STRUCTURAL PROOF THEORY

The idea of mathematical proof is very old, even if precise principles of proof have been laid down during only the past hundred years or so. Proof theory was first based on axiomatic systems with just one or two rules of inference. Such systems can be useful as formal representations of what is provable, but the actual finding of proofs in axiomatic systems is next to impossible. A proof begins with instances of the axioms, but there is no systematic way of finding out what these instances should be. Axiomatic proof theory was initiated by David Hilbert, whose aim was to use it in the study of the consistency, mutual independence, and completeness of axiomatic systems of mathematics.

Structural proof theory studies the general structure and properties of mathematical proofs. It was discovered by Gerhard Gentzen (1909–1945) in the first years of the 1930s and presented in his doctoral thesis Untersuchungen iiber das logische Schliessen in 1933. In his thesis, Gentzen gives the two main formulations of systems of logical rules, natural deduction and sequent calculus. The first aims at a close correspondence with the way theorems are proved in practice; the latter was the formulation through which Gentzen found his main result, often referred to as Gentzen's “Hauptsatz.” It says that proofs can be transformed into a certain “cut-free” form, and from this form general conclusions about proofs can be made, such as the consistency of the system of rules.