Why Do We Believe Theorems?

TL;DR

This paper examines why mathematicians trust theorems, arguing that formal derivations do not significantly enhance confidence, and explores psychological and social influences on belief in mathematical results.

Contribution

It challenges the formalist view by showing that formal derivations cannot, even in principle, increase confidence in theorems given current knowledge.

Findings

Formal derivations do not significantly increase confidence in theorems.

Psychological and social factors influence belief in mathematical results.

Formal derivations are limited by their length and complexity.

Abstract

We investigate the reasons of having confidence in mathematical theorems. The formalist point of view maintains that formal derivations underlying proofs, although usually not carried out in practice, contribute to this confidence. Opposing this opinion, the main claim of the present paper is that such a gain of confidence obtained from any link between proofs and formal derivations is, even in principle, impossible in the present state of knowledge. Our argument is based on considerations concerning length of formal derivations. We also discuss psychological and social factors that contribute to building belief in theorems.