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NUMBERS WITHOUT SCIENCE by Russell Marcus a Dissertation NUMBERS WITHOUT SCIENCE by Russell Marcus A dissertation submitted to the Graduate Faculty in Philosophy in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of New York 2007 Page ii © 2007 Russell Marcus All Rights Reserved Page iii This manuscript has been read and accepted for the Graduate Faculty in Philosophy in satisfaction of the dissertation requirement for the degree of Doctor of Philosophy. Elliott Mendelson October 26, 2006 _________________________________ Chair of Examining Committee Peter Simpson October 26, 2006 _________________________________ Executive Officer David Rosenthal (Adviser) Michael Devitt Jody Azzouni Alberto Cordero Supervisory Committee The City University of New York Page iv Abstract NUMBERS WITHOUT SCIENCE by Russell Marcus Adviser: Professor David Rosenthal Numbers without Science opposes the Quine-Putnam indispensability argument, seeking to undermine the argument and reduce its profound influence. Philosophers rely on indispensability to justify mathematical knowledge using only empiricist epistemology. I argue that we need an independent account of our knowledge of mathematics. The indispensability argument, in broad form, consists of two premises. The major premise alleges that we are committed to mathematical objects if science requires them. The minor premise alleges that science in fact requires mathematical objects. The most common rejection of the argument denies its minor premise by introducing scientific theories which do not refer to mathematical objects. Hartry Field has shown how we can reformulate some physical theories without mathematical commitments. I argue that Field’s preference for intrinsic explanation, which underlies his reformulation, is ill-motivated, and that his resultant fictionalism suffers unacceptable consequences. I attack the major premise instead. I argue that Quine provides a mistaken Page v criterion for ontic commitment. Our uses of mathematics in scientific theory are instrumental and do not commit us to mathematical objects. Furthermore, even if we accept Quine’s criterion for ontic commitment, the indispensability argument justifies only an anemic version of mathematics, and does not yield traditional mathematical objects. The first two chapters of the dissertation develop these results for Quine’s indispensability argument. In the third chapter, I apply my findings to other contemporary indispensabilists, specifically the structuralists Michael Resnik and Stewart Shapiro. In the fourth chapter, I show that indispensability arguments which do not rely on Quine’s holism, like that of Putnam, are even less successful. Also in Chapter 4, I show how Putnam’s work in the philosophy of mathematics is unified around the indispensability argument. In the last chapter of the dissertation, I conclude that we need an account of mathematical knowledge which does not appeal to empirical science and which does not succumb to mysticism and speculation. Briefly, my strategy is to argue that any defensible solution to the demarcation problem of separating good scientific theories from bad ones will find mathematics to be good, if not empirical, science. Page vi Preface The indispensability argument alleges that we have knowledge of the abstract objects of mathematics, and that this knowledge is justified by our uses of mathematics in empirical science. The evidence for mathematics is thus supposed to be empirical. In this dissertation, I defend mathematical knowledge while denying its empirical justification. Mathematics requires an epistemology independent of that for empirical science. There are at least three ways of arguing for empirical justification of mathematics. The first is to argue, as Mill did, that mathematical knowledge is knowledge of empirical objects. I will not pursue criticisms of this dead end; I refer the reader to Frege (1953). The second is to argue that while mathematical knowledge is knowledge of abstract objects, we have sensory access to such objects. I will not pursue this Aristotelian line, either. I take the fallout from Penelope Maddy’s attempt, including her own rejection of that position, to suffice.1 The currently most popular way to justify mathematics empirically is to argue: A) Mathematical knowledge is of abstract objects; B) We have experiences only with concrete objects; and yet C) Our experiences with concrete objects justify our mathematical knowledge. This is the Quine-Putnam Indispensability Argument, in its broadest form. In Chapter 1 of the dissertation I present and defend my interpretation of Quine’s version of 1 See Maddy (1980) for the position, and Balaguer (1994) for an excellent summary of problems with the position. Page vii the indispensability argument, which depends