From mathematical axioms to mathematical rules of proof: recent developments in proof analysis

Abstract

A short text in the hand of David Hilbert, discovered in Göttingen a century after it was written, shows that Hilbert had considered adding a 24th problem to his famous list of mathematical problems of the year 1900. The problem he had in mind was to find criteria for the simplicity of proofs and to develop a general theory of methods of proof in mathematics. In this paper, it is discussed to what extent proof theory has achieved the second of these aims. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

Modular labelled calculi for relevant logics

“…In addition to the newly introduced rules for →, there are also rules for R constructed through the method of conversion of frame conditions into sequent calculus rules. More precisely, we first have observed that all frame conditions are formulated either as universal axioms or geometric implications and, then, by following the methodology described in [20] (but previously also in [19,23,27]), we have transformed them into well-constructed sequent-style rules. Universal axioms are first turned into conjunctive normal form, namely,…”

Discussing Hilbert's 24th problem

“…Sara Negri & Jan von Plato used prominently the expression Hilbert's last problem in the subtitle for their book on Proof analysis [31]. In their contribution From mathematical axioms to mathematical rules of proof: recent developments in proof analysis [50] they discuss examples of Gentzen-style proof theory which address the general part of Hilbert's 24th problem concerning a theory of proof methods in mathematics.…”