Impact of Gödel's incompleteness theorems on a TOE

[PhysDrew]

Well I've searched this forum and didnt see this topic before, so I was wondering what the general consensus is on the impact of Godel's theorem (if any) on a possible TOE?
I'm still wading through it all so just wanted some other opinions...

 

[mathman]

Godel's theorem is a mathematics foundation subject. It has nothing to do with physics (TOE).

 

[PhysDrew]

Godel's theorem is a mathematics foundation subject. It has nothing to do with physics (TOE).

Mmmm but put simply physics is based on mathematics, it is written in the language of mathematics. Stephen Hawking has addressed the problem, and has conceeded that it does apply;

http://www.hawking.org.uk/index.php/lectures/91

So why has it nothing to do with physics?

 

[Chronos]

The philosophy forum is more appropriate for this thread.

 

[Chalnoth]

Well I've searched this forum and didnt see this topic before, so I was wondering what the general consensus is on the impact of Godel's theorem (if any) on a possible TOE?
I'm still wading through it all so just wanted some other opinions...

The basic argument that Hawking was making on this subject a few years back is that even if we do discover a theory of everything, physicists would still have a lot of work to do, because due to Goedel's incompleteness theorem, it would be impossible to ever discover all of the consequences of the theory of everything. Thus it's basically an argument that a discovery of a theory of everything, if it occurs, would still leave physicists with lots of work to do.

 

[Kevin_Axion]

I thought Godel's Incompleteness Theorem was only applicable to axiomatic systems. Does Physics have such systems or only pure mathematics?

 

[Chalnoth]

I thought Godel's Incompleteness Theorem was based applicable to axiomatic system. Does Physics have such systems or only pure mathematics?

I think the expectation is that a theory of everything, whatever that may be, would be an axiomatic system.

 

[Kevin_Axion]

What would justify it being an axiomatic system? Pure Mathematics relies on the notion of axioms, Physics requires experimentation.

 

[Chalnoth]

What would justify it being an axiomatic system? Pure Mathematics relies on the notion of axioms, Physics requires experimentation.

The point is that axiomatic systems are not arbitrary. Some sets of axioms produce well-behaved mathematical structures, some do not. So the sorts of mathematical structures available is limited in some sense.

The question, then, is which of these mathematical structures is isomorphic to reality. If we ever do manage to eliminate all but one mathematical structure as describing our reality, then that will be our theory of everything. And that one mathematical structure will probably be an axiomatic system.

 

[bcrowell]

There is a good book on this kind of thing: Franzen, Godel's Theorem: An Incomplete Guide to Its Use and Abuse. There are lots of different reasons why Godel's theorem is not relevant to the search for a TOE:

Physics is not axiomatic system.

We don't have a TOE, so we don't know whether Godel's theorem would apply to it, even assuming that it could be made into an axiomatic system. Godel's theorem only applies to certain types of axiomatic systems. For example, it does not apply to elementary Euclidean geometry, which can be proved to be consistent.

It is possible to prove that one axiomatic system is equiconsistent with another, meaning that one is self-consistent if and only if the other is. If we had a TOE, and we could make it into an axiomatic system, and it was the type of axiomatic system to which Godel's theorem applies, then it would probably be equiconsistent with some other well known system, such as some formulation of real analysis. Any doubt about the self-consistency of the TOE would then be equivalent to doubt about the self-consistency of real analysis -- but nobody believes that real analysis lacks self-consistency.

Finally, there is no good reason to care whether a TOE can't be proved to be self-consistent, because there are other worries that are far bigger. The TOE could be self-consistent, but someone could do an experiment that would prove it was wrong.

 

[atyy]

There is a good book on this kind of thing: Franzen, Godel's Theorem: An Incomplete Guide to Its Use and Abuse. There are lots of different reasons why Godel's theorem is not relevant to the search for a TOE:

Physics is not axiomatic system.

We don't have a TOE, so we don't know whether Godel's theorem would apply to it, even assuming that it could be made into an axiomatic system. Godel's theorem only applies to certain types of axiomatic systems. For example, it does not apply to elementary Euclidean geometry, which can be proved to be consistent.

It is possible to prove that one axiomatic system is equiconsistent with another, meaning that one is self-consistent if and only if the other is. If we had a TOE, and we could make it into an axiomatic system, and it was the type of axiomatic system to which Godel's theorem applies, then it would probably be equiconsistent with some other well known system, such as some formulation of real analysis. Any doubt about the self-consistency of the TOE would then be equivalent to doubt about the self-consistency of real analysis -- but nobody believes that real analysis lacks self-consistency.

Finally, there is no good reason to care whether a TOE can't be proved to be self-consistent, because there are other worries that are far bigger. The TOE could be self-consistent, but someone could do an experiment that would prove it was wrong.

But does the incompleteness theorem show that a TOE is not possible, even in principle?

 

[Kevin_Axion]

How can it show it's impossible if we don't know if a TOE is an axiomatic system? At this point it isn't really applicable.

 

[atyy]

Is a TOE not an axiomatic system?

 

[martinbn]

Even if TOE is an axiomatic system, why should it be incomplete? There are complete systems for example the Euclidean geometry.

 

[atyy]

Arithmetic is incomplete.

We probably need arithmetic in the TOE, no?

 

[bcrowell]

Is a TOE not an axiomatic system?

We don't have one yet, so we don't know. But, for example, the standard model has never been stated as an axiomatic system according to the definition used in Godel's theorems. The definition used in Godel's theorems is extremely strict. For example, here is a formal statement of Euclidean geometry: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.9012 Physicists never want or need to do anything with this level of formality.

-Ben

 

[D H]

PhysDrew: I take it that you are quibbling over what "everything" in "theory of everything" means. Who cares? Certainly not physicists.

All this discussion of Euclidean geometry is a bit of a red herring, for at least two reasons. #1: Even Euclidean geometry, at least where it intersections algebraic geometry, is incomplete or inconsistent. The Pythagorean theorem does it in. #2: Physics uses real numbers and more complex number systems, derivatives, all that. That places the mathematics used by physicists smack dab in the middle of a system of sufficient power so as to be subject to Godel's theorems. Once again, who cares? Physicists? No.

 

[bcrowell]

Arithmetic is incomplete.

We probably need arithmetic in the TOE, no?

It's not immediately obvious to me that we do. I know that sounds nuts, but this kind of thing is not necessarily intuitive. You would think that since the reals are a bigger, fancier mathematical system than the natural numbers, then since arithmetic is incomplete, the reals would have to be as well. But that's not the case. The elementary theory of the reals is equiconsistent with the elementary theory of Euclidean geometry, which is provably consistent. It's quite possible that a TOE could be expressed in geometrical language, without the use of any arithmetic.

 

[bcrowell]

All this discussion of Euclidean geometry is a bit of a red herring, for at least two reasons. #1: Even Euclidean geometry, at least where it intersections algebraic geometry, is incomplete or inconsistent. The Pythagorean theorem does it in.

This is incorrect. See http://en.wikipedia.org/wiki/Complete_theory for the relevant notion of completeness and incompleteness.

#2: Physics uses real numbers and more complex number systems, derivatives, all that. That places the mathematics used by physicists smack dab in the middle of a system of sufficient power so as to be subject to Godel's theorems.

This is also incorrect. See #18. The informal notion of "power" you have in mind is not the appropriate concept for discussion Godel's theorems. E.g., the reals and the complex numbers are equiconsistent, because you can model the complex numbers using the reals: http://en.wikipedia.org/wiki/Model_theory

 

[D H]

This is incorrect. See http://en.wikipedia.org/wiki/Complete_theory for the relevant notion of completeness and incompleteness.

Show me a proof of the Pythagorean theorem that does not involve multiplication distributing over addition, then.

This is also incorrect. See #18. The informal notion of "power" you have in mind is not the appropriate concept for discussion Godel's theorems.

Any theory based on the reals is, as far as I know, subject to Godel's theorems.

If you want to do physics without talking about measurement you are not really doing physics in my mind.

 

[bcrowell]

Show me a proof of the Pythagorean theorem that does not involve multiplication distributing over addition, then.

