Mathematical Theologies: Nicholas of Cusa and the Legacy of Thierry of Chartres (Oxford Studies in Historical Theology) [1 ed.] 9780199989737, 9780199384907, 9780199384914, 0199989737

Table of contents :
Cover
Mathematical Theologies
Series
Copyright
Contents
Acknowledgments
Abbreviations
Introduction
Cusanus Studies and “Modernity”
Rethinking the Mathesis Narrative
Thierry of Chartres as a Cusan Source
Some Notes on Method
PART ONE The Genesis of Neopythagoreanism
1 Platonic Transformations of Early Pythagorean Philosophy
Mathematics as Philosophy in Philolaus and Archytas
Mathematics as Mediation in Plato
Mediation and First Philosophy in the Early Academy
2 The Neopythagorean Revival: Henology and Mediation
The Origins of Henology in Eudorus and Moderatus
Henology on the Margins of Middle Platonism
Mathematical Theology in Nicomachus of Gerasa
3 The Late Antique Preservation of Neopythagoreanism
Iamblichus, Proclus, and the Legacy of Nicomachus
Augustine and the Number without number
Boethius and the Fate of the Quadrivium
PART TWO The Pearl Diver
4 Thierry’s Trinitarian Theology in Context
The Status of Mediation in Twelfth-Century Platonism
The Problem of Bernard’s Gloss
Thierry on Quadrivium and Trinity
5 The Discovery of the Fold
Attempts at a Universal Theory of Science
The Achievement of the Modal Theory
Thierry as Neopythagorean Theologian
6 Thierry’s Diminished Legacy
Confusion about Mediation
An Augustinian Censor
A Late-Medieval Refutation: Word or Number?
PART THREE Bright Nearness
7 The Accidental Triumph of De docta ignorantia
A Patchwork of Conflicting Sources
Experiments in Chartrian Theology
The Christological Double Synthesis
8 Chartrian Theology on Probation in the 1440s
An Agenda for the 1440s in Two Sermons
The Neopythagorean Counterexperiment
Two Paradigms of Mediation
9 The Advent of Theologia ­geometrica in the 1450s
The Restoration of Thierry’s Modal Theory
A New Foundation for Mathematical Theology
The Word as Number and Angle
10 Completing the Circle in the 1460s
New Impulses in the Late Works
Incarnation and Neopythagoreanism
Figurae mundi
Epilogue
Notes
Bibliography
Index

Citation preview

Mathematical Theologies

OXFORD STUDIES IN HISTORICAL THEOLOGY Series Editor David C. Steinmetz, Duke University Editorial Board Irena Backus, Université de Genève Robert C. Gregg, Stanford University George M. Marsden, University of Notre Dame Wayne A. Meeks, Yale University Gerhard Sauter, Rheinische Friedrich-Wilhelms-Universität Bonn Susan E. Schreiner, University of Chicago John Van Engen, University of Notre Dame Geoffrey Wainwright, Duke University Robert L. Wilken, University of Virginia MARTIN BUCER’S DOCTRINE OF JUSTIFICATION Reformation Theology and Early Modern Irenicism Brian Lugioyo CHRISTIAN GRACE AND PAGAN VIRTUE The Theological Foundation of Ambrose’s Ethics J. Warren Smith KARLSTADT AND THE ORIGINS OF THE EUCHARISTIC CONTROVERSY A Study in the Circulation of Ideas Amy Nelson Burnett READING AUGUSTINE IN THE REFORMATION The Flexibility of Intellectual Authority in Europe, 1500-1620 Arnoud S. Q. Visser

SHAPERS OF ENGLISH CALVINISM, 1660-1714 Variety, Persistence, and Transformation Dewey D. Wallace, Jr. THE BIBLICAL INTERPRETATION OF WILLIAM OF ALTON Timothy Bellamah, OP MIRACLES AND THE PROTESTANT IMAGINATION The Evangelical Wonder Book in Reformation Germany Philip M. Soergel THE REFORMATION OF SUFFERING Pastoral Theology and Lay Piety in Late Medieval and Early Modern Germany Ronald K. Rittgers CHRIST MEETS ME EVERYWHERE Augustine’s Early Figurative Exegesis Michael Cameron MYSTERY UNVEILED The Crisis of the Trinity in Early Modern England Paul C. H. Lim GOING DUTCH IN THE MODERN AGE Abraham Kuyper’s Struggle for a Free Church in the Netherlands John Halsey Wood, Jr. CALVIN’S COMPANY OF PASTORS Pastoral Care and the Emerging Reformed Church, 1536-1609 Scott M. Manetsch THE SOTERIOLOGY OF JAMES USSHER The Act and Object of Saving Faith Richard Snoddy

Mathematical Theologies Nicholas of Cusa and the Legacy of Thierry of Chartres

z DAVID ALBERTSON

1

1 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. Oxford New York Auckland  Cape Town  Dar es Salaam  Hong Kong  Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trademark of Oxford University Press in the UK and certain other countries. Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016

© Oxford University Press 2014 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, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by license, or under terms agreed with the appropriate reproduction rights organization. Inquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above. You must not circulate this work in any other form and you must impose this same condition on any acquirer. Library of Congress Cataloging-in-Publication Data Albertson, David. Mathematical theologies : Nicholas of Cusa and the legacy of Thierry of Chartres / David Albertson. pages cm. — (Oxford studies in historical theology) Includes bibliographical references and index. ISBN 978–0–19–998973–7 (hardcover : alk. paper) — ISBN 978–0–19–938490–7 (updf) — ISBN 978–0–19–938491–4 (online content)  1.  Religion and science.  2.  Mathematics—Philosophy.  3.  Pythagoras—Influence.  4. Nicholas, of Cusa, Cardinal, 1401–1464.  5.  Thierry, de Chartres, approximately 1100-approximately 1150.  I. Title. BL265.M3A53 2014 261.5'5—dc23 2014000014

1 3 5 7 9 8 6 4 2 Printed in the United States of America on acid-free paper

To my wife, Annie Tread softly

Contents

Acknowledgments

ix

Abbreviations

xi

Introduction: Toward a Genealogy of Christian Neopythagoreanism

1

PART ONE:  The Genesis of Neopythagoreanism: A Synopsis 1.  Platonic Transformations of Early Pythagorean Philosophy

23

Mathematics as Philosophy in Philolaus and Archytas Mathematics as Mediation in Plato Mediation and First Philosophy in the Early Academy

27 30 35

2.  The Neopythagorean Revival: Henology and Mediation

The Origins of Henology in Eudorus and Moderatus Henology on the Margins of Middle Platonism Mathematical Theology in Nicomachus of Gerasa 3.  The Late Antique Preservation of Neopythagoreanism

Iamblichus, Proclus, and the Legacy of Nicomachus Augustine and the Number without number Boethius and the Fate of the Quadrivium

40

41 45 50 60

62 68 80

PART TWO:  The Pearl Diver: Thierry of Chartres’s Theology of the Quadrivium 4.  Thierry’s Trinitarian Theology in Context

The Status of Mediation in Twelfth-Century Platonism The Problem of Bernard’s Gloss Thierry on Quadrivium and Trinity

93

95 100 107

Contents

viii

5.  The Discovery of the Fold

119

Attempts at a Universal Theory of Science The Achievement of the Modal Theory Thierry as Neopythagorean Theologian 6.  Thierry’s Diminished Legacy

121 126 132 140

Confusion about Mediation An Augustinian Censor A Late-Medieval Refutation: Word or Number?

145 149 156

PART THREE:  Bright Nearness: Nicholas of Cusa’s Mathematical Theology 7.  The Accidental Triumph of De docta ignorantia

169

A Patchwork of Conflicting Sources Experiments in Chartrian Theology The Christological Double Synthesis

175 180 190

8.  Chartrian Theology on Probation in the 1440s

199

An Agenda for the 1440s in Two Sermons The Neopythagorean Counterexperiment Two Paradigms of Mediation

202 206 217

9.  The Advent of Theologia geometrica in the 1450s

The Restoration of Thierry’s Modal Theory A New Foundation for Mathematical Theology The Word as Number and Angle 10.  Completing the Circle in the 1460s

New Impulses in the Late Works Incarnation and Neopythagoreanism Figurae mundi

222

228 236 243 253

255 261 267

Epilogue

277

Notes

281

Bibliography

407

Index

467

Acknowledgments

As Cusanus would have appreciated, writing this book has seemed an infinite task that has served to illuminate my own finitude. For that reason and for many others I am deeply grateful to everyone who supported me along the way. David Tracy, Bernard McGinn, and Jean-Luc Marion guided the doctoral work that first pointed toward this larger project, and I hope that their influence is clear enough in what follows. In the same vein I want to thank Amy Hollywood, Kathryn Tanner, Susan Schreiner, Brent Sockness, John Webster, Arnie Eisen, and Sepp Gumbrecht for their generosity and commitment as teachers, virtues that I have since learned are rare and hard won. I first sketched plans for the book at the Thomas-Institut at the Universität zu Köln in 2006–7. Andreas Speer and Hans Gerhard Senger were wonderful hosts and advisors there. In the world of Cusanus studies I  owe thanks to the members of the American Cusanus Society; to Walter Euler and the Institut für Cusanus-Forschung; and to the late H. Lawrence Bond and Morimichi Watanabe for their exemplary scholarship. Peter Casarella, Jerry Christianson, Don Duclow, and Lee Miller all provided valuable feedback on early chapter drafts. Phillip Horky of Durham University provided expert guidance through the terra incognita of early Pythagoreanism and kindly allowed me to preview chapters from his Plato and Pythagoreanism (Oxford: Oxford University Press, 2013). All mistakes, of course, remain my own. At the University of Southern California I have benefited from the Advancing Scholarship in Humanities and Social Sciences initiative sponsored by the Office of the Provost, from the various activities and financial support of the Interdisciplinary Research Group of the Center for Religion and Civic Culture, from dialogue with all of my wonderful colleagues in the School of Religion, and particularly from the support of Lisa Bitel, Don Miller, Duncan Williams, and Peter Mancall. At Oxford University Press I thank Cynthia Read, Stuart Roberts, and Michael Durnin for their dedication to the project.

x

Acknowledgments

I also thank my parents, Rick and Nancy, for more than I can record; as well as my brother, Andrew; my extended family, Jack, Chris, and Andy; and especially my children, Gabriel and Natalie. My greatest debt is to Annie for her love, encouragement, and patience over the last fifteen years. This book, at long last, is dedicated to her. Los Angeles August 2013

Abbreviations

AHDLMA

Archives d’histoire doctrinale et littéraire du moyen âge. Paris: J. Vrin, 1926–. CCCM Corpus Christianorum. Continuatio Mediaevalis. Turnhout: Brepols, 1970–. CCSL Corpus Christianorum. Series Latina. Turnhout: Brepols, 1954–. Commentum Thierry of Chartres. Commentum super Boethii librum de Trinitate. In Commentaries on Boethius by Thierry of Chartres and His School, ed. Nikolaus M. Häring, 55–116. Toronto: Pontifical Institute of Mediaeval Studies, 1971. [= Librum hunc] CSEL Corpus Scriptorum Ecclesiasticorum Latinorum. Vienna: Hölder-Pichler-Tempsky, 1866–. DC Nicholas of Cusa. De coniecturis. Opera omnia iussu et auctoritate Academiae Litterarum Heidelbergensis, vol. 3, ed. Josef Koch and Karl Bormann. Hamburg: Felix Meiner, 1972. DI Nicholas of Cusa. De docta ignorantia. In Nikolaus von Kues. Philosophisch-Theologische Werke, vol. 1, ed. Paul Wilpert and Hans Gerhard Senger. Hamburg: Felix Meiner, 2002. DM Nicholas of Cusa. Idiota de mente. Opera omnia iussu et auctoritate Academiae Litterarum Heidelbergensis, vol. 5, ed. Renate Steiger and Ludwig Baur. Hamburg: Felix Meiner, 1983. F Fundamentum naturae quod videtur physicos ignorasse. Eichstätt Cod. st 687, fols. 4r–10r. Glosa Thierry of Chartres. Glosa super Boethii librum de Trinitate. In Commentaries on Boethius by Thierry of Chartres and His School, ed. Nikolaus M. Häring, 257–300. Toronto: Pontifical Institute of Mediaeval Studies, 1971. [= Anonymous Berolinensis]

xii

H

Abbreviations

Häring, Nikolaus M., ed. Commentaries on Boethius by Thierry of Chartres and His School. Toronto: Pontifical Institute of Mediaeval Studies, 1971. IA Boethius. Institution Arithmétique, ed. Jean-Yves Guillaumin. Paris: Belles Lettres, 2002. Lectiones Thierry of Chartres. Lectiones in Boethii librum de Trinitate. In Commentaries on Boethius by Thierry of Chartres and His School, ed. Nikolaus M. Häring, 123–229. Toronto: Pontifical Institute of Mediaeval Studies, 1971. [= Quae sit] LG Nicholas of Cusa. Dialogus de ludo globi. Opera omnia iussu et auctoritate Academiae Litterarum Heidelbergensis, vol. 9, ed. Hans Gerhard Senger. Hamburg: Felix Meiner, 1998. MFCG Mitteilungen und Forschungsbeiträge der Cusanus-Gesellschaft MM Miscellanea Mediaevalia PL Patrologiae Cursus Completus, Series Latina. Paris: J. P. Migne, 1844–64. Septem (Ps.-) John of Salisbury. De septem septenis. PL 199: 945–964. TC Nicholas of Cusa. De theologicis complementis. Opera omnia iussu et auctoritate Academiae Litterarum Heidelbergensis, vol. 10/2a, ed. Heide D. Riemann and Karl Bormann. Hamburg: Felix Meiner, 1994. Tractatus Thierry of Chartres. Tractatus de sex dierum operibus. In Commentaries on Boethius by Thierry of Chartres and His School, ed. Nikolaus M. Häring, 553–575. Toronto: Pontifical Institute of Mediaeval Studies, 1971.

Mathematical Theologies

Introduction TOWARD A GENEALOGY OF CHRISTIAN NEOPYTHAGOREANISM

Nicholas of Cusa (1401–1464) was a canon lawyer, bishop, and cardinal who spent his life defending papal interests and pursuing reforms throughout the German lands. Somehow amidst his peregrinations he found time to write not only two dozen treatises on mystical theology and Platonist philosophy and nearly three hundred sermons, but also, surprisingly, a dozen books filled with speculative geometrical proofs. But unlike others gifted in both fields (one thinks of Robert Grosseteste, Nicole Oresme, or Thomas Bradwardine), Cusanus viewed his theological and mathematical explorations as belonging to the same integrated intellectual enterprise. Even in his day this was an unusual thing to do. Bernard of Clairvaux imagined his breasts swelling with milk like Mary, Julian of Norwich visualized the blackened blood of the dying Jesus, and Mechthild of Magdeburg witnessed her soul disrobing in the Lord’s chamber. By contrast, Cusanus designed austere geometrical devices to guide his contemplation, like a triangle bisected by a sweeping line or a circle rotating with infinite speed. He revered the equation 1 × 1 = 1 as a sublime name of God. Nicholas was even convinced that if he could square the circle—an ancient geometrical riddle—his solution would uncover the hidden ratio of human (linear) and divine (curved) minds. Today Nicholas’s mathematical theology may seem a bizarre chimera, but for early modern readers from Johannes Kepler to Athanasius Kircher to John Dee, the profound works of the Cardinalis teutonicus aired a new theology for a new age.1 To introduce his Parisian edition of Cusanus’s works in 1514, the great humanist Jacques Lefèvre d’Étaples concluded that “Mathesis is therefore great, but especially because it does not fail to provide a way to ascend to the divine.”2 Giordano Bruno could scarcely contain his praise: “Good God—how is even Aristotle comparable to this Cusanus, who is as greater than him as he is accessible to only a few? If his priestly outfit did not prevent me, I  would readily acknowledge his mind to be not just the equal of Pythagoras, but far superior.”3 Gottfried Leibniz may have dismissed Cusanus’s geometrical proofs out of hand, but he seconded

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his agenda: “It seems that God, when he bestowed these two sciences [arithmetic and algebra] on humankind, wanted to warn us that a much greater secret lay hidden in our intellect, of which these were but shadows.”4 On the other hand, not everyone was enthused about mathematics infiltrating theology. In his Invective against Cusanus, the famed German humanist Gregor Heimburg charged that the cardinal had sought “to demonstrate with mathematical superstitions the sacred things of the true religion.”5 And in a disputation in 1539, Martin Luther declared that “mathematics is theology’s greatest enemy of all, since there is no part of philosophy that so fights against theology.”6 Luther’s provocative opposition of the two disciplines continues under other guises today. Martin Heidegger considered mathesis universalis as the source of metaphysical oblivion in philosophy and the enemy of thinking. Conversely, Alain Badiou has proposed that the best means for completing the Nietzschean death of God is to secularize infinity through a renewed mathematical Platonism.7 Students of the history of Christianity and the history of philosophy recognize Cusanus as an indispensable figure in what Louis Dupré memorably called “the passage to modernity.”8 But because the years of his life seem to stretch from one age into another, locating Cusanus more precisely presents a challenge. Is Nicholas a late medieval author or an early modern author? On a first read the cardinal’s works do seem haunted by spirits from both eras—by Ps.-Dionysius, John Scotus Eriugena, and Meister Eckhart as much as by eerie adumbrations of Copernicus, Descartes, Spinoza, and even Kant. But after a moment’s reflection, the question loses its luster; the shorthand of historical periods is not so important. Most scholars would now agree that Nicholas is best viewed as an independent-minded late medieval author, who may resemble the Florentine Platonists in charting a new way forward for the fifteenth century, but who is rather more indebted to the Albertist school and Rhineland mysticism, even if there are also unmistakable “parallelisms” with the succession of modern German philosophers from Leibniz to Hegel.9 But to conclude that Cusanus is neither simply medieval nor modern does not yet, I have found, exhaust the question’s utility. My attempt to refine this question and hear what it is asking resulted in the present study. There is a valuable element of the original question that survives this “correct” answer unthought. Despite over a century of research there is more work to be done to situate Cusanus within the geographical and intellectual locales in which he wrote and to resist the temptation to isolate him as singularity or prophet. But even sober judges see Nicholas as the “gatekeeper of modernity” (Rudolf Haubst) standing on its “threshold” (Hans Blumenberg). When this liminal duality is not parsed into medieval and modern periods, it reappears under the guise of contending biographical types:  the loyal theologian versus the freethinking philosopher, the powerful cardinal versus the curious scientist, or the cold Rhineland

Introduction

3

shadows versus the warm Italian light. Writing about Cusanus means keeping a constant vigil against such ghosts of modernity. Naturally, different assessments of the cardinal’s epochal location emphasize different aspects of his thought. Seventeenth-century science provides a congenial background for viewing the cardinal as forerunner of the modern. Nicholas’s claim that all knowledge is mathematical encourages this interpretation, as does his passion for geometry, astronomical conjectures, and theories of measurement. On the other hand those who consider him primarily a late medieval bishop and preacher orient their readings by his frequent discussions of Trinity, Incarnation, Church, and spiritual life. The fact is that Cusanus held together a robust, Christian theological vision of the cosmos alongside the very mathematizing epistemology which, according to some theories of modernity, should have weakened, qualified, or worked against that vision. But that tension exists for us, not for him. His goal was simply to explore the theological meaning of mathematical measurement as a radically unified theme. My aim in this book is to perceive that unity and to articulate it despite the historiographical obstacles that sometimes block our view.

Cusanus Studies and “Modernity” Nicholas’s works provoke these dilemmas in part because of the circumstances of their rediscovery around the turn of the twentieth century. This was a time of instability and change in continental European philosophy. Charles Bambach has written that “the ‘legitimation crisis’ of Germany philosophy between 1880 and 1930” encourages the historian to draw parallels “between the latter stages of modernity in the postwar consciousness of crisis and the origins of modernity in the Cartesian project of scientific certitude.”10 The decades from 1900 to 1940 saw the split of the analytic and continental schools, the rise of phenomenology, existentialism, and psychoanalysis, and debates over “Theologie der Krise” in Germany and “la nouvelle théologie” in France. At the heart of these controversies lay the attempt to identify the origins of the problematic modernity that Europe inhabited and thus trace the path that had delivered it into such straits. Cusanus’s writings were first promoted by the Catholic theologian Johann Adam Möhler from the so-called Tübingen School. But things really took off among a circle of Neo-Kantian philosophers led by Hermann Cohen at Marburg and Heinrich Rickert at Heidelberg.11 Cohen sought to ground philosophical method on the mathematics of infinitesimals, but also formulated an inventive philosophy of religion studied closely by Franz Rosenzweig, Martin Buber, and Karl Barth.12 In 1914 Cohen hailed Nicholas of Cusa, not Descartes, as the true “father of modern philosophy” on the account of the cardinal’s insights into mathematical epistemology.13 Hermann Löb, a student of Cohen and Rickert, wrote that his

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teachers shared with Cusanus the conviction that “mathematics and metaphysics not only run next to each other but pass into each other.”14 Ernst Cassirer, Cohen’s most prominent disciple, studied Cartesian mathematics in his doctoral research and went on to write a magisterial history of scientific epistemology, whose first volume in 1906 praised Nicholas as the first modern philosopher.15 Around the same time, Pierre Duhem was also writing on Cusanus in the second volume of his Ètudes sur Léonard de Vinci, which along with Le système du monde demonstrated that medieval science was an indispensable precursor of the seventeenth century.16 In 1909 Duhem published a short article showing (for the first time) that Nicholas had borrowed some of his leading concepts from the twelfth-century Parisian humanist Thierry of Chartres (d. 1157).17 He did not hesitate to accuse the German cardinal of “plagiarizing” from the learned French master, as if in reprisal for Descartes’s lost title. Nationalism aside, Duhem’s discovery also curiously implied that protomodern ideas were afoot in the twelfth century. The decision to found a modern critical edition of Cusanus and enable more careful study of his works began within this Neo-Kantian network. Cohen himself was supposed to oversee the first edition, but died in 1918 before doing so. The Great War prevented further work until the mid-1920s, when two students of the Marburg Neo-Kantians, Ernst Hoffmann and Raymond Klibansky, finally began work on the edition. In one of his Heidelberg seminars on medieval Platonism in 1927, Hoffmann deputized the younger Klibansky to investigate the state of Nicholas’s manuscripts. In the same year Hoffmann’s teacher Cassirer published his brilliant introduction to Cusanus set against the background of the Italian Renaissance, where he repeated the claim that it was his “position on the problem of knowledge [that] marks Cusanus as the first modern thinker.”18 To make his case, Cassirer appended the text of one of Nicholas’s philosophical works, the 1450 dialogue Idiota de mente, even before all of the manuscripts had been collated. In so doing he linked together Cusanus’s protomodernity, his mathematizing epistemology, and the particular work De mente—a decidedly Neo-Kantian nexus that has deeply influenced modern Cusanus studies.19 While Hoffmann consulted with Rickert about the planned edition, Klibansky objected to Cassirer’s premature publication of De mente.20 Klibansky’s preliminary research on De docta ignorantia was leading him toward the conclusion that it was ill advised for Cassirer and Hoffmann to call Nicholas the first “modern philosopher,” given the cardinal’s preoccupations with medieval Platonists like Thierry of Chartres.21 Hence the initiative of the critical edition was also a means for resolving the contest over Cusanus’s epochal identity through sounder philological analysis, with a special view toward twelfth-century sources. The first volume of the edition, the great 1440 treatise De docta ignorantia, appeared in 1932. During these years, the same Neo-Kantian circles at Marburg and Heidelberg were also the matrix for a momentous rethinking of philosophical “modernity.”

Introduction

5

While Hoffmann and Klibansky set to work on the text of Cusanus’s De docta ignorantia, Edmund Husserl was busy testing a new interpretation of the historical conditions governing the emergence of the modern episteme. It is no accident that like his colleagues Cohen and Cassirer, Husserl began with the “mathematization of nature” in Galileo and Descartes.22 While not (so far as I know) a reader of Nicholas of Cusa, Husserl belonged to the same networks and shared similar interests. His earliest research concerned the philosophy of mathematics.23 When Rickert moved to Marburg to succeed Wilhelm Windelband in 1916, Husserl took his vacant chair at Freiburg and taught there until his retirement in 1928. Martin Heidegger was Husserl’s assistant there until he took the chair at Marburg in 1923, joining the Cohen protégés Paul Natorp and Nicolai Hartmann. Husserl remained in frequent correspondence with Rickert, Natorp, and Cassirer throughout his life. Finally, at the famed Davos disputation in 1929 Heidegger debated Cohen’s reading of Kant with none other than Ernst Cassirer.24 To appreciate the theoretical significance of Cusanus’s slippery epochal position, we cannot avoid a brief digression on the account of modern origins shared by Husserl and Heidegger. Husserl spent the last years of his life working intensely on a new problem. His acute concern over the future of the modern sciences and their detachment from the Lebenswelt of immediate experience first appears in an unpublished typescript fragment around 1928, “The Science of Reality and Idealization:  The Mathematization of Nature.”25 According to Husserl, empirical sciences will inevitably separate themselves from the sensible reality they analyze. Ever since the seventeenth century, grasping the world mathematically has meant constructing a priori “pure forms of generality,” namely, continua of identical units of magnitude. Such universal measures allow the sciences to anticipate sensible givens before they are experienced, and thus to hypothesize, compare, and test. But they also engender the troubling paradox that mathematical precision drives the sciences away from lived experience toward ideal geometrical models of their own creation.26 In lectures at Vienna and Prague in 1935, Husserl expanded this methodological conundrum into a broader indictment of the cultural role of the sciences in late modernity and its consequences for philosophy. These culminated the following year in Die Krisis der Europäischen Wissenschaften und die transzendentale Phänomenologie, his last book. As Husserl hurried through revisions to the Krisis, he inserted a long essay on “Galileo’s Mathematization of Nature” after the other chapters had already gone to press. This was the longest section of the Krisis and the one closest to his original essay in 1928.27 There he explains in greater detail how Galilean techniques of measurement eventually give rise to a constellation of “limit-shapes,” which progressively alienate one from the “sensible plenum” (sinnliche Fülle). Mathematical measurement ends up as the sole index of the reality of a thing, and yet the universal knowledge it promises is knowledge of a virtual world simplified

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to geometrical idealities. By focusing on Galileo, Husserl was simply following the example of the Neo-Kantians, since Cohen, Natorp, and Cassirer had already noticed the astronomer’s sympathies with Platonism.28 But Husserl may have also been influenced during this period by his former student Alexandre Koyré, who was keenly interested in Galileo’s Platonism.29 Building on Husserl’s Krisis, Koyré argued in his 1939 Études galiléennes that Galileo’s mathematical Platonism, more than his empirical experiments, was the crux of the transition to modernity.30 For it was the geometrization of the universe in Galileo and Descartes that began to dissolve the fantastical medieval cosmos and weaken its religious and social structures: It is for our purpose sufficient to describe . . . the mental or intellectual attitude of modern science by two (connected) characteristics. They are: (1) the destruction of the Cosmos, and therefore the disappearance of all considerations based on that notion; (2) the geometrization of space—that is, the substitution of the homogeneous and abstract space of Euclidean geometry for the qualitatively differentiated and concrete world-space conception of the pre-Galilean physics. These two characteristics may be summed up and expressed as follows: the mathematization (geometrization) of nature and, therefore, the mathematization (geometrization) of science.31 Koyré was quick to add that despite appearances, Galileo’s Platonist mathematics were superior to the “Neo-Pythagorean arithmology” of the Florentine Academy, indeed, he said, greater than Iamblichus and Proclus themselves.32 Yet even Koyré had to concede that Galileo affirmed some generally Pythagorean views. In The Assayer (1623), for example, Galileo famously wrote that the world is composed of number: Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.33 Less well known is another passage cited by both Koyré and Cassirer.34 In the Dialogue Concerning the Two Chief World Systems (1632), Galileo describes God as a superlative mathematician. God’s Wisdom consists in perfectly knowing all the proofs and propositions of arithmetic and geometry, intuitively and

Introduction

7

instantaneously. Human wisdom shares the same “intensive” degree of mathematical certainty of truth, but must slowly reason from one proposition to another without God’s “extensive” perfection. This entails that mathematical reasoning is the highest trace of divine presence and that the human intellect participates in God when it understands numbers.35 Like Cassirer and Husserl, Koyré drew studied comparisons between Galileo and the indisputably modern Descartes, the better to accentuate the great Paduan’s distance from the late medieval precursors praised by Duhem. It is true that when the young Descartes learned of Galileo’s condemnation in 1633, he decided not to publish his early reflections in Regulae ad directionem ingenii (1628), which only appeared posthumously in 1701. The universal mathematics (mathesis universalis) that Descartes outlined in the Regulae set the agenda for his entire philosophical project.36 First he distinguished the “outer garments” of mathematics (the particular work of geometry and arithmetic) from its “inner parts” common to both disciplines and so prior to them. This “general investigation of mathematics” or “science of pure mathematics” would be capable of producing a powerful universal method for certain knowledge in any science.37 Then follows the critical passage in Regula IV: I began my investigation by inquiring what exactly is meant by the term mathesis and why it is that, in addition to arithmetic and geometry, sciences such as astronomy, music, optics, mechanics among others, are called branches of mathematics. To answer this it is not enough just to look at the etymology of the word, for, since the word mathesis has the same meaning as disciplina, these subjects have as much right to be called “mathematics” as geometry has. . . . When I considered the matter more closely, I came to see that the exclusive concern of mathematics is with questions of order or measure and that it is irrelevant whether the measure in question involves numbers, shapes, stars, sounds, or any other object whatever. This made me realize that there must be a general science which explains all the points that can be raised concerning order and measure irrespective of the subject-matter, and that this science should be termed mathesis universalis—a venerable term with a well-established meaning—for it covers everything that entitles these other sciences to be called branches of mathematics.38 Note that Descartes defined mathesis by recalling the sciences of the medieval quadrivium. Then just as Husserl describes, Descartes’s universal method systematically traded in one kind of knowledge, the empirical sense experience of motion and quality, for another judged more certain, the ideal geometrical space of order and measure. In the fifth of his Meditationes, Descartes used this concept

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to ground all knowledge of the external world, “the whole of that corporeal nature which is the object of pure mathesis.”39 If Husserl and Koyré are “the two key-figures in the emergence of the grand narrative of mathematization of nature,” as Sophie Roux maintains, they were not its greatest publicists.40 That honor belongs to Martin Heidegger, whose critique of modern metaphysics and technology built directly upon Husserl’s Neo-Kantian narrative.41 In the same year that Husserl was revising his Galileo chapter, Heidegger lectured on Kant at Freiburg. Apparently influenced by his teacher, Heidegger added a lengthy digression on the mathematization of nature, arguing that the Cartesian mathesis, also present in Galileo and Newton, was the essence of modern science. “The mathematical,” writes Heidegger, “is that evident aspect of things within which we are always already moving . . . this fundamental position we take toward things by which we take up things as already given to us.”42 Galileo’s experiments became mathematical as soon as he viewed natural events qua calculability; and Cartesian doubt and the cogito were direct results of Descartes’s pursuit of universal mathematics in the Regulae.43 Heidegger concluded that philosophy’s captivity to metaphysics and the concomitant “loss of the gods” both arise from this elemental attitude of mathesis.44

Rethinking the Mathesis Narrative Let us call this Neo-Kantian narrative of modernity’s origins—common to Husserl, Koyré, Cassirer, and Heidegger, and centered on the dramatic leap into a mathematized or geometrized vision of the cosmos by Galileo and Descartes—the mathesis narrative for short. I have devoted a few pages to sketch its development in the 1920s and 30s for the reason that this narrative often continues to inform, sometimes unconsciously, contemporary accounts of European modernity. Even when theorists seek an index of early modernity beyond the natural sciences or technological developments, say in political formations or conceptions of the individual, it is difficult to dispense altogether with the narrative’s basic tenet: that because an altered (modern) vision of nature has discredited (medieval) religious cosmologies, the way is freshly cleared for an autonomous new foundation unhindered by the habits of the past. This is not to suggest, of course, that there is no such thing as modernity, but only to point out that the initial form of Husserl’s mathesis narrative has several defects. It is not so much incorrect as gravely incomplete, and until its vulnerabilities are addressed it seems unwise to build too highly upon its sand.45 What the mathesis narrative lacks, above all, is a greater degree of historical differentiation that might qualify the otherwise “sudden” breakthrough of the seventeenth century. For example, among historians of science, the centrality of Galileo

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and Descartes in the rise of mathematical physics has long been discredited. Husserl and Heidegger formulated their theory a decade before Anneliese Maier’s studies were published between 1949 and 1955. Maier’s groundbreaking research in the Vatican archives updated Duhem’s work on medieval natural science that had been so criticized by Koyré for underappreciating Galileo.46 Maier demonstrated and others have confirmed in detail that the mathematization of quality and motion described by Husserl was well underway among fourteenth-century scholastics at Paris and Oxford.47 Likewise, historians of philosophy have since discovered multiple antecedents of Descartes’s envisioned mathesis universalis in the previous century, when philosophers around Europe began to study Proclus’s remarkable commentary on Euclid’s Elements.48 Some have even credited late antique Neoplatonism, Neopythagoreanism, Middle Platonism, or the Old Academy itself with achieving aspects of the Cartesian mathesis avant (or après) la lettre; after all, the possibility of universal mathematics had been raised specifically by Aristotle.49 This genealogical research, of course, not only risks anachronism if one neglects semantic differences between Descartes’s mathesis and that of antiquity, but also risks mistaking the channels by which ancient Greek models actually found their way into the seventeenth century.50 Beyond these faults there is another, more serious deficit in the mathesis narrative unaddressed by historians of science and philosophy. It fails to consider the pre-Cartesian history of the notion of universal mathesis between Proclus and Galileo, and especially within the Latin Christian traditions that decisively shaped the discourse of medieval and early modern philosophy in the intervening ten centuries.51 This lacuna prevents one from seeing how mathematizing (Pythagorean) traditions were transmitted organically within the spectrum of medieval Christian Platonisms, and particularly within the Boethian tradition. Without this longue durée one can miss the subtle overlap and exchange that arose over time across theological, epistemological, mathematical, and natural scientific discourses— what Amos Funkenstein calls the “dialectical anticipation of a new theory by an older, even adverse, one; and the transplantation of existing categories to a new domain.”52 So before asserting the novelty of universal mathematical philosophies in early modernity, one should ask whether Christian traditions had made similarly universal arguments within the domain of theology at any time in the thousand years prior, and to what extent such medieval authors had contemplated the theological possibilities of mathematical discourse. It is probably safe to assume that this query did not seem germane to the Neo-Kantians, Husserl, or Heidegger. For these reasons it seems to me that Husserl’s narrative is in need of a very specific supplement:  a history of universal mathesis that does not rush back to the Greeks but turns its attention to more proximate, unexpected, even scandalous intersections with western Christian theologies between late antiquity and the Renaissance.

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Historically speaking, Descartes’s term (as he acknowledged) simply refurbished the medieval theory of the quadrivium, the “fourfold way” of arithmetic, music, geometry, and astronomy. Before its tenure as four of the seven liberal arts, the quadrivium as originally conceived for Latin Christianity by Boethius named the universal principles of first philosophy, or fundamental theology. Boethius in turn was only translating the henological ideas of Greek Neopythagoreans from the first few centuries of the present era, and in fact mathesis goes back to the leading Pythagoreans before Plato.53 The history of Pythagoreanism is the prehistory of the quadrivium; the quadrivium’s history unfolded within the long, pluriform doctrinal and pedagogical ambit of medieval Christianity; and only the latter could possibly provide the requisite context for gauging the novelty and significance of an apparently “sudden” seventeenth-century mathematization. Had they the historical resources at their disposal that we presently enjoy, Husserl and Heidegger should have attempted to study the appearances of mathesis within European philosophical and religious thought before the seventeenth century. What they should have pursued, in short, is a genealogy of Christian Neopythagoreanism. Historians of philosophy are well aware of a Christian Aristotelianism or Christian Platonism, indeed even a Christian Stoicism or Skepticism. But who has ever heard of Christian Neopythagoreanism?54 In researching this book I was surprised to discover that due to a series of contingent discursive accidents, the dialogue between this particular religion and this particular antique tradition, unlike the other permutations, never fully took place. This realization sheds light on the flatfootedness with which Christian theologians met the seventeenth century. It also suggests why Christian Neopythagoreanism sounds more preposterous prima facie than say Christian Stoicism or Christian Neoplatonism. Let us suppose for a moment that modernity were indeed distinguished by a scientific rationality that liberates the mind from the caprice of religion through the mathematization of natural knowledge. If there were an antique school that foregrounded mathematics and thus foreshadowed the modern mathematization of knowledge, and if we consider robust theological visions of the world as inherently premodern, then the most difficult eventuality for us to conceive would be a thoroughly Christian (or Jewish or Islamic) Neopythagoreanism, a theology that at its center grasps not only the world, but also the destiny of the soul and the names of God, in terms of number, quantity, and geometry. This would be the species of thought least comprehensible by the mathesis narrative because the most inherently opposed to its prejudices. There is no other premodern author who so fully took on the enterprise of a Christian Neopythagoreanism as Nicholas of Cusa (provided we understand “Neopythagoreanism” correctly). And this, I think, finally explains why the question of Nicholas’s epochal position is so vexing and ultimately unanswerable, at least in terms that oppose theology to the exact sciences. If we assume that

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modernity is constituted by the emergence of a “mathematized” method, then the question of Cusanus’s modernity would simply amount to evaluating his scientific epistemology, however medieval his mystical theology might remain. But if, under closer inspection, it turns out that Cusan theology is above all mathematical theology, then when it is given its proper due, it will resist the theory of modernity that constrains it. As Karsten Harries has written, “precisely because Cusanus straddles that threshold, he has more to teach us as we try to understand not only the legitimacy, but the limits, of modernity.”55 The very epochal divide that first attracted the Neo-Kantians to Cusanus is undermined by his writings. To appreciate the difficulties of this situation it is instructive to compare the cases of two influential German theorists of modernity, Heinrich Rombach and Hans Blumenberg. Both of their monumental books appeared in the same year, both assume the Husserlian mathesis narrative more or less, and both hinge on their respective Cusanus interpretations. Rombach understands that advances in fourteenth-century physics preceded the Cartesian mathesis universalis and searches for their Vorgeschichte in Nicholas of Cusa.56 Even more radically than Cassirer, Rombach identifies Cusanus as the instigator of modern philosophy, arguing that it was his “functionalist ontology” that enabled the mathematizing breakthrough of Galilean science.57 And yet Rombach reads Nicholas primarily as philosopher or cosmologist, entirely overlooking his many writings on Trinity or Christ. He approaches mathesis as ahistorically as Husserl and Heidegger, imposing the Cartesian usage backwards into earlier centuries even while neglecting the precedent of the medieval quadrivium. His investigation of Cusan sources begins and ends with Eckhart’s philosophy of the Word.58 In Rombach’s telling, Cusanus prefigures the Cartesian mathesis, but that achievement has nothing to do with his Christian mystical theology or his ties to medieval Pythagoreanism. Blumenberg also takes account of nominalist physics and paints a more modest portrait of Descartes.59 After exploring the problem of epochality, Blumenberg concludes in contrast to Rombach that Cusanus was the last medieval and only a precursor of the modern.60 Nicholas shared the same vision of an infinite cosmos with Giordano Bruno, Galileo, and Descartes; what separates them is the cardinal’s mystical theology of Incarnation.61 Bruno is therefore the first modern because his geometrized, infinite cosmology allows him to leave medieval Christologies behind.62 But having identified the Incarnation (quite correctly, I think) as one of the keys to the question of Cusan modernity, Blumenberg never thinks to search for possible liaisons between that doctrine and the cardinal’s equally pivotal mathematical epistemology.63 Hence we see that even those who most disagree about Nicholas’s epochal status share a common premise. The cardinal’s theology and mathematics are separable spheres, and one is foregrounded at the expense of the other. Rather than dissolve their own disciplinary or epochal categories, such interpretations

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threaten to dissolve the coherence of the historical material. By the same token, if one could reconstruct the Cusan mathematical theology in its native integrity, this would contribute eo ipso toward the task of scrutinizing the shortcomings of the mathesis narrative and reconsidering the place of the Christian religion in scientific modernity. Cusanus’s mathematical theology is by far the most fully evolved creature of its kind in the history of western Christianity and the only one to have any chance of survival. Every hair and wrinkle of its anatomy is valuable; the rest of the species has gone extinct.64

Thierry of Chartres as a Cusan Source Clemens Baeumker wrote in 1913 that if medieval Christian theology is a Gothic cathedral, and its twin spires the towers of Augustine and Aristotle, then “in that intellectual structure, next to those bright halls, there are also mystical, dark side chapels, which shine with a venerable holiness, but which also collect some dusty, worthless devices.” The shadowy side chapels, whispers Baeumker, are “Neopythagorean” theologies.65 Our understanding of Christian Neopythagoreanism has not advanced far beyond Baeumker’s hushed tones. It is well known that Pythagoreanism enjoyed a revival in the Renaissance, as Platonists like Marsilio Ficino sought esoteric alternatives to Christian scholasticism. Cusanus participated in this movement, as has long been recognized.66 But this observation alone fails to ask after the other half of the story. To begin with, what happened between Athens and Florence? Did Pythagoreanism ever cross paths with medieval Christianity? How well would Pythagorean beliefs about divine numbers have meshed with Christian teachings about the Trinity or Incarnation? Moreover, when Cusanus rejected the scholastic synthesis, he did not do so in the name of a pristine Greek or Christian antiquity, as others would in Renaissance Italy or Reformation Germany. Instead he found inspiration in the scraps and margins of past theologies, fashioning a kind of surrogate heritage for himself out of the arcana of medieval Platonism. By the turn of the sixteenth century his contemporaries had already christened him the “connoisseur of the Middle Ages.”67 Because of the cardinal’s colorful bricolage, and the notorious indeterminacy of Pythagorean traditions, it is vitally important to be cognizant of the sources one uses to explore the question of Cusan Pythagoreanism. Proclus is one such Cusan source that requires careful handling in light of the pervasive mathesis narrative. We know that it was Proclus’s Euclid commentary that inspired a Neopythagoreanizing trend among sixteenth- and seventeenth-century philosophers to mathematize philosophical method. We also know that the Neo-Kantian inquiry into Cusanus’s modernity revolves around the cardinal’s proximity to Descartes. Given this state of affairs, the discovery that

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Nicholas was deeply interested in Proclus’s works has generated great interest in Cusanus studies but also encouraged misinterpretations of the cardinal’s Neopythagoreanism. Klibansky, Haubst, and others have demonstrated that in his later years Cusanus possessed and annotated some of the first Latin translations of Proclus’s works.68 This extraordinary access to Neoplatonism has led many over the last few decades to affiliate Nicholas’s late works with the Proclian tradition. This is all well and good. Some have drawn (or insinuated) the further conclusion, however, that just as in the seventeenth century, it was Proclus or Proclianism that inspired Cusanus to mathematize his philosophical method in a Neopythagorean mode and thereby anticipate the modern breakthrough of Descartes.69 This inference is understandable, but it is simply false. In the first place it exceeds the textual evidence, which only confirms a substantive Proclian influence in the late 1450s, leaving unaddressed the cardinal’s possible sources for his most important Neopythagorean doctrines between 1440 and 1450. The fallacy also betrays a more serious historical misapprehension. Not everything in Cusanus concerns mathesis, but what does comes not from Proclus but from Boethius and Boethian traditions. Proclus was one channel for transmitting Greek Neopythagoreanism to medieval Latin Christianity, but by no means the broadest or deepest. In what follows, I show that Boethius and Augustine are equally important conduits for medieval Neopythagoreanism, and that Thierry of Chartres combined their traditions to forge his distinctive theology of the quadrivium. So it is indeed true that Nicholas anticipated aspects of the Galilean-Cartesian mathesis universalis, but his premonitions flowed rather from Boethian and Chartrian traditions than from Proclus or medieval Proclianism. This finding does not sit well at all with the mathesis narrative, which held that Greek Neopythagorean ideals such as universal mathematics expedited the early modern transition away from Christian scholasticism, and certainly not that those ideas were hidden within the most venerable medieval Christian authorities of all. In the absence of a robust, systematic account of Cusanus’s Chartrian influences to balance scholarship on Proclus, Eckhart, or Ramon Llull, there is a habitual tendency to underestimate Thierry’s significance and (as we shall see) even to associate Thierry’s ideas with Proclus or Eckhart (or vaguely “Neoplatonism”) instead.70 But to make these distinctions requires one first to perform a patient audit of the range of Neopythagorean concepts and their pathways in transit from late antique Platonism to medieval Christianity, so that their reappearance in Cusanus can be properly contextualized. If the Neo-Kantian narrative dehistoricized mathesis by leaping from Proclus to Descartes, allowing for the fantasy of a sudden seventeenth-century rupture, then by the same token a careful study of Boethius, Thierry, and Cusanus will not only restore the missing medieval millennium but also transform our narratives regarding the fate of mathesis.

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The more one studies Thierry’s ideas, the more central to Cusan thought they prove to be. Carlo Riccati has suggested that the entirety of Cusan theology is one extended meditation on Thierry’s reciprocal folding, and Bernard McGinn argues that the cardinal’s Trinitarian theology amounts to “variations” on Thierry’s arithmetical Trinity.71 Werner Beierwaltes, the great Proclus expert and historian of medieval Platonism, insists that however frequently Nicholas’s debt to Thierry is recorded, “its constitutive significance for Cusan thought is nevertheless always underestimated.”72 Beierwaltes concludes his essay on Chartrian Platonism with the following counsel: Both Augustine and Boethius stand within the reach of the Neopythagorean tradition, which sought to radicalize the Pythagorean element in Platonism. . . . When one considers that one of the chief texts of this tradition, the “Introduction to Mathematics” of the Platonizing Pythagorean Nicomachus of Gerasa, was accessible to both Augustine and Boethius, then one’s view of the sources of the Chartrians and of Cusanus is broadened. And thus a link is revealed, however narrow and indirect, from the Middle Ages and Renaissance back to Greek antiquity.73 Thierry’s treatments of Augustine and Boethius, to which Beierwaltes refers, represent the greatest resurgence of Neopythagoreanism in Christian scholastic thought. Among his contemporaries at Paris and Chartres, Thierry was praised as the leading humanist of his generation and the reincarnation of Plato himself. In his daring commentary on the book of Genesis, for instance, Thierry baldly states that “the creation of number is the creation of things,” presaging Galileo’s dictum that God wrote the universe in the language of mathematics. Given the long list of Cusan debts to Thierry—the arithmetical Trinity, the theology of divine Equality, the model of reciprocal folding, the dialectic of unity and alterity, the four modes of being—it is remarkable that we still do not know exactly how Nicholas accessed them (they are not in his library), why he was drawn to such recondite texts, how he adapted them to his own ends, or even how he conceived of Thierry as an author, since the extant texts are anonymous. Did Cusanus, for example, revisit the same passages in Augustine or Boethius that Thierry used? Did he simply reproduce Thierry’s words, or did he deploy different doctrines to different ends? My goal is not to provide an exhaustive survey of every Chartrian moment in Cusanus, but to follow the threads that tied them together at the most critical junctures and so structured the whole. Due in part to chance events, this task has remained incomplete for decades. Cusanus’s dependence on Thierry was first established by Duhem, but it was Klibansky who first understood the opportunity that discovery represented. In the

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inaugural volume of the Heidelberg edition, we find this remark from Klibansky and Hoffmann in the introduction: Regarding commentary on the sources, there is one group in particular which carry such exceptional weight that they can be simply cited: the writings of the school of Chartres and of its associates. The ancient book by Thierry of Chartres, De sex dierum operibus, and a certain anonymous commentary on Boethius’s tractate De trinitate, both of which decisively influence Cusanus’s opinions on all things derived from them, have not yet been edited. Raymond Klibansky, who discovered these writings . . . will publish a study regarding the School of Chartres, including other works of Thierry of Chartres and other commentaries of the twelfth century in an appendix, where it will be explained at greater length what relationship might exist between Cusanus’s De docta ignorantia and the philosophers of that age. (There are those who think the Cusan philosophy is cheapened if its origins are demonstrated; it is not worth the effort to refute such an error.)74 Unfortunately Klibansky’s pledge to produce an extended study of Thierry’s influence was never fulfilled. His Habilitationschrift on Bernard of Chartres and Thierry of Chartres, completed at Heidelberg in 1931, was in press at Felix Meiner in Leipzig when the publisher was bombed by the Allies on the night of December 3, 1944. All materials then in press were burned, not only Klibansky’s research on Thierry but also the plates of the nascent Cusanus edition.75 The first third of his monograph, titled Die Schule von Chartres, was to present an overview of their philosophy, and the remainder contained an edition of their selected texts. Instead, Klibansky’s oversight of the critical edition and its annotations constituted his greatest contribution to Cusanus research, and he published only one general article on Chartres.76 Dietrich Mahnke would have been another candidate to pursue this project. A professor of philosophy at Marburg since 1927, Mahnke studied mathesis universalis in Leibniz in his early work and corresponded frequently with his teacher and close friend Husserl.77 The year after Husserl finished the Krisis, Mahnke also published his final work, Unendliche Sphäre und Allmittelpunkt: Beiträge zur Genealogie der Mathematischen Mystik, which he modestly called “some minor spadework” for a larger undertaking. Adopting his title from Novalis’s ideal of “mystical mathematics,” Mahnke traces Pythagorean themes and mystical geometrical symbols from German romanticism all the way back to the Presocratics.78 He assigns a prominent role to Cusanus, but his analysis of medieval sources skips from Eckhart and Alan of Lille straight to Augustine and Boethius. From his post in Marburg, Mahnke may or may not have been aware of the work of Klibansky and

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the Cusanus edition; in any case, Klibansky was forced to emigrate to Oxford in 1933 by the German race laws. In 1952, Rudolf Haubst published his magisterial book on the Trinity in Cusanus’s theology, and four years later followed up with another on Christology.79 As the founding director of the Institut für Cusanus-Forschung, Haubst succeeded in reorienting Cusanus studies as a whole, portraying the cardinal less as protomodern philosopher than as medieval theologian of the Trinity and Incarnation. His books, among the first monographs to make full use of Klibansky’s research, focus on the arithmetical Trinity, Chartrian triads, and Christology that appear in De docta ignorantia. But because Haubst’s books unfortunately separate these three topics from each other, they systematically obscure the breadth of Thierry’s contribution.80 I  have begun from the contrary premise that the two triads and Christology should be read closely together and that the sequence in which Cusanus presented them is essential. Paul Wilpert, a later editor of the Cusanus edition, was also keenly interested in the question of Thierry’s influence. Well versed in ancient Greek philosophy and medieval Platonism, Wilpert held several doctoral seminars in 1960–62 on Cusanus and the school of Chartres. He published an important article on manuscript traditions of De docta ignorantia, and had doctoral students working on Thierry’s commentaries. But aside from a probing essay on Nicholas’s sources, Wilpert had not yet published on Thierry when he died unexpectedly in 1967.81 Since then there has been an explosion of classics scholarship on ancient Pythagoreanism, from Philolaus and Archytas through Iamblichus and Proclus up to Boethius himself. In addition, Nikolaus Häring has since edited and dated Thierry’s major commentaries, inspiring a new wave of more precise evaluations by historians of medieval philosophy. Among these are several short studies analyzing aspects of Thierry’s influence on Cusanus from which I  have benefited. But rarely are the advances across these three domains allowed to illuminate each other or are their resources pooled toward a greater end. A recent manuscript discovery has called into question some of the assumptions guiding prior scholarship on Thierry and Nicholas, potentially reshaping the terrain of future research. Since Klibansky, the second book of De docta ignorantia has been recognized as a prime instance of Cusanus borrowing Chartrian doctrines, particularly the four central chapters (II.7–10). In 1995, the Dutch historian Maarten J. F. M. Hoenen uncovered a manuscript in southern Germany (Eichstätt Cod. St. 687) containing a short philosophical treatise that follows these chapters verbatim; for textual reasons it is unlikely to postdate Cusanus’s works.82 Hoenen concludes that Cusanus must have copied long tracts from the treatise as he composed his 1440 masterwork, making a few additions and deletions. Numerous concepts essential to the theology of De docta ignorantia apparently originated with that unknown author. The Eichstätt treatise, also known as Fundamentum naturae,

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would also represent a heretofore unknown, direct mediation of Chartrian ideas to the German cardinal. If there is anything to Hoenen’s claims (as I believe there is), then suddenly the task of disentangling Thierry and Nicholas grows more complicated and its necessity more acute. Reactions to Hoenen’s discovery in the guild of Cusanus studies have been mixed but largely negative. The most common response is to suspend judgment in the hope that a “common source” behind the manuscript will eventually emerge, as if this would resolve the issues raised by the fact that an unidentified author, at whatever remove, helped to mold the structure of De docta ignorantia from within. Some have reasoned that just as Cusanus uncontroversially appropriated Thierry’s theology elsewhere in De docta ignorantia, likewise in this more dramatic case, the Eichstätt treatise simply mediates Thierry’s ideas; in essence Thierry is the common source.83 But this conclusion fails to suffice once one realizes that the manuscript does not simply transmit Thierry’s theology or celebrate it as Cusanus does, but in fact sets out to refute Thierry’s doctrines, as I have shown elsewhere.84 If this is the case, Cusanus’s enthusiasm for the contrarian treatise is a mystery that merits closer attention. And since Nicholas placed Fundamentum naturae at the center of De docta ignorantia, we should not be surprised to find that high esteem reflected in his later works as well, as I demonstrate below. To grasp what Cusanus found so compelling in the treatise, I have sought to contextualize its author’s judgments within the reception history of Thierry of Chartres. Once this is done, it becomes clear that the treatise is no pedestrian source but a crystalline recapitulation of long-term controversies over the role of number in Christian theology reaching back to antiquity. Were it anything less, how could it have fascinated the erudite Cusanus for over twenty years? By critiquing Thierry’s Neopythagoreanism from within the dominant Augustinian paradigm, the author of the Eichstätt treatise effectively delivered a map of the missing Neopythagorean landscape within medieval Christian theologies. Captivated by its unexplored territories, Cusanus could not look away.

Some Notes on Method These considerations begin to explain the unfashionably broad historical scope of this book. My intention was never to write a history of Pythagorean ideas in Christianity tout court, nor have I done so. Rather I have tried to bring to the surface a hidden interaction between two elements deep in the veins of Christian theology that Nicholas had unwittingly unearthed. To understand the rationale behind his distinctive mathematical theology, we must understand his textual motivations and their elliptical status vis-à-vis the mainstream tradition. This in turn

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requires us to investigate the inner composition of Pythagoreanizing Platonism as modified by Christianity over the centuries. Hence in this book I do not simply review “Pythagorean” passages from Cusanus’s works or chronicle the Christian “Pythagorean” ideas appearing in every century. Rather, I  examine, in a genealogical mode, how the constraints of textual access shaped the type of theological questions that Cusanus could and did pose about mathesis. In many ways my approach resembles that of Kurt Flasch in his monumental Nikolaus von Kues: Geschichte einer Entwicklung.85 Flasch, I have belatedly discovered, advocates a version of the method that I came to find necessary as I struggled to make sense of Thierry’s complex influence over time. Rather than traffic in abstract topics like number, infinity, or symbols, I have instead oriented myself to the shifting conditions facing Cusanus as a writer: what he was reading, where his interests lay, what he was not understanding well, and how he viewed his intellectual responsibilities. Flasch likewise proposes that Cusanus studies would benefit from “genetic analyses” of specific texts in their diachronic development, and from ceasing to view the Cusan oeuvre as a preformed system that generates certain identifiable doctrines. Instead, Flasch attends to the fine textures and microclimates of different regions of Nicholas’s life and to the dramatic changes that his thought underwent on several occasions, welcoming the possibility that different chapters in the cardinal’s evolution might possess different worth. Genetic analysis searches for discontinuities as much as continuities and avoids teleological narratives by seeking multiple “measures of development” (Entwicklungstadium).86 With all of this I can only heartily agree. But there are some troubling tendencies in Flasch’s execution of his method that should give us pause. One would think that Flasch’s method would encourage research into the concrete sources motivating Cusan development in different periods. Flasch often does so, yet he also warns that “in order to carry out a genetic analysis that remains close to the text and to illustrate it with even minimal adequacy I must forgo several conventions such as footnotes to ancient and medieval sources or to reception history.”87 This is a sane compromise, but it also hints at a tension in Flasch’s method between the surface of the text and other contextual factors behind or beneath, including sources. He grants relative autonomy to the “inner problematic” of Cusan thought in its own self-questioning as an immanent motor of development immune to qualification by sources.88 This leaves open the possibility that interpreters can divine Cusanus’s true development by remaining “close to the text” in an arbitrary positivism even when the meaning of a passage is inescapably codetermined by its source.89 In my experience this occurs frequently in Cusanus, particularly when he is depending on Thierry of Chartres. Without attention to sources, interpreters are left to their own devices to decide when Cusanus is taking independent strides and when he is constrained by his materials, when he is “developing” and when regressing. If Cusan development

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is fragmented into a collection of discontinuous microhistories, the interpreter is empowered to isolate and praise what she perceives to be moments of special clarity or progress. Flasch claims to have isolated “authentic” moments when the Cusan mind was temporarily freed of its medieval theological limitations. The clouds pass, the sun shines, and then the skies grow overcast again. In the tradition of Cassirer, Flasch picks out the mathematical epistemology of De mente as the most valuable work of Cusanus, and Proclus as his most important source. These perceptions happen to accord perfectly with the prejudices of the mathesis narrative. Ideally, however, one would extend the same courtesy to Cusanus’s sources as to the cardinal himself, even when this redoubles scholarly burdens. In this book I have therefore attempted to isolate a single source (Thierry of Chartres), to submit it to genetic analysis, and then to apply the fruits thereof to a more rigorously genetic reading of Cusanus—only one “measure” among others, but one that follows the red thread of a largely undervalued source. To complete this additional work has required three discrete sections. Likewise, I  have engaged current scholarship on Thierry and Nicholas extensively, not out of belligerence, but out of the recognition that I am swimming against the tide. Lacking a sturdy definition of Christian Neopythagoreanism, past scholarship in my view has frequently misclassified mathematical theologies within outdated rubrics of Pythagoreanism or number mysticism. What is the nature of Pythagoreanism in relation to Christianity? In what sense is Thierry Pythagorean, and in what sense Nicholas? Seeking to answer these questions, I  found no recent account of Pythagoreanism and Christianity that spanned Plato through Nicomachus to Boethius. Scholarship on Thierry had rarely considered how his thought might have developed dynamically before and after the arithmetical Trinity. The Eichstätt manuscript demanded a whole new line of questioning on Cusanus. In Part One, accordingly, I  examine new research on Pythagoreanism as a diverse, coherent, and recurrent tendency in Platonic traditions.90 In Chapters 1 and 2, I draw special attention to the themes of mediation and henology in the long passage from Philolaus to Nicomachus. In Chapter 3, I follow three different fates met by Neopythagoreanism in late antiquity that marginalized its influence on Christian thought. Disaggregating the different movements and innovations of Pythagoreanism provides a better heuristic for understanding the discursive conditions facing Thierry and thus for measuring the nature of his achievement. Then in Part Two, I  explain how Thierry wove his sources together in creative ways, framing new questions that would challenge his readers for centuries. In Chapters 4 and 5, I attempt to read his works chronologically as a process of development and in his twelfth-century contexts. Doing so reveals that Thierry’s true achievement is not his Trinitarian theology but his late modal theory. In Chapter 6, I  trace the misinterpretations, censorship, and critique that frustrated the

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medieval reception of Thierry’s theology, but also produced the diverse traditions that delivered it to Nicholas of Cusa. Finally, in Part Three I  try to measure the impact of this set of Chartrian sources, in complex combination with some of the ancient ones as well, as they variously motivate or limit Cusanus’s own theological development over three decades. The first task of Chapter 7 is to gain some clarity about how the cardinal handled the Chartrian traditions at his disposal, both Thierry’s works and those of Thierry’s medieval readers. I show that some major monuments of Cusan theology in De docta ignorantia are the result of his attempt to reconcile tensions among his sources. I explain in Chapter 8, however, that this first synthesis was followed by another less successful one, De coniecturis, which raised questions for Cusanus to ponder throughout the 1440s. After a time of testing, Cusanus finally reached a definitive conciliation of his Chartrian sources in 1450, as I demonstrate in Chapter 9. This breakthrough in De mente paved the way for a geometrical turn in his theology. In Chapter 10, I define the major topics of Cusanus’s mathematical theology and make the case that the late dialogue De ludo globi is an indispensable masterpiece in the cardinal’s oeuvre. Hence the paradox:  by inadvertently following Flasch’s genetic method I have come to agree with him in many particulars regarding the rhythms of Cusan development, but more fundamentally, to arrive at the opposite conclusion regarding their destination. To Flasch the learned cardinal is struggling to become a true philosopher, traveling from the darkness of mystical theology and Pythagoreanism in the 1440s to the light of henology and epistemology in the 1450s, which frees him finally from the grip of traditional doctrines of Trinity and Incarnation.91 But having traced Thierry’s genetic development and its Cusan refractions, I have come to view things quite otherwise. I see instead a dramatic but consistent evolution of Cusan mathematical theology as precisely a theology of Trinity and Incarnation, unfolding in dialogue with Thierry’s legacy from 1440 onward. I have also found myself unexpectedly excavating a lost story of Christian Neopythagorean theology and so clearing the way for others to reexamine its discursive status in late modernity. To them I commend Varro’s advice on the study of the quadrivium: “We either do not study these subjects at all, or we leave off before we understand why they should be studied; for the pleasure and usefulness of them lie in the more advanced parts, when they have been completely mastered—the elementary stages seem pointless and disagreeable.”92

PART ONE

The Genesis of Neopythagoreanism A Synopsis

Das Leben der Götter ist Mathematik. Alle göttliche Gesandten müssen Mathematiker seyn. Reine Mathematik ist Religion. Novalis

1

Platonic Transformations of Early Pythagorean Philosophy One morning around 400 bce, Eurytus the Pythagorean announced that he had discovered how to divine the essential number of any given being. His method was simple and effective. Step one: arrange pebbles into a geometrical figure tracing the creature in question, making a stone outline of the tree, horse, or man standing before you. Step two: count your pebbles. This tall tale was told by Aristotle’s student Theophrastus to confirm his teacher’s portrait of ingenuous Pythagoreans convinced, in Aristotle’s words, that “all things are numbers.”1 The historical Eurytus was indeed a younger colleague of Philolaus, the first “Pythagorean” to record his doctrines. Philolaus’s disciple, the famous Archytas of Tarentum, was one of Plato’s mentors and may have even saved his life. So when Aristotle proposed that Plato had been led astray philosophically by certain “Pythagorean” contacts, the charge stuck. Aristotle’s lengthy indictment became a dominant source for ancient and medieval readers of what “Pythagoreans” purportedly taught.2 Until very recently Aristotle’s report was taken more or less on credit, even if classicists recognized the polemical context.3 But over the last few decades, new scholarship on Pythagorean origins has upturned the term’s meaning, and left modern historians wondering if they have fallen for Theophrastus’s canard. Did the Pythagoreans really teach that all things were numbers? Is there even such a thing as a common Pythagorean doctrine or a Pythagorean school? Are Plato’s philosophical ideas about mathematics novel or borrowed? What then would it mean to say that Thierry or Nicholas—catholic Christians writing in Latin, one in Paris, one along the Rhine—are “Pythagorean”? There is no end to the confusion besetting the historiography of Pythagoreanism, whether by its own ancient chroniclers or by modern scholars looking back. Before we can weigh the prospect of a medieval, Christian Pythagoreanism, such riddles must patiently confronted. In his landmark book, Walter Burkert showed that the Old Academy, the first generation of Plato’s students, misnamed some Platonist ideas as “Pythagorean,” by which they meant “Presocratic.”4 This historical mischief was an ideological attempt to elevate the place of mathematical elements in Plato’s latest, unwritten

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teachings, an agenda supported by the first leaders of the Academy. In subsequent centuries any Middle Platonists or Neoplatonists similarly interested in mathematics followed suit, attributing their own musings to historical Pythagoreans and confounding matters further. Late antiquity witnessed a Neopythagorean revival built in no small part on such pseudepigraphical doxographies.5 This meant that the Neopythagoreans, while believing themselves to be returning to the fount of Platonism, were in fact mathematizing Plato’s philosophy to a degree surpassing even authentic Presocratic Pythagoreanism, so far as we can tell. On top of this, groundbreaking works by Carl Huffman and Leonid Zhmud have recently questioned our grasp of Philolaus and Archytas themselves.6 It now appears that they taught few of the doctrines that Aristotle attributed to his fictional “Pythagoreans,” and that Plato’s response to them involved as much critique as appropriation. Received notions about Pythagoreanism begin to look as useful as Eurytus’s pebbles. What did the earliest recorded Pythagoreans like Philolaus and Archytas share in common, if not the belief that all things are numbers? We must begin by considering Pythagoreanism in its social context as a loosely affiliated religious community.7 Since the time of Hippasus in the 450s bce, Pythagoreans divided themselves between two factions, the so-called “akousmatics” and the “mathematics.”8 Ἀκούσματα (“things heard”) were oral traditions about the life of Pythagoras, the code of ethics and diet kept by his followers, and rituals embodying beliefs in metempsychosis. Μαθήματα (“things taught”) were written doctrines about the gods, the cosmos, or the destiny of the soul, principles that the learned considered to be the heart of Pythagoreanism and that, crucially, were often expressed in arithmetical or geometrical figures.9 This distinction between practice and theory, the akousmatic and the mathematic, while first born out of intrasectarian tensions became reinforced after Plato and his Academy absorbed only Pythagorean μαθήματα into their philosophies. The rise of Platonism, and the ongoing tension between its more and less Pythagoreanized versions, therefore represents the permanent victory of the mathematical school over the akousmatical school in the history of Pythagoreanism. Within their original religious context, Pythagorean “mathematical” doctrines were not simply to be heard and obeyed, as with ἀκούσματα, but to be rigorously internalized as objective theological truths. Μαθήματα initially encompassed at least three domains. First, Pythagorean physics maintained that the center of the cosmos is a primordial fire and that the planets revolve around it in circular harmonies. Second, Pythagorean number mysticism or arithmology, beliefs often associated with Pythagorean philosophy, held that the first ten numbers (the decad) can be identified with the Olympian pantheon or with abstract properties like justice or fertility.10 Understanding the ten elements of cosmic order initiated the adherent into the cult of Pythagoras’s revelation and inspired a religious zeal

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for further studies in geometry or harmonics. But I will focus particularly on the third domain of Pythagorean philosophy, the one that provided raw material for Plato’s deliberations: namely, the programmatic deployment of mathematical concepts within philosophy and theology that goes beyond arithmological lists. Defining this third category with any precision is difficult, however, since it is just in this domain that Plato gave new meanings to concepts borrowed from Presocratic Pythagoreans. Given the historical entanglements and misrecognitions of authentic Pythagoreanism recounted above, the boundary between what we could alternatively call “mathematizing Platonism” or “philosophical Pythagoreanism” is blurry at best. Since we are less interested in recovering the original beliefs of Pythagoras than in evaluating the influence of the philosophical tradition bearing his name, it is best to view Pythagoreanism as a mathematizing moment within the larger history of Platonism, which Bruno Centrone has called an endemic “compenetration between Platonism and Pythagoreanism.”11 However inconclusive it may be, this perspective at least allows us to characterize how mathematics was used philosophically by Pythagorean philosophy before, within, and after Greek Platonist traditions. That is, we can begin to generate a durable definition of Pythagoreanism as a tendency within Platonism that could fruitfully be applied to the ancient Mediterranean and medieval Europe alike. In terms of their historical evolution, one might well expect to find Pythagorean ideas about number unfolding in several gradual steps. A primitive number mysticism in turn encourages a metaphysical search for cosmic or divine harmonies; last to come, perhaps after disappointing results like Eurytus, would be the scientific application of mathematics to the physical world. The hypothetical sequence would thus run as follows: (1) Number ontology. Numbers signify the true essence of each thing and the ultimate origin of the universe, revealing the substructure of cosmic order and the harmony of all things. (2) Mediating mathematicals. Considered in separation from the material world, these deeper mathematical realities reside at an intermediate level of being. They connect physical objects to the divine realm and anagogically direct the mind away from matter toward higher truths. (3) Transcendent henology.12 The numerically one (the monad) immediately reflects the transcendent or ineffable One (the henad). Hence the philosopher attempting to conceptualize the divine can be assisted by arithmetical exercises during the mind’s ascent. (4) Universal mathematical science. Since number reveals the truths of the physical world and eternal divinity, a discrete set of mathematical sciences, unified in themselves, can afford one ultimate knowledge of all domains of being.

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This ordering of ideas matches the standard narrative about Pythagoreanism that prevailed from Aristotle’s critique of Plato well into the 1960s. In its more recent versions, it undoubtedly reflects modern presuppositions about the resolution of nebulous mythologies into clear and distinct scientific ideas.13 But however winsome an example of the progress of enlightenment, this sequence is historically false. In point of fact, Huffman and Zhmud have shown that the “Pythagorean” philosophical signature begins with a fascinated awe before concrete mathematical sciences.14 It was the technical power of geometry and calculation in practice, not number mysticism, that inspired Philolaus and Archytas in their philosophical pursuits. Plato then substantively altered their ideas:  mathematical ideas were not for knowing the world but for leading the mind away from it, like a stairway to the Good. Plato’s successors went further by reconstructing the Good of the Republic as a transcendent One whose unity could be approached only by navigating levels of mediators. Only at the end of these developments, many centuries after Philolaus and Archytas, did Neopythagoreans revive the original focus on mathematical sciences themselves, but now for their own mystical ends. Hence the historical order of emergence ought rather to read like this: (1) Universal mathematical science: Presocratic Pythagoreans (2) Mediating mathematicals: Plato (3) Transcendent henology: the Old Academy and Middle Platonism (4) Number ontology: Neopythagoreans inspired in a new way by (i) Both Philolaus and Archytas were convinced that it was the scientific practice of mathematics, as opposed to arithmology, that could resolve philosophical problems. Devotion to concrete mathematical problems that once seemed the personal predilection of a few odd Pythagoreans now looks like the hallmark of a Pythagorean philosophical ideal that caught Plato’s attention. Plato was more interested in mathematical exercises as a method for training the eyes of the mind. But this illuminative power becomes apparent—as Philolaus and Archytas already argued, and as Plato repeated in the Republic—only when the different species of mathematical activity are assembled in concert. Their germinal notion of a systematically unified mathematical science as the starting point for philosophy fully flowered in the writings of Nicomachus of Gerasa, arguably the apex of the Neopythagorean revival. Hence the recent doubts of classicists about the identity of authentic, original Pythagoreanism have ironically had the effect of raising the profile of the very Neopythagoreanism once dismissed as frivolous.15 The Pythagorean bequest to Plato began with the four mathematical sciences of arithmetic, geometry, harmonics, and astronomy, and at least in this regard—in

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their zeal to re-mathematize Platonist concepts of the soul and the divine—the Neopythagoreans had their history right. It was precisely Nicomachus’s fourfold system of mathematical sciences that the Roman Christian official A. M. S. Boethius translated as quadrivium after studying in the Platonist schools. Philip Merlan has argued that the Boethian quadrivium, genetically inscribed from birth with such Pythagorean controversy, carries questions within itself that can trigger unforeseen effects when transplanted to other philosophical soil. “In the very idea of the quadrivium,” he writes, “the problem as to the ultimate meaning of mathematics survived.”16 For the quadrivium’s original promise was not simply to be an educational program but indeed a universal Neopythagorean cosmotheology. When this background is forgotten, according to Merlan, the quadrivium begins to lose its meaning.17 This is what happened when the Boethian quadrivium was absorbed into the curriculum of monasteries and then cathedral schools, escaping philosophical attention for centuries even while silently preserving an alien Greek theology. When Thierry and Nicholas attempted to rethink the moribund quadrivium within Christian theology, they found it, so to speak, booby-trapped with Neopythagorean ideas. Their theologies of the quadrivium, then, did not simply repristinate Boethian lore. They set in motion a dynamic confrontation between Christianity and Neopythagoreanism. Yet the story of Pythagorean traditions in medieval Christian thought is even more complex. For if the quadrivium smuggled Pythagorean elements into Latin Christianity, elements first appreciated in Thierry’s theology, this was not the first time that mathematics had inspired theological rumination, and certainly not the first commerce between Christianity and Platonism. Mathematical theologies originating within Greek Platonism proper had already raised questions about the nature of the divine and its accessibility to human experience; and Christian theologies well before Boethius had already been saturated with Neoplatonism. So it is not as if the two thought-worlds were sealed off from each other until the moment of encounter. Nevertheless, the distinctive details of the Christian Neopythagoreanism of Nicholas of Cusa can be seen only when one looks back nearly a millennium before Boethius.

Mathematics as Philosophy in Philolaus and Archytas Trying to discern the historical Plato on the topic of mathematics leaves one squinting through a fog of later interpretations. Plato wrote about mathematics in several of his dialogues, and the influence of early Pythagoreanism in those works is fairly clear. Plato alludes to Philolaus in the Philebus and to Archytas in the Republic, Laws, and Timaeus. But as Plato aged he apparently centered his philosophy less on ethics or epistemology and increasingly on mathematical ideas

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adapted from the Pythagoreans, teaching his adherents in the early Academy new doctrines that he never wrote down. After Plato’s death, both those who favored the new doctrines, the leaders of the Old Academy, and those who opposed them, namely Aristotle, at least agreed that these lost unwritten teachings were as authoritative as the written dialogues that we possess. To make matters worse, when the early leaders of the Academy tried to synthesize and systematize Plato’s written and unwritten doctrines, they accredited their interpretations by fashioning them as “Pythagorean,” when in fact they were Platonist through and through. Most of these Academic readings of Plato, however, are also lost, leaving Aristotle’s acerbic reactions to Plato’s oral teachings—polemical, tendentious and sarcastic as they sometimes are—as the most substantive source left for deducing what Plato’s final interpretation of Pythagoreanism might have been. Great caution is warranted in trying to certify the exact particulars of Plato’s philosophy of mathematics, an enterprise which for our purposes is fortunately unnecessary. More important is simply to draw a contrast between what Plato did with mathematics in his written dialogues, and what his early Pythagorean sources had done. Acquainting ourselves with the earliest Pythagorean philosophers will expose Plato’s decisive transformation, one that proved far more significant in the history of Platonism than any specific doctrine. Of the mathematical Pythagoreans predating Plato, Philolaus of Croton (ca. 470 bce–ca. 380 bce) and Archytas of Tarentum (ca. 435–ca. 360 bce) left the most substantial bodies of fragmentary writings.18 Next to nothing is known of Philolaus’s life, and he was obscure even in antiquity.19 Yet according to Huffman, Philolaus deserves praise for being “the first thinker self-consciously and thematically to employ mathematical ideas to solve philosophical problems.”20 A  contemporary of Democritus, Philolaus sought unifying cosmic principles like other Presocratics.21 Parmenides taught that knowledge requires a supreme “limit” and Anaxagoras called the elements a plurality of “unlimiteds.” But Philolaus rejected a single abstract principle like Mind or Air and instead proposed a plurality of limiters and unlimiteds.22 Huffman notes that Philolaus thereby gave number (ἀριθμός) the role Heraclitus gave to the cosmic Logos.23 “Limiters” denotes a class of structuring forms that communicate order and measurability to “unlimited” continua like matter, time, or pitch. Since limiters and unlimiteds are wholly unlike, they must be joined together by what Philolaus called “harmonies,” whether musical ratios or geometrical definitions of solids.24 In his usage, such harmonies always have specific “quantities” expressed in numbers; in fact, “numbers” for him are not abstractions for counting with but simply concrete instances of such quantitative harmonies.25 Philolaus thus neither taught that “all things are number,” as Aristotle alleged, nor equated number and limit, as modern scholars have proposed, nor indulged in arithmological speculation.26 In one notable fragment Philolaus does state that

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“all the things that are known have number. For it is not possible that anything whatsoever be understood or known without this.”27 But this may be less an ontological statement about number than an epistemological one.28 Parmenides had worried that sheer ontic multiplicity rendered certain knowledge impossible, but Philolaus saw the problem overcome by number as the principle of intelligibility. Philolaus thus commends, in Huffman’s words, the “cognitive reliability of numerical and mathematical relations,” especially the kind of “mathematics that relies on proof.”29 Philolaus discussed harmonics, geometry, and arithmetic, but he explicitly singled out geometry as the “source and mother-city [μητρόπολις]” of the rest of the mathematical sciences.30 His metaphor seems to prioritize geometry not just historically, but methodologically, on account of the certitude of geometrical demonstrations. Philolaus may have even hinted at what Huffman terms a “canonical group of mathematical sciences” with an internal hierarchy and principle of order that elevated geometry above the others.31 Although he never lists the different mathematical sciences explicitly, he implied their shared principle by referring to a “singular λόγος that arises from μαθήματα.”32 For Philolaus, then, mathematical science is by no means an afterthought, a merely secondary application of an antecedent religious arithmology. On the contrary, the kind of λόγος known in mathematics, namely calculations of harmonic quantities, is a paradigmatic mode of epistemological certainty that makes philosophy possible. To explore what Philolaus may have taught about the different mathematical sciences beyond such tantalizing fragments, we have to turn to his disciple Archytas of Tarentum, a contemporary of Plato.33 A brilliant mathematician and undefeated general, Archytas influenced not only the Pythagorean direction of Plato’s later dialogues, but also his ideal of the philosopher-king.34 All four Archytan fragments address the nature of mathematics or work out actual problems in harmonic theory. By contemporary reports Archytas was one of the leading mathematicians of Plato’s generation, renowned for multiple breakthroughs in harmonic theory and for the rigor of his geometrical argumentation.35 It is no wonder that after Socrates’s death Plato traveled to Tarentum at the southern tip of Italy to meet the man and learn mathematics from him.36 Archytas praised the mathematical sciences for “making distinctions well” (καλῶς διαγνώμεν), building upon Philolaus’s insight that mathematics guarantees certain knowledge by defining the “limiters” of continua. In one fragment Archytas compares the mathematical sciences to arithmetic, geometric, and harmonic proportions.37 In another he picks out four regions of knowledge in which mathematicians have achieved particularly “clear distinctions”:  in the movements of stars, in geometry, in numbers, and in music. Then Archytas makes the intriguing suggestion that these are not simply four examples of scientific success, but that “these sciences seem to be sisterly or akin [ἀδελφεά].”38

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Although he never systematized the sciences, he calls λογιστικά (calculation of quantities) superior to the others on account of its concreteness.39 So it seems that Archytas counted four “sibling” sciences:  astronomy, geometry, logistic, and harmonics. If Philolaus was the first to use mathematics philosophically, Archytas was first to conceive of four mathematical sciences as a unified whole.40 In hindsight, Philolaus and Archytas left a host of issues unaddressed that would drive later philosophers to formulate new questions. Philolaus glimpsed the potential of number for securing certain knowledge of the world. But do numbers come in only one variety, or many? Archytas hinted at an integrated framework for four mathematical sciences. But why were these four chosen and not others? Nicomachus and Iamblichus dove into these problems centuries later. But by that time, the reception of Pythagorean ideas had been permanently marked by the question that bothered Plato. Plato was impressed by this Pythagorean deployment of mathematics, but he was also convinced that the transcendent Good was a necessary anchor for all knowledge. Hence Plato’s question: how could a philosophical use of mathematics be harmonized with the kind of ultimate, singular Principle that Philolaus and Archytas had both avoided?

Mathematics as Mediation in Plato Philolaus and Archytas turned to number in order to focus the mind’s gaze on the cosmos itself, to sharpen and strengthen its knowledge of physical beings. But Plato (429–347 bce) found their emphasis on skillful calculation to miss the greater anagogical potential of mathematics.41 For him mathematics ought to yield a different philosophical experience altogether. It should inspire the philosopher not merely to perceive the world more clearly but to look through it, beyond it, above it. The process of abstracting a numerical quantity from a concrete harmony encouraged Plato to postulate separable, intelligible forms underlying all things, including mathematicals. Thus he invented an alternative philosophical application for mathematics that Philolaus and Archytas had never envisioned, namely as an intermediary between the visible world and the invisible realm of first principles. Mathematical being could function as a stepping-stone to assist the mind in its ascent from physics to dialectics. This innovation immediately set Plato’s project apart from the Pythagorean program. On this point even Aristotle sided with Philolaus and Archytas, agreeing that numbers were inseparable from things and therefore could have no mediating capacity.42 By reinterpreting mathematics as a necessary step toward knowing the divine Good, Plato’s modified Pythagoreanism opened up new theological possibilities. Given the importance of

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Plato’s philosophy of mathematics for our theme, we cannot avoid a brief survey of some pertinent passages in his works.43

Republic Plato first sets out his theory of mathematical mediation in Book VII of the Republic when Socrates explains the kind of education the philosopher should pursue. Only an arduous course of studies will yield the enduring μαθήματα that will make one wise (504d).44 When Socrates outlines the curriculum for philosopher-kings to pursue, he counts four discrete but interrelated τέχναι after the manner of Archytas. The first is “logistic” or arithmetical calculations, which train the mind for the rigors of dialectic (525d). Socrates also commends γεωμετρία, which forces one to reason about eternal forms (526e–527a). Next is solid geometry, including the rhythms of the stars, and finally the rhythms of music—the “sister sciences” (ἀδελφαί αἱ ἐπιστῆμαι), as the Pythagoreans say (530d). According to Socrates these four mathematical disciplines become more powerful in aggregate once they recognize their kinship and “commonality” (κοινωνία) (531d).45 These four sciences collectively occupy a mediating position within Plato’s epistemology. Plato’s hierarchy of knowledge assigns the lowest place to sense perception (αἴσθησις) that yields probable opinions (πίστις or δόξα), and the highest to reason (νοῦς) that intuits pure being at the end of the painstaking labor of dialectic. Between them stands a middle realm of understanding that Plato calls διανοία, which corresponds to geometry and her sister arts (511b). Lost in the manifold cosmos, the mind is naturally driven to search for order and unity, so that even elementary arithmetical and geometrical exercises will awaken it to contemplate the eternal in the “study of the One” (ἡ περὶ τὸ ἓν μάθησις) (525a). Through their clarity and stability mathematical objects instruct the mind about what it will encounter when it attempts knowledge of the Good itself. But just as mathematicals describe the true forms of sensible bodies, by a kind of proportion (ἀναλογία) they are themselves only shadow images of the Good (511ab). The sister arts are only dreams not fully awake to true being (533c), the prelude to what must be learned (531d). Mathematical knowledge is less scientific than dialectic, for where it relies on axioms, dialectic intuits the Good immediately (533cd). But once reason has ascended to the first principle, it knows mathematicals, too, in the same way (511cd). Plato’s strategy of absorbing the Pythagorean mathematical sciences thus introduces a deep ambivalence into his philosophy. The mediating position of mathematics in the Platonic schema is unstable, removed from its higher office among the Pythagoreans, yet still granted an indispensable anagogical function. Later Platonists had grounds both to exalt

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and to demote the station of mathematics, setting the Old Academy against the New and radical Neopythagoreans against moderates.

Philebus After the Republic Plato put his transformed Pythagoreanism to work. In Philebus he revisits Philolaus’s theory of limiters and unlimiteds and praises number as the key to certain knowledge, but in Plato’s hands these are not philosophy per se but a mediating analogy.46 In the dialogue, Socrates’s young adepts watch as his initial efforts to classify differentia founder on the problem of the One and the many. Socrates’s dialectic is overwhelmed by the irreducible multiplicity of being and faces the prospect of collapsing into an infinite regress. But then Socrates discovers a solution in number, which he calls a Promethean gift of the gods (16c). Plato explains that every instance of being conceals a combination of Limit (πέρας) and the Unlimited (ἄπειρον). Since Limit indwells every infinity, the philosopher can rely on finite numbers to order the cosmic manifold. Number performs an initial delimitation of infinity, and the limited multitude that results can then be reduced by the philosopher’s dialectic to the One. Thus number saves philosophy from sophistry by suspending an otherwise infinite disputation (16d–17a). At first this sounds like a tribute to Philolaus. But as Huffman points out, Plato transmutes his predecessor’s epistemology into a metaphysics of participation.47 In place of Philolaus’s plurality of concrete limiters and unlimiteds intersecting in quantitative harmonies, Plato proposes two singular megaforms of Limit and Unlimited. These hypostasized mediators reconcile the One and the many, bridle the untamed multiplicity of being, and therefore stabilize Socratic dialectic. Thus the distinctively Platonic signature becomes legible: mathematics serves, and conserves, philosophy by bridging the world and its Beyond.

Timaeus Even more than Philebus, the Timaeus is viewed as Plato’s most Pythagorean dialogue, given the resemblance between Archytas’s portfolio of interests and its themes. Fragments of Archytas and ancient testimonia speak to his interest in harmonic proportions as a way to define motion and its causes, his speculations about plant and animal shapes, and his theory of vision—all elements of Plato’s dialogue. But our recently improved sense of Archytas’s profile as a working mathematician changes the valence of these topical parallels. Huffman argues that Plato engages Archytas’s views in the Timaeus only in order to criticize or adjust them, as with Philolaus in the Philebus.48 Archytas denied the fundamental separation of sensible and intelligible worlds that Plato held as axiomatic and indeed pronounced with greater solemnity in Timaeus than anywhere else (27e).

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Despite its Pythagorean topoi, Plato uses the cosmogony of the Timaeus to reiterate his revisionist theory from the Republic. Philolaus and Archytas seemed to privilege mathematicals in a strictly epistemological sense. For them a given arithmetical ratio that structures a musical harmony, for instance, has no being, no history, and no intelligibility separate from the concrete harmony itself. By contrast, the Timaeus grants the world’s mathematical infrastructure a being, history, and intelligibility detachable from the physical universe. The dialogue effects this separation both chronologically, in that mathematicals provide the model for the world’s generation, and anagogically, in that mathematicals mark out a pathway for ascent back to the world’s transcendent origins. To adapt Haeckel’s phrase, ontogeny recapitulates phylogeny: the birth story of the universe in Timaeus narratively locates mathematicals in the mediating position that Plato first theorized in the Republic. Eternal patterns of number, geometry, harmonics, and astronomy reconcile the incommensurable realms of mere becoming and true being. The physical universe is built out of such proportions and rhythms, and imitating them draws one closer to eternity. A few examples from the dialogue will suffice to show how Plato articulates the mediating role of mathematicals through such cosmological and ethical patterns of mimesis. First, Plato uses the cosmogonical myth as a vehicle for defining the mathematical substructure of the world, such that the mind wishing to transcend the world must begin with knowledge of its arithmetic and geometry. In Plato’s account the divine demiurge, associated with the Good, first crafts an eternal, perfect model (παράδειγμα) of which the physical cosmos is a reflection (31a). This three-leveled vision (demiurge, model, cosmos) does not explicitly identify the eternal model with mathematicals, but it does posit an unchanging level of subdivine being in which the perfect analogue of every physical being abides. The demiurge constructs the “body” and “soul” of the world according to harmonious proportions. The four elements that comprise all solids are built out of the simplest spatial units, namely triangular planes. Each represents a different combination of numbers: fire as pyramids, earth as cubes, air as octahedrons, and water as icosahedrons (55d). The demiurge infuses soul into the world from its center to its circumference, setting the universe into rotation (36e). The animation effected by this world-soul results from the demiurge weaving together the Same (being) with the Different (becoming) in precise proportions, deriving a median between them, and then distributing the composite through a series of complex ratios. Soul generates cosmic revolutions in an unchanging outer circuit, reflecting unity, and in an inner circuit of divergent planetary orbits, reflecting number (34c–36d). Mathematical beings also mediate between time and eternity. Plato famously remarks in the Timaeus that time is the “moving image of eternity” (37d). Often overlooked is the mathematical basis of his definition. Whereas eternity rests in unity, the demiurge designed time to be equally eternal but to move according to

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number:  the bridge between time and eternity is arithmetical order. Plato envisions astronomical patterns instilling universal rhythms that render time’s motion perceptible to human experience. The movements of planets and stars are governed by the motion of the world-soul. Ensouled life forms like plants and animals participate in number when they react to diurnal cycles, learning arithmetic from the tides and the seasons (39b). Finally, in Timaeus Plato repeats the anagogical role of mathematics posited in the Republic. When the portion of the world-soul embodied in human beings encounters matter, it disrupts the harmonious balance of Same and Different and causes ignorance and confusion (43cd). But according to Plato the design of the universe includes a remedy to this problem. The mathematical patterns evident in nature—from musical harmonies to the regular courses of planets—ensure universal access to unchanging exemplars of cosmic order that can moor human life in the stability of number: God invented and gave us sight to the end that we might behold the courses of the intelligence in the heaven, and apply them to the courses of our own intelligence which are akin to them, the unperturbed to the perturbed; and that we, learning them and partaking of the natural truth of reason, might imitate the absolutely unerring courses of God and regulate our own vagaries. . . . Moreover, so much of music as is adapted to the sound of the voice and to the sense of hearing is granted to us for the sake of harmony; and harmony, which has motions akin to the revolutions of our souls, is . . . meant to correct any discord which may have arisen in the courses of the soul, and to be our ally in bringing her into harmony and agreement with herself; and rhythm too was given by [the Muses] for the same reason, on account of the irregular and graceless ways which prevail among mankind generally, and to prevail against them.49 Astronomy and music, among the other mathematical arts, train the human soul to observe and then imitate the numeric patterns built into the universe through the world-soul. The logic of the anagogical function thus implied requires a mediating position for mathematicals that we do not find in Philolaus and Archytas. Beyond these dialogues, there are several testimonies to unwritten teachings of Plato in which he further developed the theories of first principles and mathematics presented in the Timaeus and Philebus.50 In one testimony, Plato was said to have delivered a final public lecture “On the Good,” but his audience left disappointed and confused when the great teacher discoursed at length on mathematics instead.51 Reading between the lines of controversies among Plato’s students, we can reconstruct the skeleton of some oral doctrines.52 There are two

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first principles, the One (Limit, the odd, indivisible) and the Dyad (Unlimited, the even, divisible). Together these produce number, refracted in sets of primary numbers, whether four (the tetrad) or ten (the decad). Plato hinted that the eternal forms were in some way equivalent with these primary numbers. Nevertheless he seems to have distinguished “forms” and “mathematicals” as two classes of beings, and to have located the latter within the world-soul. Beyond this rudimentary sketch, little else can be known with certainty. A better strategy for exploring the consequences of Plato’s transformation of Pythagoreanism is to observe what his disciples found most stimulating or most troublesome.

Mediation and First Philosophy in the Early Academy The novelty of Plato’s theory of mathematical mediation clearly fascinated his students. But judging from the documents that survive, they seem to have been not entirely sure what to do with it, and reacted in different ways. The pseudonymous Epinomis was written to resolve the abrupt ending of the Laws, but even in antiquity Plato’s authorship was doubted by none less than Proclus.53 The short treatise was written within a few decades of Plato’s death by one of his inner circle, perhaps Philip of Opus, but it diverges from the Platonism associated with Speusippus and Xenocrates in the early Academy. The author is apparently familiar with passages from Republic and Timaeus (though little else) and repeats their list of essential mathematical sciences.54 Yet his view of the unity and purpose of mathematics differs from Plato’s. The author of Epinomis praises number as a divine gift responsible for all other good things in life, including virtue and happiness (976e–979b). The pious response to such divine favor is to study mathematics in order to perceive God better within the cosmos: first number in itself, the odd and the even (arithmetic); then incommensurable numbers with respect to surfaces (geometry) and solids (stereometry); and finally the proportions found in rhythms and melodies (harmonics) (990c–991b). Thus far the author is simply reprising the Platonic curriculum discussed above. But then Epinomis departs from Plato in two ways. First, unlike Republic, the author posits a common principle underlying all the mathematical sciences: Every diagram and system of number, and every combination of harmony, and the agreement of the revolution of the stars must be made manifest as one through all to him who learns in the proper way, and will be made manifest if, as we say, a man learns aright by keeping his gaze on unity [εἰς ἓν]; for it will be manifest to us, as we reflect, that there is one bond [δεσμός] naturally uniting all these things.55

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Geometry, arithmetic, harmonics, and astronomy are only fully understood when they are grasped in their commonality. The single “bond” named by the author appears to reference numerically measurable proportions—echoing Philolaus’s allusion to a generic “λόγος that arises from the μαθήματα” and Archytas’s suggestion of a common “kinship.” Although Plato never postulated a single foundation to unify the different mathematical sciences, Neopythagoreans like Nicomachus and Iamblichus accepted the Platonic authorship of Epinomis and read this passage as updating the Republic.56 The author’s second departure from Plato is subtler but more profound. For him the mathematical sciences were not preparatory for dialectic, as Plato makes very clear in Republic and Laws. Rather they prepared one for the cosmotheological meditations of Timaeus, where grasping the truth of the universe opens the mind to know its maker.57 Such ambiguities about the theological function of mathematics will play out in the Old Academy, in Neopythagoreanism, and well into medieval Latin Platonism, as we shall see. The first leaders of the Academy after Plato, Speusippus and Xenocrates, were both preoccupied with the status of mathematics. Plato’s final oral formulae were like a last will and testament that set the agenda for the early Academy’s approach to the dialogues. Both attempted to harmonize Plato’s written account of forms with the mathematical mediation that he found compelling in old age. Speusippus of Athens (ca. 410–339 bce), Plato’s nephew, was the first leader of the Academy after his uncle’s death, if only for a few years. Reconstructing Speusippus’s doctrines is notoriously difficult, since the only available sources stem either from his greatest antagonist, Aristotle, or from Neopythagorean enthusiasts centuries later.58 In his many lost works Speusippus tended to ascribe his own interpretations of Plato to ancient Pythagoreans, sowing the seeds of the Pythagoreanization of Plato in Middle Platonism.59 Nevertheless, Leonardo Tarán and John Dillon have worked up a relatively consistent sketch of Speusippus’s idiosyncratic Platonism. From Speusippus’s point of view, Plato’s principle of the One was so transcendent as to escape every predication. Only God was the One (τὸ ἕν); the principle of numbers was better termed the monad (μονάς). Precisely because it is the cause of being, the One must remain not only “beyond being” (ἐπέκεινα τῆς οὐσίας), as in the famous passage of Republic VI, but also beyond everything else it causes, including Intellect, Goodness, and Beauty, as if entirely cut off from philosophical apprehension. Aristotle and Xenocrates alike rejected this radical henology, insisting that calling the One beautiful and good is a nonnegotiable part of Platonism.60 Out of the transcendent One, Speusippus then generates further first principles by successive conjugations of the One with “multiplicity” (πλῆθος), also known as prime matter (ὕλη). One and multiplicity together produce both monad (the indivisible unit of number) and point (the indivisible unit of space), either sequentially or simultaneously. Monad unites with multiplicity to generate the series of numbers, just as point unites with “extension” (διάστημα) to generate space and thus the

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full range of geometrical planes and solids.61 Only with the appearance of number, and the possibility of numeric harmonies, is beauty first glimpsed in the world.62 Multiplicity unites with this nascent geometry to produce two further levels: soul (“the form of the omni-dimensionally extended” is Dillon’s evocative translation) and finally the body of the world, or physical universe.63 Speusippus’s vision takes leave of Plato in fundamental ways. First, he uses the concept of “number” in a more flexible and polysemous way than either Plato or Aristotle would countenance.64 Ordinary numbers in arithmetic are just one (lesser) instance of “number” reiterated through cascading levels of the One’s virtual manifestation within successive domains of multiplicity:  the transcendent One, the monad, the point, geometrical space, world-soul, and the physical universe. The second departure concerns this very hierarchy of beings. Where are Plato’s forms? According to Aristotle—perhaps with knowledge of the lost Speusippean work On Pythagorean Numbers, or perhaps out of polemical provocation—Speusippus simply substituted numbers for the Platonic forms and placed numbers above the world-soul.65 We should take Aristotle’s attack with a grain of salt. But it does seem that for Speusippus the forms’ function of providing a stable terminus for dialectic is taken over by the monad and the point, while the forms are collapsed into the world-soul.66 Plato’s desire to locate mathematicals between sensibles and forms is effectively inverted, with number ending up on top. Speusippean Platonism is certainly eccentric, but his vision of an ineffable One known through its unfolding in eternal mathematical structures proved enormously influential with Nicomachus and Iamblichus, making Speusippus the godfather of Neopythagoreanism.67 We know just as little about the doctrines of Xenocrates of Chalcedon (396–314 bce), the second head of the Academy.68 Xenocrates’s calm leadership over two decades allowed him to systematize Plato’s doctrines and to respond to Aristotle’s stinging critiques of Speusippus. As Dillon points out, the textbook account of Xenocrates—that where Plato held forms above mathematicals, and Speusippus mathematicals over forms, Xenocrates equated them—does not tell the whole story.69 Like Speusippus, he thought Plato’s oral doctrines did not support a sharp division between forms and mathematicals, and he struggled to fashion a coherent first principle out of the abundant raw materials of the different dialogues—the Good, the One, the demiurge, the pair of Limit and Unlimited. Compared with Speusippus, Xenocrates seems to have hewn more closely to Plato’s late teachings. For him the One is a supreme Intellect that thinks the primal numbers, or forms, not unlike Aristotle’s unmoved Mover. If the One’s thoughts are eternal, and if those thoughts define the full spectrum of beings in terms of their fundamental units, then the best name for such divine ideas would be dimensional minima like numbers, lines and planes.70 The mind of God, in other words, is wholly comprised of geometrical objects. Such views led Xenocrates to write

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books on number theory and geometrical dimensions, but also on allegories of the Olympian and Orphic myths that complemented his Pythagorean ascetic practices.71 In one fragment Xenocrates teaches that the monadic male νοῦς reigns over the dyadic female world-soul, from which the twelve Olympian deities descend.72 In his famous Timaeus commentary, Xenocrates deliberately echoed a saying of Pythagoras when he defined soul as “self-moving number” (ἀριθμός ἑαυτὸν κινῶν).73 Hans Joachim Krämer and Detlef Thiel have argued persuasively that Xenocrates not only recovered Plato’s Philolaic roots but built a monistic henology that would lay the groundwork for Eudorus and Plotinus centuries later.74 Among Plato’s immediate students, Aristotle (384–322 bce) instinctively kept the greatest distance from mathematizing interpretations of the master. As Aristotle watched Speusippus and Xenocrates labor over the enigmae of Platonic mediation, he must have had, as Dillon writes, “very little sympathy with the efforts of his former colleagues in the Academy to solve the problem they have set themselves . . . [by] introducing mystical Pythagorean flummery into the scientific discipline of mathematics.”75 We need not rehearse the Stagirite’s views about the “Pythagorean” influences that led his teacher astray or explore his own alternative philosophy of mathematics.76 It suffices to note that Aristotle closely associated his critique of Plato’s separated forms with his critique of Plato’s mathematical philosophy. In Metaphysics A, Aristotle calls Plato’s theory of participation in forms a warmed-over version of the “Pythagorean” imitation of numbers.77 Where the Pythagoreans maintain that all beings are numbers, says Aristotle: Plato states that besides sensible things and the Forms there exists an intermediate class, the objects of mathematics, which differ from sensible things in being eternal and immutable, and from the Forms in that there are many similar objects of mathematics, whereas each Form is unique.78 In Aristotle’s diagnosis Plato was seduced by unnamed Pythagoreans to misconceive the forms as if they were separate and eternal mathematicals. For our immediate purposes it does not matter if Aristotle’s account of Pythagoreanism is true, or if he is right about the coherence of Plato’s theory of forms. What is more interesting is that as an early interpreter of Plato, Aristotle too—in one voice with Epinomis, Speusippus, and Xenocrates—immediately grasped the distinction of Plato’s modified Pythagoreanism to be his deployment of mathematical beings as mediating between sensibles and intelligibles. This use of mathematics does not exist in any pre-Platonic Pythagorean source. Yet despite his severe criticisms of Plato’s three-tiered universe, even the great Aristotle found himself ultimately unable to escape its centripetal pull, at least

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when it came to the division of the sciences. In Metaphysics E, Aristotle ends up retaining the same fundamentally Platonic model of the disciplinary placement of mathematics.79 Physics studies things that move (κινητά) insofar as their forms are inseparable from matter (ἀχώριστα). First philosophy, or theology, studies unchanging things (ἀκίνητα) whose forms are absolutely beyond matter (χωριστά). Mathematical science mediates theology and physics because it shares one trait of each, although it is difficult to say which. Are mathematical beings unchanging and eternal, but always bound to concrete cases of matter? Or are they absolute from matter like the eternal forms and yet somehow subject to immaterial motion or change? This is a crucial and telling ambiguity in Aristotle’s appraisal of Plato.80 Despite its theoretical difficulties, Aristotle reluctantly acquiesces to the mediating role of mathematics, even if that makes it all the more challenging to define the boundary between mathematicals and forms, and thus to identify what makes the method of theology different from mathematics. Aristotle outlines this disciplinary riddle in Metaphysics E.81 Which of the three sciences can resolve the ambiguity about the character of mathematicals? Physics is disqualified by definition, since it studies only moving things, and mathematics is the science in question. If theology decides the question, Aristotle risks subsuming mathematics within theology, given the partial overlap between the character of forms and mathematicals—but this is the Pythagorean error he wants to avoid. So given a choice between identifying mathematics with theology or leaving it an indeterminate, intermediary science, Aristotle chooses the latter. This at least preserves, he notes, a distinct theological science to decide precisely this type of question concerning the boundaries among the three. Aristotle’s profound dilemma spawned divergent textual versions among copyists disagreeing over the right interpretation of this passage in Metaphysics. It also bedeviled subsequent medieval commentators. Boethius transmitted the problematic text to Latin Christianity without amendment in his treatise De trinitate, as we shall see. But later Boethian commentators like Eriugena, Gundissalinus, and Thomas Aquinas either ignored the issue or sidestepped it.82 Hence when Thierry of Chartres labored over this passage in De trinitate for two decades, he was not just exegeting another line of Boethius, but confronting the unresolved early Academic controversy over Plato’s revised Pythagoreanism. What is the boundary between the mathematical absolute and the theological absolute? Such was the question Thierry found himself inadvertently unearthing when he attempted to read Boethius in a new light. Historically, of course, the question did not linger unanswered until the twelfth century and had no reason to wait for Christianity’s response. On the contrary, Plato’s controversial modifications to early Pythagorean philosophies of mathematics sowed the seeds of an eventual re-Pythagoreanization of Platonism, a movement already underway in Alexandria and Rome as Christianity’s first century commenced.

2

The Neopythagorean Revival: Henology and Mediation For two centuries mathematical Platonism went dormant during the New Academy. Some decades after Xenocrates, Arcesilaus and Carneades decided to trade the elaborate hierarchies of the Old Academy for the skepticism Socrates displayed in his dialogical practice. But in the first century bce the pendulum swung back the other way under Antiochus of Ascalon (ca. 130–68 bce), who handed down a Stoic-influenced Platonism to students like Cicero and Varro.1 Antiochus thus returned the Athenian Academy to the methods, if not the views, of Speusippus and Xenocrates: the heart of Platonism was expounding the dogmas revealed in the sacred dialogues. Without sharing their conviction that Plato’s highest philosophy was mathematical, Antiochus opened the door to a Pythagorean revival, although now within the new cultural situation of the Empire in the first century bce. Rome’s dominion over the Mediterranean had collected a motley variety of religious and wisdom traditions, including Judaism and eventually Christianity. Historians of Christianity are accustomed to picking up the thread of Platonism a few centuries later. In Alexandria, a city with a long-standing Jewish population that became a leading Christian patriarchate, Platonism developed in a sustained conversation with biblical monotheism, first with Philo and then, by the turn of the third century, with Origen of Alexandria. In Rome, Plotinus’s writings from the 250s and 260s were published by Porphyry in 301 ce, translated by the Christian convert Marius Victorinus in the mid-fourth century, and then dropped into the hands of another north African in the summer of 386, the young Aurelius Augustinus. In Athens, Plutarch would lead the Academy through a Neoplatonist renaissance, which bore fruit especially with Syrianus and Proclus, who through sundry channels exerted enormous posthumous influence over medieval Christian thought. But around the turn of the first century ce such familiar developments still lay far off, when two Platonists, Eudorus and Moderatus, began to reorganize Platonist doctrine by the lights of Pythagorean traditions. In this period of Middle Platonism, well before we reach Origen, Plotinus, or Augustine

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(let  alone Proclus or Boethius), three important developments were simultaneously underway. First, the project of consolidating Platonist first principles after the manner of the Old Academy entailed naming which primary beings anchored the descending hierarchy of lesser principles. This protological or henological enterprise was inevitably a theological one as well, particularly in Timaeus exegesis that identified the demiurge with God. The search for the transcendent One led to a multiplication of intermediate divine principles, the better to isolate the supremacy of the “one” One.2 Henology and mediation worked hand in hand. Second, authors who sought continuity with original Platonism considered Pythagorean elements to indicate a given doctrine’s antiquity and authenticity. For this reason, Neopythagoreanism became an early and prominent strain in Middle Platonism. Indeed, much of what seems new in Neoplatonism can be linked to pre-Plotinian henologies authored by Neopythagoreans, as we shall see, even if later Neoplatonism, having absorbed its benefits, sought to curtail the Pythagoreanizing impulse. Finally, as is well known, it was in these same centuries that Christianity gradually distinguished itself from apocalyptic Judaism, in part as Christian theologies began to find their footing in dialogue with Hellenistic philosophies, in authors like Justin Martyr, Tertullian, or Clement of Alexandria. The rise of Neopythagorean Platonism and the rise of Christian Platonism were therefore roughly simultaneous events. This simple observation reminds us that Neopythagoreanism and early Christian theologies did not have the opportunity to interact in late antiquity, beyond a few chance encounters that I shall note below. By the time Christian intellectuals were prepared to grapple fully with Neoplatonism in the fourth to sixth centuries, the mathematizing strains of Middle Platonism had already been marginalized or coopted by the more influential Plotinian and Proclian alternatives.

The Origins of Henology in Eudorus and Moderatus During the New Academy’s turn away from Pythagoreanism, a collection of anonymous texts composed in the third or second century bce remained underground awaiting excavation by later Pythagorean sympathizers. A  trove of pseudo-Philolaean and pseudo-Archytan writings stem from this period, as do the anonymous Pythagorean Notebooks reported by Alexander Polyhistor via Diogenes Laertius that preserve the Old Academic doctrines of monad and dyad.3 The arithmological traditions that reappear in Varro, Philo of Alexandria, Nicomachus of Gerasa, Theon of Smyrna, the Iamblichean Theologoumena arithmeticae, Calcidius, Macrobius, and Martianus Capella, all stem from a third-century authority close to the Pythagorean Notebooks. This common “S” tradition associates the ten numbers of the decad with deities or divine attributes.4

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If such arithmology is not what is new in Neopythagoreanism, neither is the renaissance of “akousmatic” Pythagoreanism revived as a religious option in Rome around the time of Cicero in the first century bce.5 Cicero rarely references Pythagorean mathematics in his Dream of Scipio beyond the “music of the spheres.” Instead, he is most impressed by the moral example of the sect, whose ascetic lifestyles permitted them to amass an almost superhuman breadth of knowledge. The erudite Marcus Terentius Varro (116–27 bce), whose lost books transmitted Greek learning to Roman authors for centuries, was buried “in a Pythagorean manner” according to Pliny. Varro’s Disciplinae covered the four arts of geometry (including optics and geography), music, arithmetic, and astronomy, and his nine books on De principiis numerorum handed down arithmological lore.6 Cicero also reports that the unusual Nigidius Figulus (ca. 100–45 bce), widely distinguished for his mastery of the liberal arts, contributed to the renewal of Pythagorean ritual cult and dietary restrictions. Nigidius’s theurgical projects were recounted in his several books on dreams, astrology, and theology (also now lost).7 Instead, for the earliest stirrings of a renascent “mathematic” Pythagoreanism in the generations after Antiochus, we have to look outside Roman circles to two lesser-known figures, Eudorus of Alexandria and Moderatus of Gades.8 Eudorus (fl. ca. 50 bce), probably another student of Antiochus, identifed his Platonism with the ancient Pythagoreanism of Philolaus and Archytas, Speusippus and Xenocrates.9 Unlike his teacher, Eudorus wanted to revive not just the Old Academy’s methods, but also its enthusiasm for Plato’s oral teachings that revealed divine truths. His familiarity with Aristotle inspired him to organize the first principles of Platonist philosophy after the model of the Metaphysics.10 Toward this end, Eudorus returned attention to dialogues that the New Academy had relegated to physics or logic in what Dörrie has called its “traditionless naïvete.”11 This Eudoran brand of henological Platonism laid the foundation for Plotinus in the third century. Eudorus brokered a compromise between two opposing tendencies within Pythagoreanizing Platonism by stipulating that the supreme One stood above two lesser principles: a second One or monad (representing form) and the indefinite dyad (representing matter). On the one hand, Plato’s written works like Republic and Laws had suggested a unitary transcendent principle, which Speusippus, for example, exalted far beyond number, being, and goodness. On the other hand, the unwritten Platonic teachings that the Old Academy linked to Pythagorean traditions—but also Plato’s Philebus and Timaeus—hint at a dualism of the limited monad and the unlimited dyad, recalling the ancient Pythagorean tables of “elements” opposing male/female, odd/even and good/bad. Eudorus’s triad unites both tendencies by distinguishing the numerically one (the monad-dyad) from the theological One, but also deriving the former from the latter.12 This set in motion an elegant coordination of number and the divine principle, a new monistic

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henology whereby God’s unity is the fount of arithmetical unity. There are no clear parallels in Philolaus, Plato, or Xenocrates for Eudorus’s insight. It is neither purely Pythagorean nor purely Platonist, but something altogether new.13 Although Eudorus’s idea never caught on in the Academy, it apparently did among the profusion of Pythagoreanizing pseudepigrapha from the same period.14 Eudorus developed his triad of One-monad-dyad in connection with his rereading of the Timaeus. The demiurge uses the separated forms to shape passive matter, but Eudorus interprets this hierarchy in light of the Philebus, in which Limit and Unlimited are united by a supreme Cause. By ranking the demiurge above Limit and Unlimited, Eudorus installs a single transcendent principle that generates both: thus the One (God) stands above the monad (forms) and the dyad (matter).15 This is the first instance in Platonism of an ultimate principle uniquely transcending the opposition of lesser opposites. The method of rejecting every relative comparison in order to define an ultimate principle is the basis of negative theology in Platonism and Christian theology.16 Yet at the same time Eudorus does not entirely remove the divine One beyond all understanding, as Speusippus proposes. Rather the One is reflected, through the monad, in the structures of number. Eudorus can therefore be identified as the instigator of henological theology, the first of a line that would continue from Plotinus and Proclus to Ps.-Dionysius, Meister Eckhart, and Nicholas of Cusa. If Eudorus is the first henologist, Moderatus of Gades (fl. ca. 50 ce) is the second but perhaps the greater.17 The name Μοδέρατος Γαδειρέως refers to Cádiz in modern Spain, but the circulation of his testimonies suggests that he spent considerable time in the eastern Mediterranean and perhaps even Alexandria. Moderatus shared Eudorus’s belief in a supreme One against the dualism of some Platonists like Numenius, and he repeated the link between One and monad.18 But compared to Eudorus, Moderatus’s henology was even more explicitly connected with number. In one passage reported by Porphyry in his Life of Pythagoras, Moderatus offers an original defense of Pythagorean arithmology. Having grasped the ineffable first principles, he explains, the ancient Pythagoreans transmitted their wisdom in mathematical terms for the sake of clarity, since only numbers can express the transcendent principles in an unambiguous, universal way: And in this way the Pythagoreans called the idea of “sameness” [ταυτότης] and “oneness” [ἑνότης] and “equality” [ἰσότης], and the cause of concord [συμπνοία] and sympathy [συμπαθεία] in the universe and the cause of preservation of that the condition of which remains just the same, “one” . . . And the ideas of “otherness” and “inequality” and of all that is divisible and in change . . . they called the double ratio and the dyad.19 Porphyry considers further examples. Everything with a beginning, middle, and end is perfect; hence the triadic form (τριοειδής) represents metaphysical perfection.

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He defends the Pythagorean decad in a similar way, and speculates that the decline of the Pythagorean tradition stemmed in part from their opponents’ willful misapprehension of this sane rationale for arithmology.20 Elsewhere Porphyry recorded two remarkable doctrines of Moderatus transmitted from Simplicius.21 First, where Eudorus conceived of a single, transcendent One reflected in the monad-dyad pair, Moderatus distinguished a trinity of descending Ones. The supreme One is beyond Being, the second One is the intelligible realm of the Forms, and the third One is Soul; after Soul, sensible matter is ordered by those Ones now manifested as Quantity. Moderatus’s notion of a threefold One notably predates both the Plotinian hypostases and Christian Trinitarian doctrines.22 Harold Tarrant has even suggested that these three Ones are hinted at in the passage quoted above: the λόγος of Oneness, the λόγος of Sameness and Equality, and the cause of harmony and “conanimation” (συμπνοία).23 The second Moderatan doctrine is that matter is generated by two principles working together: a unifying or unitary Logos (ἑνιαῖος λόγος) and pure Quantity (ποσότης), which Moderatus now identifies with the χώρα of the Timaeus. This numerical Quantity is the archetype of ordinary matter, which has only a lesser degree of quantity (ποσόν)—in Tarrant’s words, quantity “qua dimension and extension rather than qua form.”24 In short, as Merlan explains, Moderatus conceived of a threefold One as a “self-contracting deity” that “releases” pure Quantity when it contracts to a lesser degree of being, namely the mathematically ordered universe.25 There is great disagreement about how to harmonize these two Moderatan doctrines. E. R. Dodds and Dillon established the dominant view: Moderatus, following Philo and en route to Plotinus, meant to assimilate the “unitary” Logos to the second One.26 Matthias Baltes contends that Moderatus rather has Timaeus in mind. The demiurge corresponds to the first One, and the activity of the unifying Logos corresponds to the third One, namely the world-soul.27 But Christian Tornau has made the most convincing case. When understood on his own terms, and not as anticipating Plotinus, Moderatus clearly equates the first One with the Logos. Like Eudorus, Moderatus draws upon the Old Academy’s monad and dyad, but converts them, respectively, into unitary Logos and contracted quantity.28 The god of Moderatus, then, is a supreme intelligent Monad that exerts the force of its unity upon matter through the structures of quantity. This entails that mathematics is what enables lesser intelligences to reason theologically—just as the Pythagoreans had always maintained. The divine One is known preeminently through number: known, because it is Logos, and through number, because it is the One. Hence with Moderatus’s configuration of the three Ones, the Logos, and Quantity, we see the first rudimentary sketch of a mathematical theology within Middle Platonism. Moderatus also discusses several theories of number that will reappear in Theon of Smyrna and Nicomachus of Gerasa in the next century. Although

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his famous triad identifies the first One as the monad, in another fragment Moderatus distinguishes the monad as the principle of number from the One as the principle of the real things numbered. He defines number as a function of monads in three ways. (1) Number is a “system [σύστημα] of monads.” (2)  Number represents a cycle of monadic movement:  “a progression [προ­ ποδισμός] of multiplicity beginning from the monad, and a regression [ἀναποδισ­ μός] ending in the monad.”29 (3) Number results from the limiting of quantity by the monad. Quantity cannot be reduced beyond the monad because the monad (μονάς) is that which remains stable (μένειν) and remains alone (μόνος). Moderatus writes:  “the monad is the limiting case of quantity [περαίνουσα ποσότης], that which is left behind when multiplicity is diminished by the subtraction of each number in turn, and which thus takes on the characteristics of fixity [μονή] and stability [στάσις]. For quantity is not able to regress [ἀναποδί­ ζειν] farther than the monad.”30 Through these definitions Moderatus embedded Pythagorean tendencies within the anatomy of later Neoplatonism. In the third definition, we hear a clear echo of Philolaus’s “limiters” as the ground of quantitative harmonies, and in the second, an early version of Plotinian emanation and even Proclus’s circuit of remaining, procession and return.

Henology on the Margins of Middle Platonism During the innovations of Eudorus and Moderatus, Middle Platonism was ramifying in directions that sometimes intersected with Mediterranean religions. Eudorus’s distinctive philosophical impulses—the search for cosmogonic myths, the speculation on first principles, the quest for saving divinization—were largely ignored by major Platonists connected either with the Athenian Academy, like Plutarch, or with the so-called School of Gaius, like Alcinous. Instead, the monistic, Neopythagorean avant-garde of Middle Platonism was closer, in both aims and influence, to such Mediterranean religions than to the Academic mainstream. Hellenistic Judaism, early Christianity, and Christian Gnosticism in the first and second centuries ce all shared similar beliefs in a transcendent divinity that liberated human beings from illness and death by mediating holy power through sacred materials, rituals, and institutions.31 At its core, the Neopythagoreanism of Eudorus and Moderatus was equally a search for reliable mediations of the divine One—not through creation myths, divine emanations, or sacred rituals, but through the primordial order, security, purity, beauty, and power of mathematical beings. Before we arrive at Nicomachus’s mature mathematical theology in the second century ce, we must first survey some of this surrounding landscape. Three new intellectual developments, shared to varying degrees by Jewish, Christian, and Gnostic theologies, each arose in interaction with the new

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Neopythagorean henologies. They also generally speaking portend a hermeneutical turn from Timaeus to Parmenides.32

Logos and Mediation The first development is the reinterpretation of the Stoic and Platonist “Logos” as a cosmic mediator between the biblical God and creation.33 This modification begins with Philo of Alexandria (ca. 20 bce–40 ce), a Greek-educated Jew who wrote an extensive series of scriptural commentaries and theological treatises a few decades after Eudorus. As David Runia notes, Philo himself was more of a “witness” to contemporary Platonisms than a participant; indeed, Clement of Alexandria (ca. 150 ce–211/215 ce) refers to Philo as a “Pythagorean,” not a Platonist.34 For Philo, Pythagoras was the essential conduit who linked the genius of Moses to the writings of Plato, which meant that the task of philosophy was to harmonize biblical monotheism, Pythagorean arithmology, and the various Platonist mediations, above all the Stoic doctrine of the cosmic Logos. Thus the cosmogony of Genesis aligns perfectly with the Timaeus, since God first created an intelligible world before the sensible one. This intelligible model represents Plato’s forms, or the mind of the Creator, which is the Logos, the location of the divine ideas. Philo’s Logos theology had far greater influence on early Christian Platonists like Clement of Alexandria than on the Athenian Academy.35 For Philo, the unnameability of the Tetragrammaton was reflected in the One’s transcendence beyond being in the Republic. Philo preserves the transcendence of the One by elaborating several different modes of mediation—by the Logos, by numbers, by the Pythagorean monad-dyad, by abstract virtues—but feels no need to organize or prune these different mediating schemata. In his commentary On Abraham, discussing the apparition of three angels at Mamre, Philo remarks that the divine One can be viewed in two ways. To lesser minds, the One appears as three, moving from the πλῆθος of numbers, through the dyad, to the monad; but to pure minds, the One remains “alone” from all things.36 Yet other principles like Wisdom, Justice, or Goodness proceed out of the Logos, manifesting correlative arithmological properties.37 Even Philo’s Logos, as John D. Turner explains, seems to have a higher and lower aspect. As God’s mind it contains the totality of forms and serves as the eternal paradigm of the cosmos. But as God’s instrument it takes on the functions of the world-soul, mobilizing the totality of numbers in order to create that cosmos.38 Forms and numbers are two faces of the same Philonic Logos. So it comes as no surprise that in passages where Philo defines the role of the divine Logos, he frequently turns to arithmetic and geometry. One classic passage in Who is the Heir of Divine Things, a commentary on Genesis 15, serves as a good example. The text concerns Abraham’s sacrifice, and Philo focuses on a verse about “dividing” the offering in the middle. Divided

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things produce equal parts, but no creature can produce equality, Philo remarks. Only God can perfectly divide a thing into equal parts with the “sharp edge” of his “all-cutting Logos” (λόγος τομεύς).39 It was this Logos that distinguished night from day and land from water in creation; likewise, only the Logos can attain absolute equality (ἰσότητος ἀνωτάτω) beyond indivisible minima. In creating the world God used four categories of equality—equality of number, magnitude, force, or proportion—that recall the four arts of arithmetic, geometry, astronomy, and music.40 Moses, whom Philo names the great “eulogist of equality,” employed arithmetical and geometrical harmonies in the Pentateuch. The monad, as an immutable source of number not a number itself, is therefore an “image of the God who is alone from the many.” Other creatures possess their own self-unity through God’s Logos, their inner “glue and bond.”41 It is only at the end of this lengthy meditation on divine unity, divine equality and the four mathematical arts that Philo pens his famous account of the Logos. As God’s “chief messenger,” the Logos “stands on the border and separates the creature from the Creator,” simultaneously as “suppliant” on behalf of humankind and as “ambassador” on behalf of God. The Logos is “neither uncreated as God, nor created as you, but midway between the two extremes, a surety to both sides.”42 But the foregoing Neopythagorean context reveals that Philo’s Logos is not only a Platonist principle of mediation between the intelligible and sensible. It is also the agent that distributes God’s transcendent equality and monadic power to the enumerated order of creation. As is often noted, Philo’s Logos theology, far more sophisticated and many-sided than we can explore here, subsequently influenced Christian authors in the Alexandrian ambit like Justin Martyr, Clement, Origen, and even Valentinus.43 Like Philo, Clement identified the Logos with the One and both with the totality of divine ideas.44 Clement also used Philo’s Logos to interpret the Christian Trinity: the Father is the absolute unknown One, but the Son is the known Logos-principle.45 For Philo, the Logos is “the measurer” who apportions God to the world; for Clement, Christ measures God to humanity and measures humanity to God.46 Justin’s Logos theology drew on Philo and Alcinous alike, but added Neopythagorean elements from Theon of Smyrna and the notion of λόγοισπερματικοί from the Stoics.47 Can we know, beyond the geographical coincidence, whether Philo was specifically influenced by Eudoran henology, as Willy Theiler and A.-J. Festugière have argued?48 Runia suggests that the question may be impossible to answer until we know more about Eudorus’s lost Timaeus commentary.49 But Mauro Bonazzi counts several significant parallels between Philo’s approach to Plato and those of early Neopythagoreans like Eudorus. They both stressed the “separateness and superiority of the first divine principle” which they name “God” or “the Divine.” They both sought out numbers as well as forms as mediating entities

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between God and matter. But they differed on the problem of mediation. Where the Neopythagoreans, including Eudorus, substituted mathematical structures in place of the forms or identified them, Philo despite his arithmological predilections chose to retain the forms.50

Neopythagoreanism and Gnosticism The mythic narratives of Valentinian and Sethian Gnosticism often incorporated Neopythagorean ideas. In the late second century, the heresiologists Irenaeus of Lyons and Hippolytus of Rome blamed Pythagorean traditions for inspiring Christian Gnostics. Given their desire to discredit doctrinal “errors” by tracing their ancient origins, such claims have rightly raised historians’ suspicions. But new research has uncovered concrete points of contact between Neopythagoreanism and Gnostic Christian theologies irrespective of sectarian polemics. As Joel Kalvesmaki has shown, while Irenaeus criticizes his adversaries for arbitrary arithmologies, his use of similar methods implies several principles for proper Christian arithmology.51 Many Gnostic authors used negative theology, some employed arithmologies, but almost none articulated henologies.52 Hippolytus’s portrait of Pythagorean tradition betrays familiarity with Eudorus’s henology, Speusippus’s Timaeus interpretation, and Philolaus’s universal mathematics.53 By decoding the ornate Valentinian cosmogonical myths into philosophical terms, Einar Thomassen has demonstrated precise correlations to Neopythagorean theories that derive matter from an originary monad.54 Since Valentinus (d. ca. 160 ce) was educated in Alexandria in the mid-second century before founding his community in Rome, it is entirely possible that Eudorus, Moderatus, and even Nicomachus were major sources of Valentinian physical theory.55 Moderatus in particular functioned as a template for Valentinian narratives of the fall of Sophia. The Moderatan three Ones correspond neatly to the Valentinian divine triad of βυθός (the supranoetic depth), πλήρωμα (the fullness of the forms), and σοφία (the world-soul). According to Moderatus, matter arises when the unitary Logos “withdraws” to negatively generate Quantity, the empty form of multiplicity; but Quantity nevertheless remains amenable to reason since its formlessness is “bounded” by magnitude. The Valentinian allegory renarrates this conceptual account as a tragedy of abandonment. Pleroma withdrew from Sophia, stripping her of reason as she declined into the sensible world, but then her descent was checked by Limit.56 Valentinian Gnosticism also adopted geometrical concepts favored by monistic Neopythagoreanism for explaining the separation of the dyad from the monad, that is, the movement of the One into plurality. Just as Moderatus defined quantity as an “extension” of the monad, Sabellians like Marcellus of Ancyra held that the One God “spreads out” into the Triad. For Valentinus, God “extends” (ἐκτείνειν) or “spreads out” (πλατύνειν) into the multitude

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of numbers, just as a point extends into a line and spreads into a plane. Some Valentinian uses are negative, such as when Sophia spreads out in her separation from the Pleroma, but some are positive and theological. In the Tripartite Tractate, the divine Son “extended himself and spread himself out” in order to give “firmness, location and a dwelling place to the Entirety” (that is, the forms). The Savior was fully extended and spread out on the cross, which in Valentinianism represents the Philolaean “Limit.” In another text the Father-Monad “spreads out” into a tetrad in a movement of self-revelation, such that the second divine person is associated with the fourfold.57

Henology and Negation The last development to consider is the rise of negative theology. This tradition does not begin with Plotinus or Ps.-Dionysius, as is sometimes said but rather in henologies connected to Neopythagoreanism. H.  A. Wolfson established that Philo was the first to describe God as “unknowable, ineffable [ἄρρητος], and unnameable,” but the Alexandrian never offers a theory of negation as a species of theological predication. That step is rather taken by Alcinous in his second-century Platonist handbook Didascalicus, around the time of Nicomachus.58 Alcinous states that God can be named in three ways: by negation, by analogy, and by causation. His examples for the second and third ways are the Good in the Republic and Diotima’s ladder to Beauty in the Symposium. But because Plato does not hold that God is ineffable, Alcinous must turn elsewhere to explain the negative way, namely Euclid’s theory of the mathematical point. “The first way of conceiving God,” he writes, “is by abstraction [ἀφαίρεσις] of these attributes, just as we form the conception of a point by abstraction from sensible phenomena, conceiving first a surface, then a line, and finally a point.”59 John Whittaker has shown that this passage draws on a preexistent Neopythagorean commentary on Euclid composed around the time of Eudorus or Moderatus.60 The unity of the One is manifested by subtracting everything not One, so that negation and henology work together. Negative theologies appear in several other second-century theologians. The Gnostic Basilides exalted God beyond ineffability.61 Clement of Alexandria repeated the “point” example but seems independently influenced by the Neopythagorean monad doctrine: “For the point which remains is a monad, so to speak, having position, from which, if we take away position, there is the conception of monad.” But, Clement adds obscurely, from the monad the Christian ascends to the higher purity of the μέγεθος τοῦ Χριστοῦ, the quantity of Christ.62 Or as Clement writes elsewhere, “God is One, both beyond the One and beyond even the monad.”63 Around the same time, the apologist Athenagoras defended Christian monotheism by allying himself with Greek philosophers who named God an “ineffable number” (ἀριθμός ἄρρητος) or the “maximal number” (ἀριθμός μέγιστος ).64

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In the next century Plotinus (204 ce–270 ce) would systematize such negative theology, substituting ἀπόφασις from Aristotelian logic in place of Alcinous’s mathematical ἀφαίρεσις, and then joining it with Moderatus’s three Ones. Like his predecessors he retained the Logos, identifying it with Moderatus’s second One. But Plotinus also simplified their various hierarchies by removing the mathematical mediation that had accompanied the Logos in Philo and Moderatus. By and large Plotinus gleaned his Pythagorean influences not from the monistic mathematical Pythagoreanism we have surveyed above but from the dualism of Numenius of Apamea (fl. ca. 150 ce), to the degree that Porphyry had to defend his teacher from charges of plagiarism.65 Although Numenius is often lumped together with Eudorus and Moderatus, in truth he belongs to a category of his own.66 Numenius thought that purifying Platonist doctrine meant returning it to its Pythagorean roots, but his few surviving fragments reveal greater interest in ascetic or “akousmatic” Pythagoreanism than in number or mathematics. In one fragment he directly attacks the monism of Moderatus.67 By contrast Numenius fused Plato’s ambivalence toward matter with Hermetic, Gnostic, and Zoroastrian traditions to produce a more dualistic variety of Platonism. This Numenian path was followed not only by Plotinus and by Calcidius after him, but also by magisterial Christian Neoplatonists like Origen, Augustine, and Gregory of Nyssa.

Mathematical Theology in Nicomachus of Gerasa In the same second-century period the Neopythagorean tradition produced its own native heir to Eudorus and Moderatus unconcerned with Jewish, Christian, or Gnostic parameters. Born in eastern Palestine, Nicomachus of Gerasa (ca. 70–150 ce) spent his life in Alexandria, where his works continued to be studied for centuries.68 Of the many works attributed to Nicomachus, only three survive. Nicomachus’s Harmonic Manual analyzes musical proportions in terms of number.69 Nicomachus’s lost Theology of Arithmetic exists only in fragments.70 Dominic O’Meara laments that while the latter treatise remains “of considerable importance” for late antique Platonism, it “has scarcely been examined as a work in itself in modern research.”71 The third book, his Introduction to Arithmetic, falls within an established genre of elementary handbooks that prepares readers to follow mathematical references in Timaeus or Republic.72 Theon of Smyrna’s Mathematics Useful for Understanding Plato, written around the same time, shares some overlapping traditions.73 But Nicomachus’s book stands out in two ways. First, as Tarán remarks, Nicomachus’s book was simply “the most influential work on arithmetic from the time it was written . . . until the sixteenth century,” and duly earned the Gerasan a measure of

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fame among late antique philosophers.74 He was known to Lucian and Porphyry as a leading mathematician and may have corresponded with Plotina, wife of the emperor Trajan.75 The book was commented on by Asclepius of Tralles, John Philoponus, and later Iamblichus, was translated into Latin by Apuleius in the fourth century and again by Boethius in the sixth, translated into Syriac in the eighth century and into Arabic twice in the ninth century (drawing the attention of al-Kindi, who offered some revisions), studied by Jewish philosophers in Spain in the twelfth century, and translated into Hebrew in the fourteenth century.76 Most significantly for our purposes, the book develops the henologies of Eudorus and Moderatus by demonstrating how mathematical discourse illuminates first principles. To appreciate the full significance of Nicomachus’s distinctive Neopythagoreanism, it is helpful to back up a few paces. In the wake of Plato’s modifications of Presocratic Pythagoreanism, Speusippus and Xenocrates had drawn attention to two problems: the identity of the One and the resemblance of forms and numbers. As we have seen, Eudorus and Moderatus returned to these topics in the moment after the New Academy, developing systematic accounts of the transcendent One. In my initial sketch (Chapter 1) of Pythagorean Platonism this corresponds to the shift from mediation (2) to henology (3). Neopythagorean undercurrents in Middle Platonism continued to fuel Logos and negative theologies in Alexandrian Judaism and Christianity, while arithmologies of the decadic numbers as gods, dating back to the pseudepigrapha of the second century bce, were gaining in popularity. Nicomachus benefited from all of these developments, but also took a step further (4). He radicalized these Pythagorean stirrings in Middle Platonism by returning not just to the Old Academy, nor even to Plato. Instead, Nicomachus returned to the source that first inspired Plato’s mathematical turn (1): namely, the Philolaean and Archytan vision of four integrated mathematical sciences. Unlike his predecessors, Nicomachus united arithmetic and geometry with first philosophy itself. This was certainly not done by Plato or his Pythagorean sources, as we have seen. If it was hinted at by Speusippus or Xenocrates, they still wrote well before the crucial theological innovations of Eudorus and Moderatus, not to mention Aristotle. As Holger Thesleff has noted, even Eudorus’s henology can be traced to Academic and Peripatetic sources, but only Nicomachus was “the first to make extensive use” of the western group of Pythagorean pseudepigrapha that aimed to expand the legacy of revered Pythagoreans like Philolaus and Archytas.77 Nicomachus thus benefited from all of the prior stages in the genesis of Neopythagoreanism. He used them in concert by returning henological theology to its ancient roots in the four mathematical sciences. As such Nicomachus’s works represents the fruition of Pythagoreanizing Platonism in Greek antiquity and the realization of what we can call a mathematical theology.

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The Four Ways In his Introduction Nicomachus breathes new life into Archytas’s vague suggestion of a “kinship” among a handful of sister sciences. He defines the singular λόγος or δεσμός unifying the four sciences that Philolaus and Archytas had only hinted at, and then he situates them within the Platonic schema of mediation. By thus elevating the role of the mathematical sciences and expanding the analytic power of numbers, Nicomachus provided a new philosophical rationale for arithmological study of the decad’s hidden meanings. This bears repeating: the Nicomachean inclusion of arithmetic within theology stems principally from a renewed enthusiasm for scientific mathematics à la Archytas, and not from an irrational, arbitrary number mysticism. I will address each of these topics in turn—the system of four mathematical sciences, their role in philosophy, and the henological functions of number—drawing on Nicomachus’s Introduction to Arithmetic as well as the extant fragments of his Theology of Arithmetic. Nicomachus begins the Introduction with a commonplace of Middle Platonism. The wisdom that philosophers seek is knowledge of what is unchanging, the eternal structures that loan matter its stability beyond the flux of appearances—in short, Aristotle’s categories.78 In the Theology, Nicomachus simply identifies the categories with number, but in the Introduction he singles out two supercategories.79 Aristotle had divided quantity (ποσόν) into two kinds. Multitude (πλῆθος) is noncontinuous divisibility, counted with numbers, while magnitude (μέγεθος) is continuous divisibility, measured in spatial dimensions. For Aristotle limited multitude is number, limited length is a line, limited length and breadth are a plane, and so forth.80 Nicomachus paraphrases Aristotle on multitude and magnitude and notes accordingly that all being can be known through number (ποσόν or ἀριθμός) and quantity (πηλίκον).81 But then he adds two distinctions that Aristotle had not: absolute versus relative number, and moving versus resting quantity. These yield the following fourfold structure, or four ways (τέσσαρες μέθοδοι) (Table 2.1).82 Among these four sciences, Nicomachus explains, arithmetic and harmonics are the highest, but arithmetic is the “origin, root, mother and nurse” of the other

Table 2.1  Nicomachus of Gerasa’s system of four mathematical sciences Number known in itself (τὸ ποσόν καθ᾽ἑαυτό) Arithmetic (ἀριθμητική)

Number known relative to other numbers (τὸ ποσόν πρὸς ἄλλο) Music or harmonics (μουσική)

Quantity known at rest (τό πηλίκον ἐν μονῄ) Geometry (γεωμετρία)

Quantity known in motion (τό πηλίκον ἐν στάσει) Astronomy or spherics (σφαιρική)

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three.83 The monad is the root of numbers and thus the principle of all arithmetic; likewise, “equality” (ἰσότης) is the principle of all harmonics.84 Right at this juncture, Nicomachus cites three passages in sequence: the great Archytan “kinship” fragment discussed above, Socrates’s defense of mathematics in education from the Republic, and the ode to mathematics in the Epinomis, which he apparently recounts from memory.85 He seems aware that his system stands within the tradition of the Pythagoreans, Plato, and the Old Academy, and yet he strives to integrate them in a new way. Despite the similarity in language, Nicomachus has gone well beyond Aristotle (and perhaps Plato) in several ways. He has explained why this particular quartet of mathematical sciences and no other delivers the “clear distinctions” that Archytas sought. He promotes arithmetic, not geometry, as the “metropole” of the mathematical domain, siding with Archytas against Philolaus. Most of all, he has provided a substantive warrant for the Academic mandate that “no one without geometry may enter” (μηδείς ἀγεωμέτρητος ἐισίτω) the higher study of first principles or theology.86 Because multitude and magnitude cover all phenomena measured through the categories, knowledge of the fourfold sciences is the necessary point of departure for every philosophy of the One. They alone provide “ladders and bridges” from sense perception to complete understanding.87 By providing a principled theory of the mathematical sciences, Nicomachus made them universal; but as universal they immediately exceed Plato’s original pedagogical function. Socrates had only required the philosopher-king to study mathematics as a prerequisite to dialectic, but Nicomachus effectively asks whether the philosopher can ever exit this mathematical initiation, or whether the highest form of thought—be it dialectics, theology, or first philosophy—can ever move beyond number. Merlan has argued that after Nicomachus, every philosophy of the quadrivium must confront the question of disciplinary priority.88 Which comes first, the theology of first principles or fundamental mathematical precepts? Traditional Platonism locates mathematics between physics and theology, but Nicomachus hints at a closer identification of mathematics and theology. One searches in vain for a programmatic statement from Nicomachus, but there is sufficient evidence to conclude that he planned to reverse the roles assigned by Plato to mathematics and dialectic.89 As Christoph Helmig has pointed out, Nicomachus seems unconcerned to follow the guidelines of the Republic and does not distinguish sharply between forms and numbers, διανοία and νόησις, or philosophy and arithmetic.90 If Plato’s mediating framework survives, it is now arithmetic that assumes the highest tier of the scientific hierarchy. If numbers and first principles are equal citizens of the eternal realm, then the meaning of dialectic must change. As a discipline, mathematics may remain in a subordinate position, at least formally; but if the first principles are themselves primal numbers, then dialectic itself becomes a kind of higher transcendental arithmetic. We can track a few different instances

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of this Nicomachean mathematical theology in action throughout the Introduction and Theology.

God and the Monad First, Nicomachean henology represents a new combination of Eudorus and Moderatus.91 Like Eudorus, Nicomachus stresses the homologies between a divine first principle, or God, and the first principle of number, or the monad. In his Introduction, the monad is “potentially” all numbers and yet therefore not a number itself, just as a point is not a line but is the possibility of all lines. The monad’s essential property is its capacity to remain itself, even when multiplied into infinitely different numbers.92 In the Theology, Nicomachus explicitly compares the monad to God: God and the monad can be compared or assimilated to each other [ἐφαρμό­ ζειν], being all things in nature in a seminal way [σπερματικῶς]. . . . [The monad] generates itself and is generated from itself, is self-ending, without beginning, without end, and appears to be the cause of enduring [διαμονῆς αἰτία], as God in the realm of physical actualities is in such manner conceived of as a preserving and guarding agent of nature.93 In the same passage Nicomachus goes on to compare the monad to the divine Mind (νοῦς) or Word (λόγος τεχνικός). In Eudorus the monad reflects the power of the divine One into the realm of number. So too for Nicomachus the monad preserves individual unities by remaining itself even throughout its multiplication, just as God safeguards the being of things while remaining transcendent. Yet in the Introduction, Nicomachus follows the Moderatan emanative model when he defines number as “a flow [χύμα] of quantity composed of monads.”94 In the Theology Nicomachus repeats Moderatus’s doctrine that material plurality results from the “unitary Logos” withdrawing from “Quantity” and then delimiting it. But interestingly he frames this activity in terms of the monad and dyad: Their [viz. the monad and dyad] congress produced the first multitude, the elements of things, which would be a triangle, whether of magnitudes or numbers, bodily or bodiless. For as rennet curdles flowing [κεχυμένον] milk by its peculiar creative and active faculty, so the unifying force of the monad advancing upon the dyad, source of easy movement and breaking down, infixed a bound, and a form, that is, number, upon the triad; for this is the beginning of actual number, defined by combinations [συστήμασιν] of monads.95

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The monad forces the formless dyad to unify itself momentarily so that the series of numbers (beginning with the triad) can evolve into an ordered structure. The monadic nature of every being, insofar as it is one, possesses the same primordial “unifying force” (ἑνωτική δύναμις) that first constrained the flow of the dyad. Thus the monad links every multitude and magnitude directly back to the moment of its origination from the divine One.96

God and Number Unlike his predecessors, Nicomachus applies henology to cosmogony. For him, number theory is sufficient to explain the divine activity that built the universe. As in the Timaeus, number preexists the world and was used by the creating god (the demiurge) as a pattern to structure the material order according to mathematical harmonies. In the Theology, Nicomachus states that the primeval chaos was “organized and arranged most clearly by number, the most authoritative and artistic form.”97 The god that made the world used the decad as a universal measure or ruler with which to frame the whole, thus rendering the cosmos intelligible for the first time by introducing immanent “ratios of concord” (λόγοι συμφωνιῶν).98 So much does not really advance beyond the Timaeus. But then in the Introduction, Nicomachus takes the step of locating such preexistent numbers within the divine Mind itself. Arithmetic, he writes, existed before all the others [viz. music, geometry and astronomy] in the mind of the creating god like some universal and exemplary plan, relying upon which as a design and archetypal example the creator of the universe sets in order his material creations and makes them attain to their proper ends.99 The pattern was fixed, like a preliminary sketch, by the domination of number pre-existent in the mind of the world-creating god, number conceptual only and immaterial in every way, but at the same time the true and the eternal essence, so that with reference to it, as to an artistic plan, should be created all these things, time, motion, the heavens, the stars, all sorts of revolutions.100 Clearly Nicomachus is glossing the Timaeus in broad strokes, but he makes a few dramatic alterations. Unlike Plato, he singles out arithmetic, the science of number, rather than harmonics or geometry. He locates this arithmetic eternally in the god’s mind and accordingly elevates such divine numbers above other numbers. Finally, he construes divine activity as the execution of an arithmetical “artistic plan” (λόγος τεχνικός). To be sure, different aspects of Nicomachus’s vision had appeared earlier and elsewhere. Plutarch, a Middle Platonist contemporary of Nicomachus, had

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already outlined a fivefold system of mathematical sciences.101 Xenocrates and Speusippus had already contemplated the similarities between preexistent numbers and preexistent forms. If we know that it was Middle Platonists who first deposited Plato’s forms in the divine mind, the details about who accomplished this, when, and how, continue to be discussed. Some scholars emphasize the role played by the Logos in unifying and organizing the ideal forms in the divine mind, and some the role of Arithmos, so to speak, or divine numbers.102 Some have focused on Philo’s debt to Posidonius as he assimilated the forms to the Stoic λόγοι σπερματικοί, others on Nicomachus’s debt to Xenocrates as he identified the forms with numbers preexistent in the divine mind, and still others on Alcinous’s debt to Aristotle as he depicted the forms as the content of divine self-knowledge.103 Seneca in his famous Letter 65 had already stated that the divine mind was populated with universal numbers: “God has these models of all things within himself, and has embraced the numbers and measure of all things which are to be accomplished in his mind. He is filled with those shapes which Plato calls ‘Ideas’: immortal, immutable, indefatigable.”104 But given the porosity of Platonist traditions, it is difficult to know whether the doctrine reached him by way of Antiochus, Eudorus, Xenocrates, or whether he was simply reading the Timaeus on his own. Even Calcidius and Macrobius repeated versions of the same idea.105 Nevertheless, judged in its entirety, Nicomachus’s program goes a great deal further than any of his predecessors in erecting the framework of an integrated mathematical theology. As Linda M.  Napolitano Valditara has argued, Nicomachus’s fourfold system establishes a mathesis universalis in the strong sense for the first time, on the strength of a double foundation (see Table 2.1). At the sensible level, arithmetic is naturally or logically anterior to the other mathematical sciences, a priority enforced by the universal categories of multitude-magnitude and rest-motion. But at the noetic level, arithmetic is also ontogonically or theologically anterior insofar as it constitutes the archetypal cosmic model in the divine mind. This doubly reinforced hegemony of number positions Nicomachus well outside the mainstream Platonist view of mathematics as a practical propaedeutic to philosophy.106 For if arithmetic is the fundamental universal science, grounded in number—the number which expresses the henological reflection of the divine One through the monadic one—then mathematics is not only a powerful tool for knowing the world, but an instrument for imitating the One’s originary number. When philosophers study arithmetic, they are studying nothing less than the constitution of the divine Mind. Since arithmetic is the foundation of the fourfold mathematical sciences, this means that for Nicomachus, the quadrivium is inherently theological: an intellectual enterprise in which one participates in the deity.

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Mediations of the One These theological passages (the only ones to mention “god” in the Introduction) raise the same questions of mediation and transcendence that had arisen among other Neopythagorean henologies. How can the One relate to plural numbers? For Eudorus, the divine One is virtually present in the numeric monad; for Moderatus, a succession of three Ones descends from pure unity through forms and soul to matter. Nicomachus by comparison has two different answers ready. First, he distinguishes “epistemic” (ἐπιστημονικός) numbers that we use to measure cosmic harmonies from the “preexistent” (προϋποστάντα), “conceptual” (νοητόν), and “immaterial” (αὔλον) numbers in the divine mind that established those harmonies.107 It may be that Nicomachus intended his Theology to address divine number through the arithmology of the primeval decad, while the Introduction concerned the concrete harmonies demonstrable in arithmetical science.108 In any case, it is clear that the numbers of the divine Mind, revealed in arithmological patterns, play an important role in mediating between the divine One and the pluralities of the world in Nicomachus’s philosophy. The divine decad is not a gauzy mystical appendix to his Platonist principles, but rather exerts a critical explanatory function.109 Second, Nicomachus refers to intradivine principles alongside the supreme One that resemble the Logos and world-soul. I examined above the interpretive problems surrounding the Logos theology of Moderatus, and Nicomachus is not much clearer. He seems to envision a Logos-principle intrinsic to the One, and his λόγος τεχνικός recalls Philo’s λόγος θεῖος ἀρχιτεκτονικός.110 In the Theology Nicomachus calls this intradivine agent the “starting-point and root of creation,” the “first-born” (πρωτόγονος) of the One, and the “impression and likeness” of the Good.111 But he also seems to assimilate the same function to the world-soul, which is the triadic “offspring” (ἔγγονος) of the monadic God and dyadic matter.112 This world-soul defines individuals by giving them their own “equality” (ἰσότης) or self-identity: “as the form of the bringing-to-completion of the universe, and in very truth number, the triad provides equality and, as it were, a removal of ‘more and less’ from all things, giving definition and shape to matter.”113 In its alternative hexadic form, Nicomachus also calls the world-soul the “form of form” (εἶδος εἴδους) through which matter is elevated to number and harmony.114 Hence in Nicomachus we discover two distinct modes of mediating the One— by the divine numbers, and by the divine Logos or world-soul. These twin mediations are left juxtaposed in Nicomachus’s extant writings, but as in Philo they do not come into conflict or competition. Indeed nothing prevents them from being harmonized. The Logos orders the material world precisely because it is the paradigmatic numeric pattern, the primeval decad or preexistent arithmetic according to which God structured the world. Nicomachean mathematical theology

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therefore synthesizes Philonic mediation and Neopythagorean mediation. Put more simply: Nicomachus is the first to harmonize Logos and Arithmos. Of course, Nicomachus had the opportunity to attempt this because of the unparalleled position he happened to enjoy within Platonist traditions as a uniquely redoubled, complex variety of Pythagoreanism. He first inherited Eudoran and Moderatan henologies, enriched by theological creativity of Middle Platonism, at the end of two centuries of the Neopythagorean revival of the Old Academy’s dogmata. But then Nicomachus performed a further iteration, influenced perhaps by pseudepigraphical claims, by returning behind the curtain of Plato to the mathematical philosophies of Philolaus and Archytas. He retrieved and revived their original conviction that actual mathematical sciences themselves yield universal knowledge, and are not only a stepping-stone toward intuiting the forms. This permitted Nicomachus two different routes of conceptual access to the mathematical ideas in Platonism:  first, through the quadrivium (epistemic numbers known through arithmetic), and second, through the decad resident within the One (divine numbers known through arithmology), which as the divine Mind, is simultaneously the Logos. This is ultimately what enables Nicomachus to harmonize the two alternative traditions of mediating the One. Only within Nicomachean mathematical theology are mediation through the set of divine ideas (as in Middle Platonism) and mediation through the decad of divine numbers (as in the Old Academy) rendered perfectly equivocal. When we speak of mathematical theology historically, then, we name (1)  a species of Neopythagorean henology, (2)  oriented around the coeval mediation of Logos and Arithmos, which assumes (3) a universal mathesis grounded in the quadrivium. Such theologies are accordingly predisposed toward subverting the neat hierarchy of three disciplines that Plato and Aristotle jointly maintained, blurring the lines between mathematics and theology not by accident or misunderstanding but deliberately and on substantive grounds. But given their historical genesis out of Logos theologies as much as out of Neopythagoreanism, mathematical theologies share, at the very least, a permanent point of contact with biblical monotheisms influenced by Middle Platonism. For the dialectic of mediation that structures those salvific theologies was also the axis of Plato’s appropriation of Pythagoreanism. This specific theological signature, generated out of local historical interactions, encompasses what is often called “number mysticism” or “arithmology,” but is clearly more sophisticated than these terms imply when stripped of their proper context. There is no reason why Nicomachus’s particular mathematical theology should exhaust the possibilities of the genre. Within the monism of Eudorus and Moderatus, Nicomachus shared the Logos theologies of Philo and Clement. Yet he did entirely without the potent henological instrument of negation that they both found so productive. Instead Nicomachus’s mathematical theology

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is purely positive:  the numbers grasped in the quadrivium fully exhaust the mystery of the divine One. Likewise Nicomachus never takes up the autonomic “spreading” or “extending” of number found in the emanatory models of Moderatus or Valentinus, at least in his extant texts. How would mathematical theology respond to these two possibilities—to negative theology and to spatialized emanation? Proclus addressed these to some degree within his account of the procession of the One into the many and its cyclical reversion. But as we shall see, Proclus deliberately downplayed the Nicomachean emphasis on quadrivium, on arithmetic and on divine numbers. I would submit that a fully conceived, spatialized Neopythagorean theology does not occur until the twelfth century. And what would happen if one added to the mathematical theology of Nicomachus the mathematical model of negative theology found in Alcinous? For an answer to this question we must wait beyond the twelfth century, until the fifteenth.

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The Late Antique Preservation of Neopythagoreanism Neopythagorean theologies are not inherently absurd or superstitious, but endure as a legitimate discursive possibility intrinsic to Platonist traditions. The enterprise of applying mathematical concepts within theological reflection in Greek antiquity was more than a ruse to embellish doctrines with the veneer of sacred arithmologies. On the contrary, as an authentic ressourcement, it returned Platonism to the founts of Philolaus and Archytas, whose aspirations were if anything more precisely scientific than Plato’s anagogical ascent beyond the world. The new henologies of Middle Platonism revived the Pythagorean traits built into the genetic makeup of Platonism that would bloom a few generations later in Nicomachus’s robust mathematical theology. To the extent that ancient and medieval Christian theologies assumed Platonism for their elaboration, Neopythagoreanism—however muted its voice—is likewise indissociable from the array of philosophical configurations that has formed and transmitted Christian beliefs down through the centuries. Examining the genesis of Neopythagoreanism allows us to observe an important incongruity. The varieties of Platonism that most influenced ancient Christian theologies were just those that most sought to restrain their own Neopythagorean tendencies. This means that ancient Christianity lost touch with Neopythagoreanism inadvertently, due to circumstances beyond its control. It is true that with Justin, Valentinus, and Clement, early Christian theologies had been initially grafted onto Alexandrian Middle Platonism.1 But it was ultimately the dualism of Numenius and not, say, the monism of Moderatus that found its way into Eusebius and Origen. Numenius influenced Plotinus through his teacher Ammonius Saccas, and through Plotinus, Porphyry and thus Augustine. Numenian dualism also influenced the text that became the primary vicar of ancient Platonism until the Renaissance, Calcidius’s fourth-century translation and commentary on the Timaeus. Proclus, as we shall see, had his own suspicions of unadulterated Neopythagorean theology. From the fourth century onward, it was largely the Neoplatonism of either Plotinus or Proclus that shaped the evolving

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contours of Christian beliefs about the Trinity, the divinity of Jesus, the sacraments, aesthetics, and mystical theology. And this is not yet to mention the even more pervasive role of Aristotelian traditions in coloring Christian attitudes toward Pythagorean theologies in antiquity and the Middle Ages. Hence it turns out that the decline of Neopythagoreanism was neither fated nor due to its inherent folly. It was rather a contingent discursive event that arose when Christian and Platonist theologies fell into an accidental alliance, one that held fast for the next millennium, if not more. In his classic essay mapping the intellectual terrain of the twelfth century, M.-D. Chenu distinguished several threads of medieval Christian Platonism. Augustinian Platonism was a Christian reinterpretation of Plotinus, Ps.-Dionysian Platonism was directly influenced by Proclian metaphysics and Iamblichean theurgy, and Boethian Platonism was intimately linked to the Timaeus reception of Calcidius. However “tangled the lines of descent,” writes Chenu, the debts owed by medieval Platonism to late antiquity could not be clearer.2 Each of these principal conduits that transmitted Platonist traditions to the Middle Ages ended up straining out Neopythagorean ideas. Augustine embraced arithmology early on, but though he quickly came to reject Neopythagoreanism, symptoms of his former enthusiasm resurface in the later works. The impact of Nicomachus’s mathematical theology upon Iamblichus and Proclus transformed the future direction of Neoplatonism. But these two titans of the late antique Academy disagreed over how to honor the great Gerasan and how to implement his vision. Boethius worked tirelessly to translate Nicomachus’s volumes into Latin and thus into western Christianity. In one stroke he recast the Neopythagorean return to Archytas and Philolaus into the durable solidity of the quadrivium. But ironically, by confining Nicomachus’s works within this new quadrivial genre, Boethius walled off their mathematical theology from his other Christian theological texts, containing their influence. These comparisons make clear that it was not only Augustine, alone in his sobriety, who perceived from afar the potential idolatry of number as a competing mediator of divine presence, as if on behalf of Latin orthodoxy to come. Rather, when he observed an overlap between the functions of Logos and Arithmos, Augustine was only repeating the hesitation felt by other Neoplatonists about the same Nicomachean and Iamblichean ideas. If Augustine emphasized the exclusivity of the Logos as the sole mediator, Proclus, too, strove to decouple Platonist theology from Arithmos. This solidarity against Neopythagoreanism— by Christian authors as much as by pagans, by the introspective Neoplatonism of Plotinus as much as by the cosmic Neoplatonism of Proclus—effectively retarded its long-term reception within Christian theologies. The fundamental homology of Logos and Arithmos within the Platonist theological heritage was never addressed at its roots, but simply sidestepped, as Christianity avoided conversation

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with Neopythagoreanism in general. Consequently the competition between these two modes of mediation remained an ineluctable, but as yet unresolved problem for Christian Platonism moving out of the late antique period. By the time of Platonism’s temporary resurgence in the twelfth century, Neopythagoreanism was still forgotten. Yet in each of these three lineages, obsolescent Neopythagoreanism remained both absent and present. With surprising consistency, mathematical theology was both engaged and suppressed by Proclus, by Augustine, and by Boethius alike, preserving it in the way a vestigial organ preserves a distant animal past, or as the strata of decomposing timber are preserved as the forest’s new floor. The lost past remained faintly visible in the contours of the alternative present enabled by its withdrawal. In this way fossilized traces of Neopythagoreanism continued to haunt each of these three major theological discourses of medieval Christian Platonism. Of course the voluminous output of these authors will preclude much attention to fine details in the following overview. But detail is not what we require in order to judge how their decisions made in reaction to mathematical theology might have constrained their respective theological projects or determined the Platonisms that they bequeathed to later centuries. Only at a bird’s eye view does a common pattern begin to emerge: a consistent, but imperfect, closure.

Iamblichus, Proclus, and the Legacy of Nicomachus To understand Proclus’s hesitations about Neopythagoreanism, we have to begin with his predecessor, Iamblichus of Chalcis (ca. 240–ca. 325 ce), born a decade after Porphyry in northern Syria.3 Despite his upper-class heritage, Iamblichus decided against the common practice of Hellenizing his Aramaic name (yam-lichu), perhaps out of pride in his descent from a dynasty of priest-kings. But Iamblichus’s choice also reflects his view that contemporary academic Platonism needed to return to the theurgical devotions of what he believed were its Egyptian and Chaldaean roots. Having studied with Porphyry, Iamblichus returned to Syria to found his own school at Apamea, and his ongoing criticisms of his teacher confirm the tensions that must have precipitated his departure. Porphyry had tried to accommodate Aristotelian logic within the Plotinian system, but Iamblichus fought against Aristotle’s critique of Pythagoreanism through a renewed focus on Nicomachus’s Theology.4 Iamblichus took issue with Plotinus for viewing numbers as abstractions from physical reality, rather than as real entities that constituted a higher degree of being.5 Instead he sought to reinfuse Neoplatonism with the very Neopythagoreanism that Porphyry and Plotinus had leached out, recovering aspects of Speusippean monism, yet (like Nicomachus) with a positive decadic

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theology in place of the negative henology shared by Speusippus and Plotinus alike.6 Iamblichus wrote amidst increasing antagonism by imperial Christianity against rural temples conserving the old rites. An unnamed official in the court of Licinius, Constantine’s foe, praised the Syrian philosopher as a divinely appointed “savior” of the Hellene religion, and the pagan emperor Julian briefly reinstated, against the cult of “the Galileans,” the hymns and rituals taught by “the divine Iamblichus” himself. Iamblichus thus sought to revive Nicomachus’s project as a major component of his program of theurgic renewal, and his turn to Pythagoras seems to represent a response to Christianity as much as to the Plotinian leanings of his teacher.7

Iamblichus and Neopythagorean Theology These priorities are reflected in works like De mysteriis, his Vita of Pythagoras, and his massive ten-volume homage to Nicomachus, On Pythagoreanism. According to O’Meara’s brilliant reconstruction of its missing books, Iamblichus organized this magnum opus according to Nicomachus’s quadrivial order and drew on the latter’s Introduction and Theology as his most important sources. He treated physics, ethics, and theology (in the Stoic division) all under the rubric of arithmetic and with arithmetical methods, so that theology literally finds its place inside the quadrivium. As O’Meara suggests, however much Iamblichus represents a turning point in the formation of Athenian Neoplatonism, he derived his agenda and methods directly from Nicomachus. Iamblichus’s views on the ontological properties of number, the transdisciplinary relevance of mathematics, and the use of arithmetic in theology all stem from the Gerasan.8 For Aristotle and Plotinus, mathematicals are abstractions from material bodies that lift the mind above matter toward the intelligible. Iamblichus disagreed. Mathematicals are rather concrete shadows of the intelligible reality above them, as images are of exemplars. “As numbers in the mathematical order can be projected downwards paradigmatically, producing physical numbers,” O’Meara explains, “so they can be projected upwards as foreshadowing higher orders, producing, in particular, divine numbers.”9 The quadrivial arts are therefore not four methods for analyzing the dimensions of multitude and magnitude, as Nicomachus had taught, but instead are themselves four hierarchical planes of numbers. Iamblichus’s theory of number determined his views on the scientific disciplines. Aristotle had distinguished physics, mathematics, and theology according to degrees of abstraction from material bodies. But for Iamblichus each of the three sciences studies number:  divine numbers, mathematical numbers, and physical numbers. Since mathematics is not the only science to treat numbers, its theorems and operations can be applied to physical and divine numbers just as

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well. Arithmetic consists of a series of abstract operations (he calls them generically ἀναφοραί, assimilations) not limited to mere addition and subtraction. These can be deployed in different scientific (that is, numeric) domains, generating different arithmetical relationships, for example, of “similarity” (ὁμοιότης), “imaging” (ἔμφασις), or “causation” (αἰτία). In this way Iamblichus outlined an arithmetical physics (Book V of On Pythagoreanism), an arithmetical ethics (Book VI) and an arithmetical theology (Book VII), before proceeding onward to the lesser mathematics of geometry, music and astronomy.10 Where Nicomachus had distinguished “scientific” numbers from “preexistent” numbers, Iamblichus interpolates new levels of number above mathematics. In several texts he refers to “self-moved” or psychic numbers, “intellectual” numbers, “essential” or intelligible numbers, and divine numbers.11 Standing above all of these is the decad of divine numbers, a unified set that (as in Moderatus and Nicomachus) flows from the first monad through the dyad to generate the rest of the divine series. These numbers are all gods, but there is also (as in Eudorus) a supremely transcendent One higher than even the decad’s monad.12 Since the One is divine and soul is a kind of number, mathematical operations for Iamblichus amount to theurgic rituals. When properly executed their activities divinize the soul and reveal divine mysteries.13 Iamblichean Neopythagoreanism is a brilliant if unrestrained extrapolation of Nicomachus’s insights. Later enthusiasts felt the need to prune its profusions, if only out of loyalty. For in seeking to rewrite Nicomachus, Iamblichus’s account of the principles ordering the four mathematical sciences is pleonastic to the point of incoherence; he proposes multiple competing principles, but none are consistently applied. Nevertheless, his On Pythagoreanism continued to shape the ascendant Athenian school after Iamblichus’s death, well into the fifth century. Syrianus (ca. 375–437 ce) passed along an interest in Iamblichus to his pupil Proclus through his commentary on Aristotle’s Metaphysics. Syrianus had learned from Iamblichus how to deploy number and mathematics in philosophy beyond mere arithmology and had grasped the potential of what O’Meara calls the “transposability of mathematicals to other domains,” evolving an arithmetical physics and theology like Iamblichus.14 But even as Syrianus defended Iamblichus’s radical Pythagoreanism, he felt its disadvantages. Where Iamblichus focused on the decad’s mediation of the One, Syrianus returned attention to the One’s supremacy. He also shifted attention away from the decad itself to a more flexible model of plural, isomorphic “henads” (ἑνάδες), which Proclus would adopt in his Elements of Theology and other works.15 Syrianus also supplemented Iamblichean psychology with a powerful proposal of his own. He posited that human minds innately contain universal ideas that, being part of our cognition a priori, make possible the epistemological experience of recollection. Mathematicals ideas are therefore “projections” (προβολαί) of the

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soul. The heuristic utility of geometrical figures is both a confirmation of our innate numbers and a sign of the mind’s weakness for requiring them to be graphed in space. The mind understands a given being when it grasps the a priori numbers in itself that are equivalent with the preexistent numbers from which a given being was originally made.16 Two students of Syrianus competed to succeed him, and their characters tell us something about Iamblichus’s success in re-Pythagoreanizing Neoplatonism. Domninus of Larissa was a mathematician whose handbook rivaled those of Nicomachus and Euclid, and Proclus Diadochus, whom Syrianus finally chose, believed himself to be the living reincarnation of Nicomachus’s soul. Like his teacher, Proclus (412–485 ce) worked within the Neopythagorean language of the Limit and Unlimited, the unfolding of the monad into multiplicity, and the search for mediations of the One. He never questioned whether mathematics should inform theology, but what kind of mathematics and how.

Proclus’s Critique of Iamblichus Despite his enthusiasm for Nicomachus’s project and his extensive knowledge of On Pythagoreanism, Proclus had serious reservations about Iamblichus’s approach. For example, Proclus adapted lengthy passages from Iamblichus’s De communi mathematica scientia into the prologues of his commentary on Euclid’s Elements. But he also edited them carefully and never named his source. Proclus evidently wanted to retain as much of Iamblichus’s Neopythagorean content as possible, yet with his own priorities and restrictions kept firmly in place.17 As O’Meara writes, “the somewhat ambivalent character of [Proclus’s] theology—both mathematical and not mathematical in form and content—is testimony to its origins as, in part at least, a critical reaction to the theologizing mathematics of Iamblichus’s Pythagoreanizing programme.”18 Proclus disagreed with Iamblichus’s mathematical theology in at least three fundamental ways. These sharply curtailed the Neopythagorean influence that Proclus would transmit to medieval Christian authors. First, Proclus wanted to restore what he saw as Plato’s original intent by ensuring that dialectic, not mathematics, remained the highest science.19 This represents a return to the Republic in lieu of Iamblichus’s Neopythagorean sources, but also to Aristotle’s desire for a more scientific mathematics. For Proclus, Plato had excelled the Presocratics chiefly because his concept of dialectic rendered philosophy scientific. If not handled correctly, he feared, the Neopythagorean enthusiasm for number would become a fatal distraction from the true tasks of philosophy. Proclus strove to preserve dialectic as first philosophy (or theology) above mathematics as a restraint on speculation. This was best understood, he thought, when one grasped the true nature of mathematical mediation.

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In the famous prologue to his commentary on Euclid’s Elements, Proclus considers what it really means for mathematical ideas to occupy a middle station between empirical knowledge and first principles.20 Mathematicals are “images” and “offspring” of the two highest principles after the One itself, the Philolaean couple of Limit and Unlimited. From the Limit they derive their stable identities, and their capacity for equality; from the Unlimited they derive their progression into multiplicity and their virtual infinity.21 Proclus reasons that one should view mathematicals neither as mere abstractions from sensible beings, nor as separable realia that precede sensibles, but instead as “projections” (προβολαί), just as Syrianus had postulated. Thus the essence of mathematical beings is their dynamic “unfolding” (ἀνελίττειν) from the reservoir of the soul’s “full complement” (πλήρωμα) of ideas, a veritable “second world-order” (διάκοσμος): “before the numbers the self-moving numbers, before the visible figures the living figures” (ζωδιακά σχήματα).22 Mathematicals are therefore projections of higher principles within soul, just as soul itself is a projection of the divine Mind (νοῦς).23 Hence mathematics can never be fused with theology or dialectic, but permanently remains a “second” science.24 Proclus argues that when the four mathematical sciences seek after their common parentage and (as in Epinomis) try to define the “bond” that unifies them, they are forced to look beyond mathematics per se.25 It is not enough to say that universal mathematics is the bond, or even that dialectic is the bond. Rather Proclus states that the highest bond of the quadrivial arts is the divine Mind itself, which “wraps up the developments of dialectical methods, binds together from above all the discursiveness of mathematical reasoning, and is the perfect terminus of the upward journey and of the activity of knowing.”26 That is to say, only in a theological horizon do the different mathematical disciplines discover their common principle; but to the same degree, this progress leads mathematics necessarily beyond itself in a way that Iamblichean theology would resist. With the priority of dialectic firmly in place, Proclus is able to countenance selected tenets of Neopythagoreanism in moderated form. He embraces, for instance, Moderatus’s principle that mathematics is the best conceptual language for expressing the ineffable.27 He also draws attention to Nicomachus’s system of four mathematical sciences.28 But where Nicomachus ordered the four sciences through the arithmetical categories of multitude and magnitude, Proclus instead views them as four aspects of the soul’s self-unfolding. Like the world-soul constructed by the demiurge in the Timaeus, the quadrivium is built by combining sameness with otherness and rest with motion. When soul recognizes the plurality of its unity, it “unfolds” number, giving rise to arithmetic; but when soul grasps the unities that harmonize those pluralities, it unfolds music. Likewise rest and motion lead soul to unfold the disciplines of geometry and spherics.29 Proclus’s second alteration follows from the first: he makes a subtle but profound adjustment to the balance among the four mathematical sciences. For

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Nicomachus and for Iamblichus, the arithmetic of the decad had provided the highest access to the mysteries of the One. Proclus too wanted to affirm the centrality of mathematics within philosophy, but to do so by focusing on the epistemology of actual mathematical techniques more than on theologies of ascending numbers.30 Toward this end he favored geometry over arithmetic as the privileged discipline, since geometry better fulfills the anagogical potential of mathematical beings.31 Proclus explains that imagination (φαντασία) is the crucial faculty connecting sense perception and pure understanding. Geometry thinks phenomena by “projecting” or “unfolding” universal ideas into the specific visualized figures of geometrical diagrams. Using the imagination like a screen, the geometer converts the soul’s invisible ideas into a graphic discourse that can dialectically (through constructions, axioms, and deductions) lead the mind to a vision of simple unity.32 Only geometrical method truly accommodates and capitalizes upon the projective nature of mathematical being. It spatializes arithmetic, renders it visible to the imagination, and thus binds number to dialectical reasoning. Hence for Proclus the priority of theology over mathematics is reinforced by the priority of geometry over arithmetic. Proclus also favors geometry because like theology it enjoys a universal domain. Geometry is competent to analyze every element of nature into its constituent points, lines, and shapes, and yet also, once freed of physical constraints, to unfold the figures of imagination in the free play of pure construction. Geometry can even demonstrate, Proclus mysteriously intones, “what figures are appropriate to the gods.”33

Proclian Mediation For these reasons Proclus also parts ways with Iamblichean theology on the nature of mediation. To connect the One to the world, both Iamblichus and Nicomachus had relied on interstitial registers of numbers to terrace the steep slope of the absolute, whether the archetypal decad of the divine mind, the physical numbers of the universe, or the psychic numbers that link them. Proclus effectively breaks with this vision of arithmetical mediation, and the two alternatives he takes up instead became hallmarks of his thought. First, Proclus mediates the gap between identity and difference by interposing a circuit of unending motion between them:  from cause to effect to cause, from One to many to One. This dynamic cycle is the Proclian triad of remaining, procession, and return, the pattern of an eternal circle repeated at the orders of being, knowing, and eternity itself.34 The triad functions effectively as mediation on the strength of the third term, remaining. A cause remains in itself even as it proceeds into its effect, and the effect remains in the cause even as it it is produced extrinsically.35 The cycle of procession and return also provides a henological infrastructure for Proclus’s philosophy of mathematics. As Glenn Morrow notes,

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mathematical beings as “unfoldings” and “projections” echo the moment of procession, and the anagogy of geometrical procedures aims to return the many to the One.36 In a similar vein the homology of the flow (ῥύσις) or unfolding (ἀνελίτ τειν) of the point into a line, and the flow of the One into many, is a Pythagorean trope that runs from Speusippus through Moderatus and Plotinus to Iamblichus and Proclus.37 While Proclus’s rhythms of emanation and return have precedents in Moderatus and Plotinus, it was the Proclian version that Ps.-Dionysius sustained as the basso continuo of patristic and medieval Christian Platonism.38 Second, in place of the sacred decad, which arithmological lore had assimilated to the Olympian gods, Proclus installed an infinite constellation of “henads” or monadic deities as the first ontic tier after the One. Like the decad, the henads function as the initial coordination of plurality and unity, the ground for the possibility that numerical pluralities below can nevertheless find a path back to the One. The henads are gods because they are “self-complete” (αὐτοτελής), and since more immediate to the One than being, life and intelligence (νοῦς), they are beyond all three.39 When Proclus resorts to henads to clarify the dynamics of participation in unity, it becomes clear that henads are effectively fulfilling the mediating function of number, yet without the numeric properties that imbued arithmetic with theological mystery in Iamblichus and Nicomachus.40 Proclus restored the original Platonic status of mathematics, championed geometry over arithmetic, and disseminated the mediating function of mathematicals into motion and into henads. These measures undo the tension between Logos and Arithmos with surgical precision, decoupling the link that Neopythagorean theology had preserved since Nicomachus. This absolved the philosophical and theological heirs of Proclus from addressing potential conflicts between Word and number, even when they adopt characteristically Neopythagorean vocabulary. Medieval Christian theologies that stem from Proclus, above all Ps.-Dionysius’s Divine Names, follow the same path.41 In a word, Proclus effectively spatializes mediation into a kind of integrated “topology” of relative orders.42 His peculiar mathematical theology seeks the epistemic certainty and spatialized ontology of geometry, yet does not require a discrete mediator. Excepted from the problematic of mediation, the role of mathematics in Proclianism—not to mention its early modern descendants—is thus converted from a substantive anagogical function to the merely procedural end of securing methodological certainty.

Augustine and the Number without number When the new bishop Augustine of Hippo (354–430 ce) drew up plans for his memoirs, he recast his life in the mold of Neoplatonist patterns, as is well known. But Plotinus and Porphyry were far from the only philosophical influences on the

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younger self depicted in the pages of Confessiones.43 Around the time of Syrianus, Augustine’s studies in Italy put him in contact with Neopythagorean influences as well. His first work on aesthetics, now lost, discussed the problem of monad and dyad. Early works like De ordine and De musica suggest traces of Nicomachean arithmetic mediated either through Varro or the lost translation of Apuleius.44 Aimé Solignac even argues that Augustine initially reached toward a “Christian Pythagoreanism.” Even if Augustine began to step away from that path by the time of Confessiones, it remains nevertheless true that the great bishop of Hippo stood squarely in the stream of mathematical theology during his formative years.45 Like Proclus, Augustine adapted Platonism to his tastes before handing it on to medieval Christianity. In his case it was not an overweening Iamblichus that needed tempering, but rather his own former predilections. After his conversion Augustine remained sanguine about the role of number in Christian theology, as we can see from several of his early works. But by the time of Confessiones, Augustine had grown suspicious of the Platonists’ interest in number, viewing it as a prideful affront to the humility of the divine Word and associating it with astrology or Manichaeanism.46 Yet well after Confessiones Augustine continued to ponder the value of his earlier Neopythagorean interests. Hence in his mature works we see not so much a rejection of mathematical structures in his theology so much as their transposition to other conceptual sites. The function of number as mediation of divine order is preserved in those later works, but those functions are assumed by other nonmathematical doctrines. Ultimately Augustine found he could no longer harmonize the parallel mediations of Logos and Arithmos, but only see them as two alternatives competing with each other, one false and one true. In this way his own intellectual development enacts the slow-motion fossilization of Neopythagoreanism within Latin Christian Platonism.

Neopythagoreanism in the Early Augustine The depth of Augustine’s early commitment to the project of mathematical theology becomes clear in works like De ordine and De musica. This was no passing affair, but a sincere if ill-fated marriage. In De ordine Augustine recapitulates several Platonist and Pythagorean themes. Well-measured harmonies lead the mind to beauty, just as number organizes the limitlessness of sound, rendering language rational.47 The modulated rhythms of music reveal the presence of numbers behind sensible experience.48 Hence, for Augustine the seven liberal arts render the world rational through number.49 When reason beholds beauty, it sees the underlying figures, then their mathematical dimensions, and ultimately the “simple and intelligible numbers” that code reality from within.50 When reason searches beyond the mathematical sciences for the ultimate ground of number, “it beg[ins] to suspect that it itself was perhaps the very number by which all things

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are numbered; or if not, that this number was there whither it was striving to arrive.”51 The soul suspects that behind the distracting images of the world lay number, “that most hidden thing by which we enumerate,” or “that hidden divine order” that gives Augustine’s book its title.52 The hidden ground of number, in turn, is the simple One.53 The mathematical sciences thus inspire theology to investigate the soul and God. Augustine’s Neopythagorean account of the disciplines in De ordine is notably ambivalent about the status of arithmetic. Arithmetic is missing from his list of the seven arts, but seems conflated with philosophy by the close of Book II. Is number, the foundation of the first six disciplines, the principal subject matter of philosophy? He contrasts dialectica (as scientia bonae disputationis) and numeri (scientia potentiae numerorum), stating principles that inform the two sets of three liberal arts.54 But he advises that the mathematical arts are more important, since even philosophy does no more than understand unity in a divine manner, and dialectic is simply a means for attaining different conceptual unities.55 The dialogue ends with repeated admonitions to praise the wisdom of Pythagoras.56 Neopythagorean topics reappear in other early works. In De immortalitate animae (387), written after Augustine’s return to Milan from Cassiciacum, the soul first recognizes its immortality when it discovers eternal truths of mathematics in the human mind. If the ratio of a circle is eternal, then the mind knowing the ratio must somehow also be eternal.57 Likewise in De quantitate animae (387–8) Augustine uses the notion of geometrical equality to define the nondimensionality of the soul. The figure exhibiting the most perfect equality is the circle, and yet the source of its geometrical perfection that equalizes the circle’s radii is the nondimensional point at its center. As a real but nondimensional entity, the soul is best understood a kind of point.58 Like Varro and Boethius, Augustine planned to compile seven handbooks of the liberal arts that drew on Greco-Roman source materials. He finished only two, for grammar and for music, and only the latter survived. The final book of De musica, finished in 391, represents another sustained exploration of mathematical theology.59 Augustine’s harmonic investigations lead him to posit five degrees of psychic numbers.60 Below psychic numbers are temporal and corporeal numbers, and psychic numbers are governed by intellectual numbers (angels) and divine unity.61 Augustine’s swelling hierarchy recalls the Iamblichean method of multiplying arithmetical orders to bridge metaphysical gaps, not to mention the Iamblichean uncertainty about the relation of soul and mathematicals. Augustine even redefines salvation as an ascent to God’s numbers. The soul is “reformed by divine numbers of wisdom” (divinis sapientiae numeris reformatur) when it turns away from numbers gained through the bodily senses.62 Likewise the soul’s elevation to rational numbers purifies the corporeal numbers in its train: “with a restored delight in reason’s numbers, our whole life is turned

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to God, giving numbers of health to the body.”63 Divine providence is visible in the ineradicability of number. No evil could ever entirely corrupt the inherent beauty of the numbers resident in things, which always preserve the possibility of return to God’s unity.64 In De musica Augustine defends an arithmetical realism reminiscent of Iamblichus and Syrianus, but on theological rather than epistemological grounds. The mysterious solidity of the progression from one to four and the isometric proportionality of the geometrical dimensions are cause for wonder. As Augustine muses in Book II of De musica, there must be a divine origin of mathematical order:  “Where, I  ask, do these things come from, if not from the highest and eternal rule of numbers, likeness, equality and order?”65 In Book VI Augustine sketches a brief theology of “eternal equality” that recalls Philo. The soul that seeks beauty in fact seeks equality, similitude, and order. But equality is determined by number, and nothing surpasses the pure equality of oneness.66 “That equality we could not find sure and fixed in sensible numbers, but yet we knew shadowed and fleeting, the mind could never indeed desire unless it were known somewhere,” he concludes. “But this could be nowhere in the spans of places and times; for those swell up and these pass away.”67 Perfect equality, in a word, must be divine. Such rich Neopythagorean soundings have been too quickly dismissed as the result of Augustine’s relative youth, even though all of these passages appear after his conversion. In this way the figure of Augustine raises the prospect for the first time of a robust Christian Neopythagoreanism. But unlike the biblical henologies vaunted by Philo, Clement, or Valentinus, Augustine had to contend with new discussions of the divinity of the Son and the humanity of the Logos. Two generations after the Council of Nicaea, the Christian vocabulary of theological mediations now included definitions of Trinity and Incarnation, even if Chalcedon was still decades off. By the turn of the fifth century, a Christian Platonist could certainly connect the Logos to the divine Son, as in the prologue of the Gospel of John. But it is far more difficult to imagine how a Christian Neopythagoreanism would operate. One would have to juggle the additional mediation of Arithmos alongside Logos, while simultaneously preserving the transcendence of the Logos as Son and affirming its Incarnation.

Wisdom and Number This is why it comes as a surprise to read Augustine not only embracing mathematical theology in his early works, but even going so far as to contemplate expressis verbis the relationship between sapientia (Logos) and numerus (Arithmos). This occurs in a remarkable passage from Book II of his philosophical dialogue, De libero arbitrio (391–5). Well beyond the indirect hints we found in Philo or Nicomachus, it is only in Augustine, the lead architect of medieval theology and

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standard of orthodoxy in the Christian west, that we first encounter explicit strategies for reconciling Logos and Arithmos—and who ultimately judges that they can peacefully coexist. As Ernst Hellgardt writes, in De ordine Augustine remained within the philosophical project of Nicomachean-Boethian arithmetic. But in De libero arbitrio the future bishop takes the step of theologizing number (numerus) as a trace of the presence of God’s Word (sapientia).68 In Book II of De libero arbitrio Augustine states that reason can prove God’s existence by its own powers, but to do so, it must discover the trace of something within that is higher than itself and eternal.69 In De trinitate Augustine will discover the Trinitarian image of memory, intellect, and will in the recesses of the soul, but here his search for a universal noetic experience leads him to the “principle [ratio] and truth of number,” since these are held in common by all who reason.70 Augustine rejects the Aristotelian abstractionist theory of number for the realism of the Neopythagoreans. Number does not derive from sense perception, but is a collection of pure “ones” (unum) known innately prior to the physical perception of composite multiples. Although one cannot find simple unities in the world, the mind can nevertheless use the unum it possesses natively to count the ubiquitous multiples.71 Number is therefore something known universally to reason and eternally certain, concludes Augustine. But then he adds that there is another: the books of Wisdom and Ecclesiastes link numerus closely with sapientia.72 So having pursued the universality of number, he now changes course and considers the universality of wisdom. Wisdom denotes the principles that lead the wise person to the Good, namely the truth that governs the happy life.73 Augustine specifically states that this sapientia is also the divine Word, the Wisdom equal to the eternal Father who leads reason to God.74 As Christoph Horn observes, for Augustine the conclusion is ineluctable: “the higher numbers stand in close connection with the second person of the Trinity.”75 In this way Augustine follows the regulae numeri and the regulae sapientiae to the same divine destination (veritas ipsa). It is striking to ponder that what Augustine has just reenacted is the very aporia that confronted Philo and Nicomachus over the two competing mediations. Augustine’s deliberations as he confronts this dilemma are worth quoting at length, clamoring as they do with the theological freight of four centuries before him, as if an unwitting parable of their history: I should very much like to know whether these two, wisdom and number, are contained in any one class, because you mention that they are coupled together in Holy Scripture. Does one depend on the other, or is one included in the other; does number, for example depend on wisdom, or is it included in wisdom? . . . When I meditate on the unchangeable truth of number, and, so to speak, its home or sanctuary, or whatever word is suitable to describe the place where number resides, I am carried far away from

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the body. Finding, it may be, something which I can think of, but not finding anything I can express in words, I return, worn out, to familiar things in order to speak, and I express in ordinary language what lies before my eyes. The same thing happens to me when I concentrate my thoughts with the fullest attention that I can on wisdom.76 At this juncture Augustine does not simply reconcile the ways of Logos and Arithmos. The more pressing challenge is to decide what basis beyond both mathematical and theological discourses could possibly adjudicate between them. What mediates the two ultimate mediations? Both operate equally well, he concedes, for making the anagogical ascent. But clearly they do not belong to the same genus, as he notes, nor is he comfortable granting priority to either one. It is difficult not to read this biographically, as Augustine thought over his Christian Neopythagoreanism in the pivotal years of the early 390s. Lacking a tertium quid Augustine nevertheless attempts three strategies to reconcile number and wisdom. First he simply identifies them.77 Second, he suggests that they fulfill complementary functions. Wisdom is the fire’s warmth heating what is close to it (reason); number is the fire’s light touching every member in the hierarchy of beings (order).78 Third, he considers that each is reciprocally prior to the other. Wisdom gives numbers to things so that they can be wisdom’s dwelling (sedes). But when we follow the trail of those numbers upward, they do not simply lead back to wisdom, but transcending our mind lead beyond wisdom into the very dwelling (manere) of truth.79 Wisdom speaks through the vestigia of numbers, but without numbers nothing could have form, and all esse is esse numerosa.80 Only when one beholds numerus sempiternus beyond space and time does wisdom shine out from its hidden dwelling.81 Unable to resolve their priority, Augustine accepts the aporetic conclusion: “Although we cannot be clear whether number is in or from wisdom or whether wisdom is in or from number, or whether both refer to the same thing, yet it is certainly plain that both are true, and true unchangeably.”82

The Humility of the Word If the younger Augustine relished this tightrope walk into the mid-390s, the tension is abruptly let out in Confessiones (397–401). In Book VII, he reflects on his reaction to the libri Platonici and its evolution from initial excitement to disappointment and rejection.83 At first Augustine is thrilled to discover their Logos doctrine:  God’s eternal Word is the agent of creation and ground of reason.84 Augustine accepted the Middle Platonist doctrine that the eternal forms are ideas in the mind of God.85 In De libero arbitrio he even held that arithmetical mediation is indistinguishable from the Christian Logos. But by the writing of Confessiones Augustine has clearly changed his mind. He now views the Logos theology of

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Neoplatonism as incurably deficient, even when it does resemble a nascent Trinity. What Platonist mediation lacks for Augustine is the tenet of divine Incarnation.86 His theology of the incarnate Logos as mediator, while not the first such patristic source, is by far the most complete and sophisticated on record.87 The libri Platonici are usually identified as works of Plotinus and Porphyry, but in light of the foregoing it seems clear that Augustine’s early Neoplatonist affinities included Neopythagorean elements as well. Indeed, just before the passage on the libri Augustine thanks God for delivering him from the “fraudulent divinations of the mathematici,” which we can translate as astronomers or scientific astrologers (ars inspectorum siderum). He is embarrassed at his former fascination with this curiosity that “enumerates” the intervals of time between constellations in order to predict the future.88 As he now sees it, the vain arts of the mathematici are the symptom of a hyperinflated reason (deliramenta) that rejects the divine Word.89 That is to say: in its pride astrology resorts to numerus instead of sapientia, because the two are now opposed. In fact the pride of the Platonist books invalidates their Logos doctrine, according to Augustine. Augustine contrasts the eternal Logos generated timelessly from the One with the incarnate Logos that assumed a frail human body, suffered, and died. He testifies that he was unable to make the Platonic ascent from beautiful bodies to immutable divine beauty until he grasped that the mediator was an embodied Logos.90 The mediation of the incarnate Logos succeeds where the Platonist Logos fails because it radicalizes the logic of mediation. Lest it fall short of one of the terms it mediates, the descent of the Logos cannot terminate until it reaches the finitude of the human condition. As he would write in De civitate dei, Christ did not mediate as Word but as human.91 This is why Augustine follows his revision of Platonist mediation with an attack on the Apollinarian Christology that supposed Jesus to be the Logos without human mind or soul.92 This was also the lesson Augustine learned from Simplicianus about Marius Victorinus himself, who had long embraced Platonism, but took the step of baptism only once he had embraced the “mysteries of the humility of the Word.”93 Clearly Book VII represents a turn from Trinity to Incarnation. But it is equally a turn away from one horn of the dilemma in De libero arbitrio to the other, from divine numerus and the eternal Logos to divine sapientia and the incarnate Logos. Augustine no longer defines divine Wisdom by the structures of numerical order, but by the posture of humility. Given Augustine’s overwhelming influence on medieval Christian culture, this turn away from Neopythagoreanism in the name of solidarity with the humility of Jesus represents not simply a dramatic intellectual pivot by one man but a paradigmatic critique of Platonist and Neopythagorean mediation sewn into the fabric of western Christian theology. Even if Augustine had once embraced Logos and Arithmos as coeval models of Christian mediation, his eventual opposition of incarnate Logos to eternal Logos

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seems to invalidate the Neopythagoreanism of De ordine, De musica, and De libero arbitrio. Augustine’s mature theology thus etches an indelible caesura between number and Incarnation, between Neopythagoreanism and orthodox Christology. Gnostic associations aside, this is why the slightest hint of mathematical theology in western Christianity remains immediately provocative.

The Number of the Creator After finishing Confessiones, Augustine embarked on new studies of Genesis and of the Trinity. In both directions we see the lingering after-effects of his turn away from Neopythagoreanism. The twelve books of De genesi ad litteram took Augustine fourteen years to write (401–415) but are his greatest hexaemeral commentary. To explain how an eternal God created a temporal world Augustine had to connect the Trinity with the structure of the cosmos. This train of thought led him back to number, as in De musica and De libero arbitrio, but as soon as Augustine began drifting toward Neopythagorean conclusions, he promptly arrested his progress and posited a different doctrine to take the place of mathematical order. In this way Augustine’s departure from mathematical theology was recapitulated and reinforced in De genesi. The same pattern can be seen in his mature Trinitarian theology as well. In Book IV of De genesi Augustine confronts some exegetical quandaries concerning the timing of creation. God foreordained everything eternally in his Word, including all the forms of creatures. Even though creatures are temporal and finite, their rationes are coeternal with the Son, living in the Son, and identical with the Son. Yet Genesis states that the eternal reasons of creatures unfolded in a finite period of six days. To solve this problem Augustine’s first instinct is to turn to Nicomachean number theory. Creation occurs in six days because the number six is perfectly proportioned (the first “aliquot” number).94 Number expresses the Creator’s activity within the limited vocabulary of temporal-spatial difference by mediating between the One and the many.95 Moreover, Scripture testifies that “You have created all things in measure and number and weight” (Omnia in mensura et numero et pondere disposuisti, Wisdom 11:21).96 But if number, measure, and weight ordered the events of creation, were they created or did they preexist? By what did God order the sources of order, if not by himself? Augustine sees no other possible conclusion than that the triad was somehow “in God” before creation.97 What this means exactly preoccupies him for several chapters. Augustine weighs two interpretive strategies, swinging back and forth as if unable to decide.98 In effect he is deliberating over the use of mathematical concepts in theology. On the one hand, Augustine stresses that the Creator cannot be said to have measure, number, and weight in the same way creatures do. Not every aspect of creation stems from within God: God gives color to creation, too, but there is

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no color in God. The three terms could assume a nonmathematical, for example ethical, meaning. All human activities need to be moderated; only the full sum of virtues perfects the soul; and the will is swayed by the weight of its loves. Or perhaps the triad only states analogically that measured, numbered, and weighed creatures exist “within” God’s providence.99 On the other hand, Augustine is fascinated by the possibility that the triad allows a glimpse of the Creator. Measure provides “limit” (modus) to creatures, number provides “form” (species), and weight provides “repose and stability” (quies et stabilitas). Likewise, God limits, forms, and orders everything (terminare, formare, ordinare).100 Furthermore, God granted the form of color, not color; unlike color or other created qualities, the triad uniquely names the foundations of created order. Measure names magnitude (augmenta et diminutiones), number names multitude (multitudo et paucitas), and weight names motion in the Aristotelian sense (levitas et gravitas). Without some notion of donated order, the idea of creation loses its meaning. Furthermore, to ask whether the triad is “inside” or “outside” God’s mind is beside the point. Indeed, Augustine adds— repeating the Neopythagorean realism of De libero arbitrio—the distinction does not even apply to human minds, since we too grasp numbers mentally before counting physical objects.101 In the end Augustine decides not to decide, just as in the aporetic conclusion of De libero arbitrio. But in this case his formulation is nonetheless richly allusive: It is a marvelous gift, granted to few persons, to go beyond all that can be measured and see the Measure without measure, to go beyond all that can be numbered and see the Number without number, and to go beyond all that can be weighed and see the Weight without weight. . . . In the realm of spirit or mind, measure is limited by another measure, number is formed by another number, and weight is drawn by another weight. But there is a Measure without measure, and what comes from it must be squared with it, but it does not come from something else; there is a Number without number, by which all things are formed, but it receives no form; and there is a Weight without weight, to which are drawn those beings whose repose is joy undefiled, and there they find their rest, but it is not drawn to any other.102 Augustine’s beautiful rhythms conceal some important theoretical implications of this tempered mathematical theology momentarily adopted in De genesi. First, Augustine grants that God is a kind of supreme numerus. Even if he qualifies that divine name as analogical or negative, he nonetheless elevates mathematical concepts to a special vocation, given that number (and measure and weight) is the primary expression of the cosmic order donated by the Creator ex definitione. In

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other words, if it means anything, “Creator” must mean “source of number.” Or as Augustine writes about the act of creation, “He has made them as He has seen them, not looking beyond himself, but He has numbered within himself all that He has made.”103 Second, Augustine hints at a link between the human experience of number and the sovereignty of divine number. God posits God’s number in such absolute freedom that it is extrinsically innumerable and hence not “number” simpliciter. Human number by contrast is conditioned extrinsically by the autonomic order of number itself, but like God it discovers numerical order planted and flourishing within itself even before empirical senses are consulted. Third, we should register the fact that Book IV brings Augustine into contact with the Nicomachean philosophy of the quadrivium. The dimensions of measure (magnitude), number (multitude), and weight (motion) comprise the foundations of the Gerasan’s table of the four mathematical sciences. But his gloss of Wisdom 11:21 as modus, species, and quies already translates those Nicomachean categories into Plotinus’s limit (ὁρισμός), form (μορφή), and rest (στάσις).104 Augustine’s compromise—God is numerus, but also sine numero—codifies his discontent with the Neopythagorean theology of his youth, despite its ongoing utility in his mature theology of creation. But by the same token the formula expresses a willingness to supersede mathematical theology if a functional alternative could be found. When Augustine finally achieves this in the next two books of De genesi, he gladly leaves his erstwhile Neopythagoreanism behind. In Book V Augustine takes up the problem of the two creation accounts in Genesis 1 and 2 and the perpetuation of creation through reproduction. These issues only restate the problem of eternal divine activity in time. Augustine’s solution in Book IV was to posit a transmathematical number in God, but he is ambivalent about returning to the Neopythagoreanism critiqued in Confessiones. When Books V and VI give him another chance to address the same problem, Augustine takes the opportunity to substitute a different answer, now grounded not in number but in the supremacy of the Word. Augustine reasons that since God created everything in his rationes aeternae, including time, the created order is not structured by time. Instead a synchronic conexio causarum stores all the “numbers” of things later to be reproduced in time.105 This solves the riddle of the double creation account in Genesis, which simply indicates these two stages of creating activity: the six days of creation that sowed the seminal potential for reproduction, and the evolving reproduction of creatures.106 Augustine then notices that the eternal creation and the two-stage creation add up to a tripartite order: God’s eternal reasons (the Word), the productions of the six days (the birth of time), and the present cycles of reproduction (in time).107 This amounts to a new hierarchical mediation that can replace the arithmetical mediation of Book IV. By Book VI, Augustine has filled in the details of this

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hierarchy. It comprises four ways in which the creation is created, stretching from the Trinity down to the potential existence of future creatures: Nevertheless, [1]‌under one aspect these things are in the Word of God, where they are not made but eternally existing; [2] under another aspect they are in the elements of the universe, where all things destined to be were made simultaneously; [3] under another aspect they are in things no longer created simultaneously but rather separately each in its own due time, made according to their causes which were created simultaneously . . . [4] under another aspect they are in seeds, in which they are found again as quasi-primordial causes which derive from creatures that have come forth according to the causes which God first stored up in the world.108 This is the locus classicus for Augustine’s renowned theory of seminal reasons.109 Augustine counts four levels of the created cosmos (universitas). Creatures exist (1) in the Word; (2) in the hexaemeral creation event; (3) in their causes temporally; and finally, (4) in their seeds resulting from (3). He notes that this schema alters how one conceives the goodness of creation, since the cosmos is both already perfected (consummata) in (1) and (3), and yet still underway (inchoata) in (2) and (4).110 Of course Augustine does not invent this theory de novo but adapts the λόγοι σπερ­ ματικοί of Plotinus and the Stoics before him.111 As Horn has pointed out, Augustine simply employs seminal reasons in Book VI to carry out the function performed by the Neopythagorean theology of number in Book IV.112 At first Augustine held that the numeri of creatures hidden in the Word are unfolded according to those same numeri in time.113 By the end of De genesi, the numbers in God have become rationes aeternae and the creaturely numbers rationes seminales. The operation of mediating the divine One is the same, but now the Neopythagorean language of arithmetical order has been removed in favor of causation linked directly to the Trinity.114 The same tacit competition between numerus and Verbum recorded in De libero arbitrio has survived intact over two decades later in De genesi ad litteram. Augustine retained the function of numerus but now Verbum carries it out. Thus the theory of seminal causation rests on the unsteady foundation of Augustine’s own ambivalence about mathematical theology.

Number and the Trinity We have now observed the same movement on two occasions. In the transition from De libero arbitrio to Confessiones, Augustine began with a Neopythagorean doctrine (divine sapientia as equivalent to numerus) and substituted a Christian

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doctrine in its place (Verbum defined by the humility of Incarnation). In De genesi ad litteram the same pattern was repeated in the theology of creation. The eternal numbers in the mind of God, who is the divine numerus sine numero, were transformed into the seminal reasons descending exclusively from the eternal Verbum. A third example concerns Augustine’s doctrine of the Trinity. In the first book of De doctrina christiana (397), Augustine teaches that only the Trinity is to be enjoyed, and all other things are to be used toward this end. In passing he meditates on the name of the Trinity for a few lines: These three have the same eternal nature, the same unchangeableness, the same majesty, the same power. In the Father there is unity, in the Son equality, and in the Holy Spirit a harmony of unity and equality. And the three are all one because of the Father, all equal because of the Son, and all in harmony because of the Holy Spirit.115 This is the lengthiest discussion of the triune persons in the whole book, and Augustine never repeats this triad again. Even in De doctrina christiana the triad is only included to illustrate a broader point about the inevitable failure of language in theology.116 However, we can probably trace Augustine’s triad of unitas, aequalitas, and concordia to one of the sayings of Moderatus preserved by Porphyry that I discussed in Chapter 2. Moderatus had listed the philosophical concepts that Pythagoreans frequently designated with the term “monad”:  sameness, oneness, equality, concord, and sympathy. In the same passage Porphyry praised the perfection of triads. Marius Victorinus may have been inspired by these lines to formulate his Trinitarian analogy against the Arians:  “God is the One that generates the monad from himself and reflects love in his unity. So too also in the Many: each and every unity has its own number since it is reflected by others beyond difference.”117 Jerome was also familiar with Moderatus and had read Porphyry’s Vita Pythagorae.118 It is possible that Augustine assembled his own triad of unitas (for ἑνότης), aequalitas (for ἰσότης), and concordia (for συμπνοία) out of this Moderatan list, placing them all under the umbrella of the most perfect “triad” (τριοειδής), the divine Trinity. Perhaps Augustine engineered this feat on his own powers, or perhaps he was following Victorinus.119 Whatever its origin, Augustine’s triad represents a Christianized Moderatan henology. Eudorus proposed a triadic henology of One, monad, and dyad, but Moderatus posited three descending Ones. Likewise for Augustine, the One processes into an equal, second One, and the unity of their joint equality generates a third One; the unity, equality, and harmony of each is possessed by each One. Yet the African bishop never once mentioned this arithmetical Trinity in his later writings, not even within the fifteen books of De trinitate composed two decades later (ca. 400?–425?).120 When he defends the absolute equality of the three persons

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in Book I, he never repeats the notion of mathematical equality from De doctrina christiana (though he does recall Wisdom 11 in passing).121 Ultimately in Augustine’s theology, the “Number without number” is not only a name for God, but a metaphor for his hesitation to define God arithmetically at all. One can only wonder to what extent Augustine’s tremendous influence trained western Christians to adopt his own unresolved ambivalence regarding the value of Neopythagoreanism for Christian theologies of Incarnation, creation, and Trinity. In discursive terms, Augustine performed the same operation within his own development that we observed in the shift from Iamblichus to Proclus. Where Proclus rejected Arithmos but retained its function as geometric spatialization, Augustine rejected Arithmos but retained its function as the Logos’s incarnation. This is what fossilization is: the preservation of a structure achieved by filling it with an alternative, more durable content.

Boethius and the Fate of the Quadrivium After the alterations and reductions made by Proclus and Augustine, the case of Ancius Manlius Severinus Boethius (ca. 475–526 ce) appears more straightforward at first. After hearing Ammonius lecture on Nicomachus in Alexandria in the decades after Proclus’s death, the Roman aristocrat decided to translate the Introduction to Arithmetic and Harmonic Manual into Latin in their entirety.122 This plan would preserve the henological passages peppered throughout Nicomachus’s treatises for Latin posterity and ensure that generations of Christian authors could scrutinize the unadulterated Neopythagoreanism that Proclus had tried to downplay. Boethius also went out of his way to reinforce the conceptual architecture that Nicomachus had given to the four sibling sciences of Archytas, and his bestselling philosophical prosimetrum, Consolatio philosophiae, depicts the cosmic harmonies of the Timaeus as a revelation of divine providence.123 At the same time, despite his Neopythagorean pursuits, Boethius’s sober analyses of Trinity and Christ in a few theological opuscula, along with his eventual martyrdom at the hands of an Arian emperor, gave his works the imprimatur of Nicene and Chalcedonian orthodoxy.124 But just as with the legacies of Proclus and Augustine, the moment of greatest proximity between Neopythagoreanism and Christian theology in Boethius became the moment of greatest distance. Boethius did not, like Augustine, systematically obscure his own debts to mathematical theologies, nor like Proclus did he deliberately counteract Neopythagorean excess. Rather, it was the shape of the Boethian corpus as a whole that introduced an accidental disjunction between mathematical and theological ideas. It is easy to overlook an obvious but salient fact that silently cooperated for centuries with the Proclian and Augustinian critiques: Boethius’s works systematically divide Christian theologies of Trinity and Incarnation from Nicomachean

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philosophies of One and number. Given the tremendous role Boethius’s texts played for centuries in medieval Christian educational institutions, the long-term discursive consequences of such generic sequestration cannot be taken for granted. By reifying Nicomachus’s fourfold system as the “quadrivium,” and then classifying his translations within that rubric, Boethius inadvertently isolated the theological potential of Neopythagoreanism within a separate discourse. The mathematics of the quadrivium and its implicit henology and anagogy were thereby cordoned off from the doctrinal theology of the opuscula and the poetic theology of the Consolatio. When Boethius discusses the divine Good in De hebdomadibus, the Trinity in De fide catholica, or the Incarnation in Contra Eutychen, he never calls on the Nicomachean resources that he knew so intimately. When he sketches the anagogical function of mathematics in Institutio arithmetica or Institutio musica, he never connects it with Christian traditions of ascent or mediation. For all its beauty the Consolatio eschews Neopythagorean arithmetic and Christian theology with an even hand, leaving his tributes to cosmic order devoid of quadrivial details and leaving the relation between Providence and Trinity ambiguous.125 If the lifeblood of earlier Neopythagorean theologies had been their deliberate inattention to the difference between mathematical and divine Ones, all of that power was dissipated when the Boethian corpus was dissected into two separate halves. Ironically, Boethius’s direct translation of Nicomachean arithmetic and harmonics ended up diminishing their theological value. To observe these divisions in the Boethius corpus one need not survey the whole, an impossible task besides. Nor must we search for him to formulate a tentative mathematical theology: in Boethius there is none to observe or describe, except perhaps (to use an apt metaphor) an infinitesimal point of tangency at the place where the two disjunct discourses just barely touch. When we read Boethius we share with the vast majority of medieval readers a tendency to abide by the apparent generic boundaries of the works, boundaries reinforced by our modern impulse to separate poetry and religion from precise science. When we follow these inclinations, the gap between quadrivium and theology in Boethius can seem natural and indeed necessary. To challenge them, to glimpse the tangent point, is to look for the mathematical questions haunting Boethius’s theology and the theological traces concealed in the quadrivial texts.126 In doing so we will also tour a few passages that dimly echo the mathematical Platonism of previous chapters and hence proved important to later students of Boethian theology, particularly those who would lift the quadrivium’s discursive quarantine in later centuries.

Henology in the Quadrivium Boethius’s quadrivial writings sometimes betray their Neopythagorean ancestry when theological vestiges spring up in the text. This first occurs when Boethius

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formulates his most influential concept, the “quadrivium.” Initially in Institutio arithmetica (ca. 515?) he simply repeats Nicomachus’s argument for mathematics in philosophy and his fourfold ordering of the mathematical sciences, much as Proclus had done in his Euclid commentary. The wisdom sought by philosophers is the immutable aspect of beings grasped in Aristotle’s categories; the primary categories are discontinuous number (multitudo) and continuous number (magnitudo); and together these structure arithmetic, harmonics, geometry, and astronomy.127 At first Boethius simply names them the “four disciplines of mathematics” (quattuor matheseos disciplinae).128 But then he adds something that Nicomachus and Proclus had lacked: a single term to encompass the four sciences and thus denote mathematizability as such, that is, an integrated and universal mathesis.129 Boethius indicates a certain “fourfold way” (quadru-vium) that leads the mind to wisdom: Among all the ancient men of authority who, following the lead of Pythagoras, have flourished in the purer reasoning of the mind, it is clearly obvious that hardly anyone has been able to reach the highest perfection of the disciplines of philosophy unless the nobility of such wisdom was investigated by him in a certain four-part study, the quadrivium, which will not be hidden to a just and penetrating mind. For this is the wisdom of things that are, and the perception of truth gives to these things their unchanging character. We say those things are which neither grow by extension nor diminish by contraction, nor are changed by variations, but are always in their proper force and keep themselves secure by support of their own nature.130 If the general argument belongs to Nicomachus, Boethius’s neologism carries two distinctive emphases. First, the quadrivium has a moral function of purifying the soul.131 Jean-Yves Guillaumin has shown that Boethius’s term builds on the Pythagorean tradition of the moral crossroads (bivium) of the figure Y, employed by Lactantius, Jerome, Varro, and Martianus Capella. The Boethian quadrivium is likewise a convergence of four ways that lead to the moral perfection sought by Pythagoreans and echoed in the four virtues. Early medieval Christians even spoke of a quadrivium virtutum. If this counts as a “discreet Christianization” of the four mathematical sciences, it would mark the sole trace of the author’s religion in his quadrivial works.132 Second, Boethius’s new term reintroduces the Platonic theme of mathematics as mediation between the world and God. “This quadrivium,” he writes, “is the mediating way [viandum] by which we bring a superior mind from knowledge offered by the senses to the more certain things of the intellect.”133 As the imperative force of the gerundive concisely suggests, the role played by mathematics

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is a necessary yet penultimate “Pythagorean discipline.”134 And yet Boethius also seems to endorse the Neopythagoreanism of Nicomachus, for whom mathematics did not merely play a supporting role as propaedeutic, but constituted first philosophy. “If a searcher is lacking knowledge of these four sciences, he is not able to find what is true,” Boethius avers. “He who spurns these, the paths of wisdom, does not rightly philosophize,” and indeed “has already shown contempt for philosophy.”135 Thus Boethius does not resolve but only perpetuates the ambiguities surrounding mediation in Neopythagoreanism that we studied above.136 The anagogical function of the quadrivium presupposes that the arithmetical structures thus discovered directly reflect God’s unity. On this point, nothing in his Christianity prevents Boethius from handing down Nicomachus’s image of numbers indwelling the mind of the demiurge. Hence for Boethius, too, arithmetic is the highest science because it is the blueprint of creation: From the beginning, all things whatsoever which have been created may be seen by the nature of things to be formed by reason of numbers. Number was the principal exemplar in the mind of the creator.137 . . . Arithmetic is prior to all not only because God the creator of the massive structure of the world considered this first discipline as the exemplar of his own thought and established all things in accord with it; or that through numbers of an assigned order all things exhibiting the logic of their maker found concord . . . .138 It is true that Boethius simply repeats Nicomachean concepts in this powerful passage. But once again the innovation of quadrivium has subtly altered the Gerasan’s mathematical theology.139 By integrating the four sciences as first philosophy, and by not translating Nicomachus’s Theology, Boethius indirectly transformed what Nicomachus meant by divine numbers. In the Boethian corpus God’s numbers are not the sacred decad that provide the basis for arithmology. Rather it is arithmetical science itself, as the content of the mind of God, that provides the basis for the other quadrivial sciences. While Boethius never stipulated this explicitly, the theological implications should be apparent: it is the difference between naming God the supreme decad and naming God the supreme mathematician. Boethius also repeats Nicomachus’s vivid henological images depicting the activity and character of the One. Although Boethius never connected such passages to his own henology in De hebdomadibus or De trinitate, they lay juxtaposed in the Boethian corpus for subsequent readers to discover. Like Nicomachus, Boethius describes unity as primarily fecund and creative, the universal mater, or the genetrix of pluralities. Unity’s procreative power is especially visible in prime numbers, the elements from which other numbers are generated.140 Unity “protects and maintains” (seruat atque custodit) the serial orders of number by separating

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each number in the progression from the next through its “divine constancy.”141 If the self-remaining of unity holds open the possibility of numbers to return to itself, as a point returns to itself in a circle, the self-equality of numbers endows them with their ligative cosmic function.142 Boethius repeats all the elements of Nicomachus’s account of number, from the Moderatan “flow” (profusus, from χύμα) of unities to the “divine power” binding even and odd until they are reconciled in harmonies.143 Like his source, Boethius had initially planned to produce four works that would span the entire quadrivium. Beyond the two completed translations in arithmetic and music, the handbooks on geometry and spherics never appeared, though later authors eagerly supplied pseudonymous works on his behalf. Yet our Roman translator had already included significant discussions of music and geometry within his arithmetic, even before one turns to Institutio musica. Like Nicomachus, Boethius states that inequalities—mathematical, but also ethical and aesthetic—are ultimately derived from pure equality, justice, and beauty, and not the other way around. Equality operates as the mater and radix from which the multifarious species of inequalities flow when it sets the “limit” (margo) to their alterity, just as in Moderatus’s monadic genesis of quantity.144 Boethius clearly recognized this as the theodical postulate that it is because he adds more vivid examples than Nicomachus had provided. As unity grounds arithmetic by preserving numbers, so equality grounds music by preserving measures or proportions.145 Numeric differences form quantities, and quantities coupled together are proportions. Multiple proportions form proportionalities related to the arithmetical, geometrical, and harmonic means.146 Boethius calls arithmetic the mother of geometry as well, since numbers are the “seeds” of geometrical figures.147 Just as unity is the principle of number but is not a number, so the point is the principle of spatial intervals but has no interval. As unity multiplied by itself remains unity, a point added to itself remains a single point. Yet unities and points, writes Boethius, can “unfold” (explicare) into numbers and lines.148

Number in the Opuscula Just as Boethius’s quadrivial writings gleam with understated theological potential, one can likewise observe Neopythagorean dogmas floating to the surface in the theological opuscula. The ostensible goal of Boethius’s compact De trinitate (ca. 519–523?) was to clarify how number can be applied to the Christian Trinity.149 But along the way Boethius found that he must address the disciplinarity of theology and the meaning of number; in other words, to explain the Trinity he must define the boundary between theology and mathematics. On the first count, Boethius repeats the Platonic model of intermediate mathematicals, but rather

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than drawing on Plato’s Republic, he paraphrases Aristotle’s tripartite schema from the Metaphysics. As he explains in Book II of De trinitate, physical science studies embedded forms in motion, mathematics studies embedded forms at rest, and theology studies static forms abstracted from material conditions. Boethius characterizes their respective methods as rationabiliter, disciplinaliter, and intellectualiter.150 Although it was likely mediated to him by Porphyry, Boethius follows the Stagirite’s account closely enough to inherit its ambiguities.151 Later in De trinitate Boethius adds that the form studied by theology is in fact the only true, divine Form. The quasiforms known to mathematics and physics are strictly speaking images (imagines) of that Form.152 Alterity, plurality, multitude, and diversity all arise within this realm of simulacra, while the true Form is the One beyond number of any kind.153 Boethius also finds he must qualify the meaning of number, distinguishing enumerative numbers from things numbered. The former includes instances where repetition does not generate plurality, such as a triple ostension of the same object or the divine Trinity.154 After a prolonged discussion of the nature of theological predication, Boethius concludes that the numerositas of the Trinity denotes only Aristotle’s category of relation, whereas the divine substance remains One.155 Note that when Boethius explains the number of the Trinity he avails himself of only one of the definitions of number that he learned translating Institutio arithmetica, namely repetitio unitatum. One can only imagine what might have transpired if Boethius had conceptualized the Trinity in terms of Nicomachus’s monadic σύστημα or χύμα. In another tractate called De hebdomadibus (ca. 519?), also known as Quomodo substantiae, Boethius confronts the riddle of the One and the many.156 How is the Good refracted into the realm of plurality? While its mysterious title (“on the sevenfolds”) suggests a connection with Pythagorean ideas, the most noteworthy aspect is its mathematical method.157 Following the example of Proclus’s Elements, Boethius organizes the work as a series of axioms from which he can deduce subsequent postulates. As he explains, “I have followed the example of the mathematical and cognate sciences and laid down terms and rules according to which I shall develop all that follows.”158 Boethius’s ultimate conclusion also has a Proclian ring: the ubiquitous intimacy of the Good as forma essendi allows beings to “flow” (fluere, defluere) from it into their native multiplicity.159 The cautious distinctions of these opuscula proceed in the spirit of Aristotle’s Organon, the books that preoccupied Boethius in his earliest writings. But one can also detect echoes of past controversies in the history of mathematical Platonisms. Like Aristotle, Boethius attempted to systematize the anagogical function of mathematics in Plato’s Republic. Like Iamblichus and Syrianus, he held that mathematical forms (numbers) are primarily reflections of a higher plane of theological forms (the One). Like Augustine, he recognized the Trinity as a kind of transcendent number, but one beyond all numbers known arithmetically.

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Like Proclus, he strove to model ontology and theology after the pattern of formal geometrical argumentation. And yet viewed as a totality, Boethius’s collection of Neopythagorean fragments never interacts in a fundamental way with his high views of the Trinity and the Incarnation. Like a cold battery, the works of Boethius store up a tremendous reserve of inert potential energy. Only when properly activated could they release the light and heat capacitated in their ancient cells. Therefore, following the familiar pattern of Proclus and Augustine, Boethius preserved elements of mathematical theology but in a manner that occluded access by later generations. Proclus did without Logos and converted Arithmos into geometrical cycles; Augustine praised Logos at the expense of Arithmos. In the same way, Boethius preserved the Nicomachean mediation by Arithmos entirely intact, but then severed it from the Logos of Trinity and Christ in his opuscula. For all their differences, these three major progenitors of medieval Christian Platonism reinforce each other on one point: they oppose Arithmos to Logos. Even if Eudorus, Moderatus, and Nicomachus contributed important components to the historical construction of Neoplatonism, those Neopythagorean elements were directly counteracted by such uniform opposition in the passage from late antique to early medieval Christian theology. To be sure, Boethius was not the first Roman intellectual in late antiquity to collect relics of Greek mathematical traditions and translate them into the chapels of Latin Neoplatonism. Macrobius, a high-ranking imperial official writing in the late fourth century, compiled sources on arithmetic, astronomy, and the harmony of the spheres. His widely read cosmographical passages may have stemmed from a lost Timaeus commentary of Porphyry. But his arithmological and henological passages added little new to Latin Neoplatonism in the realm of mathematical theology.160 The fifth-century Carthaginian lawyer Martianus Capella transmitted an influential arrangement of the four mathematical sciences nearly a century before Boethius, in the nine books of his baroque allegorical encyclopedia The Marriage of Philology and Mercury. Martianus’s geometry is the geography of Pliny, and his arithmetic passes on second-hand versions of Nicomachus and Theon of Smyrna alongside fragments of Euclid. Varro may have stood behind these, as he certainly does for the following books on music and on astronomy. But as a Neoplatonist intellectual like Macrobius, Martianus had no reason to link the quadrivium to the Christian theology of his day in the generation after Augustine.161

The Fate of the Quadrivium Only Boethius held together in one oeuvre Christian theology and his integrated vision of the quadrivium as first philosophy, even if his textual divisions worked to block the discourse of mathematical theology from even forming. But for the next five centuries after his execution only a handful of Christian authors studied the

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quadrivium as a propaedeutic to philosophical theology, and none to my knowledge in the context of the doctrinal topics of Trinity or Incarnation. By the seventh century the curricular structure developed by Cassiodorus and Isidore of Seville out of Augustine’s and Boethius’s outlines had ensured that mathematical concepts like numerability, the point, equality, or sphericity would be largely neglected by early medieval Christian philosophers and theologians in a way they had not been by Clement of Alexandria or the young Augustine.162 With only rare exceptions, the reception history of the Boethian oeuvre is divided into three different channels: the philosophical and theological works, including the opuscula sacra; the Consolatio; and, our focus here, the four arts of the quadrivium.163 The fate of the quadrivium after Boethius is the subject of some disagreement. In an influential article, Hans Klinkenberg contended that it suffered a tragic “decline” from “cosmological significance” to a “hollowed-out curriculum.” According to Klinkenberg, the sublime promise of the quadrivium from Plato to Boethius, that “mathesis is the path to knowledge of God,” was instead forced into the service of mere “propaedeutics and techniques.”164 More recently Brigitte Englisch has argued that such generalizations are difficult to sustain in detail, given the diverse trajectories of the four sciences and the different political conditions of five tumultuous centuries.165 Of course the Christian modification of the artes liberales was well underway already by the time of Augustine some two hundred years before Cassiodorus.166 Nevertheless, if we are asking specifically about the future of the Neopythagorean goods smuggled within the Boethian writings, the Klinkenberg thesis does not mislead. By the ninth century each of the four sciences was evolving independently along its own path and under its own momentum.167 Arithmetic remained inextricably linked to Institutio arithmetica even during the rediscovery of Euclid and the rise of modern calculation using a zero and erasure.168 Early medieval readers struggled with Boethius’s text, but around the turn of the eleventh century Gerbert of Aurillac revived and deepened quadrivial studies in arithmetic, music, and geometry. Gerbert not only went beyond the encyclopedists but also Carolingian arithmologies connected to Boethius.169 In the thirteenth and fourteenth centuries major commentaries on the Boethian arithmetic were still being written by Jordanus Nemorarius, Johannes de Muris, Thomas Bradwardine, and Roger Bacon. Even theologians like Bonaventure and Albert the Great read it with care.170 Yet all this time, theological appropriations of Institutio arithmetica were exceedingly rare if not extinct. By the early twelfth century the quadrivium was simply not a matter for profound speculative exploration. Take the example of Hugh of St. Victor’s highly esteemed Didascalicon, which reflects the vanguard of liberal-arts education at Paris in the 1120s–30s. Hugh’s treatment of the quadrivium consists of digests of Macrobius, Boethius, Cassiodorus, and Isidore of Seville. He warns that mathesis is one letter away from meaning “vanity,” since when pronounced

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with a hard t, the word denotes the kind of astrological superstitions condemned by Augustine.171 One reason for this was that already by the eleventh century quadrivial study entailed far more than reading Boethius’s textbooks. It meant undergoing a practical apprenticeship to learn how to handle the appropriate technologies of measure. Arithmetic had the game of rithmomachia, the abacus, or the manual computus, and music had the monochord referenced in Institutio musica. Geometers used surveyor’s instruments and astronomers the astrolabe.172 The practice of arithmetic meant calculating calendrical cycles for the liturgy or the position of stars for astrologers. Christian liturgies ensured that music enjoyed a consistent prominence in medieval arts curricula, keeping Institutio musica in constant circulation.173 If students initially preferred a practical focus on musical technique to Boethius’s austere calculations, by the later Middle Ages his harmonic theory underwent a minor renaissance.174 Likewise the medieval discipline of geometry had two faces. Speculative geometry included fragments from Euclid’s Elements (before its full Latin translation in the twelfth century and first full commentaries in the thirteenth), an assortment of sub-Euclidean traditions from late antique Roman sources, and early medieval pseudo-Boethian treatises of variable quality. But far more attention was given to practical geometry that applied its precise measurements to land surveying (gromatics), weights (statics), and vision (optics).175 Astronomy and astrology were interdependent sciences that gained steam in the twelfth century when western Christians learned from Arabic science how to calculate the positions of the planets accurately.176 By the end of the twelfth century, as Guy Beaujouan writes, the irrevocable influx of Arabic science had “profoundly altered the equilibrium of the quadrivium,” but it perdured well into Cusanus’s fifteenth century.177 Christian theology per se had little to no direct commerce with these developing sciences from the sixth to eleventh centuries. The one exception that proves the rule is John Scotus Eriugena (fl. ca. 850–870), the erudite Irish monk of the Carolingian courts. Eriugena combined as none had before the Platonist theologies of Augustine and Boethius with those of the Cappadocian fathers, Ps.-Dionysius, and Maximus Confessor. In his careful reading of the Boethian corpus, Eriugena built upon the foundations laid by Alcuin and others, who had revived the study of the mathematical artes as well as the opuscula sacra in the Carolingian courts.178 For this reason Eriugena is rightly viewed as a forerunner of Thierry of Chartres and his circle, who may have read him, as well as Nicholas of Cusa, who studiously annotated his own copies of Eriugena’s works and praised him by name.179 The Irish monk commented on the different mathematical arts in Martianus Capella and was intimately familiar with the Boethian quadrivial translations.180 He expanded upon Augustine’s reading of Wisdom 11:21.181 He theorized, in the words of Stephen Gersh, a “mathematical angelology” that parsed the celestial

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hierarchies of Ps.-Dionysius in terms of their harmonic ratios.182 As Édouard Jeauneau has shown, Eriugena adopted Augustine’s hierarchies of number from De musica, colored his biblical exegeses with arithmological figures, and studied the geometrical images in Ps.-Dionysius.183 He imagines numbers flowing eternally out of the uncreated Monad and diverging into two channels; one streams into nature and one into the created monad, human reason.184 For these reasons Werner Beierwaltes has argued that Eriugena’s cosmic vision represents the first medieval synthesis of Boethian and Augustinian theologies of divine harmony, what we might call a musical theology of unity and difference.185 Yet if we seek a medieval Christian theology that reopened the Neopy­ thagorean potentials of the quadrivium, Eriugena cannot be the final destination. The Irish monk embellished passages of Periphyseon with ideas from Institutio arithmetica, and Institutio musica certainly informs his Martianus commentary. But to my knowledge Eriugena never considers the quadrivium as such, as universal mathesis, as a resource for naming God. Eriugena follows Boethius in prioritizing arithmetic above the other sciences. But the four mathematical arts do not mediate divine order in the Platonic sense for Eriugena, any more than do the other liberal arts (like dialectic), or for that matter other symbols (like light).186 Above all, in his Neopythagorean wanderings Eriugena never ventures beyond the bounds of the theology of creation to imagine what impact Boethian arithmetic and quadrivial categories would have on doctrines of the Trinity or the Incarnation. In short, the historical development of the early medieval quadrivium did nothing to disturb its original discursive confinement in Boethius. So long as one abided by the generic and textual boundaries separating the Neopythagorean quadrivium from Christian doctrine, all was well. Hybrid Christian mathematical theologies would never arise, and the four mathematical sciences, liberated of their common theological significance, could develop separately as mundane scientific exercises unconcerned with the One. But if a medieval reader were to cross the lines and to interpret, say, Boethius’s discussions of number in De trinitate in light of the henological fragments in Institutio arithmetica, then suddenly everything would be different. For then the Boethian corpus could birth a wild new species of theology, a robust Neopythagoreanism both substantively Christian and yet unrestrained by Augustinian and Proclian moderation. One could not expect this unusual hermeneutical operation to arise frequently, if indeed at all, in the medieval centuries after Boethius. Such a task would demand a reader exceptionally familiar with the full range of humanistic and scientific learning, sensitive to the theological nuances of Platonist texts, and not a little audacious.

PART TWO

The Pearl Diver Thierry of Chartres’s Theology of the Quadrivium

And this thinking, fed by the present, works with “thought fragments” it can wrest from the past and gather about itself. Like a pearl diver who descends to the bottom of the sea, not to excavate the bottom and bring it to light, but to pry loose the rich and the strange, the pearls and the corals in the depths, and to carry them to the surface, this thinking delves into the depths of the past—but not in order to resuscitate it the way it was and to contribute to the renewal of extinct ages. What guides this thinking is the conviction that although the living is subject to the ruin of the time, the process of decay is at the same time a process of crystallization, that in the depth of the sea, into which sinks and is dissolved what once was alive, some things “suffer a sea-change” and survive in new crystallized forms and shapes that remain immune to the elements, as though they waited only for the pearl diver who one day will come down to them and bring them up into the world of the living—as “thought fragments,” as something “rich and strange,” and perhaps even as everlasting Urphänomene. Hannah Arendt, “Walter Benjamin”

4

Thierry’s Trinitarian Theology in Context The lesson of the first part of this book bears repeating. The strongest currents of Platonism feeding ancient Christian theologies had already filtered out their Neopythagorean influences. This holds whether we look to Plotinus and Augustine or to Proclus and Ps.-Dionysius. The Boethian quadrivium, on the other hand, retained Neopythagorean ideas within its very structure, but its discursive location within the practical sciences muted their theological resonance. The persistent exclusion of Neopythagoreanism from the next thousand years of Christian theology prevented the kind of dialogical exchange with this portion of Greek intellectual culture that the religion freely enjoyed with others:  with the hierarchies of Neoplatonism and its ethics of contemplation, Aristotelian natural science and political theory, Stoic psychology and natural law, and, in a later age, Skepticism. Yet given this situation, the Boethian tradition still preserved the best opportunity for Latin Christianity to access Neopythagorean theology, even in comparison to self-consciously Pythagorean revivals in the fifteenth century. When Christian mathematical theologies finally did resurface in Thierry of Chartres and Nicholas of Cusa, they originated from Boethian Platonism and its Nicomachean ancestry. Certainly Augustine was vital for Thierry, just as Proclus and Ps.-Dionysius were dear to Cusanus. But, as I will try to show, these other sources supplemented, amplified, or catalyzed conceptual goods first secured from antiquity by Boethius. The Boethian tradition therefore occupies an unusual place within the spectrum of medieval Christian theologies. On the one hand, it is the very paradigm of the Roman, Christian translation of Greek Platonism into the schools of medieval Europe. On the other hand, it remains an outlier, the lesser sibling of the dominant Augustinian and Ps.-Dionysian traditions. This paradox is captured by Remi Brague’s notion of the “eccentricity” of Roman Christianity: “to know that what one transmits does not come from oneself, and that one possesses it with difficulty,” but to embrace such permanent secondarity as a creative task.1 Boethian traditions are eccentric from the medieval mainstream in part because they harbor

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something not their own, the altogether Greek vision, not yet Christianized, of mathematical theology. Brague speculates that Latin Christianity’s double alienation from its Greek and Jewish origins spurred it to seek continual cultural rebirths. The Carolingian and Florentine renaissances of the ninth and fifteenth centuries were both marked by a resurgent Platonism, a burst of new translations, and a retrieval of the disciplines of the liberal arts. The reflorescence of Christian literature in twelfth-century France was equally marked by these events, but it was also especially preoccupied with Boethian texts, leading Chenu to name the renaissance of the twelfth century the aetas Boetiana.2 It was not just that Christian humanists in the schools read Boethius himself more fervently. More importantly, and whatever their text, their aim was to carry out Boethius’s stated program: namely, to reconcile the different quarters of Greek philosophy with each other, in order to deepen the rationality of the catholic faith. In effect, as Chenu writes, “Boethius realized in the twelfth century what he had wished to do in the sixth.”3 The very eccentricity of the “Boethian age,” however, has meant that its significance is often misunderstood. As Andreas Speer has pointed out, Chenu’s influential image of the “awakening of metaphysics” (éveil métaphysique) in the twelfth century discounts the contributions of the Boethian tradition itself.4 Aristotle had left the relation between theology and first philosophy undecided, but by the thirteenth century Aquinas had followed Avicenna in separating sacra doctrina from pure ontology or metaphysics.5 But as every twelfth-century reader of Boethius’s opuscula knew, on the contrary, theology was to be integrated with philosophy, and ontology was inseparable from the divine One.6 So when historians like Chenu depict the twelfth century as a “second beginning” of metaphysics, this not only insinuates that metaphysics was in suspension after Aristotle. It also denigrates the contributions of intervening traditions from Boethius to Eriugena to Thierry that deliberately opted for a different model of “metaphysics” altogether.7 Speer concludes that if Boethius became marginalized in medieval theologies after the twelfth century, it is chiefly because “Boethius, despite his important role in the history of metaphysics, does not fit into the master narrative at all.”8 Aquinas left his own commentary on Boethius unfinished, a potent symbol of the distance separating them; Duns Scotus and Francisco Suárez followed suit.9 Speer thus traces a “hidden heritage” of alternative Boethian traditions that collectively provide us with a unique critical resource in the present, precisely on account of their eccentricity from the Aristotelian, Augustinian, and even Ps.-Dionysian mainstreams.10 Thierry of Chartres’s rediscovery of the Boethian quadrivium in the twelfth century is one of the richest and strangest treasures of that hidden heritage. This returns us to Merlan’s prophecy about the quadrivium: when its deep Pythagorean origins are forgotten, its theological profundity is lost. But the inverse is also true. Neopythagorean effects can be accidentally reactivated whenever quadrivial

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language is placed in the proximity of theological discourse. Thierry of Chartres’s probing commentaries on Genesis and Boethius represent the moment in the history of Christianity when the latent potential of the quadrivium was retriggered in this way. Thierry began, moreover, right at the point of the tear in the fabric of the Boethian tradition, the discursive rift between number theory and doctrinal formulae, and attempted to stitch it closed. This required of him a radical hermeneutical intervention, one that is intelligible within the local contexts of his intellectual circles, but, as we shall see, one that is also already a reflection upon their common meaning.

The Status of Mediation in Twelfth-Century Platonism Thierry of Chartres was known for his keen intellect, sharp tongue, and rough edges.11 Like Peter Abelard he had left backwater Brittany for his studies, likely at Chartres, and was busy teaching in Paris by the 1120s. He gained sufficient fame through the 1130s to be picked for the chancellorship of the cathedral school at Chartres in 1141. By the late 1140s Thierry was one of a select group of theologians invited to an imperial curia at Trier and the papal consistory at Reims.12 Yet he always relished pushing the boundaries of acceptable inquiry—as in his jocular defense of Abelard at Soissons in 1121, when he mocked the errors of the presiding cardinal.13 Thierry let it be known that he would never “prostitute” his teaching to the whims of the masses, and did not hesitate to expel students from his lectures whom he judged lacking in philosophical curiosity. Amid the contentious curricular debates of the young cathedral schools, Thierry painted his opponents as “clowns” or “pharaohs,” and they declared him a “necromancer” and “heretic” in return. Yet his most loyal students praised Thierry ostentatiously. Bernardus Silvestris named him “the doctor most renowned for true eminence in learning” in the dedication to his philosophical epic Cosmographia. Clarembald of Arras called Thierry the “foremost philosopher in all of Europe” and, not to be outdone, Hermann of Carinthia hailed his former teacher as the reincarnation of the soul of Plato.14 Modern scholarship on Thierry frequently commences (as here) with a ritual listing of these tributes, a pattern that perhaps betrays the historian’s polite wonder. Such encomia from his students at least suggest that Thierry’s interests embraced an astonishing breadth of learning and that he read texts with originality and boldness. Thierry taught comfortably in all seven of the arts, a feat already becoming unusual in his day. One delicious legend tells how the brilliant young Peter Abelard had to quit the elder Breton’s quadrivium lectures on account of their difficulty. By the end of Thierry’s life, most of the Parisian masters

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teaching the trivium had been his students, and the new translators of Arabic science like Adelard of Bath and Dominicus Gundissalinus were seeking him out at Chartres.15 Like a latter-day Boethius, Thierry not only penned commentaries on trivium and quadrivium, but used his chancellorship to compile a massive encyclopedia of all seven liberal arts.16 The Heptateucon, he promised (echoing Martianus Capella), would “marry” the trivium and quadrivium and thus give birth to a nation of philosophers.17 As Jeauneau writes, “had Thierry only compiled the Heptateucon, he would nonetheless deserve high praise as ‘the most devoted explorer of the liberal arts’,” to quote the words of John of Salisbury, yet another enthusiastic student.18 Thierry states in the prologue to Heptateucon that philosophy has two instruments, the intellect and its powers of interpretation. While the trivium assists and embellishes interpretation, “the quadrivium illuminates the intellect.”19 So too in his first lectures on Boethius, Thierry remarked to his audience that “the ancients customarily learned mathematics first, so that they could attain to knowledge of divinity.”20 Histories of medieval philosophy typically reference Thierry’s hexaemeral exegesis and his “Pythagoreanism.” To modern readers, the matter-of-fact naturalism of his Genesis commentary is indeed a startling chapter of Thierry’s works. In his telling, once God set the cosmos in motion, the six days of creation unfolded solely according to materialist principles, from the concentration of protostellar matter to the germination of human bodies, entirely uninterrupted by subsequent divine intervention. But these passages in his Genesis commentary make up only a few pages out of the hundreds that Thierry wrote or taught, and one recent study has revealed the extent of his commentary’s debt to William of Conches.21 Thierry devoted many more pages to Boethius’s works. Yet the colorless density of his Boethius commentaries has kept them from being treated alongside the more vivid inventions of Hildegard of Bingen, Bernardus Silvestris, or Alan of Lille in accounts of the twelfth-century renaissance. Thierry and his peers in the schools were fascinated by Boethius. To them the Roman’s methods for applying Greek logic to problems in Christian theology seemed to anticipate the same questions that they found most pressing. Humanists in the new cathedral schools of northern Europe searched for sources both trustworthy and ancient, particularly if they had the seal of martyrdom. For such twelfth-century scholastic readers, Boethius was not the last of the Romans and first of the scholastics, in Lorenzo Valla’s phrase. Rather he was the authoritative master of that weightier portion of the liberal arts, the quadrivium, and the benchmark of orthodoxy in using the trivium to purify reason’s approach to the Christian mysteries. A twelfth-century master found in Boethius a nearly universal resource in the arts curriculum governing humanistic learning:  a library of theology, poetry, logic, harmonics, arithmetic, and geometry. As Chenu writes, “Boethius procured for the Middle Ages the verbal resources necessary for their

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speculations . . . a content impossible to inventory, and one of rare semantic dynamism.”22 Thierry is called “Pythagorean” by historians especially for comments made in his Genesis commentary, as we shall see, and for the drift of some passages in his Boethian commentaries.23 Yet the long and winding history traced in Part One makes it clear that including Thierry in this tradition only invites further questions. Which Pythagoreanism characterizes Thierry’s ideas, given the different strata of Presocratic, Platonic, Academic, Hellenistic, Middle-Platonic, and late antique iterations of that tradition? Which sources did Thierry possess and how did he reconstruct them, since precious few Greek Neopythagorean traditions had found their way to twelfth-century Paris? Thierry and his colleagues made glancing contact with some traditions through Macrobius, Calcidius, and Martianus Capella. The first two shared arithmologies with Philo and Nicomachus, while the third addressed the quadrivium. Thierry’s contemporaries, Bernard of Chartres and William of Conches, represent the apex of medieval commentaries on Plato’s Timaeus, the charter of mathematical Platonism ever since Plato confronted Archytas in that dialogue. These were all important textual conduits, but the fact is that no one channel simply delivered Thierry the raw materials of Greek Neopythagoreanism ready to hand. If they had, then Thierry would not have been the sole “Pythagorean” voice in the twelfth century.24 To determine what it would mean to call Thierry “Pythagorean,” one must examine not only what sources Thierry read, but also how he read them and, given their fragmentary nature, how he read them together. That is to say, grasping the character of Thierry’s Pythagoreanism means understanding the hermeneutical techniques he used to sift through fossilized elements of mathematical theology that remained subtly preserved in Augustine and Boethius. Thierry was no conjurer, as his most malicious detractors claimed. But the Breton master did work a kind of conceptual necromancy by congealing the rare fragments that he had collected back into a living, breathing whole, a Neopythagoreanism long expired but now revived upon the new terrain of western Christianity in the twelfth century. As I will show, Thierry’s initial hermeneutical experiment in reading Augustine and Boethius together drove him to pose a sequence of implicated questions over two decades. In answering these he slowly arrived at his mature theology. But to see why Thierry ventured this experiment at all, and why he conducted it in this way, we need to establish the contemporary intellectual conditions and Platonist sources that motivated him in the first place.

Autonomous Mediation For a relatively brief window of time, roughly the middle half of the twelfth century, the most innovative Christian authors took their cues from the second-hand

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Platonism of late antique Latin sources. Hermetic texts, Macrobius, Calcidius’s Timaeus, and the Boethian trove all drew out the theological aspects of Platonism, providing ample fodder for the masters’ rumination. In particular, the so-called “school of Chartres” exhibited a common set of interests and proclivities, if not literally a geographical center.25 As Speer has shown in his authoritative study of the rise of “natural science” in the twelfth century, this muddled Platonism was the matrix of the Aristotelian natural philosophy in the following century.26 But these years were more than a temporary scaffold for the scholastic edifice. The theological resources of Platonism spoke directly to the concerns of the twelfth century itself. The transformation of agricultural technology and commercial networks in the eleventh century delivered a period of steady urbanization that drove a revolution in education.27 The new cathedral schools produced the first institutions of relatively uniform higher learning outside of the monasteries in western Europe, servicing an urban merchant class who required skills in grammar and arithmetic. Masters who also taught theology, duly impressed by such recent social and economic successes, trained their attention on human agency—its reason, its passions, its body—fostering what Richard Southern has called a season of “medieval humanism.”28 From the start, monastic leaders were suspicious of the altered curriculum and practical methods of the schools, particularly when their curiosity wandered from the liberal arts into theological territory.29 Episcopal condemnations won by conservatives in the 1140s and then especially in 1210 ended the vogue of twelfth-century Platonism in theology and coincided roughly with the initial translations of Aristotle’s natural philosophy. To speak in the most general terms, there is a ubiquitous sense in the Christian literature of these decades that the human mind possesses a certain autonomy underestimated heretofore, an impressive array of cognitive and ethical potentials even before any interaction with the divine. Reason exercised its native freedom whenever it took as its object, not the overwhelming absolute of divine revelation, but lesser mediations of divine order that hover in between God and the physical world. The human mind can know, for example, the workings of the natura created by God separately from any knowledge of the Creator. This is possible because the cyclical rhythms of Nature reflect divine stability at one remove, so that natural causation can be sufficiently grasped by reason. One can understand the principles of weather patterns or human reproduction without needing to glimpse their invention in the mind of God. Such middle zones provided an arena for reason’s powers to test themselves on matters of fundamental importance that were not yet matters of revealed truths.30 Nature was no longer conceived, in the words of Tullio Gregory, “as a simple voluntas Dei or as sacramentum salutaris allegoriae, but as vis genitiva, ignis artifex, causarum series, qualitas planetarum, regula mundi . . . as the object of a study aiming to know the legitima causa et ratio of every natural event.”31 Human knowledge won its independence by making contact with

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these highest strata just below the deity and observing the beauty of their autonomic, self-sustaining order, knowledge of which might even communicate a share of divine stability to a consummately unsteady humankind (as Plato had proposed in Timaeus). Natura is only one example of the kind of mediations in which philosophers, poets, and scientists invested in the twelfth century. The Platonist sources studied within the new schools not only encouraged and shaped this line of thought, but also advanced potential candidates to fill such mediating roles, whether nature, seminal reasons, providence, love, number, or Plato’s anima mundi.32 The poetry of Boethius’s Consolatio is a treasury of such mediators: the persona of Lady Philosophy, the natural cycles of stars and seasons, the orders of fate and providence, or the cosmic governor lauded in the hymn O qui perpetua, to name a few. The Hermetic dialogue Asclepius, believed to reveal the Egyptian wisdom transmitted to Plato, praises the mediating function of the human as “microcosmos.” Human being cultivates the earth below in agriculture and the heavens above in worship, poised as it is between the two.33 Macrobius’s digest of Neoplatonist cosmology teaches that Nous, the divine Mind, connects God and world. In Calcidius’s commentary, the Timaeus dialogue introduces a cast of different mediators, as we have seen: the receptacle, the primal matrix of becoming that holds open the space for cosmogenesis; the archetype, an exemplary blueprint of the empirical order; the demiurge, the artifex who instructs a delegation of lesser gods to create; and the world-soul, which animates the universe with motion and life by infusing it with mathematical harmonies.34 Platonist traditions have always relied axiomatically on the necessity of such mediations shuttling between God and world. But at the same time they have suffered from a certain indecision regarding their identity. In the Old Academy Xenocrates had named soul the prime mediator, but for Speusippus it was number. Eudorus proposed his triad in order to harmonize competing theologies of mediation in Plato’s written and oral traditions. Philo and Nicomachus placed two competing schemata side by side, the Logos and the decad, and Iamblichus and Proclus differed over the status of mathematicals. Twelfth-century Christian Platonists inhabited a very different intellectual environment than Mediterranean authors centuries before. But the sources that they happened to inherit in abundance date to the period of maximal disagreement among Platonist traditions regarding the various identities and statures of such mediations, namely the second to sixth centuries ce, as we saw in Part One. I have digressed on the theme of mediation because its centrality among the most prominent twelfth-century Platonist sources points us toward the proper context for interpreting Thierry’s theology. One used to situate Thierry within the “school of Chartres,” but Southern has called into question the very existence of a unified “school” physically located at Chartres.35 Marcia Colish protests that there

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is nevertheless “a detectable family resemblance among the thinkers committed to the Chartrian project, despite their individual differences, and irrespective of whether they themselves studied or taught at Chartres.”36 Yet Southern’s deep skepticism about the cohesion of a Chartrian school of thought has persisted, in part, on account of inadequate definitions of the shared “project” to which Colish refers. It is not enough to allude to a common poetic sensibility in interpretation and composition, an openness to pagan philosophy, or a putative rationalism in natural science or theology. Rather, we can specify the family resemblances of Chartrian authors by looking to the problem of mediation bequeathed by Middle Platonist and Neoplatonist traditions and enriched by the century’s confidence in human reason. The need to name, distinguish and rank different species of mediators under different nomenclatures preoccupies all of those authors usually grouped together. In short, the unifying project of the Chartrian school was to set forth a new aesthetics of autonomic mediation, in different genres, with different sources, and to different disciplinary ends. To situate Thierry of Chartres in context, I propose, means to discover how he approached this task in connection with or distinction from his fellows.37

The Problem of Bernard’s Gloss Thierry belonged to a network of similarly minded authors in the 1130s and 1140s, particularly William of Conches, Gilbert of Poitiers, and Adelard of Bath. Their most successful students, like Bernardus Silvestris, John of Salisbury, Hermann of Carinthia, and Alan of Lille, took different cues from the methods and interests that they passed down.38 Yet William, Gilbert, and Thierry probably shared one master in common, Bernard of Chartres (1050?–ca. 1126).39 Although we know little about Bernard’s life and works, what we can piece together suggests the kind of intellectual terrain that he surveyed. Bernard is known from brief reports by John of Salisbury and from his widely copied Glosae super Platonem (ca. 1100–1115).40 He modeled a new way of reading the Timaeus that made possible a deeper philosophical confrontation between Platonism and Christianity. Both a grammarian and a philosopher, Bernard was more interested in grasping Plato’s actual words than in passing along the formulae of theological authorities from prior centuries.41 His independent appraisal of the dynamics of mediation in the Timaeus remained open to the historical difference of Greek Platonism, an indispensable prerequisite for an authentic retrieval of Neopythagorean traditions. Hence while Bernard built no monument comparable to those of later twelfth-century authors, he did secure a territory that others, like Thierry, found amenable for more ambitious constructions. Christian interest in the Timaeus had both intensified and shifted focus in the eleventh century. Commentators began to treat Calcidius and Plato as two

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distinct authors, to favor an interlinear gloss over marginal annotations, and thus to attend the metaphysics of the dialogue as much as the quadrivial information it conveyed.42 Conservatives like Manegold of Lautenbach in the eleventh century read Timaeus with Calcidius in one hand and Augustine in the other, noting their points of contact but warning Christians off from their divergence.43 For his part, Bernard of Chartres sought to do something entirely different. He dispensed with Calcidius’s Middle Platonist overlay, focused on Plato’s text on its own terms, and suspended any premature theological concerns, indeed ignoring Augustine altogether. Bernard’s insistence on a precise, patient, and above all systematic investigation of the text expressed an openness to learning matured within the cathedral schools and a conviction that grammatical analysis was the foundation of quadrivial learning. His methods not only revived some of the perennial philosophical topics raised by the Timaeus—the existence of an archetypal world, the identity of the demiurge, or the eternity of matter—but also, as Speer puts it, “catalyzed” the turn to the quadrivium in eleventh-century natural philosophy urged by Fulbert of Chartres and Gerbert of Aurillac.44

Boethian Innovations Bernard made two important proposals in his Glosae. Both illustrate the focus on mediation in twelfth-century Christian thought, and both center on Boethius. First, Bernard drew attention to the hierarchy of God, forms, and matter in the Timaeus.45 Augustine had assimilated the forms to God by locating them within the divine Word. Influenced by Numenian dualism, Calcidius had underscored the opposition between matter and God and had ignored the theological quandaries of the forms.46 Bernard refused to take either way out. Instead he postulates a secondary level of forms, formae nativae, that reside within matter.47 Paul Dutton calls these forms “active intermediary principles,” and Gangolf Schrimpf points out that they correspond to the mediating function of the world-soul.48 The formae nativae are nothing less than the ground and locus of the causal necessity that produces the autonomic regularity of nature. Reason is able to cognize natural order without divine illumination because reason grasps the formae nativae within the physical world. Accordingly, Bernard departs from the Platonic and Calcidian denigration of necessitas, disconnecting it from the disorder of ὕλη and assigning it to the operations of reason. For him, necessity is not a feature of primordial matter but of natural causation; it is a hallmark of true knowledge of the physical world.49 Bernard also proposed that the archetypal world described in the Timaeus myth cannot be known. We can only refine our grasp of the sensible world.50 This leads him to consider which discipline delivers such natural knowledge, and here, as Speer explains, Bernard makes a momentous suggestion. The two regnant divisions of science available to Bernard were the Boethian-Peripatetic

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model (theologia, mathematica, physica) and the Augustinian-Stoic model (ethica, logica, physica). Rather than simply selecting one, Bernard cannily coordinates them, such that the Boethian mathematica, the quadrivium, defines the content of the Stoic physica. Physical science thus amounts to mathematical analysis of the sensible world, making the quadrivium the primary discipline for thinking the Platonic cosmogony in the Timaeus.51 As Speer notes, this is a double breakthrough, even if in germinal form. Bernard first integrates the quadrivial arts into a unified mathematical discipline. Then he modifies its function from being a preparatory science within the liberal arts to being an applied science with a distinctive task.52 Bernard’s insights on the formae nativae and the quadrivium reinforce each another, and together they begin to address the riddles of Platonic mediation raised by Timaeus. The formae nativae provide matter with a rational infrastructure that enables precise, necessary knowledge of the physical world through the quadrivium. This means that Bernard effectively identifies number as the immanent principle of cosmic order. Number enables reason to grasp the causal necessity imbued by the formae nativae and hence to know the universe without divine illumination. As Schrimpf explains, Bernard’s formae nativae effectively mathematize the ontology of nature, construing every physical object “as a play of numbers only accessible to thought, numbers which ground its outward appearance and are expressed through it . . . as something that owes its determinacy to corresponding mathematical structures.”53 As Speer points out, if Bernard renders “all phenomena down to the totality of their smallest elements . . . as sensible expressions of mathematical givens,” then surely the mathematization of nature does not begin in the seventeenth century or even the fourteenth, but at the dawn of the twelfth.54 By carefully confronting the Platonic theology of mediation, and by linking it to the universal mathesis of the quadrivium, Bernard has rediscovered the autonomy of arithmetical mediation. Through number, it would seem, the world can be known independently of God. An aside: one could hastily conclude (Speer does not) that Bernard of Chartres was therefore the accidental progenitor of the kind of mathematizing rationalism which in early modernity seemed to render God superfluous. The apparent ease of this historical comparison, which surely informed past descriptions of the school of Chartres as “rationalist,” results from the dearth of interactions between Neopythagorean and Christian theologies in antiquity and the Middle Ages that might complicate or qualify its judgment. Yet in light of Part One above, it cannot be true that Bernard’s revival of arithmetical mediation is somehow rationalist, secularizing, or antitheological per se. As we have seen, there is on the contrary a long and differentiated history of mathematical theologies centered on arithmetical mediation. Some countenance a given Logos doctrine alongside Arithmos, and some do not. The opposition of the Christian understanding of God to the

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mathematization of nature is not an insuperable contradiction, but should be historicized, beginning with Augustine’s gradual elevation of Logos at the expense of Arithmos. So rather than ask whether Bernard anticipates the Cartesian mathesis universalis, a wiser path is to evaluate the makeup of Bernard’s inchoate mathematical theology and its relationship to Christian doctrines of Trinity or Incarnation. What ancient sources inform his theology of mediation? How does it relate to Augustine? Now, Bernard’s Timaeus gloss strategically departed from Calcidian dualism as much as from the Augustinian theology of creation in the Word. What was Bernard’s lodestone, if not these, for his pathbreaking reading of the dialogue? When we look to his immediate sources, and the common inspiration behind his two signature achievements, it becomes clear that what Bernard achieved was a Boethian reading of the Timaeus.55 After Calcidius himself, Boethius is the most frequently cited source in the Glosae, though never by name.56 Bernard references Consolatio, the theological opuscula, and the logical works, but most frequently of all, he cites Boethius’s quadrivial works on arithmetic and music, often cross-referencing the Institutio arithmetica throughout his gloss.57 Even more telling are the deep connections between Bernard’s formae nativae and Boethius’s theology in De trinitate. As Bernard prepared his gloss, he evidently ran across Calcidius’s distinction between two degrees of form, species intelligibilis and species nativa. Bernard built upon Calcidius’s idea but reinforced it with the vocabulary of Boethius. Boethius had argued that since God is the supreme form, forms embodied in matter are better thought of as images or simulacra of the one true Form. Bernard accordingly defines the forma nativae as the “simulacra” or “images of the ideas,” hidden by God within matter in the process of creation.58 Because the exemplary forms remain transcendent within God’s mind, they can imprint themselves in matter only through images of themselves. Dutton notes that Bernard not only draws upon Boethian philosophy to construct this concept, but in doing so follows the “Boethian mandate” to reconcile the forms of Aristotle with the forms of Plato.59 As Étienne Gilson has suggested, Bernard’s use of Boethius already influenced Gilbert of Poitier’s commentaries on the opuscula. Gilbert’s unusual ability to mesh Aristotelian analysis with Platonist metaphysics owed much to his former master, Bernard, as did his unusual theory of “mathematical” abstraction, his theory of universals, and his notoriously exacting style.60 In Metalogicon, John of Salisbury depicts Gilbert wishing to perpetuate the doctrine of formae nativae and links them even more directly to Boethius than Bernard’s subtle allusions suggest.61 For example, John uses Boethius’s terms from De trinitate to depict the formae nativae as reflections of the divine exemplars. He weaves in Seneca’s account of the two levels of form, which likely represents another of Bernard’s major sources.62 John even compares the exemplary forms to the immutability and

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universality of number, juxtaposing Boethius’s opening apology for mathematics in Institutio arithmetica with Augustine’s encomium to number in De libero arbitrio.63 John’s implication, apparently echoing Gilbert’s report of Bernard’s own views, is that the immutability of the divine exemplars is reflected through the semimathematical structures of the formae nativae. To make his case for this reading of Timaeus, Bernard apparently relied on Seneca, the early Augustine, and especially his cross-reading of Boethius’s doctrinal and quadrivial texts. Bernard’s goal in his Timaeus gloss was to access Plato’s cosmology on its own terms, even if that required suspending questions of Christian doctrine. This means that he deliberately left unanswered the theological questions that his deeper reading of the dialogue had provoked. Unlike his students, Bernard steadily resisted the temptation to harmonize Plato’s anima mundi with the Holy Spirit. Following Macrobius, he also carefully distinguished the transcendent God from the contents of the divine Mind, the primal ideas; even if both are eternal, Bernard maintains, only God is “coeternal” in the three persons of the Trinity.64 Bernard thus envisages a fourfold cosmic hierarchy:  God (Trinity), the eternal forms (God’s mind), the formae nativae within matter, and formless matter qua matter. Ultimately, Bernard’s gloss does not resolve, but only exposes, the uncertain status of different Platonist mediators. How should Platonist mathematical theologies—in the lineage connecting Timaeus, the Nicomachean quadrivium, and Boethius—be viewed in light of the Christian Trinity? How do Mind, anima mundi, number, or Natura relate to Word and Spirit? Before Bernard, these questions were not yet entirely visible, but now they required answering. Dutton claims that Bernard continued to tackle such problems in subsequent works, now lost.65 Yet he also suggests that despite the enormous popularity of the Glosae, Bernard’s Platonism “was quietly being put aside soon after his death, relegated by his students to a small chapter in the history of philosophy.”66 But this discounts the important case of Thierry of Chartres and forgets that in the history of ideas, influence proceeds as much by questions asked as by doctrines repeated. Bernard’s gloss named Platonist mediation as a pivotal theological issue in the new century and showed how to orient oneself to it unimpeded by Augustinian prejudice. Bernard himself made one glancing pass at answering his own question (the formae nativae), but whether or not his proposed solution stuck is finally unimportant. What matters is how long Bernard’s questions tolled unanswered in the minds of his students. According to Speer it was the encounter that Bernard first staged between the Timaeus and Boethius that allowed the wider school of Chartres to make new contributions to the Platonist Vermittlungsproblem, from Bernard’s formae nativae, to William of Conches’s elements theory, to Thierry of Chartres’s four modes of being.67 Similarly, Gregory links Thierry’s aesthetics of form to Bernard’s doctrine, and Winthrop Wetherbee has argued that the “formae nativae proved remarkably stimulating for both theological speculation and

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poetry” in the works of Gilbert of Poiters, William of Conches, Thierry of Chartres, and even Bernardus Silvestris.68 I have already briefly mentioned Gilbert. A quick glance at William and Bernardus, before moving on to Thierry himself, will help to confirm the centrality of the aesthetics of autonomic mediation among the students of the senex Carnotensis.

Two Counterexamples After studying with the renowned Bernard at Chartres, William of Conches (d. ca. 1154) would eventually attempt his own Timaeus gloss, but first he worked through Boethius’s Consolatio and Macrobius with the same comprehensive, sober analysis as his teacher.69 Although William never mentions the formae nativae, his decision to focus on theories of natural causation throughout his writing career was surely influenced by Bernard’s interests. But unlike his teacher, William actually probed the operations of natural phenomena like precipitation or human reproduction. Where Bernard had wisely demurred from the riddle of the Platonic world-soul, William dove in as eagerly as had Peter Abelard. In his Glosae ad Timaeum, William listed the challenges facing Christian readers of Plato. Where is the anima mundi located, how is it generated, and what activities does it perform? As Southern has shown, at first William boldly embraced Plato’s theology of mediation through the world-soul, but then he gradually backed away when intimidated by monastic opposition.70 This accords with Gregory’s hypothesis that as the tensions between the anima mundi and Trinity grew more evident in the early twelfth century, the mediating functions of the world-soul were transferred to natura, whose mundane identity appeared to pose fewer theological dangers.71 William’s writings are the prime evidence for this trend. The later Philosophia mundi and Dragmaticon philosophiae answered the same questions about Platonist mediation formulated in his Timaeus gloss, but now converted into a philosophy of nature.72 William’s physical theory in Dragmaticon is founded upon a distinction between elements (elementa) and “elementeds” (elementata).73 Elements are imperceptible minima known by reason but not sensation; elementeds are perceptible multiplicities, built out of elements, which construct physical reality. Informed by the cosmogony of the Timaeus, William postulates that God created the world in two stages. First, God created the primordial matter ex nihilo as nothing more than a confused wash of elements, which God then ordered by introducing cycles of exchange, redistribution, and equalization. Second, this nascent elemental order, natura, proceeded by its own momentum to generate elementata, forming the phenomenal cosmos as we know it. Consequently, nature produces only what it already has, like from like, with an autonomic regularity that requires no further activation or rejigging by the Creator.74 Nature therefore remains amenable to reason without divine illumination, insulating physics from theology. The

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physicist traces the elementeds back to their constituent elements using the laws of natural order as a guide. The task of the theologian, by contrast, is to begin with the same physical phenomena, but to trace them back past the intermediate elements and primordial chaos to their divine origin. This model by William of Conches represents one strategy to resolve the Platonic mediation problem exposed by Bernard of Chartres. William began with Bernard’s focus on the Timaeus, but when trouble arose with the world-soul as mediator, he pursued instead the mediation by natural causation intimated in Bernard’s formae nativae. William executed Bernard’s vision of a theologically neutral physics exceptionally well, along with Adelard of Bath and Hermann of Carinthia. But unlike Gilbert of Poitiers or Thierry of Chartres, William never developed a substantive doctrine of the Trinity in relation to either the mediation of world-soul or the mediation of elemental nature. In this way the encounter between Platonist mediation and Christian doctrines was obviated, not resolved. Nor did William take up the other half of Bernard’s agenda, the promotion of a unified quadrivium as a tool for natural science. Finally, William seemed to search for an adequate literary form. After glossing the old authorities as Gilbert and Bernard had, he tried his hand at philosophical dialogues in Philosophia mundi and Dragmaticon philosophiae. As Wetherbee has shown, the next generation of twelfth-century Christian Platonists chose to leave aside the strictures of commentarial traditions altogether in order to explore other aesthetics of mediation with greater nuance and complexity.75 Bernardus Silvestris of Tours, a student of both William and Thierry, invented a creative new response to the tension between Platonist and Christian mediators.76 Bernardus was aware not only of Bernard’s sources but also the new works by Bernard’s students. He patterned his allegorical epic Cosmographia after the sequence of Timaeus, but also cited Asclepius alongside contemporary Hermetic works. He borrowed heavily from William of Conches’s gloss on Consolatio and was familiar with new translations of Arabic natural science from Adelard of Bath and Constantine the African. He knew Thierry’s Genesis commmentary intimately and dedicated the work to him. As Wetherbee writes, Bernardus’s Cosmographia is “both a brilliant distillation of, and a shrewd commentary on, the achievement of Thierry, William and Bernard . . . reencoding them in a new cosmic myth.”77 Bernardus took full advantage of the flexibility of poetic discourse to suggest more complex and subtle theologies of mediation than doctrinal definitions would allow. His guiding insight was to grasp that Plato’s cosmogony—and the Chartrian theology that aimed to remain in contact with it—was made safe for Christian theology not by reducing competing mediators to Word or Spirit, as Abelard and William of Conches had tried, but on the contrary, by multiplying mediators (or rather mediatrices) with such profusion that any conclusive harmonization became impossible. What Wetherbee argued generally about Chartres applies particularly

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well to Bernardus: “poetic intuition is finally the only means of linking philosophy and theology, pagan auctores and Christian doctrine.”78 One cannot worry too much about the divinity of the world-soul when it is only one of eight or nine different semidivine figures crowding the pages of an evidently literary work. Bernardus’s decision to multiply his cast of divine emissaries in Cosmographia demonstrates that the Vermittlungsproblem continued to trouble Trinitarian theologies in the Chartres circle well into the 1150s. Other contemporaries followed in the path of Bernardus’s poetic gambit. As Barbara Newman has argued, Hildegard of Bingen and Alan of Lille depicted the mediations of Nature, Love, or Wisdom as personified goddesses, both because it was “safer to theologize about them than about the Trinity” and because they could “mediate in diverse ways between God and mortals.”79

Thierry on Quadrivium and Trinity The new hermeneutic that Thierry ventured in his Genesis commentary should be viewed in the immediate context of Bernard of Chartres’s Christian Platonism. Bernard’s scrupulous gloss reopened the most Pythagorean of Plato’s dialogues, encouraged the turn to Boethius, and focused attention on the quadrivium. Gilbert of Poitiers invested further in the Boethian opuscula and reconceived the role of mathematical abstraction in philosophy. William of Conches pursued a philosophically grounded physics, leaving others to excel in the quadrivium, and attempted (with little success) to address some of the Trinitarian riddles left in Bernard’s wake. In this light Thierry of Chartres’s intentions in glossing first Genesis and then Boethius’s De trinitate take on new meaning. It is not difficult to imagine what such a clear-eyed young master as Thierry might take up if he were, so to speak, to look around him and assess the state of Bernard’s incomplete project. As a counterweight to Timaeus, he would choose to gloss the biblical cosmogony found in the opening chapters of Genesis, employing the same disciplinary tools of physica and quadrivium commended by Bernard, and applying the same physical mechanisms recently posited by William, but now in order to address the theological quandaries of Platonic mediation that remained unresolved.80 This is precisely what we find in his early hexaemeral commentary Tractatus de sex dierum operibus. If this Genesis commentary appears “Pythagorean” (still begging further historical classification), it is not because Thierry was privately inclined to so-called number speculation; instead it was because Bernard’s study of Platonic cosmogony had left a critical task unfinished. Bernard never asked how the mathematical philosophy he encountered in Plato’s Timaeus and Boethius’s quadrivium might challenge—or even expand—the Christian understanding of the Trinity. This became Thierry’s agenda in Tractatus.

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Roads not Taken Thierry’s intentions are more easily seen when compared with those of his nearest contemporaries. Unlike Bernardus, Alan, or Hildegard, Thierry sought a constative definition of the mediating roles of Word and Spirit, not a poetic circumvention of the problem.81 Unlike Alan again—who built on Boethius’s axiomatic procedure in De hebdomadibus—Thierry did not value the quadrivial arts formally for the semantic properties or logical surety they could lend theological method.82 Unlike William, he chose to gloss scripture and not only Platonic texts with the instruments of grammar and the physics of the quadrivium. Unlike Gilbert, Thierry was willing to countenance a measure of speculative invigoration in Trinitarian theology at the expense of analytic precision, if that would help to restore the Augustinian tradition that Bernard had suspended. Hence unlike William, Adelard, and Hermann, he did not seek out fresh Arabic sources to amplify Bernard’s gloss, but rather bolstered Bernard’s use of Boethius and took pains to add Augustine, noting the reverence for number that they both shared.83 It is worth emphasizing that Thierry chose against another avenue available to him in the mid-twelfth century, namely arithmology or “number speculation.” A  rich tradition of number handbooks for scriptural interpreters went back to Isidore of Seville’s Liber numerorum, Hincmar of Reims’s commentary on Wisdom 11 in his Ferculum Salomonis, and Alcuin of York’s De comparatione numerorum.84 These stemmed primarily from Augustine’s Genesis commentary but were also deeply informed by the Boethian arithmetic. In Thierry’s own lifetime the genre was enjoying a renaissance among a circle of Cistercians, the order to which the Breton master retired and the most ardent copyists of his works.85 Inspired by the exegetical methods of his teacher, Hugh of St. Victor, Odo of Morimond (1116–61) composed Analetica numerorum et rerum, just as Thierry was revising his Boethius commentaries in the 1150s. In the 1160s William of Auberive (d. ca. 1180)  took up the sequence 3 through 12 in a careful Boethius commentary called Regulae arithmeticae. Later that decade Geoffrey of Auxerre (d. ca. 1190), who had opposed Gilbert at Reims, applied William’s methods to 13 through 20 in De sacramentis numerorum, and by the end of the century Thibaut of Langres (d. ca. 1200) had drawn up a short summa on the significations of numbers. The daring of this Cistercian arithmetical theology is evident in the reaction of their contemporaries. In the spring of 1147 the young Odo sent an early draft of Analetica to his benefactor, the archdeacon Peter of Traves, dean of the cathedral of St. Stephen in Besançon.86 In his treatise, Odo discusses the properties and poetics of numbers. Unitas defines deitas, and dualitas is the prime mediator (the Word-flesh; the God-man; God’s power-wisdom).87 Peter answered that what he took to be Odo’s central claim, that God is number, was not only obscure and

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useless speculation, but potentially heterodox.88 Odo fired off a cheeky reply exasperated with the conservative instincts of his superior: Amen, I  say unto you, Peter, that I  have not said that God is number. I  know and I  am certain what I  have written. I  stated that number and wisdom and the Word are one. What I have written, I have written. . . . At the risk of some umpire of the syllable scolding me for formulating it too directly: Augustine has clearly said that number and wisdom are one and the very same thing!89 In his defense Odo cites extracts from Book II of Augustine’s De libero arbitrio and Books III and IV of De genesi ad litteram on Wisdom 11, just as he had in his Analetica.90 These are of course the same texts that we studied in Chapter 3 in exhuming the young Augustine’s theology of number. Peter’s reaction to Odo’s Augustinian Neopythagoreanism reminds us that mathematical theology remained highly controversial even in Thierry’s day. Like Odo, Thierry was fascinated by the theological potential of number at the intersection of Boethius and Augustine. But he would combine them in a very different way. In the 1110s Bernard had glossed Timaeus and discovered the centrality of the quadrivium, but also the Vermittlungsproblem of the Trinity. In the 1120s, William had glossed the Consolatio and adduced the new Timaean physics when such topics arose in Boethius’s poetry. In the 1130s, Thierry took the next step.91 He imported William’s physics from the Consolatio commentary into the framework of a commentary on Genesis, where he would be better positioned to attempt the rapprochement between Boethius (on quadrivium) and Augustine (on Trinity) that he had in mind. Then in the 1140s Thierry wrote his own Boethius commentaries—not on Consolatio, which looked back to Timaeus, but on the treatise that spoke to the metadisciplinary questions raised by all such Platonist sources: De trinitate. When we turn to Thierry’s Genesis commentary, we do not find number symbolism. Rather, the organization of Tractatus reveals its debt to Bernard’s methods. His commentary will proceed “according to the understanding of physics” (ratio phisicorum) by attending to the literal sense of Genesis and will make three consecutive sweeps through the same verses of Genesis 1.92 In the first pass (§§2–3), Thierry proposes that the “order of causes” of creation follows a Trinitarian pattern. Deus, sapientia, and benignitas correspond to the efficient, formal, and final causes of creation; the material cause is the four elements. Thus the Father creates (efficient cause); the Son informs and arranges what is created (formal cause); and the Spirit loves and governs the arrangement (final cause).93 Here we learn that what interests Thierry about Genesis is the relevance of the Trinity to the operation of the physical cosmos. Starting over again, Thierry then outlines the

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“order of times” of creation (§§4–17), marking the elemental transformations that occur over each of six “natural days.” Following Augustine in De genesi ad litteram, Thierry holds that God “rested” after the sixth day by relying upon the immanent self-production of “seminal causes.”94 This is not only a cosmogonical statement but a discursive one. Augustine’s theology of creation comports well with Bernard’s and William’s new physical explanation of nature, and Augustinian tradition can be sustained even within the new mathematical cosmology of Timaeus. As we shall see, Thierry underscores this guiding principle by conspicuously reverting to Augustine on two other occasions in Tractatus. The third pass through Genesis 1 is the promised “literal exposition” (§§18–29). Thierry’s goal here is to demonstrate, using only a physical account of the elements, how heaven and earth are created simultaneously, how the earth was void, and how God’s Spirit moved above the waters. This leads him to the perennial topic in hexaemeral and Timaean exegesis of the world-soul. When Moses stated that the earth was formless and void, writes Thierry, he named what the Greeks call ὕλη or χάος. Is the Spirit above the waters likewise the world-soul? Thierry does not hesitate for a moment: what Christians know as the Holy Spirit is called different names, whether Plato’s anima mundi or others given by Hermes, Vergil, David, or Solomon.95 The topic of the world-soul gives Thierry the excuse he needs to take up deeper problems of mediation and Trinity in his commentary’s remaining pages (§§30–47). Why was the Spirit mentioned before the Word? How does God operate in matter? What is the role of divine Wisdom, the second person of the Trinity?96 At this juncture the text of Tractatus records several rapid leaps of thought. Thierry’s deft hermeneutical maneuvers need to be replayed at a slower place in order to observe their significance. Up to this point, Thierry has resorted to William of Conches’s physics to show that the order immanent in the four elements provides sufficient rationes for the progress of creation. Material causality has sufficed for understanding the development of created order. But once the subject of theological mediation arises with the mention of the anima mundi, the reasons of physics fall short of grasping the identity of the Spirit or Word. For Thierry to transition from material causation to the other three Trinitarian causes, he must advance to a higher level of rationes, the level of “true and holy theology.”97 Hence what follows was intended by Thierry not as poetic rumination or mere philosophical propaedeutic; it is simply the true Christian doctrine of the Trinity.

A New Strategy Thierry’s first maneuver is to pivot from physics to mathematics. In the wake of Bernard of Chartres, his own Genesis gloss secundum physicam could not help but

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shift toward an exploration secundum quadrivium, so to speak. Thierry abruptly states: There are four kinds of reasons which lead humankind to the knowledge of the Creator: namely, the proofs of arithmetic, music, geometry, and astronomy. Such tools can be used briefly in our present theology, so that the craft of the Creator in things becomes evident and so that what we have set out above can be rationally demonstrated.98 Although he does not state the term, Thierry is clearly referencing the quadrivium of Boethius. The higher reasons of mathematics can deliver the mind into knowledge of God. Just as he drew upon physics for the material cause, now he plans to use the quadrivium to probe the workings of the Trinity. From the perspective of a twelfth-century master, Boethius was associated with two achievements: his effortless anatomy of the logic of Trinitarian predication, displaying the power of the trivium, and his injunction to philosophers to seek eternal truth through study of the quadrivium. A  connoisseur of the Boethian oeuvre like Thierry might well wonder whether the quadrivium could illuminate the Trinity just as well as the trivium. To pursue this line of thought, Thierry needed only to assert the radical unity of the Boethian corpus. Note that Thierry is not actually interested in mathematics as such. In the remainder of Tractatus he refers twice to arithmetic but never mentions the other three sciences, and although he will draw repeatedly from Boethius’s Institutio arithmetica, he never moves beyond its elementary tenets.99 Rather, Thierry is drawn to the theoretical possibility that the quadrivium might inform the highest reaches of philosophy— to the very mathematizability of nature, we would say—more than actual scientific practice.100 In this he follows Bernard of Chartres, who never pretended to be a working physicist like William or a working mathematician like Adelard of Bath. Bernard instead aimed to uncover the transcendental philosophical assumptions grounding the quadrivium. After this dramatic turn to the quadrivium, Thierry quickly takes a second interpretive step, and once again, the demanding Breton master does not pause for his students to catch up. To his quadrivium citation Thierry immediately appends a reference to Augustine’s Trinitarian analogy of unitas, aequalitas, and concordia, the triad disowned by the African bishop after De doctrina christiana. This marks the second time that Thierry has conspicuously invoked an Augustinian doctrine.101 He does so again to demonstrate the potential harmony between that venerable theological tradition and Bernard’s forward-looking agenda. Like the earlier citation of causae seminales, this reference addresses the problem of mediation, but now more squarely the problem of the Trinity. Although Thierry never spells out the triad as such in Tractatus, he spends the rest of his Genesis commentary

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occupied with its three terms. In a first section, Thierry correlates the theological unitas of Augustine with the henological unitas of Boethius (§§30–36). Then in a second section he correlates the Trinitarian aequalitas of Augustine with the arithmetical and geometrical aequalitas of Boethius (§§37–47). (The case of the third term is more complicated.) This discursive pattern tells us as much about Thierry’s intentions as the particulars of his Trinitarian doctrine, for reasons I  will address in Chapter 5. Not only is he attempting to reconcile Augustine and Boethius, but also to restore the integrity of the quadrivial and theological halves of the Boethian tradition itself. As I explained in Chapter 3, the henology that Boethius passed down from late antique Platonism was broken in two, split between the theological opuscula and Consolatio, on the one hand, and the Nicomachean quadrivium on the other. But these two Boethian discourses on the divine One, however textually separate, belong to one author with one henological vision. As G. R. Evans contends in an important essay, Thierry’s true innovation lies in reading the Boethian corpus as an integrated whole. “Thierry expected to discover a general agreement among Boethius’s opinions,” she writes. “Making no distinction between Boethius’s original works and his commentaries, and his translations, he looked for unity of thought in his author and believed he had found it.”102 Thierry thus reads both halves of Boethius univocally:  the implicit mathematical henology of Institutio arithmetica along with the explicit theological henology of the opuscula, the One of number with the divine One. This identification is the very essence of Neopythagoreanism. In the first section on Augustine’s triad, Thierry shows that Boethius’s unitas possesses immediate theological value, since it defines how God is the Creator. In Institutio arithmetica Boethius had explained that the binary is the source of alterity and change, and that unity always precedes it. To Thierry’s eyes, this well explains the difference between Creator and creature. Unitas, he states, is simply another word for divinitas; the arithmetical principles governing unitas can even refine one’s use of divinitas.103 But then Thierry qualifies this arithmetical henology by adverting to Boethius’s opuscula. He adds another Boethian term from De hebdomadibus, the notion of God as the forma essendi. Just as creatures are lit by light or hot from heat, they have their being from God’s form. God’s unity is therefore the form of being.104 By reading both sides of Boethius together, Thierry shows how the quadrivium can inform Christian theological language. Thierry also suggests that the arithmetic of unity reveals God’s goodness in creating and sustaining creatures. As Jeauneau observes, for Thierry unitas is no sterile concept, but “conceals in its breast a generative power.”105 As we saw in the last chapter, Boethius repeated the Nicomachean doctrine of unity as the preserver of beings throughout Institutio arithmetica. Numbers are derivative unities that participate in the One, just as creatures depend upon their Creator to sustain

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them. In the same way, Thierry argues, unitas is not just the form of being, but the “preserver of being” (conseruatio essendi) and its “illimitable perdurance” (interminabilis permanentia).106 Arithmetic also teaches that “unity creates numbers” with unlimited power, since numbers are infinite. For Thierry, this comparison of numeration and creation is not a metaphor but a superior understanding of the essence of creation. Since creatures first exist by participating in unity, Thierry declares, “the creation of number is the creation of things.”107 By reading Boethius univocally, Thierry has effectively accessed the Nicomachean doctrines of unity as divine, as benevolent, as creative and as omnipotent. Here we finally witness Neopythagorean henology alive within medieval Christianity. In the second section on the arithmetical triad, Thierry turns to divine aequalitas, and again he urges that Augustine’s Trinity and Boethius’s quadrivium work hand in hand. Indeed one can begin with Boethian arithmetical principles and arrive at the doorstep of Augustine’s aequalitas theology. Thierry notes that Boethius contrasts two kinds of generation.108 Numbers can be generated out of themselves (2  × 2 or 3  × 3); geometrically these result in “equality of dimensions,” such as squares, cubes, circles, and spheres. Or numbers can be generated by multiplication with different numbers (2  × 3 or 3  × 2), which produces figures with an “inequality of sides.” But when unity is multiplied by itself, it yields only equality (1 × 1), and when multiplied with otherness, it yields all possible numbers (1, 2, 3 . . . ).  This means that “unity” exclusively generates equality, and it does so by itself.109 Then Thierry adds another Boethian maxim from the quadrivium: all inequalities derive from a prior equality. If this is true, then unity and equality both precede every number. But if they are both eternal, then they must be coeternal, and therefore one.110 When Thierry thus rephrases Nicene orthodoxy (the unity of Father and Son) in arithmetical terms, he is not trying to prove the triunity of God to skeptical philosophers. Rather he is pointing out to traditionalists that the implicit henology of the Boethian quadrivium is compatible with, and perhaps even implies, an orthodox Augustinian doctrine of the Trinity. To this end Thierry restates his point in terms of the prevailing academic lexicon of proprietates and personae, suggesting that the arithmetical model of the Trinity is at least as fundamental as the traditional personal model.111 The triad of unity, equality, and connection represents more than a metaphor for God in terms of number theory, as Evans indicates. It represents three “laws of mathematics” that hold for God as well as for the universe:  “scientific laws are also, in this instance, theological laws.”112 Or as Vera Rodrigues puts it, Thierry has in a mind a kind of “arithmetical realism.”113 Having shown that a theology of aequalitas can function as a point of contact between Augustine and Boethius, Thierry’s next step is to apply this to Bernard of Chartres’s theological problem. Like Augustine, Bernard had placed the forms in the divine Mind, but then, deferring to Macrobius, he broke with Augustine

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by dissociating that divine Mind from the second person of the Trinity. But since Thierry wants to build a bridge between Bernard’s Platonism and Augustine’s theology of the Word, it is no coincidence that at this juncture in Tractatus he brings up the Macrobian mens divinitatis.114 In effect he draws up a plan to mend the breach between Trinity and quadrivium in Bernard’s gloss by coordinating Boethius’s and Augustine’s respective accounts of aequalitas and sapientia. As before, we must listen for Thierry’s continuous train of thought even when some of his intuitive leaps do not make it onto the page. By drawing out the untapped conceptual resources of aequalitas, Thierry illuminates the deep connections between Boethius’s quadrivial philosophy and Augustine’s theology of creation. First, he notices that aequalitas not only names the identity of unity in self-generation (aequalitas unitatis), but also the self-identity of every particular being (aequalitas existentiae).115 When one knows the equality of a thing, one possesses its concept (notio) perfectly, avoiding every excess beyond or below its exact form.116 In this account of equality as “definition” or “determination,” Thierry is making a pointed allusion to the Boethian quadrivium. For as Boethius explained in the Institutio arithmetica, the sapientia that philosophers seek is knowledge of being that neither increases nor decreases but is always identical, that is, multitude and magnitude. Boethius accordingly defined aequalitas as that which is neither above nor below its proper mensura.117 Next Thierry suggests that the same function of equality is found within Augustinian theologies of creation. Alluding to Wisdom 11, Thierry calls aequalitas “measure” (mensura) and “weight” (pondus), as well as “mean” (modus) between excess and deficiency. All of these are terms favored by Augustine in De genesi ad litteram and elsewhere.118 In this way Thierry directly compares the mathematical concept of aequalitas in Boethius to the Augustinian account of the mathematical order of creation.119 But then Thierry points out a deeper connection. If aequalitas existentiae determines all the unchanging forms of creatures, he observes, then that aequalitas must be what “the ancient philosophers named the divine Mind, Providence or the Creator’s Wisdom.”120 By placing the quadrivial aequalitas within God, Thierry repeats the Nicomachean doctrine passed down from Boethius, that the prime exemplar of the cosmos in the divine Mind is number.121 At the same time, the notion that God orders creation through the agency of divine sapientia, the second person of the Trinity, is consummately Augustinian and even underscored by Thierry himself earlier in Tractatus.122 Hence by following the trail of the quadrivium, one is led directly toward the divine Word: If, however, the comprehension of the mind were to extend beyond or below, then falsehood would arise which has no substance, since truth is for all things the first being and prime substance. When therefore the equality of truth is of that kind described above then most evidently that

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same equality can be considered to be the divine Word. For the divine Word is nothing other than the Creator’s eternal predetermination of all things: what, what kind, and how much each of those things is, or how it holds itself in time and place.123 Something is truly known when it is known with the precision of the Creator’s Word. But this Word defines things in their equality, known through number and quantity in the quadrivium. Viewed through the prism of aequalitas, therefore, Verbum theology and mathematical theology are mutually reinforcing. Thierry thus works a kind of shuttle diplomacy between Bernard’s Boethian philosophy and Augustinian theology. His basic insight is that aequalitas can serve as an intermediary to bind the quadrivium to the Trinity. In this way he resolves his sources’ disagreement over the divine mens. From Bernard’s side, Thierry shows that the quadrivial aequalitas leads one directly toward knowledge of the eternal forms within the divine Mind—what Bernard had doubted was possible. From Augustine’s side, Thierry demonstrates that the eternal forms reposited in divine sapientia, the second person of the Trinity, are first apprehended in mathematical terms, for aequalitas essendi is none other than aequalitas unitatis. Thierry thus discovers that, contrary to appearances, a Christian Platonism armed with the triad of unitas, aequalitas, and conexio need not choose between quadrivium and Trinity.

Repercussions Thierry’s foregoing discussion of the mathematical triad in the Genesis commentary is often described as if it were a rational proof of the Trinity. But this reading risks anachronism, as if the Breton master had meant to perform an apologetic feat for the benefit of rationalist skeptics. There are at least three more significant aspects of Thierry’s arithmetical Trinity worth noting. In the first place, Thierry does not simply find in the structure of the quadrivium a proof that God is Trinity. On the contrary, he ends up concluding, more radically, that the Trinity is the source of the quadrivium. The Trinity generates number out of its inner identity, as the equalizing of unity into connected numerical series. Just as the generation of equality precedes and grounds the generation of number, so Trinitarian difference is the ineffable origin of numerical difference. The Trinity is, in Augustine’s phrase, the Number without number precisely because the Trinity is the eternal source of number. Creation can be understood through numbers not because God is numerable (Boethius had dismissed this misunderstanding in De trinitate) but because God as Creator is perpetually numerating. Conversely, this entails that mathematics is the immediate trace of divine self-numeration in the world. Since the quadrivial arts lead one most swiftly into the presence of the deity, one must

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advert to quadrivial reasons in order to clarify how Spirit and Word work within the creation. For Thierry, therefore, the mathematical triad of unity, equality, and connection is not a mere figure of speech for the Trinity using the language of the quadrivium, as perhaps it still was for Augustine. Rather, the triad is a primal name of the Trinity, one whose unchanging precision may reach further into the depths of God’s mystery than even the traditional figures of paternity and asexual generation. The difference between the symbolic and realist interpretations of the triad was explicitly discussed in the twelfth century. As I  will explain in Chapter 6, even though Thierry clearly intended the stronger, realist meaning, his revival of Augustine’s triad was interpreted symbolically in the schools.124 For Thierry the triad of unitas, aequalitas, and conexio is not an arbitrary mathematical metaphor, but a sturdy conceptual basis for grasping just what a Trinity of persons might mean. Beneath the traditional notion of generation lies the more solid foundation of number. Second, Thierry’s decision to turn to quadrivial reasons and to Augustine’s triad was itself a momentous philosophical intervention. Bernard’s historic gloss had posed a profound dilemma for western Christianity. Was there a conflict between the new mathematical physics urged by the Timaeus and older Christian notions of the Trinity indwelling the cosmos? Few of Bernard’s peers had yet grasped the question, let alone provided a possible answer, nor would many philosophers, until Aristotelian natural philosophy had been fully absorbed in the thirteenth century and then fused with the mathematization of qualities and motion in the fourteenth century. The lack of a good answer drove nominalist theologies to posit an infinite gap between the cosmos and God beyond any mediation, and here we have the beginnings of the divorce between science and religion. The important point to see is that Thierry had, in fact, answered Bernard’s question, but so rapidly, so prematurely, that no one was yet listening. The nub of the problem that such mathematical Platonism posed for Christian theology was that it promised a mediation between God and world without Word or Spirit. Bernard’s notion of an independent physics opened the way for theories of natural causation in William of Conches. But Thierry cut a different path for himself, opting for Bernard’s emphasis on the quadrivium. In effect, Thierry grasped that number underlies many of the alternative expressions of Platonic mediation. The anima mundi is coterminous with the numeric harmonies it sustains within the world, which is why Platonic traditions fluctuated between soul and number as principles of mediation. Soon natura replaced the anima mundi as the principle of the world’s rationality. But as William writes, natural causation is the force that produces similar from similar, unfolding a regular repetition of unities—leading one back again to the autonomic regularity of number. So rather than attempt to harmonize the mythic anima mundi with the Holy Spirit,

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as Abelard and William, or to poetize Natura as later twelfth-century theologians would, or to distinguish degrees of causality, as Thomas Aquinas would, Thierry’s solution strikes at the root of the problem by singling out number as the superlative idiom for conceptualizing autonomic mediation, and then directly confronting the relationship between arithmetical mediation (quadrivium) and Christian mediation (Trinity), or more simply, Arithmos and Logos. The structures of created order that reason observes are indeed numerical structures with their own immanent rationality, analyzed by the quadrivium into multitude and magnitude. Thierry of Chartres’s remarkable proposal is that such numbers are nothing but the echo of the internal numeration of the Trinity as unity, equality, and connection. God’s self-equalization is the origin of numerical form. Once Trinity is thus grasped in its mathematical basis, the Platonic and Christian paradigms of mediation can be unified without needing to contort the Timaeus into Christian strictures or to police the outer limits of physics for theological transgressions. Thierry’s views arose in response to specific textual problems in early twelfth-century schools. But they also inevitably connect him with past Platonist traditions that emphasized just such number mediation, such as the Neopythagorean revivals of Nicomachus and Iamblichus. Finally, as we have already seen, Thierry’s arithmetical Trinity is not only a doctrinal breakthrough and a philosophical solution, but also a discursive achievement that opens up new possibilities in medieval Christian thought. Augustine’s (Moderatan) mathematical triad was the code that unlocked the door to the Neopythagoreanism dormant in Boethius’s quadrivial translations. By deliberately interweaving the strands of Boethian and Augustinian Platonisms in his Genesis commentary, Thierry relaxed the tension between them. In so doing, he performed two discursive procedures simultaneously. He reconnected the two halves of Boethian theology, and he joined Boethius’s Neopythagoreanism with Augustine’s theology of the Word. We have left one task unfinished in the Genesis commentary. Thierry’s discussion of the Trinity promised three things. The divinitas of the Father works by efficient causation and is defined as unitas. The sapientia of the Son works by formal causation and is defined as aequalitas, as we have seen. But the obvious conclusion, that the benignitas of the Spirit works by final causation and is defined as the concordia of Augustine’s triad, is never stated in Tractatus. For just at this point, all versions of the Tractatus abruptly end, apparently deliberately incomplete (§47).125 Instead Thierry issues a final sentence that leaves two tantalizing clues.126 He names the third term of the triad, but in place of Augustine’s rich harmonic concept (concordia) he substitutes an arithmetical paraphrase (conexio) that returns attention to unity and equality. He also promises to correlate conexio with the “disciplinae proposed above,” which in context means the four disciplines of the quadrivium. Thierry’s cliffhanger of an ending seems to promise a reflection

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on the disciplines of the quadrivium insofar as they might inform the theological function of the Spirit. Thierry had ample reasons for circumspection when he arrived at the third term of Augustine’s triad. We should recall that in the Timaeus commentaries of Bernard and William, not to mention Abelard, the Holy Spirit’s identity was a flashpoint for controversy, evoking a contest between the anima mundi, or demiurge, and the Christian Trinity. As some scholars have speculated, this may well have given Thierry pause.127 Nor did it help matters that Thierry substituted conexio for Augustine’s concordia, since the latter would have found numerous points of contact in the Institutio musica. Yet on the other hand, his promise to return again to the disciplinae propositae despite such difficulties reflects the Breton’s intellectual swagger, and it seems out of character for Thierry to fear the monastic censures that Abelard and William had faced. But so long as we are speculating, another possibility is just as probable as fear. It could be that Thierry interrupted the ending of Tractatus for a definite reason, out of a characteristic impatience to proceed straightaway to a more penetrating insight. Perhaps an even more ambitious plan had dawned on him.

5

The Discovery of the Fold When Thierry of Chartres wanted to overhaul Bernard’s Timaeus gloss, it had made perfect sense to begin with Genesis. He wanted to explore how Christian cosmogony could be grasped in terms of physics and quadrivium, yet without retreating on the Trinity as Bernard had felt compelled to do. But once Thierry began working on this new scriptural canvas, the first few brushstrokes seem to have inspired a different vision altogether. Thierry quickly discovered that if he fused Augustine’s mathematical triad to the quadrivial foundations of arithmetic and harmonics, he could expound God’s creating activity as much from mathematical principles as from scriptural ones. This startling success at connecting quadrivium and Trinity only underscored the need to investigate, not so much the quadrivium per se, but rather the disciplinary status of the quadrivium. In the library of the twelfth-century philosopher, the premiere guidebook on theological method was Boethius’s De trinitate. Boethius had used Aristotle’s division of the sciences as a platform for rethinking the meaning of mathematical and physical form in light of the Trinity. In other words, the puzzles that Boethius had confronted in that book were the very ones facing Thierry by the end of his Genesis commentary. What procedures do theology and mathematics have in common? What sort of disciplinary structure could embrace them both and account for their similarities and differences? What he required now was a universal vision that could encompass and organize the competing rationes of physics, mathematics, and theology that he had gathered together in Tractatus. We might even provocatively call this Thierry’s search for a mathesis universalis.1 Throughout the Genesis commentary, Thierry had kept in close contact with Boethius, touching on the henology of Institutio arithmetica, the poetry of Consolatio, and the metaphysics of De hebdomadibus. But such a full-blown theology of the quadrivium demanded a fresh start, and De trinitate, despite its Aristotelian orientation, was clearly the best vehicle for further investigations. So Thierry put down Genesis and turned to Boethius. Whether or not Thierry broke off Tractatus as deliberately as I imagine, this sequence of works—the hexaemeral commentary followed by several commentaries on De trinitate—conforms to the best internal analyses of Thierry’s writings.

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Nikolaus Häring has provided ample documentation that besides Tractatus, whose authorship no one doubts, Thierry penned at least three distinct commentaries in the following order: the Commentum super Boethii librum de Trinitate (also known by its incipit, Librum hunc), the Lectiones in Boethii librum de Trinitate (or Quae sit), and the Glosa super Boethii librum de Trinitate (also known as Aggreditur propositum or Anonymous Berolinensis). Häring’s attributions and his proposed chronology have remained essentially unchallenged for forty years.2 At the same time, some contemporary readers of Thierry have struggled—like his medieval readers—to give a name to his project as a whole. When medieval readers identified points of friction between Thierry’s ideas, they questioned his orthodoxy or engineered a productive misreading. When modern readers run across the same points, they raise doubts about Thierry’s authorship3 or the sequence of his works.4 This is why attending to the proper sequence of Thierry’s works is so important. It helps us to grasp the stepwise development of his theology and to detect the guiding impulse that directed and unified that evolution. For Thierry made not one but several attempts to understand De trinitate, that is, to reconcile Boethius the theologian with Boethius the mathematician, and to harmonize them both with his Augustinian commitments. Here I  will offer a genetic account of Thierry’s development that remains sensitive to the differences among Commentum, Lectiones, and Glosa, and that interprets these differences in light of his original insights in the Tractatus on Genesis. Thierry’s doctrines, the results of many hours of reading and teaching, are not all chanted in unison by the three commentaries; far less do they form a “system.” Rather, they include false starts and sudden breakthroughs, by-products of his continuing struggle to overcome the ambiguities latent in the Boethian corpus. Likewise if one is to appreciate the complex, halting, and sometimes confused reception of Thierry’s ideas after his death, such a genetic perspective on his own writings will prove crucial. There are three benefits to this strategy. First, it helps one to navigate between two extremes of interpretation that have afflicted scholarship on Thierry of Chartres (with noteworthy parallels to scholarship on Nicholas of Cusa). Some have treated Thierry’s works as a homogeneous system, in which the individual works are mere instances of an imagined, preexistent whole.5 Others treat them as fragments too distinct to assemble, either by contesting Thierry’s authorship or by drawing too sharp a line between the Genesis and Boethius commentaries.6 The proper route between Scylla and Charybdis, in my view, is to read differences among the commentaries as signs of diachronic development. In this way we avoid ascribing to Thierry the views of one privileged work alone, or an ideal system, but instead can trace a unified but dynamic trajectory of thought. Second, by distinguishing the stages of Thierry’s intellectual development we can better avoid combining his doctrines into unrecognizable constellations that Thierry himself never intended—another foible that has afflicted medieval

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and modern readers alike. As I will show in Chapter 6, Thierry’s medieval readers struggled to piece together his various conceptual inventions into a working whole. Since the attribution of Thierry’s extant works was prolonged over several decades, modern readers have faced some of the same challenges of integration. Indeed, when Thierry’s works are not read in their proper sequence, recent scholars have unwittingly repeated the conflations, correctives, or deformations already exemplified in Thierry’s reception history.7 Third, this strategy focuses our attention on the true center of Thierry’s achievement, which is, in fact, not the arithmetical Trinity borrowed from Augustine in Tractatus.8 Ironically, the doctrine most frequently associated with Thierry, the triad of unity, equality, and connection, is only his point of departure for more ambitious plans. Across his several Boethian commentaries Thierry always pushed forward, starting from the arithmetical Trinity but then quickly venturing into new territory. Naming the center of Thierry’s theology properly will not only clarify the nature of his intellectual achievement, but will also help make sense of others’ reactions in the twelfth to fifteenth centuries. I will first study the scope of Thierry’s intentions in his first commentary, Commentum, before turning to his momentous discovery in Lectiones. As we have seen, Boethius addressed the disciplinary status of theology vis-à-vis other sciences in Book II of De trinitate. We can best observe the gradual development of Thierry’s full theology of the quadrivium by examining his evolving commentaries on this book.

Attempts at a Universal Theory of Science When Thierry reaches Book II of De trinitate in Commentum, he states that he expects to find in Boethius the “reasons” (we might say the disciplines) most appropriate for contemplating the Trinity.9 At first Thierry’s comments add very little to Boethius’s account. He simply notes for his listeners that the ancients considered mathematics to lead toward theology.10 But as we keep reading, a larger pattern begins to emerge. In his stops and starts, asides and excurses, and his varying attention to the flow of the argument of De trinitate, Thierry is reflecting on his own questions about the status of the quadrivium and about the role of number as mediator. He is seeking a new theoretical horizon, a metaprinciple to ground the connection between mathematics and theology that he uncovered in his Genesis commentary—in short, he seeks the harmony of Logos and Arithmos. But Thierry neither begins the second book of Commentum with an answer in hand, nor ends having finalized a solution. Instead he works through no fewer than five different possibilities for a unified theory of science. None of them is ultimately successful, but each takes the Breton master a step closer to his goal.

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Psychology Thierry’s first attempt (§§1–17) is to sketch an epistemology of the three sciences (theology, mathematics, and physics) on the basis of contemporary psychological theory.11 Thierry’s hierarchy of different faculties of cognition builds upon Neoplatonist distinctions developed further by Boethius in Consolatio. Boethius’s fourfold schema (sensus, imaginatio, ratio, and intelligentia) was popular in the twelfth century among Cistercians, Victorines, and Chartrians alike. Hugh of St. Victor divided Boethius’s fourth faculty into two parts, distinguishing between intellectus and the higher intelligentia named in Boethius’s commentary on Porphyry’s Isagoge.12 Now Thierry explains that each of the faculties is the result of different psychic operations.13 The soul knows either through an external medium (pro instrumento) or through itself (ex ipsa). In sensible perception, the external media are physical bodies transmitting light or motion. But imagination and reason use the external medium of spiritus, a subtle ether that passes through microscopic cells within the head. Spiritus retains sense impressions even when the corporeal body is absent and transmits these forms to the soul either confusedly (imaginatio) or coherently (ratio). Finally, the soul can also know through itself, without engaging matter. Only in this way can one grasp unchanging truth through the highest faculty of intelligentia or intellectibilitas. Thierry’s first attempt to relate mathematics and theology builds on this psychological theory. Physics is known through ratio, he proposes, but mathematics and theology are known through intelligentia, because they both know pure form in their immutable nature.14 This account helps Thierry to explain the similarities of mathematics and theology that he encountered in Tractatus, but it proves insufficient to explain their differences. He can easily separate mathematics from physics, but has a harder time clarifying how the two modes of intelligentia differ. It is not enough, he says, to define mathematical procedures as disciplinaliter and theological ones as intellectualiter, since this does no more than restate the Greek meaning of mathesis. Thierry tries to suggest that while mathematics knows the “forms of bodies,” theology knows the “true form of divinity.”15 But the inadequacy of this strategy is quickly apparent. His proposed epistemology makes no distinction between the abstract image-forms known in mathematics and the abstract exemplar-form known in theology. Instead it merely defines the operation of self-reflective intelligentia that the two disciplines share.

Hierarchy The failure of this attempt must have given Thierry pause, for at this juncture he explicitly suspends his literal commentary and launches an extended excursus (§§18–49) without parallel in the six books of Commentum. Here Thierry tests out

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three further theoretical principles. The goal of the excursus, he states, will be to grasp both “eternal and perpetual things,” in so far as they can be known by human reason.16 In other words, Thierry is focusing on the boundary between the divine and the merely sempiternal, that is, the status of mediating mathematicals that his Genesis commentary had left hanging in the air. First, Thierry pursues an ontological, rather than epistemological strategy (§§18–29). He aims at a synthesis of Platonic, Aristotelian, and Pythagorean first principles—itself a Boethian undertaking. Perhaps an exhaustive hierarchy reaching from God down to lowest degrees of form, Thierry wagers, could specify the transition from theological form to mathematical form. Aristotle had explained being in terms of act (actus) and potential (possibilitas), but there are two kinds of act corresponding to the two kinds of form in Boethius. Forms connected to matter comprise act with potential, but the divine form is pure act. Thierry counts several names for this immutable form that is the goal of philosophical contemplation. Plato called it eternitas, others simply Deus, and Pythagoreans unitas, which according to Thierry derives from onitas, a Greek (τὸ ὄν) version of entitas.17 Thierry favors his neologism entitas but especially the term necessitas, following Bernard of Chartres, as we have seen, who lifted its negative connotation in Calcidius. Having identified God, eternity, and necessity, Thierry maps three different hierarchies descending from God. In Boethius’s translation of De interpretatione, Thierry found three permutations of matter and form:  actuality without possibility, actuality with possibility, and possibility alone without actuality. Then according to Calcidius, Pythagoras taught two principles: “unity, naming God, and the binary, denoting matter.” Finally Thierry notes that Plato opposed two principles of God and matter like “two extremes,” which, following the reasoning of the Timaeus, demand two means interposed between them.18 By collating these three sources, Thierry points toward a possible ontological scaffolding for his universal theory of science (see Table 5.1). But after listing these possibilities, Thierry never draws them into a system, and indeed quickly turns his attention to his next subject, the arithmetical Trinity. This nascent schema has not yet solved Thierry’s central problem of mediation. It fails

Table 5.1  Thierry’s first attempt at a universal ontology Plato (Timaeus)

“Pythagoras” (Calcidius)

Aristotle (De interpretatione)

Deus uel necessitas Formae rerum Actualia Materia uel possibilitas

Unitas – – Binarium

Actus sine possibilitate – Actus cum possibilitate Sola possibilitas

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to provide any new models of the formae rerum, the subdivine, sempiternal forms that correspond to mathematical knowledge.

Perpetuals At this point, Thierry declares his discussion of “eternity” concluded, and he turns to the second part of the excursus on “perpetual things.”19 His exposition of the arithmetical Trinity is more succinct, confident, and complete than that of the Genesis commentary (§§30–38). At the same time, it seems that the triad no longer interests Thierry for its own sake, but rather functions as a segue to his third attempt to articulate a universal disciplinary model (§§39–43). Having already discussed divine unity in the preceding passage, he skips ahead to aequalitas, summarizing the account from Tractatus. Now in Commentum he finishes the triad by positing conexio as the love between unitas and aequalitas, since all things naturally seek unity. Thierry even remembers to link conexio back to benignitas, the Trinitarian final cause left unmentioned when Tractatus broke off.20 Beyond these amends, the Breton master also amplifies his account of the divine Son in a provocative way. Thierry now adds that the Son’s equality should not only be taken in an arithmetical sense, but in a geometrical one. In Institutio arithmetica, Boethius had taught that any number times itself produces a square, and that the first square is 1 × 1. Since this self-multiplication of unity is the model for the Son’s aequalitas, Thierry maintains that the divine Son is rightly called the First Square (tetragonus primus). When the apostle calls the Son the perfect figura of the Father (Hebrews 1:3), he is literally referring to the Son’s superlative geometry.21 After this restatement of the arithmetical Trinity, Thierry expounds a new triad for the first time. “From this holy and highest Trinity,” he writes, “descends a certain trinity of perpetuals.”22 First, although unitas is free of alterity, since alterity “descends” from unity, unity can be said to “create” matter. Second, since form denotes the “integrity and perfection” of things, such forms can be said to emanate from aequalitas. The third perpetual is finite spiritus or “substantial motion,” which descends from the divine conexio.23 The triad of matter, form, and motion reflect the pure arithmetical order of the Trinity within the physical structures of the world. Every creature’s inherent gravitation toward unity is a trace of the primordial love connecting divine unity and equality, and that Trinitarian love is the hidden impetus driving all matter toward form. Not only the Boethian quadrivium, but now Aristotelian physics too, is a corollary of Trinitarian theology. Thierry’s doctrine of the trinity of perpetuals suspended between God and material world is his simplest, boldest, and perhaps most desperate attempt yet to derive a universal theory of science. But the Breton master quickly realizes that this doctrine raises more difficulties than it solves, and he never repeats it in any

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of his known writings. Right after sketching the idea Thierry has to insist that the three creative activities do not divide the persons of the Trinity. Then he sees that the “perpetuals” of form and matter must be singular, even if physical forms and matter are not. Form in itself and matter in itself cannot yet possess plurality; it attends them only once they are combined in a given being.24 What Thierry lacks, once again, is a model of mediation that bridges the unity of “perpetual” form and the multiplicity of embodied forms.

Reflecting and Folding Thierry’s struggle to exit the dilemma he created for himself yields fascinating results (§§44–49). He reaches for two vivid images that might be able to express the mediation of One and many that he requires. The first stems from a famous passage in Macrobius on the “golden chain” of beings in descent from the Plotinian triad of God, mind, and soul. In Macrobius, the order and life of God are passed down by mind, through soul, into all of the cosmos, like a face shining through a succession of mirrors.25 Thierry adapts the image as follows: For just as one face gleaming in different mirrors is still one in itself, but through the diversity of mirrors this one is taken as another—if one can draw the comparison—so, too, the divine form shines in a certain way in all things, but only as one form-in-itself of all things, if you consider that pure and true simplicity of which the different forms are thought.26 Here the visual metaphor of “reflection” (relucere, renitere) does the work to coordinate the one divine form and the many embodied forms. A second image stems from Boethius’s account of divine providence in Consolatio. The context is especially apt, since Thierry’s organizing distinction of “eternals” and “perpetuals” comes from the same passage.27 In that chapter Boethius had distinguished providence from fate. Providence, he wrote, “embraces” (complectitur) the infinitely diverse forms of things in the simplicity of the divine Mind, while fate in a contrary but isomorphic movement is the “unfolding” (explicatio) of time. If providence is a stable, integrated (adunata) point, then fate is the mobile circle or sphere unfolding it in space (explicata).28 Hence God’s providence is “eternal,” but fate is merely “perpetual.”29 Thierry takes up this Boethian model of reciprocal “folding” as a second illustration: Therefore the very forms that the mutability of possibility (that is, of matter) unfolds [explicat] through the diversity of plurality, the divine form likewise enfolds [conplicat] into one and recalls to the simplicity of a single form in an inexplicable [inexplicabili] way.30

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Here the many forms are not simply reflections of the one form, as in Macrobius, but are reciprocally implied by each other. The divine form enfolds the lesser forms, and only those forms unfold the divine form. Like his trinity of perpetuals or his hierarchy of necessity and possibility, Thierry’s concept of folding is one more attempt to mediate between God and world and thus unify the disciplines of theology, mathematics, and physics. But Thierry seems pleased with this formulation. He can finally draw the long excursus to a close. He returns to “folding” twice more in Commentum.31

Five Modes Much later in his commentary (IV.6–10), Thierry makes one final attempt to identify foundational principles for a unified theory. In the course of explaining the predication of different categories in theology, Thierry inserts a brief remark about an additional schema that recalls previous ideas in the Commentum. According to Thierry, there are “five modes” by which one can consider the cosmos.32 Three of the modes are God, prime matter, and actualia; these recall the Platonic-Aristotelian-Pythagorean comparison ventured above. Another mode, “created spirit,” closely resembles the third perpetual. Thierry calls the final mode alternatively natura and numerus—an unusual equivalence, until we remember that Bernard of Chartres had suggested that natural philosophy culminates in the quadrivium. Thierry then reels off several sources, trying to tie together Boethius’s Consolatio, Hermes Mercurius, and Augustine’s De genesi ad litteram.33 Is the Breton master working toward some kind of synthesis late in Commentum? He briefly defines the modes of God and created spirit, but then once again breaks off his deliberations in mid-thought. A fuller treatment, writes Thierry, would exceed the limits of his present discussion. Moreover, he continues, he would first have to explain how the various observations stemming from five different cosmic modes could be unified in a rational order.34 Clearly Thierry wishes to expand his fivefold theory of modes into a universal theory of science, but for the moment he does not see the path forward.

The Achievement of the Modal Theory Thierry’s next two commentaries on De trinitate, the Lectiones and the Glosa, possess greater maturity and speculative command than Commentum. They must have been composed between 1148 and 1155, after he had become chancellor of the Chartres cathedral school.35 In Commentum, Thierry had plumbed the middle realm between God and cosmos because that is the region occupied by the mathematical forms studied in the quadrivium. He weighed five different strategies for coordinating theology and mathematics within an overarching theory of science, but all of them fell short. Now in Lectiones, returning to Boethius’s text after

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another decade, the new chancellor is finally able to answer the questions he had first raised in Tractatus and pondered in Commentum. First Thierry decided to jettison the trinity of perpetuals and the psychological epistemology. He wove together the best remaining intuitions from Commentum into a more durable and theoretically potent model. There are, he now maintains, four modes of being, structured in terms of folding, which comprise a hierarchy spanning from divine necessity down to material possibility. Boethius had famously held that the wise person proceeds within a given discipline according to its own rationes.36 Thierry infers a corollary: if such different sciences study one and the same cosmos, then the intellect must view the cosmos under so many different modes.37 This modal theory defines and unifies the respective procedures of physics, mathematics, and theology, replacing the psychological basis of his initial theory with a sturdier ontological one. It represents a masterful, “wholly unprecedented” synthesis, in the judgment of Peter Dronke, and is accordingly the central achievement of Thierry’s mathematical theology.38 In the very prologue to Lectiones it is clear that Thierry will focus exclusively on unanswered questions about the three disciplines from Commentum. He cannot resist previewing the themes he wants to discuss. The principle of theology is the Trinity, but it reaches down to angels and souls; the principle of mathematics is number, but it proceeds to proportions and magnitudes, two principles of the quadrivial order; and below these are the four elements studied by physics.39 Trinity, quadrivium, and the elements: clearly Bernard’s Timaeus gloss remains at the forefront of Thierry’s mind. Well before reaching De trinitate II in Lectiones, Thierry is already admiring how Boethius relates “theological reasons or arguments” to mathematical and physical ones.40 And once he arrives at that critical passage, in a telling departure from Commentum, Thierry postpones his customary exposition of the arithmetical Trinity until the end of Lectiones, as if to clear space for his latest invention.41

The Modes in Lectiones Like a master teacher, Thierry presents his doctrine of the four modes of being in several deliberate steps.42 First he erects a substructure of reciprocal folding (§§1–6).43 God’s unity is the “enfolding” of the entire cosmos (universitas rerum), which is the common object of the three disciplines. Likewise the cosmos is the “unfolding” of divine unity into plurality.44 This double identity of the universe as enfolding and unfolding is the key to Thierry’s solution to the problem of relating the quadrivium to theology. On this foundation he is able to construct four “universal modes” (§§7–13), four ways in which the universe simultaneously exists: This universe is in [i]‌absolute necessity in simplicity and a certain oneness of all things, which is God. And also in the [ii] necessitas complexionis in a

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certain order and progression, yet immutably. And in [iv] absolute possibility: in possibility, yet without any actuality. And also in [iii] determinate possibility: possibly and actually.45 The four modes, then, are constituted by two parallel orders of folding. The first order of being (i, ii) is act or necessity, and the second (iii, iv) is potential or possibility. But the second mode is the unfolding of the first mode, just as third is the unfolding of the fourth.46 The enfolded extremes are both “absolutes,” but their respective unfoldings differ. We can best articulate Thierry’s system of folding in terms of these two dimensions (Table 5.2). Thierry believes he has discovered in these four principles a self-evident, comprehensive system known to previous philosophers under different names. Natural philosophers (physici), for instance, have called the second mode “fate” and the fourth mode “chaos” or “prime matter.” But the real power of Thierry’s four modes is the way they immediately clarify the disciplinary structure of the sciences (§§14–34). Each of the four modes denotes a way of seeing the cosmos, a temporary phenomenal reduction, and each of the three sciences operates through a different mode: For theology considers [absolute] necessity which is also the simplicity of unity. Mathematics considers the necessitas complexionis which is the unfolding of simplicity, for mathematics considers the forms of things in their truth. But physics considers possibility, both determined and absolute. Thus absolute necessity and absolute possibility are the extremes among these, and the remaining two modes are intermediate means (which has been explained better elsewhere).47

Table 5.2  Thierry’s four modes of being as a system of reciprocal folding Complicatio Necessitas

Explicatio

Necessitas absoluta [i]‌ Deus, simplicitas Necessitas complexionis [ii] Ordo, fatum

Possibilitas

Possibilitas determinata [iii] Actualia Possibilitas absoluta [iv] Materia primordialis, chaos

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For Thierry, then, the divisions of speculative philosophy are real and not nominal, since they are generated by different ontological dimensions of the cosmos. Physics contemplates the two modes of possibility. Mathematics and theology begin with the same subject matter, divine unity or necessitas. But where theology grasps the necessity of the One in its enfolded simplicity, mathematics grasps the very same unity as it is unfolded into numerical difference.48 Finally, each discipline also represents a different faculty of knowing. Physics corresponds to sensus, ratio, and imaginatio; mathematics to intellectus; and theology to intellectibilitas or intelligentia.49 Thierry’s theory of the four modes weaves together the various mediations mooted in Commentum. His attempt to synthesize Platonic, Aristotelian, and Pythagorean ontologies now succeeds on the basis of “folding.”50 Absolute necessity and absolute possibility are now coordinated by the intervening second and third modes, which he had previously postulated but had been unable to name. Likewise, Thierry’s initial jumble of five modes is now reorganized under the aegis of folding into a principled fourfold. This in turn allows Thierry to articulate the integrated metadisciplinary perspective that he had sought first in Tractatus and then in Commentum. But it is crucial to note the theoretical fulcrum of Thierry’s success. His true innovation in Lectiones is the concept of necessitas complexionis. The second mode is the missing component corresponding to the Platonic formae rerum that his tentative protosystem of necessity and possibility had lacked (see Table 5.1 above). It resolves the ambiguity between natura and numerus that unsettled his initial five modes by defining cosmic order as the unfolded enfolding of number in the divine One. And as the mode of being that corresponds to the discipline of mathematics, it is also the key to understanding how the quadrivium relates to the Trinity, which was the query that prompted Thierry’s turn from Genesis to Boethius in the first place.51 In many ways Thierry’s necessitas complexionis simply represents a fundamental rethinking of Bernard’s formae nativae. The second mode is also the fulcrum of Thierry’s modal theory because it permanently connects reciprocal folding to its mathematical origins. As we have seen, Boethius’s account of divine providence in Consolatio was one important source for Thierry’s concept of folding.52 But another crucial inspiration was undoubtedly Boethius’s quadrivial theory. As Irene Caiazzo has recently shown, Thierry’s long-lost commentary on Institutio arithmetica contains an early reference to folding.53 Thierry remarks that arithmetic is centered on unity, and geometry on the point. Arithmetic is the foundation of the quadrivium, because its system of multitude and magnitude is grounded on the originary pluralization of unity into number. This process, Thierry proposes, occurs exclusively according to the fold. “For just as unity is the enfolding of number,” he writes, “and number the unfolding of unity, so too the point is the enfolding of every magnitude, and magnitude is the unfolding of the point.”54 This equation does not appear in

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Lectiones, but it explains perfectly why Thierry would discover his mature theology of the quadrivium by calling upon the concepts of complicatio and explicatio. What arithmetic and geometry share in common, what qualifies them to ground the quadrivium, is their homologous participation in the structure of reciprocal folding. In other words, folding is inherently mathematical, or better said, arithmetical. The spatiality of the double curve of enfolding-unfolding is originally defined by the seriality of number. The second mode, which preserves that moment of number unfolding from the One, is therefore the crux of the modal system as such and provides the subject matter of the mathematical disciplines (the quadrivium).

The Modes in Glosa Thierry’s satisfaction with the breakthrough of Lectiones can be detected in the manner in which he embellishes the modal theory in his later Glosa on De trinitate. The fact that Glosa presents little that is new signals that Thierry felt no need to alter Lectiones but merely to consolidate his achievement. Again he relegates the arithmetical Trinity to the end of the commentary.55 He also restores his psychological epistemology to its proper place in the sequence of Boethius’s text (as in Commentum), before the new agenda of Lectiones had displaced it.56 When Thierry distinguishes the three disciplines in Glosa he can afford to be more concise: he is simply restating a settled matter.57 What Thierry does add in Glosa are new names for the four modes. He now proudly lists historical cognates that connect his modal theory to the ancient schools, demonstrating that his discovery resolves perennial philosophical problems and can harmonize Platonic, Aristotelian, Stoic, Pythagorean, and Hermetic sources. For example, the first mode can also be called the “enfolding of plurality,” the “form of forms,” “providence,” or—echoing the poems of Consolatio—the divine unitas, who preserves all things in being as their “bond” (uinculum conservans). The first mode is also the God who named himself “He who is” and “I am who I am” to Moses.58 Thierry deals more briefly with the modes of possibility. He calls the third mode “defined possibility,” and notes that the fourth mode has been known as prime matter, ὕλη, silva, or chaos.59 Yet Thierry lists the most names for the second mode. His attention to necessitas complexionis in Glosa is a sign of its centrality for his modal system but also of the difficulty of defining it.60 He helpfully glosses the meaning of necessitas complexionis as necessitas determinata and necessitas ordinis.61 The first equivalent hints at the parallel structure of unfolding across the two sets of modes: the absolute enfolds the determinate, whether necessity or possibility. The second equivalent acknowledges the essence of the second mode, namely its seriality or numerability, the

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sempiternal order (second mode) that mediates God’s eternal stability (first mode) to the formal structure of the actual world (third mode). In Lectiones, Thierry had defined the second mode as the pure order known to mathematicians as number and to physicists as fate. Now in Glosa he expands the meaning of the second mode to include necessary causation: This universe which absolute necessity has enfolded into a certain simplicity in itself is unfolded into the truths of forms and of images, which we call ideas. And absolute necessity arranges them by a certain order into a series of causes, which thus exists necessarily. For a thing follows such a series when it is subordinated to something causally. This is called determined necessity or necessitas complexionis for the reason that when we happen upon some of its matter we cannot avoid the serial connection of the rest of its causes. . . . For if we do not attach its cause to the connection of causes we in no way subordinate it.62 In this passage Thierry contends that any notion of causal necessity, such as those favored in the new physics of William of Conches or Adelard of Bath, must be grounded in something resembling his second mode. The autonomous physics fomented by Bernard and his students requires a foundation (as Bernard saw) in the numerical order of the quadrivium, and this methodological requirement reaches a niveau of conceptual clarity in Thierry’s necessitas complexionis. But Thierry also acknowledges that this mediating function of the second mode had already been faintly perceived in the ancient philosophical traditions revived by twelfth-century masters. It has been known under different names as the Platonic anima mundi, the Peripatetic natura, the Stoic fatum or natural law, the Neoplatonic divine Mind (intelligentia dei), and the Hermetic εἱμαρμένη.63 Thierry envisions his second mode, in other words, as a concise reformulation of the very autonomic mediations sought by his peers, a kind of second-order metaconcept that redefines their common function. It is no accident that Thierry achieves this by repeating Bernard’s gesture of returning to Boethius, and there discovering anew the theological potential of “folding.” Indeed in Glosa Thierry seems to gain some retrospective clarity about what he had been up to in the previous Boethius commentaries. He notes here the Boethian maxim that number first exists as an “exemplar in the mind of the Creator.”64 This is the Nicomachean theologoumenon underlying Thierry’s intuition in Tractatus that the creation of number is the creation of things. Likewise Thierry again celebrates the Augustinian dictum that God is the “Number without number”—surely the best paraphrase of his arithmetical Trinity.65

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Thierry as Neopythagorean Theologian Thierry’s modal theory represents the conclusion of his dialogue with Bernard’s gloss, because it finally defines the relationship between mathematics and theology with precision.66 In Tractatus Thierry had discovered the uncanny proximity of the two disciplines, and in Commentum he sought to explain it, but failed. But the answer he ultimately reached in Lectiones deviated from his ancient sources substantially. Theology does not simply represent a higher, transmathematical degree of abstraction from matter, as Plato, Aristotle, and Boethius had all variously held. According to Thierry, mathematics stands to theology as necessitas complexionis stands to necessitas absoluta: as the unfolded to the enfolded. The two disciplines are intimately bound up as reciprocal or inverted reflections of the other, but they access one and the same divine One. Mathematics and theology pursue a univocal henology; their sole difference is the fold.67

The Meaning of the Second Mode The second mode is therefore the single most important innovation of Thierry of Chartres’s theology after the Tractatus on Genesis. But even as this raises the stakes of the concept’s intelligibility, several ambiguities persist. In the first place, it is difficult to isolate exactly what Thierry intended the term necessitas complexionis to mean. The lexical novelty of Thierry’s formulation makes it hard to specify the sense of complexio.68 Some interpreters have assimilated complexio to conexio from the arithmetical Trinity. Others have consulted how contemporary natural philosophers like William of Conches used complexio, for whom it denoted the inner constitution of the material world as the composition of the four elements. Neither of these in my judgment is very satisfactory. Although Thierry does occasionally call the second mode necessitas conexionis in an abridged version of Lectiones that survives, he most often uses conexio to denote the Spirit.69 Ultimately the fact is that Thierry chose complexio over conexio for his modal theory. Furthermore, Thierry never uses complexio to name anything resembling an elemental composition or constitution, the subject matter of physics, but instead compares it to terms beyond the material order: the divine Mind, the world-soul, the order of fate. So it helps matters little to compare the second mode to William’s name for the four elements. A better route, it seems to me, would be to rely primarily upon the placement of the second mode within the conceptual structure of folding in order to infer its meaning. This would lead us to compare complexio to complicatio, to consider necessitas complexionis a paraphrase of necessitas complicationis, and thus to translate the second mode as “the necessity of enfolding.” In his newly discovered commentary on Institutio arithmetica, Thierry seems to view complexio as roughly equivalent

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to complicatio in the sense that both order simple unity to implied multiplicity.70 One is also encouraged by Boethius’s opposition of complectitur to explicatio in the Consolatio, as well as by the common Greek root of complexio and complicatio (πλέκειν, to fold). In this interpretation, the necessitas denoted by the second mode is not the necessity of the absolute One (the first mode), but rather the necessity that inheres in the seriality of number unfolded from that One. On account of the second mode, the unfolded order known in mathematics retains, and expresses, the necessity of the original enfolding. What is accessed in the certainty of mathematics is therefore not divine necessity (properly the first mode, as in Thierry’s Trinitarian conexio), but neither is it simply the causal necessity of determinate beings (properly the third mode, as in William’s elemental complexio). Rather, the second mode names the pure, irrefragable seriality of number, the identity which perdures across the fold, but now as difference and, as Boethius wrote, which protects the possibility of return to the One. Or in Thierry’s own words, when the necessitas complexionis unfolds the simple unity of God, the result is an explicatio rerum into a crystalline ordo et progressio.71 The second difficulty with necessitas complexionis is not lexical but theological. The concept of the second mode revives a difficult Trinitarian problem. Recall that Bernard of Chartres wanted to respect Macrobius’s distinction between the supreme God and the mind of God. Hence for Bernard only the Trinity is “coeternal” with itself; the divine ideas, while somehow part of God, are merely “eternal.” Thierry’s modal theory does not intend to engage the doctrine of the Trinity. The Breton master never once states, for example, that divine aequalitas “unfolds” from God.72 But Thierry does run headlong into Bernard’s quandaries when, like Bernard, he attempts to relate the Macrobian mens divina to his newly minted second mode of being. In Commentum, even before formulating the four modes of being, Thierry already saw that the notion of a divine Mind would reinforce Boethius’s distinction between true, divine form and enmattered image-forms: “only forms which are in the divine Mind are rightly called forms.”73 Yet this only prompts the more challenging question of how the one divine Mind is composed of a plurality of ideas. In Commentum Thierry shrugged off the problem, pointing to his metaphors of reflection and folding, as we have seen.74 But in Lectiones, once Thierry’s initial account of the four modes was complete, the deeper investment in folding occasioned by that new theory required him to confront the problem of plural forms in the divine Mind (§§35–67). It is far from clear, however, that this problem can be solved within Thierry’s new modal system. In order to bring Bernard’s Platonism fully in line with Augustine, Thierry would have to collapse the distance that Bernard had placed between the mens divina and the Trinity. We have seen how Thierry took an important step toward this goal in Tractatus, and how the questions consequently raised by that accomplishment in Commentum were answered in the modal theory of

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Lectiones. But now moving deeper into his Lectiones, and reaching the passage in De trinitate where Boethius had distinguished image from exemplar (Bernard’s essential touchstone, according to Gilbert and John), Thierry finds that he can focus on mens only at the expense of the four modes. And yet paradoxically this was precisely the sort of theological problem—defining the relation between divine unity and plural forms—that the four modes were designed to solve. It seems that Thierry wants to affirm the utility of both models, both Bernard’s mens divina and his own modal theory, but he is not entirely certain that they are compatible. When he arrives at this critical passage, Thierry initially glosses Boethius’s notion of true, divine form as the forma formarum, a neologism of his own invention.75 Then he begins to connect divine form to the modal theory, contrasting actus and possibilitas.76 Thierry gets as far as necessitas absoluta, but then turns his attention to mens divina, explaining that it contains plural forms quite like the mind of a human artist: For the divine Mind generates and conceives the forms within itself, that is, the natures of things, which are called “ideas” by the philosophers. Hence divinity is nothing other than this divine Mind, which is generative of the ideas. For it conceives and holds them within itself and from it they come into possibility.77 Here the metaphors of generation and conception achieve the requisite mediation between the unity and simplicity of mens and the plurality of the forms. But if mens “generates and conceives” the forms within itself, as the Ur-form governing lesser forms, then it would seem to duplicate the function of the second mode of being. On another occasion, Thierry would even state that the mens divina “conceives, enfolds, and contains” the forms within itself.78 In his attempt to read the Macrobian mens in line with Augustine, has the Breton master rendered his own necessitas complexionis superfluous? Later in Lectiones Thierry is forced to confront the status of the second mode when he encounters a provocative line in De trinitate. Boethius had stated that the “forms” (in the plural) exist prior to matter.79 Boethius’s unusual plural gives Thierry a chance to advertise the utility of his second mode: They [viz. true forms] exist outside matter in their truth, namely in the necessitas complexionis. He said from forms in the plural since there are, there in the necessitas complexionis, many exemplars of things, which are all one exemplar in the divine Mind. For this reason Plato says in Parmenides according to Calcidius that the exemplar of all things and the many exemplars are one, in which there is neither difference nor the contrariety arising from difference, just as is said in Plato.80

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Thierry’s allusion to Calcidius is telling, for if we can trust Gilbert’s report, Bernard of Chartres cited the same text when formulating his formae nativae.81 So Thierry must have appreciated that in these matters he was face to face with the riddle that had bested the senex Carnotensis. Thierry’s argument here is that the concept of a second mode provides a new warrant for the plurality of exemplars in Platonic mediation. Just as there are many numbers that can be enfolded into unity, so too one should expect plural exemplars that can be enfolded into one exemplar through the operation of necessitas complexionis. Thus the second mode functions to coordinate the plural forms and the divine One. But clearly Thierry has already given this birthright to mens divina, which autonomously generates, conceives, and holds the many forms within itself before releasing them into matter. So which one is it? Bernard had decided this function was performed by the eternal divine mens and not by the higher “coeternal” divine Word, so that reason could know the forms without theological illumination. Augustine held that it was solely the Word or Wisdom, even if this disables the autonomy of physical science. Thierry wants to have it both ways. On the one hand he theorized the pivotal second mode as the crux of mediation through the autonomous unfolding of number. On the other hand, he wants to read the Macrobian mens as equivalent to the eternal Word and heal the rift between Bernard and Augustine.82 While Thierry does his best to shoehorn the second mode into his account of the divine Mind, in the end it is difficult to view this passage as a success. It speaks volumes that when Thierry took up the same question in Glosa, he decided to repeat most of the same maneuvers—the forma formarum, the hierarchy of act interpreted as the four modes, the Calcidius reference, and even Macrobius’s mens divina. But he silently excised any reference to the second mode.83 As early as 1130, Thierry had begun addressing Bernard’s theological questions in his Genesis commentary. By the 1150s, as he reached the end of his road, despite all of his achievements, he could not overcome this ambiguity built into necessitas complexionis. In part the second mode represents the order of number, the object of mathematical study; but if Thierry was to overcome the firewall Bernard had built between honest Platonism and faithful Christianity, the second mode must somehow also designate the activity of divine Wisdom. Like the young Augustine, Thierry had tried to maintain that number and Wisdom were coeval as early as his Genesis commentary. But as Augustine knew well, this is not as easy as it sounds. Are the divine ideas identical with the numbers flowing out of the One, or are they the wisdom of the divine Son? The conceptual precision of Thierry’s modal system and his Trinitarian theology made the habitual theological discomfort of the question sting all the more sharply. The elder Augustine had decided that the tension between Arithmos and Logos could not or should not be maintained. But if the young Augustine’s attempts were already Neopythagorean, how should we

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describe Thierry’s far more complex project? The final issue to consider, then, is Thierry’s Pythagorean identity.

Pythagorean Affinities A deep ambivalence pervades the disciplinary organization effected by necessitas complexionis. If the folding of the second mode is what links mathematics and theology, then which of these two disciplines welcomes the other and provides it a virtual home? Folding coordinates God and the mathematicals reciprocally in one term: the one unfolded is the other enfolded and vice versa. So it remains an open question whether Thierry intended the priority to run from mathematics to theology or the other way around. This is a critical dilemma that must be confronted before one can ask what it would mean to characterize Thierry’s theology as Pythagorean, or better, Neopythagorean. It could be that mathematics enjoys priority, hosting and to that extent governing theology. After all, the infinite series of numbers is what establishes the axis of difference along which unfolding takes place. If the curve of the fold were mathematically delineated, either through number or geometrical space, then Thierry would have drawn the universal vocabulary for unifying theology and mathematics from the quadrivium itself. According to this interpretation, we should understand Thierry as a kind of Christian Iamblichus. On the other hand, Thierry clearly views “enfolding” as an attribute of God, whose nature it is to gather and reconcile created difference. From this perspective, God’s folding numerates and geometrizes as a function of the divine nature. Speaking of God in terms of the quadrivium would not be a foreign restriction, because God would be the quadrivium’s source and ground. This reading accords better with my interpretation of the arithmetical Trinity in Tractatus. The creation of number is the creation of things not because the Creator is subjected to the constraints of Neopythagorean theology, but because the Trinity is itself the self-numerating engine that generates number in the first instance. Likewise in Thierry’s modal theory, God’s enfolding provides the virtual space in which the unfolding of number in quadrivial orders can arise. The divine enfolding, so to speak, has already inscribed a geometrized, and therefore enumerated, difference into the very space that shapes the eternally enfolded fold. In this way, the doctrine of a divine fold articulated into four modes in Lectiones restates the doctrine of arithmetical Trinity in Tractatus at a more profound level. In my view, this second option better expresses what is at stake with the second mode. The second mode allows Thierry to grasp how Christian Trinitarian theology opens up an autonomous space for mathematics within itself and thus bridges the divide between the two disciplines. But note that this only occurs once Thierry has exited the Platonist logic of mathematical abstraction, which

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separated the disciplines in ways Philolaus himself would have rejected. Instead Thierry has adhered to the univocal logic of the fold—a geometrical logic, but a geometry whose space is initially charted by Trinitarian difference (the in-difference of unitas-aequalitas-conexio), and only then by numerical difference. The quadrivium remains theologically significant, but in a different way. Mathematics now ranks as the primary inversion (explicatio) of the divine identity, bearing the first subdivine reflection of God’s necessity within the necessary structure (necessitas complexionis) of number’s “order and progression.” If Thierry teaches that God spatializes and enumerates out of Godself, then he is certainly some kind of Neopythagorean, but an unusually complex kind. To give a historically precise name to Thierry’s brand of Neopythagorean Christian theology requires a more sensitive hermeneutical perspective. As we have seen, Thierry of Chartres puzzled together some of the flotsam of mathematical theologies that had been drifting through Latin Christian traditions since Augustine and Boethius. To do so, he had to resort to inventive strategies of cross-reading to stitch the fragments (or “philosophemes”) into a new whole.84 To name Thierry’s Pythagorean affinities, then, is simultaneously to chart the method of his radical hermeneutical procedure. There are four sequential components to Thierry’s methods of reading and thus four corresponding facets to his Christian Neopythagoreanism. They may not follow chronologically in his texts nor necessarily reflect his own deliberations. But in an unprecedented way they recapitulate critical moments in the history of mathematical theology now transposed into the discursive limits of Latin Christianity. First, where his contemporaries indicated nature, causes, or the world-soul as the ground of autonomic mediation, Thierry rather looked to number. God imbues order into the structures of the cosmos not through causation, motion, or act, but fundamentally through multiplicity and quantity, the subjects of the quadrivium. What sounds initially Pythagorean in Tractatus is in fact Thierry’s proposal that number provides the primal mediation between world and God. If one seeks the best conceptual plane (rationes) in which to glimpse the Creator’s activity in the world, Thierry writes, one ought to scrutinize arithmetic, harmonics, geometry, and astronomy for traces of divine eternity. This notion that mathematics is the most reliable stepping-stone for ascent toward the Absolute is precisely Plato’s modification of the ancient Pythagoreans, especially as continued within the oral traditions promoted by the Old Academy. Thus the first step of Thierry’s exegesis is the eminently Platonic decision to focus on number as the site of mediation. Textually speaking, Thierry then works through his reading of Genesis by bringing together, under the aegis of mathematical mediation, the two halves of the divided Boethian corpus. The opuscula and Consolatio discuss divine unity; the Nicomachean translations discuss the unity of number. Thierry integrates both aspects by using the quadrivial texts to answer theological problems in his

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Genesis and Boethius commentaries alike. This univocal hermeneutic effectively identifies the divine One and the numeric One, and it restores the electric proximity of quadrivium to theology noted by Merlan. So Thierry’s second step is to select the quadrivium as a vehicle for henological theology. When we consult the history of Pythagoreanism, this kind of systematic mobilization of the fourfold mathematical arts for purposes of first philosophy was invented by Philolaus and Archytas and perfected by Nicomachus. Thierry’s statements that “the creation of number is the creation of things” and that numbers begin in the mind of the Creator are remarkable echoes of Nicomachean theology. In response to Bernard of Chartres’s decisions in his gloss, Thierry not only rejoins both halves of the Boethian corpus, but then tries to connect them to Augustinian views of the Trinity. In light of his revived Boethian henology, Thierry reads Augustine’s mathematical triad realistically as no one else had before. The arithmetical Trinity expresses how the unity of God self-multiplies into its own primal harmony, and from that harmony descends an infinite variety of autonomous numerical structures. On a superficial level, the triad of unitas, aequalitas, and conexio seems to echo the Philolaean triad of unlimited, limit, and their harmony. But the centrality of unitas, the divine One, in Thierry’s exposition brings him closer to the triadic henologies of Eudorus and Moderatus, for whom the linked processions of the divine Ones are the basis for the monadic progression of number. The ultimate source of Augustine’s triad, after all, was probably Moderatus himself. After turning to number, unifying Boethius, and adding Augustine, the fourth moment of Thierry’s radical hermeneutic is the transition from the arithmetical Trinity of Tractatus to the four modes of Lectiones. This final leap represents Thierry’s most sophisticated theorization of the quadrivium in the concept of necessitas complexionis. Here it must be said that Thierry has synthesized an entirely new mathematical theology. He takes up the Nicomachean vision of the quadrivium as first philosophy, but now within a different conceptual architecture, an ontology of folding that most resembles the geometrized philosophy of Proclus or the Moderatan “spreading” of the One popularized by the Valentinians. As I  noted at the close of Chapter 2, Nicomachus never addressed the spatial “extending” of number that we find in Moderatan emanation. Proclus explored the spatial cycles of the One’s procession into the many, but wanted to remove the Iamblichean emphasis on the quadrivium as a site for theology. With his modal theory, Thierry puts into play a configuration of Neopythagoreanism that builds on Greek predecessors and yet had never appeared in the ancient world.85 By the same token, however, Thierry’s breakthrough returned mathematical theology to the post-Nicomachean impasse that had divided Iamblichus and Proclus over the relative status of arithmetic and dialectic. In this way the tensions within late antique Neoplatonism over the Pythagorean legacy resemble

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some of the tensions within the medieval reception of Thierry’s theology of the quadrivium.86 As we shall see in the remainder of this book, the controversial legacy of Thierry’s reception history revolved precisely around the status of plural but eternal exemplary forms—whether to conceive of them as numbers or as the divine Mind or Word, and whether to privilege the relative roles of Christian (Trinitarian) or Platonic (mathematical) vocabularies.

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Thierry’s Diminished Legacy Thierry of Chartres crafted a remarkable Trinitarian model out of the least promising of materials. His Boethian commentaries sketched the outlines of a whole new species of Christian mathematical theology founded not on abstraction but on geometrical folding. After centuries of delay, western Christianity finally revisited and appropriated critical moments in the genesis of Neopythagoreanism for the first time. And yet, despite the widespread personal esteem of his contemporaries, Thierry’s efforts had only scant immediate influence. When we attempt to track the after-effects of his achievements in subsequent decades, the trail quickly grows faint. In part Thierry was simply the victim of shifting fashions in the schools of Paris. New scientific translations were flooding into Latin Europe from Jewish, Islamic, and Byzantine traditions, changing the subject away from the Platonism in which the school of Chartres had invested. Masters simply found these newly accessible versions of Aristotle’s tidy handbooks more pedagogically useful.1 By the turn of the thirteenth century, Chartrian Platonism already seemed naïve. Yet if we leave the story here, the eclipse of Thierry’s theological program can seem almost necessary, a planned obsolescence that cleared the way for a new era. There are other factors to consider when weighing the causes and the legitimacy of Thierry’s diminished legacy, beyond the turn to Aristotle in the thirteenth century. Some are more local, and some have a longer term, even into the present. In the first place, it must be said that Thierry picked the worst possible decade of his century to test out a novel Trinitarian analogy. Just a few years prior, a fellow Breton, Peter Abelard, had been castigated from all sides for his suggestion that the Father, Son, and Spirit could be known more originally as divine Power, divine Wisdom, and divine Goodness (potentia, sapientia, benignitas or bonitas).2 Abelard’s proposal was tied up with his idiosyncratic views of language and reason in theological method. But his soon notorious triad succeeded in provoking an unprecedented degree of investigation into the semantics of such triadic formulae. Thierry’s quadrivial reading of Augustine seemed to merit the same suspicion, and too often the same strategies for resisting or evading Abelard’s questions were simply redeployed against Thierry.

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By the 1160s, Peter Lombard’s Sententiae had begun to consolidate an emerging consensus among networks of early scholastics. Triads assigned to the Trinity, they concluded, fall into two categories.3 Some are “proper” names, and others are “appropriated” or “attributed.” Proper names denominate real differences among the three divine persons. The Father is the Generator because that is exactly what it means to be Father; the Spirit is the one who spirates. Appropriated names designate, strictly speaking, the one substance shared by all three persons, but can be provisionally parsed out according to Trinitarian differences. Hence the Son is Wisdom because it is more fitting to call the Son Wisdom than the Father; God is Love, but one may appropriate that name especially to the Spirit. On this view, Abelard’s problems began when he construed his appropriated triad as a proper triad. Thierry clearly considered his mathematical reading of Augustine’s triad to be a proper name of the Trinity. He was convinced that unity, equality, and connection are not essential attributes like beauty or goodness, but are definitions of paternity, filiation, and spiration themselves, such that the category of number (the quadrivium) can be wielded in theology as a primary means of authentically naming God. But after Thierry’s death, the most influential early scholastics— from the young Porretani Simon of Tournai and Alan of Lille, to the magisterial Peter Lombard and Richard of St. Victor—classified Augustine’s mathematical triad otherwise, lumping it together with Abelard’s triad as two classic instances of appropriated names. It was this view that influenced the giants of the thirteenth century, effectively silencing Thierry’s distinctive Neopythagorean interpretation.4 Until quite recently the minority report of those who sided with Thierry’s controversial reading was lost to history. Häring recovered two short anonymous texts from the 1140s or 1150s, known as Tractatus de Trinitate and Commentarius Victorinus, which praise the arithmetical Trinity at length.5 Composed by members of an inner circle familiar with Thierry’s commentaries, these student treatises represent the first moment of Thierry’s reception history. Both authors eagerly followed their master in applying the quadrivium within theology, spelling out what Thierry had left implicit, adding corroborating details, and demonstrating how the triad refutes heretics. Unlike their teacher they also took pains to list the authorities supporting his new approach, from the Gospel of John and Augustine to the Ps.-Dionysian “theology of negation.”6 Another student, Achard of St. Victor, penned a brilliant, lengthy exploration of the beauty of the arithmetical Trinity, De unitate dei et pluralitate creaturarum.7 But Achard’s work was already forgotten by 1200 and not fully restored until the 1980s. Alan of Lille concurred with Thierry’s reading belatedly in Regulae theologiae and De fide catholica, but only thirty years after his influential Summa “Quoniam homines” had urged many of his peers in the opposite direction. None of these four authors (with the possible exception of Achard) stepped beyond the confines of Thierry’s Trinitarian theology to consider his other innovations.8

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Even if Thierry’s doctrine of the arithmetical Trinity had not slipped through the cracks of the twelfth-century schools, his vision of a Christian Neopythagoreanism still may never have been realized in the following centuries. Thierry not only wrote right after Abelard, he wrote just before the rapidly developing exact sciences took their leave from theological oversight. The Neopythagoreanism of Moderatus, Nicomachus, and Iamblichus—and its occasional survivals in the young Augustine and Boethius—had been premised upon the immediate proximity of an integrated mathematical discourse to discourse on the gods and the One. But from the thirteenth century onward, mathematics and theology retreated from each other at record pace. This is a second, more serious challenge facing Thierry of Chartres’s legacy than the swing from Plato to Aristotle or the paucity of his supporters’ writings. The very disciplinary conjunction required by Christian Neopythagoreanism, its native territory, threatened to disappear from the map altogether. As others have explained in detail, the aging quadrivium was finally retired once mathematics had learned to expand its domain beyond static quantities. At Oxford and Paris, fourteenth-century masters like Thomas Bradwardine, Jean Buridan, and Nicole Oresme began to challenge Aristotelian natural philosophy when they discovered how to quantify motions and qualities, making it possible for the first time to render all aspects of physical phenomena in exclusively mathematical terms.9 Rather than serve the ends of contemplation as in Platonist traditions, the new mathematics sought quantitative measures of divine infinity and even divine grace.10 At the same time, the character of Christian Platonism underwent a profound transformation in the fourteenth century. In hindsight one sees just how short-lived the springtime of eccentric Boethian Platonism was in the first half of the twelfth century. It provided a rare but opportune climate for Neopythagorean ideas to flourish. But soon thereafter, Latin Christian Platonism turned to its Augustinian and Proclian lineages again, in reaction to controversies over the use of Aristotle. The conflict between radical Aristotelians and episcopal authorities at Paris in the 1270s provoked a range of different responses in academic theology over the course of the chaotic fourteenth century.11 A central part of the story concerns the genesis of so-called “nominalism.” By elevating God’s absolute freedom beyond the reach of every metaphysics of participation, William of Ockham and others worked to minimize the influence of ancient Neoplatonism on Christian theology. Augustinians like Bradwardine (again) and Gregory of Rimini found that a sharper distinction between reason and faith helped to restore a salutary focus on grace, salvation, and the Church.12 But in other quarters the Parisian condemnations encouraged a modest renaissance of Christian Neoplatonism. Dietrich of Freiburg and Berthold of Moosburg sought rather to strengthen the old alliance with Neoplatonism by investing in Ps.-Dionysius, the Liber de causis, and

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new translations of Proclus. Already in the thirteenth century, Albert the Great, Bonaventure, and Meister Eckhart had discovered that such Proclian sources could illuminate the workings of the Christian Trinity. In the fifteenth century the Albertist school in Paris and Cologne would follow the same agenda.13 As we have seen, despite their differences, the Augustinian and Proclian traditions agreed that arithmetic, and mathematics generally, could not take pride of place within Platonist theologies. So unlike the Boethian Platonism of Bernard of Chartres or Thierry of Chartres two hundred years before, these streams of thought now seized on Logos and the motion of cyclical emanation as alternatives to mediation by Arithmos. Albert linked Christ as the medium of creation to the order of the Ps.-Dionysian hierarchies. For Bonaventure and Meister Eckhart, the divine Word is the common exemplar and principium of the procession of the Trinity and the creation of the world. The soul returns to God by recognizing the Word’s mediations either hidden within created vestigia (Bonaventure) or virtually present as the indistinct One (Eckhart).14 When mathematics no longer provides a viable mediation within Christian Platonism—when the Boethian option is foreclosed and the quadrivium has no commerce with theology—then we witness Ps.-Dionysian and otherwise Proclian Christian theologies adverting to Christological mediation. This era of late medieval theology, deprived of the Boethian “hidden tradition,” exacerbated the structural oblivion of Thierry of Chartres’s thought over three centuries. But paradoxically it also provided the key to its reemergence, as will become clear in Chapter 7. A final challenge facing Thierry of Chartres’s legacy is the undue focus historians have placed on the arithmetical Trinity at the expense of other doctrines. The reception history of Thierry’s theology can be divided in three parts; to date scholars have really studied only two of them, and only one adequately. The first is Thierry’s naturalistic hexaemeral exegesis in the first part of Tractatus. Much of this text was incorporated into the Chronicon of the Cistercian Helinand of Froidmont and thus passed to Vincent of Beauvais.15 Tractatus was the most frequently copied of Thierry’s works, particularly at St. Victor in Paris and in Cistercian communities in Bavaria.16 Today Thierry’s hexaemeral exegesis has been well studied. It fits neatly into existing narratives regarding the slow rise of medieval science, the twelfth-century interest in nature, and the parallels between reading books of scripture, nature, and experience.17 The second component in Thierry’s reception history is the arithmetical Trinity. As we have seen, the doctrine is expressed more or less completely within the second half of Tractatus and is amplified slightly thereafter in Commentum. But Thierry’s arithmetical Trinity is most often approached through the lens of its reappearance later in the writings of Cusanus.18 Indeed, it was this shared doctrine that first revealed the cardinal’s debt to the Breton master.19 But the frequent comparison with Cusanus can occlude a better understanding of the fate of

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Thierry’s doctrine in the decades after its initial debut and before its eclipse in the mid-twelfth century. Students of Thierry are left wondering why the famed teacher failed to win more support for his brilliant triad; students of Cusanus, lacking evidence of anyone else engaging Thierry’s ideas before 1440, might assume that the German cardinal simply transcribed the triad into his texts, much like Thierry’s other students auditing his Parisian lectures. On the contrary, Cusanus changes the triad as it suits him and experiments with new applications within the altered parameters of late medieval theology, as we shall see. This ahistorical comparison has also encouraged some to vaguely classify Thierry’s efforts as “number speculation,” rather than as a provocative, nuanced rereading of a major patristic authority. Unfortunately the reception history of the modal theory of Lectiones and Glosa, arguably most important component of Thierry’s legacy, has been the least studied. I showed in Chapter 5 that one should not place undue emphasis on the arithmetical Trinity alone, since this was not the endpoint of Thierry’s deliberations, but their point of departure. To remedy this situation one would need to find evidence of how medieval readers interpreted not only Thierry’s triad, but also the three perpetuals, the four modes of being, reciprocal folding, and other major doctrines of the Boethian commentaries. This broader perspective on the history of Thierry’s reception would also reveal the complex potentials latent in his doctrines when combined in different permutations. To date, however, I have found only three texts that serve this purpose. (1) The author of an anonymous twelfth-century treatise, De septem septenis, hails Thierry as his teacher and combines the arithmetical Trinity with the three perpetuals and a version of the four modes of being. (2) Clarembald of Arras proudly advertises his tutelage under Thierry of Chartres and Hugh of St. Victor. Yet despite his loyalty, Clarembald’s anxiety about some of Thierry’s doctrines is painfully evident, as he weighs the relative worth of the arithmetical Trinity, the four modes and the three perpetuals. While penning his own commentaries on Boethius and Genesis, Clarembald plays the role of polite censor, tactfully softening Thierry’s sharpest edges. (3) The recently discovered Eichstätt treatise, provocatively titled On the Foundation of Nature which the Philosophers Apparently Do Not Know, was written around the turn of the fifteenth century. Its author is conversant both with Aristotelian philosophy and with several of Thierry’s doctrines, although he never addresses the Breton by name. Educated in the regnant anti-Platonism of the fourteenth century, the author of Fundamentum naturae inveighs against Thierry’s theology of the four modes, contrasting the foolishness of subdivine mediators with the wisdom of the Verbum. Viewing Thierry’s legacy through the prism of these few sources, however thin an archive, reminds us that Thierry’s contemporaries had difficulties understanding what he was up to and were uncertain about some basic features of his

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signature concepts. It also shows the great distance that lay between Thierry’s austere lectures in the 1140s and the crowded universities of northern Europe clotted with competing schools by 1400. Nor is it an abstract exercise to read these texts closely. All three sources, including the brilliant dismantling of Thierry’s ideas in the Fundamentum naturae treatise, found their way into Nicholas of Cusa’s breakthrough work, De docta ignorantia—a riddle we will have to unravel in Chapter 7.

Confusion about Mediation The moral value of the quadrivium and trivium was hotly contested in the mid-twelfth century. The new cathedral schools claimed to provide a practical education superior to that of the monasteries, but within a few decades came under criticism for the avarice and arrogance of their students.20 The treatise De septem septenis addressed the theme of the spiritual meaning of the seven liberal arts, but from the peculiar point of view of medieval Christian Hermeticism.21 The Greek text of Asclepius containing the teachings of the sage, Hermes Trismegistus, stemmed from second- or third-century Egypt. Translated into Latin around the time of Augustine, it exerted its greatest influence on Christianity during the twelfth century.22 Several of Thierry’s other students, like Hermann of Carinthia and Bernardus Silvestris, frequently referenced another contemporary Hermetic work, De sex rerum principiis.23 John of Salisbury, cultured observer of the Chartrian circle and the Parisian schools, was long believed to be the author of Septem, but this has been discredited.24 Nevertheless, the author’s concerns about the moral and exegetical value of the quadrivium fit squarely within the mid- to late-twelfth century.25 The text of Septem has seven sections, each of which lists seven terms. The author refers twice to “my teacher” (magister meus) and is obviously familiar with several of Thierry’s formulae.26 Unlike other texts by Thierry’s earliest students, Septem combines Thierry’s doctrines from several different Boethian commentaries and mixed these with a stew of other influences. This leads us to believe that the author of Septem faced the challenge of distinguishing, combining, and editing Thierry’s different concepts toward his own ends. The peculiar way that Septem ended up reshuffling different doctrines in the final section of the treatise demonstrates how imperfectly Thierry was understood by early readers. But before turning to that passage we can establish two other points of contact that betray the Breton master’s influence. First, like Thierry, the author of Septem considers how the quadrivium and trivium ought to relate to theology. He places himself within the Pythagorean lineage in pursuing scientia as a means to wisdom and virtue. The seven liberal

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arts lead one to eruditio, an integrated theoretical and spiritual knowledge of divine things.27 The author lists the quadrivium in the order assigned by Boethius and then defines the scientific lesson that each art conveys.28 But then he lists the more essential moral training acquired through the liberal arts; each via, as he puts it, should reform a different aspect of vita.29 The trivium instructs one how to distinguish true from false arguments, but the real goal is to become more truthful. The quadrivium should also grant moral dividends. What good is arithmetic, he asks, if it only fuels the merchant’s greed? Rightly understood, arithmetic provides the basic operations for reading scripture: addition (history), subtraction (allegory), division (tropology), and multiplication (anagogy). Music trains one in ethical harmony (concordia morum) amidst the dissonant voices of life. Geometry teaches measure and equality, leading the passions to the mean. Astronomy teaches wonder-filled reverence for divine providence.30 These seven disciplines, considered wisely, are seven ways for the soul to a life of integrity . . . . Therefore a well-instructed spirit gains wisdom and eloquence through these seven ways of trivium and quadrivium; in the ways and modes explained here, any soul can attain its own reformation and its passage into God. Hence through these man-made ways of the soul one perceives the universal ways of the Lord, namely mercy and truth. . . . Of this the Lord said: “I am the way, the truth and life”—the way by example, the truth by promise, and the life by reward, or again, the way to him, the life from him, the life in him.31 What first appears a moralizing homily on the liberal arts turns out to be a rather creative gloss on John 14. The viae of the quadrivium point one ultimately toward the revealed via of God, the way that is not only truth but also life. Hence for Septem the divine Word is the foundation of the quadrivium, ensuring that knowledge of the physical world leads one to theological insight. This conviction reflects the agenda of Thierry’s Genesis commentary. Second, Septem repeats Thierry’s psychological epistemology from Commentum, combining Boethian terms with a miscellany of others.32 In his discussion of intellectus, the author feels compelled to distinguish it from intelligentia, recalling the ambiguity that Thierry addressed in Boethius.33 Most importantly the treatise shares several passages on spiritus theory in common with Thierry’s Commentum.34 As I noted in the previous chapter, this likely points toward a common source, given the widespread interest in Galen’s theory of the brain among twelfth-century natural philosophers. At the end of its spiritus passage, Septem echoes Thierry closely, contrasting the disciplina or doctrina of mathematics with the intelligentia of theology.35 When he posits three types of contemplation

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(revelation, emission, and inspiration) in the sixth section, he recapitulates the Boethian anthropology of imagination, reason, and intelligence also used by Thierry.36 With these shared topics as backdrop, the author of Septem directly engages Thierry’s doctrines in the final section of the treatise, where he affirms the critique of the liberal arts found in Asclepius. Hermes Mercurius had scolded the learned philosophers for remaining ignorant of the deeper principles upon which their disciplines were founded.37 Likewise Septem writes that philosophers “seek the first principle and the other principles eagerly in the trivium, they pursue it keenly in the quadrivium, and they search after it subtly in the examinations of theology and philosophy. All seek this, but they do not find it, because the manner of their search and their ignorance of truth obstructs them.”38 To tutor such philosophers, Septem accordingly compiles seven fundamental principles and names the authorities behind each one. His emphasis on the credentials of those authorities makes him all the more deliberate in citing Thierry’s doctrines among this final “sevenfold”: According to the theologians, there is one principle, God, the creator of all things; according to the natural philosophers (physici), there are three principles: matter, form, and the created spirit (that is, nature); and according to Hermes Mercurius there are four: the law of the stars, nature, the world, and the machine of the world.39 The remainder of the treatise is devoted to expounding these seven principles in turn (the two senses of “nature” are counted as one).40 The final four principles derive from Hermetic teachings.41 But “theology” and “physics” come straight out of Thierry of Chartres. Within a few lines the author will not only cite the arithmetical Trinity, but also conflate the three perpetuals and the four modes. The first of the seven principles is that God is the eternal creator. To elaborate further, Septem layers different ancient sources upon each other, building toward the Johannine theology featured at the beginning of the treatise. Not only the Gospel of John, but Heraclitus, Hermes, Boethius, and the Sibyl have all perceived that true divine Lord is not Mars or Mercury but God’s coeternal Son.42 Then Septem apparently turns to Thierry’s lectures: Parmenides also says: “God is the one for whom being anything that is, is being everything that is.” Again he says:  “God is unity:  from unity is born the equality of unity. But the connection proceeds from unity and the equality of unity.” Whence, therefore, Augustine says:  “To all those who perceive rightly, it is clear why from sacred scripture the doctors assign unity to the Father, equality to the Son, connection to the holy Spirit.

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mathematical theologies And although from unity is born equality, connection proceeds from both; yet they are one and the same.” This is that unity of three, as Pythagoras taught, which alone deserves to be adored. . . . Indeed I would conjecture that whoever desires to consider that true unity, letting go of the mathematical point of view, must raise the soul to the simplicity of intelligence.43

For the author of Septem, the arithmetical Trinity handed down by his master was in turn bequeathed to Augustine from Parmenides, and ultimately from Pythagoras. This divine One, already worshipped by the ancients, is not opposed to mathematics, but is rather its foundation. This is precisely the view that Thierry had developed across his lectures on Genesis and Boethius. The author of Septem promised that the next three principles would derive from the physici. But once he names them, we see that they are simply Thierry’s trinity of perpetuals. The author follows Thierry’s wording in Commentum very closely: “from this high and eternal trinity descends a certain trinity of perpetuals. For matter descends from unity, form from the equality of unity, and the created spirit or nature from the connection of both.”44 This means that Septem names Thierry as a physicus as well as magister meus. Finally, he conflates the three perpetuals with Thierry’s four modes of being. Notably Septem uses the preliminary version of the four modes in Commentum (deus uel necessitas, formae rerum, actualia, and possibilitas) instead of the mature version found in Lectiones.45 These are three principles, descending from the first principle. Hence my master says that the first principle is eternity, which since it is immutable is called necessity. The second principle is matter, which because it is capable of receiving all forms is called possibility. The third principle is form, which because it determines matter in one state or another, is called finality. The fourth principle is the created spirit, who because it is the universal motion of things, is called actuality.46 Septem then attempts to link Thierry’s third perpetual (“created spirit”) to the various Hermetic principles of nature.47 Thierry’s signature in the Septem treatise is unmistakable, especially once the differential development of the Boethian commentaries has been properly taken into account. But by combining Thierry’s terms together under the pressure of his sevenfold system, the author strains well beyond his master’s theological intentions. Thierry juxtaposed the mathematical and perpetual trinities, but he never mixed either of them with the four modes, even in their nascent form in Commentum. For Thierry’s goal, even from his early meditations on Genesis, was to articulate how the Creator remains immanently within the causal or modal structures of creation. Septem’s conflation of the triads with the four modes has

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the opposite effect. He counts the first mode as a “first principle” and separates it off from the lesser “three principles,” namely the fourth, second, and third modes, which correspond to the triad of possibility, form, and created spirit. Hence the subtle effect of Septem’s combination is to lift the transcendent divine principle above the fray of the cosmic principles of the physici. The theologians alone preserve the name of the divine Son; the natural philosophers are free to juggle Hermetic principles as they see fit, since they merely pertain to the created order. This division of labor is altogether contrary to Thierry’s theological program. Nevertheless, Septem’s evident confusion about the distinct varieties of mediation in Thierry’s thought seems the unintentional by-product of his enthusiasm as a former student. The author had no qualms about passing down the doctrines he learned from Thierry without revision, particularly those from Commentum.48 Other readers of the Breton master did not agree.

An Augustinian Censor Clarembald of Arras (d. ca. 1187)  proudly counted himself as a student of both Thierry of Chartres and Hugh of St. Victor.49 He studied under Hugh before the Victorine’s death in 1141 and under Thierry between 1136 and 1146 in Paris. By 1152, Clarembald was the provost of the cathedral school at Arras, and by 1156, he was promoted to the archdiaconate, a transfer from academic to church affairs. Before his death in 1187, it seems that Clarembald enjoyed two leaves from his administrative work to lecture at the famed school of Laon. During his first stay (1157–9) he wrote two commentaries on Boethius, and during his second (1165–8) he wrote a commentary on Genesis.50 According to Clarembald, all of these were intended as tributes to Thierry’s similar works, but also as rebuttals of the theological errors of Peter Abelard and Gilbert of Poitiers.51 Judging from his citations, Clarembald evidently had access to the full complement of Thierry’s extant works. Yet Clarembald’s writings were more independent from Thierry than the early student treatises and are more conservative than the melange of esoteric sources in Septem. Because Clarembald excerpted Thierry so frequently, we can watch as he weaves together the master’s different commentaries, revealing his own interests and, at times, his discomfort. Clarembald spoke out against the controversial views of Abelard and Gilbert. Did he question Thierry’s provocative doctrines in the same way? In the first modern evaluation of Clarembald, Wilhelm Jansen contended that Clarembald “disregards those dangerous statements [by Thierry] that admit only a pantheistic interpretation.”52 Häring finds in Clarembald a sense that “after the bold sweeps of Thierry’s speculations a time of calm appraisal was clearly needed to assess their value.”53 What remains to be explored, however, is exactly how Clarembald

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amended Thierry’s signature doctrines, with what resources and by what canons. Since his commentaries relied so heavily upon Thierry’s works and covered similar territory, it is possible to peer over Clarembald’s shoulder, so to speak, as he subtly revises Thierry’s four modes of being in particular. For it was this doctrine, not the arithmetical Trinity, that Clarembald considered the most provocatively avant-garde of Thierry’s work. Among the texts we possess by Clarembald, his efforts at loyal correction had two stages.54 First, his Tractatus on Boethius’s De trinitate resituated Thierry’s doctrines safely outside their original speculative context and circumvented potentially problematic passages. Then after a short commentary on De hebdomadibus, Clarembald went out of his way to address the issue again in his Tractatulus on Genesis. This time he deployed an authoritative text from Augustine in order to buttress the orthodoxy of the four modes of being. Yet by doing so he inadvertently indicated some of the latent ambiguities in Thierry’s theology.55

Anxieties about Mediation Clarembald states that his commentary on Boethius’s De trinitate, the longest of his extant works, has two purposes. By “imitating the lectures” of his teachers, Thierry and Hugh, he will vindicate the orthodoxy of Boethius and provide a better guide than Gilbert’s notoriously difficult gloss.56 But in fact Clarembald makes selective use of Thierry’s doctrines once he arrives at the critical De trinitate II. To explain the Boethian division of sciences he follows along with Thierry’s Commentum, avoiding the fuller commentary in Lectiones. When he reaches Boethius’s difficult theory of God as form, Clarembald calls upon “my teacher” for assistance and begins to cite from Thierry’s Commentum II.18–19. But as we have seen, this was the problematic passage where Thierry had first tested (and rejected) an early ontological model of the four modes without the architecture of folding, then returned to the arithmetical Trinity, and finally added the trinity of perpetuals. At this juncture Clarembald conspicuously decides to ignore the trinity of perpetuals and to focus instead on the arithmetical Trinity. Like his teacher, he focuses on aequalitas within the triad, citing verbatim from every paragraph of Thierry’s Commentum on the arithmetical Trinity.57 But then in the midst of discussing the Son as equality, Clarembald abruptly pivots to Lectiones. He cites the mature version of the four modes with several modifications. Clarembald devotes twice as much attention to the second mode of necessitas complexionis as to the other modes. He defines the concept in his own words: “from this absolute necessity descends the necessity of enfolding or of enchaining, by which those things which are enfolded in absolute necessity from eternity are dispensed in the continuity of times as if through causes enchained together and enfolding themselves.”58 This gloss alters his teacher’s account in several ways. Clarembald envisions the

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complexio of the second mode as a causal chain (concatenatio) descending from the first mode.59 Thierry had described the descent of alterity from unity, mutability from immutability, inequality from equality, and the trinity of perpetuals from the arithmetical Trinity.60 But Thierry never applied “descent” to the four modes, whose relations were deliberately structured by reciprocal folding in lieu of such ontological subordination.61 Beyond these piecemeal adjustments, Clarembald’s decision to juxtapose Commentum and Lectiones itself modifies Thierry’s theology at a deeper level. First, it effectively decouples Thierry’s four modes from the division of sciences that had impelled Thierry to posit his modal theory in the first place. In Lectiones and Glosa, the first mode relates to the second mode as theology relates to mathematics. In Clarembald’s commentary, by contrast, the four modes are transformed from a metadisciplinary theory of science into an arbitrary ontological system. Clarembald’s editorial choices also lead him to omit (or censor) two key passages from Commentum. He skips over II.28 (the nascent four modes) and II.39 (the trinity of perpetuals) and instead focuses only on the middle section on the divine Trinity (II.30–38). If his goal was to benefit from his teacher’s unique doctrines, why avoid these passages? Given his intimate familiarity with Thierry’s commentaries, it is conceivable that Clarembald knew Lectiones to be the mature version of the four modes and observed that Thierry discontinued the trinity of perpetuals there as well. This would be more convincing if Clarembald did not return straight to both omitted passages the next time he discussed Thierry’s four modes, in his later commentary on Genesis. It seems better to ask what these two passages held in common that raised his suspicions. Clarembald apparently found something objectionable in Commentum II.28 and II.39. In the first passage, Thierry defends Plato’s theory of primordial matter, and in the second passage he describes matter and form perpetually descending from the Trinity. To Clarembald’s eyes, Thierry’s dalliance with Plato raised the specter of eternal matter and the conflation of the Holy Spirit and the world-soul. Clarembald had frankly corrected Abelard and Gilbert for their perceived doctrinal errors. So when his former master appeared to flirt with heterodox ideas, it seems that Clarembald tactfully looked the other way, passing over the offending sentences from Commentum in silence. As we shall see, Clarembald postponed his treatment of Thierry’s trinity of perpetuals until his later Genesis commentary. For by that time he had devised a way to cloak Thierry’s controversial doctrine in the mantle of Augustine. In the intervening years before that Genesis commentary, Clarembald penned a short gloss on Boethius’s difficult De hebdomadibus, in which we see him making further alterations to necessitas complexionis.62 He repeats the model of “descent” from his first Boethius commentary. But now, picking up a thread in Thierry’s Commentum, Clarembald also tries to connect the modal theory with Johannine

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Logos theology. Thierry had studiously avoided this eventuality by segregating the arithmetical Trinity from the four modes. The results of Clarembald’s experiment are somewhat awkward: For in absolute necessity all things are constituted from eternity in a certain enfolded simplicity. In it, they are what they are, as John, the greatest of theologians, witnesses: “What was made had life in him.” And since they are “life in him,” the descents [descendentia] through necessitas complexionis to εἱμαρμένη now result manifestly in definite possibility and are subject to fate.63 In this revealing passage we see that Clarembald envisions the four modes as a kind of emanation from higher to lower levels of being. Instead of relating the first two modes through reciprocal folding as Thierry had, Clarembald describes discrete tiers of vertical “descents” passing from God (the first mode) down “through” an intermediary plane (the second mode) into actual beings subject to fate (the third mode). If his intention was to domesticate the second mode by situating it within Wisdom Christology, the ultimate effect was rather to render the function of the divine Word superfluous. For in this brief account, at least, it is the second mode that mediates divine order to creation. In Clarembald’s ill-advised revisions, we can detect a trace of the contest between Logos and Arithmos in Augustine’s theology of creation and of Thierry’s struggle to resolve it.