on his method for determining ontic commitment. I also show how the most popular and promising response to Quine’s argument, Field’s dispensabilist project, fails to defeat it. In Chapter 2, I provide three criticisms of Quine’s argument. In Part 1, I deny Quine’s method for determining ontic commitment. In Part 2, I defend instrumentalism as an alternative to Quine’s method. In Part 3, I show how even if we accept Quine’s method, the indispensability argument does not yield mathematical objects. In Chapters 3 and 4, I consider other versions of the indispensability argument. In Chapter 3, I present essential characteristics of indispensability arguments, and the unfortunate consequences of these arguments. I apply these general results to structuralists who rely on the indispensability argument. Part 1 of Chapter 4 is an aside on the central role that the indispensability argument plays in Hilary Putnam’s work. In Part 2, I show how non-holistic indispensability arguments, Michael Resnik’s pragmatic argument and Putnam’s success argument, fare no better than holistic versions. The concluding Chapter 5 has four parts. In Part 1, I argue for the legitimacy of mathematics as a science in its own right. In Part 2, I argue that we should pursue an epistemology for mathematics independent of that for empirical science. I characterize several elements of this approach, which I call autonomy realism, including its dependence on mathematical intuition. In Part 3, I revisit Field’s work, as he also seeks an alternative to the indispensability argument. I show that autonomy realism is a preferable alternative to both indispensabilism and Field’s fictionalism. I conclude by indicating a few areas for further research. Page viii Acknowledgments This dissertation was begun under the advisement of Jerrold Katz. Jerry and his work continue to inspire me. I wish he were still with us, if only to tell me where my errors are. Jared Blank, Jennifer Fisher, Eric Hetherington, and Mark McEvoy have been supportive friends and helpful critics over the years. I have spoken earnestly with Eric about both my concerns about the philosophical role of formal theories and about misuses of mathematics in philosophy. I fear that my attempts to flesh out the first concern have led me to commit errors of the second kind. I am grateful to Michael Devitt for reading several drafts and making helpful comments, and also for his friendly guidance. Elliott Mendelson has been supportive all the way. I deeply appreciate Jody Azzouni’s selfless assistance. Paul Horwich generously spent hours discussing with me my work on Quine, and helped me to formulate my position more clearly. Michael Levin read drafts and made many useful comments. Errors and infelicities which remain are of course solely my responsibility. Portions of the dissertation have been presented to audiences at the CUNY Graduate Philosophy Conference, Spring 2005; the Graduate Center Philosophy Student Colloquium series, Spring 2005; the UT Austin Graduate Philosophy Conference, April 2006; the University of British Columbia Student Philosophy Conference, April 2006; and the Rocky Mountain Student Philosophy Conference, March 2006. David Rosenthal has been invaluable in many ways, not least as a trusted adviser and confidant. I owe him immeasurable gratitude and a great debt. Page ix I owe my loving family both heartfelt thanks and sincere apologies for putting their empathetic hearts through prolonged strain. My parents supported me despite wondering why I would follow my mother’s lead and leave the beginnings of a successful career to pursue a Graduate Center Ph.D. My children, Marina and Isidor, have been profound inspirations and motivations. But most of all, Emily, whose confidence in me, puzzlingly, never wavered. Page x Table of Contents Abstract ....................................................................................................................... iv Preface ......................................................................................................................... vi Chapter 1: Quine’s Indispensability Argument and Field’s Response Part 1: Quine’s Indispensability Argument §1.1: Quine’s Argument ...................................................................... 1 §1.2: A Best Theory ............................................................................ 2 §1.3: Believing Our Best Theory ........................................................ 6 §1.4: Quine’s Procedure for Determining Ontic Commitments ......... 9 §1.4.1: First-Order Logic ........................................................ 10 §1.4.2: The Domain of Quantification .................................... 14 §1.5: Mathematization ........................................................................
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