Must the Propositions of Arithmetic be Empirical?

Abstract

The empiricist theory of mathematics has gained a new champion: J. L. Mackie, in his paper 'Proof' (PAS Supp. Vol. XL, 1966). In the controversy between the empiricist theory and the analyticity theory, at least three main camps may be distinguished. There are those who, like Ayer, would argue that all the propositions of pure mathematics, including those of pure geometry, are analytic and a priori: the only territory to be conceded to the empiricist theory is that of applied geometry which, it is agreed, contains generally synthetic and empirical propositions. Then there are those who, like Frege, would contend that the field of mathematical propositions should be divided differently: only arithmetical propositions are analytic and a priori; all those of geometry are synthetic and empirical. Finally, there are the radical empiricists who, like Mill, believe that all the propositions of mathematics, of arithmetic as well as geometry are-with only a few trivial exceptions-synthetic and certifiable as true or false only by experience. It is within this third camp that Mackie clearly belongs. He cites, with evident approval, the broad strategy of argument by which Nagel (The Structure of Science, Ch. 8, R.K.P., 1961) tried to establish the empirical status of geometry, and sets out to give 'an analogous account of the epistemological status of arithmetic.' (25). Paralleling Nagel, he argues: (i) that a purely formal system for arithmetic does not contain any propositions at all but mere