Paradigm Regained In his subsequent Genesis commentary Clarembald’s concern about the respective roles of second mode and divine Word became even more pronounced.64 He intended to comment on the entire Pentateuch, but even the brief hexaemeral gloss that he completed (§§33–49) is overshadowed by his lengthier philosophical preamble (§§9–32).65 In a prefatory letter, he pays tribute to Thierry’s fame and concedes his dependency on his master’s model, and once Clarembald begins his commentary, his line-by-line debt to Thierry is obvious. But whereas Thierry dared to read Genesis according to physics and quadrivium, Clarembald reframes Thierry’s naturalistic interpretation as merely the literal sense that should ground spiritual exegesis.66 Clarembald conspicuously fails to mention Thierry’s arithmetical Trinity, which of course occupied a significant portion of his master’s Tractatus. He also says that he is writing to combat contemporary heresies. Without a proper understanding of creation, one inevitably falls into Christological errors, just as Boethius refuted Nestorius and Eutyches by exposing their philosophical failures. Thus, as Clarembald repeats several times, “on account of ignorance regarding creation” and for the sake of Christian orthodoxy, he feels compelled to set forth some philosophical principles in a protracted excursus.67

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This philosophical preamble turns out to be Clarembald’s chance to return to his unfinished business with Thierry. Armed with texts from Augustine’s De genesi ad litteram, Clarembald is now ready to revisit both of the troublesome passages from Commentum that he had previously skipped over. Apparently Clarembald had found in Augustine precisely the warrants for Thierry’s innovations that he felt were lacking. As Häring observes, compared to Thierry, Clarembald “is more anxious to quote or refer to ideas voiced by Christian and non-Christian writers of the past.”68 Clarembald accordingly applies Augustine’s concepts to shore up both the trinity of perpetuals and the four modes of being. Perhaps Clarembald had found his answers in Augustine during the interval between his writings, or perhaps Thierry had noted the source in the classroom, and Clarembald now had occasion to revisit it. As we have already seen, Thierry did draw from De genesi ad litteram both in Tractatus and in Commentum. Clarembald organized his excursus into four sections ordered according to Augustine’s theory of seminal causes in Book VI of De genesi ad litteram.69 His first three sections address what he calls “three inchoatives” (inchoativa), namely “primordial matter,” “seminal reasons,” and the motion of time. The final section discusses “four ways” (modi) that God works in creation:  in the Word, in matter, in seminal reasons, and in time. Clarembald’s project in the excursus is thus to redeem Thierry’s questionable opinions in Commentum by couching them, as he writes, in terms “gathered from the holy doctors concerning the creation of things.”70 As Clarembald discusses the trinity of perpetuals and the four modes, his anxieties about the roles of the divine Word and the second mode resurface once again. In the first place, Clarembald defines Thierry’s “three perpetuals” as “three inchoatives.”71 Augustine had drawn only a general contrast between perfected and “inchoate” creatures.72 But Clarembald renames Thierry’s triad as primordial matter, seminal reasons, and the motion of time. Most notable here is Clarembald’s attempt to spin Thierry’s second perpetual (forma) as Augustine’s rationes seminales. Unwilling to call Thierry’s perpetuals a trinitas, Clarembald can now countenance the triad provided that it is subordinated without question to the divine Son: “And we believe and declare that the Son of God is the creator of these three inchoatives. . . . The Son of God is the creator of the inchoate cosmos as well as the perfected cosmos—creator of inchoate things insofar as he is the Beginning; creator of perfected things insofar as he is the Word.”73 Clarembald goes out of his way to emphasize, as Augustine had done throughout De genesi ad litteram, that God created the world “in the beginning,” that is, in the divine Word, God’s Son.74 In the remainder of the excursus, Clarembald has very little to say about the third inchoative, and his remarks on the second simply repeat Augustine’s comments on miracles.75 So Clarembald agrees to embrace Thierry’s trinity of perpetuals on account of Augustine’s “inchoative” seminal reasons. But now he also uses seminal causation

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to justify Thierry’s four modes.76 Augustine had explained four distinct levels of creation on the basis of matter’s inherent “seeds.” Why couldn’t Thierry do the same? As Clarembald explains, the philosophers distinguish between “absolute possibility” and “definite possibility,” and in this primordial matter divine Providence works as “absolute necessity . . . from which descends necessity of enfolding or enchaining.”77 According to Clarembald, what Thierry calls the four modes of being, “Augustine signifies by other names”: All things, Augustine says, were made through him in the Word, in matter, in seminal reasons, in the creature. Thus Augustine calls absolute necessity the Word, that is, divine wisdom. He calls seminal reasons the hidden powers inserted in matter according to which some things are produced from others, each fittingly in its own time, through the necessity of enfolding. He calls absolute possibility matter; he calls definite possibility the creature.78 One has to admire Clarembald’s hermeneutical coup: Augustine’s system of seminal causation correlates beautifully with the four modes. This is not entirely accidental, of course, since it was conceivably one of Thierry’s inspirations in Lectiones. Clarembald is so content that he even admits Thierry’s favorite Calcidius passage on the monad and dyad and connects those Pythagorean terms to the four modes.79 Nevertheless, as before, Clarembald tacitly but substantively alters Thierry’s original doctrine, much in the pattern of his prior emendations. First, Clarembald carefully restricts divine activity to the two modes of necessity, something Thierry’s treatment does not contradict but certainly does not emphasize.80 Second, whereas Thierry strictly separates the four modes from the Trinity, Clarembald uses the folding of necessity to illustrate the role of the second person, divine sapientia. Returning again to John 1:4, Clarembald writes: It is divine Wisdom that works in matter. And just as all natural things subsist actually and by nature in definite possibility, so also the same things exist in divine Wisdom or Providence, enfolded through a certain simplicity. Nothing exists in divine Wisdom except divine Wisdom alone. Even John the evangelist, the greatest theologian, witnessed to this, saying: “All things have life in him.”81 Clarembald’s point is that the virtual existence all creatures have in God subsists in the Wisdom or Word of God. In his view creatures are enfolded by the Word, not by the second mode. Clarembald also repeats his wholesale reinterpretation of the second mode, first vaunted in his De trinitate and De hebdomadibus commentaries,

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as the mediator of a graded descent from God into matter. Again Clarembald uses descendentia to define necessitas complexionis, separating the cosmic function of the second mode from divine activity.82 Clarembald clearly took pride in vindicating his master’s Augustinian orthodoxy. After conspicuously avoiding texts from Commentum in his Boethius commentary, he recovered them in his Genesis commentary by offering tamer Augustinian substitutes. Yet his reliance on Augustine brings about unforeseen consequences. Clarembald deploys Augustine’s seminal reasons twice:  once to ground the trinity of perpetuals, and once to ground the four modes (see Table 6.1). This comparison of terms illustrates how “seminal reasons” performs two different functions for Clarembald. In the three inchoatives, seminal reasons are the primordial origin of the forms that determine matter, taking on the role of aequalitas in Thierry’s lexicon. In the four modes, seminal reasons account for the forms’ descent from the Word through the second mode into actuality. This double burden placed upon the rationes seminales suggests that Clarembald’s quiet corrections represent more than a conservative repetition of Thierry’s thought. By bringing the two doctrines together in order to validate them simultaneously, Clarembald accidentally exposes a lingering tension in Thierry’s mathematical theology. Thierry had taken care never to combine his different doctrines. If he had, an awkward juxtaposition would have become visible (see Table 6.2). Clarembald’s interpretive operations effectively align the divine Son with the second mode. By associating the rationes seminales with both of them, Clarembald inadvertently names their similar function. Both the Son and the second mode

Table 6.1  Clarembald’s twofold use of Augustine Tria inchoativa (trinitas perpetuorum) Thierry materia ab unitate forma ab aequalitate motus ab conexione

Augustine materia primordialis ratio seminalis principium temporalis

Clarembald informitas forma motus creaturae

Quattuor modi essendi Thierry necessitas absoluta necessitas complexionis possibilitas determinata possibilitas absoluta

Augustine Verbum / sapientia ratio seminalis opera materia primordialis

Clarembald formaliter seminaliter et complicite actualiter et reparative informaliter

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Table 6.2  Clarembald’s comparison of Thierry’s major doctrines Trinitas divina

Trinitas perpetuorum

Modi essendi

[Deus] Pater (unitas) Filius (aequalitas) Spiritus (conexio)

[Trinitas divina] Materia Forma Motus

Necessitas absoluta Possibilitas absoluta Necessitas complexionis Possibilitas determinata

possess the unchanging forms of all beings, mediating between the plurality of creatures and the unity of God. So while Clarembald may have succeeded in buttressing the orthodoxy of Thierry’s perpetuals and modes, it is a pyrrhic victory that raises troubling questions about Thierry’s theology. Is there an insuperable tension between the role of necessitas complexionis in mediating divine form (Arithmos, numerus) and the role of the Verbum in doing the same (Logos, sapientia)? Does Thierry’s modal theory, developed out of the Boethian quadrivium, remain ultimately in conflict with his theology of the Trinity? Can Thierry integrate Neopythagoreanism with Christian theology and still maintain Nicene orthodoxy? Clarembald’s reading of Thierry efficiently, if accidentally, reveals this contested zone of mediation. By translating Thierry’s ideas from their native Boethian context into the anti-Pythagoreanism of the later Augustine, Clarembald threatened to reopen the breach between Arithmos and Logos that his Breton master had struggled to heal.

A Late-Medieval Refutation: Word or Number? I have been able to locate only one other medieval author, beyond Septem and Clarembald, who collated Thierry’s major doctrines and evaluated them together.83 This is the anonymous Eichstätt treatise titled Fundamentum naturae quod videtur physicos ignorasse, recently discovered among the volumes of a little-known German Dominican from the fifteenth century. Fundamentum discusses Thierry’s four modes of being in terms reminiscent of Lectiones and Glosa, but organizes its account around a triad of matter, form, and connecting motion, recalling the trinity of perpetuals in Commentum. The arithmetical Trinity is never mentioned. Moreover, the author of Fundamentum is quite clear about his intentions not to transmit or revise Thierry’s modal theory but to reject most of it. The treatise is difficult to date precisely, but must have been composed between 1267 and 1440—that is, well after any other known medieval reader of Thierry. To date, Fundamentum has been studied only in terms of its proximity to Cusanus’s De docta ignorantia, since the two works share several pages of nearly verbatim text in

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common. I have analyzed this controversy elsewhere and explained my reasons for crediting Hoenen’s position that Fundamentum was not authored by Nicholas but rather used by him in the composition of De docta ignorantia.84 It is crucially important, however, to read the Fundamentum treatise on its own terms, and its author as an independent analyst of Thierry’s theological proposals, before considering how the text found its way into Nicholas’s book and what the German cardinal made of its ideas.85 More work remains to be done on the immediate occasion of the treatise’s composition and the contexts of its author. Yet we can already begin to situate the text by comparing Fundamentum to Thierry’s other known readers. This is not necessarily to suggest that the anonymous author had read Septem or Clarembald, nor to underestimate the centuries that lay between them. But knowing what other medieval Christian readers found distinctive, dangerous, or fruitful in Thierry’s theology can illuminate the details of Fundamentum’s particular interpretation. Both Septem and Clarembald revered Thierry as a venerable master, but the author of Fundamentum attributes Thierry’s ideas to “the Platonists” and suspects their orthodoxy even more than the anxious Clarembald. Like Septem and Clarembald, Fundamentum combines the three perpetuals and the four modes— not to link them to Hermes Trismegistus or to amend them with Augustine but rather to condemn them as theologically unsound. What has occurred in Latin Christian theology in the intervening two hundred years between Thierry’s students and Fundamentum, obviously, is the rise of Aristotelian scholasticism. Further research will have to determine the precise affiliation of Fundamentum’s author, but two things are certain even now. First, the author represents, to date, the only confirmed Aristotelian to confront Thierry’s Neopythagorean theology. His reaction is decidedly, if not surprisingly, negative. For this reason alone the terms of Fundamentum’s critique are historically valuable. Second, the author offers an astute analysis of the logic of Platonic mediation that we have tracked from Plato through ancient Neopythagoreanism into the twelfth century. So without yet mentioning the Cusan affair, the Fundamentum treatise remains one of the most significant documents of the controversial legacy of Thierry of Chartres.86 The author of Fundamentum never indicts Thierry by name, but the treatise’s title opposes itself to someone quite like the Breton master. The works concerns “the foundation of nature of which, it seems, the physici are ignorant.” The topic under consideration is prime matter (as the author later defines fundamentum naturae) and the author’s opponents are the natural philosophers (physici) who claim special insight into nature. But the actual content of the Fundamentum treatise is a systematic presentation of Thierry’s four modes, followed by a deliberately argued refutation of two of them. Hence the author conceives of the inventor of the four modes as a natural philosopher who misconceives nature (and thus incurs theological error) by attempting to understand it through a misguided Platonism.

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Thierry’s Genesis commentary interpreted the days of creation secundum physicam and discussed matter extensively. Septem’s author calls his master a physicus, and Clarembald had already worried about Thierry’s views of prime matter incurring theological error. So Thierry matches the author’s profile rather well. At the same time, the author of Fundamentum executes this critique by turning Thierry’s words against themselves. This means that although he clearly rejects the Chartrian doctrines, Fundamentum nevertheless transmits its distinctive vocabulary through its pages. A hasty reader might mistake this unrelenting deconstruction of Thierry’s thought for a summary of it.

Against Two Modes The Fundamentum treatise is organized in four sections. First the author irenically outlines “the four universal modes of being” without a hint of the criticism to come—an unpolemical preface that makes Fundamentum susceptible to misinterpretation.87 The first mode, absolute necessity, is God. It is “the form of forms, the being of beings, the reason or quiddity of things,” and in this mode all things are absolute necessity itself. The second mode is necessitas complexionis, and in it “the true forms of things exist in themselves, with the distinction and order of nature, just as in the mind.” The third and fourth modes treat determined possibility, in which things are “actually this or that,” as well as absolute possibility, in which things are able to be as such.88 Together these three lowest modes of being make up one universitas, according to Fundamentum.89 Each of the subsequent three parts of the treatise is devoted to one of them. The first part is devoted to possibilitas absoluta and concerns “the possibility or matter of the universe.”90 The second part is devoted to necessitas complexionis and concerns “the world-soul or the form of the universe.”91 The third part is devoted to possibilitas determinata and concerns “the spirit of connection or the power of the universe.”92 By the beginning of the third part, it becomes clear that this organization corresponds to materia, forma, and the motus or spiritus that binds them, that is, to what Thierry called the trinity of perpetuals. Thierry had seen no conflict between the fourth mode and the first; they both enfold all things in different ways.93 In the first part of Fundamentum the author contests this idea, arguing that the fourth mode has no existence apart from God, the first mode. According to the author, if possibilitas absoluta were to stand alongside the first mode of absolute necessity as a second absolute, it would compete with divine transcendence. His mode of argument against the fourth mode in this section relies on Aristotelian physics, draws on medieval Aristotelian handbooks, and formally resembles Aristotle’s critiques of Plato (and Pythagoreans) in Metaphysics. Like Aristotle, after reviewing the history of previous concepts of prime matter and possibility, the author contends that beings should be

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understood relative to other beings, not in terms of a separate absolute, and finds Thierry’s abstract concept to be incoherent. The author of Fundamentum wields a powerful conceptual tool against Thierry’s fourth and second modes of being, the notion of “contraction,” or delimitation from the absolute to a reduced sphere of being. Because the world is contracted, there can be no mediators between the world and God, since such mediations claim to enjoy an uncontracted status beyond the world but less than God. But in fact, for the author, God alone is “absolute” or uncontracted. This particular sense of contractio originated within the tradition of scholastic commentaries on the Liber de causis beginning with Giles of Rome around the turn of the fourteenth century.94 Contractio operates as the basic structural principle of the author’s cosmology. By his account the universe consists of infinite degrees of difference spread out along a homogeneous continuum. But each degree, however formally identical to the others, is itself a unique combination of act and potential, which are “contracted” by one another. Together these “differences and gradations” construct an ordered progression of beings in a continuum regressing symmetrically toward two opposed endpoints:  a minimum (matter, pure possibility) and a maximum (God, pure actuality).95 The cosmos is a graduated continuum of unique “degrees” of individuation that “gradually descends from the universal into the particular, and there it is contracted by means of a temporal or natural order.”96 The order of this stepwise descent is “conserved through its degrees” (servato per gradus suos).97 Hence all individuals are arranged uniquely—always singulariter, writes the author, and never aequaliter—on the continuum of discrete differences, precisely by virtue of their contraction.98 The absence of equality means that differences never repeat but relate serially in a measured progression, such that each added difference necessitates a new gradus. As soon as one accepts the author’s argument that all possibility is contracted, Thierry’s fourfold modal system collapses:  the fourth mode must either be elevated into the first mode or absorbed into the third. In effect, the author has eliminated any latent competition with divine transcendence by installing a stricter distinction between God and world: There can be no creature that is not diminished as a result of contraction, infinitely separated from the divine work. Only God is absolute; all others are contracted. Nor does a medium arise in this way between absolute and contracted, as those Platonists have imagined who thought the anima mundi to be Mind, after God but before the contraction of the world.99 When the fourth or second modes are treated as subdivine mediators between Creator and creatures, then the principle of contraction is violated and divine

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transcendence compromised. According to Fundamentum, those modes should either be identified with God (first mode) or deflated to contracted creaturehood (third mode), but they cannot hover in between after the fashion of the “Platonists.” The author mounts two strong arguments against the fourth mode of being, both of which draw inspiration from Aristotle. First, the fact that the continuum of contracted degrees is infinitely differentiated entails that there is no “simple maximum” of act or potential that can be separated from the continuum.100 Since every instance of possibility belongs within the continuum of degrees, there is no conceivable instance of a separated, “absolute” possibility. This calls to mind Aristotle’s arguments against separable forms throughout Metaphysics.101 The second argument alludes to Aristotle’s account of causation and chance in Physics.102 In a hypothetical world in which the fourth mode did exist, writes the author, all things would exist by random chance.103 For unless all possibility were contracted to a unique “difference and gradation,” there would be no reason why absolute possibility should produce any given being rather than another. The author of Fundamentum concludes that once scrutinized, the fourth mode of being leads either to God or to an absurdity. Either “absolute possibility” trivially restates the idea of God or it attempts to say something about the universe. But in the latter case, possibility truly cannot be absolute. For even if a separated, abstract possibility were to precede all actuality (the world), this would not make it identical with God, and its distance from God would remain defined as a contraction of the absolute. Only God’s eternity is absolute, and it is impossible to characterize possibility in similar terms coherently. In the author’s pithy summary: “absolute possibility in God is God, but outside of God it is impossible.”104 The second target of Fundamentum’s attack is the second mode of being. Just as Thierry devoted most of his time to the second mode, the author of Fundamentum devotes his treatise’s central part to exposing necessitas complexionis as another false mediator and another threat to divine transcendence. Again the author applies the concept of contraction in order to force a dilemma: one must reduce the second mode to the Creator or to actual creation and so remove all false mediation between them. But the author notes that the second mode goes by other names, such as mens or anima mundi.105 By enunciating these alternatives, Fundamentum’s author shows that he desires to do more than simply contest the semidivine autonomy of the second mode (as in the case of the fourth mode). Even while rejecting the notion of necessitas complexionis as such, he also wishes to identify its proper function, correctly understood, within the Christian theological economy. When the “Platonists” propose the notion of a second mode, the author believes that they name something real, but misrecognize it as the anima mundi. In his skepticism about the world-soul we already hear echoes of the twelfth-century defenders of “orthodoxy” who spoke against Abelard, William of Conches, and

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Thierry when they identified the anima mundi with the third person of the Trinity. In a similar manner Fundamentum now contrasts, as mutually exclusive alternatives, Thierry’s account of the second mode and Trinitarian orthodoxy. Thus the author declares: “the necessitas complexionis is not mens, as the Platonists suppose, lesser than the Begetter, but is the Word and the Son equal to the Father in divinity, and is called Logos or ratio, since it is the ratio of all things.”106 For Fundamentum’s author, Platonist philosophers interpose false mediators between God and the world because they do not grasp the contracted nature of creation. What they mistakenly call necessitas complexionis, he avers, is in fact the divine Word. But having replaced necessitas complexionis with Verbum, the author nevertheless defines the Word’s function in the same terms. Like the second mode, the Word unifies the forms through folding “as ordered into an order.”107 The Word is the location of the truth of forms, as the “maximal truth” and “truth of truths,” and is their ultimate ratio.108 Thus even if the Word assumes all the functions of the second mode, Fundamentum’s prohibition on mediating God and world will still hold. After all, in the mainstream Augustinian tradition, the mediating Word is itself God, and its divinity qualifies it to be the only true mediator. So the author of Fundamentum maintains that the Platonic project of mediation is not only unnecessary, since the Trinity fulfills the same functions without jeopardizing divine transcendence. It is also a sign of one’s ignorance of the Word. The Platonists’ misrecognition of the divine Verbum as a subdivine mediator is, according to the author, their fundamental error. “Indeed the philosophers have not been sufficiently instructed concerning the divine Word and the absolute maximum,” he writes. “Therefore they have considered the Mind and soul [of the world], as well as necessity, in a certain unfolding of absolute necessity without contraction.”109 Thierry of Chartres of course had intended complicatio and explicatio to operate as a reciprocal pair. But here Fundamentum’s author uses complicatio by itself to express the transcendence of the divine first mode.110 The text just cited is the sole instance of explicatio in the Fundamentum, and clearly it appears here only to indicate an undesirable error. The mistake of the “Platonists,” according to the author, was to have made room for an “unfolded” second mode outside of the singular absolute of God. Where Thierry had used reciprocal folding to provide a substructure for the four modes, Fundamentum rather deftly uses it to pit one mode against another. In the course of rejecting the second mode, the author has leaned on several of Thierry’s own ideas, from God as forma formarum to reciprocal folding. He is distraught at the notion of “plural exemplars,” an idea Thierry embraced in Lectiones, where he located them specifically in necessitas complexionis. The author is particularly bothered by the notion of a divine Mind (mens), which Thierry of course repeatedly lauded after finding it in Macrobius. Given the author’s facility for careful reading and his evident concerns about Trinitarian orthodoxy, his silence

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on one point is particularly striking. Fundamentum omits any mention of Thierry’s arithmetical Trinity or his doctrine of the divine Word as aequalitas. In several of the texts with which the author appears familiar, Thierry discussed aequalitas at length, and yet the author either overlooks or ignores such passages. Perhaps the formula was too Platonizing for the author, or perhaps he avoids Thierry’s triad because he is preparing to introduce his own alternative triad at the end of the treatise.111

Trends in Reception History Once the critical perspective of the Fundamentum treatise is fully appreciated, it seems increasingly implausible that the text could stem from Cusanus. Whatever his relation to the treatise, it is clear that Cusanus is enthusiastic about Thierry’s arithmetical Trinity and especially the doctrine of divine aequalitas. That concept drives Nicholas’s appropriation of Ps.-Dionysian negation in De docta ignorantia I, grounds the Christology of De docta ignorantia III, and becomes the subject of an entire treatise (De aequalitate, 1459). Cusanus relishes explicatio as much as complicatio, both of which pepper most of his philosophical writings over several decades. Moreover, he makes no indications outside of the suspect passages in De docta ignorantia II that he disapproves of Thierry’s four modes.112 As we shall see, he repeats the doctrine in later works. Finally, Cusanus’s thought, more than most, fits the description of the very Platonizing theology that Fundamentum set out to disassemble. Having listened carefully to the voice of our anonymous author, does it sound at all like the German cardinal? The author of Fundamentum dramatically contrasts two versions of mediation. One must choose either the eternal Word or the second mode, which in Thierry’s theology names the eternal mathematical order subtending the cosmos. The function in question is the same (the unification of plural exemplars, taken as forms or numbers), but the different agents entail different theological and discursive commitments. In this pivotal moment, when Fundamentum juxtaposes Verbum and necessitas complexionis, the treatise articulates with remarkable clarity the ongoing tension between Logos and Arithmos that we have witnessed over the centuries—in the plural mediators of Philo and Nicomachus, in Augustine’s debate with himself, and in Thierry’s attempt to perfect Bernard’s Platonism. Fundamentum thereby crystallizes the moments of confusion and hesitation over Thierry’s doctrines that we observed in Septem and Clarembald. I have found no evidence that the author of Fundamentum was aware of either precedent, but viewed together, the three sources form a consistent reception of the Breton master. As self-conscious interpreters of Thierry, the projects chosen by Septem and Clarembald both reflect the challenging question that he had left unanswered. How do the new models of mediation theorized by Thierry relate to

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the mediation of the Word? Septem used Thierry’s perpetuals alongside Hermetic principles to explore multiple, complementary mediations between God and the cosmos. But he also based his evaluation of the quadrivium on the supremacy of the Word. Clarembald singled out the Word as the sole mediator and took special pains to demote the duplicative second mode to something approximating Augustine’s seminal reasons. Similarly the author of Fundamentum exposes the awkward disjunction between Thierry’s theology of the Word and Thierry’s modal theory. Such formal continuities with the texts of twelfth-century interpreters indirectly support Hoenen’s contention that Fundamentum naturae stands as an independent text in the succession of Thierry’s interpreters. The treatise addresses many of the same themes raised by Clarembald and Septem but outstrips them in its sophisticated reading and elegant critique of Thierry’s modal theory. The very fittingness of Fundamentum’s response to theological problems lingering in Thierry’s wake argues for its autonomous existence within the Wirkungsgeschichte of the Breton master. First, the comparison to Septem and Clarembald sheds light on the title of the anonymous treatise:  Fundamentum naturae quod videtur physicos ignorasse. Since Septem attributes Thierry’s three perpetuals to the physici and later attributes the same triad to magister meus, it seems that he counts Thierry himself as a physicus, just as he believes Heraclitus the physicus to have been a source of Thierry’s doctrine.113 The correct theological interpretation of the principles of creation (fundamentum naturae) stood at the center of Clarembald’s Genesis commentary. Both Septem and Clarembald are frustrated with the ignorantia of their opponents. According to Clarembald, ancient heresies sprang “out of ignorance about the creation of things,” specifically a failure to grasp the sovereignty of the divine Verbum.114 Likewise Septem predicts that philosophers will never find wisdom so long as they remain in spiritual ignorance. He belittles the “exhausted minds” of those whose studies have taught them nothing of the true principles of nature, “because their mode of inquiry and their ignorance of the truth obstructs them.”115 Furthermore, Septem and Clarembald can help to illuminate the structure of Fundamentum. Both of them combined Thierry’s three perpetuals and four modes, something the Breton master had never done. In Septem the four interlinked modes are transformed as three principles descending from the first principle: an eternal principle of necessity from which three lesser principles descend (matter, form, and spirit; or possibility, finality, and actuality). Similarly, in Clarembald’s Augustinian reading, absolute necessity is identified with the divine Word and the lower three inchoatives correspond to the Word’s creatures (primordial matter, seminal reasons, and creaturely motion). The author of Fundamentum combines Thierry’s doctrines in much the same way. The treatise is organized to link matter, form, and motion respectively with each of three modes, which considered

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Table 6.3  Consistent trends in Thierry’s medieval reception history Septem Highest mode Necessitas Aeternitas Lesser modes (collectively)

Trinitas perpetuorum

Lesser modes Materia (distinguished) Possibilitas Forma Finalitas

Clarembald

Fundamentum

Verbum et Filius Providentia

Necessitas absoluta Deus

Tria inchoativa

Una universitas

Materia primordialis Possibilitas absoluta Informitas Possibilitas seu materia Ratio seminalis Forma

Necessitas complexionis Anima seu forma

Spiritus creatus Principium temporis Possibilitas determinata Actualitas Motus creaturae Spiritus conexionis

together make up “one universe.” Like Thierry’s prior readers, Fundamentum places the first mode above the lower three as God stands over the universe (see Table 6.3). Against the spirit of Thierry’s theology, all three authors mark off the first mode as uniquely divine, treating the latter three as lesser modes of creation. Where Septem and Clarembald count three sublevels below God, Fundamentum even more sharply separates God from creation by collectively naming the lowest three modes “one universe.” Thierry of course had consistently held that all four modes together comprise “one universe.” The whole point of the four modes was to underscore that what theology knows is the very same thing (universitas rerum) apprehended by other disciplines, just in a different way (modus).116 By combining the three perpetuals and four modes, Thierry’s readers work in perfect opposition to his intentions by dramatically intensifying God’s transcendence beyond the physical world of matter, form, and motion. The author of Septem appears oblivious of such effects amidst his enthusiasm for immanent divine principles, but Clarembald welcomes this side effect of the new Augustinian basis he gives to Thierry’s modal theory. The author of Fundamentum, however, takes on the remotion of the divine principle as the very raison d’être of his treatise. To worry about the perpetuity of prime matter, to oppose natural philosophy and orthodox belief, or to impugn the theological blindness of philosophers— these are common gestures in Christian thought influenced by Augustine. But

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they are not features of Thierry of Chartres’s understanding of creation. Thierry’s own Tractatus on Genesis, by contrast, proposes to read Genesis secundum physicam. His invocation of “philosophers” is always sanguine, commending to the reader the wisdom of Pythagoras, Plato, Aristotle, or Boethius. And at the very center of his theological interests is a fascination with mediators:  fate, nature, the world-soul, the nexus of unfolding and enfolding. Where Thierry searched for divine activity within the fabric of the created order—in causes, in numbers, in the very structure of necessity—his later readers rearranged (or in case of Fundamentum, disassembled) the balanced structure of his mathematical theology in order to elevate God beyond the need for any Platonic mediation whatsoever. This was done not only out of their common concern to safeguard divine transcendence, but because they detected in Thierry’s writings a nascent theology of the divine Word that, in their view, should have been more central. Thierry’s earliest students, such as the authors of Commentarius Victorinus or Tractatus de Trinitate, wanted to lift out in relief any statement on aequalitas that might promote a Johannine theology of the Word as mediator. Septem assembled a motley profusion of divine mediators, but also retains a Johannine focus on the unifying via et veritas of the divine Word, using Thierry’s triads to give order to the range of ancient terms. Clarembald affirmed Thierry’s modal theory but underscored, with Augustine, that the divine Word is the exclusive source of the rationes seminales. Clearly the author of Fundamentum only intensified this same line of questioning. Students of different persuasions and centuries thus identified the Word’s mediation of the exemplary forms as the critical issue looming over Thierry of Chartres’s theological legacy. Given the fragmentary nature of the evidence, this consensus is remarkable in its continuity. But equally surprising is the way in which Nicholas of Cusa, with many of the same sources in hand, overturned this reception history within the span of one book in 1440. Suddenly, Thierry’s doctrines did not threaten a theology of the Word, but, on the contrary, impelled Cusanus to draft one of the most creative Christologies in the history of medieval Europe.

PART THREE

Bright Nearness Nicholas of Cusa’s Mathematical Theology

La généalogie est grise; elle est méticuleuse et patiemment documentaire. Elle travaille sur des parchemins embrouillés, grattés, plusieurs fois récrits. Michel Foucault, “Nietzsche, la généalogie, l’histoire”

7

The Accidental Triumph of De docta ignorantia In early February 1438, Nicholas of Cusa returned to Venice by ship in the company of an esteemed delegation from Constantinople that included the Emperor John VIII Palaeologus, the Patriarch Joseph II, and Metropolitan Bessarion.1 Pope Eugenius IV had entrusted Cusanus and his colleagues the previous year with an historic and urgent mission: to open a new dialogue with the eastern churches, on the terms of the papacy rather than those of the conciliarist party. Nicholas and others had allied themselves with the papacy following the disaster of the Council of Basel. The conciliarist movement had devolved from a progressive reform platform into a shouting match, quite literally in May 1437, when the council split into two factions that promptly read out competing decrees over each other. Such intramural discord would make the reunion of the churches a practical impossibility. But even amidst this all-important trip—which did temporarily unify Christendom in 1439—Cusanus found time to pursue his intellectual passions. He made good use of his stay in Constantinople to hunt for new manuscripts, uncovering works of Basil the Great, John Chrysostom, Plutarch, and Proclus, not to mention a rare Qur’an translation, all of which he ferried home. One can imagine that during the three-month return voyage, despite its unusually cramped conditions, Nicholas might have opened some dialogues of his own, perhaps even with the future cardinal Bessarion, who emigrated permanently in 1443.2 Cusanus had been selected for the delegation as much for his political acumen as for his interests in Greek philosophy, and the journey turned out to be a pivotal moment in Nicholas’s life in several regards. He turned from conciliarism and advocating German rights to papalism and the international platform it offered, but he also set aside works on canon law and poured himself into theological and philosophical pursuits. The trip manifested a pattern that would mark the rest of his career. Even while embroiled in heated and sometimes violent legal disputes in the affairs of the church, he always found time to collect, read, and write. The son of a Rhineland merchant, Nicholas had risen within a few years from obscurity to engage the most prominent intellectual and religious leaders of his

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lifetime. He was no detached thinker in a university or monastery, leading a life of quiet study, but thrived on the political scene and never hesitated to exert the powers of his office vigorously. In 1428 and 1435, the University of Louvain offered him a chair in canon law, and twice he turned it down. Nicholas was the most prominent German official to align himself with the papacy and the only German papal nuncio advocating for Roman interests within the kingdoms in the 1440s. This notoriety won him rewards from the papacy but also several attacks on his life that led to his departure for Italy in the late 1450s. After serving as papal legate, he was made cardinal in the late 1440s, then bishop of Brixen in the Tyrol in 1450, before serving out his last years as papal vicar in Rome. (In the papal conclave of 1447, Nicholas reportedly received a few votes for the office himself.) From the 1440s onward he traveled almost incessantly, spending as much time in transit as he did at destinations—a habit that marks his thinking as well. His retinue included dozens of secretaries and support staff as he received diplomatic assignments of increasing complexity. The ruthlessly optimistic reforms that, as bishop, he often forced beyond the breaking point seemed to work better in theory than in practice. Only in the sphere of thought could the mind peer beyond the opposing perspectives that drove ecclesiastical governance.3 The same energetic, demanding cardinal was known to humanists primarily as a manuscript hunter and bibliophile.4 Cusanus won early fame in Italian circles for uncovering twelve lost comedies of Plautus, but ever the prudent lawyer, he controlled the copies himself to heighten their value. In ecclesiastical courts he surprised opponents by producing fresh manuscript evidence that he had personally retrieved from dusty codices. The humanist Aeneas Sylvius Piccolomini, future pope Pius II, related with amusement how this technique unnerved those who challenged Cusanus at the Council of Basel. As a young man in Cologne, digging through the cathedral library, Nicholas had unearthed the royal chronicle Codex Carolinus, as well as proceedings from ancient councils; in Laon in 1428, he uncovered the famous Libri Carolini. He put several of these sources to use in De concordantia catholica, his magisterial defense of Basel’s concilarism in 1433. His annotated copies of Plato, Aristotle, Ps.-Dionysius, Proclus, Eckhart, and Llull still reside on the shelves of his library in Bernkastel-Kues today.5 Reading these marginal notes makes one wonder what Nicholas might have inscribed in his own copies of Thierry’s Boethian commentaries, had they survived, and to search, in their place, for tacit remarks hidden between the lines of his own texts over three decades. After returning from Constantinople in February 1438, Cusanus slowly made his way back to the Rhineland the following year. By February 1440, he had quickly completed his first major philosophical work, De docta ignorantia. In a prefatory letter to the eminent cardinal Julian Cesarini, one of his chief patrons and a fellow dissenter from Basel, Nicholas wrote that he had been inspired to write the book

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during that long sea voyage home. A flash of divine illumination, he says, revealed a wholly new philosophy to him.6 Yet “learned ignorance” is not the only insight discussed in that work. Nicholas also covers all of the major elements of Thierry of Chartres’s theology: the quadrivium as a guide to creation, the mathematical Trinity, the triad of perpetuals, reciprocal folding, the four modes of being, and God as the form of being. He praises Pythagoras and Boethius by name as paradigms for philosophy, and he explains in detail why mathematicals provide the surest pathway to contemplating God. After centuries of silence, the mathematical theology of the Breton master suddenly washed ashore in a great resounding wave, as if following Cusanus home from his long sea voyage. Since Nicholas was remembered in subsequent centuries as a great interpreter of Ps.-Dionysius, his account of mystical insight seemed entirely apt. It also seemed almost necessary if one were to explain, as scholars have since struggled to do, how a busy lawyer with only occasional philosophical training quickly produced the wealth of Platonist theology on display in De docta ignorantia.7 As Hans Gerhard Senger observes in his study of early Cusan writings, “the fact that a 40-year-old suddenly came up with such a well thought-out and substantive philosophy is surprising.”8 Nicholas’s story also worked, perhaps not accidentally, to conceal the origins of his most recondite sources, as well as the fact that he was in fact, revelation or not, exceptionally well placed to encounter authentic Neopythagorean ideas. Like the early humanists Coluccio Salutati, Leonardo Bruni, and Poggio Bracciolini—not to mention Pico della Mirandola and Marsilio Ficino—Cusanus praised Pythagoras as a preeminent Greek philosopher and treasured all of his Platonic sources.9 Nicholas kept in contact with Bessarion, who sent him his own translations of Aristotle and with whom, after the example of the renowned Georgios Gemistos Plethon, he shared the conviction that Plato, Aristotle, and Christianity could be harmonized.10 But Cusanus stood out from many of his contemporaries not only for his rapid career ascent and passion for old books. The peculiar education that he had received put him in contact with previous moments in the history of mathematical theologies in a way that differed markedly from other contemporary Platonists, even those avowedly interested in Pythagoreanism. Unlike most other medieval theologians one might name, Nicholas did receive some modest but concrete mathematical training.11 After a year at the nearby University of Heidelberg, the promising seventeen-year-old studied for eight years at Padua, the leading center of legal studies in Europe. But Padua was also famed as the principal center for the Italian Renaissance of mathematics.12 Whether training as a physician like Paolo Toscanelli, a painter like Leon Battista Alberti, or a litigator like Cusanus, intellectual elites in fifteenth-century Italy had begun to view mathematical fluency as part of the necessary equipment for advanced learning, the lingua franca of erudition in the new age.13 At Padua Cusanus studied

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under Prosdocimo de’ Beldomandi, the leading authority on the quadrivium at the university and a great champion of Boethian studies.14 He remained in contact with Toscanelli for the rest of his life. Knowing the tastes of his German friend, Toscanelli sent Cusanus a new translation of Ps.-Dionysius’s Mystical Theology in 1443; two years later, Nicholas dedicated his first geometrical work to him in return, and then wrote him into one of his mathematical dialogues in 1457. Cusanus would go on to write a dozen different works in pure geometry aimed at solving the squaring of the circle. The great German mathematician Regiomontanus, a contemporary of Cusanus and friend of Toscanelli, quipped in a letter of 1471 that if the dilettantish cardinal must be considered a geometer, he was a geometra ridiculus. His teacher Georg von Peuerbach was also skeptical of the quality of Nicholas’s proofs, a judgment shared by historians today.15 Cusanus enjoyed a unique level of access to the major currents of Neoplatonist theology in the early fifteenth century. In 1425, the young lawyer treated himself to a year studying philosophy at the University of Cologne before taking on his first cases. Remarkably this was his only advanced training in philosophy. During his stay the junior professor Heymeric de Campo, just a few years older than Nicholas, became an informal mentor and tutor. At Heymeric’s advice, Cusanus traveled to Paris in 1428 to hunt down some manuscripts of Ramon Llull and other Neoplatonists.16 We are still unsure as to the extent of his discoveries on this trip; they may have included works of Thierry of Chartres. In any case, studying under Heymeric’s guidance must have struck Cusanus as a revelation, because the avant-garde style of philosophy headquartered at Cologne marked the young Nicholas for the rest of his life. Heymeric’s controversial agenda was to strengthen the Albertist school at Cologne as an antidote to the regnant nominalism at Paris. This meant reading the Aristotelian tradition through Neoplatonist lenses (mainly Ps.-Dionysius and the Liber de causis) after the example of Albert the Great.17 In his own Basel conciliarist treatise, written the same year as Nicholas’s De concordantia catholica, Heymeric found arguments against papal supremacy in the disciplines of arithmetic, music, geometry, optics, and astronomy.18 He designed a geometrical diagram of the Trinity to ward off the Hussite heresy.19 Heymeric thus modeled for Cusanus a way to hold together an interest in the quadrivium, Proclian henology, and theologies of Trinity and Christ.20 Through other channels Nicholas would later become familiar with other classics of Neoplatonism such as Proclus’s commentary on the Parmenides, as well as the works of the great Eriugena, his intellectual cousin in the Boethian tradition.21 The peculiar shape of Nicholas’s intellectual formation tells us two things. First, whenever and however Cusanus happened upon the manuscripts that transmitted Thierry of Chartres’s ideas, he was extraordinarily well equipped to appreciate their value. Few contemporaries could have been better prepared to make use of Thierry’s peculiar mathematical theology than was Cusanus. Second, Nicholas’s

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education had put him in touch with the currents of late medieval thought that had partially preserved authentic Neopythagoreanism—the Boethian quadrivium in Padua, as well as the Proclian and Ps.-Dionysian streams of Neoplatonism in Cologne. Once Thierry’s reinvigoration of Augustinian and Boethian fragments was added to his repertoire, Cusanus had the full range of latent Neopythagorean theologies at his disposal. From this perspective, we must qualify what it means to call Cusanus “Pythagorean.” In the first place we are now well acquainted with the range of views the term can designate. But there is a broader point worth making at the outset with respect to Nicholas’s fifteenth-century context. Cusanus is not Pythagorean in a nostalgic sense; he shows little interest in the man Pythagoras or the ancient sect, in Pythagorean physics or in arithmology. Instead, he was accidentally in contact with the Neopythagorean theologies that had survived into his lifetime under a different guise, Boethian or Proclian or Chartrian. Precisely because Nicholas was less visibly interested in Pythagoreanism as such—in contrast to Pico, Ficino, or Reuchlin—he was all the more authentically Neopythagorean. His is not an akousmatic Pythagoreanism repeating mantras from the past, but a distinctly mathematic Pythagoreanism in the ancient sense that retains only whatever goods can be dusted off and reused within contemporary learning. He praises Pythagoras, but he also doesn’t hesitate to brand his own work an inquisitio Pythagorica.22 The difference here is that between a conscious reference to the past which, given the historical distance that the reference relies upon for its novelty, must remain fundamentally ironic, and the authentic retrieval that is all the more truthful because, unwittingly, it is not a reference at all but rather an embodiment. It is what separates the trendy, fashionable revival from the real article, decades old, worn and neglected at the back of the closet. Or the Florentine humanist affecting an imaginary Greekness at his desk from the Rhinelander who endured the cold sea voyage to Byzantium and back again. The weighty tome of De docta ignorantia draws on a broad assortment of ancient and medieval sources. But as Cusanus boasts in his letter to Cesarini, his book wrests from them something exceptionally fresh and new, far from the well-worn paths of the philosophers. In the letter, Cusanus also explains the structure of the three books that comprise De docta ignorantia. Book I  discusses the simplicity of coincident contradictories, or the absolute divine maximum; Book II discusses the universe as the “contracted” maximum restricted to the conditions of finitude; and Book III discusses Jesus, the maximum who is uniquely both absolute and contracted.23 Interpreters have struggled to do justice to all three books, so crammed are they with explorations in a dizzying array of disciplines: from geometrical figures as instruments for contemplatives, to the relativity of motion and the possibility of life on other planets; from the abstract dialectics of unity, plurality, and alterity, to new models for understanding the unity of the church, the

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bodily resurrection, and the final judgment. Is this a work of theology, cosmology, or epistemology? But perhaps the greatest conundrum facing the reader is how to reconcile the cardinal’s extraordinary investment in Pythagorean mathematics in the first half of the work with his extraordinary theory of the Incarnation in the second half. What has Athens to do with Jerusalem? The recent discovery of the Fundamentum treatise lurking at the center of De docta ignorantia further complicates such efforts to identify the book’s uniqueness and to rank its constitutive parts. If there is any truth to Hoenen’s contentions, it would seem that Nicholas’s signature concepts of God as maximum and the world as contracted might be partially derived from this anonymous author, and that Fundamentum was the major vehicle for the Chartrian ideas appearing in Book II.24 But here two simple observations become crucial. First of all, Cusanus borrows from several different sources that hand down Thierry’s theology, not only from Fundamentum, already in Book I. He combines Thierry’s own works with those of his twelfth-century students, and both with the critique of Fundamentum. If it is not always exactly clear which work the cardinal had on his desk, it is beyond doubt that he was actively collating multiple Chartrian sources into his prose. All of the sources in question, however, have survived anonymously, while sharing distinct family resemblances. We can now attribute the original ideas to Thierry of Chartres, but from the cardinal’s perspective, he had before him a collection of like-minded, anonymous Boethian commentaries that shared an interest in the quadrivium and a vibrant, if unusual, Platonist lexicon. The manner in which Cusanus organizes such Chartrian sources strongly suggests that he viewed them as the product of a single author, whom (as I have argued elsewhere) he singled out for high praise in 1449.25 Cusanus grasped the common elements belonging to Thierry’s oeuvre, even if each element does not appear in every text, and he discerned a common theological voice, even if he never knew Thierry’s name. Second, even if the Fundamentum discovery helps to explain the provenance of maximitas in Book I and contractio in Book II, it does little to explain the destination toward which the whole of De docta ignorantia so deliberately aims: the novel Christology of Book III.26 The mystical theology of the Incarnation outlined in that final book has no parallel in Cusan writings before 1440, and yet versions of it reappear in his treatises and sermons throughout the rest of his life. For these reasons, in order to understand how Cusanus encounters Thierry of Chartres’s theology in De docta ignorantia it is not enough to list and expound the several doctrines he borrowed from the Breton master, nor to compare their uses of the terms they shared. Such methods can insinuate that Cusanus enjoyed the same perspective we presently enjoy on the identity, sequence and contexts of Thierry’s works. Rather, a more patient and critical assessment is required that does not underestimate the tremendous distance separating the two men. When considering which Chartrian traditions Cusanus may have used in a given line of

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De docta ignorantia, one cannot simply list textual possibilities in the abstract, as if all of Thierry’s commentaries are so many versions of the same work. Rather, we ought to take into account the diachronic differences among Thierry’s works and his reception history, and thus re-examine details of Cusanus’s compositions that might indicate his limited understanding of different Chartrian texts. These caveats hold especially true with respect to Book II and Cusanus’s doctrines of the four modes of being and reciprocal folding therein. Previous studies that have not adequately appreciated Thierry’s influence have also overlooked the legacy of those doctrines in later Cusan works.27 Since Hoenen’s discovery, discussions of Book II need to contend not only with complexities of Thierry’s influence, but also with the possibility that Fundamentum mediated between the Breton master and the German cardinal. That is, it is no longer tenable to maintain without further ado that Nicholas received Chartrian ideas about the four modes or folding in De docta ignorantia directly from Thierry’s works.28 Recent studies of Book II that neglect the Fundamentum hypothesis face challenging questions.29 Those who have addressed it have usually raised new objections, or have assumed prematurely that the matter is resolved.30 By contrast I will suggest in what follows that only in light of the Fundamentum treatise can the legacy of Thierry’s achievement in its Cusan afterlife be fully understood. We have already studied the genesis, sources, and reception of Thierry’s concepts above, stressing their gradual development from Tractatus to Glosa and their interpretation from Septem to Fundamentum. Now, in order to understand the reemergence of such Chartrian ideas in Cusan works, we must examine how Nicholas played the role of an editor with imperfect textual knowledge, weaving his sources together within De docta ignorantia with greater or lesser clarity, experimenting freely with Thierry’s concepts well beyond their original context, and often radically transforming them toward new intellectual ends some three centuries later. Finally, we will have to confront the lingering riddle of how and why Cusanus made the leap from Thierry’s dense mathematical theology to an unforeseen Christological conclusion.

A Patchwork of Conflicting Sources When Nicholas sat at his desk to compose the final version of De docta ignorantia, he evidently had several Chartrian sources before him. Some modern readers have too quickly assumed that the learned Cusanus would only have used the best versions of Thierry’s writings and the sharpest formulations of his ideas. Some continue to treat Thierry’s different commentaries as essentially alternative versions of the same book (a view I worked to refute in Chapters 4 and 5) and so have attributed unwarranted textual knowledge to Nicholas in abstraction from his

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actual words and Thierry’s actual works.31 A more careful sifting of Book I of De docta ignorantia reveals a few surprising truths. To begin with, the text that Nicholas consulted as his primary guide to navigate his collected Chartrian sources was not Thierry’s mature Lectiones or Glosa, but the second-hand Hermetic tract De septem septenis. However poor a choice this seems to us in light of Thierry’s development and reception history, Cusanus had his reasons. Only in Septem could he have found the three major doctrines that his other sources had variously mentioned all in one place: the arithmetical Trinity, the trinity of perpetuals, and the four modes of being.32 In this way Septem’s earnest conflations became the blueprint for the cardinal’s own reconstruction in Book I. He cites Septem’s praise of Pythagoras at the beginning and end of his own retelling of the arithmetical Trinity (I.7–10).33 When Cusanus turns to geometrical figures (I.11–23), he pivots off the same text.34 Finally, as we shall see, Septem’s homage to the “trinity of perpetuals” will guide Nicholas’s approach to Fundamentum in Book II (II.7–10).35 When Cusanus reinstates Thierry’s arithmetical Trinity in Book I, it is the first authentic revival of the doctrine in centuries, perhaps since Achard of St. Victor. Yet even here, Nicholas decides against citing Augustine verbatim, or Septem’s quotation of Augustine, or indeed any of Thierry’s versions passed down in the Boethian commentaries.36 Instead he attempts to present the triad as the result of his own reasoning, reconstructing the argument of Thierry’s Genesis commentary. But this is an unfortunate choice, since Tractatus is the sole text that lacks an account of conexio.37 So when Cusanus reaches this term, forced by his editorial decisions into uncharted territory, he speculates that unitas is the causa conexionis—a term that Thierry and his students would not have used.38 Besides Septem and Tractatus, Cusanus draws on other Chartrian traditions. From Commentum he repeats Thierry’s creative etymology of unitas from divine entitas.39 He states that God is the forma essendi or the precise aequalitas essendi of things.40 In another odd move Nicholas especially favors the student treatise Commentarius Victorinus. He repeats the author’s comparison of the Trinity to a threefold ostension and culls a reference to Parmenides.41 It may even be that the cardinal’s larger strategy in De docta ignorantia of connecting the Chartrian theology of the quadrivium with the Ps.-Dionysian theology of negation takes its cue from Commentarius Victorinus. After stating Augustine’s triad, the anonymous student author notes that Hilary of Poitier’s term infinitas is a negative name for God, which leads him to contrast “negative theology” and “affirmative theology” in Ps.-Dionysius.42 Likewise, Cusanus cites the same Hilarian term alongside Augustine’s triad in his discussion of Ps.-Dionysius (I.24–26).43 Having scrutinized the text of Book I  of De docta ignorantia, we can surmise that Cusanus had several different sources at his disposal. He certainly had the rare works De septem septenis and Commentarius Victorinus, as well as

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Thierry’s two best-circulated commentaries, Tractatus and Commentum. The mature works Lectiones and Glosa are thus far not central to the Cusan retrieval of Thierry’s theology, nor are the writings of Clarembald of Arras.44 Whatever accident led Cusanus to his trove of Chartrian manuscripts, it is a relatively robust collection, even before we add the Fundamentum treatise.45 One wonders what Cusanus made of them. As Gilson has written, “one cannot even imagine how completely out of date a lecture by Thierry of Chartres would have sounded at the thirteenth-century Faculty of Arts at the University of Paris”—let alone in the fifteenth century.46 We can catch a glimpse of the cardinal’s editorial mindset toward his arcane sources in I.11, where he attempts to situate his Chartrian materials in the previous chapters in historical terms. To understand the extraordinary value of this passage it is helpful to compare a similar statement the cardinal would make in Apologia doctae ignorantiae (1449). In this treatise, defending the project of De docta ignorantia against attacks, Cusanus refers to the author of a passage from Fundamentum as “a commentator on Boethius’s De trinitate—easily the most brilliant man of all those whom I  have read.”47 Notably, this statement appears in a book in which Nicholas acknowledged his debts to Ps.-Dionysius, Meister Eckhart, and other luminaries. Surely with this exorbitant praise he is not merely crediting his source for Book II, but for the arithmetical Trinity in Book I, the supreme henological mystery passed down (as Septem suggests) from Pythagoras to Augustine. Moreover, the title “commentator on Boethius” makes sense only if Cusanus intended to name not only Fundamentum, which of course has nothing to do with Boethius, but instead the whole set of Chartrian manuscripts in his possession, including Commentum and Commentarius Victorinus. In light of this retrospective remark in 1449, Cusanus’s comments in I.11 take on a new aspect. First, they represent his reflections on the diverse Chartrian sources that he has just cobbled together in the preceding chapters I.7–10. Second, they stand as our most robust record of how Cusanus understood the historical moment of the theology of Thierry of Chartres: And so in mathematics the wise ingeniously sought examples of things that the intellect was to investigate, and none of the ancients who are regarded as great undertook difficult questions by any other than mathematical likenesses. Thus, Boethius, the most learned of the Romans, maintained that without some training in mathematics no one could attain a knowledge of divine things. Did not Pythagoras, the first philosopher in name and deed, locate every investigation of truth in the study of numbers? The Platonists and our own major thinkers have followed him to such a degree that our Augustine, and later Boethius, declared that of the things to be created number was undoubtedly “the principle

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mathematical theologies exemplar in the mind of the Creator.” . . . Also the Platonist Aurelius Augustine turned to mathematics for assistance when he investigated the quantity of the soul and its immortality and other very profound subjects. Our Boethius seemed to be so pleased with this method that he constantly asserted that all true doctrine is contained in multitude and magnitude.48

If, like Cusanus, one were lacking historical particulars about Thierry, there could scarcely be a better portrait of the Breton master’s theological significance, as we have seen. Cusanus situates his Chartrian sources in a lineage from Pythagoras to the Platonists, and then to the Christians Augustine and Boethius.49 He recognizes Augustine’s early Pythagorean proclivities. He alludes to the chapters in Institutio arithmetica that passed along the Nicomachean vision of arithmetic as first philosophy and numbers filling the mind of God. Boethius and Augustine, of course, were the titans that Thierry felt bound to reconcile along the way to his own theology of the quadrivium. Such historical awareness by the German cardinal is already remarkable.50 But Nicholas also signals the direction of his thinking toward a Christian Neopythagoreanism through two further allusions. From Commentarius Victorinus he learned that when Augustine was confronted with the ineffability of God, he found refuge in mathematics.51 From Albert the Great’s commentary on the Metaphysics, he learned to connect Pythagoras and Boethius as partisans of the Platonists against a narrow Aristotelianism.52 These two allusions state precisely what the cardinal has in mind for his Chartrian sources. In Book I, Cusanus will use a Pythagorean understanding of mathematics, enunciated in his opening pages, as an instrument for coping with the conditions of Ps.-Dionysian negative theology.53 In Book II, he will use Thierry’s Pythagorean ideas to mitigate the Fundamentum’s rejection of “the Platonists.” Even the neat phrase “our Boethius” intimates the different layers of historical appropriation that Cusanus has set in motion: his identification with a long heritage of Christian Pythagoreanism, an enthusiasm he shares in common with the Chartrian commentaries from which he is building his own book, and a special solidarity with his “brilliant commentator,” Thierry, who paved the way for Nicholas to rediscover the Boethian tradition of mathematical theology. With this perspective on the Chartrian sources informing Book I, we can now compare how Cusanus composed Book II. It is immediately evident that Cusanus draws on Fundamentum in II.7–10 in a manner formally similar to his collection of Chartrian sources in I.7-10. This striking parallel leads to others. In both books Cusanus’s preliminary exposition of the maximum (I.2–6) or contraction (II.2–6) prepares the way for his major source—whom he believes stems from the same “brilliant” author—which then provide the springboard for the cardinal’s most

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unique meditations, whether on the role of geometrical figures in mystical contemplation (I.11–23), or the relativity of the earth in an infinite universe (II.11–12). Thus a definite structural pattern begins to emerge:  Cusanus gestures toward his source, then deploys it over a series of related chapters, then builds upon it through his own experimental meditations.54 I.1 

Prelude to Book I: on negation (learned ignorance) I.2–6 Cusan exposition: on the maximum I.7–10 Source material: Chartrian sources on the arithmetical Trinity I.11–23 Cusan development: geometrical meditations I.24–26  Postlude to Book I: on negation (Ps.-Dionysian naming) II.1 

II.13

Prelude to Book II: on the quadrivium II.2–6 Cusan exposition: on contraction II.7–10 Source material: the Fundamentum treatise on the four modes II.11–12 Cusan development: cosmological meditations Postlude to Book II: on the quadrivium

If we imagine Cusanus as a solitary genius struggling to record a divine illumination, the different voices at play in his book can begin to threaten the work’s integrity. But if we view him as a clever editor of a medley of Chartrian traditions—even of traditions of mixed quality that he sometimes misunderstands—then a profound order becomes visible. Cusanus organizes Books I and II around two parallel bodies of Chartrian texts and the two triads that he found within them. In Septem the cardinal discovered his key, since that text fuses the arithmetical Trinity and trinity of perpetuals to a muddled version of the four modes of being. In Tractatus the arithmetical Trinity is linked to the quadrivium and number, in Commentum to the three perpetuals, and in Commentarius Victorinus to the Ps.-Dionysian “theology of negation.” In Fundamentum Cusanus finds the four modes more distinctly without the arithmetical Trinity but still organized around the three perpetuals. The Fundamentum treatise also contained two further concepts not found in the other Chartrian materials: the maximum and contraction. Neither in Book I nor in Book II does Cusanus feel burdened to cite the Chartrian sources that he quotes verbatim.55 Cusanus’s master plan for De docta ignorantia is brilliant, but it is also entirely conceivable given his short list of source texts. Book I uses the maximum as a heuristic vehicle for considering the arithmetical Trinity. Book II uses contraction as a vehicle for examining the four modes of being discussed in Fundamentum. Septem is used to introduce both sets of source material (I.7, I.10, II.7), since it alone coordinates them all. Well trained in rhetoric, the cardinal begins and ends each book with the caesura of a common theme: negative knowledge of God in Book I (following Commentarius Victorinus), and the

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quadrivium in Book II (following Tractatus). Finally, as I  will explain shortly, since Cusanus believed that all of his sources stemmed from the same “commentator,” it made perfect sense for him, as a grateful reader, to make his own contribution by harmonizing the leitmotifs of Books I and II into a concluding Book III. Unveiling the deep structure of De docta ignorantia suggests that Thierry of Chartres, his sources, and his legacy were not one historical influence among others, like Ps.-Dionysius or Eckhart, but in fact provided the indispensable architecture of Cusanus’s new theology in 1440, upon which other sources were draped to ornament the edifice further. This is indeed the kind of intellectual debt that would merit the cardinal’s hyperbolic praise, a decade on, of the unnamed “commentator on Boethius’s De trinitate—easily the most brilliant man of all those whom I have read.”56 This synopsis, moreover, allows us to discern the areas of Cusanus’s truly original contributions in light of Hoenen’s Fundamentum discovery. Our diachronic analysis of Thierry’s mathematical theology in its genesis and reception has thus revealed the inner structure of De docta ignorantia. Now that structure in turn will allow us to perceive the topics that caught the cardinal’s attention and held it for two decades.

Experiments in Chartrian Theology If Bernard, William, and Thierry were scientists (physici) because philologists (grammatici), Cusanus was likewise a kind of an experimental scientist of the concept, an alchemist of the divine names, who tested the properties of his mercurial concoctions. So it is often less important to define exactly what his newly fabricated concepts mean than to discover why and how he combined their ingredients together in the first place. There are seven areas in which Nicholas experiments with his Chartrian sources in De docta ignorantia. Four are already evident:  the arithmetical Trinity and the geometrical figures in Book I; the cosmological speculations in Book II; and the Christology of Book III. The first three have been discussed by many others; the last I will address in detail below.57 Yet there are other moments of novelty in De docta ignorantia in which Cusanus does not simply mobilize his sources, but instead tinkers with their components before our eyes. As he struggled to make sense of his collection of Chartrian texts, Nicholas supplemented the signature concepts of Fundamentum with those of Thierry and vice versa, reading each in light of the other, deforming their original meaning, but thus transforming them into something new. These passages are especially illuminating for understanding the potential that Cusanus apparently saw in Thierry’s theology and the new directions he had in mind for its evolution in his own century.

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Cusanus’s engagement with Fundamentum thus drove him to experiment with Thierry’s theology beyond the bounds of the Breton master’s original usage. In Book I, Nicholas invents a powerful new interpretation of Thierry’s aequalitas in order to illuminate Fundamentum’s notion of the maximum. In Book II, he expands the possible applications of Thierry’s concept of folding to help him grasp Fundamentum’s difficult theory of contractio. But the cross-readings cut both ways. In order to harmonize Fundamentum with Thierry’s theology, Nicholas employs a series of subtle editorial operations that force the anonymous treatise into alignment with the target of its critique. The stress of this hermeneutical torsion reveals a great deal about the composition of De docta ignorantia.

Cusan Variations: On Divine Equality Cusanus must have quickly recognized that the majority of his Chartrian sources shared a common theology of aequalitas. Thierry had used the concept in two ways:  to name the second person in the arithmetical Trinity, and to name the universal delimiting function of the forma essendi in the cosmos. Cusanus repeats both meanings when he combines passages from Tractatus and Commentum in I.7–9. As we have seen, the term appears in Septem and Commentarius Victorinus as well. But there was one important exception, the Fundamentum treatise, which abstained from aequalitas altogether. As he worked to bridge the gaps among his sources, Cusanus ended up giving aequalitas an unfamiliar negative interpretation. This new theology of aequalitas underwrites one of the major tenets of De docta ignorantia, that mathematical knowledge is a via negativa to God, a theme that Cusanus uses to frame Books I and II.58 Nicholas’s first challenge in Book I  was to define the transcendence of the “maximum” (I.3–4).59 Here we can watch him lean on aequalitas to make sense of Fundamentum’s difficult term. In order to reject the fourth mode, the author of Fundamentum had posed an absurdity. In a universe that permitted the fourth mode to exist, there would be no contraction to “differences and gradations” of being, and this would allow maximal act and minimal potential to “coincide” with each other. But since this is impossible, the author argued, his theory of contraction must be true, and the fourth mode could not exist.60 In the same way, Fundamentum had held that there could be no such thing as maximal motion, for that would coincide with rest, which is impossible.61 Cusanus takes this counterfactual and turned it on its head. He agrees that the universe is contracted to infinite degrees of difference, but at this juncture he brings in Thierry’s idea of aequalitas. For Cusanus, the real upshot of Fundamentum’s thought experiment was that in a contracted universe, perfect equality would be unattainable. Under the conditions of finitude, as he writes, “two or more objects cannot be so similar or equal

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that they could not still be more similar ad infinitum,” which means that one only finds “equality occurring in degrees.”62 Consequently, finite minds cannot measure anything with exact precision; even if one resorts to number in mathematical measurement, aequalitas praecisa always escapes untouched. Number is simply a better means for articulating the infinite degrees of difference that frustrate perfect measurement. As Inigo Bocken observes, “one could even say that the precision of number is the precondition of the imprecision of reality.”63 For the intellect is to truth what a polygon is to a circle, as Nicholas writes. Even if a polygon’s angles were infinitely multiplied, it would never equal the flawless curve of the circle.64 Hence the true meaning of maximum, he concludes, is “maximum equality.” Beyond the opposition of human concepts, such maximum equality does in fact coincide with minima.65 Needless to say, this train of thought was never envisaged by Fundamentum. Its author was simply trying to contrast contracted and absolute possibility, and would have rejected a Neopythagorean theology of aequalitas out of hand. This Cusan redefinition of aequalitas is a wholly new construction of the term that combines Thierry’s two meanings but through the prism of Fundamentum’s cosmology of contraction. God is present within the universe of infinite contracted difference as the absent trace of equality. Such asymptotic equality, implied within every mathematical measurement of infinitesimal degrees but unattainable, is a negative name of God.66 Cusanus will repeat this idea throughout Book I.  The image can never “equal” its exemplar.67 The divine maximum is also the universally equal “measure” of all beings.68 The maximum is the one precise “equality” of things.69 God is the eternal Equality of the beings that God could make, even if God never made them.70 Cusanus fully capitalizes on his idea when he begins Book II with a discussion of the quadrivium. The mathematical measurements of the quadrivium, Cusanus explains, provide a negative index of transcendent Equality. Astronomical calculations are inherently imprecise because the motions of planetary bodies never repeat themselves but introduce perpetually new, inequal particulars. In geometry, although we may understand the equality of two triangles abstractly, their material figures never attain that precise measurement. Paradoxically, no geometer has ever actually experienced equality, but only more or less defective simulacra.71 In music, there is no limit to how precisely a harmony can be played, or a pitch perfected. Rather: “the most precise maximal harmony is the proportion obtained in equality itself, which a living person cannot hear in the flesh, since it attracts to itself the soul’s reason.”72 Because every number in arithmetic is different from every other number, no two relative differences can ever coincide. No number can ever be identified with another, but only related through proportions and harmonies.73 Hence the Boethian quadrivium is an exemplary illustration, if not a practical proof, of docta ignorantia. Arithmetic, geometry, harmonics, and astronomy

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together function, writes the cardinal, as “corollaries” of learned ignorance, manifesting the infinite horizon of human unknowing.74 By the end of Book II Cusanus drew the obvious conclusion. If God is Godself the only perfect unity and equality manifested by the quadrivial arts, then such mathematics must be originally a divine activity: In the creation of the world God made use of arithmetic, geometry, music and astronomy, which we also use when we investigate the proportions of things, including elements and motions. For through arithmetic God joined things together; through geometry God fashioned them in such a way that they receive steadfastness, stability and mobility, according to their conditions; with music God gave them such proportion that there is not more earth in earth than water in water, air in air, and fire in fire, so that no element is wholly resoluble into another.75 The elements, therefore, have been established in a wonderful order by God, who created all things in number, weight, and measure. Number pertains to arithmetic, weight to music, and measure to geometry. . . . Who would not marvel at this Artisan, who in the spheres and stars and the astral regions also employed such skill that, although without complete precision, there is a harmony of all things as well as a diversity?76 These passages clearly echo Thierry’s Tractatus, in which Boethius’s quadrivium induces the student to theology, not to mention the Augustinian favorite, Wisdom 11:21. But what is original and striking about the Cusan use of the quadrivium is that, in opposition to Thierry, mathematical order is not (as Thierry writes) “drawn upon in order that the work of the Creator within things may become apparent.”77 For Cusanus, the quadrivium does not open the door to natural theology, as Plato had first urged in the Timaeus when he admonished philosophers to model their lives after the proportion, harmony, and rhythm they discern in the world. Rather, Cusanus turned to the quadrivium because it fails to reach precise results, since (as Markus Führer sagely notes) “he envisions the quadrivium as part of apophatic theology.”78 The precision of the quadrivium can only be postulated by human practitioners, since it is never fully realized in our mathematics. The full potential of the quadrivium is achieved only in its divine use. God can attain to its impossible precision because God possesses pure equality; human beings use geometry imprecisely, but the very failure of their quadrivial experiments in the contracted universe of pure difference negatively indicates a trace of equality and identity. God is the sole mathematician, and only for this reason do human attempts at quadrivial science, in their shortcoming, provide a glimpse of what the full power of mathematical knowledge might be, were God the mathematician

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instead of us.79 As Cusanus put it in Book I, God alone is the truly “negatively infinite,” while in the contracted universe we only know the “privatively infinite.”80

Cusan Variations: On Folding Cusanus also experimented with Thierry’s signature concept of reciprocal folding.81 Thierry’s original intent had been to confine complicatio and explicatio to a few delimited spheres. He never applied folding, for example, to creation, quadrivium, or Trinity; rather, folding served chiefly to organize the four modes of being. By contrast, Cusanus eventually assigned to reciprocal folding a wide range of functions regarding divine providence, divine names, and the meaning of faith, not to mention the quadrivium and the divine Word itself.82 But at the outset, Cusanus initially introduces folding in Book II in order to solve a specific problem. After having explored how the quadrivium exposes the essential negativity of knowledge (II.1), he now weighs the consequence that the contracted universe is incomprehensible (II.2). “Who, therefore, can understand the being of the creation?” intones Cusanus no fewer than five times, like God answering Job.83 The ineffability of the Creator casts a shadow over the intelligibility of the creation. The antinomies that bedevil philosophy are generated by the divine mystery: the One and the many, eternity and time, rectitude and curvature, or invisibility and visibility. For Cusanus, Thierry’s concept of folding quickly solves all of these problems, because it provides a spatial model for reconciling such cryptotheological aporiae. Eternity is enfolded time; visibility is the invisible unfolding into sight. “God, therefore, is the enfolding of all in the sense that all are in God, and God is the unfolding of all in the sense that God is in all,” he writes.84 But Cusanus is quick to add that “the manner of enfolding and unfolding exceeds our mind.”85 Reciprocal folding coordinates Creator and creature while nevertheless preserving the unknown ratio between them mandated by learned ignorance. In Nicholas’s retooled concept, the pure reciprocity of complicatio and explicatio itself has a negative function, since the repetition of the inverse fold excludes every other link between Creator and creature outside of the fold itself. Thierry may have implied that folding has a mathematical basis, but his real purpose was to construct a general paradigm capable of embracing all of the scientific disciplines. Cusanus was far more radical as he tested the limits of Thierry’s model. In the first place, Cusanus defines folding as essentially quadrivial. Because number is, principally, a kind of unfolding, the concept of fold can organize the whole quadrivium. Number is the unfolding of unity, inequalities (proportions) are the unfolding of equality, quantity is the unfolding of the point, and motion is the unfolding of rest: his examples span the dimensions of arithmetic, harmonics, geometry, and astronomy.86 Furthermore, Cusanus explicitly

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roots the fold of number in the divine Mind:  “just as number arises from our mind because we understand as individually many that which is commonly one, so the plurality of things arises ex divina mente, in which the many exist without plurality because they exist in enfolding unity.”87 The diversity that we understand in terms of number, God understands in terms of enfolding, such that mathematics is a reflection of the divine Fold. On this basis, Cusanus concludes that all folds are ultimately univocal, embraced by the one divine Enfolding, which is unity and equality and also maximum. This transnumerical Ur-Fold in the mind of God generates all the lesser foldings that structure the being of the contracted universe and produce the conceptual oppositions that stymie the philosophers.88 Thierry, of course, never had to rank different planes of divine or cosmic or numerical folding, since he never applied the concept beyond the narrow ken of his modal theory. Cusanus therefore introduced a more sweeping and indeed more Pythagorean use of reciprocal folding than its original inventor. In part this followed from the cardinal’s agenda to name mathematics as a mode of negative theology. But it was also the result of his wish as an author to integrate the Fundamentum’s theory of contractio with his other Chartrian sources. If the contracted universe is, as the anonymous author had maintained, a universe of infinitely granulated differences or nonequal singularities, then (Cusanus suggests) the infinite unfolding of unity into numerical singularities could be the best model for thinking contractio.89 When Cusanus expands the concept of folding into a universal cosmic structure, and then grounds it in the quadrivium, he ends up depicting a universe structured by seriated numerical difference—that is, a contracted universe. This would explain why after exploring the latent potentials of Thierry’s theory of folding (II.2–3) Cusanus immediately turns his attention to Fundamentum’s theory of contraction (II.4–6).90 His conceptual experiments in this passage point in several directions, but all of them reverse the author’s original intent to brandish contractio against mediating Platonist theologies. First Nicholas invents his own examples of the distance between absolute and contracted, and even tries to define contraction as a delimitation of Thierry’s first mode of being.91 Then Cusanus turns to Meister Eckhart—just the kind of philosopher rejected by Fundamentum—in search of further analogies for contractio, musing that contraction is a kind of “emanation” of the absolute or concretization of the abstract. By the time Cusanus states that the contracted universe itself “mediates” between God and creatures (an Eckhartian doctrine) we can recognize just how far afield of Fundamentum’s theology he has strayed.92 The passages that immediately follow are significant, as we shall see below. For suddenly, amidst these labors to integrate Chartrian Platonism with the anti-Platonism of the Fundamentum treatise, Cusanus hits upon two alternative ways forward. First, he sketches the Christology that will occupy Book III, and second, he outlines the Pythagorean philosophy of

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De coniecturis.93 After this flash of insight Cusanus turns his attention to the long citation of Fundamentum he has prepared for II.8–10.

Cusan Variations: On the Four Modes We have begun to see how Cusanus’s attempt to deploy Fundamentum alongside Thierry’s ideas led him to expand Chartrian theology but also to mitigate Fundamentum’s critique. Nowhere is this clearer than in II.7, where he prefaces his Fundamentum citation with a preliminary explanation. Cusanus apparently did not notice how dramatically these few words of introduction in fact overturned the agenda of his venerated source. By comparing the minor amendments and interpolations that he made while transcribing Fundamentum into Book II, we can watch over the cardinal’s shoulder as he reacts to its claims and mutes its judgments. As we have seen, Cusanus organized his presentation of the arithmetical Trinity according to the Septem treatise. He did the same for Fundamentum in Book II. This would have made sense to him, not only because he believed Septem to be a related Chartrian text, but also since out of all of his sources, only Septem coordinates the arithmetical Trinity, four modes, and three perpetuals. Where in Book I  this rubric accented Thierry’s theology, in Book II it works against Fundamentum. Cusanus apparently observed that Fundamentum had echoed the perpetuals of matter, form, and motion in its own tripartite structure. But in content, as we have seen, its author had avoided the arithmetical Trinity fastidiously and had critiqued the four modes of being for the same errors of Platonist mediation that the doctrine of perpetuals commits. This new Septem framework in II.7 reminds us of Nicholas’s editorial originality in using his source text. Cusanus did not simply cull the choice portions from Fundamentum: even in his introduction he subtly pointed them in a new direction, a direction unintended and undesired by the author of the anonymous treatise. In II.7 Nicholas speculates that if the absolute maximum is triune then the contracted maximum must be a triad as well: the contractible substrate, the contracting agent, and their connection.94 Following the pattern of Septem, he then identifies this triad with both the descending trinity of perpetuals and the persons of the arithmetical Trinity. Contractibility is a kind of “possibility” that “descends from eternal unity.”95 Contracting is a “limiting” (terminare) or “equalizing” (adaequare) of possibility that “descends from equality of unity.”96 The third descent is the nexus of the contractibility and contracting, which Cusanus calls a kind of “motion” or “spirit of love” that descends from the Holy Spirit, the infinite connection.97 Again following Septem, Nicholas then tries to layer the four modes onto this threefold framework, such that the fourth, second and third modes correspond to the perpetuals of matter, form and connection. Obviously this Septem

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rubric obfuscates Fundamentum’s precise theological judgments. But it apparently appealed to Cusanus because it afforded him a neat table of contents for the sequence of excerpts he planned to borrow from Fundamentum: on matter and the fourth mode (II.8); on form and the second mode (II.9); and on their connection in the third mode (II.10). Hence in the prefatory II.7 we encounter a conflict between Fundamentum’s agenda and Cusanus’s attempt to appropriate the treatise. Fundamentum had attacked the illusion of “absolute possibility,” fearing that it could jeopardize the divine absolute. But influenced by Septem, Cusanus now treats possibilitas as a separate cosmic principle “descended” from God that universally “precedes” all determined beings.98 This is precisely the view of the fourth mode—hovering between the first and third modes of being—which Fundamentum had tried to reject. Next Nicholas states that the contracting agent likewise “descends” from divine aequalitas. But to make sense of aequalitas Cusanus had to move beyond Septem and consult the Boethian commentaries in his possession. Thierry’s notion of aequalitas essendi, he now suggests, implies that the universe is contracted, since the very function of equalizing (adaequans) is to apportion raw possibility into a finite continuum of discrete beings. Since for Thierry aequalitas essendi is the divine Verbum, it follows, according to Cusanus, that the contracting agent (contrahens) “descends” from the divine Word. By grounding the contraction of the universe in Thierry’s aequalitas doctrine, Cusanus has already altered Fundamentum’s meaning, since this was the Chartrian doctrine that the author had resolutely ignored. But this statement about the divine Word shows how little Cusanus shared the prerogatives of his anonymous source. Contraction, he writes, descends from the Word just as the second mode descends from the first mode: And because the Word, which is the essence, idea and absolute necessity of things, necessitates and restricts possibility through such a contracting agent [per ipsum tale contrahens], some, therefore, called that which contracts “form” or “soul of the world,” and they called possibility “matter”; others named it “fate in substance”; and still others, like the Platonists, spoke of it as necessitas complexionis, since it descends from absolute necessity to become, as it were, a contracted necessity and a contracted form, in which all forms truly exist. But this will be discussed later.99 With these words Cusanus reveals his awareness that Fundamentum wishes to substitute the Word for the second mode, in a passage that he will cite shortly in II.9. But he does not seem to grasp that defining the second mode as the agent of contraction, or the second mode as a necessary “descent” from the first mode, directly contradicts the central argument of Fundamentum. As in his treatment of possibilitas in II.7, Cusanus shows little regard for the author’s anxiety about

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divine transcendence. In the cardinal’s reading, the Word as aequalitas contracts or individuates beings by mediation of the second mode (per ipsum)—precisely what Fundamentum had worked so hard to refute.100 Cusanus’s philosophical instincts simply ran against the grain of his conservative scholastic source. Rather than discard the second mode as a phantom of Platonist imagination, he instead pondered its value as a mediator. In the same way, as one might expect, his independent exposition of necessitas complexionis in II.9 will be altogether sanguine.101 Thus Cusanus ended up affirming what his source never would have allowed: that the contracted nature of the universe itself stems from the aequalitas of the Word.102 This sort of interpretive slippage reminds us why Fundamentum’s author needed to avoid Thierry’s term. Its power resides in its univocal indication of the functions of the first mode (as aequalitas unitatis: the divine source of form) and the second mode (as aequalitas essendi: the limiting form that “equalizes” beings to themselves). As this Cusan reading well shows, the double function of aequalitas defines the mediating activity of necessitas complexionis, but it also reveals the ambiguous relations between Thierry’s modal theory and Trinitarian theology that had already troubled his student Clarembald. By recovering both dimensions of aequalitas, Cusanus effectively reconsidered Fundamentum’s rejection of the second mode. Even if he reproduces that rejection verbatim when he cites the treatise in II.9, the cardinal’s own commentary on the passage in II.7 falls considerably short of his source’s energetic censure. Likewise even if Nicholas repeated passages from Fundamentum to the contrary in Book II, his enthusiasm for Thierry’s arithmetical Trinity in Book I  reopens the mediating space that Fundamentum had attempted to foreclose. Fundamentum’s Aristotelian author, echoing Augustine’s mature reaction against Platonism, repudiated the mediating plane of perpetual forms and directed the inquirer to the Verbum instead. But as an avowed Christian Platonist (I.11), in concert with Boethius and Thierry, Cusanus was interested precisely in how the Trinity relates to such mediating levels of form.103

In Defense of Christian Platonism Cusanus’s loyalties remain on the surface throughout his long citations from Fundamentum in II.8–10. Hoenen has catalogued several of the minor but tell-tale additions made by the cardinal to his source.104 Cusanus also interpolated his own detailed expositions of Fundamentum’s ideas as he went along.105 These valuable Cusan interpolations in II.8 and II.9 arrive with exquisite timing:  they fall after Fundamentum has stated the suspect mode in question but before the author has made his argument against it. In both chapters, Cusanus thus interrupts the treatise’s critique with so rich an exposition of the mode under scrutiny that it dissipates the force of the impending attack. For example, in II.8 when

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Fundamentum condemns the “Platonists” for teaching that absolute possibility is eternal, Cusanus inserts a historical survey of the concept of possibility. He shows that many different schools besides the Platonists—notably including the Peripatetics!—made similar judgments.106 The implication is clear: if the eternity of matter is a structural feature of ancient thought, there is no sense in disparaging the Platonists in particular. The interpolation in II.9 is especially useful as a record of Cusanus’s estimation of necessitas complexionis. The cardinal never explicitly challenges Fundamentum’s judgment against the second mode, but there are definite signs of tension between author and source. To his eyes, of course, Fundamentum was an adherent of the modal theory, not its critic. Believing that the critic and critiqued were one, the cardinal felt no pressure to affirm or reject its judgments; instead, as a measure of editorial prudence, he restates his source’s views with helpful amplifications drawn from other (he thinks) sympathetic Chartrian texts. As we saw above, the second part of Fundamentum begins by stating that mens, anima mundi, and fatum were all called necessitas complexionis by “the Platonists.”107 Cusanus cites this text and then begins his interpolation in II.9. But as he repeats the charges aloud, it seems as if he begins to reconsider them for himself. In the first place, Cusanus repeats the name Platonici no fewer than twelve times within a few pages and centers his exposition on this party’s doctrines.108 Nicholas evidently believes that the Fundamentum’s author had designated with this name the provenance of the same common Chartrian doctrines that he was gathering together into De docta ignorantia. Recall that in I.11, Cusanus not only praised the same Platonici but also linked them with Pythagoras, Augustine and Boethius—a canny profile of just such Chartrian traditions. Thus we see the same pattern emerge once again of Cusanus interpreting his two bodies of Chartrian traditions with each other. Throughout his interpolation in II.9, Cusanus contends that the Platonists have made a respectable case (rationabiliter) for the second mode, even if not every Christian (or “Peripatetic”) accepts it.109 His self-appointed task is to demonstrate the plausibility of the world-soul (the second mode), if not its truth. First, Cusanus argues that the Platonists’ second mode should be understood in terms of reciprocal folding. Thierry’s complicatio and explicatio had functioned to coordinate the simplicity of the first mode with the mediating exemplars of the second. Fundamentum had permitted the complicatio of the first mode but rejected the second mode as a fantastical explicatio of divine necessity that violated the principle of contraction. But because Cusanus does not perceive (or sanction) this criticism, he takes the author’s definition as an invitation to fill out Thierry’s doctrine of folding as reciprocal. Cusanus proceeds to rebalance Fundamentum’s selective use by emphasizing explicatio more than complicatio, heedless of the damage thus inflicted on Fundamentum’s critique.110 Thus Thierry’s explicatio ends up helping Cusanus to justify the modal theory rejected by Fundamentum. Second, Cusanus

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maintains that critics of the world-soul or second mode miss the distinction between ordo naturae and ordo temporis.111 God is naturally (not temporally) prior to the second mode, just as the second mode is naturally prior to substantial forms. His point is that the second mode’s mediating function jeopardizes divine transcendence only if misconceived as temporal priority. Cusanus apparently discovered this argument within Fundamentum itself, in a passage that he does not otherwise cite in De docta ignorantia.112 His presumption is that Fundamentum’s doctrine should accord with Thierry’s modal theory. In the face of evidence to the contrary, Nicholas reestablishes their harmony himself by referencing another passage of Fundamentum. Cusanus’s final position on the “Platonist” second mode can be detected in three ways. First, his remarks emphasize the staunch opposition of the “Peripatetics,” who shift from being one ancient school among others (as in II.8) to being the prime antagonists. Aristotelian responses to the world-soul, he concludes, have been “unreasonable” and superficial.113 Second, Cusanus chooses his words carefully when he evaluates the doctrine’s value for Christian theology:  “many Christians have settled on this Platonic approach.”114 This is the canny formulation of a gifted lawyer! Personally Cusanus sympathizes with the Platonists, but his exposition exists to serve a source that does not. This tension has rightly puzzled interpreters, even to the extent that some have discerned different voices at work in the text.115 So the cardinal’s burden, delicately handled here, is to show that Christianity is at least compatible with the doctrine of the world-soul, the plurality of exemplars in the divine mind, or the second mode of being. Finally, the most revealing moment comes later in II.9 when Cusanus transcribes the momentous passage of Fundamentum in which the author directly rejects the second mode of the “Platonists.” Right at the crucial phrase, Cusanus allows himself to look the other way with an uncharacteristic slip of the pen, as he skips over the name of the accused party.116

The Christological Double Synthesis If we look back over Cusanus’s experiments with his sources and the unwitting reversals he engineered, we see ample evidence that the cardinal organized his Chartrian sources into two sets of materials and elaborated them separately in Books I and II, but also frequently used one to interpret the other. He seems unaware of any contradictions between his sources and unconcerned that he might be contravening their intentions.117 Indeed from the cardinal’s point of view, since his uniformly anonymous texts were all like-minded commentators on Boethius, his task was to synthesize them as far as possible. Given the specific works at his disposal, it would have been difficult for Nicholas to correct Fundamentum or Septem

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by the canons of Thierry’s mature commentaries. So when he listened to the voice of Fundamentum’s author, he did not hear the telltale moments of correction as attacks on Chartrian theology, but instead as the author’s lively, if oblique, contributions to the same project shared by the other Chartrian sources. The axis of controversy remained invisible to Cusanus, and he earnestly read his anonymous author accordingly. We have already seen two examples of Cusanus mistaking Fundamentum’s rhetorical tactics for positive theorems proferred by the author. Fundamentum argued that the universe must be contracted, lest maxima and minima coincide. Cusanus took this as an indication that coincidence with the minimum (his famed coincidentia oppositorum) is a hallmark of the divine maximum (I.4). For Fundamentum the height of Chartrian (“Platonist”) folly was the notion that plural exemplars (in the anima mundi, mens, or second mode) were “unfolded” immediately from God. Cusanus felt that this venerable doctrine of divine unfolding should receive greater respect from skeptical Peripatetics than it had (II.9), and he invented some new applications for the concept (II.3). A third and more consequential example of Cusan misreading arises in his response to Fundamentum’s attack on the second mode. Fundamentum’s exact words are these: “The necessitas complexionis is not mens, as the Platonists suppose, lesser than the Begetter, but is the Word and the Son equal to the Father in divinity, and is called Logos or ratio, since it is the ratio of all things.”118 When he read this, it seems that Cusanus did not detect the critical edge intended by the author. Fundamentum had juxtaposed the Word and the second mode in order to expose one of them as an impostor. But Cusanus apparently read the author’s critical substitution as a confirmation of their comparable function. As Cusanus hears it, the second mode does not only receive the name mens or anima mundi; rather, and more importantly, the divine Word itself possesses all of the mediating functions of the second mode. To him the two seem perfectly compatible. Fundamentum had provided some other names for the second mode earlier in the treatise, after all, and now the author seemed to be providing one more.119 This misreading explains the cardinal’s unusual introduction of Fundamentum’s ideas in II.7, where he commented on this crucial passage before citing it verbatim in II.9. Within the space of a few lines, as we have seen, Nicholas’s preface completely undermined Fundamentum’s critique. His primary goal was not to amplify (or even discern) the author’s charges against Chartrian theology, but rather to harmonize them with his other Chartrian texts. Cusanus wanted to bridge what he perceived to be two compatible Verbum theologies. Thierry had connected Verbum and aequalitas, and Fundamentum (in the cardinal’s reading) had linked Verbum to the second mode. Cusanus drew the obvious conclusion in II.7: aequalitas should be connected with necessitas complexionis, since both unify the plural exemplars in God. Yet when Cusanus studied aequalitas in the pages

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of Septem and Commentum, he found that these sources linked the arithmetical Trinity to the descending triad of perpetuals. And when he researched the second mode in Fundamentum, he encountered the author’s praise of contraction. Of course, in point of fact, Thierry had abandoned the perpetuals, Septem had misconstrued them, and Fundamentum had intended contractio as a critique of the second mode. But given his editorial priorities, and given his limited textual access, these nuances remained invisible to the cardinal. This hermeneutical situation of incomplete, misunderstood but nonetheless revered sources begins to explain how Cusanus came to formulate his unusual gloss in II.7. For those of us who have studied Thierry and his reception in historical context, Nicholas’s reading of Fundamentum in this passage is truly bizarre. In his view, as we have already seen, necessitas complexionis is a kind of “contracted necessity and a contracted form,” and it “descends” directly from divine aequalitas, who is the Verbum and the first mode of being.120 In short, the second mode is an instrument of contraction implemented by the Word as aequalitas. What is Cusanus doing here, if not trying to assimilate the Word and the second mode by coordinating the functions of contractio with aequalitas? This passage in II.7, I propose, represents Nicholas’s initial reaction to Fundamentum’s views of necessitas complexionis. In other words, it was his first, ultimately failed attempt to synthesize his two bodies of sources. I submit that Book III represents a second, more sustained, and ultimately successful attempt to resolve the same problem. In II.7 Cusanus had identified two concepts—aequalitas and contractio—as the major points of contact between his two bodies of sources. In Book III he will build his novel Christology around these very terms, as we shall see. But now he will add two further concepts that Fundamentum had expressly prohibited—explicatio and medium—but that will aid him in reconciling his two sets of sources in Book III. When Cusanus sets out in Book III to make his collected Chartrian sources speak with one voice, what theological resources did he have at his disposal? What lines of thought would have seemed most natural? If we examine his earlier writings, one possibility becomes clear:  his interest in Neoplatonist Christology.121 Before 1440, Cusanus’s only major theological work is De concordantia catholica in 1433. As many have noted, the opening chapters of this work of political theory are particularly striking for their embrace of Ps.-Dionysian metaphysics. The universe is a harmonious hierarchy stretching from God, through the Word, to all the gradations of creatures, in a unified circuit of procession and return from the infinite to the finite.122 Although Nicholas also discusses the Trinity at length, it is clear that Christ is the integrating principle that makes possible the harmony of the One and many, both the ecclesial concordantia of Christians and the cosmic concordantia of all beings.123 “I think of the Word from above,” writes Cusanus, “as like a magnetic stone the power of which extends through everything down to the lowest being,” reconciling every opposition in advance.124

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We also possess some twenty Latin sermons between 1430 and 1439. Given the liturgical year, Cusanus tended to preach on the Incarnation around Christmas. A quick look at his Christmas sermons from 1430–32 in Koblenz during the period in which he was composing De concordantia catholica reveals a similar Christology. On Christmas Day of 1430, Nicholas’s sermon on John 1 drew on traditional theologies of Incarnation from Anselm of Canterbury and Peter Lombard, but even more avidly on Ps.-Dionysius.125 In two Christmas sermons at Koblenz in the early 1430s, Nicholas focused increasingly on the Incarnation. He detailed the three prophecies of the Incarnation (by Caesar Augustus, by the angels, and by the divine Word) and the three births of the Son (in eternity as the Word, in time to Mary, and spiritually in the believer). He also displayed a confident ease with the distinctions of Chalcedonian Christology, the unio hypostatica of two natures in one person and the communicatio idiomatum of humanity and divinity in Christ.126 After his return from Constantinople, Cusanus again preached on the Incarnation at Koblenz. On Christmas Day in 1438, Nicholas returned to the theme of the ineffable Word in which all creatures were made, citing Ps.-Dionysius alongside the Johannine theologies of Origen and Augustine. In January 1440 he preached again on Ps.-Dionysian themes, this time a fascinating meditation on the ineffability of God and the human name “Jesus.”127 Just as in De concordantia catholica, we again witness the cardinal’s interest in high Christology situated within Ps.-Dionysian metaphysics, yet still without a hint of the distinctive concepts of De docta ignorantia: reciprocal folding, the distinction of absolute and contracted, or the arithmetical Trinity. Throughout these homilies from the 1430s, we hear a preacher interested in ancient and medieval philosophy but also on familiar terms with the conceptual possibilities of Christological doctrine. If this strand of Christian Neoplatonism was invented by Albert the Great, it was perfected in the “functional Christology” of Eckhart, Bonaventure, and Llull.128 For them the incarnate Word is the axis of the cosmic procession from and return to God, the still point of the moving world. Christ is savior, judge, and head of the church; but above all Christ is the sole nexus mediating between the divine One and the order of finite being, the goal and center of all creation. Cusanus imbibed this thirteenth-century Christology in part from Heymeric de Campo in Cologne in the 1420s.129 This Neoplatonist Christology was the instrument that Nicholas reached for in Book III to join his two sets of Chartrian sources. Now, of course, Fundamentum had stated in the strictest terms that “no medium arises between absolute and contracted,” because the eternal Word as the absolute renders mediation unnecessary.130 But Cusanus’s familiarity with Neoplatonist Christology led him in a different direction. The eternal Word, he agrees, is the absolute. But the incarnate Word, as he will write in Book III—turning yet another phrase from Fundamentum on its head—is the medium conexionis of absolute and contracted.131 From the

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cardinal’s perspective, to view the mediations of Platonism as an invitation for high Christology was not his idea alone. This was just the lesson he thought he had gleaned from Fundamentum: that when Christians want to describe how the first mode descends into the second mode, they speak of the divine Word or Son equal to the Father, the Logos or ratio of all things. Deaf to the criticism of Thierry intended by Fundamentum, Cusanus heeded these words as a Christological directive. He had read both sets of sources as reciprocal supplements throughout Books I and II, and he did so again in Book III. The Christological synthesis in Book III is arguably the most original Cusan achievement in De docta ignorantia.132 The other conceptual innovations we might list are either copied from his sources or amount to creative experiments with their terms. A  close runner-up would be Cusanus’s notion of divine aequalitas as the ground of negative theology and, consequently, his vision of geometrical figures as linear traces of divine absence that propel the contemplative toward unknowing. Here too he used Ps.-Dionysian theology to interpret his Chartrian sources, in this case Thierry. But only the Christology holds together the architecture of De docta ignorantia’s three books, applying the aequalitas theology of Book I to a higher end, mobilizing his Ps.-Dionysian meditations over the past decade, and so fashioning something wholly new. The Neo-Chalcedonian Christology of Book III is constructed out of three parts, in which the first two steps culminate in the third.133 Cusanus defines the human and divine natures of Christ and then shows how they can be united “without confusion and without composition.”134 As human, Christ is the contracted maximum; as divine, Christ is the absolute maximum; and only in him are the two realms personally united.135 This much is clear, and every exposition of Cusan Christology has followed in the cardinal’s footsteps.136 What is less often observed—indeed what cannot be observed without the Fundamentum hypothesis, and yet becomes obvious once it is granted—is that the two terms united in the Christological synthesis represent the cardinal’s two collections of Chartrian traditions. This means that the synthesis of Book III operates at both theological and documentary levels. In naming the hypostatic union, Nicholas the theologian explained the union of divine and human natures in Jesus. But in the same synthesis Nicholas the editor was perfecting the fusion of Thierry’s Boethian texts and the Fundamentum treatise toward which he had been reaching throughout Books I  and II. What Christ unites is not simply the absolute maximum and the contracted maximum. In fact, these are not the terms Cusanus uses in his final formulation. Ultimately Nicholas states that Christ unites the aequalitas essendi (from Thierry and Septem in Book I) and the universal contractio (from Fundamentum in Book II). God and human, he writes, are united “as if the universal contraction of all things were hypostatically and personally united with the equality of being all things.”137 We

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have seen how Thierry’s Platonist theology of aequalitas was anathema to the author of Fundamentum. But in Book III Cusanus achieved what Fundamentum thought impossible: he shows how the Trinitarian theology of aequalitas can illuminate, and even define, the cosmological principle of contractio. Others have analyzed the resulting Christology of De docta ignorantia on its own terms.138 For our purposes we are more interested to watch how Cusanus constructs his device out of apparently incompatible materials. Nicholas retrojected hints of the coming synthesis back into Books I and II.139 But its actual execution proceeds in three deliberate steps that neatly correspond to chapters III.1–3. In the remainder of the book Cusanus elaborates some corollaries (III.4–12), just as he had in Books I and II. All three steps, however, draw from the concluding section of the Fundamentum cited as II.10, where the cardinal’s comments are conspicuously thinner than his probing interpolations in II.8 and II.9.140 In effect, Nicholas reserved his exposition of this section of Fundamentum for Book III: on the singularity of species and degrees of contraction (III.1), on the link between contraction and enfolding (III.2); and on the medium of connection (III.3).141 Here Cusanus read these passages of Fundamentum through the lens of Thierry’s theology of aequalitas as he had developed it in Book I. This hermeneutic operation produces the three steps of the new Cusan Christology.

