A Structural Account of Mathematics

Cover......Page 1
Abstract......Page 3
Title Page......Page 5
Preface......Page 9
Contents......Page 13
1. A Nominalist\'s View of Philosophy: The Big Picture......Page 17
2. Nominalistic Reconstructions......Page 21
1. A Puzzle about Geometry......Page 24
2. Different Attitudes of Practicing Mathematicians Regarding the Ontology of Mathematics......Page 27
3. The Inertness of Mathematical Objects......Page 28
4. Consistency and Mathematical Existence......Page 33
5. The van Inwagen Puzzle......Page 35
1. The Frege-Hilbert Dispute Concerning the Axioms of Geometry......Page 43
2. Some Suggestions Regarding the Nature of Mathematics......Page 56
3. The First Puzzle......Page 57
4. The Fourth Puzzle......Page 62
5. Some Concluding Thoughts......Page 64
1. Structures......Page 65
2. My Resolution of the Puzzle......Page 68
3. The Genetics Objection......Page 72
4. The Problem of Multiple Reductions......Page 75
6. Structuralism?......Page 76
1. Shapiro\'s Characterization of Structures......Page 78
2. Mathematics Viewed as the Science of Structures......Page 79
4. Some Ways in Which My Account will Differ from the Structuralists\'......Page 81
5. Ontological Aspects of Shapiro\'s Structuralism......Page 82
6. Why Believe in the Existence of Structures?......Page 86
7. Resnik\'s Early Theses about Structures and Positions......Page 87
8. Shapiro and Thesis [2]......Page 89
9. Shapiro\'s Rejection of Thesis [1]......Page 91
10. Some Problems with Shapiro\'s ante rem Structuralism......Page 92
11. A Problem with Shapiro\'s Acceptance of Thesis [2]......Page 97
12. The Main Problem with Eliminative Structuralism......Page 99
13. Resnik on Structures......Page 100
14. Resnik\'s No-Structure Theory......Page 102
15. Problems with Resnik\'s No-Structure Theory......Page 106
16. Resnik\'s Rejection of Classical Logic......Page 107
17. The Main Problem with Resnik\'s Version of Thesis [2]......Page 111
18. A Conceptual Objection to Structuralism......Page 112
1. Gödelian Platonism......Page 115
2. Quine\'s Challenge to the Nominalist......Page 120
3. Nominalistic Responses to Quine\'s Challenge......Page 123
4. The Direct Indispensability Argument......Page 131
5. The Indispensability Argument Based on Holism......Page 135
6. Putnam\'s Version of the Indispensability Argument......Page 139
7. Resnik\'s Version of the Argument......Page 142
8. Sober\'s Objection to the Indispensability Argument......Page 143
9. Maddy\'s Objections......Page 152
10. Resnik\'s New Indispensability Argument......Page 162
11. Concluding Assessment of the Various Arguments......Page 165
1. The Burgess-Rosen Account of Nominalism......Page 167
2. Nominalistic Reconstructions of Mathematics Reexamined......Page 175
7. The Constructibility Theory......Page 185
1. A Brief Exposition of the Constructibility Theory......Page 186
2. Preliminaries of the Constructibility Theory of Natural Number Attributes......Page 189
3. Cardinal Number Attributes......Page 193
4. Shapiro\'s Objections to the Constructibility Theory......Page 200
5. Other Objections Shapiro has Raised......Page 211
6. Resnik\'s Objections......Page 223
2. Realizations without Commitment to Mathematical Objects......Page 234
3. Constructible Realizations......Page 237
4. Constructible Realizations of Mathematical Theories......Page 240
5. The Natural Number Realization......Page 241
6. Higher-Level Constructible Realizations......Page 242
7. Realization of First-Order Theories......Page 243
9. Applications......Page 245
1. Applications of Constructibility Arithmetic......Page 247
2. Peano Arithmetic......Page 256
3. Putnam\'s If-Thenism......Page 261
4. Frege and Dummett on Why Mathematical Theorems must Express Thoughts......Page 265
5. A Reexamination of Various Indispensability Arguments......Page 269
6. Shapiro\'s Account of Applications......Page 270
7. Fermat\'s Last Theorem......Page 273
8. Applications of Analysis: Some General Considerations......Page 277
9. Mathematical Modeling......Page 278
10. Albert\'s Version of the Mathematics of Quantum Mechanics......Page 281
11. The Mathematics of Quantum Physics......Page 286
12. The Fundamental Theorem of the Integral Calculus......Page 296
13. The Burgess “Tonsorial Question”......Page 302
14. Applications of Set Theory in Logic......Page 305
15. Maddy\'s Mystery......Page 307
10. If-Thenism......Page 309
1. The Third Puzzle......Page 310
2. Brown\'s “Other Avenues”......Page 313
3. The Second Puzzle......Page 320
4. Criticisms of If-Thenism......Page 323
5. The Main Stumbling Block of the Eliminative Program......Page 331
11. Field\'s Account of Mathematics and Metalogic......Page 333
1. Why I Should not be Called a “Fictionalist”......Page 334
2. Field\'s Metalogical Theorems......Page 335
3. Field\'s Justification for Accepting Standard Metalogical Results......Page 337
4. Field\'s Other Justifications of his Conservation Principle......Page 345
5. Field\'s Arguments that Good Mathematics is Conservative......Page 348
6. A Comparison between Two Views of Mathematics......Page 355
7. The Fundamental Theorem Revisited......Page 361
Appendix A. Some Doubts About Hellman\'s Views......Page 365
Appendix B. Balaguer\'s Fictionalism......Page 372
Bibliography......Page 379
C......Page 389
E......Page 390
H......Page 391
M......Page 392
N......Page 393
P......Page 394
S......Page 395
Z......Page 396