A Centenary Companion to Principia Mathematica

Abstract

russell: the Journal of Bertrand Russell Studies n.s. 35 (summer 2015): 71–94 The Bertrand Russell Research Centre, McMaster U. issn 0036–01631; online 1913–8032 c:\users\arlene\documents\rj issues\type3501\rj 3501 061 red.docx 2015-07-10 4:07 PM oeviews A CENTENARY COMPANION TO PRINCIPIA MATHEMATICA Graham Stevens Philosophy / U. of Manchester Manchester m13 9pl, uk graham.p.stevens@manchester.ac.uk Nicholas Griffin and Bernard Linsky, eds. The Palgrave Centenary Companion to Principia Mathematica. Basingstoke and New York: Palgrave Macmillan, 2013. Pp. xxvii, 458. isbn: 978-1-137-34462-5. £65; us$120. famous anecdote of Russell’s neatly summarizes the impact of Principia Mathematica in Russell’s lifetime: I used to know of only six people who had read the later parts of the book. Three of these were Poles, subsequently (I believe) liquidated by Hitler. The other three were Texans, subsequently successfully assimilated. (MPD, p. 86) That the later parts (dealing with purely mathematical concerns) were somewhat neglected, was a disappointment to Russell and Whitehead. However, the same could not be said for the early parts of the work (dealing with philosophical and mathematical logic).The influence of these parts has been profound . Logicism is a simple enough thesis: mathematics (to a greater or lesser degree , depending on which version of logicism—Frege’s or Russell’s—we are talking about) is part of logic. Every mathematical truth is really just a logical truth.This deceptively simple philosophical claim, however, can only be taken seriously if confirmed by supporting evidence. Logicism is a philosophical thesis that requires logical demonstration. Both Frege and Russell began by stating the philosophical case for logicism, and then attempting to demonstrate the thesis formally in subsequent work. Frege’s attempt famously floundered after Russell discovered its inconsistency. After the best part of a decade spent trying to remove that inconsistency, Russell and Whitehead produced Principia Mathematica, which was their attempt to demonstrate logicism on, in the words of the editors of this volume, “a truly epic scale” (p. xvi). ^= 72 Reviews c:\users\arlene\documents\rj issues\type3501\rj 3501 061 red.docx 2015-07-10 4:07 PM Much appeared to have changed during those years in which PM was constructed . Russell’s original philosophical statement of logicism in The Principles of Mathematics was a paradigm of elegance, at least with regard to its reduction of number theory to the calculus of classes. But in PM, things look very different.The logic of PM is stratified into a theory of types, something only tentatively considered in an appendix to the Principles; the classes to which numbers were reduced in the Principles are absent from the “no-classes” theory of PM; the much celebrated theory of quantification first presented in “On Denoting” is incorporated into PM. Additionally, Russell andWhitehead claim at some points to have rejected any commitment to propositions as entities , replacing them with a new theory of judgment to explain “propositional ” content. After protracted attempts to digest these many changes and innovations to Russell’s logic, most commentators remained unconvinced that the demonstration of logicism was successful. It is a powerful testament to the lasting importance and value of PM that, in the century following its publication, the fact that the orthodox view of the work was that it had failed in its intended purpose posed no obstacle to its dramatic impact on the development of both mathematical logic and analytical philosophy. PM’s enormous influence in mathematical logic is largely due to subsidiary achievements made in the service of the logicist enterprise, rather than a reflection of that enterprise. PM gave the first accessible axiomatization of propositional and predicate logic (Frege had previously given an arguably more rigorous axiomatization, but his notation made the work far less accessible) which became the point of reference for subsequent work in the development of metamathematics, culminating in Gödel’s famous incompleteness theorems which proved the incompleteness of any consistent formalization of arithmetic based on PM’s formal system. Furthermore, for generations of logicians, PM was the only book available in which to study mathematical logic. From our current perspective...