Equality, Contraction, Folding Fundamentum envisions the contracted universe as a continuum of unique “degrees” (gradus) of difference descending “gradually” (gradatim) from universals down to particulars, and thus preserving the distinctness of “species” and individuals.142 “All things are moved singularly [singulariter] . . . such that nothing exists equally [aequaliter] just as another,” writes the author, “yet each thing contracts the motion of another by its own motion and participates in it mediately or immediately.”143 In this passage the author elaborates more fully the concept of contractio that he had deployed against Thierry’s fourth and second modes earlier in his treatise. At the beginning of III.1, Cusanus immediately alludes to this definition of contracted difference from Fundamentum. All creatures are distinguished by gradus into species and individuals; no contracted thing participates in the same degree of contraction as another.144 Then he focuses on the author’s contrast between singularity and equality: There is nothing in the universe that does not enjoy a certain singularitas that cannot be found in any other thing. Therefore, nothing prevails over all others in every respect or over different things aequaliter, just as there can never be anything in every way equal to another. Even if at one time

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The cardinal’s insight in this passage is the fulcrum of his Christology and to that extent of his enterprise in De docta ignorantia as a whole. Perfect aequalitas is the ground of contractio: what it means for the universe to be contracted is that equality is always implied in creaturely difference yet always absent. Such nonequality is precisely what gives rise to a continuum of uniquely differentiated degrees of being. As Thomas Leinkauf explains, “difference is the necessary expression of the unfolding of the One, the non-Different; it is none other than the boundary between unfoldings that distinguishes different unique identities from each other and thereby places them in relation.”146 This must be what Cusanus meant by his cryptic statement in II.7 that the activity of contrahens “descends” from aequalitas essendi. The agent of contraction is the divine Equality, which (Cusanus proposes) operates “through” the second mode to delimit creatures with form. Cusanus’s intuition in II.7 and III.1 complements his Ps.-Dionysian interpretation of Thierry’s aequalitas in Book I. There he said that perfect Equality is implied by but never attained in the quadrivium, so that number leaves a negative trace of divine absence. Therefore it is unsurprising to find Cusanus returning to similar ideas in III.1. Without yet spelling out the Christological synthesis, he postulates a singular, divine “limit” (terminus) that all beings approach but never reach.147 He identifies the degrees of differentiated species as “a series of sequentially progressive numbers” that one can “count,” up to God and back down again.148 In such passages, well beyond the intentions of Fundamentum, Cusanus is beginning to radicalize the contracted nature of the universe by mathematizing it. In his version, the infinite grains of creaturely difference are tantamount to numbers flowing out of the divine One, and the absent Limit or Equality that they reflect is nonetheless the principle of their origination, stability and singularity.149 This is something that Philolaus, Moderatus, or Nicomachus could have written. But it is only the first step of the cardinal’s Christology. Earlier in Fundamentum the author had used complicatio to define the supremacy of the “universal form” (the divine Word) over “contracted forms.” In the treatise’s final section the author also explained the workings of contractio as itself a kind of complicatio. In light of Aristotle, Fundamentum takes it as given that what gives order to the contracted universe is the natura or motus uniting potential and act. But then he states that this natura that suffuses the universe preserves individual differences within the continuum of contraction by “enfolding” all things together.150 In III.2, Cusanus appropriates this vague link between contraction and enfolding. If there were a contracted maximum, writes the cardinal, it would exist

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in a given species. As the universal “limit” and “equality” of its species the contracted maximum would “enfold” all the individuals within it.151 Cusanus uses the remainder of III.2 to foreshadow the Neo-Chalcedonian logic of the Christology to come. Absolute and contracted maxima will be united like the two natures of Jesus.152 The cardinal’s preferred terms cannot conceal the fact that these first two steps have recapitulated his struggles throughout Books I and II to harmonize his sources. Nicholas argues in III.1 that an absolute maximum would be defined negatively through number, as the Equality absent in the contracted universe. But this was precisely his strategy for reading Thierry’s arithmetical Trinity in line with Fundamentum in I.2–6. Likewise in III.2 he suggests that a contracted maximum would be defined in terms of its enfolding power. As I have shown, the project of II.2–6 was to connect contraction with Thierry’s folding. From the perspective of III.1–3, it seems that the cardinal’s master plan for Books I and II might well have stemmed from the final section of Fundamentum. As we shall see, several key terms used by Cusanus to complete the synthesis in III.3 originated in the same section.

Intersection After rejecting the fourth and second modes, Fundamentum accepted the third, possibilitas determinata, which states nothing more than the Peripatetic notion of matter delimited by form. In essence, the contracted universe is one in which everything, save God, is the third mode. Fundamentum calls this third mode the conexio or medium between form and matter, or simply the medium conexionis of potential and act.153 Likewise in III.3 Cusanus explains that the absolute and contracted maximum must be a “middle nature, which is the medium conexionis of the lower and the higher,” so that it can “enfold” all natures in itself.154 This medium is human nature, which Cusanus describes in terms of his Christological sermons from the 1430s. As in Asclepius, human being is the “microcosmos” because it coordinates the intelligible and sensible orders. As Bonaventure and Eckhart had taught, the “medium” of Christ is the center point of the cosmic circuit in Ps.-Dionysian metaphysics. Hence, writes Cusanus, all things descend into contraction and ascend to the absolute “through the same medium . . . who is the beginning of their emanation and the end of their return.”155 Cusanus’s first step in his Christology was to define the absolute maximum as the aequalitas absent from the contracted universe. The second step defined the activity of the contracted maximum as complicatio. In both cases he discovered points of contact between Thierry’s theology and the final section of Fundamentum.156 Now he states his Christological conclusion:

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mathematical theologies First, therefore, there is God as creator. Second there is God as both God and human, whose created humanity has been assumed into unity with God, as if the universal contraction of all things were hypostatically and personally united with the equality of being all things. And therefore, in the third place, through the most absolute God and by the mediating of the universal contraction, which is humanity, all things come forth into contracted being so that they could thus be what they are in the best possible order and manner.157

Cusanus chooses precise terms here with care; note that when he restates the synthesis in III.4, he retains the same three steps.158 He does not simply state that the Incarnation represents the union of absolute and contracted maxima. What Nicholas actually writes is that what mediates (medians) between God and world is the unity of aequalitas essendi and contractio universalis.159 On the theological level this means that the Verbum enfolds the forms of all contracted things, as he had awkwardly foreshadowed in II.7, thus performing the function of the second mode or anima mundi. The Incarnation becomes the medium that connects the absolute and the contracted, the first mode and the third mode—an interpretation of Fundamentum that might have disquieted its author.160 But on the documentary level, the cardinal has also fused the most valuable and distinctive concepts from his two bodies of sources stemming (in his eyes) from the same Boethian commentator.161 This double achievement in 1440—theological and editorial—sets up a double task for Cusanus that we will now trace through the 1440s, 1450s, and 1460s.

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Chartrian Theology on Probation in the 1440s In one stroke Cusanus’s Christological synthesis resolved longstanding tensions among the traditions stemming from Thierry of Chartres and dispelled the Christological doubts hanging over the Breton master’s legacy. The constitutive problem of mathematical Platonism, the mediation of God and world, had been resolved by Thierry in two ways. Rather than pursue substitutes for the anima mundi like Bernard’s formae nativae, Thierry had postulated the arithmetical Trinity (Logos as aequalitas) and the fold of the four modes (Arithmos as necessitas complexionis). Subtle tensions between these two mediations confused Septem and worried Clarembald. The Fundamentum’s author, crystallizing their unease into a systematic critique, insisted that the divine Word is the sole mediator in a finite universe structured by contraction, rejected necessitas complexionis as one more idolatrous mirage of Platonism, and ruled out the Platonizing theologies of aequalitas and reciprocal folding. But Cusanus, reading Fundamentum alongside Thierry and Septem as well, sought to construct as many links between them as possible. The texts spread before him encapsulated centuries of disagreement over the status of Platonic mediators in medieval Christian theology. His years of legal practice, we should recall, had refined his skills at negotiating between opposed points of view. So in his effort to harmonize what he believed were univocal sources, as he began to compose De docta ignorantia Nicholas also unwittingly began to reconcile Logos and Arithmos.1 In effect, Cusanus did heed Fundamentum’s complaint about Thierry’s theology, that Christian Platonism neglects the Word. But because he misunderstood his source, his cure was to reinstate Thierry’s aequalitas theology, and yet to alter it in two ways. First, he reinterpreted transcendent aequalitas as the infinite regress of divine precision approached in mathematical knowledge; number became an instrument of negative theology. This doctrine had the benefit of accentuating Thierry’s Neopythagoreanism, but in a manner that accorded with Fundamentum’s ontology of an infinite continuum of contracted difference.2 Second, Cusanus transposed Thierry’s Trinitarian mathematical theology into a Christological

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mathematical theology. Anxieties about the Breton master’s Platonism, if unjustified, were exacerbated by his silence regarding the humanity of the Word. The humility of the Word’s Incarnation, after all, was precisely what had motivated Augustine to become frustrated with Platonism and Pythagoreanism. Cusanus in so many words suggested that the center of a Christian mathematical theology cannot be the eternal Word, nor the merely incarnate Word, but rather the eternally incarnate Word.3 The cardinal’s accidental but breathtaking insight is that the singular mediation of the Logos (aequalitas essendi) is best reflected and expressed through the serial singularities of Arithmos (contractio universalis). If finite contractio and infinite aequalitas are inversely related, that means that they are both removed from each other beyond proportion (satisfying Fundamentum’s anti-Platonism), but also immanently present as each other’s reciprocal ground (satisfying Thierry’s Platonism). Their common singularity—one by its unique supremacy, one by its degrees of infinite difference—is the only possible point of contact for their unity. This common singularity, this intersection point, is the “concept of Jesus,” as Cusanus calls it in the prologue to Book III.4 If mathematical theologies are preoccupied with mediation, then the surest mathematical theology in the Christian idiom is not a henology, as Cusanus discovered, nor even an arithmetical theology of the Trinity. It is a mathematical theology of the divine medium, the incarnate Word. In short, Nicholas mathematized both human and divine natures in order to unite them. Human nature is the maximal enfolding of the sets of all numbers of species. The Word is the aequalitas of all number-species that preserves them in their identity. The hypostatic union is therefore the point of intersection of the transcendent ground of number and the infinitely differentiated numerical array. Cusanus’s Neopythagorean Christology therefore represents a substantive solution to the problem of mediation in Christian Platonism, from Augustine and Boethius through Bernard and Thierry, and even into the fourteenth-century Proclians. In order to correct the Platonist conflation of divine Logos with the anima mundi or necessitas complexionis, one cannot remain in the register of Trinitarian theology. As we have seen in Thierry’s reception history, this ultimately leads to competition between ambivalent mediators. The compromises of Platonism are only overcome once the Word’s uniqueness is expressed in the fine grain of contracted being, that is, as an incarnate Word whose materiality is measured in the infinitesimal ontic granules of numbers, but also as a Word whose divine absoluteness is the principle of such arithmetical granularity. This is what the hypostatic union of aequalitas essendi and contractio universalis ultimately means. Of course Nicholas did not set out to write a mathematized Christology, but merely to do justice to the rich sources he had collected. The Cusan triumph in De docta ignorantia was therefore an accidental triumph. It may have sounded as odd

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to him at first as it does to us. But the history of Platonism is rife with productive misunderstandings. Aristotle found clarity in his critique of Plato by misconstruing Philolaus’s views of number. Moderatus and Plotinus misread Parmenides as a sober theology of the One. For some seven centuries Ps.-Dionysius was thought to be not a Syrian Neoplatonist but an Athenian adherent of St. Paul. The Liber de causis, built mostly from Proclian excerpts, was revered by the strictest Peripatetics as Aristotle’s theology. To this list we can now add Cusanus’s supposition that Fundamentum and Thierry’s commentaries were written by the same brilliant author, and thus invited his attempt to reimagine Thierry’s quadrivial theology in new ways, not least by venturing a complex theory of the Incarnation. As Dodds has written of Plotinus, even if the great man is now viewed less as an innovator than a culminator, this is only to praise a different species of genius that can construct marvelous syntheses out of less than promising materials.5 Or as Jeauneau once mused, “if Nicholas of Cusa’s De docta ignorantia effectively prolonged the effort of the Chartrian masters, who could dare to think that Thierry of Chartres had labored in vain?”6 Our next task is to measure the potentials and limits of that synthesis and especially its stability over the next twenty-five years as Nicholas continued to write. His first instincts were not to repeat the same formula but to test out alternative readings of his dossier of Chartrian sources. Already in De docta ignorantia Cusanus had hinted at plans in the works for a sequel called De coniecturis. This book is difficult to date, but it must have been underway by 1439 and was revised for the last time in 1445.7 In De docta ignorantia we can only glimpse the cardinal’s new direction, but the timing of his references to the work in progress signal that he thought of his new book as another response to the Fundamentum treatise.8 With only one exception, Cusanus mentions De coniecturis exclusively within Book II. At the end of each of the Fundamentum chapters (II.8, II.9, II.10), he promises to continue his deliberations in De coniecturis.9 Three times he states that De coniecturis will further treat the continuum of graduated, contracted differences conceptualized in III.1.10 The most important preview occurs in II.6—which of course just precedes the critical preface to Fundamentum in II.7. Here Nicholas announces that in De coniecturis he will fully elaborate a system of what he calls four “unities.” Absolute, divine unity is contracted to one universe, whose intrinsic plurality is comprised by three lesser unities linked through chains of reciprocal folding: the decadic root, its square, and its cube.11 In its final version De coniecturis swelled to the same size as De docta ignorantia. Yet unlike its forerunner, the theologies of Trinity, Incarnation, and Church have no place in the book, while number and mathematics grow far more prominent. It may be tempting to classify the two works as products of the theological and philosophical imagination, respectively, but this ignores the fact that both books wrestle with the same ideas culled from Thierry’s commentaries, from Septem, and from the Fundamentum treatise, as I will demonstrate.

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Puzzling over the connections between De docta ignorantia and De coniecturis is only the first whisper of a larger dilemma. How does De docta ignorantia relate to the rest of the Cusan corpus penned over the next two decades? Is it, as some scholars have proposed, the indispensable charter for Cusanus’s later works, or is it a temporary flash of light sparked by its unique sources? Will the cardinal continue thinking within Thierry’s peculiar language or will other sources supersede it? Will he retain an interest in Fundamentum? Will he continue his research into other unmined quarries of Thierry’s texts, sources, and readers? Will his Christological solution prove durable as he steps beyond the editorial demands of composing De docta ignorantia and turns to other philosophical subjects? All of these questions are answered in the developments of the 1440s, 50s, and 60s. The argument I  will sustain in this chapter is twofold. First, Cusanus kept Thierry’s theology on probation in the 1440s, in both senses of probare. He kept it on trial so that he could test it further. Cusanus remained uncertain whether, or how, to invest further in the mathematical theology that had led him from the clear vision of the 1430s into the dark mystery of De docta ignorantia, and just for this reason he wanted to weigh other approaches to his Chartrian sources. Second, this is in essence what transpires in De coniecturis: Nicholas weighs an entirely different combination of Thierry and Fundamentum. His first attempt had yielded the Christology (and with it the overall structure) of De docta ignorantia. The second attempt now returns to the Boethian-Nicomachean foundation of the quadrivium. The Fundamentum treatise had rejected Thierry’s four modes in the name of preserving a Christian theology of the Word. One way around this, the solution pursued by De docta ignorantia, is to mathematize Christology. The alternative solution of De coniecturis is more obvious. If the Verbum conflicts with the mathematical mediation of necessitas complexionis, then one could simply dispense with Verbum and explore the operations of the autonomic mediation of number, reframing Thierry’s four modes in purely mathematical terms. This is the Cusan gambit in De coniecturis: to write a comprehensive Chartrian theology of the quadrivium without recourse to the Christological synthesis.

An Agenda for the 1440s in Two Sermons The fact that De coniecturis represents Cusanus’s second attempt to come to grips with the Fundamentum treatise is already witnessed in De docta ignorantia itself. Having completed his exposition of contractio (II.2–5), Cusanus juxtaposes two possible paths forward for himself before he turns to the work of citing Fundamentum (II.7–10).12 One path is the Christological synthesis outlined in Book III but here presented in nuce (II.5). The other path is that of De coniecturis, which Cusanus indicates by previewing its major innovation, the four contracted unities (II.6).

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Other hints of a new direction come almost a year later in some sermons preached amidst Nicholas’s laborious revisions of De coniecturis. Cusanus had completed De docta ignorantia in February 1440. In Christmas 1440 he preached two sermons at Augsburg in which, for the first time outside of De docta ignorantia, he reprised Chartrian ideas from Thierry and Fundamentum. Sermons XXII and XXIII thus allow us to peek into Cusanus’s mind as he transitions from De docta ignorantia to De coniecturis.13 Sermon XXII was given on Christmas Day, and like many longer Cusan sermons it is divided into parts like a mini-treatise. In the first part, Nicholas tours the landmarks of De docta ignorantia and then sketches another similar way to coordinate Thierry and Fundamentum.14 It seems that after the experience of handling Thierry’s triads, Cusanus was inspired to posit that creatures are one, distinct, and connected. To name God is to ascend from the “contracted unity” of indivision, distinction, and connection to the “absolute unity,” which is infinite indivision (or unity), infinite distinction (or equality), and their infinite connection.15 Here we see another coordination of ideas from Fundamentum (contraction) and from Thierry (the arithmetical Trinity). But next Cusanus tries something brand new, a fresh synthesis of his Chartrian sources that he will pursue further in De coniecturis by returning to the foundations of the quadrivium. In De docta ignorantia Nicholas had conspicuously praised “our Boethius,” endorsing the philosopher’s view in Institutio arithmetica that first philosophy is impossible without the categories of multitude and magnitude.16 These are the pillars of the Nicomachean quadrivium:  multitude (or number), which grounds arithmetic and harmonics, and magnitude (or quantity), which grounds geometry and spherics. (Notably Thierry had not discussed multitude and magnitude in his extant commentaries on De trinitate, but only in his recently discovered commentary on Institutio arithmetica discussed by Caiazzo.) Cusanus must have decided after De docta ignorantia to revisit the topic of the Boethian arithmetic, since he now announces his intention in Sermon XXII to discuss the quadrivium as well as his plans to treat the subject “elsewhere” in greater detail. Nicholas preaches that if we wish to see God, “then we should see how all rational intelligence is encompassed [claudi] by multitude and magnitude, for reason grasps nothing outside of multitude and magnitude.”17 The principle of multitude is unity, and the principle of magnitude is threefold dimensionality, in points, figures, and solids. This means that the first principle of the cosmos must be triune, because the quadrivium itself is inherently triune. The subject of his next work (De coniecturis), Cusanus mentions, will be this very “trace” (vestigium) of the divine Tri-unity as found at the heart of the quadrivium, in the nonopposition of number and quantity, or simplicity and composition.18 Then Cusanus quite appositely repeats Thierry’s arithmetical Trinity.19

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Having mentioned Thierry’s theology of the Trinity, Cusanus’s instincts are, as always, to bring it in line with the Fundamentum treatise. So in the next breath in Sermon XXII he turns to Fundamentum’s vision of the supreme Verbum enfolding all forms without mediation. The Son of the Father is like a mental word or intellect, the sapientia, ars, or ratio of God.20 Nicholas defines the Son as ars by marching through the passage in Fundamentum where the author explains how the Verbum enfolds the forms of things. God’s Word or Art is not less than God but equally eternal.21 The Art is infinite reason that enfolds the differences of forms within its eternal unity.22 Only in God’s Art do beings attain precise equality, since in their material form they are only images.23 Hence the whole world of forms, as enfolded in God’s Art, is God.24 All of this comprises the first half of Sermon XXII. In the second half, Cusanus preaches straight out of De docta ignorantia III. God assumes human nature because it is the medium that enfolds all creatures universally.25 “Christ is the Lord,” writes Cusanus, “because above every creature he is conjoined to the absolute maximity; for nothing can be greater than the one in whom infinite power is completed and perfected, since he is God and is the Art itself or infinite form of all the things that are.”26 The slight difference in emphasis when compared to De docta ignorantia—here Christ is less aequalitas than ars—might be due to the demands of a Christmas sermon on Christ’s human nature, but it may also hint that the Christology itself is drifting in a new direction away from the mathematical theology of De docta ignorantia.27 Let us review the moves made by Cusanus in this dense sermon. First, he tests out another triad in connection with Thierry’s arithmetical Trinity. Second, he sketches a new approach to the arithmetical Trinity by way of the quadrivium, discovered independently of Fundamentum and Thierry alike, apparently through his own study of Boethius. Third, he returns to Fundamentum’s account of the divine Word. Finally, Cusanus repeats the theology of Incarnation worked out in De docta ignorantia III, as a discrete by-product now separable from its sources. The quadrivium and arithmetical Trinity are one thing; the Christology is another. This structure is not accidental. Its different strata hint at a methodological divide that begins to enter the cardinal’s deliberations throughout the 1440s. The following week, Cusanus preached his next sermon on the first day of January 1441, the feast of the circumcision of Jesus, which happily for this enthusiast of Ps.-Dionysius raised the topic of the divine names. The cardinal’s train of thought in Sermon XXIII shows signs of the same new outlook. Cusanus preaches that Christians seeking God’s name are not limited by books, but like the sages Socrates and Pythagoras can also turn away from human writings to read the book of the world written by the finger of God.28 All phenomena in that visible world are ordered by their “multitude,” their “inequality,” and their “division.” Multitude originates from unity, inequality from equality, and division from connection. So

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the visible world implies a transcendent arithmetical Trinity that, itself invisible, grounds every appearance.29 This immanent triad of multitude, inequality, and division in Sermon XXIII resembles the indivision, distinction, and connection of Sermon XXII. Yet Nicholas’s new triad points more unambiguously toward the quadrivium. According to Boethius, multitude is the principle of arithmetic, inequality the principle of harmonics, and division the principle of geometry.30 So in essence Cusanus proposes in Sermon XXIII that studying the cosmos with the tools of the quadrivium reveals the arithmetical Trinity as the name of the Creator. He will explore this notion further throughout the 1440s. Cusanus also maintains in Sermon XXIII that the arithmetical Trinity can be reached in another way:  not through the quadrivium, but through the name of Jesus. In this he builds upon Sermon XX, but now from the new horizon of De docta ignorantia. Since the wisdom of Christ enfolds all human knowledge, the name of God can be heard within the speech of Jesus.31 Thus Nicholas undertakes a kind of philosophical commentary on the Lord’s Prayer, in which Jesus names God. “Our Father” reveals the unity of God; “hallowed be thy name,” the equality; and “thy kingdom come,” the love or connection.32 Cusanus underscores the ineffability of the divine name, citing Ps.-Dionysius and the Hermetic motto that God is known by all names and no names. But he also connects the doctrine with the epistemology of De coniecturis, since all human language is, for this theological reason, inherently imprecise or conjectural, operating under the shadow of Ps.-Dionysian unknowing. “Not only the name of God is unnameable,” Nicholas preaches, “but also the precise name of any kind of thing. The name of God is reflected in all names as in an image; and just as the Word of God is the infinite Name, thus also is it the infinite Speech, and all languages and all speech are its unfolding.”33 As the manifold cosmos unfolds the One, so the manifold of language unfolds the one unspeakable Word.34 This infinite Name has nonetheless been given in the concrete name “Jesus” by means of the Incarnation, a divine self-naming revealed on the day of circumcision.35 In Sermon XXIII, Cusanus thus discovers that the Incarnation leads him to the same negative theology that he had grounded on number, equality, and mathematical precision in De docta ignorantia. Just as in Sermon XXII, Christology and quadrivium are two routes to the same goal. What we have detected in these two sermons are the tremors that presage a rift within Cusan theological discourse after De docta ignorantia. In effect, Cusanus has dissolved the mathematized Christology of Book III, decoupling Thierry from Fundamentum, and then has reconstituted two fresh possibilities in its place: the purely quadrivial theology of De coniecturis and a Verbum theology separated off from mathematical mediation. De coniecturis will use some of Fundamentum’s vocabulary, but attempt to think mathematical theology without Christology. Likewise, the Christology left over will have no more commerce with number,

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measure, or the quadrivial arts, but only (as in mainstream Neoplatonism) with the world’s emanation or the intellect’s ascent. And yet both new directions will be haunted by the absent ghost of the other as Cusanus’s works unfold in the 1440s. De coniecturis will have an awkward relationship with Thierry’s texts, and alongside the Christological opuscula of the 1440s a distinct geometrical project will arise. It used to be said that De docta ignorantia was the charter document for all of Cusanus’s later works. This grand theory seemed tempting only until the diversity of those later writings were better understood. But a more modest version has some claim to truth. In the immediate wake of De docta ignorantia, these sermons set the agenda for some of the major developments of Cusan theology in the decade of the 1440s, because they mapped out a new way for Nicholas to coordinate his discordant Chartrian sources.

The Neopythagorean Counterexperiment The first monument to the cardinal’s new approach after De docta ignorantia is De coniecturis. Josef Koch has argued that De coniecturis is the fruit of a prolonged rewriting of the 1440 treatise that supersedes the latter’s mystical preoccupations by erecting a new henology worthy of Plotinus or Proclus.36 But this dense book is not a sequel in the sense that it takes a step forward in an inevitable evolution of thought, or even less that it changes the subject from theology to philosophy, disavowing Nicholas’s earlier doctrinal concerns. Rather, De coniecturis should be understood as an alternative resolution of the same concrete problem of the Chartrian sources that the cardinal first engaged in 1440. As Bocken has pointed out, the hints at a forthcoming sequel already in De docta ignorantia “show that Nicholas wishes, with his second major work, to treat a problem already given in De docta ignorantia, but which is of such a nature that it demanded a new, autonomous book.”37 De coniecturis marks a radically new approach to what are nevertheless similar themes: epistemological humility (whether conjectures or learned ignorance), an anthropological focus (whether the human intellect or the perfect intellect of Jesus), and the arithmetical coordination of the One and the many.38 As a second try at the same feat, De coniecturis is no less dependent on Thierry and Fundamentum. In De docta ignorantia Cusanus had retained the organization implied by the Fundamentum treatise itself, adapted Thierry’s contributions accordingly, and sought to suture them together in a third book. In De coniecturis Cusanus again wished to harmonize Thierry’s Neopythagoreanism with Fundamentum, but now with some distance he is prepared to frame the synthesis on his own terms. Nominally, of course, De coniecturis explores how human knowledge, restricted to learned ignorance, operates through “conjectures” which approach precise truth but never achieve it. But in substance, the treatise offers more

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than an epistemology of conjecture, just as De docta ignorantia contains far more than an epistemology of learned ignorance. Such titles provided Nicholas cover for the challenging editorial project, executed twice over, of thinking through the meaning of his Chartrian sources. What one finds in De coniecturis is a radically Neopythagorean theology, far beyond what Thierry had dared. This new mode of thought results directly from Cusanus’s new tactic for harmonizing Thierry’s theology and Fundamentum’s critique. In De docta ignorantia, Cusanus had borrowed Thierry’s concept of aequalitas essendi and had linked it to the arithmetical Trinity to define the divine Word. Then he borrowed contractio from Fundamentum to define the humanity of Jesus as the universal contraction. The synthesis of these two produced the mystical theology of the Incarnation in Book III. In De coniecturis one finds a familiar pattern. First Cusanus focuses on the foundations of the Boethian quadrivium (multitude and magnitude) and connects them with Thierry’s arithmetical Trinity. We have just seen this fresh reading of Thierry foreshadowed in the two Christmas sermons. Then Cusanus adapts the four modes of being from Fundamentum but dresses them up as four interconnected “unities,” a maneuver first tested in De docta ignorantia II.6. As in 1440, these two readings of Thierry and Fundamentum are woven together in De coniecturis. But now Cusanus achieves his synthesis not through a mathematized Christology, but rather through an integrated henological vision (or “monistic idealism”) that echoes not a few elements of Neopythagorean theologies past.39

Pythagorean Motifs Let us begin with Nicholas’s new interpretation of the arithmetical Trinity in De coniecturis. As we have seen, Thierry connected the quadrivium to the Trinity in unprecedented ways, particularly in his arithmetical analogy. In De coniecturis Cusanus followed the example of the Breton master when he argued that the principles of the quadrivium themselves stem from the Trinity. When viewed up close, the cardinal’s radical teaching on the quadrivium outstrips not only De docta ignorantia in its Neopythagoreanism but even Thierry’s original doctrine. For Cusanus now suggests that the mind’s mathematical operations of unity, equality, and connection reflect its divine exemplar, the arithmetical Trinity. Thierry had implied that the divine Trinity was the source of all numeration, but now Cusanus takes the further step of explaining why. Since the mind’s operations reflect a Creator who intrinsically mathematizes, as Thierry held, Nicholas suggests that an analogous mental trinity is the source of mathematicals in the created realm, such that the human mind enfolds within itself the foundations of the quadrivium. “The mind’s unity enfolds within itself all multitude,” he explains, “and its equality enfolds all magnitude, just as its connection enfolds all composition.”40 Just

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as unity generates equality within the arithmetical Trinity, so in the mental triad multitude generates magnitude.41 Hence the mathematical structure of cognition derives from the Trinity and reveals the mind’s eternal origin in God’s number, in whose mathematical activity it participates. Number is also the “first exemplar” of the human mind itself, in its every rational activity. Just as number is composed of nothing but itself (as Boethius teaches in Institutio arithmetica), so too the mind measures nothing but itself when it reasons mathematically. Like Philolaus, Cusanus conceives of number as prior to all things, since even the basic categories of unity and alterity are arithmetical functions; there is no access to beings before their number.42 This doctrine of the Trinitarian origins of the quadrivium is not a mystical embellishment but carries important consequences. First, by citing from Boethius’s Institutio arithmetica in order to modify Thierry’s arithmetical Trinity, Cusanus signals once again that he understands just where to turn to amplify the Breton master’s thought. Despite not knowing Thierry’s identity, and despite confusing him with his critic, Cusanus has nevertheless grasped the quadrivial heart of his mathematical theology. Just as in his perspicacious portrait of Christian Neopythagoreanism in De docta ignorantia I.10, Cusanus has again zeroed in on those passages in Boethius that most directly telegraph the mind of Nicomachus of Gerasa. Moreover, by using the Trinitarian language of generation to define number and quantity, Cusanus radicalizes Thierry’s Boethian traditions. For if multitude “generates” magnitude, then number enfolds not only the sciences of arithmetic and music, but also geometry and astronomy, consolidating the double foundation of the quadrivium in Nicomachus and Boethius into one. Yet this potent, arithmetized quadrivium is itself a projection of the human mind, according to Cusanus. In this the cardinal echoes not just Philolaus and Archytas, who suggested that all mental activity is inherently mathematical, but Syrianus and Proclus, who held that the mind projects the four mathematical sciences out of itself by virtue of its own numerical composition.43 Cusanus apparently read the Fundamentum treatise differently in De coniecturis. The centerpiece of the new book is a fourfold hierarchy of unities that coordinates different planes of numbers into one cosmic system. These four unities provide the governing rubric to which the cardinal constantly returns in his various investigations throughout De coniecturis. Cusanus uses several different schema to define the four unities that can be summarized as follows (Table 8.1).44 He explains that since the number four contains the decad (1 + 2 + 3 + 4 = 10), the progression of numbers from unity is completed in the fourfold.45 Each unity is a descending degree of contraction from the divine absolute (the first unity), and each successively unfolds the enfolding of the prior unity. The first unity, God, is transnumeric, and the subsequent three unities correspond to epistemological faculties of intellect, reason, and sense, moving from higher degrees of

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Table 8.1  Cusanus’s four unities in De coniecturis First unity

Second unity

Third unity

Fourth unity

Simple [1]‌ 1 God Simple number Ineffable  numbers Truth itself Purely absolute

Denary 1 10 Intellect Root numbers Intelligible  numbers The true Absolute/  contracted Enfolds third [Unfolds first?]

Centenary 12 100 Reason Square numbers Rational  numbers Verisimilitude Contracted/  absolute Enfolds fourth Unfolds second

Millenary 13 1000 Body Cubic numbers Perceptible  numbers Confusion Purely contracted

Enfolds second –

– Unfolds third

certainty and truth to lower. As the unities progress further from the divine absolute, they grow increasingly contracted. By all appearances this schema of the “four unities” filled in Nicholas’s sketch in De docta ignorantia II.6. The fully extrapolated version in De coniecturis reminds one all the more of the four modes of being theorized by Thierry. Even if some details have been altered, Thierry’s basic pattern—a fourfold universal hierarchy of being organized by reciprocal folding—is difficult to deny. We can also discern, overlaid atop Thierry’s original account, the hallmarks of the version preserved in Fundamentum: one singular divine instance and three lesser reflections, ordered through the opposition of absolute and contracted. Of course, unlike Fundamentum, the second and fourth modes are not dismissed in the four unities. Nicholas might have misunderstood the force of Fundamentum’s critique, as I considered above; or he may have rejected it, as he appears to do in later works. Or, as seems to me most likely, Nicholas may be testing the utility of that controversial doctrine in modified form in De coniecturis precisely because he has not decided whether to discard or embrace it. For now, the fourfold hierarchy remains on probation. Now against this view, Koch has argued that the four unities in De coniecturis represent in fact Cusanus’s greatest departure from the more rudimentary De docta ignorantia.46 For they allowed him to leave aside the Seinsmetaphysik of 1440 in favor of Neoplatonist henology (Einheitsmetaphysik).47 Given the influence enjoyed by Koch’s theory, it is important to consider briefly the strength of

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his argument.48 To begin with, his certainty about the philosophical novelty of De coniecturis requires him to rule out prima facie the possibility that the book’s henology flowed from Books I and II of De docta ignorantia.49 Instead Koch posits that Cusanus must have discovered a “Platonic or Neoplatonic text” that inspired him to discard De docta ignorantia entirely: in the intervening time [Cusanus] has concerned himself intensively with the study of Platonic or Neoplatonic texts. I can only formulate this indeterminately for the simple reason that I do not know at this point what the new source is. Cusanus himself does not inform us. In De docta ignorantia he cites all sorts of authors; in De coniecturis one finds not one explicit citation! If one then considers how often Cusanus was inspired by his readings to attempt new philosophical endeavors, one can also perceive in this text a fertilization from the outside.50 Koch’s case accordingly turns on the identity of the putative Platonist source. It seemed at first that his hypothesis had been stunningly prescient, for a few years later Rudolf Haubst published his discovery that Cusanus had indeed annotated Latin fragments of Proclus while composing De coniecturis.51 But upon closer examination the Proclus excerpts prove insufficient to explain the complex interconnections of the four unities in De coniecturis. Koch himself admitted this when a few years later he tried to supplement Haubst’s discovery with the additional Neoplatonist source of Meister Eckhart.52 But as I will show, neither Proclus nor Eckhart illuminate the inner structure of the four unities so well as Thierry’s reciprocal folding or Fundamentum’s contraction.53 As several scholars have realized, nothing precludes Koch’s missing Neoplatonist source from being both the Proclus excerpts and Cusanus’s challenging Chartrian texts, including Thierry and Fundamentum. For Bocken the question is not only whether Cusanus had access to Proclus, but how he appropriated Proclian ideas for his own usage.54 Haubst came to conclude that the philosophical signature of De coniecturis comes from Nicholas combining the Proclian excerpts on unity with Thierry’s structure of reciprocal folding.55 So it makes the most sense to imagine these influences working together.56 If, as I have proposed, Nicholas wrote De coniecturis as a second interpretation of Fundamentum, he would have welcomed an authoritative Neoplatonist framework to help ground his independent recasting of the four modes. Perhaps the event that encouraged Nicholas to set out on his own in De coniecturis—not as a contradiction of Book II of De docta ignorantia but as a renewed engagement with its constitutive ideas—was precisely the Latin Proclus excerpts that he fortuitously discovered around 1440. Taken together, Cusanus’s new approaches to Thierry and to Fundamentum comprise the major contribution of De coniecturis. As the image of the Trinity the

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mind is the source of quadrivial dimensions, and such universal quadrivial understanding is structured through a fourfold system of unities. This means that De coniecturis represents as much of an encounter with Chartrian traditions as De docta ignorantia was. But the cardinal’s editorial efforts in the sequel were not yet another experiment but in fact a counterexperiment, for this time he assembled his puzzle of sources with an altogether different end in mind. Cusanus’s self-consciously mathematical theology in De coniecturis exalts number as the prime mediator, not the incarnate Word—a synthesis of Arithmos in lieu of Logos. By accident or design, Nicholas’s new book resurrects several classical Pythagorean topics, but in each case his Chartrian sources shine through. So pace Koch we ought not interpret the intensification of Neopythagoreanism in De coniecturis as if Nicholas had wanted to replace the project of De docta ignorantia with a more refined philosophical method. It was simply his redoubled, alternative attempt to make sense of the same Chartrian dossier as in 1440. For example, when Nicholas glosses Thierry’s dialectic of unity and alterity as the odd and the even, he effectively revives the ancient Pythagorean columns of opposites (συστοιχίαι).57 Eventually De coniecturis investigates other dyadic oppositions such as male/female, soul/body, nature/art, actual/potential, and universal/specific.58 But the double-pyramid Figure “P” that graphs such oppositions relies on Chartrian ideas.59 Unity cannot be participated maximally, minimally, or equally, but only through a double mean connecting two extremes “as if in four simple and distinct modes of being.”60 Cusanus also explores ancient arithmological traditions in De coniecturis, particularly the tetrad and decad. Figure “U” depicts the universe as three worlds of three interlocking circles with God at the center, making a perfect decad.61 God is the supreme unity whose “theophanic descent” into the decad and subsequent numerical progressions orders the universe.62 But it is Nicholas’s four unities (or four modes) and not the Pythagorean τετρακτύς that accounts for that decadic organization.63 When he tries out other arithmologies, Cusanus confines his arguments to his four unities rather than “S” traditions shared by Philo, Nicomachus, and Macrobius.64 Most surprising is Cusanus’s revival of the hierarchies of number-species found in Iamblichus and in the young Augustine. Returning to his favorite line from Boethius (“in the mind of the Creator number is the first exemplar of things”), he now states that such divine numbers through which God created the world are “real, ineffable numbers,” and that the human mind understands that world through its own “rational numbers.” The mind expresses itself in number, just as God expresses God in the Word.65 Like Iamblichus, who held that lower numbers are the images of higher ones, Cusanus holds that numbers only count things of their own or lesser degrees in the fourfold hierarchy of divine (simple), intelligible (root), rational (square), and sensible (cubic) unities.66

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In contrast to his Greek predecessors, Cusanus grounds such Pythagorean-sounding doctrines on the four unities in De coniecturis, which to all appearances are a dressed-up version of Fundamentum’s paraphrase of Thierry’s four modes. But as we have seen, this doctrine is precisely the point of greatest tension among his different Chartrian sources. Thierry of course had postulated the four modes for a definite reason, namely, to define the intimate relationship between mathematics and theology. After failed attempts in Commentum, it was finally the enfolded hierarchy of four modes in Lectiones and Glosa that explained the quadrivium’s pertinence to Trinitarian theology. When the author of Fundamentum rejected “Platonist” mediation, he likewise rejected the modal theory that had grounded Thierry’s mathematical theology, and was glad to have done so. But in De coniecturis Cusanus wants, so to speak, to have it both ways, and in the end this makes it difficult for him to specify the relationship between mathematics and theology. It would have made sense to build a purely Neopythagorean treatise on the basis of Thierry’s system from Lectiones, which had resolved that disciplinary question. Or he might have drafted a mathematical theology by his own lights that left the fourfold system unmentioned and leaned directly on Boethius and the arithmetical Trinity. But what drove Nicholas first to adapt Fundamentum’s account of the four modes in a Neopythagorean direction, and then to use this thinly veiled account of four unities to wrestle with the very disciplinary questions that Lectiones had already answered? It makes one wonder whether Nicholas had read Lectiones at all before De coniecturis. Lest their similarities appear merely coincidental, I will first show why it seems very likely that Cusanus used Fundamentum’s four modes as the basis for his four unities. Then I will examine what happens to the disciplinary configuration in De coniecturis built atop this shaky scaffolding.

The Four Modes Reprised It is rather easy to find clues that Cusanus modeled his four unities in De coniecturis on the four modes of being from 1440. To begin with, he constructed his system of unities on the same pattern from the Timaeus that Thierry had first tried in Commentum. Between the absolute extremes of the first unity (God) and fourth unity (solids), two means are required.67 The unities are four ways to view the same cosmos, just like Thierry’s modal system; in the first unity, all things are the first unity, and in the second the second.68 On two occasions Nicholas even let down his guard and called the fourfold system “modes” instead of unities.69 But in the main, he followed the presentation of the four modes in Fundamentum. The four unities are integrated by relations of reciprocal folding, but unlike Thierry’s model, the Cusan system (like Fundamentum) proceeds unilaterally

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from the highest, absolute unity to the lowest contracted one.70 He even alludes to Fundamentum’s distinctive image of reciprocal ascent and descent through degrees of contraction.71 Fundamentum had deviated from Thierry by marking off the first unity from the lesser three, which comprise one contracted universe. Likewise, Cusanus assigns the first unity to God and the other three to levels of human cognition.72 The first unity enjoys all the divine titles in De docta ignorantia: the arithmetical Trinity, absolute unity, and even “absolute necessity,” the first mode of being.73 Nicholas maintained this apparent subterfuge that the four unities are not the four modes, but something new of his own invention. Yet he also mentioned Thierry’s four modes in passing later in De coniecturis. It is odd that he included them and odder still that he buried them deep among some incongruous, minor applications of the four unities themselves. In that passage Cusanus likens the four modes to degrees of “conjectures” that correspond to the four unities. True conjectures ascend from possibility toward necessity through four modes: “shadowy possibility,” “actual possibility,” “second necessity” or “necessity of consequence,” and finally “absolute necessity.”74 Note that Cusanus has closely obeyed the dictates of Fundamentum and censored Thierry’s original terms for both the second and fourth modes. Moreover, he has converted the modes’ ontological meaning into a logical one and erased the reciprocal folding that structured the modes in Thierry’s account and indeed in Fundamentum. Conveniently this circumvents the controversies among his sources concerning the second mode as mediator, the plural exemplars, and the anima mundi. In this way Nicholas seems to have confined the original source of his governing theory to a minor list of examples illustrating that theory’s utility—a possible symptom of either confusion or anxiety over the status of the four modes. Cusanus was apparently drawn to the Platonic harmony of their fourfold structure and considered them part of the textual heritage of his Boethian commentator. But he could not ignore the treatment they had received at the hands of Fundamentum, and in particular the mediating second mode that had, apparently, so fascinated him in De docta ignorantia. If De coniecturis represents an alternative attempt to harmonize his Chartrian sources, then naturally Cusanus desired to retain as much as possible from Fundamentum. But his solution shows signs of a bad conscience, as he smuggles the four modes into the treatise and protests, too much, that in fact they illuminate the central concept of conjecture. There is another curiosity in De coniecturis regarding the cardinal’s fraught relation to the four modes. Throughout the entire treatise the second mode is never properly named as necessitas complexionis, that is, in terms of its inner connection with folding.75 This is true even though in many ways Cusanus’s invention of the four unities of De coniecturis extrapolates Thierry’s notion of necessitas complexionis to other modes of being. What I mean is that the function of the second mode in

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Thierry’s mature thought is precisely the paradigm of what each of the four unities is designed to be, and that the domain known uniquely in Thierry’s second mode, mathematics, is the universal vocabulary of the Neopythagorean endeavor of De coniecturis. It is as if Cusanus took Thierry’s account of the second mode of being and universalized it as a general model for the four unities. Thierry had defined the second mode as the unfolding of divine simplicity into order and progression of number and hence he made it the exclusive domain of mathematicae rationes.76 Cusanus effectively duplicated that function in his account of the second, third, and fourth unities. Unfolding the higher number and enfolding the lower—arithmetical mediation by reciprocal folding—is precisely what constitutes each of the four unities.77 The cardinal’s fascination with the second mode is so prevalent that at first it is difficult to see. But if we revisit his preview of De coniecturis in De docta ignorantia, it becomes clear that from the beginning Cusanus centered his promised sequel on the “second unity,” the decadic mediator that bridges God and world.78 If the function of necessitas complexionis were indeed assumed by each of Cusanus’s four unities in De coniecturis, this would confirm our suspicions that the cardinal remained especially interested in the second mode. For example, he uses the arithmetical term “root” (radix) to distinguish the special office of the second unity. After the first divine unity, the second unity is uniquely among the others the unitas radicalis or “root unity,” possessing the unity and simplicity of God in virtual form. This is so, Cusanus continues, because the second unity is nothing other than the primordial decad that has no root, since it is the insuperable source of all numerical difference.79 By a “theophanic descent” the first divine unity “progresses” into the decad just as the decad makes a simultaneous return to unity.80 In Cusanus’s fourfold system, then, the second unity uniquely mediates divine unity to the lesser planes of number, and it does so as the decad without root. This closely resembles Nicomachus’s elevation of the decad to equivalence with the Logos and anima mundi. Rather than replace the second mode with the divine Verbum as Fundamentum had done, and as Cusanus had faithfully repeated in the Christology of De docta ignorantia, now the cardinal quietly christens the primordial decad as the heir to necessitas complexionis, one that continues its office of “unfolding” divine unity and thereby commences a progression of further numerical unfoldings into the lesser unities of soul and body. One can only imagine how the author of Fundamentum would have greeted this Neopythagorean recension of his treatise. Such questions about the four unities disrupt the epistemological and disciplinary distinctions that Cusanus attempted to make in De coniecturis. As Jean-Michel Counet has observed, the whole fourfold system hinges on the second unity, yet this is the most ambiguous of the four.81 Recall that the original function of Thierry’s four modes was to answer Bernard of Chartres’s question: how did the

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quadrivium relate to Christian doctrine? Thierry’s breakthrough came when he coordinated theology and mathematics through folding, specifically at the fulcrum of the second mode in Lectiones and Glosa. The mediation of necessitas complexionis established number as the first unfolding of divine simplicity and carved out a distinct space for mathematical science that was not yet theological. It seems that in De coniecturis Cusanus wanted to embrace the mathematical method of his Chartrian sources even more fully than he had in De docta ignorantia. But one of those sources (Fundamentum) demanded the suppression of necessitas complexionis, the very mode that the cardinal’s new hyper-Pythagorean project would most require. So Cusanus shunted Thierry’s four modes into a corner of the book and instead foregrounded his own arithmetical simulacrum of them, the four unities. The contradiction between his sources forced Nicholas to do without the very modal theory that would justify his desired revival of Thierry’s mathematical theology in De coniecturis. To make matters worse, by universalizing the function of the second mode in his schema of unities, Cusanus did away with the original modal theory’s critical disciplinary function.

Dissonance One can predict the result—Cusanus will likely have difficulty distinguishing the boundary between theology and mathematics—and this is precisely what one finds. As Table 8.1 suggests, the cardinal’s lowest three unities correspond to the faculties of intellect (intelligentia), reason (ratio or anima), and perception (sensus). Had Cusanus retained Thierry’s terminology, intellect would correspond with the second mode, which in Thierry’s view names the domain of mathematical science. But in De coniecturis, Cusanus assigned the mathematical domain to ratio. This demotion of the epistemological status of mathematical knowledge does not sit comfortably with the prominence of number and Pythagoreanism in the book as a whole. It also makes it difficult to tell whether the activity of intelligentia (the second unity) is itself theological or mathematical. On the one hand, Nicholas wants to say that the second unity of intellect is the highest mode of human knowledge. It mediates divine unity to the other three by converting the simplicity of the One into the primal decad of intelligible numbers. Whether this mediation is achieved through unfolding or some other mechanism is unclear. Cusanus seems to regard the sequence of descending unities as homo­ logous, and the lower unities descend through unfolding.82 But he never explicitly names the movement from first to second unity as unfolding, perhaps influenced by Fundamentum’s condemnation of the second mode as a false explicatio. On the other hand, Cusanus characterizes the second unity as supramathematical. Intellect is the simple “root” that enfolds reason, which unlike reason can enfold opposites.83 But the disjunction of opposites is what enables reason

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to posit multitude (the difference of number) and magnitude (the difference of quantity).84 “The root of all rational assertions is that a coincidence of opposites is not attainable,” writes the cardinal. “Hence every number is either even or odd; hence, there is numerical progression; hence, there is numerical proportion.”85 In other words, the inferiority of reason is what makes it able to think mathematically.86 Just as divine precision is superior to intellect, so intellect possesses a “precision” that reason always seeks in mathematics but never wins.87 Reason separates unity’s procession from its return from alterity; only intellect grasps the full circuit as a single bond and so knows the truth beyond unity and alterity.88 Intellect recognizes the “inadequacy” and “failure” of reason’s terms and ultimately “rejects” them.89 Intellect is not itself quantitative and uses mathematical concepts analogically.90 Nicholas finds himself in a zero-sum situation: the higher that he elevates the second unity as the highest degree of theological insight, the more he marginalizes the mathematics known by the third unity. Mathematical discourse is now a symptom of the mind lingering in a lesser domain. God is no longer the missing precision of mathematics, as in De docta ignorantia; mathematics illuminates theology only indirectly, since its precision is found entirely within the human intellect. Truly theological vision, according to De coniecturis, turns one away from the myopia of multitude and magnitude and dissolves number in the absolute One. As one ascends up the four unities toward God, “one sees every number of reason resolved into the most simple Unity.”91 The closer the ascent to the One along the axis from sense to reason to intellect, the less mathematics has to do with theology. The form of number is always penultimate and must finally be escaped in order to pass into the numberless intellect and the formless One. In De docta ignorantia Cusanus had repeated Fundamentum’s critique of the four modes, then sought to redress the rejection of the second mode elsewhere. Since his goal was not pure Neopythagoreanism in 1440, this shift into Christology was tenable. But in De coniecturis Nicholas handled the second mode (as second unity) quite differently, even while he enthusiastically amplified Thierry’s theology of the quadrivium into an integrated henological system. By organizing his four unities in terms of degrees of contraction rather than reciprocal folding— in deference to Fundamentum—Cusanus inadvertently dissociated number and mathematics from the second mode. The result, as we have seen, is a surprising denigration of quadrivial thought, in spite of the Neopythagoreanism that floods the pages of De coniecturis. Number was no longer a sure trace of the inner divine life, as Thierry had maintained, for now number might also represent an illusion relative to one’s current degree of contraction that should ultimately be left behind. In place of Thierry’s hard-won quadrivial theology, we find in De coniecturis a pervasive Neopythagorean idiom that yet remains deeply uncertain of its theological purpose.

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All agree that De coniecturis was self-consciously a revision of De docta ignorantia. But I  have tried to show that in this endeavor Cusanus was particularly aware of the Fundamentum treatise and was still busy trying to broker a new compromise among his unruly Chartrian sources. His plan in De coniecturis to embrace Thierry’s Neopythagoreanism but do so without the suspect four modes may have run aground. Nevertheless the book demonstrated the range of insights that mathematical thinking could provide the German cardinal and allowed him space to wield Thierry’s powerful concepts of folding without the aid of his wonted high Christology. Cusanus never repeated anything like the extreme Neopythagoreanism of De coniecturis again. For our own appraisal of the cardinal’s mathematical theology it is crucial to observe this reluctance and even tacit disavowal. In its place Cusanus turns to actual mathematical exercises of the kind he had merely reflected upon in De coniecturis, pursuing the ascent from reason to intellect no longer in theory but in practice. The path forward for his theological writings, as we shall see, thus divides in two for the next decade. Only years later, once Cusanus had returned to his Chartrian sources again, would he be able to exorcise the ghost of De coniecturis and construct a more durable foundation for mathematical theology.

Two Paradigms of Mediation In February 1440 at Kues, Nicholas must have been satisfied at his success in harmonizing his clamorous Chartrian sources. But he was already hard at work on a sequel that assembled the puzzle of his sources in a very different way. By the mid-1440s, the German cardinal had arrived at not one but two editorial solutions, not one but two theological paradigms for stabilizing Platonic mediation. But having invented two competing paradigms, Cusanus now had to contend with both. If the mathematized Incarnation of De docta ignorantia momentarily fused Logos and Arithmos, the Neopythagorean alternative in De coniecturis dissolved that unstable compound into its two components, so to speak, which then reacted with each other in Cusan thought over the next two decades. As Schwarz observes, what we find in Cusanus is neither a continuous philosophical system, nor an absolute caesura between periods, but rather a “permanent dispute” (I would say productive tension) between different horizons of thought playing out across the cardinal’s entire oeuvre.92 This was not a division between doctrinal theology and secular philosophy—a distinction that Cusanus, like Ps.-Dionysius, Bonaventure, or Eckhart, would not have readily granted—but rather between autonomic mediation by number and heteronomic mediation by the incarnate Word. In this light Cusanus’s trajectory in the 1440s becomes intelligible in a whole new way. We see two major developments in this decade. First, Cusanus penned

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a series of four impressive short works between 1444 and 1446 concerned with Ps.-Dionysian negation and with the kind of cosmic Incarnation broached in De docta ignorantia. Direct lines can be drawn between the Christmas sermons of 1440–41 and the themes and sources of those opuscula.93 Second, in 1445 Cusanus launched an entirely new intellectual project that sought to solve theological problems with geometrical proofs. As Fritz Nagel has written, in the wake of De coniecturis Cusanus decided to pursue a mathematics that was “independent from the implications conveyed by its instrumentalization for the theologia mathematica of docta ignorantia.”94 More investment in Christology and more investment in mathematics: these are the two threads running through the 1440s, as Cusanus grappled with his two different paradigms for mediation. Both developments are unthinkable without the mathematized Christology of De docta ignorantia, but Nicholas neither repeated that solution nor returned to the Neopythagorean option of De coniecturis. In the early 1440s, Cusanus continued to search for new applications for Thierry’s fecund vocabulary. As he had preached in Sermon XXIII, “lest one err, one should approach understanding all things with respect to their enfolding and their unfolding.”95 In a reflective letter to archdeacon Rodrigo Sánchez de Arévalo in 1442, Nicholas used folding to explain the priority of the papacy in the church.96 In these years he was busily attending imperial diets in Mainz, Frankfurt, Nuremberg, and Aschaffenburg as papal legate for Eugenius IV. But he also preached extensively while working to complete the four opuscula. In sermons from March and April of 1444 Cusanus called the Word the divine ars infinita and began to sketch his concept of filiatio.97 In May and June he revisited themes from Thierry’s theology including the arithmetical Trinity and reciprocal folding.98 On Christmas Cusanus shifted back to the Incarnation and preached on Christ as the eternal ground of creation, the end of creaturely motion, and the enfolding of all things.99 The first two opuscula, Dialogus de deo abscondito (1444) and De quaerendo deum (1445), returned to the Ps.-Dionysian theology of divine names found in Sermons XXII and XXIII.100 But in the second two, De filiatione dei (1445) and De dato patris luminum (1446), Nicholas followed his own admonition in Sermon XXIII to discover Christ hidden in the intellect and in creation. Thus in De filiatione dei Cusanus revisited soteriological questions left unanswered by the Christology of De docta ignorantia.101 “Filiation” is the cardinal’s term for deification or theosis attained through divine Sonship.102 Through philosophical reflection on difference and unity in the world, the intellect ascends to union with God by grasping its own virtual likeness to the Filius. All of creation works like a “school” to tutor the human mind to exercise its potentially universal knowledge.103 The manifold of created difference incites the mind to think God through the “ablation” of otherness, the “resolution” of multitude, and the “transfusion” of unity.104 The mind

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is like a living mirror that stretches and purifies itself to better mirror the infinite divine Mirror (the Son or Word), in whose reflection it glimpses an image of itself beholding all the fragmentary mirror-shards that form the created order.105 The lessons learned by the intellect on the path to filiation in the Word are thus essentially henological—discerning unitary refractions of the One amidst the many. Curiously, however, Cusanus never commends mathematical knowledge as a strategy for attaining the resolution of difference through numerical abstraction, as one might have expected from the author of De docta ignorantia and De coniecturis. Where Christological mediation reigns, there is no need for number. In the next work, De dato patris luminum, Cusanus describes how all creatures manifest God through the supreme manifestation, the incarnate Son. Since creation descends from the Father after the pattern of the Son, the generation of the Son discloses the theophanic purpose of all creation.106 World and Christ, creation and incarnation, are thus bound together in mutual coordination. The double origin of the world in eternity and time reflects their singular intersection in the incarnate Word. Because creation is Christomorphic, it can be defined with all the paradoxes that mark the classic Christological formulations.107 Hence there is a deep thematic unity between De filiatione dei and De dato patris luminum. If the first explains how the intellect ascends to God through the world, the second explains the descent of the world to the intellect. If the first maps out a Christomorphic path of ascent, the divinizing journey of filiation, the second plots the vector of an equally Christomorphic descent, as God gives Godself in sensible form, not only in the body of Jesus but in the body of the entire emanated cosmos.108 In these works Cusanus clearly refocused on the Neo-Chalcedonian Christology of De docta ignorantia. But occasionally Pythagorean motives still rise to the surface, as if number haunted the mediation of the Word. In De filiatione dei, for example, Nicholas once compares the ubiquity of the unattainable One to the supranumerical monad present in every decad.109 In De dato patris luminum, creatures descend from the creator in the way that an infinite series of numbers descends from God’s unity. Just as the Creator used the quadrivium in De docta ignorantia, now “the Creator’s creating is comparable to reason’s calculating or numbering.”110 Yet in these same years, while Cusanus was expanding the Incarnation theology of De docta ignorantia in new directions, he was also busy completing the first of his dozen geometrical works, all of which were devoted to the task of squaring the circle.111 In September 1445 he dedicated De transmutationibus geometricis to his mathematician friend from Padua, Paolo Toscanelli. Nicholas never achieved his goal, and his methods have since been shown to have serious flaws. But in his mind the stakes could not have been higher. The human mind was like a polygon that tried through reasoning to approach the divine perfection of the circle, as the cardinal had written in De docta ignorantia. No matter how many sides one adds to the polygon, even an infinite amount, its straight lines can never be perfectly

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converted into circular curvature. But, Nicholas surmised, if someone could formulate the universal, mathematical ratio of the straight to the curved, solving the quadrature of the circle, it would be tantamount to gazing upon the mind of God. Cusanus had praised the utility of mathematical symbols in De docta ignorantia, but now De transmutationibus geometricis exalts actual mathematical proofs. Despite the labors and achievements of past mathematicians, he writes, who uncovered much that was formerly hidden to reason, the one thing lacking was a demonstration of the equality of square and circle. This appeared impossible until his discovery of the coincidence of opposites: Since no rational proportion obtains between [the straight and the curved], the secret must lie hidden in a certain coincidence of extremes. Since the coincidence takes place in the maximum (as recounted elsewhere), and the maximum is the circle that is unknown, it will be shown here that one must seek the coincidence in the minimum, namely in the triangle.112 Given the limits of the present study, we need not probe the mathematical value of the several quadrature works.113 It suffices to note that the cardinal’s dedication to geometrical proofs—neither the mathematized Christology of De docta ignorantia nor the henology of De coniecturis, but actual working mathematics—is a new component of his larger intellectual program, side by side with the theology of Incarnation and perhaps even flowing from the same source. Two years would pass before Cusanus wrote another theological work. Nominally De genesi (1447) is a commentary on Genesis, and Cusanus does address the eternity of the world and the perils of literal exegesis.114 But mostly the cardinal builds on Proclus’s interpretation of the One as the “Same” (idem) in the Parmenides and thus meditates on the problem of the One and the many. This treatise, a sequel to De filiatione dei, would seem the perfect opportunity for Neopythagorean henology.115 But instead Nicholas explains at the outset that he wants to avoid mathematical discourse, since the divine Same is not subject to number. The Same is “unmultipliable” (immultiplicabile) and cannot be reduced to merely numerical identity.116 In place of mathematical concepts, Cusanus reverts (like Proclus) to figures of dynamic movement and reflection. For example, the genesis of the universe is movement of assimilatio “toward” (ad-similatio) the Same itself.117 Paradoxically, the Same is known only through the multitude of creatures; far from being opposed to the Same, plurality and difference are proof of its assimilative power.118 Thus we see that by De genesi, even while continuing his efforts toward the quadrature of the circle, Cusanus remained ambivalent about the role of mathematical concepts within theology. In a marked departure from De coniecturis, he no longer leaped at the chance to order his henology in terms of number and quadrivium.

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This cursory glance through the 1440s suggests that Cusanus struggled over the legacy of De docta ignorantia. The focal point of the tension among his Chartrian sources was the four modes of being, which Thierry had theorized precisely in order to resolve the ambiguous proximity of mathematics and theology. After Cusanus had approved that doctrine’s removal by Fundamentum, but nevertheless also embraced Thierry’s theology, we watch him oscillate between two disciplinary poles:  an extreme mathematizing without adequate foundations in De coniecturis, followed by an uncertain return to Christology in the opuscula, culminating in a new devotion to geometrical proofs. In this period Cusanus still “evidently operates with two different conceptions of science,” one theological and one mathematical.119 From the standpoint of the 1440s, it seems clear that if Cusanus could either make peace with the four modes despite Fundamentum, or somehow access them in Thierry’s original formulation, he would take a great step forward in stabilizing his own nascent mathematical theology.

9

The Advent of Theologia ­geometrica in the 1450s In ancient and modern geometry every curving arc is defined by its sagitta, the vertical line that extends from the height of the arc down to the midpoint of its base. The sagitta marks the arc’s topmost point as it curves up and then down again, which means it also measures the space the arc defines. The taller the sagitta, the more capacious the curve. Most readers of Nicholas of Cusa agree that the sagitta of the cardinal’s intellectual odyssey is the 1450 dialogue Idiota de mente (“The Layman: On Mind”). Here Cusanus attains a new degree of clarity about the role of mathematics in his thinking, a palpable self-assurance that lasts into the next decade. But this scholarly agreement about the centrality of De mente in the development of Cusan thought masks a host of disputes among modern interpreters about the character of that development and its motivations. The position of the sagitta is fixed; only its height remains unknown. As I suggested in the Introduction, the notion of a Christian theology oriented toward mathematical concepts is difficult to imagine so long as we are constrained by historiographical narratives linking the rise of modernity with the mathematization of nature. If quantitative accounts of world order are inherently secularizing forces that deflate myths and dispel religious cosmologies, then one would have little to say about (and so might evade or dismiss) counterfactual cases like the sanguine Christian Neopythagoreanism of Thierry or Nicholas. Given this blind spot, there has been a tendency in modern scholarship on these figures to overlook or misconstrue, unwittingly but systematically, the moments of direct interface between mathematics and theology. In the case of Thierry, as we have already seen, his Genesis commentary is praised for its protoscientific naturalism, but his arithmetical Trinity (in the same work) is lumped in with medieval “number mysticism”—a convenient extraction of his mathematizing physics from his mathematical theology. The modal theory where Thierry integrated these disciplines with each other is then largely ignored. The vast Cusan corpus presents similar difficulties on a more challenging scale. Of all of his works, De docta ignorantia is in some ways the easiest case to handle.

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Nicholas revives Thierry’s Trinity, adverts to geometrical models, and lauds the quadrivium like a new Boethius, all the while hastening toward his Christology. Even to the naked eye unaware of the filaments of Chartrian sources woven into the text, Cusanus is evidently mixing ancient Pythagoreanism with Ps.-Dionysian mystical theology and reaching toward what Dietrich Mahnke has called mathematische Mystik. But De docta ignorantia is only the first Cusan text to engage mathematics, leaving ample time for Nicholas to “develop” in other directions, as the philosophical focus of De coniecturis seems at first to suggest. These two works set the stage for the crucial, more difficult question. Will Nicholas continue to pursue the mathematical theology of Trinity and Incarnation on display in 1440? Or will he find that the project cannot be sustained and seek different avenues for his mathematical interests? Because of its pivotal status in the cardinal’s development, De mente has played a central role in this controversy. Every interpretation of the dialogue is codetermined by a particular evaluation of De docta ignorantia and by a judgment as to which texts after 1450 are the most important. As I stated above, Cassirer connected Cusanus’s modernity with his mathematical epistemology and both with De mente, a tradition continued today by Kurt Flasch. In this reading, De mente signifies the moment when Nicholas reveals his plan to cast off the constraints of Christian doctrine and to aim instead at a purely philosophical henology. He steps away from the mystical darkness and Pythagorean enthusiasm of De docta ignorantia toward a more sober epistemology based on the mind’s self-production of number, not unlike the Proclian projectionism that presaged Descartes’s mathesis universalis or Kant’s a priori categories. Whenever interpreters quietly downplay Cusanus’s medieval sources or his energetic explorations of Trinity and Christology after 1450, they betray the same premise. If one expects that the great cardinal will eventually, inevitably, leave behind his late medieval assumptions, then it is natural to anticipate that his protomodern breakthrough will be mathematical in character, and therefore, the reasoning goes, less theological. In this manner, interpretations of Cusanus’s modernity have too often relied upon the opposition of theology and mathematics, even though his agenda was clearly not to separate the two but to marry them. Such interpretations of De mente accord so neatly with the modern mathesis narrative discussed above as to invite our suspicion. Against them one can point to good evidence that the new mathematical epistemology in De mente was already previewed in De coniecturis, that Cusanus returned to Ps.-Dionysian mysticism and Christology already in the early 1450s, and that he never stopped writing about the Trinity. In 1453 he even designed a two-volume work that conspicuously integrated mathematics and theology. But to respond with these data alone fails to address some more fundamental problems in modern interpretations of Nicholas of Cusa’s mathematical theology. As we peer over the edge of the 1450s, these must be addressed to clear the way for the next two chapters.

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A few scholars have taken up the task of defining Nicholas’s mathematical theology, but these tend to float above the details of historical context and development. For example, Julia Inthorn and Michael Reder maintain that Cusanus deploys two complementary Denkformen, the binary (mathematical) and the dialectical (philosophical).1 Both obey the principle of noncontradiction, but where the binary thinks in terms of precise differentiation, and hence treats quantities, the dialectical treats relations through synthetic metaphors. But having stated this, it proves difficult for them to explain how to integrate the competing forms of thought.2 Gregor Nickel tries to bridge the two disciplines by considering the structural homologies shared by theology and mathematics:  abstraction, recursion, equality, infinity, coincidence of opposites, and freedom.3 Both of these studies, unmoored from the cardinal’s concrete development, imagine what a pristine mathematical theology could look like hypothetically rather than chronicling Nicholas’s actual achievements in their untidy historical details.4 At the same time, several prominent interpreters who have delved into those particulars have doubted whether Cusanus’s mathematical theology still stands after the revolution of De mente. One common view is that in 1450 Cusanus effectively abandoned his earlier Pythagorean projects. This can be partially traced back to Theo van Velthoven’s otherwise excellent study of Cusan epistemology, which chronicles the cardinal’s shift from the “weakness” of knowledge in De docta ignorantia to the creative “productivity” of knowledge in De mente. Given his interests, Velthoven naturally considers the 1440 treatise to be less about Trinity or Christ than about learned ignorance.5 But if one is not careful, this narrative can perpetuate a whiggish reading of De mente as the dawn of reason putting a halt to the cardinal’s medieval fantasies. If De docta ignorantia were chiefly concerned with negative knowledge, then the entire work could indeed be superseded by a productive epistemology in 1450. Admittedly Velthoven repeats a long-term prejudice in Cusanus studies for which he is not solely culpable. Yet he does introduce three misunderstandings that place unnecessary obstacles in the way of modern interpreters of De mente and of Cusan mathematical theology generally. First, since all agree that Nicholas’s new epistemology centers on the mind’s folding power, one expects Velthoven to name Thierry of Chartres, the inventor of reciprocal folding, as the silent partner behind the cardinal’s 1450 breakthrough. Instead he credits Proclus as the chief source for folding.6 This attribution is simply false:  as we have seen, folding began with Thierry, and then Nicholas made extensive modifications. Yet by awarding credit to Proclus instead, Velthoven conveniently cuts the link between medieval scholastic theology and Cusanus’s mature, protomodern epistemology. Second, Velthoven maintains that Cusanus’s new mathematical epistemology in 1450 is essentially a rejection of Pythagoreanism:  “From this [Cusan] conception of mathematics follows its irreconcilability with the Pythagorean notion that the mathematical (above all,

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numbers) is the intelligible core of material reality.”7 This view has proven unusually durable in the literature on Cusanus’s views of mathematics.8 Yet Velthoven’s reasoning does not hold up under scrutiny.9 These misapprehensions come to a head when Velthoven confronts the Cusan doctrine of God as a mathematician. Here his reasoning takes a peculiar turn. Velthoven concedes that Nicholas does often depict God using the quadrivium and that he often repeats Boethius’s notion of number in God as the paradigm for creation.10 But on rather dubious grounds he deems it incredible that Nicholas could have intended this doctrine literally.11 Given the prominence, nevertheless, of the cardinal’s Boethian affinities, Velthoven feels compelled to introduce an interpretive distinction between “mathematical number” and “substantial number” into Cusan thought, lest it appear self-contradictory or incoherent.12 As he sorts Cusanus’s statements into these two categories, it becomes clear that while the former indicates human mathematical projections, the latter encompasses both the physical numbers in the world as well as divine numbers in God’s mind. According to Velthoven, we should take all references to physical and divine “substantial numbers” metaphorically. Thus when Cusanus speaks of “numbers” in creatures, this merely connotes the Creator’s gifts of order and structure; when Cusanus calls God a “mathematician,” he is simply naming the Creator symbolically with reference to human mathematical abilities.13 In the aphorism of Karl Jaspers that Velthoven quotes at the outset of his analysis, “mathematical thinking is human thinking.”14 This approach to interpreting Cusan mathematical theology has several problems. It excludes the possibility of historical difference such that Cusanus could sincerely affirm the Boethian-Chartrian theology. If one has already decided that the Cusan epistemology is protomodern, and hence not Neopythagorean, then one must erect a firewall between mathematics and theology on the cardinal’s behalf. This approach also suggests that Nicholas’s beliefs in the Trinity as the self-numbering One or the Word as Equality incarnate (left unmentioned by Velthoven) have nothing to do with his interest in mathematics, which seems implausible. Furthermore it brings to light a deeper assumption that can hinder modern readers of De mente. Velthoven infers that because mathematicals are projections of the mind in De mente, they cannot provide the foundation of world order as the “Pythagoreans” taught. If they are mental products, for that very reason they cannot be literally ascribed to God. But this logic relies on a post-Kantian sentiment that categories of perception are strictly either in our mind or in the world.15 Boethius, Thierry, and Nicholas did not share this view. For them, so long as the number in God’s mind was simultaneously the exemplar of human mathematizing and the exemplar of creation, there could be no opposition between mathematicals as human products and mathematicals as cosmic order. There is no need to leave behind mathematicals qua mathematicals in order to ascend to

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God, if God is the highest mathematical. Troubles arise only when one removes the tenet of God as exemplary mathematician, a doctrine that of course Cusanus repeatedly invoked. But Velthoven specifically rejects this theological linchpin as a dispensable metaphorical embellishment. If we go down the other road not taken, and accept that Cusanus considers God to be a mathematician literally, then we see that the anthropological and theological registers of number in De mente are not simply univocal (as Velthoven paints Boethius), nor equivocal (as he paints Cusanus in contrast), but analogical. In sum, De mente appears to reject Neopythagoreanism only if one fails to take seriously the Boethian and Chartrian sources of Cusanus’s epistemology. But if that dialogue did not overturn the mathematical theology of the 1440s, it begins to look like the moment when that project found its footing. In a similar vein, Kurt Flasch has described De mente as a departure from the mathematical theology of De docta ignorantia. In an early essay Flasch warned that “to the superficial reader the role of mathematics appears greater in the Cusan philosophy than it truly is.”16 Mathematical concepts, he argues, are what Cusanus calls aenigmata, metaphorical likenesses that point beyond themselves. Such concepts can never lead to theology since they are products of the human mind, as Nicholas describes in De mente. Flasch even adds that Cusanus tried to dissociate the mathematical One from the divine One, and that he used complicatio and explicatio to do so.17 In his book on Cusan development, Flasch leaves these points unspoken, but he retains the same suspicion of Neopythagoreanism and Chartrian sources as so many impediments to the cardinal’s philosophical maturation to be overcome. Flasch’s masterful narrative can be read as an alternative version of the story I have presented over the last two chapters.18 The differences through the looking glass are enlightening. For Flasch the regrettable “mathematico-theology” of Book I does not overwhelm the true philosophical innovation of the text, which is its resolution to begin from the limited condition of human knowledge. Accordingly he foregrounds the epistemological aspects of Nicholas’s coincidence of opposites and his negative theology.19 Flasch has very little to say about Books II and III.20 He too finds Sermon XXII to be an important moment for Nicholas’s treatment of Chartrian concepts. But according to Flasch, the sermon augurs a turn to the Proclian Logos-philosophy that will preoccupy the cardinal’s late works.21 The sermon accelerates the philosophizing of Trinity and Incarnation underway in De docta ignorantia, in which Cusanus discovered epistemological and ontological models within the language of Christian revelation.22 Regarding De coniecturis, Flasch seeks to update Koch’s argument and emphasize the new Proclian influences.23 For Flasch, that work not only critiques De docta ignorantia but marks a fresh start for an authentic Cusan philosophy of mens. As in my account, the opuscula of the 1440s bear witness to a burgeoning “methodological division” between geometry and theology.24

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Flasch hails the dialogue De mente as the “key” (Schlüsseltext) to his account of Cusan development.25 Here Nicholas perfects his attempt in De coniecturis to build a platform for autonomous philosophy by reconceiving the way that mens knows the world. Flasch agrees with Velthoven that the dialogue’s innovation is its portrait of the human mind as active and productive self-knowledge.26 According to Flasch, this is why De mente seals the demise of the negative theology of De docta ignorantia and its passivity towards transcendence.27 Returning repeatedly to Nicholas’s definition of mens as enfolding, Flasch acknowledges the link to Thierry, yet searches instinctively for connections to Plotinus or Proclus.28 When Cusanus revisits Thierry’s arithmetical Trinity in De mente, Flasch reads it not as a moment of continuity with De docta ignorantia but as an assertion of philosophy’s prerogative over dogma.29 As in my account, Flasch views De theologicis complementis as carrying out the mathematical mandate of De mente (although we differ as to what that is).30 Flasch prioritizes the later Cusan works focused on Proclian philosophy, such as De beryllo, De li non aliud, or De principio. I propose a different way to approach the knot of questions encircling De mente and Cusanus’s development in the 1450s. The dialogue is indeed the fulcrum of Cusan development, but not for the reasons usually given. To articulate its novum one must explain three things: the agenda of De mente as it differs from De docta ignorantia, the sources of Nicholas’s new insights, and the methodological character of the breakthrough in De mente worth sustaining in other works. I would decide these differently than Velthoven and Flasch, not in retreat from Flasch’s genetic method but by way of intensifying it. To avoid being misled by modern prejudices against Neopythagoreanism, it is especially crucial for our study of mathematical theology to follow Flasch’s prescriptions with care. We must begin with Nicholas’s actual medieval sources, explain his textual compositions as each proceeds from those sources, and remain suspicious of synoptic abstractions like number or infinity that introduce subtle biases. For Flasch and Velthoven, the core of De docta ignorantia is its apophatic mysticism. I have argued otherwise in Chapters 7 and 8. The essential event in De docta ignorantia is rather Nicholas’s struggle to deploy his conflicting Chartrian sources, an effort he only redoubles in De coniecturis. If this is the case, then the status of De mente in Cusan development ought to be rethought. Rather than weighing degrees of fidelity or departure from the method of docta ignorantia, we should keep our eyes trained on Cusan sources and query how the balance of Chartrian texts in the 1440s is retained, altered, or discarded in 1450. Both Flasch and Velthoven attempt to tie De mente to Proclus, yet as I  will show, a careful reading of De mente in light of the Fundamentum hypothesis reveals that Nicholas returned therein to his Chartrian sources, and met with greater success than he had in the 1440s. This leads us to the question of a new method broached in De mente. If one assumes that mathematics is irreconcilable with Christian

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doctrines, it is sensible to expect that De mente heralded Proclus as Nicholas’s new philosophical muse. But likewise if the major innovations of 1450 can be traced to the twelfth-century Breton master, then one must accept that Christian doctrines remained just as germane to the program of De mente as they were to De docta ignorantia. Both De mente and De coniecturis should be understood as standing in continuity with the theology of De docta ignorantia, since they all labor over the same Chartrian source problem. Like Bernard and Thierry before him, Nicholas aimed to think the Trinity in harmony with the quadrivium, Logos with Arithmos. Contrary to Flasch’s account of development, what actually occurs in De mente is not the purification of the cardinal’s philosophical impulses but the deeper integration of Neopythagoreanism with Christian theology.

The Restoration of Thierry’s Modal Theory For his decade in service to the papacy, Cusanus was named cardinal in December 1448 and invited to Rome to accept the red hat from Nicholas V, his friend Tommaso Parentucelli. He arrived in January 1450 and stayed through December, enjoying in that Jubilee year a rare period of protracted calm and leisure to write, especially in his summer retreats to a monastery near the country town of Fabriano.31 The result was a trilogy of masterful philosophical dialogues—completed in July, August, and September, but surely underway for months—that refreshed Nicholas’s entire intellectual program and opened up new avenues for exploration throughout the 1450s. The three works center on a professional Orator interviewing an Idiota, or unlettered Layman, who despite his simple vocation as spoonmaker shares insights that astonish the Orator as well as the Philosopher they later meet. The Idiota trilogy is ordered according to measure, number, and weight from Wisdom 11:21.32 In De sapientia the Layman tutors the Orator how to conceive of divine Wisdom as the “absolute exemplar” or “infinite form,” that is, the universal measure of things. In De mente the Layman explains that the natural activity of the mind is mathematical measurement and hence through number the mind can ascend toward knowledge of God. In De staticis experimentis the Layman resolves several riddles of medicine and geometry through various techniques of weighing, not unlike an early computer technology.33 Even more than De coniecturis, the treatise De mente foregrounds the autonomy and dignity of mathematical reasoning. As a living number, the human mind naturally measures; in every measurement, it is fundamentally measuring itself; and the mathematical projections by which it grasps the cosmos demonstrate its own divine origin. It is this nexus of mathematical knowledge as self-knowledge and as absolute knowledge that makes De mente read more like a product of the seventeenth century than of the fifteenth. Nicholas’s provocative elevation of Layman

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over Philosopher, moreover, reflected his humanist disdain for school philosophy and paid tribute to Socrates’s egalitarianism (recall his declaration in the Meno that even slaves understand mathematics). Yet throughout the 1440s Cusanus had been unable to decide whether pure Neopythagoreanism or cosmic Christology would reign supreme in his thought. As we have seen, what connected the two major efforts of De docta ignorantia and De coniecturis was the cardinal’s ongoing deliberations over the four modes. The fact that in 1450 Cusanus retreads the same terrain of Thierry’s modal theory speaks volumes about the meaning of his trilogy. For if Nicholas found De coniecturis unsatisfactory, then De mente appears to be the corrective.34 It is no accident that De mente is the first text after De docta ignorantia in which Cusanus named the second mode of necessitas complexionis expressly; the only other occasion is De ludo globi. What new breakthrough allowed the cardinal to start the project of De coniecturis afresh, so to speak, in 1450, and to amend its faults? What kind of insight inspired not one work but a trilogy? The answer, as I will show, is that in De mente Cusanus turned for the first time to Thierry’s original account of the four modes.35 (Granted, it is difficult to know why Cusanus worked through Glosa, as the evidence suggests, and not Lectiones, and why only in 1450; but these are separate questions for further research.36) By embracing an authentic model of the modal theory—not Fundamentum’s abridgment, and not his derivative four unities—Nicholas found he could better articulate the nature of his integrated mathematical theology. Tellingly he also returned in De mente to De septem septenis and Thierry’s Commentum, the two Chartrian texts most favored in Book I of De docta ignorantia. Thus we ought to view De mente as the cardinal’s third major attempt to reconcile his Chartrian sources, or better, as the successful revision of his second attempt in De coniecturis. To celebrate the occasion, Cusanus penned a trilogy of works around that most Pythagorean scripture, Wisdom 11:21, and conspicuously installed his discovery of Thierry’s authentic modal theory at its very heart, in the central three chapters of the trilogy’s central book. After all, it was in October 1449, while preparing to depart for Rome and perhaps beginning to conceive the Idiota trilogy, that Nicholas praised Thierry as “easily the most brilliant man of all those whom I have read.” Cusanus was clearly fascinated by the second mode in De coniecturis, but out of persisting regard for Fundamentum had banished its name even while depending on its function. In De mente things look very different. Cusanus now explores the dimensions of necessitas complexionis with the passion of a new discovery, savoring its meaning on no fewer than ten occasions throughout the treatise. Indeed he arranges the order of his chapters and his cullings from Septem and Commentum around this conceptual treasure pried from Glosa. His retrieval of Thierry’s authentic modal theory for the first time had dramatic consequences for the cardinal’s theological development. The resolution reached in De mente showed Cusanus a way to exit the standoff between his two competing paradigms

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of mediation. After 1450, we witness a productive rapprochement between the cardinal’s Verbum theology and his geometrical pursuits, as I will detail below. Three specific features of De mente mark it as a critical turning point in 1450. First, already in De sapientia and early in De mente, there are signs that Nicholas has Fundamentum on his mind. But this time, he will address the treatise’s concerns about mediation in a different way than he had in the 1440s. Second, Cusanus again weaves together various Chartrian sources with the same editorial care he displayed in 1440. But now he reads the four modes as an account of the human mind, not as universal ontology or as Platonist temptation. If the second mode is not a semidivine mediator competing with the divine Word, but a stage intrinsic to the mind’s ascent, then the cardinal has grounds to reverse Fundamentum’s judgment and to embrace necessitas complexionis within his Trinitarian theology. Yet this does not mean, third, that Cusanus simply takes over Thierry’s term without modification. As in 1440, by deploying the second mode to his own ends, Nicholas introduces some modifications to Thierry’s original meaning. The cardinal’s new clarity about the disciplinarity of mathematics and theology enables him, in turn, to formulate a more intimate connection between number and God, clearing the way for a new stage in his development after 1450.

