Sources and Studies in the History of Mathematics and Physical Sciences Editorial Board J.Z. Buchwald J. Lu¨tzen G.J. Toomer Advisory Board P.J. Davis T. Hawkins A.E. Shapiro D. Whiteside Gert Schubring Conflicts between Generalization, Rigor, and Intuition Number Concepts Underlying the Development of Analysis in 17–19th Century France and Germany With 21 Illustrations Gert Schubring Institut fu¨r Didaktik der Mathematik Universita¨t Bielefeld Universita¨tstraße 25 D-33615 Bielefeld Germany [email protected] Sources and Studies Editor: Jed Buchwald Division of the Humanities and Social Sciences 228-77 California Institute of Technology Pasadena, CA 91125 USA Library of Congress Cataloging-in-Publication Data Schubring, Gert. Conflicts between generalization, rigor, and intuition / Gert Schubring. p. cm. — (Sources and studies in the history of mathematics and physical sciences) Includes bibliographical references and index. ISBN 0-387-22836-5 (acid-free paper) 1. Mathematical analysis—History—18th century. 2. Mathematical analysis—History—19th century. 3. Numbers, Negative—History. 4. Calculus—History. I. Title. II. Series. QA300.S377 2005 515′.09—dc22 2004058918 ISBN-10: 0-387-22836-5 ISBN-13: 978-0387-22836-5 Printed on acid-free paper. © 2005 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring St., New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, com- puter software, or by similar or dissimilar methodology now known or hereafter developed is for- bidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (EB) 987654321 SPIN 10970683 springeronline.com Preface This volume is, as may be readily apparent, the fruit of many years’ labor in archives and libraries, unearthing rare books, researching Nachlässe, and above all, systematic comparative analysis of fecund sources. The work not only demanded much time in preparation, but was also interrupted by other duties, such as time spent as a guest professor at universities abroad, which of course provided welcome opportunities to present and discuss the work, and in particular, the organizing of the 1994 International Graßmann Conference and the subsequent editing of its proceedings. If it is not possible to be precise about the amount of time spent on this work, it is possible to be precise about the date of its inception. In 1984, during research in the archive of the École polytechnique, my attention was drawn to the way in which the massive rupture that took place in 1811—precipitating the change back to the synthetic method and replacing the limit method by the method of the quantités infiniment petites—significantly altered the teaching of analysis at this first modern institution of higher education, an institution originally founded as a citadel of the analytic method. And it was in a French context, so favorably disposed to establishing links between history and epistemology, that I first presented my view that concept development is culturally shaped; it was at the third Ecole d'Eté de Didactique des Mathéma- tiques of July 1984 in Orléans that I presented my paper “Le retour du réfoulé— Les débats sur ‘La Méthode’ à la fin du 18ème et au debut du 19ème siècle: Condillac, Lacroix et les successeurs.” When the work was eventually completed in 2002, it was accepted as a Habilitationsschrift by the Mathematics Department of the University of Bielefeld. I am grateful to Jesper Lützen, editor of the series Sources and Studies in the History of Mathematics and Physical Sciences, and to the publishing house of Springer, for publishing the script in their series. The relatively independent former Chapter C, investigating the context of the 1811 switch in the basic conceptions at the École polytechnique is published separately as “Le Retour du Réfoulé—Der Wiederaufstieg der synthetischen Methode an der École polytechnique” (Augsburg: Rauner, 2004). Its principal results are summarized here in Chapter IV.3. VI Preface In an effort to accelerate the publication of this volume, the sheer size of the manuscript led me to organize its translation from German into English, practically as a collective endeavor. I am indebted to the commitment shown by Jonathan Harrow and Günter Seib, Chris Weeks, and Dorit Funke as translators. I wish to thank the following archives for their kind permission to reproduce documents: Archiv der Berlin–Brandenburgischen Akademie der Wissenschaften (Berlin), Archives de l’École polytechnique (Paris/Palaiseau), Archives de l’Académie des Sciences (Paris). In conclusion, I should like to make two points. Firstly, all the translations of non-English quotations—both from the original sources and from publications— are