Mathematical Proof Between Generations

TL;DR

This paper explores the evolving role of proofs in mathematics, emphasizing the potential of computer proof assistants to bridge the gap between formal and intuitive understanding, and discusses the implications for mathematical practice.

Contribution

It analyzes the differences between theoretical and practical definitions of proofs and advocates for integrating computer-assisted verification to enhance mathematical rigor and accessibility.

Findings

Computer proof assistants can verify complex proofs beyond human capacity.

Mathematical proofs are becoming increasingly sophisticated and formalized.

The paper encourages collaboration between mathematicians and computer scientists.

Abstract

A proof is one of the most important concepts of mathematics. However, there is a striking difference between how a proof is defined in theory and how it is used in practice. This puts the unique status of mathematics as exact science into peril. Now may be the time to reconcile theory and practice, i.e. precision and intuition, through the advent of computer proof assistants. For the most time this has been a topic for experts in specialized communities. However, mathematical proofs have become increasingly sophisticated, stretching the boundaries of what is humanly comprehensible, so that leading mathematicians have asked for formal verification of their proofs. At the same time, major theorems in mathematics have recently been computer-verified by people from outside of these communities, even by beginning students. This article investigates the gap between the different definitions of a proof and possibilities to build bridges. It is written as a polemic or a collage by different members of the communities in mathematics and computer science at different stages of their careers, challenging well-known preconceptions and exploring new perspectives.