The Philosophy of Mathematics

Abstract

We have already commented on the formative role the study of mathematics, particulary arithmetic and algebra, played in the development of Berkeley’s theory of signs. Mathematics modeled, in his view, some of the crucial aspects of all language. Three elements of this paradigmatic function are, we believe, worth noting, (a) Mathematics reveals more clearly than ordinary language the contingent character of the relation between sign and designatum. This “contingency,” of course, is not unique to mathematics; all “conventions of reference” share this trait. In Berkeley’s hands, the metaphor of the “ostensive definition,” applied to the “causal” relation emphasizes the lack of necessity in this relation. The observed connections in nature are viewed as invariable correlations of types of events established by divine fiat, (b) Mathematics, again, particularly algebra, emphasizes the importance of order, or syntax over reference, to the point, in algebraic “games” where reference plays no role at all and we have a pure formalism. From the syntactical point of view, one can judge the “correctness” (not “truth”) of linguistic utterances in term of whether the rules for the combination of “signs” have been correctly followed. Even where reference is important, the importance of syntax emerges in Berkeley’s critique of the “Lockean” theory of communication, which appeared to require as a necessary condition for “understanding” language, that each non-logical expression be simultaneously associated with its referent.