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Vladimir_Nesov comments on What independence between ZFC and P vs NP would imply - Less Wrong Discussion
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Please don't confuse math with "reality". Math as about axioms and proofs (well-formed finite strings), and is often a useful tool in mapping the territory, but it is just that, one of the tools.
Quantum Mechanics is a classic counterexample, as far as we know, in a sense that there is no deeper underlying theory that would predict an outcome of a measurement when QM says it cannot be determined.
Your formalist ideology is noted, but please don't state its (disputable) claims as clear truths.
Not sure what you mean by "formalist", or " claims as clear truth", but feel free to provide a different definition of mathematics acceptable to a mathematician.
Mathematics seemingly studies mathematical structures, guided by rather elusive criteria for what is worth studying, with axioms and proofs not obviously constituting the whole of its focus. In philosophy of mathematics, asserting that only formal statements and proofs make sense is known as formalism: