Principia Mathematica at 100

Abstract

May 14, 2011 (10:00 am) C:\Users\Milt\WP data\TYPE3101\russell 31,1 078 red 002.wpd russell: the Journal of Bertrand Russell Studies n.s. 31 (summer 2011): 5–8 The Bertrand Russell Research Centre, McMaster U. issn 0036-01631; online 1913-8032 reface PRINCIPIA MATHEMATICAz AT 100 Nicholas Grif{f{in Russell Research Centre / McMaster U. Hamilton, on, Canada l8s 4l6 ngrif{f{in@mcmaster.ca Bernard Linsky Philosophy / U. of Alberta Edmonton, ab, Canada t6g 2e5 bernard.linsky@ualberta.ca A zll the papers in this special issue of Russellz are connected with the zPMy@z100 conference held at the Bertrand Russell Research zCentre at McMaster University to celebrate the centenary of the publication of the Wrst volume of Whitehead and Russell’s Principia Mathematica. The conference took place 21–24 May 2010, and over 60 people from ten diTerent countries attended to hear more than 30 papers coveringtheorigins,impact,philosophicalmotivationandmathematical content of that vast and still relatively unexplored work. All but two of the nine papers in this collection were presented at the PMy@z100 conference . Of the two exceptions, Anellis’s paper was included in the conference programme, but in the event Anellis was unable to attend and so the paper was never presented. Blackwell’s paper, by contrast, was presented as an after-dinner talk, with logicians in attendance, to the 2010 annual meeting of the Bertrand Russell Society which was held at McMaster over the same weekend at the PMy@z100 conference. The nine papers published here exemplify the range of the conference itself. Graham Stevens’ paper considers the view, much discussed by contemporary philosophers of language (cf.z notably Neale 1990, 1993, 2008), that all noun phrases are either semantically structured quantiWers or semantically unstructured singular terms. This view seems like a natural extension of the sharp distinction Russell made in his theory of descriptionsbetweendeWnitedescriptions ,whicharehandledquantiWcationally, May 14, 2011 (10:00 am) C:\Users\Milt\WP data\TYPE3101\russell 31,1 078 red 002.wpd 6 nicholas grif{f{in and bernard linsky and logically proper names, which are genuine singular terms. And yet there is a diUculty, which Stevens addresses in his paper. One reason Neale reformulates Russell’s theory of descriptions in terms of binary or restricted quantiWers is to narrow the gap Russell famously opened between the logical form of a sentence, as revealed after its analysis by the theory of descriptions, and the grammatical form that the unanalyzed sentence has in natural language. The outcome of this seems to be that Russell’s claim that deWnite descriptions are incomplete symbols is lost in the reformulation. Stevens argues that this is not the case, but only if “incomplete symbol” is properly understood, not as a symbol having no meaning, but as one which does not contribute an object to a proposition . Ray Perkins deals with another aspect of the very same issue, but with a quite diTerent approach. Perkins is concerned with a famous argument in Principiaz designed to show that deWnite descriptions are incomplete symbols. The argument has been frequently criticized, starting immediately with the book’s publication (Jones 1910–11), precisely because of its dependence on an alleged ambiguity in the notion of meaning it deploys. Perkins defends the argument against this charge and, addressing the concerns of the historical Russell rather than his modern followers , goes on to explore the implications of the argument for Russell’s epistemology, metaphysics and his conception of analysis. The theory of descriptions is one of Russell’s two best-known and most important contributions to Principiaz; the theory of types is the other. Yet the theory of types was never considered, even by its creator, as the triumphant success that the theory of descriptions was. It brought with it too many problems. One of them was the axiom of reducibility, discussed in Russell Wahl’s paper. The axiom was essential if the logicist project was to go forward; without it, for example, least upper bounds in real number theory would fall outside the sets they bounded. And yet the axiom’s status as an axiom of logic is very much in doubt. It requires the existence of very...