# Proof Theory: History and Philosophical Significance

**Proof Theory**

** History and Philosophical Significance
**Edited by

Vincent F. Hendricks

Stig Andur Pedersen

Klaus Frovin Jørgensen

Synthese Library Series, Kluwer Academic Publishers 2000

Abstract | Contributing Authors | Table of Contents | Cover

Review – Bulletin of Symbolic Logic, volume 8, 2002 | Review – Annalen für Mathematique

This volume in the Synthese Library Series is the result of a conference held at the University of Roskilde, Denmark, October 31st – November 1st, 1997. The aim was to provide a forum within which philosophers, mathematicians, logicians and historians of mathematics could exchange ideas pertaining to the historical and philosophical development of proof theory. Hence the conference was called Proof Theory: History and Philosophical Significance. To quote from the conference abstract:

Proof theory was developed as part of Hilbert’s Program. According to Hilbert’s Program one could provide mathematics with a firm and secure foundation by formalizing all of mathematics and subsequently prove the consistency of these formal systems by finitistic means.

Hence proof theory was developed as a formal tool through which this goal should be fulfilled. It is well known that Hilbert’s Program in its original form was unfeasible mainly due to Gödel’s incompleteness theorems. Additionally it proved impossible to formalize all of mathematics and impossible to even prove the consistency of relatively simple formalized fragments of mathematics by finitistic methods. In spite of these problems, Gentzen showed that by extending Hilbert’s proof theory it would be possible to prove the consistency of interesting formal systems, perhaps not by finitistic methods but still by methods of minimal strength. This generalization of Hilbert’s original program has fueled modern proof theory which is a rich part of mathematical logic with many significant implications for the philosophy of mathematics.

Although a completely secure justification of mathematics is impossible it is, however, possible to achieve many fundamental partial results concerning relative consistency of theories, concerning the strength of axiomatic systems and finally concerning the relationship between constructive, predicative and classical systems of analysis.

The purpose of this meeting is to track the history of proof theory and its role in the analysis of the philosophical foundations of mathematics from its first primitive form in Hilbert’s original Program to its modern highly articulated form. Accordingly, the emphasis will be on historical and epistemological important episodes in the development of proof theory, not on technical aspects. All lectures will be of such a nature that they can be followed by mathematicians and philosophers without any professional training in proof theory butprovided with general knowledge of fundamental issues.

In order of apperance:

**Solomon Feferman** is Professor of Mathematics and Philosophy and Patrick Suppes Family Professor of Humanities and Sciences at Stanford University. He is the author of In the Light of Logic and editor-in-chief of the Collected Works of Kurt Gödel. He is noted for his many contributions to logic and the foundations of mathematics, in particular in the areas of proof theory and predicative systems of mathematics.

**Leo Corry** teaches at the Cohn Institute for History and Philosophy of Science, Tel Aviv University. His main research field is the history of mathematics in the early 20th century.

**David E. Rowe** teaches history of mathematics and exact sciences at Mainz University. His principal research interests center around mathematics in Germany in the period from 1800 to 1945. Alongside foundations issues he has been studying parallel developments in the foundations of physics connected with relativity theory. In connection with the latter, he has worked as a contributing editor for the Einstein Editorial Project at Boston University since 1998.

**Wilfried Sieg** is Professor in the Department of Philosophy at Carnegie Mellon University, Pittsburgh. He has been pursuing proof theoretic issues ever since his graduate studies at Stanford University: consistency proofs for subsystems of analysis, characterization of provably total functions for fragments of arithmetic, and automated search for natural deduction proofs in logic. The mathematical work has been complemented by historical and philosophical analyses of the foundational work of Hilbert and Bernays, but also of the emergemce of the concept of effective calculability.

**Dirk Van Dalen** is Professor in the History of Logic and Philosophy of Mathematics at Utrecht University. He is in charge of the Brouwer Project for preparing and editing the unpublished manuscripts and correspondence of Brouwer. He has published the first volume of the Brouwer Biography.

**Moritz Epple** presently holds a Heisenberg fellowship in the History of Mathematics. After graduating in physics and philosophy, he received a PhD in mathematical physics from Tuebingen University. Since then, he has been teaching history of science and history of mathematics at the universities of Mainz and Bonn.

**Erhard Scholz** is Professor of History of Mathematics at the Department of Mathematics, University of Wuppertal. His main research interests include the history of the 19th and 20th centuries. He contributes to the Hausdorff edition (Bonn). Additional research interests concentrate on the historical interrelation between mathematics and physics, centered around the work of Hermann Weyl.

Preface

Introduction: Vincernt F. Hendricks, Stig Andur Pedersen and Klaus Frovin Jørgensen

Part I – Review of Proof TheoryChapter 1. Solomon Feferman: Highlights in Proof Theory

Part II – The Background of Hilbert’s Proof Theory.Chapter 2. Leo Corry: The Empiricist Roots of Hilbert’s Axiomatic Approach

Chapter 3. David Rowe: The Calm before the Storm: Hilbert’s Early Views on Foundations

Chapter 4. Wilfried Sieg: Toward Finitist Proof Theory

Part III – Brouwer and Weyl on Proof Theory and Philosophy of MathematicsChapter 5. Dirk Van Dalen: The Development of Brouwer’s Intuitionism.

Chapter 6. Moritz Epple: Did Brouwer’sIntuitionistic Analysis Satisty its own Epistemological Standards?

Chapter 7. Solomon Feferman: The Significance of Hermann Weyl’s Das Kontinuum.

Chapter 8. Erhard Scholz: Hermann Weyl on the Concept of Continuum.

Part IV – Modern Views and Results from Proof TheoryChapter 9. Solomon Feferman: Relationships between Constructive, Predicative and Classical Systems of Analysis.

Index