Russell’s Conception of Mathematics

Abstract Logicism is the thesis that mathematics can be derived from logic. In both The Principles of Mathematics and Principia Mathematica, Russell sets out to demonstrate that mathematics can be deduced from a small number of “indefinable” concepts and “primitive propositions.” But until now, little attention has been paid to the conception of mathematics induced by this logicist approach: What mathematics were the logicians seeking to reduce? How did they treat the differences between mathematical disciplines (as, for example, the differences between arithmetic, algebra, geometry, etc.)? Did they consider that the demarcations between fields within mathematics had to be derived logically, or did they consider, on the contrary, that this was unnecessary? This chapter addresses this question and shows, by examining the organization of both The Principles and Principia Mathematica, that the ambition of Russellian logicism was not only to deduce the content of mathematics but also to account for its architecture.