ours, except in cases where English translations already existed, which are so indicated. In a few cases, the original quotation is preserved when the context makes its meaning sufficiently clear. The second point concerns terminology. The French reflections on foundations constitute the major focus of this study; consequently, the “triad” of basic concepts used in French mathematics to “span” the contemporary concept fields—namely quantité, grandeur, and nombre—is also used here, as the configuration of basic terms. Since quantité was understood to be the key foundational concept (cf. the Encyclopédie defining mathematics as the science of quantité), and since my English terms are intended to recall the French understanding of the time, quantité is rendered here throughout as “quantity,” and grandeur as “magnitude.” University of Bielefeld Gert Schubring September 2004 Contents Preface v Illustrations xiv I. Question and Method 1 1. Methodological Approaches to the History of Science 1 2. Categories for the Analysis of Culturally Shaped Conceptual Developments 8 3. An unusual Pair: Negative Numbers and Infinitely Small Quantities 10 II. Paths Toward Algebraization – Development to the Eighteenth Century. The Number Field 15 1. An Overview of the History of Key Fundamental Concepts 15 1.1. The Concept of Number 16 1.2. The Concept of Variable 19 1.3. The Concept of Function 20 1.4. The Concept of Limit 22 1.5. Continuity 25 1.6. Convergence 27 1.7. The Integral 30 2. The Development of Negative Numbers 32 2.1. Introduction 32 2.2 An Overview of the Early History of Negative Numbers 35 From Antiquity to the Middle Ages 35 European Mathematics in the Middle Ages 39 2.3. The Onset of Early Modern Times. The First “Ruptures” in Cardano’s Works 40 2.4. Further Developments in Algebra: From Viète to Descartes 45 VIII Contents 2.5. The Controversy Between Arnauld and Prestet 49 A New Type of Textbook 49 Antoine Arnauld 50 Jean Prestet 52 The Controversy 54 The Debate’s Effects on Their Textbook Reeditions 57 2.6. An Insertion: Brief Comparison of the Institutions for Mathematical Teaching in France, Germany, and England 61 Universities and Faculties of Arts and Philosophy 61 The Status of Mathematics in Various Systems of National Education 64 New Approaches in the Eighteenth Century 66 2.7. First Foundational Reflections on Generalization 67 2.8. Extension of the Concept Field to 1730/40 73 2.8.1. France 73 2.8.2. Developments in England and Scotland 88 2.8.3. The Beginnings in Germany 95 2.9. The Onset of an Epistemological Rupture 99 2.9.1. Fontenelle: Separation of Quantity from Quality 99 2.9.2. Clairaut: Reinterpreting the Negative as Positive 102 2.9.3. D’Alembert: The Generality of Algebra—An Inconvénient 104 2.10. Aspects of the Crisis to 1800 114 2.10.1. Stagnant Waters in the French University Context 114 2.10.2. The Military Schools as Multipliers 121 2.10.3. Violent Reaction in England and Scotland 126 2.10.4. The Concept of Oppositeness in Germany 132 2.11. Looking Back 149 III. Paths toward Algebraization—The Field of Limits: The Development of Infinitely Small Quantities 151 1. Introduction 151 2. From Antiquity to Modern Times 153 Concepts of the Greek Philosophers 154 3. Early Modern Times 157 Contents IX 4. The Founders of Infinitesimal Calculus 161 5. The Law of Continuity: Law of Nature or Mathematical Abstraction? 174 6. The Concept of Infinitely Small Quantities Emerges 186 7. Consolidating the Concept of Infinitely Small Quantities 192 8. The Elaboration of the Concept of Limit 206 8.1. Limits as MacLaurin’s Answer to Berkeley 206 8.2. Reception in the Encyclopédie and Its Dissemination 209 8.3. A Muddling of Uses in French University Textbooks 213 8.4. First Explications of the Limit Approach 220 8.5. Expansion of the Limit Approach and Beginnings of Its Algebraization 228 9. Operationalizations of the Concept of Continuity 238 10. A Survey 255 IV. Culmination of Algebraization and Retour du Refoulé 257 1. The Number Field: Additional Approaches Toward Algebraization in Europe and Countercurrents 257 1.1. Euler: The Basis of Mathematics is Numbers, not Quantities 257 1.2. Condillac: Genetic Reconstruction of the Extension of the Number Field 260 1.3. Buée: Application of Algebra as a Language 265 1.4. Fundamentalist Countercurrents 268 Klostermann: Elementary Geometry versus Algebra 276 2. The Limit Field: Dominance of the Analytic Method in France After 1789 279 2.1. Apotheosis of the Analytic Method 279 2.2. Euler’s Reception 283 2.3. Algebraic Approaches at the École Polytechnique 286 3. Le Retour du Refoulé: The Renaissance of the Synthetic Method at the École Polytechnique 295 3.1.