Trends in the History of Science Vincenzo De Risi Editor Mathematizing Space The Objects of Geometry from Antiquity to the Early Modern Age Trends in the History of Science Trends in the History of Science is a series devoted to the publication of volumes arising from workshops and conferences in all areas of current research in the history of science, primarily with a focus on the history of mathematics, physics, and their applications. Its aim is to make current developments available to the community as rapidly as possible without compromising quality, and to archive those developments for reference purposes. Proposals for volumes can be submitted using the online book project submission form at our website www.birkhauser- science.com. More information about this series at http://www.springer.com/series/11668 Vincenzo De Risi Editor Mathematizing Space The Objects of Geometry from Antiquity to the Early Modern Age Editor Vincenzo De Risi History of Science Max Planck Institute Berlin Germany ISSN 2297-2951 ISSN 2297-296X (electronic) Trends in the History of Science ISBN 978-3-319-12101-7 ISBN 978-3-319-12102-4 (eBook) DOI 10.1007/978-3-319-12102-4 Library of Congress Control Number: 2014957136 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 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 Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.birkhauser-science.com) Foreword The present book collects the essays presented at the conference on Space, Geometry and the Imagination from Antiquity to the Early Modern Age, jointly organized by the Max Planck Institute for the History of Science (Berlin) and the Centro Matematico ‘Ennio De Giorgi’ of the Scuola Normale Superiore (Pisa), and hosted by the Max Planck Institute in August 27–29, 2012. The conference was part of the activities of the MPI Research Group on Modern Geometry and Space, directed by Vincenzo De Risi. The Group’s principal aim is to study the inter- connections between the history of geometry and the philosophy of space in the pre-Modern and Early Modern Age. In particular, several investigations of the fellows of the Research Group were directed toward understanding the complex epistemological revolution that transformed the classical geometry of figures into the modern geometry of space. The above-mentioned conference represented one of the high points of this research, and we are grateful to Birkhäuser for accepting to publish the proceedings of the meeting. We acknowledge the generous support of the two institutions, the Max Planck Institute and the Scuola Normale, for organizing the conference. We would par- ticularly like to thank the Director of the Centro ‘Ennio De Giorgi’ in Pisa, Mariano Giaquinta, whose early involvement in the organization was crucial for the reali- zation of the project. We are especially grateful to the distinguished scholars who took part in the conference and accepted to discuss their views with us on space and geometry, and for their willingness to contribute to the present volume. Their intellectual gener- osity made it possible for a real scientific exchange to take place that enlightened the workshop in Berlin and still shines in their written essays. We also thank all other participants in the conference, fellows of the Max Planck Institute, students, and visiting scholars, whose points of view and objections played an important part in enriching the discussion. We finally thank Chiara Fabbrizi and Fred Sengmueller for their help in editing the present volume. v Contents Introduction ........................................... 1 Vincenzo De Risi What’s Location Got to Do with It? Place, Space, and the Infinite in Classical Greek Mathematics ............................. 15 Henry Mendell A Note on Lines and Planes in Euclid’s Geometry ............... 65 Jeremy Gray Theon of Smyrna and Ptolemy on Celestial Modelling in Two and Three Dimensions .................................... 75 Alexander Jones Proclus’ Conception of Geometric Space and Its Actuality.......... 105 David Rabouin Subject, Space, Object: The Birth of Modernity ................. 143 Franco Farinelli On Natural Geometry and Seeing Distance Directly in Descartes ..... 157 Gary Hatfield Hobbes’s Theory of Space ................................. 193 Douglas Jesseph Mathematics and Infinity in Descartes and Newton ............... 209 Andrew Janiak vii viii Contents Leibniz’s Transcendental Aesthetic........................... 231 Daniel Garber Hume’s Skepticism and Inductivism Concerning Space and Geometry .......................................... 255 Graciela De Pierris Kant on Geometry and Experience........................... 275 Michael Friedman Index ................................................ 311 Contributors Graciela De Pierris Department of Philosophy, Stanford University, Stanford, CA, USA Vincenzo De Risi Max Planck Institute for the History of Science, Berlin, Germany Franco Farinelli Dipartimento di Filosofia e Comunicazione, Università di Bologna, Bologna, Italy Michael Friedman Department of Philosophy, Stanford University, Stanford, CA, USA Daniel Garber Department of Philosophy, Princeton University, Princeton, USA Jeremy Gray Department of Mathematics and Statistics, The Open University, Buckinghamshire, UK Gary Hatfield Department of Philosophy, University of Pennsylvania, Philadelphia, PA, USA Andrew Janiak Department of Philosophy, Duke University, Durham, NC, USA Douglas Jesseph Department of Philosophy, University of South Florida, Tampa, FL, USA Alexander Jones Institute for the Study of the Ancient World, New York University, New York, USA Henry Mendell Department of Philosophy, California State University, Los Angeles, CA, USA David Rabouin Université Paris 7—CNRS Laboratoire SPHERE UMR 7219, Paris Cedex 13, France ix Introduction Vincenzo De Risi Today the definition of geometry as the science of space is generally accepted by both epistemologists and mathematicians. The history of modern geometry is entirely built around the mathematical notion of space, and different approaches to this science, from Gauss’ studies of intrinsic curvature to the Erlangen Program, from the discovery of General Relativity to the most recent developments in topology (take, for instance, Thurston’s geometrization conjecture and its proof) rely on a general understanding of mathematical space that remains constant through different perspectives and offers a common ground for regarding all these developments as parts of a single enterprise. Modern geometry is simply incon- ceivable without the notion of space. Nonetheless, the definition of geometry as the science of space, however stan- dard, is properly speaking modern. Should we go through the thirteen books of Euclid’s Elements, or in fact the entire corpus of ancient mathematics, we would find almost no occurrence of spatial concepts or terms. Were we to follow the millennial development of Classical geometry in the Middle Ages or the Renais- sance, we would still not find any reference to space. The first (and quite rhetorical) mention of spatium in a geometrical essay does not predate the last decades of the sixteenth century. To see spatial notions effectively employed in geometrical rea- soning, we have to wait for another one hundred years. Leibniz’ work on geometry (the analysis situs) is probably the first attempt in this direction, and in any case it ushered in a general discussion about the object of geometrical investigation. The eighteenth century debated whether geometry had to be regarded as a science of space, and this new idea initially attracted more opponents than supporters; by the end of the century the spatial backers won their battle, and the nineteenth century V. De Risi (&) Max Planck Institute for the History of Science, Boltzmannstraße 22, 14195 Berlin, Germany e-mail: [email protected] © Springer International Publishing Switzerland 2015 1 V. De Risi (ed.), Mathematizing Space, Trends in the History of Science, DOI 10.1007/978-3-319-12102-4_1 2 V. De Risi declared that space was indeed the object of geometry. The mathematization of space was then complete: classical geometry came to an end and modern geometry was born. In fact, even though the divide between ancient and modern geometry may be arbitrarily demarcated into several historically distinct episodes (such as the birth of algebraic geometry or the discovery