Revisiting the Source Problem If Nicholas’s discordant collection of Chartrian sources bothered him after De coniecturis, he had little time to devote to the problem, distracted by his responsibilities as papal legate to the imperial diets. But right at the outset of the Idiota trilogy, Cusanus’s train of thought suggests that the problem was on his mind. In De sapientia, the Layman exhorts the Orator to delight in the sweet taste of divine Wisdom, and the Orator asks if “Wisdom” is the same as “Word.” In his reply the Layman explains that God created all things in Wisdom in the sense that God is unity, equality, and connection.37 To define Wisdom further the Layman lists a series of divine names (“infinite Form,” “Exemplar of every figure,” “most adequate Measure enfolding forms”) that precisely recapitulate the titles given by Fundamentum to the divine Word.38 Just as in De docta ignorantia, the cardinal has juxtaposed Thierry’s arithmetical Trinity with Fundamentum’s praise of the Word. In the second half of De sapientia Cusanus alludes to Fundamentum again, applying its doctrine of maximum and minimum to depict Wisdom as the “absolute Exemplar.”39 Hence on two separate occasions in De sapientia Cusanus meditated on Fundamentum passages that attack the second mode. This emerging pattern in De sapientia resurfaces at the opening of the next part of the trilogy. In the first five chapters of De mente Cusanus appears to retrace the steps of his reading of Fundamentum in 1440. Let us recall how the cardinal initially reacted in De docta ignorantia II.9 when Fundamentum turned to the subject

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of necessitas complexionis. (1) First Nicholas cited Fundamentum’s statement on the second mode, which identifies it as forma universi, anima mundi, mens, and intelligentia (§§141–142). (2) Upon reading such allusions to ancient philosophy, he broke off his citation and interpolated his own summary of the debate between Plato and Aristotle on universals, siding with the Platonists (§§142–148). Then Cusanus returned to Fundamentum’s text to cite the remainder of its critique, which names three reasons why the second mode cannot exist. Namely: (3) The second mode represents plural exemplars, and there is only one divine Exemplar of all images (§148). (4) It counterfeits the proper function of the divine Word (§149). (5) It violates divine transcendence by purporting to “unfold” the divine absolute directly; rather, the absolute “enfolds” all things (§150). Remarkably, this sequence of argument from De docta ignorantia II.9 is paralleled step by step in the initial chapters of De mente. (1) In De mente I, the Layman defines mens as that which measures (mensurare), and then explains that measurement is governed by the second mode of being.40 This places necessitas complexionis at the heart of the mysterious link between mens and number.41 (2) In De mente II, the Layman compares different opinions in the debate over universals, and the Philosopher praises his command of both Peripatetic and Platonist positions.42 (3)  The Layman responds that such philosophical oppositions are only overcome in the one infinite divine Exemplar, and then he repeats the passage on the Exemplar from De sapientia in which he had paraphrased the Fundamentum treatise.43 (4) Then the Layman adds that the true identity of the Exemplar is the divine Word. As in Sermon XXIII, since the Word cannot be named, no creature can be named precisely.44 Thus far the pattern holds:  new names for the second mode, Plato versus Aristotle on universals, the singular Exemplar, and the divine Word. (5) The last step of Fundamentum’s critique hinged on the opposition of enfolding and unfolding. Likewise, the Layman’s complex account of mens that follows in De mente III–IV is grounded precisely on reciprocal folding. God enfolds all beings, and the human mind is an image of God. Thus the Layman declares that “mind is the primary image of divine Enfolding, which enfolds all images of enfolding.”45 The Layman stresses his next point. The enfolding power that the mind enjoys vicariously as divine image, however, is not itself an unfolding of the divine Enfolding. Rather, it stands above all lesser unfoldings (e.g., the unfolding of unity into multitude, or the unfolding of point into magnitude). The Layman sternly warns the Philosopher against the grave philosophical error of confusing “image” and “unfolding.” The mind is the supreme imago of divine complicatio, not its explicatio.46 For example, says the Layman, the image of unity is equality, but the unfolding of unity is plurality. So too the mind is the image of divine Enfolding, not its unfolding, for there can be no such unfolding of divine Enfolding. By rejecting the legitimacy of every direct explicatio of God, the Layman repeats one

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of Fundamentum’s most distinctive arguments against Thierry’s use of reciprocal folding.47 Given this sequence of parallel arguments, it seems that at some level Cusanus is revisiting the Fundamentum treatise within the pages of De mente. In De docta ignorantia, Cusanus had acknowledged Fundamentum’s reasons for suspecting the function of the second mode (taken as mens or anima mundi) as being too similar to the divine Word. Having mistakenly read the author’s critique as equivalence, Nicholas ended up pursuing a theology of the Word. In De coniecturis he looked to number as a new instrument to weld his Chartrian sources together, but stumbled over the different faculties of the mind. Now in De mente Cusanus returns to the problem again. He still concurs with Fundamentum that the Platonist divine mind and world-soul are illegitimate unfoldings of God. But now this humanist cardinal of the quattrocento probes further. Fundamentum had rejected mens because it pretended to be a divine mediator. But what if instead of that numinous semideity the mediator was the human mind, imbued with all the dignity that Cusanus had bestowed upon the faculty of intelligentia in De coniecturis? This human mens would be an image of God, and as image (not unfolding) it could reflect divine Enfolding at a lesser level. Mind would then take up a unique intermediate position, both enfolding beings and “assimilating” their unfoldings.48 In both De docta ignorantia and De coniecturis, Cusanus was drawn to the power of number and the potential of quadrivial devices in theology. But until he encountered Thierry’s original modal theory it could not have occurred to him that number was the domain of the second mode. That essential link between Thierry’s Neopythagoreanism and his concept of necessitas complexionis had been censored in Fundamentum and was missing in Commentum, Tractatus, and Septem, the sources that Nicholas seems to have been using in the 1440s. He lacked textual access to the one doctrine he most required to understand Thierry’s mathematical theology coherently, namely that the second mode corresponds to number and quadrivium. Once Cusanus grasped this connection in De mente, all the pieces fell into place. It is therefore no exaggeration to say that in De mente Cusanus definitively solved the riddle of retaining Thierry’s Platonism alongside Fundamentum’s anti-Platonism. However it occurred, this new access to Thierry’s full theology of the quadrivium finally allowed him to relax his suspicions of the modal theory insinuated by Fundamentum. And it was only fitting that the dialogue which achieved this textually would be centered on the meaning of mind, de mente.49

A Third Redaction, a Second Synthesis These considerations begin to explain why Nicholas’s book on mens focuses inordinately on Thierry’s theology and other Chartrian sources. At the heart of De

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mente one finds a textual synthesis of two Chartrian doctrines: a theory of mind (Commentum and Septem) with an account of the four modes (Glosa).50 This resembles the synthesis in De docta ignorantia that bound together Thierry’s ideas in Tractatus, Commentum, and Septem with the four modes in Fundamentum. We saw how in De docta ignorantia Cusanus used passages from Septem to frame selections from other works like Tractatus or Commentarius Victorinus, and then linked these texts to the Boethian quadrivium. The same pattern holds for the central chapters of De mente. First, Cusanus uses Septem’s account of mens to frame his rediscovery of Glosa on the four modes (chapters VII–VIII). Then he uses the quadrivium to frame his own newfound insights into the deep meaning of mensurare (chapters IX–X) (see Table 9.1). To begin with, can we be sure that Cusanus is using Glosa and not another Chartrian text? With relative certainty, yes.51 But Thierry handled his doctrines very differently than Nicholas does in De mente. Thierry would never have combined

Table 9.1  Probable Chartrian sources of De mente VII–X Sources of De mente VII–VIII: Septem as frame for Glosa and Commentum 97–99 100–102 102–107 108–110 111 112–115

Theme: on mind and the four modes Spiritus theory of mind: Septem 952D–953C Glosa II.12–23 on four modes and Commentum II.3–6 on self-measure [Digression on conceptio: cf. De sapientia II (34–35)] Glosa II.1–9 on mind and disciplina Spiritus theory of mind: Septem 952D–953D

Sources of De mente IX–X: Boethian quadrivium as frame for Cusan synthesis 116–121 122–123 124 125 126–128

On multitude and magnitude: Institutio arithmetica II.4 Synthesis of second mode (Glosa II) and self-measure (Commentum II) [Digression on facies: cf. De sapientia II (39–40)] Recapitulation: on mind and the four modes On multitude and magnitude as disciplina: Institutio arithmetica I.1

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the spiritus theory of mind with the four modes of being, because he viewed the latter as a much-improved substitute for the former. In Commentum Thierry had proposed that the soul knows things either through the instrument of an extrinsic body (the ether-like fluid or “spirit” that fills cells in different regions of the head), or intrinsically by “using itself as an instrument” instead of the fluid spirit.52 Reason operates extrinsically in physics, and intellect operates intrinsically in mathematics and theology. With this theory Thierry could not yet distinguish the operations of mathematics and theology until the breakthrough of Lectiones and Glosa. In the new modal theory, the first mode corresponds to theology, the second to mathematics, and the third and fourth to physics.53 With this solution in hand in Glosa, Thierry alluded to the problem of the faculties but never mentioned spiritus psychology again.54 But when Nicholas wanted to combine the four modes with the spiritus theory of mind, he reached for Commentum or Septem where the outdated psychology was still on view. Why would he want to do this? The answer is that Cusanus is still struggling to make sense of the second mode. Here it is crucial that one distinguish between Thierry’s different Boethian commentaries. Given the evidence we possess, it appears that Cusanus did not have access to the fully developed modal system articulated in terms of reciprocal folding, which Thierry had explained in detail only in Lectiones and did not bother repeating again in Glosa. Working from Thierry’s digest in Glosa, Nicholas now at least understands that the mode of necessitas complexionis corresponds to the discipline of mathematics and to the faculty of intelligentia. This was already an improvement on De coniecturis, where his attempt to build on Fundamentum’s defective abridgement of the four modes led to a contradictory account of mathematics. But now even with Glosa in hand, Nicholas knows only that, and not why, the second mode is mathematical. In De mente I–V, as we have seen, he recalled that Fundamentum linked the second mode to mens. But what Cusanus still lacked was a conceptual link between mind and mathematics that could illuminate the meaning of necessitas complexionis. To solve this new hermeneutical problem in De mente, Cusanus apparently turned to his trusted guide from De docta ignorantia, the treatise De septem septenis. There he found a lengthy account of how soul reasons through self-measure, similar to Commentum.55 In Septem Nicholas read that for higher forms of knowledge, the soul “uses itself as an instrument” (utens se ipsa pro instrumento). Ever the prudent editor, he transformed this into a maxim that solved his problem. Mind (mens) is mathematical because it always, natively, self-measures (mensurare).56 With this tool Cusanus could construct a bridge between Thierry’s discarded spiritus theory in Commentum and his mature modal theory in Glosa. The second mode is the domain of mathematical thought because that is the highest activity of mens; for mens perpetually self-measures, and this measuring is inherently mathematical. Thus in De mente Cusanus

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describes mens as sovereign living measure (viva mensura). Like a compass come to life, mens scales its own dimensions and so remains incalculable by multitude or magnitude.57 Yet as the origin of these quadrivial categories, mind is ineluctably arithmetical in its every cogitation, a “living, divine number.”58 In this indirect way Cusanus’s reasoning in De mente circumvents the need for the Lectiones model of the four modes as structured by reciprocal folding. Nicholas’s rediscovery of self-measure in Septem and Commentum allowed him to embrace Thierry’s modal theory for the first time in De mente. The link between mensura and modi, as Table 9.1 demonstrates, lies at the very core of the dialogue and is the source of its remarkable novelty. Cusanus discusses the modal theory on three occasions. The first is simply a list of the four modes.59 The second defines their different operations in terms of mind’s self-measuring activity. Mind assimilates itself to the fourth mode in order to measure all things in their possibility, to the first mode to measure things in their simple unity, to the second mode to measure things “in [mind’s] own being” (in proprio esse), and to the third mode to measure things as they actually exist.60 Cusanus’s third account of the modal theory is much lengthier, and merits its own treatment in my next section.61 By embracing Thierry’s four modes of being, Nicholas can now in turn clarify the muddled disciplinary boundaries in De coniecturis. In De mente VII–X Cusanus finally grasps that the second mode defines mathematics as proximate to theology yet distinct from it. As if conscious of this breakthrough, Cusanus frames his discovery of Glosa with two passages on the nature of the quadrivium in Boethius’s Institutio arithmetica—just as he had framed his citation of Fundamentum in De docta ignorantia II with parallel discussions of the quadrivium. In the first passage, he defends the Neopythagorean doctrine that all things are known through number and quantity and that without the quadrivium there is no philosophy.62 In the second, he explains that mind generates numbers and geometrical points out of itself; because all things are known by multitude and magnitude, mind is the universal measure of all things.63 Cusanus also focuses on passages where Thierry had distinguished separate faculties for the two disciplines.64 Following the Breton master he assigns intelligentia or intellectibilitas to knowledge of the first mode (theology), and disciplina or doctrina to knowledge of the second mode (mathematics).65 Then later in De mente Cusanus explicates the four disciplinae of geometry, astronomy, “magnitude” (harmonics), and “number” (arithmetic), which he defines as four ways of determining the categories of discretio (multitude) and integritas (magnitude).66 Hence we see the first consequence of Cusanus’s retrieval of the second mode: an extensive reinvestment in the Boethian quadrivium, indeed the greatest since De docta ignorantia.

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A New Foundation for Mathematical Theology Cusanus’s breakthrough in De mente was not a brilliant stroke of untutored insight, but rather the modest work of judiciously gathering others’ ideas into a new synthesis, just as he had in 1440. Nor does the cardinal’s achievement in 1450 represent a turning away from doctrinal theology to philosophy, or from mysticism to sober mathematics. It was nothing more or less than a relatively more successful redaction of the same Chartrian sources that he had used to build De docta ignorantia. That said, Cusanus was by no means repristinating Thierry’s modal theory, even that of Glosa. Operating from limited textual knowledge and guided by his own agenda, the cardinal’s reconstruction of the four modes in De mente made profound changes to Thierry’s doctrine, and to necessitas complexionis in particular.

Cusan Variations: The Second Mode First and foremost, the Cusan four modes are not ordered by a system of reciprocal folding as in Lectiones. Instead they represent four stages of a stepwise Platonic ascent from the fourth mode up to the divine first mode. Cusanus still agrees with Fundamentum that the fourth mode of pure possibility does not exist in the same way as the others.67 Nevertheless, the “phantasms or images” known in the two lower modes “incite” the mind to seek their more beautiful exemplars.68 Rising from the third mode to the second, mind reaches the certainty of abstract mathematical knowledge in which true forms exist in a necessitas complexionis. In this second mode, the mind now uses itself as an instrument (utens se ipsa pro instrumento) by looking to its own immutability.69 Yet mind remains unsatisfied because it does not yet behold absolute precision. The mathematical necessity belonging to the second mode is still marked by composition and alterity.70 So now mind must look again, not to its own immutability but to its simplicity, as if turning its gaze, Nicholas writes, from magnitude to a point. When it does this, mind can glimpse the divine necessity of the first mode: Assuredly he would see, beyond a determined necessitas complexionis, all the things that he previously saw in variety now apart from variety in a most simple necessitas absoluta, without number and magnitude and without otherness. Now, in this highest mode mind uses itself insofar as it is the image of God, and God, who is all things, shines forth in mind when mind, as a living image of God, turns to its own Exemplar by assimilating itself thereto with all its effort.71 To be sure, Cusanus reads the difference between the second and first mode as an ascent:  from mathematics to theology, from composition to simplicity,

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from alterity to unity. But it is not another ascent like the fourth to the third, or third to second. Instead the final step progresses beyond that abstractive procedure altogether to the mystical, visionary experience of beholding oneself as God’s image. The mind “turns” (convertere) to observe its mathematical activity in the second mode as an image in which God “shines forth” (relucere). This posture of reflexive self-visualization induces the mind to generate “theological speculations.” Nicholas’s first alteration of Thierry’s modal theory implies two more. Thierry had intended the four modes to be four lenses for interpreting the cosmos. But Cusanus singles out the second mode as the epitome of the mind’s native activity and favors spatial terms to describe it. The second mode is the realm (locus) or zone (regio) inhabited by the mind.72 The mind has the power to withdraw forms from matter and reposit them in an “unchanging domain” (regio invariabile).73 Nicholas’s subtle spatialization of the second mode underscores his conception of mens as essentially coterminous with necessitas complexionis. Mind is the proper sphere of mathematical operations, a base from which one can venture theological sorties toward the first mode. Rather than ordering the second and first modes through reciprocal folding like Thierry, Cusanus adopts the Boethian aesthetics of image and exemplar. If the divine first mode is the universal Enfolding, then mind reflects it as a lesser enfolding. Cusanus writes: All enfoldings are images of the Enfolding of infinite simplicity, not its unfoldings but its images, and they exist in the necessitas complexionis. And mind, the first image of the Enfolding of infinite simplicity, enfolding by its power the power of those enfolding, is the place or domain [regio] of necessitas complexionis.74 That is to say, the second mode is not an unfolding of the first mode, as Thierry had maintained, but rather an image of the first mode, as Nicholas had already conspicuously underscored in De mente IV.75 In all of its measurements, mind never reaches perfect divine precision, since its enfolding power is only the image of the divine Exemplar.76 In one important regard, however, Nicholas did follow the lead of the Breton master. As we have seen, Thierry’s reciprocal folding had broken with Plato’s model of deriving mathematicals from abstractive ascent. Back in De docta ignorantia Cusanus still retained that view when he explained how geometrical figures raised the mind above the flux of matter. In that work he had proposed that mathematical figures provide the purest images for contemplation on account of their abstraction from matter:

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In this passage from De docta ignorantia, Cusanus teaches mathematicals mediate between the sensible world and the divine. Although they are the best tools available to the theologian, all mathematical beings are nevertheless finite. Hence one must “leap beyond” (transilire) mathematical figures in a three-step ascent: first “considering” the finite figure in its visual details, then “translating” it into an infinite figure by subtracting its fixed dimensions, and finally “transuming” the infinite figure to divine simplicity, experienced as mystical darkness.78 It may seem at first that Nicholas reinstated this same model from 1440 in De mente, since he presents the four modes as stages of increasing abstraction and even uses the same term for ascent (transilire).79 But closer inspection shows that this is not the case.80 In De mente, on the contrary, the mathematical abstraction of form reaches its epitome and terminus in the second mode. From that point forward, the final step toward the first mode is not a process of abstraction, but of a different visual mechanism altogether. Once more the mind looks to itself, according to Cusanus, but now not as an instrument of self-measure, but rather as an image of the divine Exemplar. And in this way, by combining the mathematicality of the second mode with the tradition of imago dei, Nicholas happened upon a powerful theological surmise. For if the mind amidst its native activity in the realm of necessitas complexionis remains an icon of God, then God’s native activity must be mathematical to an exemplary degree. In its egress from the second to the first mode, mind envisions the self-measurement of God, of which mind’s own self-measurement is an image. This divine self-measurement is precisely what Thierry’s arithmetical Trinity had defined when it named the eternal self-equality of the One that generates numerical difference. The implication is unavoidable: God is the prime mathematician, and the mind fulfills its destiny when it grasps its own mathematical categories as reflections of God’s mathematizing self-measure. The mind’s self-measure is intrinsically mathematical because its divine Exemplar, as Trinity, is the autonomous fount of numerical self-measure. By the same token, theological predication flows from visualizing one’s participation in the eternal divine μάθησις. In this way, De mente provides a unique account of the disciplinary relation between mathematics and theology. For Cusanus, theology does not think God by exercising a right of exception from the universal reach of mathematical knowledge

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(as in most Christian theology after the seventeenth century). Nor must theology apply the mathematical frame retroactively to authenticate traditional theologoumena (as in the method more geometrico of Alan of Lille, Descartes, or Spinoza). Nor should it simply intensify the abstractive procedure beyond mathematicals to grasp toward an altogether formless One (as in Speusippus or Plotinus). Rather, theological thinking is nothing more than the ecstasy of mathematical thinking, immanent to it but on occasion drawn up into a higher mode—a mysticism inside of mathematics, as it were. The self-measurement that provides the basis of number and quantity is a reflexive turn to the mind’s immutability; the self-measurement that enters into the vision of God is an iconological turn to the mind’s simplicity, revealing the virtual identity of one’s own (mathematical) self-measurement with God’s own (mathematical) self-measurement. Cusanus had named this divinizing process filiatio in 1445, without yet grasping the deep mathematical basis of what we might call his theory of arithmetical theosis.

Number as Icon We saw in De docta ignorantia how Cusanus first organized his Chartrian sources into Books I and II before developing them further through his own meditations. In De mente something similar occurs. Cusanus first reconsidered his reading of Fundamentum (De mente I–V), retrieved Glosa in tandem with Septem (VII– IX), and linked these to other Boethian traditions he knew (X on the quadrivium, and XI on the arithmetical Trinity). The final chapters of De mente represent a kind of omnibus appendix covering various controversies between Platonism and Aristotelianism surrounding mens: on the universal agent intellect (XII), on Plato’s anima mundi (XIII), on the faculty of intellectibilitas (XIV) and on the immortality of the soul (XV). Of course the cardinal did not necessarily compose the chapters of De mente in their present sequence. If his first goal was to address the problem of Chartrian sources, it is more likely that his own meditations followed his editorial work, as in De docta ignorantia. This supposition appears to be confirmed when we turn to the one chapter I have not yet mentioned. For in De mente VI, Cusanus articulated a theology of number that perfectly reflects the new mystagogical ascent of the mind through the four modes. With the influence of Fundamentum mitigated by his new approach to Thierry’s texts, Cusanus finally lays hold of what Thierry had reached toward in Lectiones. Number in theology need not be an autonomic mediation that competes with Logos; Arithmos too can be an icon of the Trinity. As in De coniecturis, one encounters in De mente VI a compendium of Pythagorean number theories. In the voice of the Layman, Nicholas presents Aristotle’s notion that the Pythagoreans held numbers to be the principia of things and then defends the Pythagorean position.81 He cites Boethian definitions of

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number as composed of even and odd, and number as unities expressing harmonies, ideas that go back to Philolaus.82 He repeats Xenocrates’s teaching that the soul is self-moving number and Moderatus’s suggestion that the Pythagoreans used numbers to denote ineffable principles.83 And once again Cusanus quotes the Nicomachean doctrine that number is the first exemplar in the mind of the Creator. Even though the Pythagoreans first stated this theologoumenon, the Layman declares, Christians should follow Boethius in affirming it.84 After so many Neopythagorean testimonies within the space of a single chapter, the Layman admits with disarming frankness, “I do not know whether I’m a Pythagorean or something else”—an allusion to Jerome’s confession and a wink from the cardinal himself.85 Nicholas’s newfound access to Thierry’s mature theology clears the way for him to build something new. He has reconceived the role of necessitas complexionis in terms of an anagogical ascent through the four modes and in light of the biblical doctrine of imago dei. The mind as a mathematical self-measure reflects the Trinity’s own self-measure. These adaptations of Thierry’s theology point toward a bold and singularly Cusan idea: [The Pythagoreans] were speaking symbolically and plausibly about the number that proceeds from the divine Mind—of which number a mathematical number is an image. For just as our mind is to the infinite, eternal Mind, so number from our mind is to that Number. And we give our name “number” to number from the divine Mind, even as to the divine Mind itself we give the name for our mind.86 Human number is an image of divine Number. In this passage Nicholas finally hits upon the simple equation that loomed over Thierry’s theology, everywhere implied but nowhere stated. Mind generates number constitutively; without mind, there is no number.87 This means not only that human reason is inherently mathematizing, but more importantly that our quadrivial categories are a distant echo of the Number that is God, the “innumerable” number that, Cusanus now writes, is “no more a number than not a number.”88 If despite the deficiencies of language, God can be named Mind, God can even more surely be named Number, the eternal Numerus sine numero whom Augustine praised in his Genesis commentary. Cusanus’s new insight leads him to discuss four corollaries that, although they are somewhat compressed in the text of De mente VI, merit special attention. The first concerns the quadrivium. If human mathematics is an image of God, then the quadrivial sciences that structure mathematical reasoning are no longer foreign disciplines, but essential tools of every theology. For this reason Nicholas finds himself turning at once from arithmetic to music and geometry.

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He compares the divine Mind to a musician who renders his mental harmonies sensible in order to bring forth their audible beauty for his listeners. By this conceit the Word that God speaks is the harmonia divina, whose vibrations resonate within the living number of the human mind. Likewise, the divine Exemplar can shine through geometrical figures because their proportions, the locus of their forms, are defined numerically.89 A second corollary addresses the possible conflict between human and divine numbers. If number begins in the Trinity before it is reflected in the mind’s self-measure, does this mean that the apparent plurality of creatures is simply a figment of the mind’s projection? That is, does the theological origin of number jeopardize the integrity of human mathematics? Nicholas astutely responds that in fact his number theory is tethered to the physical world precisely through its divine referent. Even if human minds do project the plurality that they count in the world, that plurality is a true image of the real (Trinitarian) plurality in the divine Exemplar. There is, so to speak, a preestablished harmony between the plurality of number proceeding from God’s mind and the plurality of number projected by human minds.90 The one real enumeration in God is made possible by God’s unique capacity to grasp the singular in infinitely different modes. Hence there is no conflict between the multiplicity of number and the unity of God, so long as God’s mind contains the primal numbers of things, as in the Boethian dictum that Cusanus proceeds to cite.91 The cardinal’s third corollary indicates the necessary basis of this solution: the arithmetical Trinity. God’s capacity for plurality is rooted in the Trinity, which undoes every competition between identity and difference. The reason why God can grasp the singular in different modes, Nicholas explains, is that God is the original aequalitas unitatis.92 Divine self-equality is what endows human creatures with the native activity of self-measure and hence mathematical thought. Just as God is unity, equality, and connection, our minds operate according to a parallel arithmetical pattern. Boethius held that God’s mind is the exemplar rerum, but Cusanus adds that the human mind is the exemplar notionum.93 Whenever and whatever mens measures using the quadrivial arts of arithmetic and geometry, it comes to realize its hidden identity as the “living, uncontracted likeness of infinite Equality.”94 That is, the quadrivium is not merely a trace of God, but more specifically manifests the arithmetical Trinity itself. The fourth corollary carries the greatest consequences for Nicholas’s ongoing dealings with the Fundamentum treatise. At the heart of his new vision in De mente is the radical intimacy of divine Number hidden within human number. As Cusanus states later in the dialogue, “no creature can escape the number in the divine Mind.”95 The mathematical signatures of creatures flow immediately out of God’s self-numeration. But this means not only that “the numbers of things are the things themselves,” as in the old Pythagorean maxim, but also, as Nicholas

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now asserts, that “between the divine Mind and things there is no actually existing intervening number.”96 Once mental number becomes an unmediated reflection of divine Number, the category of number as such loses its mediating function. If God mathematizes naturally as Trinity, then the mathematicals are no longer extrinsic to God and hence cannot operate as mediators between God and world in the same way as before. Since the natural, defining activity of the human mind is to measure and thus to participate in divine numeration, the project of mathematical mediation loses its raison d’être. In De mente Nicholas has restored Thierry’s original modal theory in contravention of Fundamentum’s critique. But as we have seen, the cardinal also transformed that theory in the process, even to the point where he can now broker a compromise with Fundamentum on the question of mediation. Fundamentum had maintained that “between absolute and contracted there is no medium,” and so rejected necessitas complexionis.97 Having condemned the second mode in De docta ignorantia and avoided it in De coniecturis, Nicholas lavished attention on the concept in De mente like a prodigal returned. As much as Fundamentum had rejected the second mode to prevent its Platonist influence, so too Cusanus now retained it to ensure the proximity of mathematics and theology. But we must always keep in mind that the cardinal’s chief priority was to discern the concordance between Fundamentum and Thierry’s commentaries. Wherever possible Nicholas tries to preserve the principles expressed in the treatise. We saw one striking example already in De mente when Cusanus affirmed Fundamentum’s critique of mens-as-explicatio and substituted mens-as-imago in its stead. Another is the gap between the second and first modes in the mind’s ascent, which subtly preserves Fundamentum’s sequestration of the highest divine mode from the lesser three cosmic modes. This last corollary is another such example. Since all number belongs to God, just as there is no mediation between the absolute and the contracted, so too “there is no actually existing intervening number between the divine Mind and things.” Despite his radically Neopythagorean theology, Cusanus can still retain Fundamentum’s proscription of Platonist mediators threatening the sovereignty of the Word. For from Nicholas’s new perspective in De mente, Logos is not opposed to Arithmos. Fundamentum was right to reject the mediation of necessitas complexionis, but only given a concept of God that was insufficiently mathematized. Once God is grasped as the innumerable Number, Arithmos is no less immediate to God than is Logos. When it performs quadrivial measures, the human mind discovers itself to be a similitudo incontracta of the Trinity.98 The marriage of mathematics and theology no longer provokes scandal. De mente has long been celebrated as a pivotal work in the Cusan corpus, in part because it is reportedly the prime occasion for observing Nicholas’s protomodernity. In my view such praise is warranted but for quite different reasons.

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First, it is in De mente that Cusanus fully embraces Thierry’s modal theory. The treatise connects a new theology of number with a spatialized interpretation of necessitas complexionis as the native domain of the mind and thus the primary field of theological vision. Second, it is surprising to recognize that in De mente, for the first time in the Cusan corpus, Nicholas explicitly theorizes the quadrivium per se in the full extent of Nicomachus’s theory of multitude and magnitude.99 This does not take place in De docta ignorantia, despite his enthusiasm for the quadrivium, nor in De coniecturis, despite the overwhelming Pythagoreanism. In sum, De mente marks the lifting of the cardinal’s initial probation of Thierry’s theology of the quadrivium. The recovery of Glosa placed Cusan mathematical theology on a solid foundation comparable to, but distinct from, Thierry’s original model. Henceforth Cusanus could abandon the tortuous glosses and quiet modifications he had pursued throughout De docta ignorantia and De coniecturis and instead could seek to move forward confidently within the discourse of mathematical theology. If De mente comes across to some as protomodern, this must be explained in light of the dialogue’s doubling down on Boethian and Chartrian Neopythagoreanism, not its departure from them.

The Word as Number and Angle At the end of his quiet year in Rome, now as bishop of Brixen and papal legate, Cusanus embarked on a taxing journey of 2,800 miles from Rome around the German empire, preaching and legislating in the name of Nicholas V.  Rarely were the reforms of liturgy, monasteries, or cathedral chapters that he instituted obeyed for long, if at all. Nevertheless the cardinal traveled diligently from Austria to Germany to the Netherlands and then back south to Brixen in the Tyrol, on the road nearly continuously from December 1450 to April 1452.100 Yet all the while it seems Nicholas was working steadily on a new approach to the quadrature of the circle. Inspired by a new Latin translation of Archimedes by Jacob of Cremona given to him by Nicholas V himself, Cusanus worked studiously on a lengthy work he named De mathematicis complementis in 1453.101 The book integrated some of Archimedes’s insights and tested a new deductive method. When he sent it to Toscanelli, his friend agreed to pass it along to Peuerbach and Regiomontanus, but also warned Nicholas of some major flaws. In the same year Cusanus also finished a companion piece, De theologicis complementis, designed to unlock his first volume’s inner meaning.102 In this Complement, he wrote in an important letter that same year, “I have translated mathematical figures to theological infinity.”103 When the episcopal legation took him through Leuven in 1452, Cusanus likely met up with his old colleague Heymeric, now an esteemed doctor of theology. After arriving back at Brixen the following year, Cusanus wrote to Heymeric and

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sent along three works freshly completed:  a treatise on religious tolerance, De pace fidei, and the double-volume of “the Complement.” Heymeric acknowledged that he received the three works in a letter to the cardinal’s secretary, Dietrich von Xanten, in January 1454. He mentions them again in his great masterwork of the 1450s, Centheologicon. That book chronicles the “hundredfold” strategies used by different philosophical schools and religious sects to name God, suggesting that God’s simplicity lay well beyond them all. As Ruedi Imbach has shown, there are definite reminiscences of Cusan theology in Centheologicon, and in particular of the books Heymeric received in 1453. Among his hundredfold varieties, Heymeric includes the species of “ignorantly learnèd theology” (alluding to De docta ignorantia), “theology conjecturally triune” (De coniecturis), and “theology harmonizing various sects” (De pace fidei). He also refers to a certain theologia geometrica (“geometrical theology”), the name he invents for the project of De theologicis complementis and its matching mathematical volume. Under this rubric Heymeric alludes not only to Cusanus’s quadrature works but also to the geometrical images nestled within De docta ignorantia, which Heymeric had left unmentioned in his discussion of theologia ignoranter docta. While Heymeric never lauds Nicholas by name, he refers with admiration to a certain mathematicus theologus, a “mathematical theologian.”104 Heymeric’s motto, theologia geometrica, is an especially apt description of the broader Cusan enterprise in the 1450s. It encapsulates why the cardinal’s Complement, long undervalued in modern Cusanus scholarship, might represent the destination toward which his theology had been reaching since 1440. Consider the placement of the work in the context of the 1450s. Energized by De mente Nicholas redoubled his efforts toward the quadrature of the circle after 1450. After finishing the Idiota trilogy he devoted the majority of his intellectual efforts to mathematics, writing six consecutive works on geometry. Seven of his twelve mathematical treatises are compressed into the four years of 1450 to 1454. Geometry was for him a kind of mathematical laboratory for speculative discoveries, or better, a kind of playground where he could observe his mind’s movements and exercise it for theological tasks. Yet during the same period up until De beryllo in 1458, the only theological writings that Cusanus produced are De theologicis complementis, De visione dei, and De pace fidei. The last was written in response to the Ottoman capture of Constantinople, and De visione dei (as Nicholas himself explained) originated directly from meditations on measure and vision at the conclusion of the Complement.105 So prima facie the best place to witness the doctrinal consequences of the cardinal’s new clarity about his Chartrian sources would seem to be De theologicis complementis. By the time he wrote the Complement in 1453, Cusanus was well poised to realize the potential of the theologia geometrica that Thierry had begun to

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name toward the end of his life. In many ways Cusanus was recapitulating the arc of the Breton master’s own development. Thierry had struggled to surpass the limitations of Commentum but finally reached the new paradigm of Lectiones and Glosa. Nicholas reenacted the same struggle in De mente as he shifted sources from Commentum and Fundamentum to Glosa, which in turn enabled him to surpass the limitations of De coniecturis. Thierry’s theology of the quadrivium had become possible once he had grasped divine unity as the source of number (the arithmetical Trinity) and divine simplicity as the source of the geometrical fold (the four modes). Cusanus addressed the former in De mente when he considered human number the image of divine Number. But because of his idiosyncratic revisions to the four modes in De mente, he really addressed the latter only in De theologicis complementis, as we shall see. Thierry’s breakthrough of reciprocal folding had already begun to geometrize his theology nascently. Under this influence, the cardinal’s next work after De mente was a theological meditation on geometry.106 The central image informing the theological Complement is the quadrature of the circle attempted in the mathematical Complement, where the cardinal postulates a single measure able to encompass the rectilinear and the curved.107 This leads Cusanus to plumb the dialectics of infinite “isocircumferential” circles (a triune infinity of Center, Radius, and Circumference) and even of circles rolling on tangent lines.108 On several occasions he reviews the steps necessary to convert a particular geometrical figure lodged in the mind’s eye into a transfigural vision of divine infinity.109 But I will focus on two particular accomplishments that mark De theologicis complementis as the zenith of the cardinal’s mathematical theology.110 First, Cusanus now steps beyond the modest analogies of previous works and portrays God as the geometer par excellence. If God self-measures, then God’s instrument in his geometrical constructions must be a kind of divine Angle. Second, Cusanus also explains in detail why geometrical theology must be both visual and mystical. In this way the Complement is the rightful heir to De docta ignorantia and the primary matrix of the remarkable De visione dei written later that year. As Nicholas had already intimated in De mente, theological vision is an ecstasy of mathematical procedures, in which one glimpses the Exemplar of one’s own image whilst in the act of performing geometrical operations. Hence the cardinal’s new maxim: “If something is true in mathematics, it will be even more true in theology.”111 After discussing these two feats I  will draw some brief comparisons to bring out the distinctiveness of the Complement, first retrospectively to De docta ignorantia, and then prospectively to De beryllo. The lines that connect them confirm that we have indeed entered a new phase in the development of Cusan thought.

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The Geometry of the Trinity We saw in De docta ignorantia that Cusanus imagined the Creator wielding the quadrivium in the fabrication of the cosmos. In the Complement, Cusanus explores what it would mean to take this doctrine seriously. He begins by considering the method used to construct regular polygons. The geometer drawing a triangle does not conceive a triangle of definite shape or size. Instead she visualizes an “infinite exemplar” of the triangle absolute from any given multitude or magnitude.112 Just as the geometer looks to a transquantitative exemplar, so too the Creator looks to Equality, the divine Word, when constructing creatures. “When the Creator creates all things,” writes Nicholas, “he creates all of them while he is turned toward himself, because he is that infinity which is equality of being.”113 By looking to himself, the Creator generates oneness (or being) and equality (or form) and their union.114 Later in the Complement, Cusanus outlines a quadrivial cosmogony in these terms. First God created the point (almost nothing) and unity (almost God), the principles of multitude and magnitude respectively. Then God united these into one point, enfolding all the polygons and circles that give form to creatures.115 God’s geometrical constructions thus begin outside the ordinary conditions of the quadrivium, in unity beyond multitude and equality beyond magnitude. But it is the Creator’s transquantitative mathematics that endows the human mind with the capacity to imitate God in the mundane quadrivium. Echoing Bonaventure, Cusanus calls the enfolding power of the divine Word the Creator’s “infinite fecundity.”116 In the act of creation God transfers this enfolding power to the creature. “Looking unto himself and his infinite fecundity,” Nicholas writes, “the Creator himself creates the fecund essence of creatures. In this essence is present an enfolding beginning of power, which is the creature’s center, or being; this latter enfolds within itself the creature’s power.”117 The cardinal had already compared the enfolding power of divine and human minds in De mente, but now, crucially, he connects it to the quadrivium. By creating the human mind through divine Equality, the Trinity imbues it with the Word’s enfolding power. This is what originally enables the human geometer to unify (projecting multitude) and to equalize (projecting magnitude), and thus to generate her own finite quadrivial categories (to mathematize). After describing God’s complex relation to the quadrivium, Cusanus takes the more radical step in De theologicis complementis of naming God as the “infinite Angle” and “infinite Number.” Nicholas reasons that an infinite angle would enfold all opposing angles, maxima and minima. It cannot be simply maximal as an infinite quantity might be, but rather must be infinite in a transquantitative manner. Just as the geometer uses angles to transform any geometrical shape into another, so too the Creator uses the infinite Angle to transform any creature into

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another.118 Yet since this Angle used by God must also be God, the Angle represents an intra-Trinitarian relation of self-measure: By means of himself God works whatever he wills to, even transforming one thing into another. Moreover, for different transformations, it is not necessary that God have (as it is necessary that a geometer have) different angles and different instruments; instead, God transforms all things by means of a single infinite Angle. And because God is this Angle and because the will of God is God—and so, the will of God is this unqualifiedly maximal Angle—it follows that God, by his own will alone, transforms and transmutes all things.119 God measures by measuring Godself, and God creates by using Godself as a geometrical instrument. Since God relates to God as the Word (the will of God), Cusanus effectively depicts the Word as an infinite Angle. God is therefore the preeminent geometer on account of the fact that God is the preeminent geometry. This is nothing more than the principle of mens as mensura from 1450 taken to the ultimate degree, the divine instance. Next Cusanus moves from geometry to arithmetic and repeats the same form of argument in terms of number.120 God is an infinite Number that enfolds all numbers in itself and measures all things, but is itself innumerable: Thus, you behold the incomprehensible, infinite, and innumerable number, which is both maximal and minimal, and which reason attains only in a shadow and in an obscuring mist, because it is disproportional to every numerable number. And you see that God, who is called the Number of all things, is Number without discrete quantity, just as he is great without continuous quantity. And the infinite Angle is the same thing as the infinite Number, so that God himself, being most simple, most simply numbers and measures and transforms each and every thing.121 In a distinct echo of Augustine, Cusanus finally states in the most express terms possible the chief consequence of Thierry’s arithmetical Trinity. God is number without multiplicity and magnitude without quantity. By arithmetizing and geometrizing naturally, as the unmeasured Number and Angle, God establishes the respective foundations of the quadrivium in multitude and magnitude. The quadrivium is therefore, as Thierry was trying to say, an immediate trace of the Trinity. By combining Augustine’s “Number without number” with the Boethian analysis of the quadrivium, Cusanus has effectively restored the original discursive nexus that yielded Thierry’s mathematical theology.

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In light of this Neopythagorean doctrine, the cardinal’s remarks on Greek number theory in the Complement become especially noteworthy.122 Cusanus asks rhetorically whether number is a substance, following the Pythagoreans, or an accident, following the Aristotelians. His clever answer divides number into two kinds, such that each philosophical school apprehends the truth in part. The key is to distinguish between number in “created mind” and number in “uncreated mind.” Divine number originates in God’s mind, the divine Word, as the only truly substantial or natural number. Ordinary human numbers, by contrast, are accidental and artificial.123 That is to say: if one remains within the Aristotelian horizon of his Fundamentum source, then Pythagoreanism is indeed a foolish error, as the Stagirite said. But if one views the Pythagoreans as so many unwitting theologians—or translated their henology into a Christian idiom—then their doctrine of number would be exactly right.

The Geometry of Contemplation Cusanus’s second breakthrough is to reconnect the mathematical meditations of De mente with the mystical contemplations of De docta ignorantia. The doctrines of God as geometer, Angle, or Number may have provided the presuppositions of the Cusan geometrical theology, but its method and character are essentially visual and ecstatic. These two traits are only briefly mentioned in De mente when Nicholas describes the mind’s experience of glimpsing divine simplicity within the “space” of mathematical thinking. Yet both elements reappear in greater detail in the Complement, underscoring the remarkable continuity between 1450 and 1453. Cusanus begins and ends De theologicis complementis with discussions of visuality.124 First he argues that mathematical measure is a mode of intellectual vision that strengthens the mind because it shows the mind to itself. Mathematics is a reflexive activity because it is visual, and visual because it is reflexive.125 Nicholas writes that he wants readers of his book “to behold with mental sight how it is that in the mirror of mathematics there shines forth that truth . . . not only in a dimly remote likeness but also with a certain bright shining nearness.”126 Then in the Complement’s final chapter he adds a kind of epilogue on “seeing” God, the germ of the idea that will inform his famous De visione dei. Here the vision of God follows from the portrait of God as geometer. When God measures Godself in creation, “measuring and being-measured coincide.”127 But God’s self-measure is primarily a self-seeing, a divine visual activity: Simultaneously with seeing himself God sees also all created things; God does not at all see himself and other things in different ways. And simultaneously with seeing created things God sees also himself. For created

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things, because they are created, are not seen perfectly unless their Creator is seen. . . . His seeing himself is his being seen by himself, and his seeing creatures is his being-seen in creatures. One can answer in the same way if he is asked about creating; for in God, creating is seeing.128 Cusanus explains that God “sees” creatures into being when God enumerates or unfolds them using the Word as Measure. God’s arithmetic enumerates the world and God’s geometry spatializes the world, but the primary event is God’s self-seeing in the Word, a self-measure that visualizes the world into being. Even if Nicholas let the geometrical idiom drop away, the dialectics of divine measurement and visualization in the Complement clearly paved the way for his next work, De visione dei. Geometrical theology in De theologicis complementis is not only visual in nature but also ecstatic. In De docta ignorantia Cusanus had suggested that the human mind knows God like a polygon approaching circularity:  no matter how many sides are added, the polygon still remains an infinite distance away.129 Now in the Complement he adds a new detail. “By the grace of the Creator, the mind is caught up from angular capacity to circular capacity,” when it is illuminated and perfected by God’s light.130 Just as readers pass from particular books to scholarly mastery, so the contemplative passes from particular signs in the material world (the polygonal) “through rapture, to the perfection of mental capacity” (the circular).131 The common term here (raptus) is the word used most frequently in Latin Christian mystical texts to denote ecstatic experience.132 Nicholas had referred to ecstasy in other contexts, but now in the Complement he transplants the concept into the unfamiliar soil of the quadrivium and specifically geometry.133 Geometrical theology, in which one progresses from the polygonal human mind to the circular divine mind, must be a mystical theology, for nothing bridges the infinite divide between the two species of geometrical figures. As Cusanus stressed in De mente, when human and divine minds are both numbers, then number no longer serves to mediate between them.134 Therefore God must intervene to seize the human geometer beyond herself. The moment of transition from the second to the first mode, as the mind in the mathematical realm exceeds impossibly beyond number to glimpse the Number without number, is necessarily a gift of ecstasy.

Measures of Development As Nagel has observed, there is a special connection between De docta ignorantia and De theologicis complementis that encourages us to draw some comparisons between the cardinal’s initial encounter with Thierry in 1440 and his later accomplishments.135 The usual custom in considering the arc of Cusan intellectual development is to attend primarily to De coniecturis and De mente as departures

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from the cardinal’s former theological preoccupations. Flasch contrasts the negative mystical theology of De docta ignorantia with the positive mathematical philosophy of De mente. But what if, on the contrary, De mente were only a turning point that facilitated the continuation of large portions of the agenda of De docta ignorantia into De theologicis complementis and beyond? If I am right that in De coniecturis and De mente Nicholas developed, above all, in his textual understanding of the Chartrian sources, then the effects of that newfound clarity would become visible primarily after 1450, not in 1450. Only in the Complement can we measure how far Nicholas’s mathematical theology has come since 1440. First, Cusanus returned for the first time in De theologicis complementis to the project of naming God through geometrical figures that he had begun in Book I of De docta ignorantia. In 1440 he was still constrained by Fundamentum at the heart of Book II and the resulting Christology of Book III. Such compositional demands in De docta ignorantia prevented Nicholas from correlating his geometrical images with other themes like Thierry’s reciprocal folding, God as quadrivial artist, or the identity of the divine Word. After the détente reached with Fundamentum in 1450 on the second mode, Cusanus was free to mix together geometrical images with the divine Word, and both with reciprocal folding and quadrivium. This was precisely what he undertook in the Complement, as we have just seen. Even while he invoked Ps.-Dionysius on the necessity of theophanies and the leap into darkness, his argument in De docta ignorantia remained beholden to the Platonic (and Aristotelian) method of gradual ascent through increasing mathematical abstraction. In De mente that model was replaced by the iconic function of mathematics. By the time of De theologicis complementis Cusanus had expanded upon his new model by returning to Ps.-Dionysian elements from 1440, now unbound by the restrictions of the mediating Platonic schema. In place of the mind’s stepwise leap (transilire, transumere), the Complement prefers the immediacy of divine raptus, and in place of the mind’s darkness, the “bright shining nearness” of divine visio.136 One can also compare the theological function of the quadrivium in both works. In 1440 Cusanus had depicted God creating the cosmos with the tools of the quadrivial arts. In 1453 he repeated the image, not as a charming conceit, but now intended literally and argued for rigorously. The Creator is the preeminent geometer, and the act of creation is a geometrical act: the projection of spatial folds and the visualization of arithmetical patterns heretofore concealed in the divine Mind. In 1440 God’s perfect precision had revealed the negative character of human knowledge, and Nicholas had considered the quadrivium a sign of our exile to the realm of finitude, proportion, and relativity. But in 1453 the self-measuring of the human mind in the quadrivium pointed in the other direction, toward the divine self-measure in the Word as Angle. That is, the quadrivium is now an icon of the Trinity.

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Finally, in the wake of De docta ignorantia the opposition of Logos and Arithmos had continued to disturb Cusan theology throughout the 1440s, as we saw in Chapter 8. These were reconciled in principle if not yet in deed when Nicholas restored Thierry’s second mode in De mente. But it is only in De theologicis complementis that Cusanus fully worked out the solution. By describing God’s self-measure and self-regarding gaze not only as Word but as Number, Nicholas embraced the equivalence of Logos and Arithmos in the most definitive terms possible. In this way Arithmos was harmonized with the Trinity, at least with the Logos as Verbum increatum. For the last and most difficult feat—reconciling the disembodied quadrivium with the flesh of the Verbum incarnatum—one must follow the cardinal’s path into the 1460s. This new plateau reached in 1453 continued to inform the cardinal’s theological writings well beyond De visione dei later that year. In 1458 Cusanus redoubled his efforts in the rigorously philosophical De beryllo. Composed amidst a flurry of purely geometrical works in 1457–9, this treatise repeats some major themes of De theologicis complementis, such as God’s self-measuring through angles and the enfolding of geometrical space.137 It reframes major principles of Cusan mathematical theology like the inherent negativity of quadrivial knowledge and the transquantitative status of the quadrivial categories applied to God.138 But it also does something new. In De beryllo Cusanus conspicuously turns his attention back toward De docta ignorantia, and specifically toward the philosophical problems generated by his appropriation of Fundamentum. The new clarity of his post-1450 geometrical theology had provided him with a superior vantage point from which to inspect two unresolved issues. The first problem that Cusanus had overlooked until De beryllo was the tension between Plato and Aristotle generated in Book II of De docta ignorantia. As we have seen, in 1440 Nicholas shoehorned a Peripatetic critique of Platonism into an otherwise profoundly Platonist work, compelling him to mitigate some of his source’s diatribes against the Platonici and in one case to censor that term of abuse. In De beryllo he surprises us by attributing both Thierry’s arithmetical Trinity and his modal theory to Plato. What Cusanus considers Plato’s triad is in fact the supreme god, demiurge, and world-soul of the Timaeus.139 But then he assimilates these to the first, second, and third “modes of being,” as well as to Aristotle’s prime mover. He cites his prior discussion in De docta ignorantia by name and repeats Fundamentum’s judgment that there is no world-soul or universal intellect on account of the distinction between absolute and contracted.140 A few pages later Nicholas even attempts to collapse those theological and modal triads into Aristotle’s form, matter and their union.141 Note that when Cusanus attempted to reconcile Plato and Aristotle in De beryllo, his first instinct was to focus on the charges that Fundamentum had raised against Thierry. Indeed, the very strategy of linking Thierry’s doctrines to ancient philosophies was initiated by the author of Fundamentum himself.142

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The second philosophical debate that Cusanus tries to settle is the controversy over Pythagorean number theory in Plato and Aristotle. Cusanus had characterized De docta ignorantia as a “Pythagorean” project (inquisitio pythagorica), and in De mente the Layman quipped that he did not know if he was a Pythagorean or not. Could Cusanus answer this question for himself? In De theologicis complementis the cardinal researched the differences between Aristotelian and Pythagorean theories, concluding that the Pythagoreans only erred when they failed to grasp the theological nature of their enterprise. Now in De beryllo the cardinal sides with Aristotle against Plato and the Pythagoreans alike. He would have known from reading Albert’s commentary on the Metaphysics that Aristotle charged Plato with being seduced by Pythagorean folly. Cusanus concludes that whereas Plato wrongly assumed that mathematicals would convey the mind to a still higher abstraction in the forms, the Pythagoreans mistook mathematicals for ontological principles when they are only epistemological.143 As we have seen, this passage has been interpreted as the cardinal’s ultimate break with Pythagoreanism.144 But this is rather difficult to square with the praise he continues to shower on Pythagoras and Pythagoreanism well into the 1460s in De ludo globi, as we shall see. In the context of his evolving dialogue with Fundamentum, this passage in De beryllo ought rather to remind us of the difficulties Cusanus encountered by holding fast both to the Christian Platonism he shared with Boethius and Thierry and to Fundamentum’s anti-Platonist attack on Chartrian theology. By 1458 Nicholas had already cast his lot with the Platonici, especially after embracing Thierry’s modal theory definitively (even if he remained sufficiently confused about its provenance to hope that it represented the universal philosophy of Plato and Aristotle alike). Yet it remained unclear what theologia geometrica portended for the lasting significance of De docta ignorantia, and not least for that treatise’s extraordinary Christology. It is at least certain that by these years Cusanus had given up his geometrical experiments for good. Following a string of works after De theologicis complementis, the last major text, De mathematica perfectione in 1458, was considered by Cusanus to be his best. It sliced up the quadrature problem into partial lines and curves and leaned on the notion of visio intellectualis detailed in De beryllo. The following year Cusanus penned a few new pages and named them “the golden proposition.”145 In their final lines Nicholas alludes mysteriously to the Trinity as he draws one last geometrical figure. If three lines proceed from the same point, then no matter how they are bounded by curves or lines of varying lengths, their relative ratios to each other will remain the same. Hence their aequalitas, writes Nicholas, becomes the medium that binds contrary to contrary. For there will be no separation among this “trinity of lines” (trinitas linearum), but only a “one simple length” (una simplex longitudo) to behold. “The highest speculation of wisdom,” reads the final sentence, “abides with the triune principle and the things that flow from it.”146

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Completing the Circle in the 1460s We have watched as different species of mathematical theology survived in different skins and climes since the turn of the first millennium. Five centuries after Boethius, Thierry of Chartres posed the question, really for the first time, of whether they could thrive within Latin Christianity as well. His works, like seeds, found their way to Cusanus to replant and hybridize with other strains. After fifteen centuries of colorful Neopythagorean varieties, what would it mean at this juncture to characterize Nicholas of Cusa as a mathematical theologian? We have already counted three distinctive Cusan achievements around 1440 that stand out against earlier Neopythagorean theologies and set the course for a decade of writing. (1) Cusanus was the first to demonstrate the potential of the Nicomachean quadrivium as an intrinsically negative theology. For him mathematical knowledge is fundamentally an encounter with an absent divine perfection, equality, or precision. Thierry had only hinted at this possibility, but Nicholas fully realized it in De docta ignorantia. By combining Thierry’s theology with Ps.-Dionysius, Cusanus effectively revived the original foundation of apophaticism in Alcinous and Clement. For them, numerical concepts are inherently subtractive, such that every proper negative theology must be mathematical to some degree. Yet unlike his predecessors, Nicholas executed this vision specifically through the Nicomachean quadrivium, the four arts structured by multitude and magnitude that best preserved the authentic doctrine of Philolaus and Archytas. The novelty and power of this Cusan synthesis are difficult to overestimate. The theologies of the quadrivium found in Nicomachus, Boethius, and Iamblichus are all positive henologies, consummated in the latter’s arithmetical theurgy. Only Cusanus manages to bind the quadrivium back to Alcinous’s original sense of number as negation, because only he had access to Thierry’s aequalitas concept— itself an echo of the Philolaean “Limit” and the Philonic image of divine Equality as “all-cutting Logos.” (2) Nicholas’s next achievement was a fully mathematical theology of the Incarnation, or a mathematized Christology. In this too he took a step beyond Thierry’s Trinitarian theology by confronting the problem of contending mediations. In his zeal to root autonomic arithmetical mediation directly in God, Thierry

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had failed to preserve the identity and supremacy of the Word in the eyes of several of his readers; he tipped the balance too far from Logos to Arithmos. Nicholas accidentally solved this problem by adopting Fundamentum’s Peripatetic theory of contraction that equated individuation with enumeration. When he combined this with Ps.-Dionysian Logos Christologies of the thirteenth century, he could define Christ as the unique intersection of the Equality of the One and the singularity of numerical series. With these tools in hand Nicholas relieved the pressure between different Platonic mediations in ways that Thierry in his day could not. (3) One should not discount the Neopythagorean achievement of De coniecturis on its own terms. Whatever its value for the future development of Cusan thought, that book integrated within one theological vision a range of different Neopythagorean voices as few had done before. Nicholas combined the Nicomachean theory of multitude and magnitude with the arithmological decad of Philo, Theon, and Macrobius. He reimagined the cascade of unities descending from the One that Eudorus and Moderatus had envisioned, but then aligned these with Iamblichus’s hierarchy of number-species. What De coniecturis curiously lacked was the mediation of the Logos, whether in Platonist or Christian form. When mediation is secured by Arithmos alone, there is no tension, no contest of paradigms. This explains why Nicholas’s impressive treatise became less important as he focused on harmonizing the two competing mediations throughout the 1440s. Looking beyond the first decade of Cusanus’s evolution, we registered three further innovations that flowed from his restoration of Chartrian theology around 1450. (4) Nicholas explored the anthropological implications of Thierry’s arithmetical Trinity and tested out his discarded spiritus theory of mind. When he put these together in De mente, the result was a powerful recasting of mathematical theology for the fifteenth century that outstripped many Neopythagoreans of the past with its boldness and coherence. God is the Number without number who self-measures as the Trinity of unity, equality, and connection. The human mind that is God’s image is therefore inexorably mathematical and self-measuring, a virtual participant in the eternal divine μάθησις. As it ascends through Thierry’s four modes of being, the same mind can turn inward and perceive its own activity iconically. This ecstasy of its ordinary quadrivial procedures seizes the mathematician up into a vision of God: a mysticism not opposed to mathematics but within it. (5) Besides the arithmetical Trinity and modal theory, Cusanus also radically expanded the purview of Thierry’s signature theology of the spatial fold. Thierry had confined reciprocal folding to the task of organizing the four modes. Nicholas never used the doctrine in his version of the modal theory, but he also enlarged the domain of folding to include every possible interface between unity and plurality, God and world, eternity and time. In De docta ignorantia Cusanus meditated on geometrical figures even as he tinkered with new applications of complicatio and explicatio. But only in 1450, after he had come to embrace necessitas complexionis,

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the highest fold this side of God, did his theology fulfill the promise of Thierry’s geometrized or spatialized theology. This began in earnest in De theologicis complementis, a treatise on what we could very well call the space of God, the divine capacitas infinita. Cusanus explained the mystical meaning of infinite circles, drew connections between the visualization practices of the geometer and the contemplative, and named God the primal Geometer who is also an Angle. In De ludo globi, such spatializing effects continued to reconfigure how Nicholas thought about the “roundness” of the Trinity. (6) Nicholas grasped from the beginning that the Boethian quadrivium was intimately linked to his Chartrian sources. Over two decades, he took every opportunity he could to supplement Thierry’s Neopythagoreanism with elements from Boethius—a hermeneutical practice that effectively intensified the Nicomachean influence already coursing through Cusan theology from Thierry. In De docta ignorantia Cusanus compared Boethius and Augustine on their esteem for mathematics and situated the mysterious Fundamentum between two discussions of the quadrivium. In the 1440 sermons and De coniecturis he scrutinized the mechanics of multitude and magnitude to a degree matched only by Nicomachus. Although Thierry never dwelt on these categories in the same way, Nicholas found that he could ground them in the arithmetical Trinity itself. Like both Boethius and Thierry, he was fond of repeating the Nicomachean maxim that number was the first exemplar in the mind of the Creator. In the end it was the restoration of Thierry’s modal theory that entailed a restoration of the foundations of the Boethian quadrivium: first in De mente, then in De theologicis complementis, and once more, in a very different key, in De ludo globi. Nicholas clearly felt himself in solidarity with the man he called “our Boethius,” another statesman who knew mathematics as well as theology. This distinguishes Cusan Neopythagoreanism as arguably the most authentic variety of his day, at least when compared with Renaissance peers distracted by the red herring of Pythagoras legends. For it was through the actual historical channels that had first siphoned mathematical theology into Latin Christianity—Boethius and his “brilliant commentator”—that Nicholas made contact with the spirits of Nicomachus and Moderatus, Philolaus and Archytas. These six achievements, along with a seventh to be named in this chapter, allow us to define without anachronism the kind of mathematical theology that Cusanus could achieve within the discursive limits of his century and his religion.

New Impulses in the Late Works Near the end of his long life, Nicholas felt the same retrospective impulse that seems fitting to us here. The late 1450s had been trying years. As his reform initiatives

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waned, his episcopacy devolved into a violent standoff with the local prince, Duke Sigismund. By 1458, driven out of his own diocese by Sigismund’s forces and left fearing for his life in Bruneck castle, Nicholas decided to depart for Rome. He would never return to his native land. Life in the curia was another disappointment, and he pleaded with Pius II to relieve him of his duties. Nevertheless, the late 1450s were exceptionally fruitful period for Nicholas, and his works shine with fresh colors and new energy. But after a prolonged intestinal illness almost took his life in 1461, the cardinal decided to get his affairs in order.1 He commissioned a new manuscript of his collected works and authorized a last will confirming the foundation of a charitable hospital in Kues. In 1463 he wrote the lengthy retrospective De venatione sapientiae. In lieu of constructing new names for God, this work surveyed ten “fields” in which he had hunted for divine Wisdom throughout his past works.2 If Cusanus began to take stock of his intellectual accomplishments in the early 1460s, this must have included reflecting on his Chartrian sources. In De beryllo he had already wondered if the new geometrical theology of 1453 might reduce the tensions between Platonism and Aristotelianism that had troubled Book II of De docta ignorantia. The next and more important question to ask concerned the theology of Incarnation in Book III, the most significant result of his engagement with Chartrian theology from 1440. Hence Cusanus’s final task in the 1460s was to connect his first synthesis with his final synthesis, the mathematized Christology of 1440 with the geometrical theology of 1453. This is achieved over the course of the lengthy and complex Dialogus de ludo globi (1463). Finished a year before the cardinal’s death, the dialogue stands as a kind of second, parallel last testament alongside De venatione sapientiae. But instead of being a retrospective account of the names for God that Cusanus had invented throughout his writings, De ludo globi is a more intimate memoir of his own authorial struggles over two decades of meditating on the Chartrian texts. In De ludo globi Cusanus also discusses, for the last time, both the four modes of being and the quadrivium together in one text. This had occurred previously only in De docta ignorantia and De mente. It is no accident that the concluding section of the work is a tribute to Thierry as well as to Pythagoras, whose modus philosophandi, writes the aging cardinal, not even Plato superseded.3

Return to Christology Cusanus’s attempt to bridge Christology and quadrivium in De ludo globi benefited  from the momentum he had built up over the previous few years. Since the late 1450s, he had been focusing his energies away from purely mathematical works and instead on two new arenas of interest. The first was a renewed attention to the theology of the Word flowing out of his intensive preaching between

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1454 and 1457 as bishop of Brixen. Nicholas had formerly preached only on major holidays, but he now came to view sermons as central to his episcopal vocation.4 As Walter Euler has demonstrated, the sermons from this period are staunchly Christocentric and revive major themes from the 1440s. The Incarnation represents the medium and goal of creation, the manifestation of the ineffable God, and the perfect enfolding of the human intellect.5 “If Christ is known,” Cusanus preached at a diocesan synod in 1457, “then all things are known in him. If Christ is possessed, then one possesses all things in him.”6 In the summer of 1459, Nicholas collected these Brixen sermons into a new edition. He appended to the same volume two short discourses putatively on the Johannine theology of the Word, but also intensely philosophical: De aequalitate on John 1:4, and De principio on John 8:25.7 In the first treatise Cusanus contended that the self-knowledge that the soul naturally seeks finds its basis in the self-equality of the divine Logos. Here Nicholas followed the thread in De filiatione dei and De mente on the reflective character of knowledge, where mind as self-measure is an image of the Word’s sovereignty.8 He tried to connect this epistemology to Thierry’s aequalitas, implying that both give expression to the Johannine Logos.9 In De principio Nicholas constructs a henological account of the divine Logos as absolute beginning (principium) and leans heavily on Proclus’s Parmenides commentary.10 The unusual character of the treatises as philosophical addenda to sermons illuminates the cardinal’s intellectual situation at the end of the decade. We saw that after the theological successes of 1453, Cusanus used De beryllo to address lingering philosophical questions from De docta ignorantia. In these two treatises Cusanus once again looked back, but now he sought to link the high Christology of 1440, newly revived in his Brixen preaching, to his post-1450 philosophy. But if Nicholas’s goal was to draw connections between his Christocentric homilies and his epistemological explorations, neither treatise is completely successful. The wheels are spinning but not gaining traction. With the benefit of hindsight Cusanus seems constricted by his decision to use the arithmetical Trinity as the only bridge between Logos Christology and mathematizing epistemology. For that triad alone does not enable Nicholas to speak to the crux of the problem: not the eternal Word per se, but the material dimension of the Word’s embodiment in Jesus. How does the quadrivium relate to the corporeal solidity of the Incarnation at the center of the physical cosmos, such as in De docta ignorantia III? This omission would have to be rectified if Cusanus was to unify the achievements of the 1450s and the 1440s. Around the same time, Nicholas was also increasingly interested in the theological potential of possibilitas. Of course Fundamentum had dismissed the mode of absolute possibility out of hand, and even when Cusanus accepted the second mode in De mente, he tried to minimize the role of the fourth mode.11 But from

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1460 forward, Cusanus took step after step toward naming God in just these forbidden terms. God is the singular intersection of sheer posse and real existence, the eternal Can-Be (posse-est: Trialogus de possest, 1460). Indeed, understood rightly, God is nothing other than absolute possibility (posse absolutum:  De ludo globi, 1463) or possibility itself (posse ipsum: De apice theoriae, 1464).12 It was not until 1450 that Cusanus embraced the second mode as a name for mens. Likewise not until 1460 did he begin to speculate that the fourth mode could name God. Closely connected with his fascination with possibilitas was the cardinal’s new preference for dynamic models in which God is glimpsed through movement and life, as opposed to static geometrical figures. In the Trialogus de possest Cusanus goes so far as to state that God is not simply a mathematical Trinity, but a “true and living” Trinity that can be mathematically understood.13 Of course in the Boethian and Chartrian orders of the sciences, possibility and motion are the subject matter of physics. The new appeal that such categories held for Cusanus appears to be more than a cultural or linguistic turn in the cardinal’s late works. Rather, it represents a disciplinary sea change from theologia geometrica toward a kind of theo-physics.

A Physics of Divine Possibility This methodological shift began with De possest in 1460 when Cusanus squarely addressed the theological status of possibilitas at the beginning of the treatise. Not surprisingly, he appears to do so with reference to Fundamentum. Nicholas posits that God is easily known as actualitas absoluta. But what about absoluta possibilitas? He reasons that since actuality and possibility imply each other, neither one ultimately precedes the other; hence they are coeternal. This leads him to formulate a Trinitarian analogy: absolute potential, absolute actuality, and the nexus of both.14 This triad, which Nicholas repeats often throughout the treatise, validates possibilitas as an equally warranted divine name. In fact the titular neologism posse-est is nothing more than a provocatively condensed version of the same triad.15 The notion of possest, he later explained, is sufficient by itself as the “unitary principle for all modes of being.”16 Nicholas’s argument plays out on Fundamentum’s territory in several regards. First, Thierry had never used the term actualitas absoluta in his formulation of the four modes, but Fundamentum readily switched between necessitas and actus. The very idea that God can be named as the coincidence of act and potential is formulated in Fundamentum’s polemic against the fourth mode.17 In fact Cusanus’s Trinitarian formula verbally resembles the concluding triad proposed by the author of Fundamentum.18 Moreover, in making his case in De possest, Nicholas examines the status of possibilitas by asking whether it was “coeternal” with God, “preceded” the actuality of the world, or existed “before” it. Since all three terms

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were used together by Fundamentum in order to demote the fourth mode, it made sense for Nicholas to focus on these charges that he wished to see overturned.19 Fundamentum had defined the divine absolute both in terms of its enfolding power and as the maximum motion coinciding with its minimum, perfect rest. Likewise, after defining his new triad, Nicholas links it to enfolding and to maximal and minimal motion—a comparison that leads him directly to the image of a spinning top.20 There are several indications that Cusanus was consulting Commentum as well.21 For these reasons it seems likely that in De possest Nicholas returned to Fundamentum to sharpen his provocative arguments in favor of the divinity of the fourth mode. Unlike the static figures of the 1440s and 50s, those in De possest introduce genuine physical motion into geometrical space.22 Cusanus called on such mobile images to illustrate the paradox of the possest. He describes the game of a spinning top that rotates faster the more forcefully one pulls the top’s thread, ultimately so fast that it appears to stand motionless. Nicholas envisions the upper surface of the top as one circle (eternity) that projects an identical circle on the ground beneath it (time). If the top spins with “infinite velocity,” then any two points on the circumference of the top’s surface would be co-present (in eodem puncto temporis) with any given point on the fixed circle. Two opposite points on the upper circle, he explains, would come before or after a single point on the lower circle when the top was moving at a finite velocity.23 But at “infinite velocity” every point on the rotating circle would occupy every point on the static circle, leaving no motion or succession detectable, so that “the maximal motion would at the same time also be the minimal motion and no motion.”24 This coincidence of motion and rest appears in one of the passages of Fundamentum that had been copied into De docta ignorantia.25 Cusanus delights in this mobile figure as an illustration of coincidentia oppositorum. From the spinning top, he notes, one can grasp other truths. A top that is motionless and “dead” can be brought to life when the player knows how to apply the right motion. By pulling the string one can “impress” (imprimit) movement into the resting top “beyond its nature” (supra naturam). The “moving spirit” thus applied lifts the heavy top from its own center and makes it rotate like the celestial spheres, remaining “invisibly present” so long as the impressio of the imparted force lasts, like the motion of the heavenly zodiac enlivening nature. In the same way, Nicholas writes, the Creator gives the “Spirit of life” to the dead.26 With the term impressio the cardinal alludes to the impetus theory of projectile motion that held sway before the early modern theory of inertia.27 By using a dynamic image from physics to indicate both the zodiac and the Creator, Cusanus tests the limits of the Boethian disciplinary configuration. In De docta ignorantia, mathematics had functioned as a waystation for mystical ascent because static geometrical figures minimized material flux and imitated divine

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stability. When its images become mobile, mathematics cannot play the same mediating role. But this is a development we have already seen underway in the cardinal’s new geometrical theology. Number is relieved of its duties of hierarchical mediation and instead operates as an icon or trace of divine presence in the world, as Cusanus mapped out in De mente and then De theologicis complementis. In De possest the beauty of the physical cosmos, teeming with the life of creation, is a sensible phenomenon (ostensio) that manifests the invisible God.28 The absolute is reached immediately not by mathematical abstraction, but by transcending contracted experience, whether that means the quadrivial category of magnitude (quantitas sine quantitate) or the phenomenal richness of the senses (odor sine odore, sonus sine sono).29 Cusanus seems well aware of the tension between his mathematical pursuits in the 1450s and his new physics of Possibility. Toward the end of De possest, as if to clear the air, the cardinal conspicuously reiterates the Boethian division of sciences that had organized Thierry’s commentaries. Physica examines nature and enmattered forms with the senses and reason; the highest inquiry, performed by the faculty of intellectualitas, is to theologize about motionless form.30 In between these, Nicholas draws attention to mathematica performed by intellect and imagination, a science also known as disciplina or mathesis, as he carefully records. With these words Nicholas references the passages from Thierry’s commentaries that had helped him articulate the new disciplinary model of De mente.31 Now in De possest Cusanus restates the special theological vocation of mathematical concepts as aenigmata: visual symbols that participate in the mathematizing divine reality that they signify. Whatever knowledge we have of God, the divine Precision, is imprecise and known indirectly through the aenigma or analogical mirror of our own mathematical thinking.32 “We have no certain knowledge except mathematical knowledge,” writes Cusanus, “and this is an aenigma for searching into the works of God.”33 Or as he later concludes: “I think that a quite close aenigma can be made for theology in accordance with an accurate understanding of mathematics.”34 Every important philosophical doctrine is based upon a mathematical similitude. According to Nicholas, even the Christian Trinity is seen in the aenigma of quadrivial foundations (principium mathematicae), since oneness symbolizes multitude (discrete quantity) and threeness symbolizes magnitude (continuous quantity).35 Hence as we turn to the 1460s, Cusanus gradually turned away from the abstractions of mathematical thought toward matter and possibility, toward the concrete and tangible, toward culture and language—toward the same dimensions of Incarnation that he had been rediscovering in his Christological Brixen sermons.36 It is as if in the cardinal’s last years, having exhausted his investigations at the nexus of theology and mathematics, he turned to the third science in the Boethian division, namely physics. Or we can think of this shift in terms of the quadrivium

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itself. If Boethius had conceived divine unitas in terms of arithmetic, Thierry had discovered a path to harmonics (the proportions of aequalitas) and protogeometry (the space of complicatio and explicatio). Taking the baton, Nicholas advanced into geometry proper (universal folding) and finally, in his last years, to the fourth quadrivial art of “spherics,” or solids in motion. The great De ludo globi built upon these themes of the 1460s, as if bending the arc of the cardinal’s development in the 1450s until it reconnected with his 1440 starting point, completing the circle.

Incarnation and Neopythagoreanism Ever since De docta ignorantia, Cusanus had struggled to access Thierry’s theology freely despite the obstacles thrown up by the Fundamentum treatise. He weighed its prohibitions in the 1440s, upended one critique in the 1450s (the second mode), and discarded another in the 1460s (the fourth mode). Yet it was this dialectical confrontation with his sources that drove Nicholas to explore, test, internalize, and reimagine the full range of Chartrian doctrines. Those difficult texts prodded his thought like the sand in an oyster. On the other side of his labors, De ludo globi stands as the irenic summa of the cardinal’s mathematical theology, and its pearl. It contains the entire library of Neopythagorean doctrines developed since 1440. The dialogue’s import is not manifest on its lustrous surface, but hidden beneath many layers. The dialogue comprises two books of conversations with John and Albert, the young dukes of Bavaria. It manifests both of the new themes from the 1460s we have just examined. On the one hand, the central image, a “game of spheres” or children’s ball game, begins by delineating a geometrical space.37 Players roll a spherical wooden ball on a gameboard composed of nine concentric circles. Each circle receives a number, marking the degree of its distance from the center, which is the number ten. But then Cusanus animates this geometry with materiality and motion. The difficulty of the game arises because the spherical ball has a smaller hemisphere scooped out of one side, making it wobble into a spiral when rolled. The goal is to master the ball’s irregular rotation and make it come to rest as close to the center as possible, winning points according to the circle where the ball lands. On the other hand, Nicholas deploys several major Christological passages at critical junctures, as the mystical figuratio of the game’s hidden meaning unfolds.38 In Book I the central point of the gameboard is marked by the “sphere of Christ” (globus Christi). The first to score thirty-four points, the years of Christ’s life, wins the game. Book II considers the meaning of the gameboard’s flat surface. The ring of graduated circles is also a map of the world: nine choirs of angels and nine levels of life encircle and cycle around the decadic Christ-point like planets orbiting the sun.39

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Cusanus also continues the trend from De possest of using physics to construct his aenigmata. As John notes, there are two factors that determine the paths taken by the rolling ball, and that ensure that no two throws are the same: the conditions of the surface “medium” on which the ball rolls, and the “impetus” transferred to it.40 Nicholas insists that one cannot abstract from material irregularities in the figure of the ball game such as uneven pavement, obstructing pebbles, dirt on the ball, fissures in its construction, or the inherent randomness of each throw.41 This emphasis on creaturely difference is consonant with his conception of precision and divine Equality in De docta ignorantia. No two rolls of the ball are exactly alike, and no two balls can be equidistant from the Christ-center.42 Attending to material irregularities within the geometrical symbol represents, however, a remarkable methodological volte-face compared to the cardinal’s enthusiasm for mathematical abstraction in the 1440s and 1450s. Impetus theory also helps Cusanus to explain how the ball’s uneven shape makes it roll in a “helical” or “spiral” movement.43 According to late medieval physics, the more impetus that is applied, the straighter the line will be, and as the level of impetus recedes, the ball returns to its native tendency to spiral inward to a halt. So too in human affairs, the unsteady ball (or weak will) tilting toward the Christ-center can be forced by a stronger throw (or exercise of virtue) to overcome its tendency to veer off path.44 And if a ball were perfectly round, Nicholas notes, it could attain a perpetual self-motion.45

Christ as Center point Unlike De possest, the new dialogue takes advantage of the Christological impulses of the late 1450s.46 There are three major passages to consider. The first concerns Christ as the center of the gameboard and comes at the end of Book I. Cusanus explains that Christ is not only the divine center point, but like every other human life is a sphere. Yet the sphere of Christ lacks the scooped-out hemisphere that causes all other human lives to wobble in their transit across the gameboard. Therefore Christ became the center of the gameboard by rolling the ball of his life in an exemplary way: This game, I say, represents the movement of our soul from its kingdom to the kingdom of life in which is peace and eternal happiness. Jesus Christ, our king and the giver of life, presides in its center. Since he was similar to us, Christ moved the sphere of his person so that it came to rest in the middle of life, leaving us the example so that we would do just as he had done. And our sphere would follow him although it would be impossible that another sphere attain peace in the same center of life where the sphere of Christ rests.47

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As human, Christ is the exemplary ball, and as divine, the center of the board. Throwing one’s gameball thus represents the not only the soul’s movement toward its center, but also the desire to conform one’s own curvings to the straight line of Christ.

Christ as Motion A second, longer passage arrives near the beginning of Book II.48 Here Nicholas again takes up the image of Christ as the center point of the gameboard’s concentric rings. The nine circles of the gameboard signify what Cusanus calls the “kingdom of life.” If the central point is the divine king, the circles mark out the infinite degrees (gradus) of the living cosmic kingdom. Their roundness signifies perpetual movement around the center; like planets around the sun, the closer each is to Christ, the faster it moves. The motion which is the life of the living is therefore circular and central. The closer the circle is to the center the more rapidly it can orbit around the center. Therefore the circle that is also the center can orbit instantaneously. Therefore it would be infinite motion. The center is a fixed point. It will therefore be the greatest or infinite motion and likewise the smallest, where the center and the circumference are the same. And we call it the life of the living, which enfolds all possible motion of life in its fixed eternity.49 In this complex figure, Christ enfolds both cosmic space and cosmic motion. In terms of geometry, Christ the center point is at the same time the greatest circle (as human) and the non-space of the point (as divine). In terms of physics, Christ is infinite stillness (as divine) and yet, as the circle infinitely nearest the center, Christ also moves with infinite velocity (as human). Christ is therefore the personal union of point and circle and of rest and motion. Cusanus thus repeats the image of coincident rest and motion in the infinite velocity of the top from De possest a few years before. But the notion of the two natures of Christ as the personal reconciliation of cosmic oppositions and the theme of God as cosmic center are most reminiscent of the Christology of De docta ignorantia III. Nicholas is hearkening back toward the Neo-Chalcedonian theology that he deployed to synthesize his Chartrian sources. But now the cardinal is testing it within the new coordinates of the 1460s, namely a geometrical space animated by real physical motion. Our suspicions are confirmed when Cusanus nods to the language of Chalcedon a few pages later: For here the center of the life of the creator and the circumference of the creature are identical. For Christ is God and man, creator and creature; he is the center of all blessed creatures. . . . [A]‌ll the blessed represented

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through the circumference of the circle come to rest in the circumference of Christ which is similar to the created nature. And they attain their end on account of the hypostatic union of the circumference of the created nature with the uncreated nature than which nothing greater can be. . . . [Christ] is the unique mediator through whom access to the living life can be had.50 As the unio hypostatica and unicus mediator of center and circumference, Christ reconciles geometrical oppositions by virtue of being himself the principle of the theological space inscribed in De ludo globi. By moving in their own circumferential circles, the creatures in their infinite diversity rest in Christ’s created nature. By following Christ’s trajectory toward the divine center that he, too, inhabits, creatures are drawn into the infinite velocity of the Creator. Hence the space opened by the distance of center to circumference—or the transition from movement to rest—is indelibly Christomorphic. This remarkable sequence represents the pinnacle of the cardinal’s theologia geometrica after De theologicis complementis. Since then Cusanus has apparently rethought the spatial basis of that divine geometry and now wishes to ground it in the Incarnation. In De docta ignorantia Nicholas had called God the universal center and circumference of the cosmos, since the infinite universe can have no fixed center or circumference of its own contracted nature.51 Now in De ludo globi those predicates of the infinite sphere are transferred specifically to Christ. Christ is the sphere in which the centers of Creator and cosmos are juxtaposed and united. When they are so rendered, the physics of motion and geometry of space become themselves authentic avenues to theological contemplation. No longer a hierarchical mediator bridging time and eternity or contracted and absolute, Christ is now a spatial medium, a ubiquitous center disseminating networks of geometrical order, the pathways for return, throughout the entire theocosmic topography.

Christ as Facial Image The final Christological passage comes at the close of Book II. Cusanus concludes De ludo globi with an unusual discussion of monetary “value” (valor), where Christ is compared to the face stamped on coins. Just as minting adds value to coins, so the goodness of being adds “value” to being. On account of its divine source, the absolute Good is the only valuation and yet one beyond all value (mirroring the dialectic of divine names). Even if value is not reducible to the intellect’s valuation, the intellect is the only being that perceives value.52 Cusanus therefore proposes the metaphor of a coinmaker (monetarius:  God) who mints an infinite variety of money out of his inexhaustible treasury. Each coin is appraised by a banker (nummularius: the intellect) who is empowered by the mint with extraordinary freedom to discern the “value, number, weight, and measure” of each coin. But what guarantees

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the monetary value of each coin, notes Cusanus, is not its gold or silver material, but rather the “image or stamp” (signum) of the mint, the seal of the coinmaker.53 In this allegory, the inexhaustible wealth of the divine Coinmaker is revealed exclusively through this sign, the “likeness of his face,” by which one discerns each coin’s value. But the image of God the Father is the Son: Now, the image is nothing other than an inscribed name. Accordingly, Christ asked, “of whom is the coin’s image and inscription?” They answered: “Of Caesar.” Therefore, the Minter’s Face and Name and the figure of his substance and his Son are the same thing. Therefore, the Son is the living image and the substantial form and the splendor of the Father through whom the Father or coiner makes money and puts his sign on everything. Since it would not be money without such a sign, that which is signed on all coins is the unique exemplar and the formal cause of all the coins.54 Although it may seem unusual at first, the figure of the coinmaker builds upon several themes in Cusan theology from the 1440s and 1450s. Nicholas’s allusion to Hebrews 1:3 recalls Thierry’s favored verse in Commentum and Tractatus, and his discussion of the mystical name of Christ recalls Sermon XXIII.55 In the opuscula of the 1440s Nicholas had held that the lesser theophanies of creatures participated in the one great theophany of the Incarnation. During the 1450s he explored how the intellect’s mathematical ability is an image of God’s self-numbering. Moreover, the peculiar conceit of coinage quite cleverly unites two Platonist theological vocabularies. Cusanus defines God’s presence (valor) both as impressed sign and as calculated weight or measure—that is, both divine form as image and divine form as number.56 The “single face” of Christ in all coins reveals their common origin in the divine mint.57 But the intellect also numbers, weighs, and measures (Wisdom 11:21) the coins to discern their value.58 Enumerating the quantity of the coin’s value, then, is not other than reading the name of Christ written upon it. The mathematized Christology of De docta ignorantia has cycled back into the cardinal’s thought in De ludo globi, but it returns, so to speak, in a helical manner, completing the circuit at a level higher than where he had started twenty years before. For as we shall see shortly, when this last Christological passage appears in the dialogue, it is imbricated within Thierry’s four modes of being—for which, in the intervening years, the cardinal had gained a newfound appreciation.

Taking Inventory We have seen how Cusanus used De ludo globi to revisit the physics of De possest and to revise the Christology of De docta ignorantia. Yet he also took the opportunity,

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especially in Book II, to sift through his complete collection of Thierry’s doctrines in a way that he had not done since 1440. These moments make the dialogue stand out as the cardinal’s most evidently Neopythagorean work since De coniecturis, but now with a degree of deliberation and self-awareness that the latter seemed to lack.59 To begin with, Nicholas’s choice of a game of spheres as his setting transforms the entire work into a product of Pythagorean culture. Certain sciences, he explains at the outset, are linked to “instruments and games,” such as rithmomachia for arithmetic or the monochord for music.60 The former, an ancient game of “battling numbers” (a cross between chess and Sudoku), was known as the “philosopher’s game,” and Cusanus compares it to the eponymous ludus of the dialogue itself. Throughout the Middle Ages and during its Renaissance revival, rithmomachia was closely tied to the Boethian quadrivium. No medieval cultural object could have been better chosen to indicate the joint wisdom of Boethius and Pythagoras than this game.61 What makes the ludus globi a fitting vehicle, Cusanus explains, is the way the game exercises the soul’s arithmetical powers by converting the quadrivial arts into didactic play.62 This is why De ludo globi should be viewed as ludus on account of the centrality of number and quadrivium for the dialogue’s argument, and not formally as a “ludic” meditation in general. Having crafted a unique Pythagorean framework for De ludo globi, however, Cusanus appears to simply recapitulate the particular Chartrian doctrines he had already discussed over the decades. As in De docta ignorantia, he includes Thierry’s arithmetical Trinity, reciprocal folding, and the four modes of being.63 As in De coniecturis, he meditates on the hidden power of numbers and the preeminence of the decad.64 As in De mente, he defines the soul’s native activity as self-measuring through number.65 As in De theologicis complementis, he explores the meaning of the quadrivium and its basis in the one and the point.66 So at first blush it can seem that the only special quality of the Cusan Neopythagoreanism on display in De ludo globi is its sheer comprehensiveness. In no other work outside of De mente are all such doctrines collected into one text. Yet now, quite unlike that treatise, they are also spliced together with the cardinal’s Christological insights. So does De ludo globi contain anything new? Interpreters have often considered the work less original and hence less valuable than De li non aliud the year before or De apice theoriae the year after. Some have suggested that Cusanus does little more in De ludo globi than repeat ideas from his past works.67 Most readers are frustrated with its rambling lack of organization—which some are tempted to attribute to the author’s advanced age (despite evidence to the contrary in other late works). It is true that unlike other Cusan works of comparable length, the dense dialogue lacks any kind of chapter divisions. In search of a coherent theme, several different interpreters have suggested that Cusanus viewed philosophy, theology, or human life as a game or form of play. But such abstract ludic readings, as

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their diverse emphases suggest, can become disconnected from the particulars of the text and shed little light on its order.68 Other readers have discovered a putative theme in one corner of the capacious dialogue only to find it absent in another.69 So it is difficult to find interpretations of the structure of the two books of De ludo globi that both attend to the work’s actual content and articulate its originality.70 Nicholas himself complains of fatigue in the dialogue, as if aware that his own gameball was drifting toward its rest. At one point he concedes that he has written better about the same matters elsewhere, since now his powers are fading and his memory slower to respond. Yet he also cryptically states that his “proposed aim” (propositus meus) in writing De ludo globi was “to reduce to an order useful to our purpose this newly devised game.”71 That is, Cusanus gave the game a certain ordo that expressed his purpose in writing the book, which lends a distinctive value to De ludo globi in excess of the particular doctrines recounted in its pages, since he admittedly expressed them better in their original setting. The purpose of the dialogue is therefore to create the space of the game, because this space allows Cusanus to reconfigure old doctrines into a fresh constellation.

Figurae mundi This is the puzzle of De ludo globi: nothing new happens, except its structure, but at the same time that structure is devilishly hard to discern. But this situation is just what one would expect if my argument about the development of Cusan mathematical theology were correct. I have suggested that De ludo globi is the moment when Nicholas looks back from the successful restoration of Thierry’s modal theory in 1450 to the Christology that brokered the initial settlement in 1440. If Cusanus is content with the geometrical theology of 1453, and is only searching for a home for his Incarnation doctrine within it, then his remaining task is to bring these two together. We should not expect any innovation in De ludo globi, because the tensions that drove the cardinal’s creativity on these Chartrian topics have been abated. Rather we should anticipate a new, overarching construction in which past themes can assume new coherence and collective meaning—what Cusanus in other contexts called the figura universi or figura mundi.72 The originality of De ludo globi does not stem from another discrete aenigma to unravel, since the game is played out across the entire textual space of the dialogue’s two books. Nor does it pretend to autonomy from past Cusan works, since on the contrary Nicholas gathers older ideas together deliberately. Instead, by coordinating multiple figures within an integrated spatial field, the dialogue intensifies the tendency toward geometrization initiated in 1453.73 In fact there are not one but three overlaid patterns that organize the text of De ludo globi. Each has its own rationale, and each occasionally occludes the others.

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As in a palimpsest, we can dimly perceive three superimposed spaces that collectively produce a text admittedly jumbled in its richness. The three patterns correspond to different perspectives on the cardinal’s post-1440 development. Viewed in itself the dialogue is best understood through the prism of the Hermetic triplex mundus. This was likely the intention of Nicholas himself. But viewed in light of its composition, De ludo globi, like other Cusan works, manifests traces of his manipulation of different Chartrian sources that rise to the surface at a few junctures. Finally, viewed in light of Cusanus’s total development, De ludo globi links quadrivium and Incarnation more radically than ever before. Whether intended or not by the cardinal, this third option explains more details of the dialogue than the others and deals in particulars rather than abstractions.

A Threefold Space According to the Hermetic Asclepius, the human soul is a microcosm of the macrouniverse, and together they reflect the divine being.74 Cusanus alludes to this doctrine in De ludo globi when he states that “the world is threefold: a small world that is man, a maximal world that is God, and a large world that is called universe.”75 Each of these three “worlds” is visible within the others. God is manifest within the physical universe, which in turn shines through into the human microcosmos.76 “The stable oneness of the large universe is unfolded quite perfectly,” writes Cusanus, “in such a variegated plurality of many small transient worlds that succeed one another.”77 So one way to unlock the structure of the dialogue is to view it in light of this Hermetic notion of triplex mundus.78 For we can see that

Table 10.1  Structure of De ludo globi as threefold world-space Book I: The game in motion LG 1–7 LG 8–19 LG 20–43 LG 44–49 LG 50–60

Playing the game: rolling the ball World: the ball’s roundness as the world’s eternity Soul: the ball’s motion compared to the soul’s motion God: the ball’s source in God’s creative motion Meaning of the game: Christ as the soul’s rest

Book II: The game at rest LG 61–67 LG 68–72 LG 73–89 LG 90–103 LG 104–121

Playing the game: the circles of the gameboard Meaning of the game: Christ as the cosmic center World: degrees of life ordered by number Soul: the soul’s powers of enumeration God: divine unity exemplified in numbers

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each of three worlds—God, cosmos, and the soul—is repeated in the two books of De ludo globi. In Book I the worlds are in motion, in Book II at rest, so that between them the sphere expresses every dimension of its meaning (see Table 10.1). In the first book Cusanus compares the ball’s roundness (rotunditas) to the infinite, invisible roundness of the world that is a trace of its eternity. As it rolls the ball’s roundness signifies the perpetual self-motion of the soul, and as a sphere crafted with a rotary lathe the ball exemplifies the Creator’s form-giving activity. At the end of Book I, Nicholas explains that Christ himself is a ball, the globus Christi. In Book II, as we have seen, the rings of the gameboard represent the cosmos as a graduated continuum with Christ as center point. Through the exercise of the quadrivium the soul can integrate the cosmic manifold, which leads it toward the origin of number in God. The image of the ball on the gameboard thus generates two iterations of the triplex mundus—one spherical and mobile, one static and circular. When “played” by the reader, the text dynamizes the image, setting it in motion and casting light on the interior features of its triplex space.79 Yet this geometrical space differs qualitatively from the geometrical images that pepper the other Cusan writings examined above. Those are discrete images in the text that can be, as it were, lifted out and separately contemplated by the reader. What is original in De ludo globi is that the geometrical image has been extended to envelop the whole work, inhering within the very structure of the text, such that the fine details of the organizing figure and their mystical depth have become coextensive.80 The space of the dialogue is structured by three superimposed spheres, simultaneously emanating from their shared Christic center—an intersection that enables the cardinal to treat God, cosmos, and soul univocally.81 By turns, for example, Cusanus measures the degrees of the sphere’s extension from its center, defines the sphere’s rotunditas absoluta as an invisible tangent point, and argues for the perpetual motion of a perfect circle.82 These spatial operations are at once theological investigations, cosmological speculations, and illustrations of the soul’s mathematizing essence.83 As early as De docta ignorantia Cusanus had already discussed the concept of the infinite sphere, whose center is everywhere and whose circumference nowhere.84 But the subtle differences between 1440 and 1463 show how far the cardinal had traveled. In De docta ignorantia Nicholas had compared the infinite sphere to God in Book I and to the cosmos in Book II.85 Since the Incarnation unites the absolute God with the contracted cosmos, Christ too resembles an infinite sphere, as Cusanus states in Book III.86 Thus in De docta ignorantia the three applications of the infinite sphere metaphor are distinct and sequential. By contrast, in De ludo globi the human intellect has become one of the three inextricable spheres of the triplex mundus, following De mente. Christ is now the geometrical center that grounds and integrates the cosphericity of God, cosmos, and soul. Moreover, the geometry of the infinite sphere now comprises the space of the entire dialogue.

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This example illustrates the nature of De ludo globi’s novelty: even in a moment of obvious repetition, the dialogue does not merely repristinate the old ideas of De docta ignorantia and De mente, but represents an even more Christocentric, geometrical synthesis than was conceivable in the 1440s and 1450s. Uncovering the Hermetic basis of De ludo globi restores order to the work while respecting the cardinal’s own sense of its structure. Perhaps Nicholas intended it for public consumption. Yet as an interpretation it remains ultimately disappointing, because it fails to illuminate the two most important features of the dialogue in light of Cusan development. It does nothing to explain why the extended Christological passages appear where they do, and why three times, and it has no use for the Chartrian theology collected throughout the text.

