Bertrand Russell on his paradox and the multiplicative axiom. An unpublished letter to Philip Jourdain

The bulk of the contribution that Bertrand Russell (1872-1970) made to mathematical logic and the foundations of mathematics was carried out between the autumn of 1900, when he was inspired by the work of G. Peano to make a new approach to the subject, and the autumn of 1910, when he and A. N. Whitehead completed their Principia Mathematics [19]. During much of this time he conducted a correspondence with Philip Jourdain (1879-1919)r on a wide variety of problems in which they were both interested. Recently I found a large number of Russell’s letters to Jourdain in the Institut Mittag-Leffler at Djursholm near Stockholm (see [4]). I hope to prepare an edition of these letters [5], because they give in a number of ways a fresh view of foundational researches at that time; and in this paper I give the text of one of them to show the type of discussion that took place between the two men. The correspondence that I found was in the form of two thick notebooks, in which Jourdain had either drafted (or had copied) his own letters and pasted in the replies that he received. These notebooks were used during the period 1901-1910, and the principal collections of letters are from Russell, G. Cantor (now published in [3]), and G. H. Hardy. The letter that I publish below was written in March, 1906. It was a crucial time in Russell’s work, for, after several years of sterile activity failing to solve the paradoxes, he was beginning at last to make substantial progress. The theory of denoting, which he published in 1905 [13], was beginning to make its impact on his logical programme; the commitment to the theory of types was becoming firm; and the project of Principia Mathematics, which was mainly written out between 1907 and 1910, was taking shape. But the letter is of outstanding interest not only for its timing, but also because of the range of topics which it treats. Today we have a very partial viw of the developments to which Russell and his contemporaries contributed; we have heard about the paradoxes