Plato’s Problem Also by Marco Panza ANALYSIS AND SYNTHESIS IN MATHEMATICS (co-edited with M. Otte) ISAAC NEWTON NEWTON ET L’ORIGINE DE L’ANALYSE, 1664–1666 NOMBRES: Eléments de Mathématiques pour Philosophes DIAGRAMMATIC REASONING IN MATHEMATICS (co-edited with J. Mumma and G. Sandu) Also by Andrea Sereni ISSUES ON VAGUENESS (co-edited with S. Moruzzi) Plato’s Problem An Introduction to Mathematical Platonism Marco Panza CNRS (IHPST, Paris), France and Andrea Sereni San Raffaele University, Italy © Marco Panza and Andrea Sereni 2013 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 authors have asserted their rights to be identified as the authors of this work in accordance with the Copyright, Designs and Patents Act 1988. First published 2013 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number 785998, 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 10010. 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 978–0–230–36548–3 hardback ISBN 978–0–230–36549–0 paperback 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. 10 9 8 7 6 5 4 3 2 1 22 21 20 19 18 17 16 15 14 13 Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne Contents Preface vii Acknowledgements xii Terminological Conventions xiii Introduction 1 Platonism in the Philosophy of Mathematics 1 Nominalism in the Philosophy of Mathematics 9 The Indispensability Argument 14 1 The Origins 16 1.1 Plato as a Platonist? 17 1.2 Aristotle Between Platonism and Anti-platonism 27 1.3 Proclus: The Neoplatonic Interpretation of Euclid’s Geometry 32 1.4 Kant: The Transcendental Interpretation of Classical Arithmetic and Geometry 36 2 From Frege to Gödel (Through Hilbert) 45 2.1 Frege’s Logicist Platonism 45 2.2 Russell and the Separation of Logicism and Platonism 66 2.3 Set Theory 69 2.4 The Problem of Foundations 73 2.5 Gödel’s Platonism and the Rise of Mathematical Intuition 90 3 Benacerraf’s Arguments 99 3.1 What Natural Numbers Could Not Be (According to Benacerraf) 101 3.2 Benacerraf’s Dilemma 107 3.3 A Map of Responses to Benacerraf’s Dilemma: Contemporary Solutions to Plato’s Problem 110 4 Non-conservative Responses to Benacerraf’s Dilemma 112 4.1 Field’s Nominalism: Mathematics Without Truth and Science Without Numbers 112 4.2 Mathematics as Fiction: Field and Yablo 125 v vi Contents 4.3 Eliminative Structuralism and its Modal Version 136 4.4 Maddy and the Cognitive Origins of Set Theory 144 5 Conservative Responses to Benacerraf’s Dilemma 149 5.1 Neo-logicism: A Revised Version of Frege’s Programme 149 5.2 Linsky, Zalta and ‘Object Theory’: Mathematics and Logic (or Metaphysics) of Abstract Objects 165 5.3 A First Version of Non-eliminative Structuralism: Ante Rem Structuralism 177 5.4 A Second Version of Non-eliminative Structuralism: Parsons and the Role of Intuition 187 6 The Indispensability Argument: Structure and Basic Notions 196 6.1 Four Versions of IA 197 6.2 The Quine-Putnam Argument and Colyvan’s Argument 201 6.3 (In)dispensability 203 6.4 Quine’s Criterion of Ontological Commitment 210 6.5 Naturalism 212 6.6 Confirmational Holism 214 6.7 The Dispensability of Naturalism and Confirmational Holism 215 7 The Indispensability Argument: The Debate 217 7.1 Against Indispensability 217 7.2 Against Ontological Commitment 224 7.3 Against Naturalism and Scientific Realism 235 7.4 Against Confirmational Holism 241 Concluding Remarks 250 Notes 255 References 273 Index 296 Preface The aim of this book, a revised and partially enlarged edition of the Italian original Il Problema di Platone published in 2010 (Carocci Editore, Roma), is to offer an introduction to the philosophy of mathemat- ics addressed to all those wishing to familiarize themselves with the subject, and even to those lacking any previous acquaintance with it. We have done our best to organize the discussion so as to avoid reliance on previous knowledge. An introduction, however, cannot cover everything, and the philosophy of mathematics is a vast domain with a proliferation of close connections to several other domains: mathematics itself and its history, logic, philosophy of language, history of philosophy, just to take some obvious examples. The reader will thus unavoidably encounter remarks for which full understanding requires acquaintance, if only at basic levels, with different disciplines. Several other introductions to the philosophy of mathematics are available. Mentioning a few of them, and limiting ourselves to books in English, we can point to Brown (1999), Potter (2002), Shapiro (2000a), George and Velleman (2001), Giaquinto (2002), Mancosu (2008a), Bostock (2009), Colyvan (2012). Our book will obviously discuss many topics that are covered also in these introductions, but it is intended to approach them with a particular focus. Introductions to the philosophy of mathematics have often aimed at offering a comprehensive account of the discussion in the field since at least the end of the nineteenth century. In some cases, the his- torical background is kept to a minimum, in order to leave room for a theoretical discussion of the main philosophical options available in the contemporary debate. We have followed a different strategy. We have focused on a single problem and a single option that has been offered as a solution of it; we then trace the main historical development of this latter option (Chapters 1–2), discuss the debate it has engendered from the 1960s to the present day (Chapters 3–5) and finally focus in more detail on a currently widely discussed argument that has been advanced in that debate (Chapters 6–7). The problem we have focused on is that of the ontology of mathematics, the problem that in our title we call ‘Plato’s problem’ (the reasons for this will be