Titles include: Stewart Candlish THE RUSSELL/BRADLEY DISPUTE AND ITS SIGNIFICANCE FOR TWENTIETH- CENTURY PHILOSOPHY

Transcription

1 History of Analytic Philosophy Series Editor: Michael Beaney Titles include: Stewart Candlish THE RUSSELL/BRADLEY DISPUTE AND ITS SIGNIFICANCE FOR TWENTIETH- CENTURY PHILOSOPHY Annalisa Coliva MOORE AND WITTGENSTEIN Scepticism, Certainty and Common Sense George Duke DUMMETT ON ABSTRACT OBJECTS Sébastien Gandon RUSSELL S UNKNOWN LOGICISM A Study in the History and Philosophy of Mathematics Gregory Landini FREGE S NOTATIONS What they Are and What they Mean Sandra Lapointe BOLZANO S THEORETICAL PHILOSOPHY An Introduction Omar W. Nasim BERTRAND RUSSELL AND THE EDWARDIAN PHILOSOPHERS Constructing the World Douglas Patterson ALFRED TARSKI Philosophy of Language and Logic Graham Stevens THE THEORY OF DESCRIPTIONS Nuno Venturinha (editor) WITTGENSTEIN AFTER HIS NACHLASS Pierre Wagner (editor) CARNAP S LOGICAL SYNTAX OF LANGUAGE Pierre Wagner (editor) CARNAP S IDEAL OF EXPLICATION AND NATURALISM Forthcoming: Andrew Arana and Carlos Alvarez (editors) ANALYTIC PHILOSOPHY AND THE FOUNDATIONS OF MATHEMATICS Rosalind Carey RUSSELL ON MEANING The Emergence of Scientific Philosophy from the 1920s to the 1940s

2 Giusseppina D Oro REASONS AND CAUSES Causalism and Non- Causalism in the Philosophy of Action Anssi Korhonen LOGIC AS UNIVERSAL SCIENCE Russell s Early Logicism and its Philosophical Context Franz Prihonsky THE NEW ANTI-KANT (trans. Sandra Lapointe) Consuelo Preti THE METAPHYSICAL BASIS OF ETHICS The Early Philosophical Development of G.E. Moore Erich Reck (editor) THE HISTORIC TURN IN ANALYTIC PHILOSOPHY Maria van der Schaar G.F. STOUT The Psychological origin of Analytic Philosophy HISTORY OF ANALYTIC PHILOSOPHY Series Standing Order ISBN (hardback) Series Standing Order ISBN (paperback) (outside North America only) You can receive future titles in this series as they are published by placing a standing order. Please contact your bookseller or, in case of difficulty, write to us at the address below with your name and address, the title of the series and one of the ISBNs quoted above. Customer Services Department, Macmillan Distribution Ltd, Houndmills, Basingstoke, Hampshire RG21 6XS, England, UK

3 Russell s Unknown Logicism A Study in the History and Philosophy of Mathematics Sébastien Gandon Professor of Philosophy, Clermont Université, Université Blaise Pascal, Clermont-Ferrand, France

4 Sébastien Gandon 2012 Foreword Michael Beaney 2012 Softcover reprint of the hardcover 1st edition All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6 10 Kirby Street, London EC1N 8TS. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The author has asserted his right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act First published 2012 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number , of Houndmills, Basingstoke, Hampshire RG21 6XS. Palgrave Macmillan in the US is a division of St Martin s Press LLC, 175 Fifth Avenue, New York, NY Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world. Palgrave and Macmillan are registered trademarks in the United States, the United Kingdom, Europe and other countries. ISBN ISBN (ebook) DOI / This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin. A catalogue record for this book is available from the British Library. A catalog record for this book is available from the Library of Congress

5 Contents List of Figures and Tables Acknowledgements Series Editor s Foreword vii ix xi Introduction 1 1 Projective Geometry Projective and metrical properties The status of the fundamental theorem of projective geometry Russell s twofold analysis Russell s choice Space as an incidence structure Conclusion 47 2 Metrical Geometry Logicism, if-thenism and non-euclidean geometries The projective definition of a metric Metrical geometry, distance and stretch Poincaré and Russell on the perception of distance Metrical geometry as an empirical science Conclusion 76 3 Geometry, Logicism and If-Thenism Russell s if-thenism (I): definitions Russell s if-thenism (II): existence-theorems Topic specificities and psychologism The types of relations and the architecture of mathematics The interaction between logic and mathematics: a persistent difficulty? Conclusion Quantity in The Principles of Mathematics Magnitude and quantity The concept of a measurable magnitude The absolutist theory and the concept of a kind of magnitude The relational theory of distance Conclusion Quantity in Principia Mathematica Dedekind, Burali-Forti and Frege on real numbers Russell and Whitehead s theory of numbers 141 v

