BR TO LOUIS COUTURAT, 15 MAY 1906
BRACERS 53284. ALS(X). La Chaux-de-Fonds Bib. SLBR 1: #139
Edited by N. Griffin

Providence House Clovelly
15 May 1906

Dear Sir,

I was very glad to receive a letter from you again, but I’m sorry to learn that you are tired by your course. Don’t be afraid to speak in your course about things that I have passed on to you; to begin with, as you say, it will not lead to indiscretions, and then, even if it did, that would upset me very little.1 I have retired home alone for two months, in order to get through my work a little faster than usual. I still believe that my solution to the contradictions is good, but it seems to me that it must be extended to propositions, that is to say that the latter, like classes and relations, cannot replace ordinary entities.2 To say, for example, that the principle of excluded middle is not red, is to utter a nonsense, and not a truth. I will follow your advice in replying to M. Poincaré.3 For this reason, I will not reply quickly, because I want to get into order what I have to say about the solution of the contradictions. As regards Peano, I am completely of your opinion.4 I intend to compare my reform to the exclusion of infinitesimal from the differential calculus, which has not, however, ruined the work of Leibniz.5

I learn with great regret that you and Mme Couturat will not be coming to visit us this year. I hope that you will next year.

With my best wishes to you and Mme. Couturat

Bertrand Russell

• 1

  Don’t be afraid to speak … upset me very little. Couturat was conducting a course on mathematical logic and was concerned that Russell might think he was plagiarizing ideas Russell had passed on to him in correspondence. Russell was admirably untouched about such matters.

• 2

  my solution to the contradictions … cannot replace ordinary entities The extension of the substitutional theory to propositions was no easy matter, and, in fact, when Russell made the extension the contradiction reappeared in a new form. The discovery of the contradiction in the substitutional theory caused Russell to withdraw from publication, after it had been set in type, the paper he had read to the London Mathematical Society on 10 May. It caused him also to return to the theory of types, first suggested in The Principles of Mathematics but very much elaborated for Principia Mathematica, as a solution to the paradoxes. None the less the line of thought about propositions expressed in this letter to Couturat still appears in Principia.

• 3

  I will follow your advice in replying to M. Poincaré. Poincaré had published a general attack on mathematical logic, “Les Mathématiques et la logique” (Revue de métaphysique et de morale, May 1906), paying special attention to Russell’s article “Some Difficulties in the Theory of Transfinite Numbers and Order Types” in the Proceedings of the London Mathematical Society (March 1906). Russell wrote a long rejoinder, “Les Paradoxes de la logique” (Revue de métaphysique et de morale, September 1906). Russell’s original English version, “On ‘Insolubilia’ and Their Solution by Symbolic Logic”, was published in Essays in Analysis (1974). The dispute, in which many other logicians including Couturat played a part, had a significant influence in the development of the philosophy of mathematics, in that it helped to lay out the early demarcation lines between the three main philosophies of mathematics: logicism, formalism, and intuitionism.

• 4

  As regards Peano, I am completely of your opinion. Poincaré acknowledged some of Peano’s achievements but denied that Peano’s mathematical logic was one of them or had played any part in the others. Russell and (by now) Couturat thought that Peano’s logic was his crowning achievement. Couturat had asked Russell to include a defence of Peano in his reply to Poincaré.

• 5

  I intend to compare … work of Leibniz. See Essays in Analysis, pp. 192–3.