Does the reproducibility crisis affect mathematics?

Abstract

Reproducibility crisis in science accepted by academia as acute issue (including the problem of funding). The goal of this article is to discuss how the phenomenon and the crisis of repro­ducibility is manifested in mathematics, and how it perceived by the mathematical commu­nity. We argue that traditional approaches to the analysis of the proof in mathematics presup­pose its visibility, the possibility of fundamental verification of all steps of the proof by competent members of the scientific community. The meaning of the mathematical proof seen in its aim to convince community members of the correctness as a whole, and validity of all its components. By presenting a proof, its author takes on the (moral) responsibility that the statement (theorem) she formulates is correct, and everyone can repeat the path that leads to its justification. The increasing complexity of mathematical proofs in the course of its his­torical development and, above all, the expansion of computers as important elements of the proof, leads in some cases to the loss of its visibility. Thus, the shift of the reception of the proof to indirect signs is rather evident (confidence in the correctness of algorithmic pro­cedures and provers). All this leads to the need to reconsider views on the degree of reliability of mathematical proofs and their assessment not as reliable, but only as plausible. This is the basis for characterizing the new era in the development of mathematics as “post-rigor­ous”, which raises serious problems related to comprehension and analysis of reproducibility in mathematics, and the status of proof in this era. These problems especially relevant in the context of expansion into the sphere of mathematical creativity of computer-based sim­ulation and computers as a tool of discourse.