clear in § 1.1). The following is a very simple way of presenting it: granted that the statements of mathematics are about vii viii Preface something, what are they about? The answer we focus on goes under the name of ‘platonism’. It claims, broadly speaking, that these statements are about a domain of abstract objects, which they describe. We are well aware that this way of proceeding can only offer a very partial representation of past and present philosophy of mathematics, which has dealt with several other problems and answers. Rather than presenting a broad-brush summary of this vast range of topics, we prefer to limit the scope of our introduction so as to make it possible to go into much more detail on the option being discussed and the specific argument in its support. Chapters 6 and 7 are devoted to this latter, the so-called ‘indispensability argument’. Our hope is that the reader will be guided through a significant part of present-day discussion in the philosophy of mathematics, possibly paving the way for further studies. In order to achieve this, the book’s chapters have been written in three different styles. Chapters 1 and 2 contain an (inevitably partial) historical recon- struction, with authors and topics considered in chronological order. Connections among specific topics are emphasized. The aim of these chapters is to offer the historical background that seems to us required in order to understand why the contemporary debate is so relevant. Philosophy, even in its most technical and specific regions, is largely motivated by its history and tradition, and no philosophical discussion can really be appreciated when these are wholly disregarded. Chapters 3, 4 and 5 offer a synchronic reconstruction of several options, some supporting platonism and some opposing it. We have cho- sen to arrange them depending on which sort of response they offer to the dilemma advanced by Paul Benacerraf in the early 1970s, a dilemma that in our opinion can easily be seen as a modern version of Plato’s problem. The aim of these chapters is to present an articulated summary of the current discussion on this problem, one that the philosophy of mathematics, as the previous chapters show, inherits from its tradition. Chapters 6 and 7 present a systematic reconstruction of a particular argument, aiming to show its assumptions, motivations and difficulties, together with the necessary and/or sufficient conditions for its conclu- sions. We have opted for a detailed, and often explorative, analysis of the various theoretical ingredients involved in one single argument, an argument that has been suggested during the course of the past fifty years by one of the most authoritative thinkers in the empiricist tradi- tion, Willard van Orman Quine. The contrast between this tradition and the platonic one, that finds a point of intersection in the indispen- sability argument, is one of its most interesting traits. Preface ix We have chosen to devote two chapters to this argument since it involves several crucial issues in contemporary philosophy of math- ematics. This is not to say that we intend to appeal to the argument (once appropriately stated) in order to convince the reader of the cor- rectness of the platonist option in the version it supports. Quite the contrary, we believe this argument to have various limitations and difficulties, many of which will be discussed. Our only purpose is to familiarize the reader with the intricacies of contemporary philosophy of mathematics through the consideration of a specific example among the current debate. Even the choice of treating Plato’s problem is partial. Other problems differ from it not only as regards their content, but also as regards their nature. Roughly speaking, we can single out four kinds of problems pertaining to the philosophy of mathematics. First, there are foundational problems concerning the best way of founding, justifying and organizing the edifice of mathematics or at least some relevant parts of it. Next, there are general interpretative issues relating to the nature of mathematics itself, such as Plato’s prob- lem and other problems variously related to it, such as the problem of the nature of mathematical knowledge. (Is there mathematical knowl- edge, and if so, what kind of knowledge is it?) or that of the logical character of the truths or theorems of mathematics. (If there are any truths in mathematics, what is their source? Are they analytic or syn- thetic? A priori or empirical? And if there are none, what legitimates the theorems of mathematics? Just to give an example, if ‘3 + 5 = 8’ is not a true statement, in what way does it differ from ‘3 + 5 = 9’?) There are also more specific interpretive problems, concerned with particular mathematical theories, or with mathematical practice as it has developed over time. Mancosu (2008a) offers an excellent survey of some of these problems, such as that of the availability of a criterion for acknowledging when arguments are explanatory, or that of visualiza- tion, or more generally of diagrammatic reasoning in mathematics, or that of the possibility of offering a principle of purity, selecting some proofs or theories as being better than others. Last, there are problems relating to the applicability of mathematics, the justification for this applicability and how it takes place, to the role mathematics has in empirical sciences and in our everyday lives. Many philosophers of mathematics believed, and still believe, that their main task is to celebrate the beauty of mathematics. Carl Jacobi – a great mathematician of the first half of the nineteenth century maintained – against Joseph Fourier – that the only aim of m athematics