Is it undecideable? Just because we cannot solve the question yet, it doesn't mean it cannot be solved. Take Fermat's theorem, there was no proof for a long time. It turned out to be true.

There is actually a double layer of uncertainty in mathematical proofs: Given a mathematical system (such as ZFC set theory), there are statements that are neither provable nor refutable in ZFC. It would be nice to have natural examples, but often proving that something is unprovable is as hard as proving.

Fermat's last theorem is an example. You might be able to prove it. Or you might be able to disprove it. But you could never prove that it's undecidable, because such a proof could be turned into a proof that it is true:

If it's false, it's provably false. Turning that around, if it's not provably false, then it's true. So if it is undecidable, then it must be true.

So if it's undecidable, you could never prove that it is undecideable.

 

[friend]

It's an example of the type of problem that is unsolvable: Do all elements of this infinite set have property P?

Every specific yes/no question is potentially solvable someday. But there are sets of yes/no questions such that it is provable that it is impossible to solve every problem in the set.

I've already given one example: Does such and such polynomial equation have an integer solution? For any method you choose to answer such questions, there will be questions that your method gets wrong.

Mathematically, there is no function which given an equation (written in ascii, for definiteness) will return True or False depending on whether the equation has an integral solution.

Your examples of unprovability would have been well known before Godel so that if they were definitively shown to be precises unprovable, then Godel would have never had need to construct the GIT. For there would be the examples of unprovability without Godel, and everyone would have know it. So your examples don't mean a thing. They only show that we don't know how... yet.

 

[stevendaryl]

Your examples of unprovability would have been well known before Godel so that if they were definitively shown to be precises unprovable.

Before Godel, it was not known whether there were any undecidable statements (neither provable nor disprovable). So I don't know what you are talking about. Some people (most people?) believed that any meaningful ([edit] mathematically precise) statement could either proved or disproved.

This thread is sounding argumentative for no purpose. Is there actually an issue to discuss, or are we just arguing?

 

[Drakkith]

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