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Wayne State University Wayne State University Dissertations 1-2-2013 Nominalism In Mathematics - Modality And Naturalism James S.j. Schwartz Wayne State University, Follow this and additional works at: http://digitalcommons.wayne.edu/oa_dissertations Recommended Citation Schwartz, James S.j., "Nominalism In Mathematics - Modality And Naturalism" (2013). Wayne State University Dissertations. Paper 795. This Open Access Dissertation is brought to you for free and open access by DigitalCommons@WayneState. It has been accepted for inclusion in Wayne State University Dissertations by an authorized administrator of DigitalCommons@WayneState. NOMINALISM IN MATHEMATICS: MODALITY AND NATURALISM by JAMES S.J. SCHWARTZ DISSERTATION Submitted to the Graduate School of Wayne State University, Detroit, Michigan in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY 2013 MAJOR: PHILOSOPHY Approved by: Advisor Date c COPYRIGHT BY JAMES S.J. SCHWARTZ 2013 All Rights Reserved DEDICATION For my mother ii ACKNOWLEDGEMENTS Special thanks are due to the chair of my dissertation committee, Susan Vineberg, who has carefully reviewed far too many drafts of each chapter. Without her guidance this project would not have been possible. Many thanks are of course due to the other members of my committee: Eric Hiddleston, Michael McKinsey, and Robert Bruner. I am also indebted to Sean Stidd (my unofficial “fifth committee member”), John Halpin, and Matt McKeon for discussion along the way, as well as to various other philosophers, colleagues, and friends: David Baxter, Sharon Berry, Daniel Blaser, Marcus Cooper, John Corvino, Michael Ernst, Travis Figg, David Garfinkle, Herbert Granger, Geoffrey Hellman, Timothy Kirschenheiter, Teresa Kouri, Øystein Linnebo, Lawrence Lombard, Penelope Maddy, Gonzalo Munevar,´ Gregory Novack, Christopher Pincock, Lawrence Powers, Daniel Propson, Mark Rigstad, Bruce Russell, Scott Shalkowski, Stewart Shapiro, and Daniel Yeakel. An abridged version of the second chapter was read at a Philosophy Department Colloquium at Wayne State University in April of 2012 and at the 2012 Meeting of the Society for Exact Philosophy in October of 2012. An earlier version of the fourth chapter was the topic of a Natural Philosophy Workshop at Oakland University in March of 2012, and an abridged version of this chapter was presented at the 4th Annual Graduate Exhibition at Wayne State University in March of 2013. I thank those in attendance at the above events for comments and discussion. Chapter three has improved greatly from extensive comments from Eric Hiddleston and from Michael McKinsey. The chapters on naturalism (four and five) are the result of many rounds of comments from Susan Vineberg. Their influence on these sections of the dissertation is warmly acknowledged; I fear they do not receive enough credit in the footnotes—especially for helping me to think more clearly about which ideas were worth developing and which were not. Finally, I would to thank my mother, Cynthia Johnson, whose encouragement and sup- iii port throughout my entire education has made possible my completion of this dissertation. iv TABLE OF CONTENTS Dedication . ii Acknowledgments . iii Introduction . 1 I Thesis . 1 II The Basics . 2 II.1 What is the Philosophy of Mathematics? . 2 II.2 What is Nominalism? . 3 III Modality . 6 III.1 Modality in Philosophy of Mathematics . 6 III.2 Shapiro’s Challenge to Nominalism . 8 III.3 Modality and Reduction . 14 IV Naturalism . 18 IV.1 Reflections on Burgess . 20 IV.2 Reflections on Maddy . 22 V Conclusions . 26 Part 1: Modality . 28 1 Modality in the Philosophy of Mathematics . 28 1.1 Introduction . 28 1.2 Chihara’s Constructibility Theory . 30 1.2.1 The Constructibility Theory . 30 1.2.2 A Closer Look at Modality in Constructibility Theory . 34 1.3 Hellman’s Modal Structuralism . 44 1.3.1 A Closer Look at Modality in Modal Structuralism . 47 1.4 Field’s Fictionalism . 50 v 1.4.1 A Closer Look at Modality in Field’s Fictionalism . 54 1.4.2 Field and Justifying Modal Assertions . 58 1.5 Recent Developments . 67 1.5.1 Balaguer’s Fictionalism . 67 1.5.2 Leng’s Fictionalism . 70 2 Shapiro’s Challenge to the use of Modality in Nominalist Theories . 71 2.1 Introduction . 71 2.2 Modality and Ontology . 74 2.2.1 The Emperor’s New Epistemology . 75 2.2.2 The Emperor’s New Ontology . 83 2.3 First Reply . 86 2.4 The Paraphrase Response . 