Hilbert's Proof Theory and its modern Development

Seminar paper from the year 2021 in the subject Mathematics - Miscellaneous, grade: 1,0, University of Hagen, course: Philosophy of Mathematics, language: English, abstract: David Hilbert first dealt with proofs as independent mathematical objects during the foundational crisis in mathematics at the beginning of the 20th century. Hilbert wanted to dispel all doubts about classical mathematical reasoning by a theory that makes mathematical proofs themselves to its objects (Hilbert, 1923). We examine the reasons and aims of Hilbert's proof theory and show how it came to a surprisingly sudden end.

Gerhard Gentzen continued proof theory in the spirit of Hilbert. We will see that Gentzen's system is more closely related to mathematical practice and get an outline how he succeeds in proving the consistency of number theory by means of new methods.

Attempts to grasp the real essence of proofs started afterwards. First we show how the important question of proof identity evolved in General Proof Theory. Second, how formal proofs can be represented in a new language by mathematical category theory and the lambda calculus to derive new identity criteria.