Final theory in Physics: a mathematical existence proof?

Question

Some time ago, I read something like this about the issue of "a final theory" in Physics:

"Concerning the physical laws, we have several positions as scientists

1. There are no fundamental physical laws. At the most elementary level, the Universe/Multiverse is essentially chaotic and anarchical. There are no such laws.

2. There are a continuous sequence of more and more precise theories, but there is no a final theory. Physics will be always evolving from one approximate theory to another bigger and more accurate. In the end, we will be also able to find a better theory and additional levels of complexity or reality.

3. There is a final theory explaining everything, and we will find if and only if:

   i) We are clever enough to find such a theory. ii) We make good and sophisticated enough mathematics. iii) We guess the right axioms/principles/ideas. iv) We interpretate data correctly and test the putative final theory with suitable instruments/experiments. "

Supposing 3) is the right approach...

Question: How could we prove the mere mathematical existence of such a theory? Wouldn't it evade the Gödel's incompleteness theorem somehow since, as a Theory of Everything, it would be explain "all" and though it should be mathematically self-consistent? How could a Theory of Everything be a counterexample of Gödel's theorem if it is so, or not?

Note: The alledged unification of couplings in supersymmetric theories is a hint of "unification" of forces, but I am not sure if it counts as sufficient condition to the existence of a final theory.

Complementary: Is it true that Hawking has changed his view about this question?

SUMMARY:

1') Does a final theory of physics exist? The issue of existence should be tied to some of its remarkable properties (likely).

2') How could we prove its existence or disproof it and hence prove that the only path in Physics is an infinite sequence of more and more precise theories or that the Polyverse is random and/or chaotic at the most fundamental level?

3') How 1') and 2') affect to Gödel's theorems?

I have always believed, since Physmatics=Physics+Mathematics (E.Zaslow, Clay Institute) is larger than the mere sum that the challenge of the final theory should likely offer so hint about how to "evade" some of the Gödel's theorems. Of course, this last idea is highly controversial and speculative at this point.

Answer 1

This is an interesting questionning, rather than question, as I am not sure what is the question stated, because too many issues are questionned or questionnable

I think it is different, but very much related to a previous question regarding the physical provability of the Church-Turing thesis stating that any computing device that can be built will compute no more than what is computable by a Turing machine.

One issue with the Church-Turing thesis is that the concept of proof in an axiomatic theory is fundamentally the same as the concept of a computer program, i.e., a Turing Machine. This is not in the sense that a program can churn out theorems and proofs, as in Ron Maimon's presentation of Gödel's proof, but because a program can be "read" as a proof of its specification ("Given a value x such that P(x) holds, there is a result y such that Q(x,y) holds.") and conversely a proof may be read as a program that actually computes whatever is stated by the theorem. This presentation is of course a very simplified version of a result by Curry and Howard (1980) which is still researched.

Hence one major issue with the possible limitations of calculability, whether there are such physical limitations or not, and whether we can prove or not the existence of such limitations, is that mathematical proofs are directly concerned by the same limitations. One crucial aspect of such limitations, which I will address below is the denumerable nature of intellectual processes.

We can assume, we must assume, that our way of doing mathematics, including physical theories, is consistant. This is really (to me at least) a physical view of it: we do find inconsistencies, usually called paradoxes, but there are resolved (have been so far) like experimental inconsistencies in physics, by refinements of the théories and evolution of concepts to overcome the difficulty and state the problems appropriately.

Assuming mathematics is essentially consistent is essential, because whatever we can prove should not be questionned by future extensions, if any are physically possible, of the concepts of calculability or provability.

Now some results make deep assumptions that are not always obvious to interpret. In the case of Gödel incompleteness result, one major aspect is that logical formulae, theorems and proofs can be encoded as integers. This means that our logical deduction systems are fundamentally denumerable entities (like Turing machines). If it turned out that a breakthrough in physics allowed us to deal effectively with non-denumerable systems, then results relying on this denumerability would be in question. This is precisely the case for Gödel incompleteness result, as stated (possibly it could come back in another form).

I addressed this denumerability aspect in my answer to the question on the physics provability of Church-Turing thesis. At the time this answer was entirely based on my informal understanding of these issues. Intending to improve a bit on my answer, I looked for some litterature, and the topic is currently actively researched. While my knowledge of this literature remains more than shallow, it seems that my intuition was correct, that a proper handling of the fundamentally discrete (or denumerable) character of calculability, and provability, is essential in the current state of the art, for deriving Church-Turing thesis from the laws of physics, and that continuity or real numbers are a major issue.

One approach I have looked at (I am limited to papers in open access on the web) relies on assuming a specific property of the physical world, presented as dual of the limitation on the speed of light and information, which is a limitation on the space density of information, both limitations together ensuring density limitation in space-time. The translation of this new law in physical terms can actually be subtle to account for various existing physical laws. This apparently excludes unregulated use of real numbers.

If this limited density law is actually verified, I think it would also mean that Gödel theorem is also a consequence of physical laws.

Whether such a limited information density law is actually verified is another matter. If it is not, then doors remain open for an extension of the concepts of calculability and provability.

In such a case, assuming that our mathematics are otherwise consistent, all provable results would remain provable, but we might be able to prove new theorems that were true but not provable in the classical denumerable setting.

