Applied Proof Theory: Proof Interpretations and their Use in Mathematics

Ulrich Kohlenbach presents an applied form of proof theory that has led in recent years to new results in number theory, approximation theory, nonlinear analysis, geodesic geometry and ergodic theory (among others). This applied approach is based on logical transformations (so-called proof interpretations) and concerns the extraction of effective data (such as bounds) from prima facie ineffective proofs as well as new qualitative results such as independence of solutions from certain parameters, generalizations of proofs by elimination of premises. §The book first develops the necessary logical machinery emphasizing novel forms of Gödel's famous functional ('Dialectica') interpretation. It then establishes general logical metatheorems that connect these techniques with concrete mathematics. Finally, two extended case studies (one in approximation theory and one in fixed point theory) show in detail how this machinery can be applied to concrete proofs in different areas of mathematics.This is the first treatment in book format of proof-theoretic transformations - known as proof interpretations - that focuses on applications to ordinary mathematics. It covers both the necessary logical machinery behind the proof interpretations that are used in recent applications as well as via extended case studies carrying out some of these applications in full detail. This subject has historical roots in the 1950s. This book for the first time tells the whole story.This is the first treatment in book format of proof-theoretic transformations -- known as proof interpretations -- that focuses on applications to ordinary mathematics. It covers both the necessary logical machinery behind the proof interpretations that are used in recent applications as well as via extended case studies carries out some of these applications in full detail. §This subject has its historical roots in pioneering work of G. Kreisel going back to the 1950s but was developed more systematically only during the past 15-20 years, mainly by the author and his collaborators in numerous paper. The main direction in this work is to apply proof transformations that originally had been developed in the course of foundational studies (erg. consistency proofs and Hilbert's program) as well as new versions and extensions thereof to concrete pieces of mathematics. This work so far only existed in the form of research papers that either developed the logical machinery and were published in logic journals or that presented concrete applications (mainly in analysis) and were published in analysis journals on the expense of dropping most of the logical background. §The present book for the first time tells the whole story: the logical theory, how to connect this theory up with ordinary mathematics and, finally, concrete applications in approximation theory and fixed point theory.