Proof and Knowledge in Mathematics

Michael Detlefsen0-203-97910-9

Proof and Knowledge in Mathematics tackles the main problem that arises when considering an epistemology for mathematics, the nature and sources of mathematical justification. Focusing both on particular and general issues, these essays from leading philosophers of mathematics raise important issues for our current understanding of mathematics. Is mathematical justification a priori or a posteriori? What role, if any, does logic play in mathematical reasoning or inference? And how epistemologically important is the formalizability of proof?Michael Detlefsen has brought together an outstanding collection of essays in a volume which will be essential for philosophers and historians of mathematics who are interested in the nature of reasoning and justification. A companion volume, Proof, Knowledge and Formalization is also available from Routledge.

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Table of contents :
BOOK COVER……Page 1
HALF-TITLE……Page 2
TITLE……Page 3
COPYRIGHT……Page 4
DEDICATION……Page 5
CONTENTS……Page 6
NOTES ON CONTRIBUTORS……Page 7
PREFACE……Page 9
I. INTRODUCTION……Page 12
II. REFORMULATING THE TRUTH/PROOF PROBLEM……Page 13
III. HOW PROOFS ARE USED AND WHERE THE TRUTH/PROOF PROBLEM ARISES……Page 15
IV. HOW WORKING PROOFS WORK……Page 17
V. CONTACT WITH MATHEMATICAL OBJECTS……Page 19
VI. BEYOND DOT PROOFS……Page 25
VII. SUMMARY……Page 26
NOTES……Page 27
REFERENCES……Page 28
I……Page 29
III……Page 31
V……Page 32
VI……Page 33
VIII……Page 34
IX……Page 35
X……Page 36
XII……Page 37
XIII……Page 38
XIV……Page 42
XV……Page 43
XVI……Page 44
XVII……Page 45
NOTES……Page 46
REFERENCES……Page 48
SUMMARY……Page 49
I……Page 50
II……Page 52
III……Page 53
IV……Page 55
V……Page 57
VI……Page 59
VII……Page 66
VIII……Page 67
IX……Page 68
NOTES……Page 72
REFERENCES……Page 74
I. INTRODUCTION……Page 76
II. PROOFS AND EXPERIMENTS: THE DISTINCTION……Page 77
III. THE PROOF/EXPERIMENT DISTINCTION: SUPPORTING EVIDENCE……Page 80
IV. AN ATTACK ON APRIORISM……Page 84
V. APRIORISM: WITTGENSTEIN’S DEFENSE……Page 86
NOTES……Page 90
REFERENCES……Page 91
I. ELEMENTARY SYNTHETIC VERSUS AXIOMATIZED AND ANALYTIC GEOMETRY……Page 92
II. THE PYTHAGOREAN PROGRAM AND EUDOXUS’ PROPORTION THEORY……Page 93
III. FURTHER DEVELOPMENTS IN THE ARITHMETIZATION OF GEOMETRY……Page 95
IV. LOGICAL ANALYSIS OF ABSTRACT ENTITIES……Page 96
V. WHAT ARE IDEAL GEOMETRICAL FORMS?……Page 97
VI. PLANE SURFACES AND RECTANGULAR SOLIDS……Page 99
VII. WHY ARE GEOMETRICAL TRUTHS SYNTHETIC A PRIORI?……Page 102
VIII. MATHEMATICAL MODELS OF A BOUNDED (“FINITE”) SPACE—TIME……Page 103
NOTES……Page 104
REFERENCES……Page 105
SUMMARY……Page 106
NOTES……Page 111
REFERENCES……Page 113
SUMMARY……Page 115
I. VARIATIONS AND METAPHORS……Page 116
II. FOUNDATIONS AND PSYCHOLOGISM……Page 119
Rationalism……Page 122
The semantic conception……Page 123
IV. MARRIAGE: CAN THERE BE HARMONY?……Page 125
Joint custody……Page 127
Rationalism denied……Page 128
Deductive systems without rationalism……Page 130
VI. LOGIC AND COMPUTATION……Page 131
NOTES……Page 134
REFERENCES……Page 136
I. PRECIS……Page 138
II. POINCARE’S CONCERN……Page 139
III. CLASSICAL EPISTEMOLOGY……Page 143
IV. BROUWERIAN EPISTEMOLOGY……Page 146
V. INTUITIONISTIC LOGIC……Page 153
VI. CONCLUSION……Page 157
NOTES……Page 158
REFERENCES……Page 163
INDEX……Page 165