Mathematics, Models, and Modality: Selected Philosophical Essays

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 11
Source notes......Page 13
ABOUT ‘‘ REALISM’’......Page 17
AGAINST HERMENEUTIC AND REVOLUTIONARY NOMINALISM......Page 19
AGAINST FICTIONALIST NOMINALISM......Page 21
FOUNDATIONS OF MATHEMATICS : SET THEORY......Page 23
FOUNDATIONS OF MATHEMATICS: LOGICISM......Page 26
MODELS AND MEANING......Page 28
MODELS AND MODALITY......Page 29
MODALITY AND REFERENCE......Page 30
HERMENEUTIC CRITICISM OF CLASSICAL LOGIC: RELEVANTISM......Page 32
REVOLUTIONARY CRITICISM OF CLASSICAL LOGIC: INTUITIONISM......Page 34
PART I Mathematics......Page 37
1 REALISM VS NOMINALISM......Page 39
2 BIGFOOT......Page 41
3 NUMBERS......Page 42
4 REALISM VS NOMINALISM REVISITED......Page 44
1 IN STRUMENTALIST NOMINALISM......Page 47
2 SCIENTIFIC DISPENSABILITY AND NON EXISTENCE......Page 49
3 HERMENEUTIC NOMINALISM......Page 50
4 REVOLUTIONARY NOMINALISM......Page 52
5  NOMINALISM, ONTOLOGICAL AND EPISTEMOLOGICAL......Page 55
A Chihara’s modal nominalism......Page 57
B Field’s spatiotemporal nominalism......Page 59
1 ‘‘NOMINALISM’’ AND ‘‘REALISM’’......Page 62
2 LITERARY GENRES......Page 64
3 NON–LITERAL LANGUAGE......Page 67
4 CONTRASTING CASES......Page 71
5 UNDECIDABLE QUESTIONS......Page 73
6 INTERMINABLE DEBATE......Page 75
7 MEANINGLESS QUESTIONS......Page 79
1 TWO SENSES OF ‘‘FOUNDATIONS OF MATHEMATICS’’......Page 82
2 QUINIANISM VS PLATONISM......Page 85
3 QUINIANISM VS ‘‘FICTIONALISM’’......Page 88
4 QUINIANISM VS CARNAPIANISM......Page 91
5 QUINIANISM VS CARNAPIANISM, BIS......Page 93
6 QUINIANISM VS INTUITIONISM......Page 96
1 A LOGICIAN LOOKS AT NOMINALISM......Page 101
2 REVOLUTIONARY RUMBLINGS......Page 103
3 HERMENEUTICAL HI-JINKS......Page 106
4 READING GOD’S MIND OR IMPOSING A SCHEME ON THE WORLD?......Page 108
5 ABSTRACT SKEPTICISM VS CONCRETE CREDULITY......Page 112
6 TALKING OF OBJECTS – OR NOT......Page 113
7 THE DARK SIDE......Page 117
1 THE ORIGIN OF SETTHEORY......Page 120
2 PLURAL LOGIC......Page 123
3 EXTENSIONALITY......Page 125
4 LIMITATION OF SIZE......Page 130
5 THE REFLECTION PRINCIPLE......Page 133
6 DEDUCING EXISTENCE AXIOMS OF ZF......Page 135
7 EXTENSIONALITY REVISITED......Page 137
8 THE AXIOM OF CHOICE......Page 140
9 SET-THEORETIC MODELS FOR PLURAL LOGIC......Page 145
1 NEO-NEO- LOGICISM......Page 151
2 LITE LOGICISM......Page 156
PART I I Models, modality, and more......Page 163
1 DEFINABILITY INDISREPUTE......Page 165
2 MATHEMATICAL DEFINITION VS LINGUISTIC ANALYSIS......Page 168
3 TARSKI’S TRIUMPH AND TRESPASS......Page 170
4 MODAL MUDDLES: SPURIOUS COMMITMENTS......Page 173
5 4 MODAL MUDDLES: UNWARRANTED COMPLACENCY......Page 176
6 FROM TARSKI TO DAVIDSONIANISM......Page 178
7 THE ‘‘SEMANTIC’’ PARADOXES......Page 182
1 THE QUESTION......Page 185
2 SOUNDNESS......Page 188
3 COMPLETENESS FOR VALIDITY LOGIC......Page 190
4 DEMON STRABILITY AND PROVABILITY......Page 193
5 AGAINST McK......Page 196
6 AGAINST S4.2......Page 198
1 THE DISCOVERY PRINCIPLE......Page 201
2 INEFFABLE TRUTHS......Page 203
3 EPHEMERAL TRUTHS......Page 204
4 ARE FORMULATION......Page 205
6 EXAMPLES......Page 207
7 ‘‘NOW’’......Page 209
8 DE RE ATTITUDES......Page 210
9 AN IMPERFECT ANALOGY......Page 212
10 COMBINING TEMPORAL AND MODAL FEATURES......Page 214
11 VERIFICATIONISM......Page 216
1.1 Quine and his critique......Page 219
1.2 Non-trivial de re modality......Page 220
1.3 Strict necessity......Page 222
1.4 ‘‘Aristotelian essentialism’’......Page 224
1.5 The mathematical cyclist......Page 226
1.6 The morning star......Page 227
1.7 Coda......Page 229
2.1 Quine and his critics......Page 230
2.2 Potpourri......Page 231
2.3 Smullyanism or neo-Russellianism......Page 235
2.4 Quine’s rebuttal to neo-Russellianism on descriptions......Page 237
2.5 Neo-Russellianism on names......Page 239
2.6 Quine’s rebuttal......Page 240
3.1 Hints from Quine for the formal logic of modalities......Page 242
3.2 A hint from Quine for the theory of reference of names......Page 243
3.3 Formal differences between logical and metaphysical modality......Page 245
3.4 New alternatives in the theory of reference for names......Page 247
3.5 Have the lessons been learned?......Page 249
1 MILLIANISM AND ANTI -MILLIANISM......Page 252
2 ANTI - ANTI - MILLIANISM AND TRANSLATION......Page 254
3 ANTI- ANTI - ANTI - MILLIANISM......Page 256
4 ANTI - MILLIANISM AND ANTI-DESCRIPTIVISM......Page 259
5 ENVOI......Page 260
1 INTRODUCTION......Page 262
Example 1. Analysis......Page 265
Commentary......Page 266
Example 2b. Argument......Page 267
Example 2b. Analysis......Page 268
3 CONCLUS ION......Page 269
1 TEXTS......Page 272
2 DUMMETT’S CASE AGAINST PLATONISM......Page 273
3 DUMMETT’S CASE AGAINST FORMALISM......Page 283
4 SUMMARY......Page 292
‘‘Consistency proofs in model theory’’ (Burgess 1978a)......Page 293
‘‘Equivalence relations generated by families of Borel sets’’ (Burgess 1978b)......Page 294
‘‘What are R-sets?’’ (Burgess 1982a) ‘‘Classical hierarchies from a modern standpoint, parts I & II’’ (Burgess 1983a and b)......Page 295
‘‘A remark on Henkin sentences and their contraries’’ (Burgess 2003b)......Page 296
‘‘The unreal future’’ (Burgess 1979d) ‘‘Decidability and branching time’’ (Burgess 1980c)......Page 297
‘‘Quick completeness proofs for some logics of conditionals’’ (Burgess 1981b)......Page 298
‘‘The completeness of intuitionistic propositional calculus for its intended interpretation’’ (Burgess 1981a)......Page 299
References......Page 300
Index......Page 313