6 vi Contents 5.3 The relational theory of vector family Hale s Fregean axiomatic definition versus Russell and Whitehead s relational theory Conclusion Application Constraint in Principia Mathematica Whitehead s conception of applied mathematics Real analysis and real numbers The architectonic use of Applic Applic in neo-logicism Applic and the distinction between logical and apparent forms of propositions Conclusion Russell s Universalism and Topic-Specificity Pre-logicized and logicized mathematics Russell s positive universalism Logicism and the definition of the mathematical content Russell s logicism and practice-based philosophy of mathematics Conclusion 210 Notes 213 Bibliography 245 Index 259

7 List of Figures and Tables Figures 1.1 Two planar figures in perspective ABCD and A B C D are in perspective from centre O Desargues s theorem Pappus s theorem The image of E on Δ by the perspectivity (AB, A B ) is the point at infinity E A representation of the real projective plane as the set of vectorial lines in R The segment (ACB) Von Staudt s quadrilateral construction The introduction of (integral) coordinates on a projective line The descriptive plane represented as a convex part of the projective plane Aristotelian classification and cross-classification Fundamental points: the various cases The Absolute as a real conic: the hyperbolic case The Absolute as a degenerate conic: the Euclidean case The transformation of a conic The structure of PM VI Frege s doctrine 154 Tables 2.1 A summary of Russell s distinctions Correspondence between the parts of PoM and the relational types Summary of the different positions of Hale, Shapiro and Wright 177 vii

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9 Acknowledgements Some of the ideas expressed in this book I have already expounded in print in different papers. In particular, in writing Chapters 1, 2 and 3, I have freely drawn upon four of my articles: Russell et l Universal Algebra de Whitehead: la géométrie projective entre ordre et incidence ( ), Revue d histoire des mathématiques, 10, pp (2004); Which Arithmeticisation for which Logicism? Russell on Quantities and Relations, History and Philosophy of Logic, 29(1), pp (2008); Toward a Topic- specific Logicism? Russell s Theory of Geometry in the Principles of Mathematics, Philosophia Mathematica, 17(1), pp (2009); Logicism and Mathematical Practices: Russell s Theory of Metrical Geometry in The Principles of Mathematics (1903), in C. Alvarez and A. Arana (eds), Analytic Philosophy and the Foundations of Mathematics, Basingstoke: Palgrave Macmillan, forthcoming. As I found it necessary to undertake a complete revision of the ideas presented in these papers, the reader should be warned that they do not correspond neatly with the chapters of the present book. I am grateful to the Société Mathématique de France, the Taylor & Francis Group, Oxford University Press and Palgrave Macmillan for having allowed me to use this material. Several pages from Chapter 5 are reprinted from an article which will be included in the volume to be edited by N. Griffin and B. Linsky for the centenary of Principia Mathematica, and I am grateful to them for permission to reproduce this material. The intellectual debts I incurred in writing this book are many and varied. While studying for my PhD (on Wittgenstein s Tractatus Logico- Philosophicus), I benefited from the advice of Christiane Chauviré and François Schmitz. During this period, several discussions I had with Jerôme Sackur on Wittgenstein, Russell and Frege proved indispensable I owe him no less than my way of viewing Russell. More recently, my work on the history of analytic philosophy has benefited from interactions with many different people. I am especially indebted for comments, conversations and criticism to Patricia Blanchette, Nicholas Griffin, Gregory Landini, James Levine, Igor Ly, Mathieu Marion and Jean-Philippe Narboux. Having obtained my PhD, I began exploring the philosophy and history of mathematics. I have greatly benefited from the intense activity one finds in Paris, and particularly from the friendly reception my work received at the REHSEIS research centre. I particularly appreciate the unique mixture of philosophers and historians of mathematics one encounters in this lab. I would like to thank Karine Chemla, Philippe Nabonnand, Marco Panza, ix

10 x Acknowledgements David Rabouin and Ivahn Smadja for having introduced me to the delights of the history and philosophy of mathematics. My recent works in philosophy (and, in particular, this book) have been markedly influenced by the numerous discussions I have had with Brice Halimi. Certain questions that Brice raised with the most innocent expression on his face kept me awake at night. I hope in the near future to be able to return the favour. I have the good fortune to be a member of the supportive department of philosophy at the Université Blaise Pascal in Clermont-Ferrand, and also a member of the PHIER (the research centre associated to it). I am particularly grateful to my colleagues Emmanuel Cattin, Laurent Jaffro and Elisabeth Schwartz for having created in Clermont a place open to philosophical research, which combines the French tradition in the history of philosophy with a deep interest in analytic philosophy and the philosophy of science. For the last four years, I have, with Sébastien Maronne, co-organized a seminar in the Maison des Sciences de l Homme on philosophy and the history of mathematics. This has been a further opportunity for me to interact with historians and mathematicians. I am also indebted to the Institut Universitaire de France, by which I was accepted as a junior member in 2008 and without whose support I would have been unable to commit the time involved in writing this book. SÉBASTIEN GANDON

