Mathematical Thought and Its Objects

Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also analyzes the concept of intuition and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite. 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 Half-title 3 Title 5 Copyright 6 Dedication 7 Contents 9 Preface 13 Sources and Copyright Acknowledgments 21 1 Objects and Logic 23 搂1. Abstract objects 23 搂2. The concept of an object in general: Actuality 25 搂3. Intuitability 30 搂4. Logic and the notion of object 32 搂5. Is whatever is an object? 35 搂6. Being and existence 45 搂7. Abstract objects and their concrete representations 55 2 Eliminative Structuralism and Nominalism 62 搂8. The structuralist view of mathematical objects 62 搂9. The concept of structure 65 搂10. Dedekind on the natural numbers 67 搂11. Eliminative structuralism and logicism 72 搂12. Nominalism 78 搂13. Nominalism and second-order logic 83 搂14. Structuralism and application 95 3 Modality and Structuralism 102 搂15. Mathematical modality 102 搂16. Modalism 114 搂17. Difficulties of modalism: Rejection of eliminative structuralism 118 搂18. A noneliminative structuralism 122 4 A Problem About Sets 139 搂19. An objection 139 搂20. Ontological conceptions of set 141 搂21. The iterative conception of set 144 搂22. "Intuitive" arguments for axioms of set theory 146 搂23. The replacement and power set axioms 152 5 Intuition 160 搂24. Intuition: Basic distinctions 160 搂25. Intuition and perception 165 搂26. Objections to the very idea of mathematical intuition 170 搂27. Toward a viable conception of intuition: Perception and the abstract 174 搂28. Hilbertian intuition 181 搂29. Intuitive knowledge: A step toward infinity 193 搂30. The objections revisited 201 6 Numbers as Objects 208 搂31. What are the natural numbers? 208 搂32. Cardinality and the genesis of numbers as objects 212 搂33. Finite sets and sequences 221 搂34. Sets and sequences, intuition and number 227 搂35. Difficulties concerning intuition of finite sets 233 搂36. Well, then, what are the numbers? Structuralism put in its place 240 搂37. Intuition of numbers denied 244 搂38. Appendix 1: Theories of sets and sequences 247 搂39. Appendix 2: Relative substitutional semantics for the language of hereditarily finite sets 253 7 Intuitive Arithmetic and Its Limits 257 搂40. Arithmetic as about strings: Finitism 257 搂41. The elementary axioms 266 搂42. Logic and intuition 269 搂43. Induction 274 搂44. Primitive recursion 276 搂45. The limits of intuitive knowledge 282 搂46. Appendix 284 8 Mathematical Induction 286 搂47. Induction and the concept of natural number 286 搂48. The problem of the uniqueness of the number structure: Nonstandard models 294 搂49. Uniqueness and communication 301 搂50. Induction and impredicativity 315 搂51. Predicativity and inductive definitions 329 9 Reason 338 搂52. Reason and "rational intuition" 338 搂53. Rational intuition and perception 347 搂54. Arithmetic 350 搂55. Set theory 360 Bibliography 365 Index 387 "In Mathematical Thought and Its Objects, Charles Parsons examines the notion of object, with the aim of navigating between nominalism, which denies that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a "nature" than that confers on them." "Parsons also analyzes the concept of intuition and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite. An intuitive model witnesses the possibility of the structure of natural numbers. However, the full concept of number and knowledge of numbers involve more that is conceptual and rational. Parsons considers how one can talk about numbers, even though they are not objects of intuition. He explores the conceptual role of the principle of mathematical induction and the sense in which it determines the natural numbers uniquely." "Parsons ends with a discussion of reason and its role in mathematical knowledge, attempting to do justice to the complementary roles in mathematical knowledge of rational insight, intuition, and the integration of our theory as a whole."--BOOK JACKET In Mathematical Thought and Its Objects, Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a "nature" than that confers on them. Objects And Logic -- Structuralism And Nominalism -- Modality And Structuralism -- A Problem About Sets -- Intuition -- Numbers As Objects -- Intuitive Arithmetic And Its Limits -- Mathematical Induction -- Reason. Charles Parsons. Includes Bibliographical References (p. 343-363) And Index.

in Mathematical Thought And Its Objects, Charles Parsons Examines The Notion Of Object.