This entire comment is arguing against an opinion I believe nobody even stated. But why would we waste time trying to discover the metaphysics of an ultimately incorrect description of physical reality?
If we think our models are right, we will push them until we find out where they are wrong. That's what we do, over and over again. Pushing classical physics to its breaking point spawned quantum mechanics. We know there's a breaking point for present day physics too.
Models in Physics involve experimentation, validation, interpretability
With mathematics, we can describe hypotheses of reality, but there may be details and nuances to reality which we're simply not able to observe and thus not able to empirically consider.
I said “description.” You’re adding additional constraints that may or may not be warranted.
All existing attempts to describe physics have encountered some sort of incompleteness. Newtonian physics yields ordinary differential equations, which can encode Turing-complete problems. [0][1] Quantum mechanics yields Hamiltonian operators, which can encode Turing-complete problems too. [2][3] Both of these paradigms predate Turing's work; physics was Turing-complete a long time before you came along to look at the problem.
I can empathize with your original point: Surely reality exists! It seems so real! But reality can be real without existing. Ultimately, our models are only hypotheses about reality, and what they have shown us here is that even our models are so complex as to admit problems that are not going to be solved by mere assertion of existence.
[0] https://link.springer.com/chapter/10.1007%2F11494645_21
[1] https://sapientia.ualg.pt/bitstream/10400.1/1008/1/05-GCB-st...
[2] https://www.nature.com/articles/nature16059
[3] https://eprints.ucm.es/38062/1/spectral-gap_supplementary.pd...
That’s a bald assertion, not an answer. And one can certainly imagine descriptions of physics that don’t require a formal system.
Here’s one: “every particle remains at rest for all t.” Now, that doesn’t describe our universe very well. But it might be a perfectly good description of a single-electron universe.
There’s no a priori reason to assume that a valid description of physics is axiomatizable in a formal system. Aruguments to the contrary are necessarily based on our own ignorance, not on anything fundamental.
Interesting definition of “while”. Even relative to human civilization (which itself is a drop in the bucket) we haven’t been searching all that long.