ABSTRACT
In his Principles of Mathematics (1903) Russell arrived at a view of arithmetic which was strikingly similar to that of Frege but he came there by a different route. As he relates, he was initially concerned with the foundations of dynamics, and was led to thinking about geometry and thence to numbers and arithmetic via the work of Cantor (1955) and Peano (1889). Cantor developed a theory of infinite numbers as part of the general project of nineteenth century mathematicians to construct a numerical model of the geometrical continuum (see Tiles 1989). Peano is known principally for his axiomatization of arithmetic, but the logical symbolism now familiar through Russell's work was adapted from Peano. Before writing the Principles of Mathematics Russell had published a shorter book on the foundations of geometry (Russell 1897) in which he subjected the Kantian view of geometry and the whole idea that there could be synthetic a priori truths to searching scrutiny. His Principles of Mathematics is a non-formal and wide ranging discussion of foundational issues in which he argues, in a programmatic way, for a logicist position. Most of the book was written before he had read Frege's work and a brief discussion of Frege is added as an appendix. The formal completion of the logicist programme, whose outlines had for the most part been
sketched in the Principles, was undertaken in collaboration with Whitehead and became the three-volume Principia Mathematica (Whitehead and Russell 1910--13). The philosophical upshot of this mammoth undertaking is presented in a non-formal and very accessible form in Russell's Introduction to Mathematical Philosophy (Russell 1919) which, because of its accessibility, has probably been the vehicle through which Russell's logicist view of mathematics has been most widely known. However, its order of presentation does not give many clues as to the route by which Russell arrived at his views, since it starts with a consideration of the natural numbers. This leads to a greater assimilation of his views to those of Frege than is strictly warranted.
