Defending mathematical apriorism

pp. 41-70

in: Janet Folina, Poincaré and the philosophy of mathematics, Basingstoke, Palgrave Macmillan, 1992

Abstract

The claim that mathematics is synthetic a priori is most commonly and most famously attacked by the logicists. Logicists usually argue against the syntheticity of mathematics in order to argue against Kant's thesis that mathematics possesses an extra-logical subject matter. Frege, Russell, and others endeavoured to show that Kant was wrong about the content of mathematics by showing how all true mathematical statements can be derived, once certain primitives (e.g., for Frege, for arithmetic, " number", "immediate predecessor", "0", and the "ancestral" relation) have been defined in logical terminology alone.¹ This is generally taken as an attempt to show that mathematics is analytic a priori. (This label is not accurate for Russell's logicism, as he believed even logic to be synthetic. However, the following general point still stands.) If successful, logicism would have shown mathematics to be independent of "intuitions' or extra-definitional content, for its foundation would essentially be that of logic. Logicism and the related foundational programme of set theory are addressed below, in the following chapters.