Reply to Demopoulos

Abstract

IN HIS REVIEW I of the first of the Synthese special issues on Russell's early philosophy, William Demopoulos raises three objections to my contribution2 to that volume. He complains, first, that I misunderstand Russell's account of mathematical propositions in The Principles of Mathematics (1903); second, that I am mistaken in supposing that between the Principles and Principia Mathematica (1910) Russell switched from metaphysical to an epistemological account of the apodicticity of mathematical propositions, coming to emphasize their certainty rather than their necessity; and, third, that, contrary to what I said in my paper, Russell's type theory is not context-sensitive. Only the second of Demopoulos's complaints is just, however, and that only partly. It is indeed true that the evidence Demopoulos quotes (Principia, 2nd ed., I: 12-13, 59) refutes my injudicious remark that Russell sought to demonstrate the certainty ofmathematics by deduction from axioms which are certain (Gp. u8), as does Russell's transcendental procedure for arriving at an appropriate axiom set for mathematics. However, it does not refute, as Demopoulos seems to think, my wider claim that Russell was more concerned in Principia and later writings with the certainty of mathematics rather than its generality, claim which is, I think, well documented in my paper. For one thing, with the introduction of type theory, the concept ofgenerality was not available to Russell in the form in which he had used it in the Principles. For another, it is not unreasonable to suppose that Russell was attempting to establish the certainty of mathematics through transcendental deduction of its axioms. Demopoulos's other charges, however, seem to me completely without foundation. He charges me with a rather unusual account of Russell's 'if-thenism' (Dp. 164), namely, that mathematical propositions are conditionals with antecedents designed to protect the consequent against falsifying assignments of values to its variables. But to describe this as an account ofRussell's if-thenism is already to beg the question, for it was precisely my point that Russell was not an if-thenist, at least in