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Adopted from Pdflib Image Sample ON NUMBERS by LINDA ELIZABETH WETZEL B.A. City College of New York (1975) Submitted to the Department of Linguistics and Philosophy in Partial Fulfillment of the Requirements of the Degree of DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 1984 @ Linda E. Wetzel The author hereby grants to M.I.T. permission to reproduce and to distribute copies of this thesis document in whole or in part, Signature of Author: Department of Linguistics and Philosophy January 13, 1984 Certified by : .-_ - Richard L, Cartwright Thesis Supervisor Accepted by: v Richard L. Cartwright Chairman, Departmental Graduate Committee ON NUMBERS Linda E, Wetzel Submitted to the Department of Linguistics and Philosophy of October 21, 1983 in partial fulfillment of the requirements for the degree of Doctor of Philosophy ABSTRACT We talk as though there are numbers. The view I defend, the "popularn view, has it that there -are numbers. However, since they clearly are not physical objects, we reason that they must be abstract ones. This suggests a realm of non-spatial non-temporal objects standing in numerical relations; arithmetic knowledge is then knowledge of this realm. But how do spatio~temporal creatures like ourselves come to have knowledge of this realm? The problem ("Benacerrafls probleni") can be avoided by arguing that there are no numbers. In "What Numbers Could Not Bew Benacerraf himself took such a route. In chapter one, I discuss three of Benacerrafls arguments, showing that the first is circular, that the second involves a consideration that can be explained by less drastic means than supposing there are no numbers, and that the third would, if successful, show that neither sets nor expressions exist either. Yet despite the lack of success of arguments purporting to show that there are no numbers, accounts of arithmetic which make no reference to numbers might be thought preferable to the popular view on the ground that they manage to avoid the epistemological problem. Accordingly, in chapter two I examine three quasi-formalist accounts that llreducen number talk to talk about other sorts of objects, to see whether this is so. I show that each of then1 involves commitments to other familiar mathematical objects, and hence makes no headway on the epistemological problem. In view of the failure of quasi-formalist positions, the next logical step for someone anxious to avoid the epistemological problem by denying there are numbers is to retreat t; full-blooded formalism, maintaining that there are no mathematical entities, only linguistic ones. The presupposition here seems to be that unlike mathematical entities, expressions are epistemologically "accessible" (and therefore not subject to the epistemological problem.) By way of justifying the presupposition, I believe that a formalist would adva-nce the fcllowlng grounds: (1) Expressions are concrete whereas mathematical entities, including numbers, are not, (2) Even if expressions are abstract objects, we can know about them on the basis of their tokens, which -are concrete objects; but since numbers have no tokens, this sort of explanation is not available. In section one of chapter three I examine (I), siiowing that the claim that expressions are concrete is just not tenable, Then I examine (2), arguing that it, too, is false, because in whatever sense we can know about expressions on the basis of interacting with their tokens, we can know about numbers, too, on the basis of interacting with pairs, trios, quadruples, etc. Thesis Supervisor: Richard L, Cartwright Title: Professor of Philosophy A plurality is not an instance of num- ber, but of some particular number. A trio of men, for example, is aninstanoe of the number 3.. .. This point may seem elementary and scarcely worth mention- ing; yet it has proved too subtle for the philosophers, with few exceptions. --Bertrand Russell ACKNOWLEDGEMENTS I owe an enormous debt fo Richard Cartwright and to George Boolos. In the course of many discussions, Richard Cartwright patiently guided me through fhe jungle of ideas that beeame this thesis. His advice and detailed criticisms were invaluable. No less a debt do I owe George Boolos. Besides providing much needed guidance and encour- agement, he first sensitized me to,the difficulties with nominalism in philosophy of mathematics--albeit through his devastating but elegant criticisms of an early paper I wrote. My intellectual\ development was greatly aided by exposure to the rigorous ways and keen insight of these two people. To Sylvain Bromberger I am.also deeply beholden. He not only played a substantial role in my education, especially as regards philosophy of language, but by generously bringing his philosophical acumen to bear upon chapter 111, greatly improved.it. My heartfelt gratitude goes to my friend Lon Berk. Always supportive, he discussed with me each argument, and read every word I wrote. Not only would this thesis have suffered without his valuable criticisms, but it might not have been at all. It was his tips on how-to- write-a-thesis that eventually carried the day. For general discussions