92 2.4.1 Nominalist Paraphrase as Synonymy . 95 2.5 Reply to the Paraphrase Response . 100 2.6 The Structuralist Response . 105 2.6.1 Resnik and Patterns . 106 2.6.2 Shapiro and Structure . 114 2.7 Reply to the Structuralist Response . 125 2.7.1 Withering Coherence . 127 2.8 Shapiro’s Challenge: What Exactly is the Problem? . 137 3 Reducing Modality as a Solution to Shapiro’s Challenge . 146 3.1 Introduction . 146 3.2 Reduction and Shapiro’s Challenge . 149 3.2.1 Mission Planning . 154 3.3 Justifying Modal Assertions Under Lewis’s Reduction . 156 3.3.1 Lewis’s Reduction . 156 vi 3.3.2 Lewis and Justifying Modal Assertions . 158 3.4 Possible Worlds and Functional Roles . 164 3.5 Lessons for Shapiro and Set Theory . 166 3.5.1 The Set-theoretic Reduction . 166 3.5.2 Justifying Modal Assertions Under the Set-Theoretic Reduction . 169 3.6 Living With Primitive Modality . 174 Part 2: Naturalism . 178 4 Reflections on Burgess . 181 4.1 Introduction . 181 4.2 The Master Argument Against Nominalism . 185 4.2.1 Burgess’s Naturalism . 186 4.2.2 The Arguments . 193 4.3 Replies in the Literature . 200 4.3.1 The Scientific Merits of Nominalistic Reinterpretation . 201 4.3.2 Scientific Merits Redux: Chihara and the Attitude-Hermeneuticist . 206 4.3.3 The False Dilemma Reply . 208 4.4 My Reply . 211 4.4.1 The Tonsorial Question . 213 4.5 Why Burgess is not a (Moderate) Platonist . 227 5 Reflections on Maddy . 234 5.1 Introduction . 234 5.2 Second Philosophy of Mathematics . 239 5.2.1 Two Provisional Second-Philosophical Objections to Modal Nomi- nalism . 245 5.3 The Method-Affirming Objection to Modal Nominalism . 249 5.3.1 A Miscellany of Objections . 250 vii 5.3.2 Method-Affirming as a Prophylactic . 254 5.3.3 Method- and Result-Rejecting: The Good and The Bad . 257 5.3.4 The Method-Affirming Objection Reconsidered . 258 5.3.5 The Method-Affirming Objection: Coda . 265 5.4 The Method-Contained Objection to Modal Nominalism . 266 5.4.1 Thin Realism . 268 5.4.2 Arealism . 272 5.4.3 There is no Difference Here . 274 5.4.4 The Method-Contained Objection: Coda . 277 5.5 How Much Naturalism is Too Much Naturalism? . 279 5.6 Conclusion: Naturalism and Modal Nominalism . 283 Bibliography . 286 Abstract . 297 Autobiographical Statement . 299 viii 1 Introduction I Thesis This dissertation is a defense of certain modal nominalist strategies in the philosophy of mathematics against two varieties of criticism. My primary thesis, which is defended in Part I (chapters one, two, and three), is that modal nominalists do not encounter uniquely challenging difficulties in regards to justifying the modal assertions that figure in modal nominalist theories of mathematics. My secondary thesis, which is defended in Part II (chapters four and five), is that modal nominalism does not engender any serious conflict with the practice of mathematics, and thus is consistent with the naturalistic impulse to respect science and mathematics. In defending these theses I hope to establish modal nominalism as a viable and attractive philosophy of mathematics, one that can provide satisfactory answers to many of philosophy of mathematics’ longstanding questions. My goal in this introduction is to explain the provenance of these theses by examining the following questions: What is the philosophy of mathematics? What is nominalism? What is modal nominalism? What has modality got to do with nominalism in mathematics, and what, if anything, should a modal nominalist find troubling about this? What is naturalism, and why should anyone suppose that naturalism and modal nominalism are incompatible? 2 II The Basics II.1 What is the Philosophy of Mathematics? The philosophy of mathematics is a broad-ranging subdiscipline of analytic philosophy tasked with answering questions such as: “What is mathematics about?” “What is the best characterization of what the practicing mathematician does?” “What is the methodology of mathematics, be it pure or applied?” “How do mathematicians decide whether to adopt new theories or axioms?” “Do there exist mathematical objects?” “What kind of a logical foundation is necessary for doing mathematics?” “Why is mathematical reasoning so successful in scientific applications?”. and the list could go on. Nevertheless, the picture that emerges is one according to which philosophers of mathematics are interested in the nature of mathematics as well as the.
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