Do you have evidence in favor of the idea that, say, there exists a finitely-axiomatized formal system which describes reality? We've been searching for a while, and evidence so far suggests that it's not going to be that easy. Tarski's Undefinability is a non-trivial barrier.
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'm merely helping you catch up on the past century of improvements in logic. What would your description consist of, if not a finite listing of axioms? I bet that you'll complain if my description includes a susbystem for arithmetic, in accordance with Gödel's Incompleteness.
It necessarily includes a formal system. That much has been hammered into the ground already, and you'll discover it for yourself as soon as you attempt to formalize your description.
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.
> It necessarily includes a formal system. That much has been hammered into the ground already, and you'll discover it for yourself as soon as you attempt to formalize your description.
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.
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.