Philosophy of mathematics

Cover......Page 1
Title Page......Page 5
Copyright Page......Page 6
General Preface......Page 7
Contents......Page 9
Preface......Page 11
Contributors......Page 17
Les Liaisons Dangereuses......Page 19
Acknowledgements......Page 51
Realism and Anti-Realism in Mathematics......Page 53
1.1 Mathematical Realism......Page 54
1.1.1 Anti-platonistic realism (physicalism and psychologism)......Page 55
1.1.2 Mathematical platonism......Page 58
1.2 Mathematical Anti-Realism......Page 62
2.1.1 The Epistemological Argument Against Platonism......Page 68
2.1.2 The Non-Uniqueness Objection to Platonism......Page 79
2.1.3 Responses to Some Recent Objections to FBP-NUP......Page 88
2.2.1 Introduction: The Fregean Argument Against Anti-Platonism......Page 94
2.2.2 Critique of Non-Fictionalistic Versions of Anti-Realistic Anti-Platonism......Page 96
2.2.3 Critique of Realistic Anti-Platonism (i.e., Physicalism and Psychologism)......Page 99
2.2.4 Indispensability......Page 102
2.3 Critique of Platonism Revisited: Ockham's Razor......Page 105
3 Conclusions: The Unsolvability of the Problem and a Kinder, Gentler Positivism......Page 108
3.1 The Strong Epistemic Conclusion......Page 109
3.2 The Metaphysical Conclusion......Page 112
Bibliography......Page 116
1 Introduction......Page 121
2 The Aristotelian Realist Point of View......Page 123
3 Mathematics as the Science of Quantity and Structure......Page 128
4 Necessary Truths About Reality......Page 133
5 The Formal Sciences......Page 141
6 Comparison with Platonism and Nominalism......Page 145
7 Epistemology......Page 154
8 Experimental Mathematics and Evidence for Conjectures......Page 159
Bibliography......Page 169
1 Introduction......Page 175
2 Aristotle......Page 178
3 John Stuart Mill......Page 186
3.1 Mill on geometry......Page 188
Appendix: non-Euclidean geometry......Page 191
3.2 Mill on arithmetic......Page 193
4.1 Kitcher against apriorism......Page 197
4.2 Kitcher on arithmetic......Page 204
4.3 Maddy on arithmetic......Page 209
5 Quine, Putnam and Field......Page 214
5.1 The indispensability of mathematics......Page 215
5.2 How much mathematics is indispensable?......Page 217
5.3 Digression: Nominalism......Page 220
5.4 Doing Without Numbers......Page 225
6 Logic and Analysis......Page 231
6.1 Analysis......Page 232
6.2 Logic......Page 234
6.3 Coda......Page 243
Bibliography......Page 244
A Kantian Perspective on the Philosophy of Mathematics......Page 249
1 Mathematics, Science of Forms......Page 250
2 Individual Objects — Why Mathematics Cannot Be Reduced to Logic......Page 252
3 Formal Rules — Why Mathematics Cannot Be Reduced to Manipulation of Marks on Paper......Page 255
4 Rules and Forms of Representation — Hilbertian Formalism......Page 257
5 Axiomatization and Structures — Changing the Object of Mathematics......Page 259
6 Does Set Theory Provide a Pure Theory of Manifolds?......Page 261
7 Ordinal, Cardinal and Two Kinds of Infinite......Page 262
8 Intuition and the Theory of Pure Manifolds......Page 269
9 Manifolds as Aggregates......Page 272
9.1 Aggregates as Manifolds......Page 274
10 Maxima, Minima — Totalities and Quantifiers......Page 277
11 What Is a Kantian Approach?......Page 279
A Non-Euclidean Geometry and Einstein's Relativity Theories......Page 280
Bibliography......Page 286
1 What Is Logicism?......Page 289
2 What Is Mathematics?......Page 290
3 What Is the Logic of Logicism?......Page 292
4 Frege the First Logicist......Page 294
5 Frege's Logic of Quantifiers......Page 295
6 Defining Real Numbers......Page 296
8 Axiomatic Set Theory vs. Logicism......Page 297
