Pythagoras Revived This page intentionally left blank Pythagoras Revived Mathematics and Philosophy in Late Antiquity Dominic J. O'Meara CLARENDON PRESS · OXFORD Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi São Paulo Shanghai Taipei Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Dominic J. O'Meara 1989 The moral rights ofthe authors have been asserted Database right Oxford University Press (maker) First published 1989 First issued in paperback 1990 All rights reserved. No part ofthis publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing ofOxfordUniversity Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope ofthe above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer British Library Cataloguing in Publication Data Data available Library ofCongress Cataloging in Publication Data O'Meara, Dominic J. Pythagoras revived. Bibliography: p. Includes index. 1. Pythagoras and Pythagorean school. 2. Mathematics—Philosophy—History. I. Title. B243.046 1989 119′.09′015 88–25246 ISBN 0–19–823913–0 (Pbk.) PATRI DILECTISSIMO SACRUM This page intentionally left blank Preface Research begun many years ago with the support ofa Junior and Visiting Fellowship at the Dumbarton Oaks Center for Byzantine Studies brought me to the texts and theses explored in this book. A sabbatical grant and generous leave ofabsence from The Catholic University ofAmerica and a grant provided in ideal circumstances by the Alexander von Humboldt-Stiftung allowed me to develop these theses. Completion of the book was made possible by the support made available by the Université de Fribourg. I am also much indebted to friends and colleagues for their assistance. Henri-Dominique Saffrey made many helpful suggestions and corrections for which I am grateful, as I am to Oxford University Press's reader for constructive and detailed comments. Ilsetraut Hadot also proposed improvements to the book, which is indebted to her research as it is to the work ofPierre Hadot, a longstanding source ofinspiration for me. Leendert Westerink and John Duffy were unfailing in their expert assistance. I am particularly grateful to Werner Beierwaltes: he put me on the track ofIamblichus and gave me invaluable advice and help during my stay in Freiburg- im-Breisgau and Munich. The deficiencies that remain in the book are ofcourse mine. My wifeCarra gave my work her full encouragement and support throughout many years. Amy Eiholzer-Silver prepared the typescripts of the final revisions ofthe book with intelligent care. In an effort to simplify footnotes as far as possible, I have adopted the practice of referring to modern works by the name ofthe author and the year ofpublication: further details may be found in the second part ofthe Bibliography. The texts ofancient authors are cited by page and line in the editions listed in the first part ofthe Bibliography (for fragments, by fragment number and line). In cases where this seemed appropriate, I have added a chapter number before the page reference (e.g. I ch. 2, 3=vol. I, chapter 2, page 3). Translations are mine, unless otherwise noted. The following standard abbreviations are used: CAG Commentaria in Aristotelem Graeca, Berlin, 1870 ff. DK H. Diels, W. Kranz, Die Fragmente der Vorsokratiker, Berlin, 1952. viii PREFACE LSJ Liddell, Scott, Jones, Greek—English Lexicon. RE Pauly, Wissowa, Kroll, Realencyclopädie der classischen Altertumswissenschaft. SVF Stoicorum veterum fragmenta, ed. H. von Arnim, Leipzig, 1905–24. This book was completed early in 1986. I have tried to include mention ofwhat has been published since, but have only been able to take account to a limited extent in particular ofthe important Paris colloquium on Proclus: Proclus, lecteur et interprète des anciens, ed. J. Pépin, H.