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Parsons Ch-Math.Pdf This page intentionally left blank P1: JZP CUNY1138-FM CUNY1138/Parsons 978 0 521 45279 3 November 8, 2007 21:58 MATHEMATICAL THOUGHT AND ITS OBJECTS 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 mathe- matical knowledge of rational insight, intuition, and the integration of our theory as a whole. Charles Parsons is Edgar Pierce Professor of Philosophy, Emeritus, at Harvard University. He is a former editor of the Journal of Philosophy.Heisthe author of Mathematics in Philosophy and co-editor of the posthumous works of Kurt Godel.¨ i P1: JZP CUNY1138-FM CUNY1138/Parsons 978 0 521 45279 3 November 8, 2007 21:58 ii P1: JZP CUNY1138-FM CUNY1138/Parsons 978 0 521 45279 3 November 8, 2007 21:58 Mathematical Thought and Its Objects Charles Parsons Harvard University iii CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521452793 © Cambridge University Press 2008 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008 ISBN-13 978-0-511-37867-6 eBook (NetLibrary) ISBN-13 978-0-521-45279-3 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. P1: JZP CUNY1138-FM CUNY1138/Parsons 978 0 521 45279 3 November 8, 2007 21:58 For Marjorie v P1: JZP CUNY1138-FM CUNY1138/Parsons 978 0 521 45279 3 November 8, 2007 21:58 vi P1: JZP CUNY1138-FM CUNY1138/Parsons 978 0 521 45279 3 November 8, 2007 21:58 Contents Preface page xi Sources and Copyright Acknowledgments xix 1Objects and logic ................................1 §1. Abstract objects1 §2. The concept of an object in general: Actuality3 §3. Intuitability8 §4. Logic and the notion of object 10 §5. Is whatever is an object? 13 §6. Being and existence 23 §7. Abstract objects and their concrete representations 33 2Structuralism and nominalism ......................40 §8. The structuralist view of mathematical objects 40 §9. The concept of structure 43 §10. Dedekind on the natural numbers 45 §11. Eliminative structuralism and logicism 50 §12. Nominalism 56 §13. Nominalism and second-order logic 61 §14. Structuralism and application 73 3Modality and structuralism ........................80 §15. Mathematical modality 80 §16. Modalism 92 §17. Difficulties of modalism: Rejection of eliminative structuralism 96 §18. A noneliminative structuralism 100 vii P1: JZP CUNY1138-FM CUNY1138/Parsons 978 0 521 45279 3 November 8, 2007 21:58 viii Contents 4Aproblem about sets ............................117 §19. An objection 117 §20. Ontological conceptions of set 119 §21. The iterative conception of set 122 §22. “Intuitive” arguments for axioms of set theory 124 §23. The Replacement and Power Set axioms 130 5Intuition ....................................138 §24. Intuition: Basic distinctions 138 §25. Intuition and perception 143 §26. Objections to the very idea of mathematical intuition 148 §27. Toward a viable conception of intuition: Perception and the abstract 152 §28. Hilbertian intuition 159 §29. Intuitive knowledge: A step toward infinity 171 §30. The objections revisited 179 6Numbers as objects .............................186 §31. What are the natural numbers? 186 §32. Cardinality and the genesis of numbers as objects 190 §33. Finite sets and sequences 199 §34. Sets and sequences, intuition and number 205 §35. Difficulties concerning intuition of finite sets 211 §36. Well, then, what are the numbers? Structuralism put in its place 218 §37. Intuition of numbers denied 222 §38. Appendix 1: Theories of sets and sequences 225 §39. Appendix 2: Relative substitutional semantics for the language of hereditarily finite sets 231 7Intuitive arithmetic and its limits ....................235 §40. Arithmetic as about strings: Finitism 235 §41. The elementary axioms 244 §42. Logic and intuition 247 §43. Induction 252 §44. Primitive recursion 254 §45. The limits of intuitive knowledge 260 §46. Appendix 262 P1: JZP CUNY1138-FM CUNY1138/Parsons 978 0 521 45279 3 November 8, 2007 21:58 Contents ix 8Mathematical induction .........................264 §47. Induction and the concept of natural number 264 §48. The problem of the uniqueness of the number structure: Nonstandard models 272 §49. Uniqueness and communication 279 §50. Induction and impredicativity 293 §51. Predicativity and inductive definitions 307 9Reason .....................................316 §52. Reason and “rational intuition” 316 §53. Rational intuition and perception 325 §54. Arithmetic 328 §55. Set theory 338 Bibliography 343 Index 365 P1: JZP CUNY1138-FM CUNY1138/Parsons 978 0 521 45279 3 November 8, 2007 21:58 x P1: JZP CUNY1138-FM CUNY1138/Parsons 978 0 521 45279 3 November 8, 2007 21:58 Preface The present work is largely concerned with a limited number of themes in the philosophy of mathematics. The first is the notion of object as it is deployed in mathematics. I begin in Chapter 1 with a general discus- sion of the notion of object, not on the whole focused on mathematics. One of the motives of this discussion is to defuse too-high expectations of what the existence of objects of some mathematical type such as numbers would entail. We proceed to discuss issues surrounding the structuralist view of mathematical objects, which has had a lot of currency in the last forty years or so but has much earlier roots. The general idea of this view is that mathematical objects do not have a richer “nature” than is given by the basic relations of some structure in which they reside. The problem of giving a viable formulation developing this idea is not trivial and raises a lot of issues. That is the concern of Chapters 2 and 3. Chapter 2 is mainly devoted to pursuing a program that uses the structuralist idea to elimi- nate explicit reference to mathematical objects. Along the way, I discuss some questions about nominalism, about second-order logic, and about how structuralism understands the application of mathematics. Some difficulties of the eliminative program call for using modal notions, and their use in mathematics is a subject of Chapter 3. But in the end even the modal version of the eliminative program is rejected, and in §18 a version of structuralism is sketched that takes the language of mathematics much more at face value. Chapter 4 responds to an objection to the application to set theory of the version of structuralism I defend. Along the way, it considers other questions about the concept of set and the axioms of set theory. The second main theme is a particular notion of mathematical intu- ition, which has its origin in the thought of Brouwer and Hilbert about the most basic elements of arithmetic but whose original inspiration comes xi P1: JZP CUNY1138-FM CUNY1138/Parsons 978 0 521 45279 3 November 8, 2007 21:58 xii Preface from Kant. In Chapter 5, I lay out some basic distinctions concerning the notion of intuition, about which writers who use the notion (even to crit- icize it) are often unclear. But the main point of the chapter is to explain the particular conception of intuition that concerns me, develop some of its implications, and reply to some possible objections to it. Intuition so conceived offers part of the entry of mathematical thought into the infinite. The structure of natural numbers is shown to be witnessed by what can be called an intuitive model. Chapters 6, 7, and 8 all concern the arithmetic of natural numbers, and it is in the first two of these that the work done by the conception of intuition of Chapter 5 is visible. Chapter 6 discusses the role of natural numbers as finite cardinals and ordinals and considers in a rather ideal- ized way how language referring to natural numbers might originate. This genetic method is inspired by W. V. Quine’s Roots of Reference.Aconclu- sion of the chapter is that in the sense of Chapter 5, there is not intuition of numbers properly speaking. We also explore theories of finite sets and discuss the question of intuition of such sets, with the conclusion that the analogy with perception that such intuition requires would be too much stretched if it is claimed that the theory of hereditarily finite sets rests on intuition in the sense in which Hilbert and Bernays claimed that finitary arithmetic does.
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