The Infinite in Mathematics

1. Front Matter

   Pages i-xvii

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2. The Infinite in Mathematics and its Elimination

   1. Front Matter

      Pages 1-5

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   2. Analytic Table of Contents

      • Felix Kaufmann

      Pages 6-9

   3. Introduction

      • Felix Kaufmann

      Pages 10-13

   4. Basic Facts of Cognition

      • Felix Kaufmann

      Pages 15-41

   5. Symbolism and Axiomatics

      • Felix Kaufmann

      Pages 42-67

   6. Natural Number and Set

      • Felix Kaufmann

      Pages 68-90

   7. Negative Numbers, Fractions and Irrational Numbers

      • Felix Kaufmann

      Pages 91-113

   8. Set Theory

      • Felix Kaufmann

      Pages 114-150

   9. The Problem of Complete Decidability of Arithmetical Questions

      • Felix Kaufmann

      Pages 151-157

   10. The Antinomies

       • Felix Kaufmann

       Pages 158-164

3. Remarks on the Controversy about the Foundations of Logic and Mathematics

   1. Remarks on the Controversy About the Foundations of Logic and Mathematics

      • Felix Kaufmann

      Pages 165-187

4. Questions of Logical Principle in the Investigation of the Foundations of Mathematics

   1. Questions of Logical Principle in the Investigation of the Foundations of Mathematics

      • Felix Kaufmann

      Pages 188-224

5. Back Matter

   Pages 225-237

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The main item in the present volume was published in 1930 under the title Das Unendliche in der Mathematik und seine Ausschaltung. It was at that time the fullest systematic account from the standpoint of Husserl's phenomenology of what is known as 'finitism' (also as 'intuitionism' and 'constructivism') in mathematics. Since then, important changes have been required in philosophies of mathematics, in part because of Kurt Godel's epoch-making paper of 1931 which established the essential in­ completeness of arithmetic. In the light of that finding, a number of the claims made in the book (and in the accompanying articles) are demon­ strably mistaken. Nevertheless, as a whole it retains much of its original interest and value. It presents the issues in the foundations of mathematics that were under debate when it was written (and in some cases still are); , and it offers one alternative to the currently dominant set-theoretical definitions of the cardinal numbers and other arithmetical concepts. While still a student at the University of Vienna, Felix Kaufmann was greatly impressed by the early philosophical writings (especially by the Logische Untersuchungen) of Edmund Husser!' He was never an uncritical disciple of Husserl, and he integrated into his mature philosophy ideas from a wide assortment of intellectual sources. But he thought of himself as a phenomenologist, and made frequent use in all his major publications of many of Husserl's logical and epistemological theses.

• Cardinal number
• arithmetic
• constructivism
• set theory

• The Queen’s College, Oxford, England

  Brian McGuinness

Policies and ethics