Truth After Proof: Formal Epistemics in Modern Mathematics

This work examines how the relationship between proof, verification, and truth has changed in modern mathematics. As mathematical arguments have grown in scale, abstraction, and formal sophistication, the classical ideal—where a proof is a finite object that can, in principle, be fully inspected by an individual—no longer captures how mathematical certainty is actually established. The manuscript introduces formal epistemics as a framework for analysing this shift. Rather than questioning the truth of mathematics, it investigates the architecture of justification: where certainty resides, how it is distributed, and what its limits are. Three canonical case studies structure the analysis: Fermat’s Last Theorem, illustrating exact truth supported by large-scale, distributed verification and transitive trust. The Prime Number Theorem, illustrating asymptotic truth whose content is provable but never fully instantiated at any finite stage. Gödel’s incompleteness theorems, illustrating principled limits on provability, where true statements exist that cannot be proved within the formal systems that express them. Formal verification and proof assistants are treated not as solutions to these tensions, but as tools that make them explicit by relocating certainty and exposing dependencies. The work argues that modern mathematics has not lost rigor, but has diversified the forms that rigor takes. The manuscript is intended for mathematicians, logicians, philosophers of mathematics, and scholars interested in the epistemology of formal systems. This Zenodo release constitutes Version 1.1, considered stable and archival.