Though the effects of Russell’s paradox on the early foundations of mathematical logic are well known, only during the last decade has it been feasible to study its origins in detail. Much of Bertrand Russell’s writings leading to the publication of his [Principles of mathematics (Cambridge University Press, Cambridge) (1903)] were unpublished until [Philosophical papers, 1896-1899 (The collected papers of Bertrand Russell, Vol. 2, edited by {\it N. Griffin} and {\it A. C. Lewis}, Unwin Hyman, London) (1990)] and thought not to be relevant anyway in few of Russell’s own dismissal of this supposedly murky, neo-Hegelian period: the latter had to be left behind in order to make headway with “real” philosophical problems in mathematics. The author in his 1983 doctoral dissertation was one of the first to look carefully at these unpublished manuscripts. His book builds on his earlier analysis and uses recent findings that illuminate the role of Georg Cantor, C. Burali-Forti, and others in the story. He adheres closely to the theme of the set-theoretic paradoxes while not dwelling on related matters that have been adequately treated elsewhere such as the role of paradoxes and contradictions in Russell’s pre-1900 philosophy. Thus the book focuses on “the way in which Russell discovered the ‘contradiction’ of the greatest cardinal number, known today as the ‘Cantor paradox’; how Russell discovered his own inconsistency of the class of all classes which are not members of themselves; and how he presented the elements that would give rise to the ‘contradiction’ of the greatest ordinal number, today called the ‘Burali-Forti paradox’ (p. 81). The main historical tool is the reconstruction of the complex stages of preparation, over six years, that led to the final draft of ‘The principles of mathematics’. Though the texts of these manuscripts are not quoted at length here, the main stages are sufficiently summarized and represented by the various tables of content that helped keep Russell himself organized. One of the results of this investigation is the discovery that Russell’s paradoxes or contradictions “did not originate from the discovery of the nascent ‘paradoxes’ of Burali-Forti and Cantor, which, in fact, were not recognized as inconsistencies at the time” (p. 152). Furthermore, the author maintains that the works of George G. Berry, E. Zermelo, Jules König and Alfred Cardew Dixon show Russell’s direct involvement in the discovery of some of the non-mathematical, semantic, paradoxes. Though there are references to how the “mathematical community” reacted, no case is made that the set-theoretic paradoxes were of interest to a larger mathematical community at the turn of the century beyond the few individuals mentioned here, such as G. H. Hardy, E. H. Moore, G. Cantor, and E. W. Hobson. How this work became relevant to mathematicians is largely another story. Some fifty pages are devoted to transcriptions of letters, mainly to and from Russell, with correspondents that include his first wife, Alys, G. G. Berry, A. N. Whitehead, G. H. Hardy, G. E. Moore, David Hilbert, and Burali-Forti. This account is written for nonspecialists in mathematical logic but its extensive footnotes and 45-page bibliography guide the reader into the current literature. It also prepares the way for a study of the next stage: how Russell’s contradiction led him from the proposed second volume of ‘The principles of mathematics to work instead’ with A. N. Whitehead on [Principia mathematica (Cambridge University Press, Cambridge) (1910-1913)].
Reviewer: A.C.Lewis (Hamilton)