9 Principia Mathematica and its Aftermath......Page 298
10 Logicism vs. Metamathematics......Page 300
11 The Transformation of Logicism......Page 301
12 Correcting Frege's Theory of Quantification......Page 303
14 Logicism Vindicated?......Page 304
Bibliography......Page 306
1.1 Problem of Definition......Page 309
1.3 Working Mathematicians......Page 310
2.1 Contentless Manipulation......Page 311
2.2 Frege's Critique......Page 312
3.2 Implicit Definition and Contextual Meaning......Page 314
3.3 Dispute with Frege......Page 315
4.1 Logicism's Waterloo and other Paradoxes......Page 317
4.2 Self-Restriction......Page 319
5.1 Preparations......Page 320
5.3 Finitism......Page 321
5.4 Syntacticism and Meaning......Page 323
6 Gödel's Bombshell......Page 324
7.1 Proof Theory......Page 325
8 Conclusion......Page 326
Bibliography......Page 327
1 Introduction: Varieties of Constructivism......Page 329
1.1 Crucial Statements......Page 330
1.3 Constructions......Page 331
1.4 Constructive Logics and Proof Conditions......Page 332
2.1 Paul du Bois-Reymond......Page 334
2.2 Leopold Kronecker......Page 337
3 Intuitionism and L. E. J. Brouwer......Page 338
4 Heyting and Formal Intuitionistic Logic......Page 345
5 Markovian or Russian Constructivism......Page 347
6 Bishop's New Constructivism......Page 350
7 Predicativism......Page 351
8 Finitism......Page 355
Bibliography......Page 357
Fictionalism......Page 363
1 Kinds of Fictionalism......Page 364
1.1 Truth......Page 365
1.2 Interpretation......Page 366
1.3 Elimination......Page 368
2 Motivations for Fictionalism in the Philosophy of Mathematics......Page 369
2.1 Benacerraf's Dilemma......Page 370
2.2 Yablo's Comparative Advantage Argument......Page 372
3 A Brief History of Fictionalism......Page 374
3.1 William of Ockham: Reductive Fictionalism......Page 375
3.2 Jeremy Bentham: Instrumentalist Fictionalism......Page 376
3.3 C. S. Peirce: Representational Fictionalism......Page 378
3.4 Hans Vaihinger: Free-range Fictionalism......Page 381
4 Science Without Numbers......Page 384
5 Balaguer's Fictionalism......Page 391
6 Yablo's Figuralism......Page 392
7.1 Truth in a Fiction......Page 395
7.2 Constructive Free-range Fictionalism......Page 396
7.3 Definitions......Page 398
7.5 Minervan Constructions......Page 401
7.6 Marsupial Constructions......Page 402
7.7 Open Models......Page 404
7.8 Modal Translations......Page 406
Bibliography......Page 407
Set Theory from Cantor to Cohen......Page 413
1.1 Real Numbers and Countability......Page 414
1.2 Continuum Hypothesis and Transfinite Numbers......Page 416
1.3 Diagonalization and Cardinal Numbers......Page 419
2.1 Axiom of Choice and Axiomatization......Page 423
2.2 Logic and Paradox......Page 428
2.3 Measure, Category, and Borel Hierarchy......Page 430
2.4 Hausdorff and Functions......Page 433
2.5 Analytic and Projective Sets......Page 436
2.6 Equivalences and Consequences......Page 438
3.1 Ordinals and Replacement......Page 441
3.2 Well-Foundedness and the Cumulative Hierarchy......Page 443
3.3 First-Order Logic and Extensionalization......Page 445
3.4 Relative Consistency......Page 447
3.5 Combinatorics......Page 452
3.6 Model-Theoretic Methods......Page 455
4.1 Forcing......Page 459
4.2 Envoi......Page 464
Bibliography......Page 465
1 The Naïve Notion of Set......Page 479
3 Sets and Membership......Page 480
4 First-Order Versions......Page 481
6 Solution Routes......Page 482
8 The Limitation of Size Doctrine......Page 483
10 Topology and Indiscernibility......Page 484
11 Indiscernibility as a Lightning Discharger (?)......Page 485
