The problem of proof identity, and why computer scientists should care about Hilbert's 24th problem

Abstract

In this short overview article, I will discuss the problem of proof identity and explain how it is related to Hilbert's 24th problem. I will also argue that not knowing when two proofs are 'the same' has embarrassing consequences not only for proof theory but also for certain areas of computer science where formal proofs play a fundamental role, in particular, the formal verification of software. Then I will formulate a set of four objectives that a satisfactory notion of proof identity should obey. And finally, I discuss Hughes' combinatorial proofs and argue that they can be seen as a first step towards a possible solution to the problem of proof identity. This article is part of the theme issue 'The notion of 'simple proof' - Hilbert's 24th problem'.