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Logic, Epistemology, and the Unity of Science 28 Fabrice Pataut Editor Truth, Objects, Infinity New Perspectives on the Philosophy of Paul Benacerraf Logic, Epistemology, and the Unity of Science Volume 28 Series editors Shahid Rahman, University of Lille III, France John Symons, University of Texas at El Paso, USA Editorial Board Jean Paul van Bendegem, Free University of Brussels, Belgium Johan van Benthem, University of Amsterdam, The Netherlands Jacques Dubucs, CNRS/Paris IV, France Anne Fagot-Largeault, Collège de France, France Göran Sundholm, Universiteit Leiden, The Netherlands Bas van Fraassen, Princeton University, USA Dov Gabbay, King’s College London, UK Jaakko Hintikka, Boston University, USA Karel Lambert, University of California, Irvine, USA Graham Priest, University of Melbourne, Australia Gabriel Sandu, University of Helsinki, Finland Heinrich Wansing, Ruhr-University Bochum, Germany Timothy Williamson, Oxford University, UK Logic, Epistemology, and the Unity of Science aims to reconsider the question of the unity of science in light of recent developments in logic. At present, no single logical, semantical or methodological framework dominates the philosophy of science. However, the editors of this series believe that formal techniques like, for example, independence friendly logic, dialogical logics, multimodal logics, game theoretic semantics and linear logics, have the potential to cast new light on basic issues in the discussion of the unity of science. This series provides a venue where philosophers and logicians can apply specific technical insights to fundamental philosophical problems. While the series is open to a wide variety of perspectives, including the study and analysis of argumentation and the critical discussion of the relationship between logic and the philosophy of science, the aim is to provide an integrated picture of the scientific enterprise in all its diversity. More information about this series at http://www.springer.com/series/6936 Fabrice Pataut Editor Truth, Objects, Infinity New Perspectives on the Philosophy of Paul Benacerraf 123 Editor Fabrice Pataut FRE Sciences, Normes, Décision, CNRS Paris France ISSN 2214-9775 ISSN 2214-9783 (electronic) Logic, Epistemology, and the Unity of Science ISBN 978-3-319-45978-3 ISBN 978-3-319-45980-6 (eBook) DOI 10.1007/978-3-319-45980-6 Library of Congress Control Number: 2016950889 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland © Fellows of the American Academy of Arts and Sciences Acknowledgements In many ways, the articles “What Numbers Could Not Be” and “Mathematical Truth” by Paul Benacerraf have dominated the philosophy of mathematics for about fifty years. If you take into account the early version of “Mathematical Truth” that Benacerraf has kindly agreed to publish in this volume for the first time, some of his main ideas have circulated in the philosophical community since the late sixties. Many seminal articles and important books have argued in favor of views about what truth in mathematics amounts to and about what numbers could or could not be. Most of them have done so by taking a stand on these two timeless papers; at the very least, this is where they started from. The philosophy of mathematics might be philosophy “in a pure state, stripped of all worldly appendages; austere; philosophy without the sugar coating of a pretense to Relevance to Life” (Benacerraf 1996b1: 10), yet everyone working today in philosophy tout court has read them. And this is only, if I may say so, the tip of the iceberg. “Frege: The Last Logicist,” from 1981, and Benacerraf’s very first publication, “Tasks, Super-Tasks, and Modern Eleatics,” from 1962, have also shaped, or reshaped, some key areas of the discipline. Because of them, people have looked at Frege’s logicist project of reducing mathematics to logic, and at the vexed puzzle of how an infinite number of operations might be performed in a finite time, in a different way. As far as Frege is concerned, we should perhaps travel back to the spring of 1961, when Benacerraf presented his 1960 dissertation on the “fundamentally mistaken” doctrine of logicism in a course on the philosophy of mathematics he gave at Princeton (Benacerraf 1960: iii, 255). George Boolos, then a student, was “irritated” by Benacerraf’s reading of it, judged it “perverse” but later on came to agree with it (Boolos 1996: 