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.

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

From the reviews:

"This book covers … from proof theory to a rich set of applications in areas quite distinct from mathematical logic: approximation theory and fixed point theory of nonexpansive mappings. … Almost every chapter has a detailed … informative final section with exercises, historical comments and references to related work. … In summary, this book is a very welcome addition to the proof theory literature." (H. Schwichtenberg, Mathematical Reviews, Issue 2009 k)

• FB Mathematik, Technische Universität Darmstadt, Darmstadt, Germany

  Ulrich Kohlenbach

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