Here is what we know for sure:

There can be no enumerable list of axioms for the true statements of mathematics. No computational procedure could exist to determine whether propositions are valid, provable, or even equivalent. And no matter how you formulate the number-theoretic axioms, a mathematician would always have insights (for instance, about whether a Diophantine equation has a solution) that are both clearly “true” and obviously unprovable. This holds true for all digital systems.

Here is what we don’t know for sure:

The metaphysical implications.

Your distinction between science and philosophy is incorrect. Science is inductive and abductive. It can’t “prove” things deductively. That’s mathematics and philosophy.

Philosophy also determines the formal systems we use as a basis of reasoning, for instance, in science.