143–144)2. It is striking that Benacerraf’s rejection of logicism, mainly on the ground that concepts of mathematics cannot be defined in terms of concepts of pure logic (Chaps. I and II of Benacerraf 1960), and that we do not accept (or reject) mathematics on extra-mathematical grounds (Chap. III of Benacerraf 1960), has stimulated an interest in the logicist outlook, e.g., in sub- programs of logicism offering reductions of fragments of arithmetic, in neo-logicism and, more generally, in a discussion of Hume’s principle. vii viii Acknowledgements As for supertasks, Benacerraf and Putnam pointed as early as 1964, in the wake of the 1962 paper, that “[i]f we take the stand that ‘nonconstructive’ procedures—i.e., procedures that require us to perform infinitely many operations in a finite time—are conceivable,3 though not physically possible (owing mainly to the existence of a limit to the velocity with which physical operations can be performed), then we can say that there does ‘in principle’ exist a verification/refutation procedure for number theory” (Benacerraf and Putnam [1964] 1983: 20). The point here is that the notion of truth in number theory is a bona fide notion only insofar as the notion of a completed actually infinite series of operations is one itself. Intuitionist worries about the actual infinite, Dummettian antirealist worries about the non-constructivity of classical proofs, and, a fortiori, strict finitist worries about the permissiveness of the “in principle” clause at work in the claim, all take us back to the notion of truth, a unifying, perhaps the unifying theme of Benacerraf’s work.4 Then, last but not least, there is the anthology of texts in the philosophy of mathematics, the Selected readings that generations of students have used as a sourcebook—indeed as a textbook—edited, twice prefaced, and introduced in collaboration with Hilary Putnam (Benacerraf and Putnam [1964] 1983). Few philosophers trigger such a fresh start on perennial philosophical problems and do so much to teach their discipline. The idea of an international meeting devoted to a critical evaluation of Benacerraf’s work in the philosophy of mathe- matics, with its wide bearings on the philosophy of language, the philosophy of logic, and epistemology, seemed a natural one. The idea came in the form of a workshop rather than in the form of a glorifying celebration, with a large part of the time to be set aside for discussion. The prospect of benefiting from Paul’s partic- ipation certainly gave it more than a tinge of excitement. The workshop eventually took place in Paris at the Collège de France on May 10 and May 11, 2012. The participants were Jody Azzouni, Jacques Dubucs, Bob Hale, Brice Halimi, Sébastien Gandon, Mary Leng, Andrea Sereni, Stewart Shapiro, Claudine Tiercelin, and myself. The present volume includes some of the papers read at the workshop and five additional essays. Unfortunately, the papers by Jacques Dubucs, Bob Hale and Claudine Tiercelin are missing from the volume. Hale’s “Properties and the Interpretation of Second-Order Logic” has since appeared in Philosophia Mathematica, Vol. 21 (2), 2013, pp. 133–156; it was first published online by the journal on August 3, 2012. The five essays that were not presented at the workshop are, respectively, by Justin Clarke-Doane, Jon Pérez Laraudogoitia, Antonio León-Sánchez and Ana C. León-Mejía, Marco Panza, and Philippe de Rouilhan. The fourteen essays have been conveniently organized into four parts. Mine serves as an introduction. The first part includes those that directly address what is known today as “Benacerraf’s dilemma.” The second gathers contributions directly involved with issues in the philosophy of mathematics and in particular with that part of it concerned with arithmetic and number theory. The third is devoted to supertasks. The fourth part contains the ancestor of “Mathematical Truth,” a draft from January 1968 that Benacerraf started writing in 1967 with the joint support of the John Simon Guggenheim Foundation and of Princeton University, and Acknowledgements ix “Comments on Reduction,” a lecture he gave in a graduate seminar at Princeton around 1975 to discuss the views on reductions of natural numbers within set theory that he previously expounded in print in “What Numbers Could Not Be.” Many thanks to Paul for having allowed the publication of these two essays.

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