MATHEMATICS AND THE DIVINE: AHISTORICAL STUDY Cover illustration by Erich Lessing MATHEMATICS AND THE DIVINE: AHISTORICAL STUDY Edited by T. KOETSIER Vrije Universiteit, Amsterdam, The Netherlands L. BERGMANS Université de Paris IV – Sorbonne, Paris, France 2005 Amsterdam • Boston • Heidelberg • London • New York • Oxford • Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo ELSEVIER B.V. ELSEVIER Inc. ELSEVIER Ltd ELSEVIER Ltd Sara Burgerhartstraat 25 525 B Street, Suite 1900 The Boulevard, Langford Lane 84 Theobalds Road P.O. Box 211, 1000 AE San Diego, CA 92101-4495 Kidlington, Oxford OX5 1GB London WC1X 8RR Amsterdam The Netherlands USA UK UK © 2005 Elsevier B.V. All rights reserved. This work is protected under copyright by Elsevier B.V., and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. 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British Library Cataloguing in Publication Data A catalogue record is available from the British Library. ISBN: 0-444-50328-5 ∞ The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands. This book is dedicated to our children Quod mathematica nos iuvet plurimum in diversorum divinorum apprehensione (Mathematics helps us greatly in understanding various divine truths) Cusanus, De Docta Ignorantia I, XI This page intentionally left blank Preface The idea that is at the basis of the present book was born several years ago. A compre- hensive historical study of the relations between mathematics on the one hand, and meta- physics, religion and mysticism on the other hand, did not exist. Because of the expanse of the subject we decided to combine the expertise of numerous experts. Although it proved to be impossible to give attention to everything in the present volume, a wide range of topics was covered, as is evident from the titles in the list of contents. We believe that at least the major aspects of the subject are covered. Wherever possible, sources of quotations are ack- nowledged. Every effort has been made to discover the original source of the illustrations used and to obtain permission to include them. We are very grateful to the scholars who contributed the articles. Moreover, we gratefully acknowledge the following individuals who have assisted us in various ways in making this book a reality: Marc Bergmans, Henk Bos, Vera Brauns, Sébastien Busson, Georgia Gauley, Ineke Hilhorst, Jan Hogendijk, Jan Willem van Holten, Bas Jongeling, Jan van Mill, Mickaël Robert, Karel Schmidt Jr., Arjen Sevenster, Eric-Jan Tuininga. Luc Bergmans (Paris) Teun Koetsier (Amsterdam) October 2004 vii This page intentionally left blank List of Contributors Adamson, D., Goldsmiths’ College, London and Wolfson College, Cambridge, retired (Ch. 21) Beeley, P., Westfälische Wilhelms-Universität, Münster, Germany (Ch. 23) Bergmans, L., Université de Paris IV – Sorbonne, Paris, France (Ch. 29) Breger, H., Leibniz Archiv, Hannover, Germany (Ch. 25) Breidert, W., Universität Karlsruhe (TH), Karlsruhe, Germany (Ch. 26) Charrak, A., Université de Paris I Pantheon – Sorbonne Paris, France (Ch. 19) Counet, J.-M., University of Louvain, Louvain-la-Neuve, Belgium (Ch. 14) De Gandt, F., Université Charles de Gaulle, Lille, France (Ch. 32) Demidov, S.S., Institute of the History of Science and Technology, Russian Academy of Sciences, Moscow, Russia (Ch. 31) de Pater, C., Vrije Universiteit, Amsterdam and Universiteit Utrecht, Utrecht, The Nether- lands (Ch. 24) Garcia, H., Pfarrhaus, Zillis, Switzerland (Ch. 12) Hayoun, M.