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.