Proof and Knowledge in Mathematics

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. 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 This volume of essays addresses the main problem confronting an epistemology for mathematics; namely, the nature and sources of mathematical justification. Attending to both particular and general issues, the essays, by 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 of what epistemological importance is the formalizability of proof? The editor, Michael Detlefsen, has brought together an outstanding collection of essays, only one of which has previously appeared. It will be essential for philosophers and historians of mathematics, as well as philosophically inclined logicians and philosophers interested in the nature of reasoning and justification. A companion volume entitled Proof, Logic and Formalization, edited by Michael Detlefsen, is also available from Routledge These questions arise from any attempt to discover an epistemology for mathematics. This collection of essays considers various questions concerning the nature of justification in mathematics and possible sources of that justification. Among these are the question of whether mathematical justification is a priori or a posteriori in character, whether logical and mathematical differ, and if formalization plays a significant role in mathematical justification,