Contents -- Preface -- Introduction -- 1. A Nominalist's View of Philosophy: The Big Picture -- 2. Nominalistic Reconstructions -- 1. Five Puzzles in Search of an Explanation -- 1. A Puzzle about Geometry -- 2. Different Attitudes of Practicing Mathematicians Regarding the Ontology of Mathematics -- 3. The Inertness of Mathematical Objects -- 4. Consistency and Mathematical Existence -- 5. The van Inwagen Puzzle -- 2. Geometry and Mathematical Existence -- 1. The Frege-Hilbert Dispute Concerning the Axioms of Geometry -- 2. Some Suggestions Regarding the Nature of Mathematics -- 3. The First Puzzle -- 4. The Fourth Puzzle -- 5. Some Concluding Thoughts -- 3. The van Inwagen Puzzle -- 1. Structures -- 2. My Resolution of the Puzzle -- 3. The Genetics Objection -- 4. The Problem of Multiple Reductions -- 5. An Intuitive Understanding of Set -- 6. Structuralism? -- 4. Structuralism -- 1. Shapiro's Characterization of Structures -- 2. Mathematics Viewed as the Science of Structures -- 3. Mathematical and Ordinary Structures -- 4. Some Ways in Which My Account will Differ from the Structuralists' -- 5. Ontological Aspects of Shapiro's Structuralism -- 6. Why Believe in the Existence of Structures? -- 7. Resnik's Early Theses about Structures and Positions -- 8. Shapiro and Thesis [2] -- 9. Shapiro's Rejection of Thesis [1] -- 10. Some Problems with Shapiro's ante rem Structuralism -- 11. A Problem with Shapiro's Acceptance of Thesis [2] -- 12. The Main Problem with Eliminative Structuralism -- 13. Resnik on Structures -- 14. Resnik's No-Structure Theory -- 15. Problems with Resnik's No-Structure Theory -- 16. Resnik's Rejection of Classical Logic -- 17. The Main Problem with Resnik's Version of Thesis [2] -- 18. A Conceptual Objection to Structuralism -- 5. Platonism -- 1. Gödelian Platonism -- 2. Quine's Challenge to the Nominalist.

3. Nominalistic Responses to Quine's Challenge -- 4. The Direct Indispensability Argument -- 5. The Indispensability Argument Based on Holism -- 6. Putnam's Version of the Indispensability Argument -- 7. Resnik's Version of the Argument -- 8. Sober's Objection to the Indispensability Argument -- 9. Maddy's Objections -- 10. Resnik's New Indispensability Argument -- 11. Concluding Assessment of the Various Arguments -- 6. Minimal Anti-Nominalism -- 1. The Burgess-Rosen Account of Nominalism -- 2. Nominalistic Reconstructions of Mathematics Reexamined -- 7. The Constructibility Theory -- 1. A Brief Exposition of the Constructibility Theory -- 2. Preliminaries of the Constructibility Theory of Natural Number Attributes -- 3. Cardinal Number Attributes -- 4. Shapiro's Objections to the Constructibility Theory -- 5. Other Objections Shapiro has Raised -- 6. Resnik's Objections -- 8. Constructible Structures -- 1. A Problem for the Structuralist -- 2. Realizations without Commitment to Mathematical Objects -- 3. Constructible Realizations -- 4. Constructible Realizations of Mathematical Theories -- 5. The Natural Number Realization -- 6. Higher-Level Constructible Realizations -- 7. Realization of First-Order Theories -- 9. Applications -- 1. Applications of Constructibility Arithmetic -- 2. Peano Arithmetic -- 3. Putnam's If-Thenism -- 4. Frege and Dummett on Why Mathematical Theorems must Express Thoughts -- 5. A Reexamination of Various Indispensability Arguments -- 6. Shapiro's Account of Applications -- 7. Fermat's Last Theorem -- 8. Applications of Analysis: Some General Considerations -- 9. Mathematical Modeling -- 10. Albert's Version of the Mathematics of Quantum Mechanics -- 11. The Mathematics of Quantum Physics -- 12. The Fundamental Theorem of the Integral Calculus -- 13. The Burgess "Tonsorial Question".

14. Applications of Set Theory in Logic -- 15. Maddy's Mystery -- 10. If-Thenism -- 1. The Third Puzzle -- 2. Brown's "Other Avenues" -- 3. The Second Puzzle -- 4. Criticisms of If-Thenism -- 5. The Main Stumbling Block of the Eliminative Program -- 11. Field's Account of Mathematics and Metalogic -- 1. Why I Should not be Called a "Fictionalist" -- 2. Field's Metalogical Theorems -- 3. Field's Justification for Accepting Standard Metalogical Results -- 4. Field's Other Justifications of his Conservation Principle -- 5. Field's Arguments that Good Mathematics is Conservative -- 6. A Comparison between Two Views of Mathematics -- 7. The Fundamental Theorem Revisited -- Appendix A. Some Doubts About Hellman's Views -- Appendix B. Balaguer's Fictionalism -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- V -- W -- Y -- Z.