Truth After Proof: Formal Epistemics in Modern Mathematics

This work examines how the relationship between proof, verification, and truth functions in contemporary mathematics. As mathematical arguments have grown in scale, abstraction, and formal sophistication, the classical ideal—according to which a proof is a finite object that can, in principle, be fully surveyed by an individual—no longer provides an adequate description of how mathematical certainty is actually established in practice. The manuscript introduces formal epistemics as a descriptive framework for analysing this situation. Rather than questioning the objectivity or rigor of mathematics, it investigates the architecture of justification: where mathematical certainty resides, how it is justified and distributed, and what principled limits constrain it. The analysis is structured around three canonical case studies: Fermat’s Last Theorem, illustrating exact truth sustained by large-scale proofs, distributed verification, and transitive trust within an expert community. The Prime Number Theorem, illustrating asymptotic truth, where a rigorously proved universal claim has content that is expressed through limiting behaviour rather than finite instantiation. Gödel’s incompleteness theorems, illustrating a principled boundary of formal closure, where true statements exist that cannot be proved within the formal systems that express them. Formal verification and proof assistants are treated neither as remedies for these features nor as sources of epistemic crisis, but as tools that clarify them by making dependencies, assumptions, and boundaries explicit. The work argues that modern mathematics has not lost rigor, but that rigor now takes multiple, structurally distinct forms. The manuscript is intended for philosophers of mathematics, logicians, mathematically literate philosophers of science, and mathematicians interested in the epistemology of proof and formal systems. This Zenodo release constitutes Version 1.2, considered stable, submission-ready, and suitable for archival reference. Future versions, if any, will be limited to minor typographical or presentational corrections.