Why Do Informal Proofs Conform to Formal Norms?

Kant discovered a philosophical problem with mathematical proof. Despite being a priori, its methodology involves more than analytic truth. But what else is involved? This problem is widely taken to have been solved by Frege’s extension of logic beyond its restricted (and largely Aristotelian) form. Nevertheless, a successor problem remains: both traditional and contemporary (classical) mathematical proofs, although conforming to the norms of contemporary (classical) logic, never were, and still aren’t, executed by mathematicians in a way that transparently reveals why these proofs—written in the vernacular to this very day—succeed in conforming to those norms.