-R., Université de Strasbourg, Strasbourg, France (Ch. 10) Harmsen, G., Rijksuniversiteit Groningen, The Netherlands, retired (Ch. 22) Ho Peng-Yoke, Needham Research Institute, Cambridge, UK, director, retired (Ch. 1) King, D.A., Johann Wolfgang Goethe University, Frankfurt am Main, Germany (Ch. 8) Koetsier, T., Vrije Universiteit, Amsterdam, The Netherlands (Ch. 15, 30, 34) Knobloch, E., Technische Universität Berlin, Berlin, Germany (Ch. 17) Lohr, C., Albert-Ludwigs-Universität Freiburg, Freiburg, Germany (Ch. 11) Mattéi, J.-F., University of Nice–Antipolis and Institut Universitaire de France, France (Ch. 5) Mueller, I., The University of Chicago, Chicago, IL, USA (Ch. 4) Netz, R., Stanford University, Stanford, CA, USA (Ch. 3) Nicolle, J.-M., Université de Rouen, Mont-Saint-Aignan, France (Ch. 20) O’Meara, D.J., University of Fribourg, Fribourg, Switzerland (Ch. 6) Pinchard, B., Université Jean Moulin Lyon 3, France (Ch. 33) Plofker, K., University of Utrecht, Utrecht, The Netherlands (Ch. 2) Probst, S., Leibniz Archiv, Hannover, Germany (Ch. 23) Reich, K., Universität Hamburg, Germany (Ch. 15) Remmert, V.R., Johannes Gutenberg-Universität Mainz, Mainz, Germany (Ch. 18) Schneider, I., Universität der Bundeswehr München, Neubiberg, Germany (Ch. 16) Sylla, E.D., North Carolina State University, Raleigh, NC, USA (Ch. 13) Terrien, M.-P., University of Le Mans, France (Ch. 7) Thiele, R., Universität Leipzig, Leipzig, Germany (Ch. 27, 28) van der Schoot, A., Universiteit van Amsterdam, The Netherlands (Ch. 35) Wallis, F., McGill University, Montreal, Canada (Ch. 9) ix This page intentionally left blank Contents Preface vii List of Contributors ix Introduction 1 T. Koetsier and L. Bergmans 1. Chinese number mysticism 45 Ho Peng-Yoke 2. Derivation and revelation: The legitimacy of mathematical models in Indian cosmology 61 K. Plofker 3. The Pythagoreans 77 R. Netz 4. Mathematics and the Divine in Plato 99 I. Mueller 5. Nicomachus of Gerasa and the arithmetic scale of the Divine 123 J.-F. Mattéi 6. Geometry and the Divine in Proclus 133 D.J. O’Meara 7. Religious architecture and mathematics during the late antiquity 147 M.-P. Terrien 8. The sacred geography of Islam 161 D.A. King 9. “Number Mystique” in early medieval computus texts 179 F. Wallis 10. Is the Universe of the Divine dividable? 201 M.-R. Hayoun 11. Mathematics and the Divine: Ramon Lull 213 C. Lohr 12. Odd numbers and their theological potential. Exploring and redescribing the arithmetical poetics of the paintings on the ceiling of St. Martin’s Church in Zillis 229 H. Garcia 13. Swester Katrei and Gregory of Rimini: Angels, God, and Mathematics in the fourteenth century 249 E.D. Sylla xi xii Contents 14. Mathematics and the Divine in Nicholas of Cusa 273 J.-M. Counet 15. Michael Stifel and his numerology 291 T. Koetsier and K. Reich 16. Between Rosicrucians and Cabbala—Johannes Faulhaber’s mathematics of Bib- lical numbers 311 I. Schneider 17. Mathematics and the Divine: Athanasius Kircher 331 E. Knobloch 18. Galileo, God and Mathematics 347 V.R. Remmert 19. The mathematical model of Creation according to Kepler 361 A. Charrak Color figures 375 20. The mathematical analogy in the proof of God’s Existence by Descartes 385 J.-M. Nicolle 21. Pascal’s views on mathematics and the Divine 405 D. Adamson 22. Spinoza and the geometrical way of proof 423 G. Harmsen 23. John Wallis (1616–1703): Mathematician and Divine 441 P. Beeley and S. Probst 24. An ocean of truth 459 C. de Pater 25. God and Mathematics in Leibniz’s thought 485 H. Breger 26. Berkeley’s defence of the infinite God in contrast to the infinite in mathematics 499 W. Breidert 27. Leonhard Euler (1707–1783) 509 R. Thiele 28. Georg Cantor (1845–1918) 523 R. Thiele 29. Gerrit Mannoury and his fellow significians on mathematics and mysticism 549 L. Bergmans 30. Arthur Schopenhauer and L.E.J. Brouwer: A comparison 569 T. Koetsier 31. On the road to a unified world view: Priest Pavel Florensky—theologian, philosopher and scientist 595 S.S.