Table Of ContentMATHEMATICAL INTUmON
SYNTHESE LIBRARY
STUDIES IN EPISlEMOLOGY,
LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Managing Editor:
JAAKKO HINTIKKA, Florida State University. Tallahassee
Editors:
DONALD DAVIDSON, University ofCali/ornia, Berkeley
GABRIEL NUCHELMANS, University ofL eyden
WESLEY C. SALMON, University of Pittsburgh
VOLUME 203
RICHARD L. TIESZEN
MATHEMATICAL
INTUITION
Phenomenology and
Mathematical Knowledge
KLUWER ACADEMIC PUBLISHERS
DORDRECHT/BOSTON/LONDON
Library of Congress Cataloging in Publication Data
Tieszen. Richard.
Mathematical Intuition phenomenology and mathematical knowledge
by Rlehard lleszen.
p. eN. -- (Synthese library; v. 203)
Bibliography: p.
Inc I udes Index.
1. Mathematics--Philosophy. 2. Intuition. 1. Tit Ie.
II. Series.
QA8.4.T53 1989
510·.1--de19 88-37514
ISBN-13: 978-94-0 I 0-7529-9 e-ISBN-13: 978-94-009-2293-8
DOl: 10.1007978-94-009-2293-8
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For Nancy
TABLE OF CONTENTS
PREFACE xi
ABBREVIATIONS FOR WORKS OF HUSSERL xv
CHAPTER 1. THE CONCEPT OF INTUmON IN MATHEMATICS 1
1. Introduction 1
2. Knowledge, Evidence, and Intuition 2
3. Intuition "of' and Intuition "that" 5
4. Some Recent Views of Mathematical Intuition 6
5. Hilbert and Bemays 6
6. Parsons 8
7. Brouwer 12
8. Some "Extended" Proof-Theoretic Views 13
9. GOdel on Sets 14
10. Platonism and Constructivism 16
11. Mathematical Truth and Mathematical Knowledge 17
12. Principal Objections to Mathematical Intuition 18
CHAPTER 2. THE PHENOMENOLOGICAL VIEW OF INTUITION 21
1. Introduction 21
2. Intentionality and Intuition 22
3. Intuition of Abstract Objects 25
4. Acts of Abstraction and Abstract Objects 31
5. Acts of Reflection 36
6. Types and Degrees of Evidence 38
7. Comparison with Kant 43
8. Intuition and the Theory of Meaning 45
viii TABLE OF CONTENTS
CHAPTER 3. PERCEPTION 48
1. Introduction 48
2. Sequences of Perceptual Acts 49
3. The Horizon of Perceptual Acts 51
4. The Possibilities of Perception 56
5. The "Determinable X" in Perception and Indexicals 58
6. Perceptual Evidence 61
7. Phenomenological Reduction and the Problem of Realism /
Idealism 63
CHAPTER 4. MATHEMATICAL INTUITION 66
1. Introduction 66
2. Objections About Analogies Between Perceptual and
Mathematical Intuition 67
3. Objections Based on Structuralism 71
4. Objections About Founding 75
5. A Logic Compatible With Mathematical Intuition and
the Notion of Construction 79
6. Is Classical Mathematics to be Rejected? 89
CHAPTERS. NATURAL NUMBERS I 92
1. Introduction 92
2. The Concept of Number Cannot Be Explicitly Defined 93
3. The Origin of the Concept of Number 96
4. Intuition of Natural Numbers 99
5. Ordinals 101
6. Ordinals and Cardinals 105
7. Constructing Units and the Role of Reflection and Abstraction 111
8. Syntax and Representations of Numbers 116
CHAPTER 6. NATURAL NUMBERS II 119
1. Introduction 119
o
2. and 1 121
3. Numbers Formed by Arithmetic Operations 122
4. Small Numbers and Singular Statements About Them 124
5. Large Numbers and Mathematical Induction 128
6. The Possibilities of Intuition 131
7. Summary of the Argument for Large Numbers 135
8. Further Comments on Mathematical Induction 137
TABLE OF CONTENTS ix
9. Intuition and Axioms of Elementary Number Theory 140
CHAPTER 7. FINITE SETS 143
1. Introduction 143
2. A Theory of Finite Sets 145
3. The Origin of the Concept of Finite Set 147
4. Intuition of Finite Sets 153
5. Comparison with GMel and Wang 156
6. Unit Sets, the Empty Set, and Mereology vs. Set Theory 161
7. Large Sets and a Hierarchy of Sets 164
8. Illusion in Set Theory 169
9. Concluding Remarks 170
CHAPTER 8. CRITICAL REFLECTIONS AND CONCLUSION 172
1. Introduction 172
2. Summary of the Account 172
3. Areas for Further Work 175
4. Platonism, Constructivism, and Benacerraf's Dilemma 177
NOTES 183
BIBLIOGRAPHY 194
INDEX 201
PREFACE
"Intuition" has perhaps been the least understood and the most abused term in
philosophy. It is often the term used when one has no plausible explanation
for the source of a given belief or opinion. According to some sceptics, it is
understood only in terms of what it is not, and it is not any of the better
understood means for acquiring knowledge. In mathematics the term has also
unfortunately been used in this way. Thus, intuition is sometimes portrayed
as if it were the Third Eye, something only mathematical "mystics", like
Ramanujan, possess. In mathematics the notion has also been used in a host
of other senses: by "intuitive" one might mean informal, or non-rigourous, or
visual, or holistic, or incomplete, or perhaps even convincing in spite of lack
of proof.
