## Phenomenology, Logic, and the Philosophy of Mathematics

This book is a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics. The first of the three parts, “Reason, Science, and Mathematics,” contains a general essay on Husserl’s conception of science and logic, an essay on mathematics and transcendental phenomenology, and an essay on phenomenology and modern pure geometry. Part II is focused on Kurt Gödel’s interest in phenomenology. It explores Gödel’s ideas and also some work of Quine, Penelope Maddy, and Roger Penrose. Part III deals with elementary, constructive areas of mathematics – areas of mathematics that are closer to their origins in simple cognitive activities and in everyday experience. This part of the book contains essays on intuitionism, Hermann Weyl, the notion of constructive proof, Poincaré, and Frege.

Richard Tieszen is Professor of Philosophy at San Jose State University.

*To my parents,*

*James D. and Beverly J. Tieszen*

# Phenomenology, Logic, and the Philosophy of Mathematics

RICHARD TIESZEN

*San Jose State University*

CAMBRIDGE UNIVERSITY PRESS

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Cambridge University Press

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© Richard Tieszen 2005

This book is in copyright. Subject to statutory exception and

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the written permission of Cambridge University Press.

First published 2005

Printed in the United States of America

*A catalog record for this publication is available from the British Library.*

*Library of Congress Cataloging in Publication Data*

Tieszen, Richard L.

Phenomenology, logic, and the philosophy of mathematics / Richard Tieszen.

p. cm.

Includes bibliographical references and index.

ISBN 0-521-83782-0

1. Mathematics – Philosophy. 2. Phenomenology. 3. Logic, Symbolic and mathematical. 4. Constructive mathematics. 5. Intuitionistic mathematics. I. Title.

QA8.4.T534 2005

510′.1 – dc22

2004062888

ISBN-13 978-0-521-83782-8 hardback

ISBN-10 0-521-83782-0 hardback

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## Contents

Acknowledgments |
page vii |
|||

Introduction: Themes and Issues | 1 | |||

PART I. REASON, SCIENCE, AND MATHEMATICS |
||||

1 | Science as a Triumph of the Human Spirit and Science in Crisis: Husserl and the Fortunes of Reason | 21 | ||

2 | Mathematics and Transcendental Phenomenology | 46 | ||

3 | Free Variation and the Intuition of Geometric Essences: Some Reflections on Phenomenology and Modern Geometry | 69 | ||

PART II. KURT GÖDEL, PHENOMENOLOGY, AND THEPHILOSOPHY OF MATHEMATICS |
||||

4 | Kurt Gödel and Phenomenology | 93 | ||

5 | Gödel’s Philosophical Remarks on Logic and Mathematics | 112 | ||

6 | Gödel’s Path from the Incompleteness Theorems (1931) to Phenomenology (1961) | 125 | ||

7 | Gödel and the Intuition of Concepts | 149 | ||

8 | Gödel and Quine on Meaning and Mathematics | 177 | ||

9 | Maddy on Realism in Mathematics | 201 | ||

10 | Penrose on Minds and Machines | 215 | ||

PART III. CONSTRUCTIVISM, FULFILLABLE INTENTIONS, AND ORIGINS |
||||

11 | Intuitionism, Meaning Theory, and Cognition | 227 | ||

12 | The Philosophical Background of Weyl’s Mathematical Constructivism | 248 | ||

13 | Proofs and Fulfillable Mathematical Intentions | 276 | ||

14 | Logicism, Impredicativity, Formalism: Some Remarks on Poincaré and Husserl | 294 | ||

