“If Numbers Are to Be Anything At All, They Must Be Intrinsically Something”: Bertrand Russell and Mathematical Structuralism

Bertrand Russell was one of the first philosophers to recognize clearly the philosophically innovative nature of Richard Dedekind’s philosophy of arithmetic: a position we now describe as non-eliminative structuralism. But Russell’s response was deeply negative: “If [numbers] are to be anything at all, they must be intrinsically something” (Principles of Mathematics, §242). Nevertheless, Russell also played a significant positive role in making possible the emergence of structuralist philosophy of mathematics. This chapter explains Russell’s double role, identifying three positive contributions to structuralism, while laying out Russell’s objections to Dedekind’s non-eliminative structuralism.