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

[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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