Applied Proof Theory: Proof Interpretations and Their Use in Mathematics

  • Ulrich Kohlenbach

Part of the Springer Monographs in Mathematics book series (SMM)

About this book

Introduction

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.

 

Keywords

Arithmetic Computational Mathematics Finite Mathematical logic Proof Interpretations Proof Mining calculus function geometry mathematics proof proof theory theorem

Authors and affiliations

  • Ulrich Kohlenbach
    • 1
  1. 1.FB MathematikTechnische Universität DarmstadtDarmstadtGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-540-77533-1
  • Copyright Information Springer-Verlag Berlin Heidelberg 2008
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-540-77532-4
  • Online ISBN 978-3-540-77533-1
  • Series Print ISSN 1439-7382
  • About this book