The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

I. ONTOLOGY, MODELS, AND INDETERMINACY

Indeterminacy and logical consequence are two key issues of Part I. Both are discussed in the context of model theory. Hartry Field is concerned with the indeterminacy which we encounter in our use of certain fundamental mathematical notions such as the notion of set, and scrutinizes in this connection Hilary Putnam's model-theoretic argument in "'Models and Reality'".2 Stewart Shapiro and Charles Chihara argue in defence of a model-theoretic account of the notion of logical consequence, and reject some objections raised by John Etchemendy in his book The Concept of Logical Consequence.3 There is also a link tying Paul Benacerraf essay 'What Mathematical Truth Could Not Be--I' to Field's paper 'Do We Have a Determinate Conception of Finiteness and Natural Number?' In the final section of his essay, Benacerraf comments on Putnam's and Crispin Wright's reflections on the Löwenheim-Skolem Theorem. In what follows, I shall first comment on Benacerraf's twofold challenge to Platonism and shall then briefly summarize Etchemendy's critique of Tarski's model- theoretic account of the concept of logical consequence.

Mathematical Platonism is the doctrine that numbers, functions, sets, and the like are abstract (that is, non-spatial, non-temporal, causally inert) entities existing independently of the human mind. The advocate of this doctrine, the Platonist, characteristically holds that mathematical statements are either true or false, regardless of whether or not we are capable of determining their truth-value.

In his influential paper 'What Numbers Could Not Be',4 Benacerraf questioned the legitimacy of mathematical Platonism on ontological grounds. He argued that numbers cannot be sets, because there is no good reason to hold that any particular number is some particular set. Plainly, there is no unique set-theoretic specification of, for example, the natural or the real numbers. The best we can achieve is to determine the numbers of either kind up to isomorphism. Not only does the identification of numbers with sets appear to be undermined by the arbitrariness of our actual choices, but also by the arbitrariness of our taking sets to be the basic objects of mathematics. It is possible, for instance, to regard ordinal numbers as basic and to define sets in terms of them. By way of extending his argument from sets to objects in general, Benacerraf concludes that numbers cannot be objects at all. Only by appeal to non-structural properties of numbers could we pick out a number as a determinate object. Yet in stating the properties of numbers which are necessary and jointly sufficient for the study of arithmetic, we merely characterize an abstract structure. According to this view,

____________________
2
Journal of Symbolic Logic, 45 ( 1980), 464-82.
3
Harvard University Press, Cambridge, Mass., 1990.
4
Philosophical Review, 74 ( 1965), 47-73.

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