Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true. Chihara builds upon his previous work, in which he presented a new system of mathematics, the constructibility theory, which did not make reference to, or resuppose, mathematical objects. Now he develops the project further by analyzing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings. A Structural Account of Mathematics will be required reading for anyone working in this field.
Author(s): Charles S. Chihara
Edition: First Edition
Publisher: Oxford University Press, USA
Year: 2004
Language: English
Pages: 395
Contents......Page 12
Preface......Page 8
1. A Nominalist's View of Philosophy: The Big Picture......Page 16
2. Nominalistic Reconstructions......Page 20
1. A Puzzle about Geometry......Page 23
2. Different Attitudes of Practicing Mathematicians Regarding the Ontology of Mathematics......Page 26
3. The Inertness of Mathematical Objects......Page 27
4. Consistency and Mathematical Existence......Page 32
5. The van Inwagen Puzzle......Page 34
1. The Frege–Hilbert Dispute Concerning the Axioms of Geometry......Page 42
2. Some Suggestions Regarding the Nature of Mathematics......Page 55
3. The First Puzzle......Page 56
4. The Fourth Puzzle......Page 61
5. Some Concluding Thoughts......Page 63
1. Structures......Page 64
2. My Resolution of the Puzzle......Page 67
3. The Genetics Objection......Page 71
4. The Problem of Multiple Reductions......Page 74
6. Structuralism?......Page 75
1. Shapiro's Characterization of Structures......Page 77
2. Mathematics Viewed as the Science of Structures......Page 78
4. Some Ways in Which My Account will Differ from the Structuralists'......Page 80
5. Ontological Aspects of Shapiro's Structuralism......Page 81
6. Why Believe in the Existence of Structures?......Page 85
7. Resnik's Early Theses about Structures and Positions......Page 86
8. Shapiro and Thesis [2]......Page 88
9. Shapiro's Rejection of Thesis [1]......Page 90
10. Some Problems with Shapiro's ante rem Structuralism......Page 91
11. A Problem with Shapiro's Acceptance of Thesis [2]......Page 96
12. The Main Problem with Eliminative Structuralism......Page 98
13. Resnik on Structures......Page 99
14. Resnik's No-Structure Theory......Page 101
15. Problems with Resnik's No-Structure Theory......Page 105
16. Resnik's Rejection of Classical Logic......Page 106
17. The Main Problem with Resnik's Version of Thesis [2]......Page 110
18. A Conceptual Objection to Structuralism......Page 111
1. Gödelian Platonism......Page 114
2. Quine's Challenge to the Nominalist......Page 119
3. Nominalistic Responses to Quine's Challenge......Page 122
4. The Direct Indispensability Argument......Page 130
5. The Indispensability Argument Based on Holism......Page 134
6. Putnam's Version of the Indispensability Argument......Page 138
7. Resnik's Version of the Argument......Page 141
8. Sober's Objection to the Indispensability Argument......Page 142
9. Maddy's Objections......Page 151
10. Resnik's New Indispensability Argument......Page 161
11. Concluding Assessment of the Various Arguments......Page 164
1. The Burgess–Rosen Account of Nominalism......Page 166
2. Nominalistic Reconstructions of Mathematics Reexamined......Page 174
7. The Constructibility Theory......Page 184
1. A Brief Exposition of the Constructibility Theory......Page 185
2. Preliminaries of the Constructibility Theory of Natural Number Attributes......Page 188
3. Cardinal Number Attributes......Page 192
4. Shapiro's Objections to the Constructibility Theory......Page 199
5. Other Objections Shapiro has Raised......Page 210
6. Resnik's Objections......Page 222
2. Realizations without Commitment to Mathematical Objects......Page 233
3. Constructible Realizations......Page 236
4. Constructible Realizations of Mathematical Theories......Page 239
5. The Natural Number Realization......Page 240
6. Higher-Level Constructible Realizations......Page 241
7. Realization of First-Order Theories......Page 242
9. Applications......Page 244
1. Applications of Constructibility Arithmetic......Page 246
2. Peano Arithmetic......Page 255
3. Putnam's If-Thenism......Page 260
4. Frege and Dummett on Why Mathematical Theorems must Express Thoughts......Page 264
5. A Reexamination of Various Indispensability Arguments......Page 268
6. Shapiro's Account of Applications......Page 269
7. Fermat's Last Theorem......Page 272
8. Applications of Analysis: Some General Considerations......Page 276
9. Mathematical Modeling......Page 277
10. Albert's Version of the Mathematics of Quantum Mechanics......Page 280
11. The Mathematics of Quantum Physics......Page 285
12. The Fundamental Theorem of the Integral Calculus......Page 295
13. The Burgess "Tonsorial Question"......Page 301
14. Applications of Set Theory in Logic......Page 304
15. Maddy's Mystery......Page 306
10. If-Thenism......Page 308
1. The Third Puzzle......Page 309
2. Brown's "Other Avenues"......Page 312
3. The Second Puzzle......Page 319
4. Criticisms of If-Thenism......Page 322
5. The Main Stumbling Block of the Eliminative Program......Page 330
11. Field's Account of Mathematics and Metalogic......Page 332
1. Why I Should not be Called a "Fictionalist"......Page 333
2. Field's Metalogical Theorems......Page 334
3. Field's Justification for Accepting Standard Metalogical Results......Page 336
4. Field's Other Justifications of his Conservation Principle......Page 344
5. Field's Arguments that Good Mathematics is Conservative......Page 347
6. A Comparison between Two Views of Mathematics......Page 354
7. The Fundamental Theorem Revisited......Page 360
Appendix A. Some Doubts About Hellman's Views......Page 364
Appendix B. Balaguer's Fictionalism......Page 371
Bibliography......Page 378
C......Page 388
E......Page 389
H......Page 390
M......Page 391
N......Page 392
P......Page 393
S......Page 394
Z......Page 395