My aim in this book is to sweep all of this aside, to argue that there is a
perfectly coherent, philosophically respectable notion of mathematical
intuition according to which intuition is a condition necessary for mathemati
cal knowledge. I shall argue that mathematical intuition is not any special or
mysterious kind of faculty, and that it is possible to make progress in the
philosophical analysis of this notion.
This kind of undertaking has a precedent in the philosophy of Kant. While
I shall be mostly developing ideas about intuition due to Edmund Husser!
there will be a kind of Kantian argument underlying the entire book. The
style of argument is clearly present in HusserI's later philosophy. Simply put,
it is to ask how mathematical knowledge is possible; in particular, to ask how
knowledge of number or of other mathematical objects is possible. In
Husserl's philosophy this is an especially interesting question, for Husserl
wanted to be a kind of platonist or realist about. mathematical objects, and yet
to develop an epistemology that was compatible with being a realist. To
succeed in any measure at this task would be to make progress with one of
the main problems in the philosophy of mathematics.
The main positive argument of the book can be looked at as comprised of
two subarguments: first, arguments to show that mathematics is about
abstract objects, objects that are eternal, unchanging, causally inert, and
outside of spacetime. HusserI gives arguments to the effect that sense
experience does not contain abstract mathematical objects, and that mathe
matics has a kind of universality and necessity not found in sense experience.
xii PREFACE
I discuss these arguments, and I think there is much to be said for them, but
in this book I do not pursue them as far as I think they could be pursued.
Secondly, I consider arguments that purport to show how it is possible to
have knowledge about such objects, where knowledge requires intuition,
based on the Kantian strategy. Thus, the idea is to determine the kinds of
cognitive structures and processes which are neccesary for knowledge about
numbers and finite sets, where these objects are understood as abstract This
part of the argument is taken up in some detail and the consequences are
explored for a variety of major issues in the philosophy of mathematics.
Chapter 1 starts with a brief statement of how the notions of knowledge,
evidence and intuition are connected and then surveys recent views on
mathematical intuition. It is interesting to look at these views as more or less
adequate attempts to answer the Kantian question. The main problems about
mathematical intuition that I would like to address in the book are then
reviewed: problems about the analogousness of mathematical intuition and
perception, about mathematical structuralism, and about the sense in which
mathematical knowledge is supposed to be "founded" on mathematical
intuition.
In Chapter 2 I turn to HusserI' s writings to describe the phenomenological
conception of intuition. In deference to HusserI scholarship I have quoted
extensively from the texts to establish my interpretation on this issue. At the
center of the phenomenological view is the concept of intentionality, and it is
by approaching intuition via intentionality that I believe it is possible to
obtain new insights into mathematical intuition and mathematical knowledge.
Chapter 3 is devoted to an examination of perceptual intuition and is intended
to set the stage for the analogies I wish to draw with mathematical intuition
later in the book.
Chapter 4 gives a detailed statement of objections to the notion of mathe
matical intuition, along with some preliminary indications about how they
will be met. Toward the end of the chapter I present the framework for the
account of mathematical intuition to follow. The main idea here is to set up
logic for different domains of objects in such a way that it does not abstract
from intuitability conditions. Intuition is understood in terms of fulfIllment
(or fulfillability) of intentional acts, and following Heyting, Martin-Lfif and
others, I say that an intentional act whose content is expressed by S is
fulfilled (or is fulfillable) if and only if there is a construction for (or proof
of) S. On our view, a theory of constructions, suitably understood, is a theory
of mathematical intuition.
Chapters 5 and 6 work through many of the details of this kind of view in
