How would anyone know if they saw the equation of everything?

Question

Given this answer by user34445 to the question: Final theory in Physics: a mathematical existence proof? https://physics.stackexchange.com/questions/76058/final-theory-in-physics-a-mathematical-existence-proof

How would anyone know if they saw the equation of everything?

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'.

so once again, does anyone think they would know if they saw the equation of everything and how would they know?

Answer 1

Options 1 to 3 are based on a fundamental misconception, since physical theories cannot be proven mathematically for correctness. At best, they can be proven mathematically as self-consistent. A "good" physical theory provides predictions which could be falsified by appropriate experiments, if one of the assumptions (axioms) is inconsistent with nature.

We know, that the two most successful theories, general relativity, and quantum theory, aren't compatible. A mathematically consistent theory covering quantum theory and general relativity in all aspects regarding consistency with experiments and observation, and predicting correctly yet unknown phenomena neither of the two generally accepted theories is able to predict correctly, and describing all known cosmological parameters, and constants of nature, would be accepted as an intermediate theory of everything. If it also predicts the results of all experiments and observations to be carried out in future, it would become the canon, as long as no contradicting observation is made.

For this, it would be necessary, to show, that the axioms of quantum theory, and the axioms of general relativity occur as approximations of the new theory for parameters which have been confirmed experimentally thus far.

Hence option 4 of the options gets closest to what can be expected.

Consistency and completeness at the same time aren't necessarily ruled out by Gödel's incompleteness theorem, since the theorem just applies to sufficiently complex formal systems, like arithmetics, or set theory. Any finite structure, as an example, can be complete and consistent at the same time.

A "final theory" would be unexpectedly simple, and obviously not possible to be simplified even more. It would be capable to predict phenomena, which wouldn't be expected to be predictable by such a simple theory, such that incidential correctness would be exceedingly unlikely.

But even then "they" wouldn't know for absolutely sure.