The Cambridge companion to

Cambridge University Press, Cambridge, UK and New York, US, xvii + 550 pp.— therein: Ivor Grattan-Guinness. Mathematics in and behind Russell’s logicism, and its reception. Pp. 51–83. Richard Cartwright. Russell and Moore: 1895–1905. Pp. 108–127. Peter Hylton. The theory of descriptions. Pp. 202–240. Martin Godwyn and Andrew D. Irvine. Bertrand Russell’s logicism. Pp. 171–201. Michael Beaney. Russell and Frege. Pp. 128–170. Gregory Landini. Russell’s substitutional theory. Pp. 241–285. Alasdair Urquhart. The theory of types. Pp. 286–309. The above papers have been selected from The Cambridge companion to Bertrand Russell as those likely to be of greatest interest to readers of this journal. Together they cover the logically most productive period of Russell’s life, which may be taken to run from 1900 to 1910. The major events of this period include: the discovery of the paradox (May 1901), the publication of Principles of mathematics (May 1903), the discovery of the theory of descriptions (June 1905), the discovery of the ramified theory of types (1907), and the completion (with Whitehead) of Principia mathematica (1910). There is a wealth of material from this period that remained unpublished at the time but is becoming available in The collected papers of Bertrand Russell, the relevant volumes being Toward the “Principles of mathematics”, 1900–02, edited by Gregory H. Moore, London: Routledge, 1993, Foundations of logic, 1903–05, edited by Alasdair Urquhart with the assistance of Albert C. Lewis, London: Routledge, 1994, and Toward “Principia mathematica”, 1906–08, which has yet to be published. The papers under review provide a useful guide to this important chapter of Russell’s career, in many cases making use of formerly unpublished material to shed light on Russell’s logical development. The papers may be grouped as follows: The first sets the historical stage and provides an overview of Russell’s work in logic while the remaining pairs deal respectively with the theory of denoting, the nature of logicism, and the resolution of the antinomies. Historical. Grattan-Guinness locates the work of Whitehead and Russell at the end of a long tradition of rigorization and reduction in mathematics. This tradition began with a rigorous account of the notions of limit, continuity, and derivative (by Bolzano, Cauchy, Weierstrass and others) and ultimately led to the reduction (via informal set theory) of the reals to the rationals and the rationals to the natural numbers (by Dedekind, Cantor and others). Dedekind and Peano then gave a rigorous account of the natural numbers and Cantor both developed set theory into an independent discipline and extended it into the transfinite. Whitehead and Russell brought all of this material into a single general framework, isolating several axioms of set theory (for example, infinity and choice) and pressed the reduction further with the aim of showing that ultimately even the arithmetical notions could be reduced to set theoretic notions and that these in turn could be reduced to logical notions. The theory of denoting. Russell’s conception of logic is richer than the modern conception both in terms of the principles it embraces and the kinds of entities these principles involve. During the time of the Principles logic involved the extensional notion of class and the intensional notions of proposition, propositional function and denoting concept. These four notions play a central role in what follows and given that Russell’s conception of the three intensional notions is not widely known it will be useful to start with a brief account, following the excellent exposition of Cartwright.