12 Introduction......Page 486
13 Partial Sets......Page 487
14 Positive Sets......Page 490
15 Paradoxical Sets......Page 492
16 Double Sets......Page 494
17 Introduction......Page 495
18 Towards Modal Set Theory......Page 496
19 Proximity Structures......Page 497
20 Proximal Frege Structures......Page 499
21 The Ortholattice of Exact Sets......Page 501
22 Models of PFS......Page 502
23 On the Discernibility of the Disjoint......Page 504
25 Conclusion......Page 506
Bibliography......Page 507
2 Variables......Page 511
4 Sentences......Page 513
5 Interpretations and Distinctions......Page 514
6 Frequency......Page 515
7 Propensity......Page 516
8 Chance......Page 517
9 Bayesianism......Page 518
10 Chance as Ultimate Belief......Page 520
12 Subjective and Objective Bayesianism......Page 521
13 Objective Bayesianism Outlined......Page 522
15 Motivation......Page 524
16 Language Dependence......Page 526
17 Computation......Page 528
18 Qualitative Knowledge......Page 530
19 Infinite Domains......Page 533
20 Fully Objective Probability......Page 535
21 Probability Logic......Page 538
22 The Role of Interpretation......Page 544
23 The Epistemic View of Mathematics......Page 546
24 Conclusion......Page 547
Bibliography......Page 548
1.1 Foundational contexts......Page 553
1.2 Overview......Page 555
1.3 Connections......Page 557
2 Decidability and Calculability......Page 558
2.1 Decidability......Page 559
2.2 Finitist mathematics......Page 562
2.3 (Primitive) Recursion......Page 567
2.4 Formalizability and calculability......Page 573
3 Recursiveness and Church's Thesis......Page 579
3.1 Relative consistency......Page 580
3.2 Uniform calculations......Page 582
3.3 Elementary steps......Page 587
3.4 Absoluteness......Page 590
3.5 Reckonable functions......Page 593
4 Computations and Combinatory Processes......Page 595
4.1 Machines and workers......Page 597
4.2 Mechanical computors......Page 602
4.3 Turing's central thesis......Page 606
4.4 Stronger theses......Page 609
4.5 Machine computability......Page 611
5 Axioms for Computability......Page 614
5.1 Discrete dynamical systems......Page 615
5.2 Gandy machines......Page 618
5.3 Global assembly......Page 620
5.4 Models......Page 624
5.5 Tieferlegung......Page 626
6 Outlook on Machines and Mind......Page 627
6.1 Mechanical computability......Page 628
6.2 Beyond calculation......Page 631
6.3 Beyond discipline......Page 635
6.4 (Supra-) Mechanical devices......Page 640
Acknowledgments......Page 641
Bibliography......Page 642
1 Introduction: The Paradoxes......Page 649
2 The Role of Logic......Page 650
3 Pure Mathematics......Page 654
4 Geometry......Page 657
5 Applied Mathematics......Page 658
6 Logicism and Foundationalism Revisited......Page 660
7 Revisionism and Duality......Page 662
8 The Role of Text......Page 663
9 Conclusions......Page 664
Bibliography......Page 665
Mathematics and the World......Page 669
1.1 Realism and Anti-realism in Mathematics......Page 670
1.2 Indispensability Arguments......Page 671
2 What Is it to Be Indispensable?......Page 677
3 Naturalism and Holism......Page 681
3.2 Quinean Naturalism......Page 682
3.3 Holism......Page 685
4 The Hard Road to Nominalism: Field's Project......Page 687
4.1 Science without Numbers......Page 688
5.1 Maddy......Page 694
5.2 Sober......Page 700
6 The Unreasonable Effectiveness of Mathematics......Page 707
6.1 What is the Puzzle?......Page 708
6.2 Is the Puzzle Due to a Particular Philosophy of Mathematics?......Page 711
7 Applied Mathematics: The Philosophical Lessons and Future Directions......Page 715
Bibliography......Page 716
Index......Page 721