D.H. MELLOR

#### 1 Mathematics

The British philosopher Frank Plumpton Ramsey graduated in mathematics from Trinity College Cambridge in 1923, became a Fellow of King’s College in 1924 and a University Lecturer in Mathematics in 1926. In his short life he did work in mathematics and economics which created new branches of those subjects. But his vocation and greatest achievements were in philosophy, influenced by and influencing his Cambridge colleagues, especially Russell, Wittgenstein and Keynes. His first major work, on ‘The Foundations of Mathematics’ (1925) (this can be found, as can all his other works cited in this entry, in Ramsey (1990a)), makes two key improvements to the attempted reduction of mathematics to logic in Whitehead and Russell’s Principia Mathematica (see Logicism). First, Ramsey tightens Principia’s definition of mathematical propositions as purely general by requiring them also to be tautologies in the sense of Wittgenstein’s Tractatus. Second, he shows that it need deal only with purely logical paradoxes, like that of the class of all classes which are not members of themselves, not semantic ones like ‘this is a lie’, which depend on the meanings of words like ‘lie’. This lets Ramsey simplify Principia’s complex hierarchy of propositions, since it no longer needs to make ‘this is a lie’ ill-formed by making ‘p’ and ‘p is a lie’ propositions of different types. This in turn lets him drop the ‘axiom of reducibility’ (of propositions of higher types to equivalent ones of lower types) that Principia needs to validate ‘many important mathematical arguments’ (1990a: 190). Dropping this axiom is essential to logicism, since ‘there is no reason to suppose [it] true; and if it were true, this would be a happy accident and not a logical necessity, for it is not a tautology’ (1990a: 191) (see Theory of types; Semantic paradoxes and theories of truth).

In ‘Mathematical Logic’ (1926) Ramsey defends his logicism against ‘the Bolshevik menace of Brouwer and Weyl (1990a: 219) (see Intuitionism) and the formalism of Hilbert (see Hilbert’s programme and formalism). The former he attacks for denying that all propositions are either true or false – ‘Brouwer would refuse to agree that either it was raining or it was not raining, unless he had looked to see’ (1990a: 228) – and the latter for reducing mathematics to ‘a meaningless game with marks on paper’ (1990a: 233). But he remains embarrassed by logicism’s need to assume that there are infinitely many things; and R.B. Braithwaite, introducing his edition of Ramsey’s papers in 1931, reports that in 1929 Ramsey ‘was converted to a finitist view which rejects the existence of any actual infinite aggregate’. Despite this, and logicism’s general rejection by mathematicians, Ramsey’s version of it remains of great interest to logicians (Chihara 1980).