Table Of ContentPhilosophy of Mathematics
and Mathematical Practice in
the Seventeenth Century
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Philosophy of Mathematics
and Mathematical Practice in
the Seventeenth Century
PAOLO MANCOSU
New York Oxford
OXFORD UNIVERSITY PRESS
1996
Oxford University Press
Oxford New York
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and associated companies in
Berlin Ibadan
Copyright © 1996 by Paolo Mancosu
Published by Oxford University Press, Inc.
198 Madison Avenue, New York, NY 10016
Oxford is a registered trademark of Oxford University Press
All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise,
without the prior permission of Oxford University Press.
Library of Congress Cataloging-in-Publication Data
Mancosu, Paolo.
Philosophy of mathematics and mathematical practice in the
seventeenth century / Paolo Mancosu.
p. cm.
Includes bibliographical references and index.
ISBN 0-19-508463-2
1. Mathematics Europe—Philosophy—History—17th century.
I. Title.
QA8.4.M36 1996
510'.9'032- -dc20 95-47024
1 3 5 7 9 8 6 4 2
Printed in the United Slates of America
on acid-free paper.
To my parents, Angela and Porfidio
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PREFACE
The present book is the result of several years' work in the history of the
philosophy of mathematics in the seventeenth century. While doing research
for it I have come into contact with several scholars whose help and
encouragement it is a pleasure to acknowledge here. I began work on this
subject while I was a graduate student at Stanford University. At that time I
had the opportunity to talk a great deal with Wilbur Knorr, Nancy Cartwright,
and Ezio Vailati. Ezio kindly agreed to my using a joint article in chapter 5 of
this book. I also owe a great deal to my Ph.D. advisor, Solomon Feferman.
Although we rarely spoke about the seventeenth century (we were too busy
discussing theories of operations and classes) his point of view on the
foundations of mathematics has greatly influenced my outlook on the subject.
And that influence goes far beyond purely technical knowledge; Sol has been
a model as a scholar and as a human being.
A great deal of the archival research for the book was made possible by a
stipendiary junior research fellowship in the history and philosophy of science
and mathematics at Wolfson College, Oxford. During those three years
(1989-91) I had access to the treasures of the Bodleian Library. In addition to
the joy of being at Oxford, I also experienced the wonderful friendship and
intellectual support provided by Daniel Isaacson.
The book was completed in Germany thanks to an Alexander von
Humboldt-Stiftung fellowship and a Morse fellowship (from Yale), which
allowed me to take time off from my teaching duties at Yale University. During
my stay at the Institut fur Philosophic, Wissenschaftstheorie, Wissenschafts-
und Technikgeschichte of the Technische Universitat in Berlin, Eberhard
Knobloch has been a most wonderful host and a critical reader of my work.
Other scholars have also helped me with the project or have invited me to
give seminars. I will simply list them in alphabetical order, confident that they
are aware of my gratitude even if I cannot express their merits individually:
Ruth Barcan-Marcus, Philip Beeley, Patricia Blanchette, George Boolos, Fabio
Bosinelli, Herbert Breger, Peter Cramer, Lisa Downing, Michael Dummett,
Luciano Floridi, Sergio Galvan, Donald Gillies, Giulio Giorello, Anthony
Grafton, Karsten Harries, Jonathan Lear, Ernan McMullin, Giuseppe Micheli,
Siegmund Probst, Neil Ribe, Carlos Sa.
viii Preface
I want to thank my friend and colleague Gyula Klima for having provided
the translation of Biancani's work published in the appendix to the book. I
owe very special thanks to Amy Rocha. She has not only discussed with me
several parts of this work but has been for years a wonderful companion and
friend.
I am grateful to the University of Chicago Press for the permission to use
the article "Torricelli's infinitely long solid and its philosophical reception in
the seventeenth century" (Isis, 82, pp. 50-70, © 1991 by the History of Science
Society, Inc. All rights reserved) which I coauthored with Ezio Vailati. It is also
a pleasure to acknowledge that Mancosu (1989, 1991, 1992a, 1992b, and 1995)
parts of which are used in this book (see notes) have respectively appeared in
Historia Mathematica (© 1989 by Academic Press, Inc.), Synthese (© 1991 by
Kluwer Academic Publishers, Inc.), Studies in History and Philosophy of Science,
Gillies (1992) Revolutions in Mathematics (© 1992 by Oxford University Press),
and Conway and Kerszberg (1995) The Sovereignty of Construction: Studies in
the Thought of David Lachterman (© 1995 by Rodopi). The relevant parts are
here reprinted by permission of the publishers.
Finally, I want to thank Wolfson College, the Whitney Griswold Faculty
Research Fund, the Morse Foundation, and the Alexander von Humboldt-
Stiftung for their generous financial support.
Berlin, Germany P.M.
May 1994
CONTENTS
1. Philosophy of Mathematics and Mathematical Practice in the Early
Seventeenth Century, 8
1.1 The Quaestio de Certitudine Mathematicarum, 10
1.2 The Quaestio in the Seventeenth Century, 15
1.3 The Quaestio and Mathematical Practice, 24
2. Cavalieri's Geometry of Indivisibles and Guldin's Centers of Gravity, 34
2.1 Magnitudes, Ratios, and the Method of Exhaustion, 35
2.2 Cavalieri's Two Methods of Indivisibles, 38
2.3 Guldin's Objections to Cavalieri's Geometry of Indivisibles, 50
2.4 Guldin's Centrobaryca and Cavalieri's Objections, 56
3. Descartes' Geometrie, 65
3.1 Descartes' Geometric, 65
3.2 The Algebraization of Mathematics, 84
4. The Problem of Continuity, 92
4.1 Motion and Genetic Definitions, 94
4.2 The "Causal" Theories in Arnauld and Bolzano, 100
4.3 Proofs by Contradiction from Kant to the Present, 105
5. Paradoxes of the Infinite, 118
5.1 Indivisibles and Infinitely Small Quantities, 119
5.2 The Infinitely Large, 129
6. Leibniz's Differential Calculus and Its Opponents, 150
6.1 Leibniz's Nova Methodus and L'Hopital's Analyse des Infiniment
Petits, 151
6.2 Early Debates with Cliiver and Nieuwentijt, 156
6.3 The Foundational Debate in the Paris Academy of Sciences, 165
Appendix: Giuseppe Biancani's De Mathematicarum Natura, 178
Translated by Gyula Klima
Notes, 213
References, 249
Index, 267
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