11 Series Editor s Foreword During the first half of the twentieth century, analytic philosophy gradually established itself as the dominant tradition in the English-speaking world, and over the last few decades it has taken firm root in many other parts of the world. There has been increasing debate over just what analytic philosophy means, as the movement has ramified into the complex tradition that we know today, but the influence of the concerns, ideas and methods of early analytic philosophy on contemporary thought is indisputable. All this has led to greater self-consciousness among analytic philosophers about the nature and origins of their tradition, and scholarly interest in its historical development and philosophical foundations has blossomed in recent years, with the result that history of analytic philosophy is now recognized as a major field of philosophy in its own right. The main aim of the series in which the present book appears, the first series of its kind, is to create a venue for work on the history of analytic philosophy, consolidating the area as a major field of philosophy and promoting further research and debate. The history of analytic philosophy is understood broadly, as covering the period from the last three decades of the nineteenth century to the start of the twenty-first century, beginning with the work of Frege, Russell, Moore and Wittgenstein, who are generally regarded as its main founders, and the influences upon them, and going right up to the most recent developments. In allowing the history to extend to the present, the aim is to encourage engagement with contemporary debates in philosophy, for example, in showing how the concerns of early analytic philosophy relate to current concerns. In focusing on analytic philosophy, the aim is not to exclude comparisons with other earlier or contemporary traditions, or consideration of figures or themes that some might regard as marginal to the analytic tradition but which also throw light on analytic philosophy. Indeed, a further aim of the series is to deepen our understanding of the broader context in which analytic philosophy developed, by looking, for example, at the roots of analytic philosophy in neo-kantianism or British idealism, or the connections between analytic philosophy and phenomenology, or discussing the work of philosophers who were important in the development of analytic philosophy but who are now often forgotten. Bertrand Russell ( ) is the most central figure of all in the emergence of analytic philosophy. His rebellion with Moore against British idealism at the turn of the twentieth century is one of the founding events of analytic philosophy. No less important was the logicist project that both xi

12 xii Series Editor s Foreword Frege and Russell pursued, in their different ways, in the thirty or so years around the turn of the twentieth century. Frege s aim was to show that arithmetic is reducible to logic, but his attempt foundered on the paradox that bears Russell s name. Russell s aim was to show that geometry as well as arithmetic is reducible to logic, and he had to find a way of solving the paradox as well as providing richer logical resources to deal with geometry. Russell s first attempt was in The Principles of Mathematics of 1903, and his second was in Principia Mathematica, co-authored with Whitehead and published in three volumes in 1910, 1912 and These are the key works that Sébastien Gandon considers in the present book. The literature on these works is now extensive, but, as Gandon points out, most of the attention has focused on Russell s theory of logic and his philosophy of arithmetic rather than his theory of quantity and his philosophy of geometry. This is reflected in the topics that are familiar in the history of analytic philosophy: Russell s concern with relations and series, his 1903 conception of denoting, his 1905 theory of descriptions, his definition of the natural numbers as equivalence classes of classes, his theory of types, to name just some. According to Gandon, however, this leads to a distorted view of the kinds of analysis that are characteristic of Russell s work, and, given the centrality of that work, of analytic philosophy itself. Focusing on the relatively unknown aspects of Russell s logicism, Gandon argues, enables us to explore an issue that raises profound questions concerning the nature of analysis: the relationship between pre-logicized and logicized mathematics, as Gandon puts it. The philosophical problem here is essentially the paradox of analysis: How can an analysis be both correct and informative? In the case of Russell s logicism, the problem can be stated as follows. If the logicist reconstruction captures everything that is present in pre-logicized mathematics, then how can it be informative? If, on the other hand, it reforms pre-logicized mathematics, then how can it be correct? Gandon identifies two extreme responses to the problem, which he calls the Fregean and the Wittgensteinian. On the former, the logicist is taken as not beholden to pre-logicized mathematics. On the latter, the working mathematician is allowed to reject logicized mathematics if it diverges. Whether or not these responses are justly described as Fregean and Wittgensteinian, Gandon is right to seek an intermediate path, and he builds a convincing case for interpreting Russell as offering such a path. Gandon s detailed account of Russell s philosophy of geometry and theory of quantity provides the means to illuminate this path. In opposition to the first response, Gandon argues that pre-logicized mathematics provides constraints on logicist reconstruction, which need to be explained in the work of analysis. In opposition to the second response, Gandon argues that there is too much disagreement among working mathematicians for it to make sense to talk of capturing everything. Russell, according to Gandon, was not only sensitive to these disagreements but also sought to reconcile them in

13 Series Editor s Foreword xiii his logicist reconstruction through careful articulation. Logicism can itself contribute mathematically, clarifying and delineating mathematical content by integrating it in a logical framework. Logicization should be seen as a dynamic process, appreciated by recognizing all stages in the process, its value lying in the reflective equilibrium it establishes between the ex ante and ex post perspectives. The Russellian solution to the paradox of analysis that Gandon develops in this book is, in my view, essentially right; it is also superbly illustrated in the book s own methodology. The textual and historical data are carefully selected, discussed and structured in a fruitful interpretive framework. It is both insightfully about, and an excellent exercise in, Russellian analysis. MICHAEL BEANEY