15 | The Philosophy of Arithmetic: Frege and Husserl | 314 | ||

Bibliography |
337 | |||

Index |
349 |

## Acknowledgments

The essays collected here were written over a period of fifteen years. In preparing them for publication in this volume I have modified them in a few places, mostly for clarity and for continuity with other chapters in the collection. I have also cut some material from a few of the essays. Some overlap or repetition remains here and there, but the trade-off is that such overlapping allows the essays to be read independently of one another. In any case, I think that a little repetition is not onerous. It may even be helpful to some readers. The Bibliography has been standardized, and in the case of Husserl’s work in particular I have provided references to the original publications, the relevant *Husserliana* editions, and the English translations. Since Husserl’s works are usually composed in short sections I have adopted the convention in the essays of referring to quotations from Husserl’s works by providing the title of the work and the section number. This method puts the reader in touch with the relevant texts but allows for choice in consulting the different editions and languages. In the case of Gödel’s writings I have followed the citation style used in *Kurt Gödel: Collected Works.* The Bibliography for the present volume includes very few works that were not cited in the original essays. I have included a few new references where there has been some clear line of development of an argument or point in the essays.

Chapter 1 was written for the volume *Continental Philosophy of Science*, edited by Gary Gutting (Oxford: Blackwell, 2004). It appears here with the permission of Blackwell Publishing.

Chapter 2 originally appeared under the title “Mathematics” in *The Cambridge Companion to Husserl*, B. Smith and D. Smith (eds.) (Cambridge: Cambridge University Press, 1995), pp. 438–462. It is reprinted here with the permission of Cambridge University Press.

Chapter 3, “Free Variation and the Intuition of Geometric Essences: Some Reflections on Phenomenology and Modern Geometry,” will appear in *Philosophy and Phenomenological Research*. It is reprinted here with the permission of the editors.

Chapter 4 appeared as “Kurt Gödel and Phenomenology,” *Philosophy of Science* 59, 2 (1992), pp. 176–194. It is reprinted here with the permission of the Philosophy of Science Association.

Chapter 5 was originally published as “Gödel’s Philosophical Remarks on Logic and Mathematics: Critical Notice of *Kurt Gödel: Collected Works*, Vols. Ⅰ, Ⅱ, Ⅲ,” *Mind* 107 (1998), pp. 219–232. It is reprinted by permission of Oxford University Press.

Chapter 6 appeared as “Gödel’s Path from the Incompleteness Theorems (1931) to Phenomenology (1961),” *Bulletin of Symbolic Logic* 4, 2 (1998), pp. 181–203, © copyright 1998 Association for Symbolic Logic. It is reprinted here with permission of the Association for Symbolic Logic.

Chapter 7 appeared as “Gödel and the Intuition of Concepts,” *Synthese* 133, 3 (2002), pp. 363–391, © copyright 2002 Kluwer Academic Publishers. It appears here with kind permission of Kluwer Academic Publishers.

Chapter 8 appeared as “Gödel and Quine on Meaning and Mathematics,” in *Between Logic and Intuition: Essays in Honor of Charles Parsons*, R. Tieszen and G. Sher (eds.) (Cambridge University Press, 2000), pp. 232–254. It is reprinted with the permission of Cambridge University Press.

Chapter 9 was originally published as “Review of *Mathematical Realism*, by Penelope Maddy,” *Philosophia Mathematica* 3, 2 (1994), pp. 69–81. It is reprinted here with the permission of the editor, Robert S. D. Thomas.

Chapter 10 was originally published as “Review of *Shadows of the Mind: A Search for the Missing Science of Consciousness*, by Roger Penrose,” *Philosophia Mathematica* 4, 3 (1996), pp. 281–290. It appears here with the permission of the editor, Robert S. D. Thomas.

Chapter 11 appeared as “Intuitionism, Meaning Theory and Cognition,” *History and Philosophy of Logic* 21, 3 (2001), pp. 179–194. It is reprinted here with the permission of Taylor & Francis Limited (http://www. tandf.co.uk).

Chapter 12 was published as “The Philosophical Background of Weyl’s Mathematical Constructivism,” *Philosophia Mathematica* 3, 8 (2000), pp. 274–301. It is reprinted with the permission of the editor, Robert S. D. Thomas.