in aid of the present project, not to mention the accompanying good times, I thank Jerrold Katz and Raymond Smullyan. My debt to them extends beyond that, however, for it was Jerrold Katz who intro- duced me to philosophy of language, and Raymond Smullyan who ffrsh taught me.to relcfsh logic and paradoxes. For specific discussions in aid of the present project, a special thanks goes to Paul Benacerraf. Along with the members of his spring 1981 seminar in philosophy of mathe- matics at Princeton, he endured a presentation of many of the ideas contained in chapters I and 11. His insightful response to those ideas proved most useful. Thanks also to Harold Hodes, whose cheerful espousal of nominalism in part prompted chapter 111. Others whose conversations were helpful and encouragement timely include Ken Albert, John Bacon, (belatedly) Joseph Bevando, Ned Block, Jack Cobetto, Judi DeCew, James Higginbotham and Joseph Ullian. Finally, I would like to thank my parents, William and Elizabeth Wetzel, brothers Stephen and John, and good friends Deborah Bowen-Weil, Eleanor Druckman and Valerie Johnson, for the support, good humor and kind words they lavished on me throughout the duration. TABLE OF CONTENTS ACKNOWLEDGEMENTS ................................ 5 INTRODUCTION ................................... 7 CHAPTER IIBENACERRAF ........................... 18 1 . First Argument .......................... 18 2 . Second Argument ......................... 36 3 . Third Argument .......................... 49 CHAPTER II/SOME RIVALS OF THE POPULAR VIEW .... 78 1 . Introduction ........................... 78 2 . Harman's Account ....................... 84 3 . White's Account ........................ 92 4 . Benacerrafvs Account ................... 99 5 . A Final Word ........................... 117 CHAPTER III/FORMALISM. ....................... 126 1 . Introduction ........................... 126 2 . First Formalist Claim .................. 127 3 . Second Formalist Claim ................. 162 SELECTED BIBLIOGRAPHY ......................... 214 7 INTRODUCTION My concern is whether there are numbers, especially natural numbers. I believe th~tpopular opinion, not only with the public (if introductory philosophy students are any guide), but with mathematicians and probably scientists as well, has it that there are numbers. Monk,' for example, has estimated that the mathematical world is populated with 65% platonists, 30% formalists, and 5% intuitionists.' There seems little doubt that we talk as though there are numbers. Why then am I concerned with whether there are numbers? Why not just conclude that there are? Because the popular view may run, into insuperable epistemological difficulties, difficulties 2 detailed by Benacerraf in "Mathematical Trutht'. Here is how those difficulties might be seen to arise. To the philosophically uninitiated it is natural to suppose not only that there are numbers, but also that we are quite familiar with at least some of them. Every- one can add and divide them (although not always too well); quantities having nuaber are all around us, The more mathematically sophisticated among us can prove 1. J. Donald Monk, Mathematical Logic (New York: Springer- Verlag, 1976) p. 3. 2, Journal of Philosophy 70 (1973): 661-679. esoteric truths about them. But when asked: what -is a number? the philosophical novice is likely to point to numerals. Ah, but numbers cannot be identical to numerals, it will be said, because although 5 + 7 = 12, for example, it is not true that '5 + 7' = '12'. So numerals must refer -to numbers. This routine initiation into philosophy leads to what I will call the popular view. According to it, the following sort of procedure is unproblematical. Suppose a bare uninterpreted number calculus is given, using the symbols lo', 'Sf, etc. where the usual concatenation rules are employed, so we can get the sequence lo1, 'SO1, lSSO1, ... The semantics can then be given in the fo1,lowing. , terms: '0' refers to 0, 'SO1 refers to 1, lSSO1 refers to 2, and in general '0' preceded by -n lSWsrefers to the number -n. If it is asked: what does '2' refer to, though, or ltwol, or 'seventeen1? the answer would be: why, the numbers two, two, and seventeen, respectively, of course. The basic idea behind this view is that the mathe- matical terms in question are already a part of English, which is an interpreted language. Hence these terms already refer. To what? To numbers. What's the problem? So far there is no problem, When queried as to the ontological status of numbers, an advocate of the view I am considering will say: numbers are abstract objects. Having traipsed this far into realist territory, when pressed for further details he or she may say: numbers are non-spatial and non-temporal; they are members of a platonic 9 realm of mathematical objects that stand in mathematical relations to each other. It is a realm of mathematical facts, as it were, whereby every mathematical statement is either truo or false. The mathematician's task is to go out there and, not invent, but discover the principles, or laws, we could even say, governing this realm.
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