Skolem and the Skeptic

structure that all progressions have in common in virtue of being progressions. It is not a science concerned with particular objects-the numbers. The search for which independently identifiable particular objects the numbers really are (sets? Julius Caesars?) is a misguided one.1 The argument which precedes these conclusions is well known. Suppose that someone has received satisfactory definitions of'l' (or '0'), 'number', 'successor', '+', and 'X' on the basis of which the laws of arithmetic can be derived. Let him also have confronted, if you like, an explicit statement of Peano's axioms. Suppose further that he has been given a full explanation of the 'extra-mathematical' uses of numbers2-principally, counting -and has thereby been introduced to the concepts of cardinality and of cardinal number. Then it is possible to claim that, conceptually at least, the subject's arithmetical education is complete: he may be ignorant of all sorts of aspects of advanced (and less advanced) number theory, but his deficiencies, if any, are not in his understanding--at least not if he has followed his training properly. Yet the striking fact is-the argument runs-that someone who in this way perfectly understands the concept of (finite cardinal) number has no basis for (non- arbitrary) identification of the numbers which any objects given in some other way. Benacerraf makes the point vivid by comparing two hypothetical logicist-educated children, each of whom takes zero to be A but one of whom identifies successor ' P. Benacerraf, 'What Numbers Could Not Be', Philosophical Review 74 (1965). The above quotation is taken from the reprint in P. Benacerraf and H. Putnam, eds., Philosophy of Mathematics, 2nd edition, Cambridge University Press 1983, p. 291. 2Cf. Benacerraf, loc. cit., p. 277. This content downloaded from 157.55.39.106 on Mon, 25 Apr 2016 06:33:12 UTC All use subject to http://about.jstor.org/terms 122 II-CRISPIN WRIGHT (following Zermelo) with the unit set operation while the other (following Von Neumann-Berneys-Gbdel) identifies the suc- cessor of a number with the set consisting of that number and all its elements. Each of the set-theoretic frameworks is perfectly adequate for the explanation of the arithmetical primitives, and the derivation of the Peano axioms, and supplies the background against which the applications of arithmetic can be satisfactorily explained. Yet a dispute between the two children as to the true identity of the numbers is intractable. The moral is that the concept of number has no content sufficing to resolve such disputes, has indeed no content sufficing genuinely to individuate the numbers at all. When the explanations, formal and informal, are in, a good deal will have been said to characterise the structure which the numbers collectively exemplify, and which, in Benacerrafs view, is the real object of pure number-theoretic investigation; but nothing will have been said to enable a subject to know which, if any, sets the numbers are-or which, if any, objects of any sort they are. However if the numbers really were objects of some kind, surely someone who perfectly understood the concept of number should be able, at least in principle, to identify them. Since we do not have the slightest idea how such an identification might be defended, we ought to contrapose. Whence Benacerraf's anti- platonist conclusion. I have tried to give reason elsewhere3 for thinking that the force of this argument is qualified both by certain internal weaknesses-the concept of finite cardinal number is deter- minate in ways the argument overlooks-and by the company it keeps-for instance Frege's 'permutation' argument and the various Quinean arguments for inscrutability of reference. But what is the point of reminding you of this argument here? Simply that it may be contended to furnish somewhat strange company for the direction Benacerraf would have us take in response to his 'Skeptic'. He writes The meaning of 'e' and the range of the quantifiers must constrain the class of permissible interpretations if the 3 C. Wright, Frege's Conception of Numbers as Objects, Aberdeen University Press 1983, especially Section xv, pp. 117-29. This content downloaded from 157.55.39.106 on Mon, 25 Apr 2016 06:33:12 UTC All use subject to http://about.jstor.org/terms SKOLEM AND THE SKEPTIC 123 formalised version is to retain the connection with intuitive mathematics with which set theory began-if it is to be a formalization of set theory. (this volume, p. 106)