Yet we use it to reason about things, so formalizing and finding inconsistencies leaves less space for exploiting it. Also even though math is backed by a form

Yet we use it to reason about things, so formalizing and finding inconsistencies leaves less space for exploiting it. Also even though math is backed by a formal system in theory, in practice no one has time to completely verify mathematicians' work bottom-up from axioms (at least until we become better at proof assistants). Ultimately it comes down to intuition and trust, i.e. these people understand the proof, and I trust them, so I think the proposition holds. And similarly in principle a large group of people can decide that they don't trust the axioms (they are arbitrary in a sense, just like laws), yet it formal methods in math are still useful.