Euclid's original proof doesn't refer to numbers or multiplication at all. In any case, it makes a difference whether you're talking about addition and multiplication of natural numbers or of reals; only the latter would be used in a proof of the Pythagorean theorem.

You seem to be asserting that Alfred Tarski's life work is fundamentally flawed. http://en.wikipedia.org/wiki/Tarski's_axioms

Any theory based on the reals is, as far as I know, subject to Godel's theorems.

This is incorrect.

If you want to do physics without talking about measurement you are not really doing physics in my mind.

Nobody said anything about doing physics without measurement.

 

[atyy]

We don't have one yet, so we don't know. But, for example, the standard model has never been stated as an axiomatic system according to the definition used in Godel's theorems. The definition used in Godel's theorems is extremely strict. For example, here is a formal statement of Euclidean geometry: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.9012 Physicists never want or need to do anything with this level of formality.

-Ben

Yes, but the standard model is not a TOE. My point of view is that there is no TOE. "The physical theory that can be formulated cannot be the final ultimate theory ... The unformulatable ultimate theory does exist and governs the creation of the universe (http://books.google.com/books?id=1f...ook_result&ct=result&resnum=1&ved=0CC4Q6AEwAA)"

But there certainly is a distinguished line of thought that a TOE exists, and hunting it down is the goal of some sub-discipline of physics.

 

[bcrowell]

Yes, but the standard model is not a TOE.

My point was simply that most likely no broad physical theory has ever been formulated as an axiomatic theory in the sense defined in Godel's theorems, and probably none ever will be. (A possible exception is that I did read somewhere that someone had formalized all the propositions in Newton's Principia and worked on checking them with a computerized proof system. Whether this formalization constitutes a physical theory, or just one aspect of it, is a different matter)

 

[friend]

Science seeks an explanation for all things and assumes that all facts are consistent and reasonable. So it is tempting to think that a TOE can be derived from logic alone. As I understand it deductive logic and predicate logic have been proven to be complete. Now if it happens to be that math is introduced as a way to parameterize the spaces used to construct the topologies involved in the unions and intersections of logic, then does the system become incomplete because we coordinatized the spaces involved? Or is it more correct that since the underlying topologies are independent of which coordinates are used, that the math is incidental and should not be used to judge the consistency of the system?

 

[bcrowell]

Here's the book on the formalization of part of the Principia: Jacques Fleuriot, A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton's Principia, https://www.amazon.com/dp/1852334665/?tag=pfamazon01-20 Google books will also let you peek through a keyhole at it.

 

[ordered_chaos]

These proofs of geometry are relative, not absolute proofs. Also, simple systems isolated from interference can indeed be proven consistent and complete. But as soon as things need more robustness, I think that's where Godel's theorem kicks in.

Regardless, physicists seem not give flying fart about it. I do find this strange. They have a lot faith in the language they are using. But I think the closer you get to the truth, the harder it to tinker with less you disturb your experiments.

As physics gets more fundamental, i.e. information theory, I believe these questions surrounding the foundation of math will need to be considered.

 

[bcrowell]

These proofs of geometry are relative, not absolute proofs.

Could you explain what you mean by this?

Also, simple systems isolated from interference can indeed be proven consistent and complete. But as soon as things need more robustness, I think that's where Godel's theorem kicks in.

And this?

 

[yossell]

On the question of whether geometry is subject to Godel's theorems...

I think it's not very hard to effectively embed arithmetic in geometry. Using some fairly simple geometric constructions, you can effectively define + and x geometrically, and then prove incompleteness. I think you need at least two dimensions to do this, but it can be done.

On another issue, I'm just intrigued as to what people think the alternatives to an axiom system is. I don't see axiom systems as abstract constructs of mathematical logic. Euclid had an axiom system for geometry way before the mathematical logicians hit the scene. I just think of axiom systems as an explicit list of the principles that constitute a theory. I would have thought that, without some such list, the theory wouldn't be well defined.

 

[Kevin_Axion]

The thing is that physics in general doesn't require axioms to have a well-defined theory.

 

[yossell]

The thing is that physics in general doesn't require axioms to have a well-defined theory.

Ok - I'm willing to believe it. But I'm interested in how you do get a well defined theory without, at some point, appealing to axioms.