Review of Principia Mathematica

Review of Principia Mathematica Giuseppe Peano and Giovanni B. Ratti (bio) Introduced, translated and edited by GIOVANNI B. RATTI As all historians of logic know, Russell met Peano for the first time at the 1900 International Congress of Philosophy in Paris. He was so impressed by the way Peano and his school clearly articulated their ideas and held the upper hand in every debate that he famously recalled this revelation in two different places, namely in My Philosophical Development and the Autobiography.1 Along these lines, Hans Freudenthal once wrote End Page 87 of the Paris congress that "In the field of the philosophy of sciences the Italian phalanx was supreme: Peano, Burali-Forti, Padoa, Pieri absolutely dominated the discussion. For Russell, who read a paper that was philosophical in the worst sense, Paris was the road to Damascus."2 From that moment on, Russell began to develop his own conception of analytic philosophy. After the Paris Congress, Russell claims, he mastered Peano's logical-mathematical notations in a few months and, equipped with such powerful technical tools, began his remarkable journey into the foundations of mathematics at an incredible pace.3 The "epiphany" of Russell's first meeting with Peano is a milestone in the history of mathematical logic and is often recalled in the history of philosophy. However, subsequent relations between the two are usually neglected. In the general literature, it is sometimes implied that Russell adopted the new symbolism he so urgently needed for his logical project from Peano and then went his own way without any further contact with the Italian mathematician. This impression has probably been created by the fact that Peano's "decline" coincided with Russell's rise in the field of mathematical logic. The Paris Congress can be seen as a "passing of the baton", so to speak.4 This metaphorical reading is nevertheless somewhat misleading as far as historical facts are concerned. Peano and Russell met several more times and exchanged letters over a number of years.5 It is not always easy to reconstruct End Page 88 this correspondence, still less the entire philosophical relationship between the two men, as some important pieces of evidence are missing. Evidence that the relationship between the two was particularly intense in the period between the Paris Congress (1900) and the publication of Russell's Principles of Mathematics (1903) can be traced relatively easily. Russell was keen to contribute (including financially) to Peano's publishing efforts and sent two essays, in which he mainly extended Peano's system to the logic of relations, to Peano's Rivista di matematica (Revue de mathématiques).6 Letters and notes were therefore frequently exchanged between Turin and England, although the evidence here is one-sided as Russell's letters to Peano are no longer available and are probably lost for good. As mentioned, this period ends with the appearance of Principles of Mathematics,7 which Peano hails as marking a new epoch in the philosophy of mathematics in a letter to Russell dated 27 March 1903 (ra1 710.054224). Peano reiterated this opinion in 1904 at the Academy of Sciences in Turin, saying that Russell's book was the best work on the subject. Another important exchange took place in 1906, when two letters from Peano dealt with the axiom of choice (which he himself discovered—and rejected—in 1890, fourteen years before Zermelo) and logical antinomies, respectively. Again, Russell's replies are not extant. In the same year, Peano, in what was probably his last original contribution to logic,8 published a paper dealing with logical antinomies in which he provided "perhaps the most penetrating commentaries on Richard's paradox",9 systematized Russell's paradox as a generalization of the Burali-Forti's paradox, and made a groundbreaking distinction between End Page 89 linguistic and mathematical antinomies.10 Early in April 1908 Peano and Russell were both in Rome for the Fourth International Congress of Mathematicians, although there is no record of a meeting between them. However, Ernst Zermelo mentions in a letter to Georg Cantor written in late July that he had had friendly conversations with both men.11 The last known correspondence and...