-D. Saffrey, Paris 1987. D.J.O'M. Fribourg, Switzerland Christmas 1987 Table of Contents Introduction Part I The Revival OfPythagoreanism In the Neoplatonic School 1. Varieties ofPythagoreanism in the Second and Third Centuries AD 9 1. Numenius ofApamea 10 2. Nicomachus ofGerasa 14 3. Anatolius 23 4. Porphyry 25 2. Iamblichus' Work On Pythagoreanism: Title, Plan, the First Four Books 30 1. The Title and Overall Plan 32 2. Pythagoras (Book I: On the Pythagorean Life)35 3. Pythagorean Philosophy (Book II: the Protreptic)40 4. Pythagorean Mathematical Science (Book III: On General Mathematical Science)44 5. Arithmetic (Book IV: On Nicomachus' Arithmetical Introduction)51 3. On Pythagoreanism V – VII: The Excerpts in Michael Psellus 53 1. Michael Psellus, Philosophical Excerptor 53 2. Psellus' Excerpts from On Pythagoreanism V – VII 57 3. On Pythagoreanism V 60 (i) General Plan 60 (ii) Physical Number 62 (iii) Formal and Material Causation 62 (iv) Efficient Causation 64 (v) Change 64 (vi) Place 66 (vii) The Void 67 (viii) Conclusion 68 x CONTENTS 4. On Pythagoreanism VI 70 (i) General Plan 70 (ii) The First Principles ofEthics 71 (iii) The Powers ofthe Soul 72 (iv) Virtue 73 (v) Particular Virtues 74 (vi) Conclusion 75 5. On Pythagoreanism VII 76 (i) General Plan 76 (ii) Divine Number 79 (iii) The Approach by Analogy 81 (iv) Conclusion 84 4. Iamblichus' Work On Pythagoreanism: General Conclusions 86 1. On Pythagoreanism: A BriefReview 87 2. The Relation of On Pythagoreanism to Iamblichus' Other Works and Interests 91 3. Iamblichean Pythagoreanism and its Predecessors 101 Part II Iamblichean Pythagoreanism In the Athenian School 5. Hierocles 109 1. The Introduction ofIamblichean Philosophy at Athens in the Fourth Century 110 2. Hierocles on the History ofPhilosophy 111 3. Pythagoreanism in Hierocles' Commentary on the Golden Verses 114 4. Conclusion 118 6. Syrianus 119 1. The History ofPhilosophy in Syrianus 119 2. Philosophy as Revelation in Hermias' Commentary on the Phaedrus 124 3. Mathematics and Philosophy in Syrianus' Commentary on the Metaphysics 128 (i) Syrianus' Commentary and Iamblichus' On Pythagoreanism 128 (ii) The Tripartite Structure ofReality in Syrianus 131 CONTENTS xi (iii) Number and the Soul 132 (iv) ‘Physical Number’ 134 (v) Number in Metaphysics 135 (vi) Syrianus and Iamblichus 138 7. Proclus: Some Preliminary Issues 142 1. The Rivalry with the Mathematician Domninus ofLarissa 143 2. Plato and Pythagoras 145 3. The Theory ofSuperior Souls 149 4. Learning, Discovery, and Revelation 152 8. Proclus on Mathematics 156 1. General Mathematics 157 (i) Iamblichus' On Pythagoreanism III and Proclus' On Euclid Prol. I 157 (ii) Proclus' Revisions ofIamblichus 164 2. Arithmetic and (or?) Geometry 166 3. Geometry 170 (i) The Composition of On Euclid Prol. II 170 (ii) Euclid as Platonist 170 (iii) The Method ofGeometry 171 (iv) The Transposition ofGeometry 173 4. Conclusions 175 9. Mathematics and Physics in Proclus 177 1. Aristotle's Physics Geometricized 177 2. Plato's Timaeus as ‘Pythagorean’ Physics 179 3. The Geometrical Method ofPlato's Physics 181 4. Is Physics a Science? 182 5. The Application ofMathematical Truths in Physics 185 6. Concluding Comment 192 10. Mathematics and Metaphysics in Proclus 195 1. The Elements ofTheology : A Geometrical Metaphysics? 196 2. The Science ofDialectic 198 3. Mathematical Theology and Dialectic 204 4. Concluding Comment 208 Conclusion 210 xii CONTENTS Appendices I. The Excerpts from Iamblichus' On Pythagoreanism V – VII in Psellus: Text, Translation, and Notes 217 II. The Arabic Commentaries on the Golden Verses attributed to Iamblichus and Proclus 230 Bibliography 233 1. Ancient Authors 233 2. Modern Authors 236 Index 247 Introduction Plotinus had been dead some thirty years when his pupil Porphyry published on the threshold ofthe fourth century AD the definitive edition ofhis master's works. Porphyry's edition opened with a biography ofPlotinus, the Vita Plotini, that was intended, among other things, to introduce the reader to Plotinus as the figure ofthe ideal philosopher. Inspired by this model and by the wisdom and otherworldly serenity that it promised, the reader would move on to read the Plotinian tractates that followed. These tractates Porphyry divided and grouped in six sets of nine works (‘Enneads’) and arranged the sets in such a way that their sequence led the mind ofthe reader up from the material world, through ever higher levels, to the source ofall reality, the One. Thus Porphyry's edition, taken as a whole ( Vita and Enneads), constituted a systematic introduction, initiation, and path to the highest truths ofphilosophy. Porphyry's edition could give his contemporaries every reason to feel confident in the strength and promise oftheir ancient philosophical culture: Plotinus had brought together with considerable success the many philosophical tendencies ofpreceding centuries into a whole that came astonishingly close to Plato, ifnot