Chapter 13 was originally published under the title “What Is a Proof ?” in *Proof, Logic and Formalization*, M. Detlefsen (ed.) (London: Routledge, 1992), pp. 57–76. It appears here with the permission of Routledge.

Chapter 14 was originally published under the title “Logicism, Impredicativity, Formalism,” in *Henri Poincaré: Science and Philosophy*, J. L. Greffe, G. Heinzmann, and K. Lorenz (eds.) (Berlin: Akademie Verlag and Paris: Albert Blanchard, 1996), pp. 399–415. It is reprinted with the permission of Gerhard Heinzmann and Akademie Verlag.

Chapter 15 appeared as “The Philosophy of Arithmetic: Frege and Husserl,” in *Mind, Meaning and Mathematics*, L. Haaparanta (ed.) (Dordrecht: Kluwer Academic Publishers, 1994), pp. 85–112, © copyright 1994 Kluwer Academic Publishers. It is reprinted here with kind permission of Kluwer Academic Publishers.

Over the years I have discussed ideas about phenomenology, logic, and mathematics with many friends and colleagues. I have included the original acknowledgments in the chapters themselves, but there have been many other people who also deserve to be thanked for discussion, comments, and suggestions.

My interest in the relationship of phenomenology to the exact sciences goes back to my days in graduate school and college. During this period Ⅰ benefited most from interactions with Charles Parsons, Wilfried Sieg, Howard Stein, Shaughan Levine, and Isaac Levi at Columbia University; Robert Tragesser at Barnard; and J. N. Mohanty and Izchak Miller at the Graduate Faculty of the New School for Social Research. Soon after I arrived at Columbia, Charles Parsons gave a seminar on Husserl’s *Logical Investigations*. This allowed me to deepen the study of Husserl’s works that I had already begun at the New School and in college. At the New School, J. N. Mohanty gave a year-long seminar on Husserl’s logical works. It was this kind of in-depth seminar that made the Graduate Faculty unique. As an undergraduate I benefited from studies of phenomenology with Robert Welsh Jordan and of modal logic and other systems of logic with Fred Johnson. It was during my time at Columbia that I met and began the first of many discussions with Hao Wang about Gödel’s philosophical views. Wang was one of the people I knew who most encouraged thinking about mathematics from a phenomenological standpoint. I also met Dagfinn Føllesdal while I was a graduate student, and although he was never formally my teacher I have been discussing Husserl’s work with him since that time. Several other people were especially helpful and encouraging early on: Michael Resnik, Gian-Carlo Rota, Bill Tait, and Bill McKenna come to mind. I am indebted to Mike Resnik in particular.

As I began to focus even more on constructive mathematics I benefited from discussions with Dirk van Dalen, Per Martin-Löf, Anne Troelstra, Dag Prawitz, and Göran Sundholm. Van Dalen has been especially helpful. After I arrived in California I began to attend many of the logic events at Stanford. These were often organized by Sol Feferman, and I have profited from many exchanges with him over the years.

In addition to all of those mentioned, I would like to thank many other people with whom I have discussed my work: Mark van Atten, Michael Friedman, David Smith, Barry Smith, Ed Zalta, John Corcoran, Albert Visser, Karl Schuhmann, Robin Rollinger, Paolo Mancosu, Thomas Ryckman, Karl Ameriks, Guglielmo Tamburrini, Ernan McMullin, Gary Gutting, Charles Chihara, Jerrold Katz, Paul Cortois, Grisha Mints, Dieter Lohmar, Claire Hill, Jairo da Silva, Peter Hadreas, and Kai Hauser. (It is possible that as I write this I have not remembered everyone who deserves to be mentioned.)

Finally, I thank my wife, Nancy, for her patience and her meditative ways. *Nam Mô A Di Ðà Ph⃞t*. I dedicate this book to my parents, James D. and Beverly J. Tieszen.