Towards a Philosophy of Real Mathematics

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 11
1.1 REAL MATHEMATICS......Page 13
1.2 THE CURRENT STATE OF PLAY......Page 15
1.3 THE FOUNDATIONALIST FILTER......Page 17
1.4 NEW DEBATES FOR THE PHILOSOPHY OF MATHEMATICS......Page 22
1.5 TOWARDS A PHILOSOPHY OF REAL MATHEMATICS......Page 30
1.6 RELATIONS WITH THE PHILOSOPHY OF PHYSICS......Page 34
1.7 FACTORS GOVERNING THE WAY MATHEMATICS PROCEEDS......Page 37
1.8 OUTLINE OF THE BOOK......Page 43
PART I Human and artificial mathematicians......Page 47
2.1 INTRODUCTION......Page 49
2.2 AUTOMATED MATHEMATICAL REASONING......Page 51
2.3 THE ARGONNE PARADIGM......Page 57
2.4 THE ROBBINS PROBLEM......Page 60
2.5 REFORMULATION......Page 63
2.6 CONCLUSION......Page 67
3.1 INTRODUCTION......Page 69
3.2 COMBINATORIAL ENUMERATION......Page 73
3.3 NUMERICAL IDENTITIES AND ENUMERATION......Page 76
3.4 INDUCTIVE LOGIC PROGRAMMING......Page 78
3.5 THE PROBLEM OF THE HOMOTOPY GROUPS OF THE SPHERES......Page 80
3.6 RESULTS AND DIAGNOSIS......Page 84
Independent sources......Page 87
Language......Page 88
3.7 CONCLUSION......Page 90
4.1 INTRODUCTION......Page 92
4.2 THOUGHTS ON ANALOGY......Page 95
4.3 THE ANALOGY BETWEEN NUMBER AND FUNCTION......Page 102
4.4 A WATERSHED IN THE USE OF ANALOGY......Page 108
4.5 CONCLUSION......Page 110
PART II Plausibility, uncertainty and probability......Page 113
5.1 INTRODUCTION......Page 115
5.2 PROBABILITY THEORY AS LOGIC......Page 117
5.3 QUANTITATIVE BAYESIANISM......Page 122
5.4 WHAT MIGHT BE ACHIEVED BY BAYESIANISM IN MATHEMATICS......Page 130
Analogy......Page 131
Strategy......Page 134
Enumerative induction......Page 135
5.5 CONCLUSION......Page 140
6.1 INTRODUCTION......Page 142
The stability of the solar system......Page 144
The onset of turbulence......Page 147
Quantum field theory......Page 149
6.3 MATHEMATICS AND SCIENTIFIC MODELS......Page 151
6.4 THE PROBLEM OF OLD EVIDENCE......Page 156
6.5 CONCLUSION......Page 160
PART III The growth of mathematics......Page 161
7.1 INTRODUCTION......Page 163
7.2 LAKATO\'S STAGES OF DEVELOPMENT......Page 164
7.3 THE METHOD OF PROOFS AND REFUTATIONS......Page 167
7.4 THE RESEARCH PROGRAMME STAGE......Page 173
7.5 LAKATOS ON RIGOUR......Page 175
7.6 THE JAFFE–QUINN DEBATE......Page 183
7.7 CONCLUSION......Page 185
8.1 INTRODUCTION......Page 187
8.2 PUTTING THE METHODOLOGY OF MATHEMATICAL RESEARCH PROGRAMMES TO THE TEST......Page 191
8.3 DEDEKIND VERSUS KRONECKER......Page 200
8.4 OTHER ATTEMPTS TO CONSTRUCT A METHODOLOGY OF MATHEMATICAL RESEARCH PROGRAMMES......Page 206
8.5 KOETSIER AND THE METHODOLOGY OF MATHEMATICAL RESEARCH TRADITIONS......Page 208
8.6 MADDY\'S NATURALISM......Page 212
8.7 CONCLUSION......Page 214
9.1 VALUES IN MATHEMATICS......Page 216
9.2 WHAT IS A GROUPOID?......Page 220
9.3 HOW GROUPOIDS COMPARE WITH GROUPS......Page 223
9.4 THE FULL EXPLOITATION OF ONE-DIMENSIONALITY......Page 226
9.5 GROUPOID ALGEBRAS USED TO COMPENSATE FOR BAD SPACES......Page 230
9.6 APPLICATIONS AND OLD CONJECTURES......Page 232
9.7 NEW PROSPECTS: HIGHER-DIMENSIONAL ALGEBRA......Page 234
9.8 THE CONCEPTUAL AND THE NATURAL......Page 235
9.9 CONCLUSION......Page 242
PART IV The interpretation of mathematics......Page 245
10.1  INTRODUCTION......Page 249
10.2 CATEGORIFICATION: MAKING DISTINCTIONS......Page 252
10.3 HIGHER-DIMENSIONAL ALGEBRA AS AN ORGANISING LANGUAGE......Page 259
10.4 DIAGRAMMATIC CALCULATION......Page 264
10.5 APPLICATIONS......Page 274
10.6 CONCLUSION......Page 281
FIELD......Page 283
CATEGORY......Page 284
MORITA EQUIVALENCE......Page 285
Bibliography......Page 286
Index......Page 298