Mathematical Thought and Its Objects

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
Sources and Copyright Acknowledgments......Page 21
§1. Abstract objects......Page 23
§2. The concept of an object in general: Actuality......Page 25
§3. Intuitability......Page 30
§4. Logic and the notion of object......Page 32
§5. Is whatever is an object?......Page 35
§6. Being and existence......Page 45
§7. Abstract objects and their concrete representations......Page 55
§8. The structuralist view of mathematical objects......Page 62
§9. The concept of structure......Page 65
§10. Dedekind on the natural numbers......Page 67
§11. Eliminative structuralism and logicism......Page 72
§12. Nominalism......Page 78
§13. Nominalism and second-order logic......Page 83
§14. Structuralism and application......Page 95
§15. Mathematical modality......Page 102
§16. Modalism......Page 114
§17. Difficulties of modalism: Rejection of eliminative structuralism......Page 118
§18. A noneliminative structuralism......Page 122
§19. An objection......Page 139
§20. Ontological conceptions of set......Page 141
§21. The iterative conception of set......Page 144
§22. \"Intuitive\" arguments for axioms of set theory......Page 146
§23. The replacement and power set axioms......Page 152
§24. Intuition: Basic distinctions......Page 160
§25. Intuition and perception......Page 165
§26. Objections to the very idea of mathematical intuition......Page 170
§27. Toward a viable conception of intuition: Perception and the abstract......Page 174
§28. Hilbertian intuition......Page 181
§29. Intuitive knowledge: A step toward infinity......Page 193
§30. The objections revisited......Page 201
§31. What are the natural numbers?......Page 208
§32. Cardinality and the genesis of numbers as objects......Page 212
§33. Finite sets and sequences......Page 221
§34. Sets and sequences, intuition and number......Page 227
§35. Difficulties concerning intuition of finite sets......Page 233
§36. Well, then, what are the numbers? Structuralism put in its place......Page 240
§37. Intuition of numbers denied......Page 244
§38. Appendix 1: Theories of sets and sequences......Page 247
§39. Appendix 2: Relative substitutional semantics for the language of hereditarily finite sets......Page 253
§40. Arithmetic as about strings: Finitism......Page 257
§41. The elementary axioms......Page 266
§42. Logic and intuition......Page 269
§43. Induction......Page 274
§44. Primitive recursion......Page 276
§45. The limits of intuitive knowledge......Page 282
§46. Appendix......Page 284
§47. Induction and the concept of natural number......Page 286
§48. The problem of the uniqueness of the number structure: Nonstandard models......Page 294
§49. Uniqueness and communication......Page 301
§50. Induction and impredicativity......Page 315
§51. Predicativity and inductive definitions......Page 329
§52. Reason and \"rational intuition\"......Page 338
§53. Rational intuition and perception......Page 347
§54. Arithmetic......Page 350
§55. Set theory......Page 360
Bibliography......Page 365
Index......Page 387