Home – Home – Jornals – Philosophy of Sciences 2023 number 1

One of the main requirements for mathematical proofs is that of rigor. Rigor guarantees that there is a logical connection between the proposition being proved and the assumptions or axioms being accepted. Traditionally, the rigor of a mathematical proof is thought to be ensured by appealing to formalization. However, this understanding does not correspond to mathematical practice. The vast majority of proofs (both in the past and at present) are given informally and accepted by the community of expert mathematicians without any formalization. The article presents an alternative position regarding the rigor of mathematical proof proposed by S. De Toffoli, which is supposed to be consistent with practice. The article is divided into four parts. The first part considers the traditional understanding of proof as a surveyable and formalizable construct used for persuasion; it discusses the structure of proof and the standard understanding of rigor provided by formalization. The second part of the article discusses the practice of referring to “manipulative imagination” with respect to topology and shows the special status of this ability in “filling in” gaps in informal proofs. It is argued that this ability is developed through an appropriate training and involves only such manipulations that are justified from a strictly formal point of view. The third part discusses an alternative to the traditional notion of proof as a shareable, transferable and formalizable construct, components (“steps”) of which can be enumerated. Also, it introduces the notion of simil-proof which is a means of explaining the mathematical practice of accepting incomplete or error-containing proofs. In the fourth part of the article, the final points are formulated concerning the commitments of accepting relativism and anti-realist ontology when explaining the notions of proof and rigor.