So the answer to the question whether a theory of Everything would evade Gödel's incompleteness theorem is very much dependent on what Everything is, since it actually determines the context in which calculability or provability have to be defined. Would a theory of Everything include a law limiting information density.

Note that even with Gödel's result being valid, there could be a possibility of a theory of Everything, in which all true facts regarding the universe would indeed be true. It is just that you would not be able to prove it (so that the universe would keep some mystery for us to wonder in starry nights).

On the other hand, there could be no such theory of everything. But, what would the ultimate definition of a theory if physics allowed us to question the discrete nature of the language in which they are expressed ?

For the rest, other than taking 42 as ultimate answer, I can only suggest leaving the matrix to get the Truth about our world, or reading Simulacron-3.

Answer 2

There exists the platonic/pythagorian view, that mathematics exists in an idea space that nature fulfills. In this case mathematical proofs have a meaning for observations also.

On the other hand there exists the view that mathematical theories modeling nature can never be proven correct, they can only be validated by data or falsified even by one datum.

The answer to your summary depends on the philosophical orientation of the answerer.

SUMMARY:

1') Does a final theory of physics exist? The issue of existence should be tied to some of its remarkable properties (likely).

At some point, when I was learning quantum mechanics I was of the platonic school: that mathematics is a matrix that nature will fulfill. After years doing experiments I am of the second view , that the more we dig the more we find that cannot be modeled by the latest theory; it is a never ending task.

2') How could we prove its existence or disproof it and hence prove that the only path in Physics is an infinite sequence of more and more precise theories or that the Polyverse is random and/or chaotic at the most fundamental level?

In my view a theory can never be proven, only validated by data, so this question has no answer.

3') How 1') and 2') affect to Gödel's theorems?

In my view , since it is an open ended search of mathematical theories to describe data the theorem is fulfilled.

Answer 3

My idea of a final theory is that all possible experimentally doable situations will have a calculable answer from basic principles in a logically manner. I don't think that this would violate godel's theorem in any way. For example in any situation in plane geometry will have a precisely calculable answer in that sense it is complete.

Answer 4

Physics presupposes Mathematics. So any "Final Theory of Physics" would also need to presuppose Mathematics for it to have efficacy. Therefore, because this 'Final Theory must presuppose Mathematics, it cannot prove (or disprove) the validity of Mathematics. Likewise, Mathematics presuppose the validity of Logic, so Mathematics cannot therefore prove (or disprove) the validity of Logic. Therefore Physics cannot prove (or disprove) the validity of Logic or Math. Since Physics cannot prove (or disprove) all metaphysical laws (such as Logic or Mathematics), therefore Physics is incomplete.

Now you might say, bah, that is merely math and logic - metaphysics not physics, which is true, except that anything physical is necessarily also metaphysical since anything that actually exists, can be thought to exist. Physics is a subset of metaphysics. We can imagine many things, even impossible things, that do not actually exist, but nothing that actually exists cannot be 'not imagined', whether or not we experience it directly. The domain of all things physical must be a subset of the domain of all things metaphysical. So if physics is incomplete metaphysically, it must also be incomplete physically, since physics is a subset of metaphysics. Therefore, we have good reason to believe that a 'Physics Final Theory' must be incomplete.

Notwithstanding these very good reasons for believing 'The Final Theory in Physics' must be incomplete, lets assume, for the sake of argument, that such a theory did exist, and was both consistent and complete, contradicting both of Gödel's Incompleteness Theorems; such a theorem could then prove its own consistency, the question would become 'How complex would proofs be involving this theorem?' Would proving the consistency or completeness of this 'The Final Theory of Physics' be an NP-complete problem? What about proving both consistency and completeness?

The following are possible:

Option 1. A 'Final Theory of Physics' exists that is both consistent and complete.

Although we've already shown that it could not be complete, we've assumed otherwise. We know that there are physical problems in Quantum Mechanics that are NP-Complete hard, so if it were a Final Theory and complete, proving this would also be an NP-Complete hard problem since proving some of its parts are. Similar reasoning shows that proving it's consistency would also be NP-Complete, so proving this theory to be both complete and consistent will take beyond the age of the universe.

Option 2. A 'Final Theory of Physics' exists that is consistent, but is not complete - with the following possibilities:

2.1. Proving the consistency of this theory is an NP-complete problem, so effectively not provable (at least not in the age of the universe)

2.2. Proving the consistency of this theory is not an NP-complete problem and is provable in some finite amount of time.

Looking at 2.2, we've already shown that because NP-Complete problems exist in Quantum Mechanics, which would be parts of the Final Theory, proving the consistency of the theory is an NP-complete problem, which contradicts our presupposition in 2.2. So Option 2.2 is impossible.

Option 3. A 'Final Theory of Physics' exists, that is complete, but not consistent.

Because the theory in this option is not consistent, we could both prove and disprove that it is complete, a contradiction. Any theory that contradicts itself is useless as a theory. This option's 'Final Theory' is useless.

Option 4. A complete 'Final Theory of Physics' does not exist, but better and better aggregate approximations exist which are consistent, but not complete, which give us better understandings of the laws of physics over time (I am assuming the laws of physics are universal since the non-universality of physical laws is another topic entirely).

What can we conclude from this? We can conclude that even if such a Final Theory exists that is consistent, it is going to be impossible to prove that it is the 'Final Theory', whether or not it is complete. So there is really no way to know once we've discovered the 'Final Theory'.