A Memoir of Chartrian Theology There is evidence of another pattern taking shape within De ludo globi, not on the well-ordered surface of the text, but symptomatic of deeper and more tumultuous currents beneath. For it seems as though Cusanus used De ludo globi to ponder his long engagement with Chartrian texts, especially his deliberations over Fundamentum’s attack on the four modes of being. Nicholas had discussed Thierry’s modal theory only three times in the two decades before 1463:  first abiding by Fundamentum’s proscription in De docta ignorantia and De coniecturis, then reversing himself in De mente. It cannot be a coincidence that now, for the last time, he returns to Thierry’s four modes on two separate occasions in De ludo globi. This is the first clue that something is afoot in the dialogue as the cardinal reflects on his intellectual journey, in what he knew were the last years of his life. In short, there are signs that Cusanus used the dialogue’s two books to revisit ideas that he had gleaned from Fundamentum, both in De docta ignorantia and in De mente. As if strolling through memories of composing his magnum opus two decades prior, Cusanus now echoes in Book I, in exact sequence, themes from the three chapters of De docta ignorantia in which he had cited Fundamentum. For Fundamentum, the possibilitas of the world “is not coeternal with God” since only God is absolute (II.8). Now in De ludo globi Cusanus remarks that the essence of the world-sphere’s roundness is its possibilitas essendi, a maximal but not absolute roundness, since it is an image of God’s eternity and not eternity itself.87 Again, Fundamentum had listed a series of cognate names for the second mode (II.9). Cusanus now remarks that anima mundi is also called natura, spiritus universorum, necessitas complexionis, or fatum in substantia.88 Finally, just as in De docta ignorantia (II.10), Cusanus connects the modes of being with fortune and fate.89 Then in Book II of De ludo globi, as Cusanus examines the gameboard, he apparently returns to Fundamentum again. First he reopens the question of plural

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exemplars that he had addressed in De sapientia and De mente. Nicholas now writes that despite the “plurality of exemplifications” (i.e., the mystical circles of the gameboard), there is a single divine Exemplar that unifies them.90 The cardinal calls this principle the central lesson of De ludo globi, which he carefully restates in the dialogue’s final sentence: “there is only one true, precise and most sufficient Form.” Yet this statement simply repeats Fundamentum’s own words against the anima mundi and second mode.91 Similarly Cusanus recollects De mente by name after recounting numerous themes from that work: the arithmetical Trinity, the soul’s projection of the quadrivium, and the theological functions of number.92 When we view these scattered repetitions of past works in concert (see Table 10.2), we catch another glimpse of what Cusanus is up to in De ludo globi. For if the dialogue were indeed a final reflection on his factious collection of Chartrian voices, then it would not be surprising to find him revisiting a select group of past writings as he wrote. Nor would one be surprised to find the text rambling at times, like a ball wobbling uneasily toward its center. The dialogue functions then as a retrieval operation in which the cardinal remembers the brightest flashpoints over the years from the controversy between Thierry and Fundamentum.93 This synopsis suggests a reason why Cusanus would return to the four modes of being on two separate occasions in the dialogue. In fact these correspond to his two major engagements with Thierry’s modal theory in De docta ignorantia and De mente. In Book I Cusanus alludes to the four modes when he describes how God creates the sphere of the world like a gameball.94 John, the cardinal’s interlocutor,

Table 10.2  Structure of De ludo globi as memoir of Chartrian controversies Book I: Themes from Chartrian texts in De docta ignorantia LG 14–18 LG 36–41 LG 49–50 LG 55–59

On the eternity of the world: DI II.8 (140) On anima mundi and mens divina: DI II.9 (142, 148, 150) Digression: On the four modes of being: DI II.7 (130–131) On divine motion and fate: DI II.10 (151)

Book II: Themes from Chartrian texts in De mente LG 62–66 LG 82 LG 90–103 LG 104–109 LG 116–119

On plural exemplars: DM II (67–68) On the arithmetical Trinity: DM XI (131–133) On soul and quadrivium: DM X (126–128) On number: DM VI (89–90) Coda: On the four modes of being: DM VII (97), DM IX (122–125)

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states that there are four different ways in which things exist. All things exist in God as enfolded and in motion as unfolded, in the way that a point unfolds into a circle. But all things also exist as possibility and as determinate possibility.95 This sounds more than a little like the first, second, fourth, and third modes of being. Immediately following this passage, the character of Nicholas interjects (seemingly to the reader herself) that this topic of the four modes strays from the theme of the dialogue, and yet has nonetheless crept into the conversation in an unexpected way.96 Evidently the problems of Fundamentum have resurfaced in the cardinal’s thinking. Then in Book II, the four modes make a second appearance in the coinmaker conceit, which we can now examine more closely. In this passage Cusanus references the modes in three distinct ways. First he explicitly names the four elements of the metaphor as so many modi essendi: the omnipotent Art of the coinmaker (first mode), the instruments used to mint coins (second), the minted coins (third), and the mintable material (fourth).97 But then Nicholas invents a parallel “rational” fourfold by which God’s Face is impressed on the coins.98 For reason discerns the value of the coin through the Christological signum, and this sign appears in the coin in four ways that correspond to Thierry’s four modes.99 Finally, Cusanus connects the four modes with the dialectic of visibility that we have seen in De docta ignorantia and indeed in De theologicis complementis. Just as the sun and the point cannot be seen directly, so too God and matter, the first and fourth modes, are invisible to the intellect.100 Putting these all together we can tabulate the cardinal’s new account of the four modes in De ludo globi (Table 10.3). Cusanus likens the first mode of being to the Son’s true face, considered in itself before its impression onto the coin. This is the invisibility of God known only through negation. The second mode is an image of that invisible Face that

Table 10.3  The four modes of being in De ludo globi Mode of being

Mode of sign-image

Mode of visibility

Necessitas absoluta   (= Infinita actualitas) Necessitas complexionis  (= Contrahens necessitatem in complexum) Possibilitas determinata  (= Elevans possibilitatem in actum) Absoluta possibilitas   (= Infinita possibilitas)

Ante signum   (= Veritas ante figuram) In signo  (= Veritas in imagine: primum in signo) In signo  (= Veritas in imagine: signum in ultimo) Post signum   (= Signatum a signo)

Videre negative Videre affirmative

Videre affirmative

Videre negative

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perfectly reflects its truth, but now in the first degree of visibility: the incarnation of the Son’s image into the metal of the coin. The third mode considers the stamped coin qua coin (rather than qua sign) as an image of the Face at one further remove. In the fourth mode, the theophanic dimension of the coin’s raw matter is practically imperceptible when considered apart from its divine sign. Note what this Christological modulation of Thierry’s theory entails: the second mode of necessitas complexionis corresponds to the conceptual moment of the Son’s Incarnation. The mediations of Arithmos and Logos are not in tension; they are coeval. At first Cusanus’s meditation on coins can seem an odd if not arbitrary way to end an important work. But upon closer scrutiny it turns out to have been a brilliant vehicle for rethinking Thierry’s four modes of being one last time. Nothing else would have tied together Nicholas’s struggles over Fundamentum from De docta ignorantia to De mente and encompassed his twenty years of meditation on the theology of the Breton master. Tellingly his final formulation of the four modes in De ludo globi draws heavily from Thierry’s original model in Commentum and Glosa of two extremes connected by two means.101 In lieu of reciprocal folding (as in Lectiones), Cusanus now orders the modes by the aesthetics of image and exemplar, visibility and invisibility—precisely the inherent visuality of geometrical theology that the cardinal had foreshadowed at the close of De theologicis complementis. All told, this second account of the structure of De ludo globi clarifies several things. It explains why the dialogue appears to wander, since Nicholas’s original Hermetic plan is frequently interrupted by his recollection of topics from Fundamentum. It explains why the four modes erupt unaccountably, as Cusanus himself admits, toward the end of each book and why the cardinal is preoccupied with the divine “exemplar” at the beginning and end of the second book:  both the four modes and exemplarity play a prominent role in the argument of Fundamentum. But this account has its shortcomings as well. It disregards the obvious themes in the text and does not explain what Fundamentum, De docta ignorantia, and De mente have to do with the roundness of the ball or the circle of the game board. What we require is a means of bridging the Hermetic, geometrical patterns on the surface of De ludo globi with the troublesome sources roiling beneath.

A Christology of the Quadrivium We can now approach a third possible interpretation of the dialogue’s structure. Unlike any text since De mente, Chartrian and Pythagorean themes are foregrounded in De ludo globi, from the quadrivium to the arithmetical Trinity, and from the praise of Pythagoras to the game of rithmomachia. A strong interpretation of the dialogue should be able to explain this recurrence. But it should also find a home for the three dense Christological passages, accounting for their

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respective timing and their distinct arguments. From scrutinizing past works we know that the cardinal is capable of constructing intricate textual architectures, particularly when he is struggling to moderate between his Chartrian sources. A superior reading of De ludo globi would do justice to these criteria. Cusanus makes it clear that the reader’s progress from Book I to Book II is a transition from one kind of contemplation to another, let us say from physics to geometry. In Book I, Nicholas discovered that the spherical figure suggested by the Hermetic triplex mundus could accommodate the general themes of his commentary on Fundamentum in De docta ignorantia: the world’s eternity vis-à-vis God, the status of the anima mundi, and the nature of fortune. By the end of the book, the moving sphere has come to rest. In Book II, as he reworked the lessons of De sapientia and De mente, the intellect ascends from plural images to the singular divine exemplar. Consequently this book travels from the outer rings of the decadic circles to the henological point that enfolds their continuum. Notice the sequence: from moving sphere to resting sphere, and from the planar figures to the center point. These transitions both within and between the books of De ludo globi unfold in a sequence of decreasing dimensionality, from rolling spheres to rotating circles to fixed centers to discrete units. This progression should immediately bring to mind the structure of the Nicomachean and Boethian quadrivium. Arithmetic and music treat number (discontinuous multitude), while geometry and astronomy study quantity (continuous magnitude) (see Table 2.1). If one begins with astronomy (or “spherics”) and progresses toward arithmetic, a series of spatial dimensions are successively collapsed as one progresses toward pure unity. Astronomy is quantity in motion, but when motion is removed geometry remains. Geometry organizes continuous magnitude, but when converted to multitude its arcs and lines in space become harmonic proportions denoted numerically. Music is the study of such relative numbers, but number viewed in itself is arithmetic, which leads the mind toward the simple One. The fourfold quadrivium is thus organized around three transitions:  from motion to rest, from magnitude to multitude, and from relative to absolute unities. The text of De ludo globi manifests a very similar structure. It is organized as an ascent through the four arts of the quadrivium, from spherics all the way to arithmetic, which accordingly passes through three transitions—a reduction to a useful order, as the cardinal described his ultimate aim in the dialogue. However, in Nicholas’s version of the quadrivial progression, each transition is warranted by a specific Christological principle. In each case, the hypostatic union of the Incarnation bridges the opposition that each reduction traverses (see Table 10.4). In Book I, the ball of Christ makes the winning roll, a perfect motion that provides the resting center point of the gameboard, and hence is also the terminus of all motion. Since the two natures of Christ have thus reconciled motion

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Table 10.4  Structure of De ludo globi as reduction of quadrivial antinomies Book I (1)  Reduction from spherics to geometry: from motion to rest (a)  Spherical gameballs as astronomical bodies (LG 1–49) (b)  Christ as center brings motion to rest (LG 50–60) Book II (2) Reduction from geometry to harmonics: from figure (magnitude) to number (multitude) (a)  Degrees of circumference around center (LG 61–72) (b)  Christ as harmony of center and circumference (LG 73–75) (3) Reduction from harmonics to arithmetic: from relative number to absolute unity (a)  Patterns of numeric harmony ordering cosmos and intellect (LG 76–109) (b)  Christ as principle of divine unity beyond degree (LG 110–121)

and rest, the second book can proceed to the static geometry of the gameboard. Early in Book II the hypostatic union overcomes another opposition. As Nicholas notes, when the Christ-circle is also the center, the center and circumference are the same. Without such quantitative magnitudes, geometrical space is collapsed and converted into pure numerical order. This reduction to arithmetic accounts for the sudden shift to Pythagoreanism and Thierry’s theology midway through Book II. The final transition is from the harmonies of number to absolute unity. In the quadrivium such unity denotes arithmetic, but in the context of the dialogue it also signifies the divine One. This brings us to Nicholas’s Christological coda on the “sign” of the Son as refracted through the four modes of being. Through the Incarnation, which so to speak impresses the sign of the Father, the Son reconciles the concordant multitude of creaturely images to the singularity of the divine Exemplar. Was this stunning order deliberately engineered by our cardinal? Or is it the accidentally beautiful side-effect of his decision to fuse the physics of motion (1460s), the space of geometry (1450s), and the anagogy of number (1440s) with his Neo-Chalcedonian Christology from 1440? Either way, the literary form of De ludo globi is a remarkable testament to the promise of his Neopythagoreanism. Not simply a reworking of older themes, the dialogue achieves a new constellation of past components that Cusanus had never quite harmonized. Since De mente, mathematics had no longer played a simple mediating role in Cusan theology but instead an iconological one, with the result that his theology was essentially

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geometrized in De theologicis complementis. But Nicholas remained uncertain about the relevance of his Christological convictions to his new mathematical outlook. In De mente, the ascent to divine Exemplar was not Christomorphic at all but rather passed stepwise through the four modes of being. Similarly, in the 1440s Nicholas discovered that the Incarnation could be used to harmonize the world’s descent in emanation and the mind’s ascent in filiation, but these had nothing to do with mathematics or geometrical space. The antinomies inherent in Nicomachus’s fourfold structure—between motion and rest, multitude and magnitude, relation and simplicity—prevented the quadrivium from operating anagogically. The text of De ludo globi finally demonstrates how the Cusan theology of Incarnation could overcome the Nicomachean antinomies and, in the final analysis, fulfill Bernard of Chartres’s vision of undoing the competition between the new mathematizing cosmology and the older Christian one. Once the quadrivial antinomies were dissolved, Thierry’s mature theology of the quadrivium as formulated in the four modes of being was also necessarily transfigured. For if the Incarnation is the hidden basis of every mathematical dimension, then reciprocal folding is not the sole possible foundation of the Breton master’s modal theory. In its place Cusanus installs an eminently appropriate alternative in the dialogue’s coda: an aesthetics of the manifold divine Face, the figure of the Invisible. Cusanus had first reacted to Thierry’s mathematical theology with his own Christological response. The components of both—quadrivium and Incarnation— were then disassembled and reassembled for the next twenty years. Only in De ludo globi does Cusanus enunciate what he seems to have intuited from the start, namely that the overcoming of every opposition in the Incarnation grounds the mathematical order of the cosmos as it is rendered visible in the quadrivium. If the Creator spontaneously and primally mathematizes, then of course the arithmetical Trinity is a superlative name of God. But this finally cannot be separated from the Christian doctrine of the divine Word made flesh. The Incarnation must somehow or another matter to the theological function of the quadrivium explored in pre-Christian and in Christian Neopythagoreanism. The Cusan model in De ludo globi explained how, for the first time. Far from being opposed, nothing prevents Logos and Arithmos from being complementary within Christian theologies.

Epilogue

At the beginning of the book I suggested that writing the history of Christian Neopythagoreanism would be an important step toward critiquing the mathesis narrative and supplementing what it had to say about the origins of modernity and Christianity’s status within it. We can now ask what remains valid in the narrative and what does not. From our vantage point in late modernity the notion of mathesis seems in many ways more germane today than ever. Contemporary American (and to some extent global) culture is driven by forces of quantification that are accelerating at an unprecedented pace. However imprecise their historical claims proved to be, Husserl and Heidegger astutely perceived in the 1930s that we are beginning to live and think within a homogeneous, numerical grid stretching out into infinite space, whose unreal dimensions threaten to overshadow or impoverish the sensible plenum of the Lebenswelt. But if their generation worried about mass production, commodification, and the reproduction of images, in our century the forces of mathematization are colonizing further regions of life: the calculation of the human genome; the automated algorithms driving financial markets; the invisibility of social goods not monetarily quantifiable; the state’s pursuit of total informational awareness; and our mania for transforming every sensual experience into a simulacrum of ones and zeroes. All of its own momentum, mathesis is becoming ever more universal. This situation separates us as much from Greek Pythagoreans as from Latin Christians. Whatever they believed about universal mathematical structures invisibly ordering their lives, they were nonetheless dealing in theories of number. We have less need for such theories because mathematization is increasingly a compulsory mode of practice, a matrix in which human lives are contained. The wealthiest among us are busy converting our identities, creations, and relations into a single, timeless Cloud, a concrete mathesis universalis if ever there were one, whose digital fecundity seems more real and desirable than the material order. It does not

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take a theological mind to wonder what political, ethical, and indeed religious commitments are concealed in such a total dominion of number. I would submit that the mathesis narrative remains valuable today less for indexing modernity’s origins, as in the past, than for opening up theoretical horizons to understand the intersection of number and transcendence in late modernity. That is, a genealogy of Christian Neopythagoreanism is an indispensable tool for thinking under the new conditions of universal mathematization. No one doubts that something new happened in the seventeenth century. Galileo, Descartes, and others began to describe the universe more consistently and uniformly in terms of mathematical quantity. Like all centuries, this represented a new development, and one whose scientific and technological effects continue to shape our epoch decisively. It is less prudent, however, to assert that this was a sudden, singular event, or a historical watershed that establishes self-evident conceptual divisions—between religious authority and scientific autonomy, between tradition and innovation, between a poetic cosmology and a mechanistic one, or between supernatural faith and secular reason. As I  have shown, there were significant moments in the twelfth and fifteenth centuries in which indubitably medieval Christian theologians embraced just such a mathematized view of the cosmos, but did so precisely in the name and in the ancient categories of their own religion’s sacred teachings. A genealogy of ancient and medieval Christian mathesis thus argues for greater continuity between late medieval and early modern epistemes and between religion and science. Mathematization, in short, is not ipso facto secularization. The history of medieval Christian Neopythagoreanism can also add a footnote to accounts of scientific and religious change in the seventeenth century by helping to explain the failure of European Christianity to adjust to the new world picture coming from advances in the natural sciences. In light of the fate of Arithmos after late antiquity, we can now see that the particular aesthetic of number and geometry promoted by seventeenth-century science touched western Christianity right at its weakest point, addressing the least developed sector of the entire Christian engagement with late antiquity, namely Pythagorean ideas. The mathematization of nature invited a response not just from any ideal Christianity, but from the particular western Christianity produced by the discursive history that I have traced. If Galileo saw the universe written in the language of number, what could this possibly have to do with the language of the Word? But now we can see that this mutual incomprehension was tautological:  they had nothing to say to each other because they had divided what could be said. Because Christianity ended up adopting Logos as the singular valid mediation and excising Arithmos out of any constitutive role in theology, the religion disenfranchised itself from meaningful dialogue with those ideas when they took on new life in early modernity. To the extent

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that Neopythagorean-type convictions informed the early modern program of mathesis universalis, that cosmological and epistemological vision was inherently incompatible with western, Augustinian theology. By the turn of the twentieth century, one could be forgiven the impression that the Christian religion had nothing to do with the exact sciences, or that modern science had in the seventeenth century seized the liberty that was legitimately its own all along, an apparent rupture that cleared the way for secular politics. Of course, seventeenth-century mathematization was in fact a secularizing force, and it is incontrovertible that historically speaking the autonomy of number can prove corrosive to religious cosmologies. But in the case of early modern Christianity, we must remember that this was an accident of discourse, not a logical necessity. The counterexamples of Thierry of Chartres and Nicholas of Cusa demonstrate that there is nothing inherent in Christian beliefs which makes them definitively incompatible with mathesis universalis. The very contingency of the saga of Arithmos in Christianity, its many twists and turns, suggests the possibility of radical alternatives. If today Christian theology and the natural sciences have anything to say to each other, surely it would be concealed within the fruits of Cusanus’s labors to reconcile Arithmos and Logos. For Cusanus in his day performed what Christian theology since the seventeenth century has always had to do: to think Incarnation and Trinity within a mathematized cosmos that seems to operate autonomously and hence in which they seem unnecessary. We live in an age of unprecedented mathematization of a quality that Nicomachus and Iamblichus saw only in dreams. But unlike the thousands of pages spent reconciling Aristotelianism, Neoplatonism, nominalism, Kantian liberalism, Darwinism—even Marxism and existentialism—with its beliefs, western Christianity has really only on two brief occasions engaged Pythagorean ideas on a deep level. This is not to suggest that the endeavor is ultimately advisable, but only to state that it has not been attempted often or understood well, and to observe that such a discursive dislocation is not, in principle, impossible to remedy. The Cusan example of a robust mathematical theology hints at a way beyond the limitations shared by liberal (modern) and postliberal (antimodern) theologies, confined as they are by the boundaries that the mathesis narrative has imposed. Cusanus, as if curiously a-modern, skirted those restrictions five centuries ago. For a theology like his, already thoroughly mathematized, the onset of more intense mathematization would have required no fundamental changes. Indeed such an event might have conveyed even greater opportunities for complexity and relevance than it had previously enjoyed.

Notes

In t roduct ion 1.

See Meier-Oeser, Präsenz des Vergessenen. I follow the convention of referring to Nicholas as “Cusanus,” his preferred humanist moniker, or simply as “the cardinal,” even in years before his ascension to that post. 2. “Mathesis igitur magna est, sed tum maxime cum modus ad divina surgendi non abest.” Lefèvre d’Étaples, Opera, 1; cited in Meier-Oeser, Präsenz des Vergessenen, 48. 3. “Deus bone, ubi illi Cusano adsimilandus, qui quanto maior est, tanto paucioribus est accessibilis? Huius ingenium si presbyteralis amictus non interturbasset, non Pythagorico par, sed Pythagorico longe superius agnoscerem, profiterer.” Bruno, Oratio valedictoria, 17; cited in Catana, Concept of Contraction, 140. 4. “videtur Deus, cum has duas scientias generi humano largitus est, admonere nos voluisse, latere in nostro intellectu arcanum longe majus, cujus hae tantum umbrae essent.” Leibniz, Philosophische Schriften VII, 184; cited in Breidert, “Mathematische und symbolische Erkenntnis,” 118. 5. “Mathematicis superstitionibus putas verae religionis sacra demonstrare.” Cited in Flasch, “Nikolaus von Kues: Idee der Koinzidenz,” 249. On Heimburg, see Watanabe, Nicholas of Cusa: A Companion, 142–148. 6. “Mathematica est inimicissima omnino theologiae, quia nulla est pars philosophiae, quae tam pugnat contra theologiam.” Luther, Disputatio De sententia:  Verbum caro factum est §16 (WA 39); cited in Velthoven, Gottesschau und menschliche Kreativität, 131. 7. See, e.g., Badiou, “Philosophy and Mathematics.” 8. See Dupré, Passage to Modernity, 186–189; cf. Blumenberg, Legitimacy of the Modern Age, 483–484. 9. See Hirschberger, Stellung des Nikolaus von Kues, 123–124; cf. Stallmach, “Ansätze neuzeitlichen Philosophierens.” More recent evaluations include Hopkins, “Nicholas of Cusa”; Cubillos, “Nicholas of Cusa”; and especially Senger, Wie modern ist Cusanus? 10. Bambach, Heidegger, Dilthey, 30, 37.

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11. See Watanabe, “Origins of Modern Cusanus Research.” 12. See Fisher, Revelatory Positivism, 7–71; and Dreyer, Idee Gottes. 13. See Bruckstein’s comments on Logik der reinen Erkenntnis in Cohen, Ethics of Maimonides, 85. 14. Löb, Bedeutung der Mathematik, 49; cf. Nagel, Nicolaus Cusanus und die Entstehung der exakten Wissenschaften, 173–183. 15. See Watanabe, “Origins of Modern Cusanus Research,” 28–32; cf. Cassirer, Erkenntnisproblem, 52–77. 16. See Duhem, Ètudes sur Léonard de Vinci, 2:  97–279; cf. Duhem, Système du Monde, 10:  247–347 (on Thierry and Nicholas, 269–275). Duhem discusses Thierry’s Tractatus in Système du Monde, 3: 184–193. 17. See Duhem, “Thierry de Chartres et Nicolas de Cues.” 18. Cassirer, Individuum und Kosmos, 10 (my translation). In this passage, Cassirer is speaking literally of De docta ignorantia, but clearly from the point of view of De mente. Hankins suggests that the Neo-Kantianism of Cassirer and Hoffmann also influenced the work of two other leading historians of Renaissance philosophy, Eugenio Garin and Paul Oskar Kristeller (see “Garin and Paul Oskar Kristeller,” 489–492). 19. See the fine analysis by Benz, “Nikolaus von Kues: Wegbereiter neuzeitlicher Denkweise”; cf. Benz, Individualität und Subjektivität, 37–45. 20. See Cubillos, “Nicholas of Cusa,” 239. 21. See Watanabe, “Origins of Modern Cusanus Research,” 26–35; cf. Hans Gerhard Senger, “Raymond Klibansky, 1905–2005,” xi–xxviii. This set the agenda for Klibansky’s important work The Continuity of the Platonic Tradition. 22. For a summary of the problems posed by this phrase, see Roux, “Forms of Mathematization,” 324–329. 23. For an overview of Husserl’s early work and his relationship with Georg Cantor, see Gray, Plato’s Ghost, 204–209; cf. Sfez, “L’hypothétique influence.” 24. See Friedman, Parting of the Ways; and Gordon, Continental Divide. The last chapter in this story that remains to be written is the role played by Maurice de Gandillac (d. 2005), who mediated Cusanus to French philosophy in his 1941 thesis written under Étienne Gilson. Gandillac, who attended the Davos debate along with Emmanuel Levinas, went on to advise the doctoral work of Louis Althusser, Michel Foucault, Gilles Deleuze, Jacques Derrida, and Jean-François Lyotard. 25. See “Realitätswissenschaft und Idealisierung. Die Mathematisierung der Natur,” in Krisis, ed. Biemel, 279–293 (Abhandlung I); trans. Carr, 301–314. On this period see Mohanty, Edmund Husserl’s Freiburg Years, 387–419; and Carr in Husserl, Crisis, xvi–xix. 26. Husserl further examined the intrasubjective linguistic bases of mathematical objects in an important fragment left out of the Krisis, “Der Ursprung der Geometrie,” ed. Biemel, 365–386 (Beilage III); trans. Carr, 353–378.

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Derrida’s first published work was a translation and commentary on this text (see Edmund Husserl’s Origin of Geometry). Baring has shown that in doing so Derrida quickly ran up against theological questions surrounding the Infinite (in Baring’s words, the “God of mathematics”) that date back to Kant (see Young Derrida, 170–180). 27. See Husserl, Krisis §9 (“Galileis Mathematisierung der Natur”), ed. Biemel, 20–60; trans. Carr, 23–59. On this section see Gurwitsch, “Husserlian Perspectives”; Gasché, “Universality and Space”; and Buckley, Husserl, Heidegger, 37–54. De Gandt contests Husserl’s interpretation of Galileo (see Husserl et Galilée). On Husserl’s problematic interpretations of mathesis universalis, see Olivo, “L’Évidence en règle”; and Rabouin, “Husserl et la projet Leibnizien.” 28. See De Gandt, Husserl et Galilée, 97–98. Cf. Natorp, “Galilei als Philosoph” (from 1882); and Cassirer, Erkenntnisproblem, 289–339 (from 1906). Cassirer later responded to Koyré’s interpretation in “Galileo’s Platonism.” 29. Carr raises this possibility (see Husserl, Crisis, xix), but it is contested by De Gandt, Husserl et Galilée, 99–104; and Buckley, Husserl, Heidegger, 42–43. Koyré became known in Anglophone circles for his lectures on infinity in early modern cosmology in Cusanus, Galileo, and Descartes (see From the Closed World, 5–27, 88–109). On Koyré, see further Geroulanos, Atheism that is not Humanist, 80–99; Harries, Infinity and Perspective, 1–40; and especially Jorland, La science dans la philosophie, 248–310. 30. See Koyré, Galileo Studies. In 1943 Koyré published his findings in two programmatic English essays, later gathered in Metaphysics and Measurement (see “Galileo and the Scientific Revolution”; and “Galileo and Plato”). 31. Koyré, “Galileo and Plato,” in Metaphysics and Measurement, 19–20. 32. See Koyré, “Galileo and the Scientific Revolution,” in Metaphysics and Measurement, 14. 33. Galilei, Assayer, in Discoveries and Opinions, trans. Drake, 237–238. 34. See Natorp, “Galilei als Philosoph,” 204–205; Cassirer, Individuum und Kosmos, 171–172; and Koyré, “Galileo and Plato,” in Metaphysics and Measurement, 40–41. 35. See Galilei, Dialogue, trans. Drake, 117–121. On similar trope in Descartes, see Marion, Sur la théologie blanche, 161–203. 36. See Marion, Sur l’ontologie grise; and Marion, Sur la théologie blanche, 203–227. Cf. Olivo, Descartes et l’essence de la vérité, 72–140. 37. See Descartes, Regulae IV, ed. Adam and Tannery, 377; trans. Cottingham, 1: 19. Cf. Regulae VI, ed. Adam and Tannery, 385; trans. Cottingham, 1: 23. 38. Descartes, Regulae IV, ed. Adam and Tannery, 377; trans. Cottingham, 1: 19. On Descartes’s mathesis, see Marion, Sur l’ontologie grise, 55–69; as well as Marion’s notes in his edition, Règles utiles et claire, 144–164, 302–309. 39. See Descartes, Meditationes V, ed. Adam and Tannery, 65–71; trans. Cottingham, 2: 45–49. Descartes repeats this formula (subjectum purae matheseos) several

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times in the sixth meditation as well. See further Marion, On Descartes’ Metaphysical Prism, 14–20, 44–63. 40. Roux, “Forms of Mathematization,” 319. 41. On the contexts of Heidegger’s appropriation of seventeenth-century science, see Bambach, Heidegger, Dilthey, 30–55; and Gordon, Continental Divide, 43–86. On mathematics in Heidegger, see Glazebrook, Heidegger’s Philosophy of Science, 14–64; and especially Roubach, Being and Number. Heidegger and Koyré were not the only students of Husserl to continue in this direction. Oskar Becker celebrated the “contemporaneity of Pythagorean thinking” (see “Aktualität”) but later took up a Nietzschean defense of Nazi race ideology (see De Gandt, Husserl et Galilée, 27–30). Jan Patočka wrote several excellent studies of the Krisis; see, e.g., “Cartesianism and Phenomenology,” in Kohák, Jan Patočka, 285–326. On Dietrich Mahnke, another Husserl student, see below. 42. Heidegger, Frage nach dem Ding §18(b), ed. Jaeger, 76; trans. Krell, 277–278. 43. On Galileo, see ibid. §18(e), ed. Jaeger, 89–95. On Descartes, see ibid. §18(f), ed. Jaeger, 98–108. 44. See ibid. §18(f), ed. Jaeger, 96–98. Heidegger elaborates further in his 1938 lecture on technology, “Die Zeit des Weltbildes.” 45. See, for instance, Badiou’s assertion of the “Galilean rupture” of mathematicized nature as a self-evident overcoming of medieval cosmology—even to the point of citing Koyré on Galileo’s Platonism (see Being and Event, 3, 123–125, 142–145; cf. Badiou, Manifesto for Philosophy, 43). Foucault makes extensive use of mathesis universalis in his analysis of the modern “episteme” (see Order of Things, 50–76). To his credit, Foucault broadens the meaning of mathesis beyond mathematics per se and also complicates the association of mathesis and modernity by distinguishing a separate “Classical” epoch. At the same time, his account of mathesis remains essentially tied to the quadrivium, and the decisive moment of mathematization remains the seventeenth century. 46. See Maier, Vorläufer Galileis; and Maier, Zwei Grundprobleme. Maier’s most important studies are translated in Sargent, Threshold of Exact Science. On Maier and Duhem, see further Murdoch, “Pierre Duhem and the History of Late Medieval Science.” A. C. Crombie’s Augustine to Galileo (1952), the first major history to build on Maier’s findings, argued accordingly for deep continuities between medieval and early modern science (see Eastwood, “On the Continuity of Western Science”); cf. Koyré’s critical review in “Origins of Modern Science.” 47. For some general references, see my discussion at the beginning of Chapter 6. 48. Strong’s remarkable but little-known book, his 1936 dissertation, was one of the first to broach the subject (see Procedures and Metaphysics); cf. Cassirer’s critical review, “Mathematische Mystik.” The canonical guide to these works is Crapulli, Mathesis universalis; see also Mittelstrass, “Philosopher’s Conception of Mathesis Universalis”; Helbing, “La fortune des Commentaires”; and Rabouin, Mathesis Universalis (for a tidy Forschungsbericht, see ibid. 13–21).

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49. See Napolitano Valditara, Le idee, i numeri, l’ordine; but cf. the methodological caveats of Bechtle, “How to Apply Modern Concepts.” On Aristotle, see especially Rabouin, Mathesis Universalis, 37–129. 50. Bechtle rightly notes in this regard the centrality of Boethius and his twelfthcentury commentators like Gilbert of Poitiers (see “How to Apply Modern Concepts,” 145–149). 51. Strong, Crapulli, and Rabouin all confine their attention to the mobilization of Proclus in the sixteenth and seventeenth centuries. This problem dates back to the Neo-Kantians. In the same year that Cassirer first wrote on Cusanus’s modern epistemology, Nicolai Hartmann published his study, Des Proklus Diadochus:  Philosophische Anfangsgründe der Mathematik nach den ersten zwei Büchern des Euklidkommentars (1909) in a Neo-Kantian series edited by Cohen and Natorp. Hartmann praised Proclus as the philosophical conscience of Platonism, a modern before his time who displayed a precision and rigor worthy of Kant and indeed foreshadowed Kant’s schematism of the understanding (see Breton, Philosophie et mathématique, 175–179). 52. Funkenstein, Theology and the Scientific Imagination, 14. Funkenstein interprets seventeenth-century mathematization as a suspension of the rule against metabasis, that is, the collapse of the quadrivial distinction between multitude and magnitude (ibid., 299–317). 53. “Pythagoreanism” is not an especially useful term, but one I  find necessary in order to gesture toward the received notion of an undifferentiated ancient school. Like many others I  use “Neopythagoreanism” to name the sophisticated henology on display from Eudorus and Moderatus in the first century bce through Nicomachus in the second century ce. I hope readers hear in these terms rough analogies of “Platonism” and “Neoplatonism,” including the difficulties that sometimes attend their use. Likewise, rather than using the vague and deprecatory “number mysticism,” I designate views concerning the mystical powers of individual natural numbers as “arithmology,” following Délatte, Études sur la littérature pythagoricienne. 54. The origins of the term “Neopythagorean” are revealing. It was first formulated as a pejorative by the Jesuit theologian Melchior Inchofer, during his examination of Galileo’s philosophical writings in 1633. Inchofer warned that whoever followed Galileo’s teachings was no longer Christian but “Neopythagorean.” The modern usage begins around 1815 (see Dörrie, “Neupythagoreismus,” 756). 55. Harries, Infinity and Perspective, xi; cf. Blumenberg, Legitimacy of the Modern Age, 476–77. For this reason some have detected similarities between Cusan themes and elements of French poststructuralism: see, e.g., Hoff, Kontingenz, Berührung, Überschreitung. Hoff has recently argued that Cusan thought resists modern “nihilism” and “narcissism” by overcoming the separation of religious and scientific language, a theme quite consonant with my own (see Analogical

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Turn). Where I  have focused on Chartrian traditions and mathematics, however, Hoff focuses on Ps.-Dionysius and perspective, with special attention to De visione dei. 56. See Rombach, Substanz, System, Struktur, 135–150. 57. See ibid., 150–178. Rombach’s interpretation has strongly influenced Jacobi, Methode der cusanischen Philosophie, 240–308; and Nagel, Nicolaus Cusanus und die Entstehung der exakten Wissenschaften, 6–25. Well before Rombach, Volkmann-Schluck had already drawn a comparison between Cusanus’s use of mathematics and Descartes’s mathesis universalis (see Nicolaus Cusanus, 159–190). 58. See Rombach, Substanz, System, Struktur, 179–212. 59. On nominalism, see Blumenberg, Legitimacy of the Modern Age, 152–177. Blumenberg esteems Descartes for the self-assertion of reason (ibid., 183– 198), but criticizes his cosmogony for remaining thoroughly medieval (ibid., 206–214). 60. See ibid., 457–480; cf. Brient’s astute critique in Immanence of the Infinite, 139– 143. Like Rombach, Brient appeals to Eckhart as the most relevant source for determining Cusan epochality. 61. See Blumenberg, Legitimacy of the Modern Age, 469, 516–517, 542–557. 62. See ibid., 560–563. 63. See ibid., 526–538. 64. What I am attempting, in short, is the inverse of Hollywood, Sensible Ecstasy; and Holsinger, Premodern Condition. Where they document the influence of medieval studies on the genesis of contemporary theorists like Bataille and Lacan, I try to complete the historical research implied yet fatally missing in Husserl and Heidegger. 65. Baeumker, “Pseudo-hermetische ‘Buch der vierundzwanzig Meister’,” 196. 66. See, e.g., Joost-Gaugier, Pythagoras and Renaissance Europe; and Aragón, “Influencia pitagórica.” Cf. Blum, “Nicholas of Cusa and Pythagorean Theology” (a study of Petrus Bungus, d. 1601). Already in 1897 Willmann discussed Cusanus’s “Pythagorean speculations” at some length: see Geschichte des Idealismus, vol. 3, §87 (“Der Pythagoreismus der Renaissance”), 24–34; cited in Schwaetzer, “Intellektuelle Anschauung,” 247. 67. See Cassirer, Individuum und Kosmos, 36. 68. For a survey of this scholarship, see Senger in Nicholas of Cusa, Cusanus-Texte III, 11–16. See further Klibansky, Proklos-Fund; and Beierwaltes, “ ‘Centrum totius vite’.” I discuss the matter in detail in Chapters 8 and 9. 69. This can sometimes be done very subtly. See, e.g., the easy passage from Plato to Proclus to Descartes in the otherwise fine introduction by Moran, “Cusanus and Modern Philosophy,” 174–175. 70. On Proclus, see my discussion of Velthoven and Flasch in Chapter 9. On Eckhart I  have in mind Counet’s otherwise insightful Mathématiques et dialectique.

Notes

71.

72.

73. 74.

75.

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Although he readily acknowledges Thierry’s influence, Counet’s study of mathematics in Cusan theology focuses instead on Anselm of Canterbury (ibid., 19–40), Eckhart (ibid., 44–49), Eriugena (ibid., 140–153), and John Duns Scotus (ibid., 167–169). The book is organized around a passage in Eckhart that Cusanus may have never read, rather than those in Thierry that we know he studied closely. See Riccati, “Processio” et “explication”, 115; and McGinn, “Unitrinum Seu Triunum,” 96. Hopkins illustrates McGinn’s point in “Verständnis und Bedeutung.” Beierwaltes, “Einheit und Gleichheit,” 372. Schulze likewise regrets Cusanus’s “noch immer nicht genügend gewürdigten Beitrag zur Geschichte des Pythagoreismus” (see Harmonik und Theologie, 5). To his credit, Schulze intuited a connection between Logos and number in his stimulating overview of ἀναλογία in Presocratic Pythagoreans through Boethius and Augustine (see Zahl, Proportion, Analogie, 6–23). His plan to sketch Cusanus’s “harmonic theology,” however, was limited by its inattention to the sequence of the cardinal’s works and their medieval sources. See Schulze, Zahl, Proportion, Analogie, 128– 144; and Schulze, Harmonik und Theologie, 27–32. Schulze’s primary model is Haase, Geschichte des harmonikalen Pythagoreismus. Beierwaltes, “Einheit und Gleichheit,” 380. “In fontium commentario unum genus, quod rei eximium affert pondus, leviter tantum significari potuit: Sunt scripta scholae Carnotensis eiusque asseclarum. Alter liber Theoderici Carnotensis De sex dierum operibus et commentarius quidam anonymus in Boethii De Trinitate libellum, quorum uterque magni discriminis est omnibus Cusani sententias a capite arcessentibus, nondum editi sunt. Haec scripta R. Klibansky, qui ea in cod. camerac. lat. 339, cod. paris. lat. 3584, cod. berol. lat. fol. 817 repperit, una cum ceteris Theoderici Carnotensis operibus aliisque XII saeculi in appendice commentationis de scholae Carnotensis doctrina foras dabit, ubi quae ratio intercedat inter Cusani Doctam ignorantiam et illius aetatis philosophos, uberius explicabitur. (Sunt, qui putent philosophiam Cusanam vilioris venire, si origines eius demonstrantur; huiusmodi errorem extorque nullum operae pretium est.)” De docta ignorantia, ed. Hoffmann and Klibansky, xii. In a letter to the journal Isis in 1936, Klibansky referenced “my edition of the works of both Bernard and Thierry of Chartres, which will appear in the course of this winter” (“Standing on the Shoulders of Giants,” 147). See Watanabe, “Origins of Modern Cusanus Research,” 40. Even the Deutsches Literaturarchiv in Marbach that preserves many of Klibansky’s writings holds no duplicate version. The edition plates included De docta ignorantia, Apologiae doctae ignorantiae, the Idiota trilogy, and De concordantia catholica. I  thank Dr.  Hans Gerhard Senger for an informative conversation on Klibansky and Wilpert (Cologne, November 30, 2006). See further Senger, “Raymond Klibansky, 1905–2005”; and Senger, “In Memoriam Raymond Klibansky.”

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76. See Klibansky, “School of Chartres.” Klibansky’s other great contribution to Quellenforschung, his pioneering essay on Cusanus’s use of Proclus, had far more influence on Cusanus studies (see Proklos-Fund). 77. See Biller, “Mahnke”; and Mancosu, Adventure of Reason, 346–356. 78. See Mahnke’s treatment of Cusanus in Unendliche Sphäre, 140–178. Mahnke distinguishes “exact mathematicians” and “speculative mathematicians” from “mathematical mysticism” (Unendliche Sphäre, 83). Haubst has complained that the latter term cannot apply to Cusanus, since his Ps.-Dionysian mysticism leaves every symbol behind, even purified mathematical concepts, “in order to behold the divine truth itself without mediation” (Bild des Einen, 291–293). 79. See Haubst, Bild des Einen; and Haubst, Christologie. These excellent studies are still worth consulting, as I frequently do below. 80. Haubst separated the arithmetical Trinity (see Bild des Einen, 231–254) and the cosmic trinity (see ibid., 99–144) on opposite ends of his first book and reserved Christology for his second book. 81. See Wilpert, “Philosophiegeschichtliche Stellung.” See further Zimmerman, “Im Memoriam Paul Wilpert.” Stollenwerk remarks in her appended Lebenslauf (see “Genesikommentar”) that upon Wilpert’s death Zimmerman advised her dissertation on Thierry. 82. See Hoenen, “ ‘Ista prius inaudita’.” 83. For an overview and analysis, see Albertson, “Learned Thief.” Cf. recently Mandrella, “Rara et inaudita,” 247–248; Beierwaltes, “Nicolaus Cusanus:  Innovation durch Einsicht,” 351; Moritz, Explizite Komplikationen, 234–235; Rusconi, “Commentator Boethii”; and Rusconi, “El uso simbolíco,” 38–47. I discuss some of these further in Chapter 7. For the record, after Duhem first discovered Thierry’s influence on Cusanus, the initial reaction was the same: wait for a common source! Here is Vansteenberghe’s counsel in 1920, before Klibansky’s research took hold: “Il est possible, cependant, que l’un et l’autre [Thierry and Nicholas] aient utilisé une source commune. En tous cas, Cusa ne présente pas sa théorie comme originale, mais plûtot comme quasi classique, et il y a quelque exagération à parler ici de «plagiat»” (Le Cardinal Nicolas de Cues, 411 n. 7). 84. See Albertson, “Late Medieval Reaction.” 85. See Hoye’s justly critical review (“Review”; cf. Hoye, Mystische Theologie). 86. Flasch, Nikolaus von Kues: Geschichte einer Entwicklung, 12. 87. Ibid., 14. Cf. Flasch’s diatribe against excessive Quellenforschung (ibid., 291–292, 307–308). 88. Ibid., 43. 89. On this methodological problem, see Stadler, Rekonstruktion, 8–20. Stadler counts three major models for sophisticated, nonsystematic Cusanus interpretations:  the genetic (Cusan thought as differential development), the aporetic (Cusan thought as constituted by conflicting sources), and the epochal

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(Cusanus as protomodern) (see Rekonstruktion, 8–12). Against the specter of “einer unauflösbaren immanenten Aporetik,” Stadler urges a “Strukturanalyse” that studies how Cusanus alters his sources and uses them variably across different works (ibid., 18–20). 90. A note on translations: In Part One I have gladly deferred to authoritative translations of Greek and occasionally Latin ancient sources. The translations in Part Two on Thierry of Chartres, his circle, and his readers, are all my own. Regarding Cusanus’s works in Part Three, I have worked from the Latin edition to verify or correct all of the translations that I  use, and often make my own translations. But I have also deliberately chosen to reference the standard existing translations wherever possible to avoid the needless multiplication of versions. 91. See Flasch, Nikolaus von Kues: Geschichte einer Entwicklung, 42. 92. Cited in Rawson, Intellectual Life, 157.

chapt er 1   1. See Riedweg, Pythagoras, 86. Eurytus was likely demonstrating the Pythagorean τετρακτύς, the power of the fourfold, with the visual aid of pebbles. Since 1 + 2 + 3 + 4 = 10, the number 4 represents the essential link between the monad (1) and the decad (10). When one arranges rows of 1, 2, 3, and 4 stones, they form an equilateral triangle, the first perfect polygon. See further Burkert, Lore and Science, 71–73.  2. See Metaphysics 987a13–988a10.   3. See Cherniss, Aristotle’s Criticism; and Cherniss, Riddle of the Early Academy.   4. See Burkert, Lore and Science. (The English translation revises and supersedes the German original.) Burkert controversially argued for the authenticity of several Philolaus fragments (ibid., 218–277); cf. Huffman, Philolaus, 17–18, 55–56. For a sound introduction based on scholarship after Burkert, see Kahn, Pythagoras and the Pythagoreans.   5. See Thesleff, Introduction to the Pythagorean Writings; and Pythagorean Texts, ed. Thesleff.  6. Huffman and Zhmud published two programmatic articles simultaneously that largely agree. See Huffman, “Role of Number”; and Zhmud, “ ‘All Is Number’?” Zhmud clarifies areas of agreement and disagreement in “Some Notes on Philolaus.” Against Huffman and Zhmud, see Schibli, “On ‘the One’ in Philolaus.”  7. I  leave aside the question of the search for the historical Pythagoras; see Riedweg, Pythagoras.  8. See Burkert, Lore and Science, 192–208; Zhmud, Wissenschaft, Philosophie und Religion, 93–104; Huffman, Philolaus, 10–12; and Horky, Plato and Pythagoreanism, 3–35. (Zhmud’s Wissenschaft translates the Russian original

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from 1994, a new English translation of which has now been published as Pythagoras and the Early Pythagoreans.)   9. On the earliest meanings of μάθημα and μάθησις, see Snell, Ausdrücke, 72–81. 10. On the term “arithmology,” see Délatte, Études. 11. Centrone, “Cosa significa essere Pitagorico,” 141–142. See further Cornelli, In Search of Pythagoreanism. 12. The term “henology” was introduced by Gilson in 1948 to describe the enterprise of Plotinus’s philosophy of the One (τὸ ἕν) in distinction from a philosophy of being: “ce ne sont pas deux ontologies qu’il s’agit ici de comparer entre elles, mais une ‘ontologie’ et, si l’on peut dire, une ‘énologie’ ” (L’être et l’essence, 42). Wyller and Beierwaltes popularized Henologie as a synonym for what Koch had called Neoplatonist Einheitsphilosophie (see Aertsen, “Ontology and Henology,” 120–121; Beierwaltes, Denken des Einen, 11; and Wyller, “Henologie”). Wyller has sketched a thought-provoking survey in “Zur Geschichte der platonischen Henologie.” I discuss the concept in Chapter 2. 13. For instance, see Kirk, Raven, and Schofield on fifth-century Pythagoreanism: “In the project of mathematicizing science they scarcely got beyond numerological fancy, despite the boldness and ingenuity of some of their thinking. But the idea at the heart of their project—if it is legitimate to regard it as the same idea as the Pythagoreans conceived—has now borne astonishingly abundant fruit” (Presocratic Philosophers, 350). Even Burkert derides Philolaus’s putative number mysticism (see Lore and Science, 218–298). 14. It is important to distinguish the application of mathematical concepts and procedures to philosophical problems from purported innovations in mathematics itself, for “the question is not who invented mathematics, but who connected mathematics with philosophy” (Burkert, Lore and Science, 413). Before Burkert, scholars like Erich Frank, B. L. Van der Waerden, and Paul-Henri Michel were often preoccupied with awarding or denying Pythagoras and Pythagoreans credit for pre-Euclidean mathematical breakthroughs. But Zhmud argues that if the achievements of Eudoxus or Hippias in the fifth and fourth centuries bce are counted within the penumbra of Pythagoreanism, then “a new stage in Greek mathematics does begin with Pythagoras” (Wissenschaft, Philosophie und Religion, 167). 15. See John Dillon, “ ‘Orthodoxy’ and ‘Eclecticism.’ ” 16. Merlan, From Platonism to Neoplatonism, 188. 17. Ibid., 8, 85. 18. Other candidates like Hippasus of Metapontum, Eurytus of Metapontum, Epicharmus of Syracuse, and Empedocles of Agrigentum are discussed in Horky, Plato and the Pythagoreans. 19. On Philolaus, see also Kahn, Pythagoras and the Pythagoreans, 23–38. 20. Huffman, Philolaus, 55; cf. Horky’s discussion of Epicharmus of Syracuse in Plato and the Pythagoreans, 131–137. Huffman supports Burkert’s arguments for the authenticity of Fragments 1–7, 13, and 17.

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21. Huffman views the “Pythagoreanism” of ancient Greek authors more as a private religious affiliation than as philosophical school with definite doctrines (Philolaus, 8–11). 22. On πέρας and ἄπειρον, see Huffman, Philolaus, 37–53; and Heidel, “Peras and apeiron in the Pythagorean Philosophy.” Burkert compares Philolaus’s use of mathematics to Zeno’s philosophy (Lore and Science, 285–289). 23. Huffman, “Role of Number,” 22. 24. See Fragment 6, Fragment 6a, and Testimonium A24. On harmonies in Philolaus and Archytas, see Barker, Science of Harmonics, 263–307. 25. See Huffman’s discussion of contemporary understandings of ἀριθμός in Philolaus, 173–176, and on the uncertain status of harmony in ibid., 54–77. 26. Aristotle may have had Philolaus in mind when criticizing Plato’s Pythagorean sources (see Huffman, Philolaus, 57–64, 176). Huffman points out that Burkert’s characterization of Philolaus’s philosophy as number mysticism should be seen in light of his decision to accept the authenticity of the arithmological Fragment 6b, Testimonium A14, and Testimonium A26 (see ibid., 18, 56). 27. Fragment 4, trans. Huffman, Philolaus, 172. 28. Against Huffman, see Nussbaum, “Eleatic Conventionalism and Philolaus”; and Horky, Plato and Pythagoreanism. 29. Huffman, Philolaus, 72. 30. Testimonium A7a, trans. Huffman, Philolaus, 193ff. 31. Huffman, Philolaus, 198–199. 32. Testimonium A29, trans. Huffman, Philolaus, 199. Zhmud speculates that “Philolaus was brought up in within the framework of the Pythagorean mathematical quadrivium” (Origin of the History of Science, 64). 33. Important studies of Archytas before Huffman include Bowen, “Foundations of Early Pythagorean Harmonic Science”; Lloyd, “Plato and Archytas”; and Barker, “Ptolemy’s Pythagoreans.” 34. See Huffman, Archytas, 5–19, 32–42. Archytas may have also concurred with the Socratic rejection of the Sophists: see Huffman, “Archytas and the Sophists.” 35. Huffman calls Archytas’s solution to doubling the cube a “tour de force of the spatial imagination” (Archytas, 46); cf. Mueller, “Greek Arithmetic,” 312 n. 23. 36. See Huffman, Archytas, 40–41. As the dominant mathematical authority of his generation, Archytas likely influenced Plato and Aristotle alike as they transformed science from τέχνη to an exact and certain ἐπιστήμη (Zhmud, Origin of the History of Science, 71). 37. Fragment 2, trans. Huffman, Archytas, 162ff. 38. Fragment 1, trans. Huffman, Archytas, 103ff. Burkert’s doubts about authenticity (Lore and Science, 379 n. 46) are addressed in Huffman, “Archytas Fr. 1”; and Huffman, Archytas, 65–67.

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39. Fragment 4, trans. Huffman, Archytas, 225ff; cf. Huffman, Archytas, 68–72. For Archytas ἀριθμητικά may have connoted nonscientific preoccupations by some Pythagoreans with classes of numbers. Plato appears to follow Archytas in this unusual use of λογιστικά (ibid., 240–244). See further Fowler, Mathematics of Plato’s Academy, 154–157. 40. See Huffman, Archytas, 129; cf. Theaetetus 145a. Archytas’s theory assumes a well-developed tradition of mathematical activity going back to the Pythagoreans Theodorus and Hippasus three generations prior. Zhmud even suggests that “the formation of the quadrivium either took place under the direct influence of Pythagoras or dates back to the man himself” (Wissenschaft, Philosophie und Religion, 170). Cf. Zhmud, Origin of the History of Science, 63–65. 41. See Huffman, Archytas, 40–42, 57–59, 84–88. 42. See Huffman, Philolaus, 21; and Burkert, Lore and Science, 15–97, 401–485. 43. On the role of mathematics in Plato, see Ross, Plato’s Theory of Ideas, 176–220; Wedberg, Plato’s Philosophy of Mathematics; Annas, Aristotle’s Metaphysics, 3–26; and especially Gaiser, Platons ungeschriebene Lehre. On the difficult question of Plato’s own mathematical acumen and the achievements of the Academy, which I will not address, see Lasserre, Birth of Mathematics; Fowler, Mathematics of Plato’s Academy; and Mueller, “Mathematical Method.” Contra Fowler, cf. Zhmud, Origin of the History of Science, 82–116. 44. On the mathematical curriculum of the Republic, see Gaiser, “Platons Zusammenschau”; Mueller, “Ascending to Problems”; Mueller, “Mathematics and Education”; and Burnyeat, “Plato on Why Mathematics is Good for the Soul.” 45. A similar educational program is prescribed in Laws 809b–810c, 817e–818d. 46. See Huffman, “Philolaic Method”; and Striker, Peras und Apeiron. 47. Huffman, Philolaus, 39–40. 48. Huffman, Archytas, 85–88. 49. Timaeus 47b–47e; trans. Jowett in Hamilton, Collected Dialogues, 1175. 50. Plato’s late dialogue Parmenides was also used extensively by Middle Platonists and Neopythagoreans, as we shall see, but represents a special case. The dialogue lacks references to Presocratic Pythagoreans (as one finds in Republic, Phaedrus, and Timaeus), number, and mathematics (beyond a passing reference to multitude at 144a). Plato seems to have intended the dialogue as an exercise in his method of division (as in the companion dialogue the Sophist) and a testing of Socrates’s theory of forms. But since the time of Eudorus and Moderatus, it was rather interpreted as a solemn exposition of henological principles and their articulation through negative theology (see Chapter 2). 51. For a good overview see Gaiser, “Plato’s Enigmatic Lecture”; but cf. the criticisms by Ilting, “Platons ‘Ungeschriebenen Lehren.’ ”

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52. See Dillon, Middle Platonists, 2–11; and Dillon, Heirs of Plato, 16–29. This line of interpretation is emphasized today by the so-called Tübingen school, which emphasizes the importance of Plato’s unwritten teachings. Since Plato’s late doctrine moved in the direction of mathematics and transcendent first principles, this method effectively stresses the deep connections between Plato’s thought and the development of Neopythagoreanism and Neoplatonism, making it particularly sympathetic to the present study. Major works include Gaiser, Platons ungeschriebene Lehre; and Krämer, Ursprung der Geistmetaphysik. On the claims and method of the Tübingen school, see further Gaiser, “Quellenkritische Probleme”; Krämer, “Die grundsätzlichen Fragen”; and now especially the volume of translations by Nikulin, ed., The Other Plato. 53. See Tarán, Academica, 3–13; and Tarán, “Old Academy,” 600–607. 54. See Tarán, Academica, 92–97, 140–153. 55. Epinomis 991e–992a; trans. Lamb, Plato, Vol. 12, 485. There is some disagreement about how to translate this crucial passage; cf. Tarán, Academica, 201. Here is Harward’s alternative (Epinomis, 108):  “Every geometrical diagram, every related group of numbers, every combination of a musical scale, and the single related system of the revolutions of all the heavenly bodies, must needs be revealed to the man who learns by the right method, and they will be so revealed if, as we say, a man learns aright, with his eye fixed on one point [εἰς ἓν]; for there will be revealed to them, as they reflect, a single bond [δεσμός] of nature binding all these together.” 56. See Tarán, Academica, 43, 166, 345–347. For example, Nicomachus quotes this passage at Introductio arithmetica I.3.5–7, ed. Hoche, 7–9. 57. See Tarán, Academica, 30–32. 58. On Speusippus generally, see Dillon, Heirs of Plato, 40–64. Speusippus’s fragments can be found in Tarán, Speusippus of Athens. 59. See Burkert, Lore and Science, 15–96; and Isnardi Parente, Studi sull’Accademia Platonica Antica, 11–151. As Dillon writes, “Speusippus developed Platonism in a direction which was legitimate, perhaps, but which was to find no other partisans, so far as we can see, until Plotinus, unless we may trace certain Neopythagorean and Gnostic speculations to his ultimate inspiration” (Middle Platonists, 12). 60. See Merlan, From Platonism to Neoplatonism, 96; and Dillon, Middle Platonists, 13, 16. 61. See Mueller, “On Some Academic Theories of Mathematical Objects,” 118. The theoretical unfolding of solid geometry from number thus begins not with early Pythagoreans but with Speusippus and Xenocrates intepreting the Timaeus. See Philip, “ ‘Pythagorean’ Theory of the Derivation of Magnitudes.” 62. See Dillon, Middle Platonists, 12; Merlan, From Platonism to Neoplatonism, 86–118. 63. Dillon, Heirs of Plato, 45.

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64. See Tarán, Speusippus of Athens, 13–32. 65. Metaphysics 1069a33ff., 1028b18ff. 66. See Dillon, Heirs of Plato, 49, 52. 67. Halfwassen has contended that Speusippus’s henology might stem from an ontological reading of Parmenides (far earlier than that of Moderatus). See Halfwassen, “Speusipp und die metaphysische Deutung von Platons ‘Parmenides’ ”; cf. Dillon, “Speusippus and the Ontological Interpretation of the Parmenides.” 68. Xenocrates’s fragments can be found in Isnardi Parente, Senocrate—Hermodoro; on Xenocrates generally see Thiel, Philosophie des Xenocrates. 69. Dillon, Heirs of Plato, 108; cf. Burkert, Lore and Science, 22. As Dillon notes, “neither Speusippus nor Xenocrates liked the distinction between Ideas and Mathematicals, and, as we have seen, each abolished it in different directions” (Middle Platonists, 47). 70. See Dillon, Heirs of Plato, 101–107, 120–121. 71. See ibid., 153–154; as well as Hübner, “Die geometrische Theologie.” Huffman notes that the material Hübner analyzes goes back to Xenocrates, not Philolaus (Philolaus, 391). 72. See Baltes, “Theologie des Xenocrates”; and Thiel, Philosophie des Xenocrates, 265–285. 73. See Dillon, Middle Platonists, 23–29; and Dillon, Heirs of Plato, 121–122. 74. Krämer, Ursprung der Geistmetaphysik, 45–62; and Thiel, Philosophie des Xenocrates, 420–421, 464–466. 75. Dillon, Heirs of Plato, 112. 76. On Aristotle’s views of mathematics in Metaphysics M and N, see above all Annas, Aristotle’s Metaphysics; as well as Cleary, Aristotle and Mathematics. 77. Metaphysics 987b10–987b14. 78. Ibid. 987b14–987b18; trans. Tredennick, 45. 79. Ibid. 1026a11–10266a16. 80. See Merlan, From Platonism to Neoplatonism, 56–59. Merlan argues that the textual problem began with a conceptual dilemma stemming from Aristotle’s attempt to combine a principle of being (κινητά/ἀκίνητα) with a principle of knowledge (χωριστά/ἀχώριστα). Against Merlan’s reading, cf. Cleary, Aristotle and Mathematics, 425–438. 81. Metaphysics 1026a22–1026a31; cf. ibid. 1063b36–1064b14. 82. See Merlan, From Platonism to Neoplatonism, 71–75.

chapt er 2   1. See Dillon, Middle Platonists, 52–62.   2. See Bonazzi, “Towards Transcendence,” 241–242.

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  3. On Alexander Polyhistor, see Long, “Eclectic Pythagoreanism.” Thesleff studies the post-Platonic pseudonymous Pythagorean corpus that survives in several dozen fragmentary works of varying length and dialect. He rejects past attempts to locate them around Alexandria in the first century bce and instead breaks them into two differentiated groups of doxographical handbooks (Introduction, 30–71). The first (eastern) group is about or attributed to the historical Pythagoras and originated in the eastern Mediterranean around Alexandria or Athens as early as the fourth or third century bce. The second (western) group, attributed to a host of lesser Pythagoreans, was concentrated in southern Italy in the third and second centuries bce (ibid., 99–116). Thesleff’s conclusions argue for a more continuous and to that extent authentic Pythagorean tradition feeding into the Neopythagorean monism of Eudorus, Philo, and Moderatus in the first century bce (ibid., 110–120). Many of these pseudonymous works are translated in Pythagorean Sourcebook, ed. Fideler.   4. See Robbins, “Posidonius”; and Robbins, “Tradition of Greek Arithmology.”  5. See Kahn, Pythagoras and the Pythagoreans, 86–93; and Rawson, Intellectual Life, 30–33.   6. On Varro’s lost books on the quadrivium, see Rawson, Intellectual Life, 156–169; and especially Hagendahl, Augustine and the Latin Classics, 589–649.   7. On Nigidius see Rawson, Intellectual Life, 291–294.   8. Thesleff challenges Eduard Zeller’s hypothesis of an Alexandrian Pythagorean renaissance around 100 bce (Introduction, 46–50), but his arguments pertain to the date of the Hellenistic pseudepigrapha, not to Eudorus, Moderatus, or Philo themselves. See Szlezák, Pseudo-Archytas, 14–15.   9. On the deep connections between Eudorus, Moderatus, and Speusippus, see Halfwassen, “Speusipp und die metaphysische Deutung.” 10. See Bonazzi, “Pythagoreanising Aristotle”; and Chiaradonna, “Platonist Approaches to Aristotle” 41–50. 11. Dörrie, “Erneuerung des Platonismus,” 160. Eudorus also wrote extensively on ethics: see Bonazzi, “Eudorus’ Psychology”; and Sedley, “Ideal of Godlikeness.” 12. See Dillon, Middle Platonists, 126–128. 13. In Mansfeld’s judgment:  “Eudorus’s approach, according to which the Pythagorean Hen (not:  Monas) is the most high God beyond the opposites which spring forth therefrom, is original. Proof for a pre-Eudorean (Neo-) Pythagorean meta-dualistic sole principle of all things which would be God I, for my part, have not found” (“Compatible Alternatives,” 102–103). Cf. Bonazzi, “Towards Transcendence,” 242. On monism in Eudorus and subsequent Neopythagoreanism, see also Turner, Sethian Gnosticism, 349–355; and Rist, “Monism.” Against this line of interpretation, Staab views dualistic elements in Eudorus and Moderatus as their philosophical signature, but this may simply reflect his decision to include Numenius (see “Kennzeichen des neuen Pythagoreismus”).

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14. See Merlan, “Pythagoreans”; and Kahn, Pythagoras and the Pythagoreans, 78–83. 15. See Dörrie, “Der Platoniker Eudoros,” 304, 307; and Bonazzi, “Eudorus of Alexandria,” 370–373. The latter is an abridged version of Bonazzi, “Eudoro di Alessandria e il Timeo di Platone.” 16. See Whittaker, “Transcendent Absolute.” 17. See Dillon, Middle Platonists, 344–351. On Moderatus’s origins, see Jurado, “Moderato de Gades.” 18. Eudorus is likely among “the Pythagoreans” to whom Moderatus attributed a new theological reading of “the One” in the Parmenides, well before Plotinus. See Whittaker, “EPEKEINA NOU KAI OUSIAS,” 97–98; and Dodds, “Parmenides of Plato,” 139–140. Tarrant attempts to excavate Moderatus’s sources and agenda in further detail (Thrasyllan Platonism, 150–176). 19. Porphyry, Vita Pythagorae 48–50, in Opuscula selecta, ed. Nauck, 43–44. Robbins translates the passage in Nicomachus, trans. D’Ooge et  al., 103. Cf. Guthrie’s translation in Pythagorean Sourcebook, ed. Fideler, 133. 20. Porphyry, Vita Pythagorae 51–53, in Opuscula selecta, ed. Nauck, 45–46. O’Meara has shown that Porphyry’s discussion of unity in this passage has roots in Nicomachean harmonics. Nicomachus defines concordant intervals as “different in magnitude” yet “single in form” (ἑνοειδής), a harmonic term widely adopted in subsequent antique Neoplatonism (see “Music of Philosophy,” 138–139). 21. The passage in Porphyry is replete with hermeneutical difficulties. See the alternative readings with annotations and analysis in Merlan, “Pythagoreans,” 91–94; Dillon, Middle Platonists, 348; Turner, Sethian Gnosticism, 368–372; and Turner, “Platonizing Sethian Treatises,” 161. 22. See the important analysis in Dodds, “Parmenides of Plato.” Dillon summarizes: “We have in the metaphysics of Moderatus, and perhaps in that of the Pythagorean movement even before his time, a great part of what has been conventionally taken to be the distinctive contribution of Plotinus” (Middle Platonists, 351). Dodds’s argument altered the landscape of scholarship, but against his views see the questions raised by Rist, “Neoplatonic One and Plato’s Parmenides”; Dillon, “Speusippus and the Ontological Interpretation of the Parmenides”; and Hubler, “Moderatus.” 23. Tarrant, Thrasyllan Platonism, 160–161. 24. Ibid., 156. 25. Merlan, “Pythagoreans,” 94. 26. Dodds, “Parmenides of Plato”; and Dillon, Middle Platonists, 347–349. 27. Baltes, Platonismus, 481. 28. Tornau, “Prinzipienlehre,” 216–218, 206. Tornau notes that attributing intelligence to the first One is problematic only within the Plotinian framework of an absolutely transcendent first principle: “The highest One of Moderatus was indeed beyond being, yet not beyond intellect, but rather was itself the highest

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form of intellectual knowing. . . . For Moderatus, as for Numenius, the notion of a God without knowledge and consciousness would have been blasphemous” (ibid., 218). 29. Trans. Dillon, Middle Platonists, 350. 30. Ibid. 31. See Dörrie, “Erneuerung des Platonismus,” 161–63. Cox examines how the “Middle Platonic intermediary doctrine” of Eudorus, Moderatus, and Philo was used by all three traditions (By the Same Word, 28–55). Whittaker notes that while Alcinous in the Didascalicus “seems shy of associating himself with anything that might smack even slightly of Neopythagorean influence,” Philo, Justin, Clement, and Eusebius had no such qualms (“Platonic Philosophy,” 118). 32. See Turner, “Platonizing Sethian Treatises,” 134–139; and Turner, “Gnostic Sethians and Middle Platonism.” 33. On the Stoic Logos doctrine, see Aall, Der Logos, 1: 98–167; and Hahm, Origins of Stoic Cosmology, 136–139. 34. See Runia, “Witness or Participant?”; and Runia, “Clement of Alexandria.” 35. For a survey of Philo’s Logos theology, see Runia, Philo of Alexandria and the “Timaeus”, 446–451; Aall, Der Logos, 1: 184–231; and Heinze, Die Lehre vom Logos, 215–297. On its influence, see Runia, Philo in Early Christian Literature. 36. On Abraham XXIV: 119–123, in Philo, vol. 6, trans. Colson, 63–65. 37. Philo’s treatise On Numbers has not survived, but has been partially reconstructed in Staehle, Zahlenmystik. Robbins surmises that the lost book of Philo, unlike that of Nicomachus, “was not an elements of arithmetic . . . but an arithmology” in the “S” tradition (“Arithmetic in Philo Judaeus,” 360–361). 38. See Turner, Sethian Gnosticism, 356–361. Cf. Krämer, Ursprung der Geistmetaphysik, 266–281. 39. On the sources of Philo’s λόγος τομεύς, see Hay, “Philo’s Treatise on the Logos-Cutter,” contra Goodenough, “Neo-Pythagorean Source.” 40. Who is the Heir 140–145, trans. Colson, 353–355. On the Pythagorean background of the λόγος τομεύς, see Krämer, Ursprung der Geistmetaphysik, 269–272. 41. Who is the Heir 187–190, trans. Colson, 377–379 (modified). 42. Ibid. 205–206, trans. Colson, 385–387. 43. See the overview by Cox, By the Same Word, 87–123; and generally Runia, Philo in Early Christian Literature. 44. See Lilla, Clement of Alexandria, 199–212. 45. See Runia, Philo in Early Christian Literature, 132–156; Hägg, Clement of Alexandria, 180–207; and Aall, Der Logos, 2: 396–427. On the related question of Clement’s two-stage or one-stage Logos theology, see Edwards, “Clement of Alexandria,” contra Wolfson, “Clement of Alexandria.” 46. See Runia, “Clement of Alexandria,” 275–277. 47. See Andresen, “Justin und der mittlere Platonismus.” 48. See Theiler, “Philo von Alexandria”; and Festugière, Révélation, 19–25.

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49. See Runia, Philo of Alexandria and the “Timaeus”, 499–501; and Runia, “Beginnings of the End.” 50. See Bonazzi, “Towards Transcendence,” 236–245. 51. See Kalvesmaki, Theology of Arithmetic, 103–124. 52. See respectively Williams, “Negative Theologies and Demiurgical Myths”; Turner, Sethian Gnosticism, 628–630; and Brankaer, “Is there a Gnostic ‘Henological’ Speculation.” For examples of Valentinian arithmology, see Kalvesmaki, Theology of Arithmetic, 27–102. 53. See Mansfeld, Heresiography in Context, 171–187. 54. Thomassen, Spiritual Seed, 269–294; and Thomassen, “Derivation of Matter in Monistic Gnosticism.” Turner finds parallels in other kinds of Gnosticism as well (Sethian Gnosticism, 368–372). 55. See Thomassen, Spiritual Seed, 417–422, 491–494. According to Festugière, some Hermetic texts such as Poimander also transmitted a modified version of Moderatan theology (Révélation, 18–31). 56. See Thomassen, Spiritual Seed, 273. 57. See ibid., 275–283. “This double usage of the terminology of extension and spreading out—partly to describe the unfolding of the deity himself, and partly to account for the negative plurality in Sophia as a source of matter—undoubtedly reflects the ambiguities inherent in the Neopythagorean attempts to derive plurality from the oneness of the Monad” (ibid., 277). 58. See Wolfson, “Albinus and Plotinus”; Festugière, Révélation, 92–102; Hadot, Porphyre et Victorinus, 278–283; Hägg, Clement of Alexandria, 93–133; and Turner, Sethian Gnosticism, 380–385. Following Whittaker’s Budé edition, Dillon and others refer to the author of Didascalicus as Alcinous, not Albinus (see Handbook of Platonism, trans. Dillon, ix–xiii). Almost nothing is known of Alcinous beyond his book. 59. Alcinous, Didascalicus 10.5 (165), trans. Dillon, 18. See Dillon’s commentary on Didascalicus 10 (Handbook of Platonism, trans. Dillon, 100–111). 60. Whittaker, “Neopythagoreanism and Negative Theology,” 112, 115. 61. See Wolfson, “Negative Attributes”; and Whittaker, “Basilides on the Ineffability of God.” 62. Clement of Alexandria, Stromateis V.11.71 (2–3), ed. Boulluec, 143; trans. in Wolfson, “Negative Attributes,” 148 (modified). The allusion is to Ephesians 3:18: see further Osborn, Clement of Alexandria, 124–125. On Clement’s negative theology, see Lilla, Clement of Alexandria, 212–226. On his Pythagorean sympathies, see Afonasin, “Pythagorean Way of Life”; and Kalvesmaki, Theology of Arithmetic, 128–151. 63. Paedagogus I.8 (71), PG 8: 336A. 64. Legatio pro Christianis VI, PG 6: 901A. Athenagoras cites a certain “Lysis and Opsimus.” 65. For example, Plotinus’s doctrine of multiplicity (Ennead VI.6) combines Numenius’s theology of divine stasis with Moderatus’s second definition of

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kinetic number. This maneuver allows Plotinus to entirely circumvent the more obvious source for a third-century Platonist theory of number, the famed arithmetic of Nicomachus. See Slaveva-Griffin, Plotinus on Number, 43–53; and Charles-Saget, L’architecture du divin, 151–185. 66. On Numenius see Frede, “Numenius”; Baltes, “Numenios von Apamea”; and O’Meara, Pythagoras Revived, 12–14. Frede argues that Numenius should be considered a “Pythagoreanizing Platonist” rather than a “Neopythagorean,” since he clearly repudiates the monism common to Eudorus, Moderatus, and Nicomachus (“Numenius,” 1047, 1052). Whittaker (“Platonic Philosophy,” 119) and Thomassen (Spiritual Seed, 270) both agree. 67. See Dillon, “ ‘Orthodoxy’ and ‘Eclecticism’.” 68. On Nicomachus’s life, see Tarán, “Nicomachus of Gerasa”; and Haase, “Untersuchungen,” 34–119. There is some controversy surrounding the date of his death: cf. Dillon, “Date for the Death of Nicomachus of Gerasa?”; and Criddle, “Chronology of Nicomachus of Gerasa.” 69. Nicomachus’s Manual of Harmonics is edited by Karl von Jan and translated by Flora R. Levin. See Levin’s study, Harmonics of Nicomachus. 70. Some caveats are in order regarding the text of Nicomachus’s Theology. Significant fragments of this lost work are preserved in an anonymous text by the same name (Theologoumena arithmeticae) that combines extracts from Nicomachus’s Theology with others from Anatolius’s On the Decad. In his edition of Theologoumena, Ast attempted to isolate the authentic Nicomachean passages. Robbins defends Ast’s selections and then translates several of them in his essays on Nicomachus (see Robbins in Nicomachus, trans. D’Ooge et al., 79–87, 88–123). Later studies of Nicomachus by Dillon and O’Meara used the more recent edition by De Falco that largely follows Ast on the question of Nicomachus. In his review of De Falco’s edition, however, Oppermann challenged this consensus view, positing an intervening source between the Nicomachean and Anatolian originals and the anonymous Theologoumena, namely the lost Theologoumena arithmeticae of Iamblichus himself, such that the anonymous Theologoumena combined passages of Nicomachus, Anatolius, and Iamblichus all together. Oppermann’s solution is not without its own problems, and for our purposes, does not materially affect the Nicomachean attributions cited by Ast (see Tarán, Speusippus of Athens, 291–297; and O’Meara, Pythagoras Revived, 15). In what follows I use only passages from the anonymous Theologoumena that have been identified and translated by Dillon in Middle Platonists (following De Falco) or by Robbins in Nicomachus, trans. D’Ooge et al. (following Ast, but I cite to De Falco for consistency). Waterfield has since made corrections to De Falco’s edition (see “Emendations of [Iamblichus]”). 71. O’Meara, Pythagoras Revived, 19. 72. I  use Martin Luther D’Ooge’s translation of the Introductio arithmetica based on the Hoche edition. D’Ooge’s translation, published posthumously, has been

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revised and supplemented with excellent studies by Frank Egleston Robbins and Louis Charles Karpinski. The French translation by Janine Bertier includes a fine introduction and glossary. For a provocative rethinking of Nicomachean number theory, see the recent work by Radke, Theorie der Zahl; but cf. the cautious reviews by O’Meara and Helmig. 73. See Theon of Smyrna, Exposition des connaissances, ed. Dupuis. Robbins compares the two works in Nicomachus, trans. D’Ooge et al., 37–43. 74. Tarán, Asclepius of Tralles, 5. See also Karpinski in Nicomachus, trans. D’Ooge et al., 138–145. 75. See McDermott, “Plotina Augusta and Nicomachus of Gerasa.” 76. See Zonta and Freudenthal, “Nicomachus of Gerasa in Spain”; and Mansfeld, Prologomena Mathematica, 82–91. 77. Thesleff, Introduction, 48, 120. 78. Introductio arithmetica I.1.1–I.2.1, ed. Hoche, 1–3; trans. D’Ooge et al., 181–82. 79. “The original universe, lacking order and shapeless and totally devoid of the things that give distinction according to the categories of quality and quantity and the rest, was organized and arranged most clearly by number as the most authoritative and artistic form, and gained a share in a harmonious exchange and flawless consistency in accordance with its desire for and its receiving the impression of the peculiar properties of number.” Theologoumena arithmeticae VI, ed. De Falco, 44; trans. Robbins in Nicomachus, trans. D’Ooge et al., 98. 80. Metaphysics 1020a7–1020a15. See Robbins in Nicomachus, trans. D’Ooge et al., 112–113. 81. Introductio arithmetica I.2.4–5, ed. Hoche, 4–5; trans. D’Ooge et al., 183–184. 82. Ibid. I.3.1–2, ed. Hoche, 5–6; trans. D’Ooge et al., 184. Cf. Proclus, Euclid I.12, ed. Friedlein, 35–37; trans. Morrow, 29–30. Huffman argues that the reference to two primary forms of being which make the four sciences “akin” is not originally Archytan but rather Nicomachus’s gloss (Archytas, 115–124, 154–155). 83. Introductio arithmetica I.4.1, ed. Hoche, 9; trans. D’Ooge et al., 187. Cf. ibid. I.5.3, ed. Hoche, 11. 84. Ibid. II.1.1–2, ed. Hoche, 73-74; trans. D’Ooge et  al., 230. On unity, see ibid. I.8.1–2, ed. Hoche, 14. On equality, see ibid. I.17.2, ed. Hoche, 44; and ibid. I.23.6, ed. Hoche, 65. 85. See Robbins’s caveats in Nicomachus, trans. D’Ooge et al., 185 n. 3. 86. On the sources of this saying, see Saffrey, “AGEŌMETRĒTOS MĒDEIS EISITŌ”; and Fowler, Mathematics of Plato’s Academy, 197–202. 87. Introductio arithmetica I.3.3-6, ed. Hoche, 6–8; trans. D’Ooge et al., 184–186. 88. Merlan, From Platonism to Neoplatonism, 81–82. 89. See O’Meara, Pythagoras Revived, 16–17, 21–22. 90. Helmig, “Relationship between Forms and Numbers.” 91. See Turner’s sketch of Nicomachean henology in Sethian Gnosticism, 377–378. 92. Introductio arithmetica II.6.3, ed. Hoche, 84; trans. D’Ooge et al., 237–238.

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 93. Theologoumena arithmeticae I, ed. De Falco, 3; trans. Robbins in Nicomachus, trans. D’Ooge et al., 95–96 (modified).  94. Introductio arithmetica I.7.1, ed. Hoche, 13; trans. D’Ooge et al., 190. Nicomachus cites two other definitions of number. The first, number as a “system” (σύστημα) or combination of monads, is at least as old as Thales and appears in Moderatus as well. The second definition, stemming from the mathematician Eudoxus, a student of Archytas, is “limited multitude” (πλῆθος ὡρισμένον); compare Aristotle’s similar definition (πλῆθος πεπερασμένον) at Metaphysics 1020a7ff. These two more common definitions also appear in Euclid (“the multitude composed out of monads” [τὸ ἐκ μονάδων συγκείμενον πλῆθος]) and in Theon of Smyrna. See Robbins in Nicomachus, trans. D’Ooge et al., 114–115.  95. Theologoumena arithmeticae II, ed. De Falco, 8–9; trans. Robbins in Nicomachus, trans. D’Ooge et al., 117.   96. Ibid.   97. Ibid. VI, ed. De Falco, 44; trans. Robbins, 98.   98. Ibid. IX, ed. De Falco, 78–79; trans. Robbins, 107.  99. Introductio arithmetica I.4.2, ed. Hoche, 9; trans. D’Ooge et al., 187. 100. Ibid. I.6.1, ed. Hoche, 12; trans. D’Ooge et al., 189. 101. Plutarch’s hierarchy was built upon a cumulative chain of phenomenal dimensions:  arithmetic (ποσόν), geometry (ποσόν and μέγεθος), stereometry (μέγεθος and βάθος), astronomy (στερεόν and κίνησις), and harmonics (κινουμένον and φωνή). See Napolitano Valditara, Le idee, i numeri, l’ordine, 409–411. 102. On Logos, see Wolfson, “Extradeical and Intradeical Interpretations.” On Arithmos, see Krämer, Ursprung der Geistmetaphysik, 23–26, 281. While Krämer discusses Logos at length alongside Arithmos, Wolfson circumvents the question of number altogether, neatly obviating the possibility of Neopythagoreanism interacting with early Christian doctrines of the Trinity. 103. See Rich, “Platonic Ideas,” 125–127. These three positions are defended respectively by Wolfson, “Extradeical and Intradeical Interpretations”; by Krämer, Ursprung der Geistmetaphysik; and by Jones, “Ideas as Thoughts of God.” 104. “Haec exemplaria rerum omnium deus intra se habet numerosque universorum, quae agenda sunt, et modos mente conplexus est: plenus his figuris est, quas Plato ideas appellat, inmortales, inmutabiles, infatigabiles.” Seneca, Ad Lucilium 65 (7), ed. Gummere, 448; trans. Gersh, Middle Platonism, 190. Cf. the freer translation in Seneca, Selected Philosophical Letters, trans. Inwood, 11. First noted by Theiler (Vorbereitung des Neuplatonismus, 1–38), the possible Middle Platonist sources of Seneca’s Letters 58 and 65 are now broadly discussed; see Inwood’s overview in Seneca, Selected Philosophical Letters, 107–155. 105. “Ordo autem sine harmonia esse non potest, harmonia demum analogiae comes est, analogia item cum ratione et demum ratio comes indiuidua prouidentiae reperitur, nec uero prouidentia sine intellectu est intellectusque sine mente non est. Mens ergo dei modulauit ordinauit excoluit omnem continentiam

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corporis; inuenta ergo est demum opificis diuina origo. Operatur porro opifex et exornat omnia iuxta uim rationabilem maiestatemque operum suorum; opera uero eius intellectus eius sunt, qui a Graecis ideae uocantur; porro ideae sunt exempla naturalium rerum.” Calcidius, Timaeus 304, ed. Waszink, 305– 306. “Haec monas [viz. unitas] initium finisque omnium, neque ipsa principii aut finis sciens, ad summum refertur deum eiusque intellectum a sequentium numero rerum et potestatum sequestrat, nec in inferiore post deum gradu frustra eam desideraveris. Haec illa est mens ex summo enata deo, quae vices temporum nesciens in uno semper quoad adest consistit aevo, cumque utpote una non sit ipsa numerabilis, innumeras tamen generum species et de se creat et intra se continet.” Macrobius, Commentarii in Somnium Scipionis I.6.8-9, ed. Willis, 19–20. See further Krämer, Ursprung der Geistmetaphysik, 275–279. 106. Napolitano Valditara, Le idee, i numeri, l’ordine, 413–434, esp. 426–429. 107. Robbins in Nicomachus, trans. D’Ooge et al., 98–99. Cf. Introductio arithmetica I.6.2–4, ed. Hoche, 12–13; trans. D’Ooge et al., 189–90. As O’Meara rightly cautions, this distinction is not as clear in the text as Robbins suggests (Pythagoras Revived, 18 n. 33). 108. Robbins in Nicomachus, trans. D’Ooge et al., 99. 109. For a summary of Nicomachus’s arithmological doctrines, see Robbins in Nicomachus, trans. D’Ooge et al., 104–107; and Mattéi, “Nicomachus of Gerasa.” 110. Robbins, in Nicomachus, trans. D’Ooge et al., 95–96; cf. Heinze, Die Lehre vom Logos, 179–181. 111. Theologoumena arithmeticae VII, ed. De Falco, 57–58; trans. Dillon, Middle Platonists, 357. 112. Ibid. III, ed. De Falco, 19; trans. Dillon, 358. 113. Ibid. III, ed. De Falco, 17–18; trans. Dillon, 356–357. 114. Ibid. VI, ed. De Falco, 45; trans. Dillon, 358.

chapt er 3   1. Kalvesmaki finds that an early fluidity in Christian attitudes toward Pythagorean arithmologies had mostly disappeared by the fourth century (Theology of Arithmetic, 170–172).   2. Chenu, “Platonisms of the Twelfth Century,” 49. Chenu counts a fourth tradition of “Islamic” Neoplatonism comprising the Liber de causis and Avicenna; the former is a florilegium of Proclus’s Elements, and the latter does not influence medieval Christianity until the thirteenth century. Klibansky provides a fuller sketch of the connections among Platonist traditions within the Byzantine, Arabic, and Latin spheres; within Latin Platonism he too names Augustine, Boethius, and the Proclianism of Ps.-Dionysius (in that order) as the three major conduits to medieval Christianity (Continuity of the Platonic Tradition, 21–27).   3. On Iamblichus’s life, see generally Dillon, “Iamblichus of Chalcis.”

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  4. See O’Meara, Pythagoras Revived, 80.   5. See O’Meara, “La question de l’être,” 412–414.   6. See Halfwassen, “Das Eine als Einheit und Dreiheit”; cf. Kalvesmaki’s instructive comparison of Theodore of Asine and Iamblichus (Theology of Arithmetic, 158–169).   7. See Shaw, Theurgy and the Soul, 1–4; and O’Meara, Pythagoras Revived, 213–214.  8. See O’Meara, Pythagoras Revived, 32–35. Iamblichus wrote a commentary on Nicomachus’s Introduction (see In Nicomachi Arithmeticam introductionem, ed. Pistelli) and discussed Nicomachus’s fourfold hierarchy of mathematics in De communi mathematica scientia (see De communi mathematica scientia, ed. Festa), a work that would enjoy a wide diffusion via Proclus’s Euclid commentary. According to O’Meara, these two volumes correspond to On Pythagoreanism, Books IV and III.  9. O’Meara, Pythagoras Revived, 84. 10. Ibid., 33–34, 47–49. See further Romano, “Le rôle de la mathématique.” Staab argues that Iamblichus may have been following the example of similar multivolume compendia (now lost) by Moderatus and Nicomachus (Pythagoras in Spätantike, 77–92). 11. On soul and mathematicals in De communi mathematica scientia, see Merlan, From Platonism to Neoplatonism, 8–29. 12. See O’Meara, Pythagoras Revived, 79–85. 13. See Shaw, “Geometry of Grace”; and Shaw, “Eros and Arithmos.” 14. On Syrianus’s philosophy of mathematics, see O’Meara, Pythagoras Revived, 132–141; O’Meara, “Le problème de la métaphysique”; and Mueller, “Syrianus.” 15. The term originates in the Philebus as a synonym for “monad”; Plotinus states that the Pythagoreans call the forms “henads.” Theon of Smyrna ranks the henads near the One, and Syrianus identifies them with the gods in his commentary on the Metaphysics. See further Proclus, Elements, ed. Dodds, 258–259. There is a long-standing controversy over whether to credit Syrianus (thus Saffrey and Westerink) or Iamblichus (thus Dillon) with inventing Proclus’s sense of the term. Among recent discussions, see Bechtle, “Göttliche Henaden und platonischer Parmenides”; and Clark, “Gods as Henads.” 16. See O’Meara, Pythagoras Revived, 131–133. Although this proto-Kantian concept of a priori intramental geometry is usually attributed to Syrianus, O’Meara suggests that aspects of the psychic projection of numbers are already present in Iamblichus (“La question de l’être,” 415). See further Charles-Saget, L’architecture du divin, 191–201; and Bechtle, “Über die Mittelstellung und Einheit.” 17. See Mueller, “Iamblichus and Proclus’ Euclid Commentary”; and O’Meara, “Proclus’ First Prologue to Euclid.” 18. O’Meara, Pythagoras Revived, 207. 19. See ibid., 164–166; and Lernould, “La dialectique comme science première.” 20. On mathematicals as mediation in Proclus, see Breton, Philosophie et mathématique, 137–149.

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21. Euclid I.1–2, ed. Friedlein, 4–6; trans. Morrow, 3–5. 22. Ibid. I.6, ed. Friedlein, 16–17; trans. Morrow, 14–15. 23. Without digressing into the complex status of soul in Proclus, suffice it to say that the ψυχή that unfolds mathematicals is itself also an “image” and “unfolding” of νοῦς, “a tablet that has always been inscribed and is always writing itself and being written on by νοῦς.” Euclid I.6, ed. Friedlein, 16; trans. Morrow, 14. 24. Euclid I.10, ed. Friedlein, 32; trans. Morrow, 26. 25. Ibid. I.14, ed. Friedlein, 42; trans. Morrow, 35. 26. Ibid. I.14, ed. Friedlein, 44; trans. Morrow, 36–37. 27. Ibid. I.8, ed. Friedlein, 22; trans. Morrow, 18–19. 28. Strictly speaking, Proclus simply juxtaposes two accounts of the different mathematical sciences, describing Nicomachus’s as that of “the Pythagoreans,” but then neither affirming nor denying its validity. See Euclid I.13, ed. Friedlein, 38–42; trans. Morrow, 31–35. 29. Euclid I.12, ed. Friedlein, 36–37; trans. Morrow, 30. On number in Proclian physics, see O’Meara, Pythagoras Revived, 185–189. 30. See Mueller, “Mathematics and Philosophy.” 31. On the functions of geometry in Proclus, see Breton, Philosophie et mathématique, 33–83; and O’Meara, Pythagoras Revived, 167–176. 32. Euclid II.1, ed. Friedlein, 54–56; trans. Morrow, 44–45. See Breton, Philosophie et mathématique, 123–132; and Nikulin, “Imagination and Mathematics in Proclus.” As Mueller explains, “Iamblichus put forward the doctrine of projectionism as an account of Pythagorean mathematics, which he glorified at the expense of ordinary mathematics; he was followed by Syrianus, but Proclus transformed projectionism into an account of ordinary mathematics to which he restored its Platonic role” (“Aristotle’s Doctrine of Abstraction,” 480). 33. Euclid II.3, ed. Friedlein, 62; trans. Morrow, 50. See Steel, “Proclus on Divine Figures.” 34. See Elements, Prop. 25–39, ed. Dodds, 28–43. See further Beierwaltes, Proklos, 165–239; and Gersh, Kinesis Akinetos. 35. See Elements, Prop. 26, ed. Dodds, 30–31; and ibid., Prop. 30, ed. Dodds, 34–35. 36. Morrow, Commentary, lxii; see also Bechtle, “Über die Mittelstellung und Einheit,” 31–40. 37. See Vinel, “La rhusis mathématique.” 38. See Gersh, From Iamblichus to Eriugena. 39. See Elements, Props. 113–115, ed. Dodds, 100–103. 40. See ibid., Prop. 135, ed. Dodds, 120–121. Even if the term originates with Iamblichus or Syrianus, Proclus puts it to more robust use. Yet Dodds notes that Proclus’s decision to identify the henads with Olympian gods (explored further in Platonic Theology) “is no doubt to be understood as a last desperate attempt to carry out the policy of Iamblichus and maintain the united front of Hellenic philosophy and Hellenic religion against the inroads of Christianity” (Elements, ed. Dodds, 259).

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41. There is no original treatment of arithmetic, geometry, or the mathematical arts in Ps.-Dionysius to speak of. In the final chapter of Divine Names there is an unremarkable henology that places the One beyond every number (see 977c–980d, ed. Suchla, 227–229). The author occasionally uses geometrical images that were common in contemporary philosophical schools. Numbers proceed from the monad like radii proceeding from the unifying center of a circle (ibid. 821a, ed. Suchla, 185). Divine intelligences and soul move in a hybrid of circular and linear motion, namely an ascending spiral (ibid. 704d–705b, ed. Suchla, 153–154; and ibid. 916cd, ed. Suchla, 213). The procession and return of the Good is like an eternal circle (ibid. 713d, ed. Suchla, 162). See Gersh, From Iamblichus to Eriugena, 251–252. 42. See Charles-Saget, L’architecture du divin, 271–296; and Siorvanes, Proclus, 247–256. 43. On Augustine’s classical sources, see the still valuable overview, Courcelle, Late Latin Writers, 165–196. On Latin sources see O’Donnell, “Augustine’s Classical Readings.” On Greek sources of Augustine’s views on the liberal arts, the major works are Svoboda, L’Esthétique; Solignac, “Doxographies et manuels”; and Hadot, Arts libéraux et philosophie. 44. Svoboda makes this suggestion (L’Esthétique, 70), and Solignac proposes no fewer than six specific points of contact between Augustine and Nicomachus’s Introduction (“Doxographies et manuels,” 133–137). See further the notes by Agaësse and Solignac, La Genèse au sens littéral, 633–635; and Beierwaltes, “Augustins Interpretation,” 56–58. But Hadot finds evidence in Letter 3 that the source could be Plotinus (see “ ‘Numerus intelligibilis infinite crescit’ ”). 45. Solignac, “Doxographies et manuels,” 114, 131. See for this section the exemplary study by Christoph Horn, “Augustins Philosophie der Zahlen.” Two other fine essays are hidden in Ladner, The Idea of Reform, 212–238 (cf. 449–462); and Hellgardt, Zum Problem, 157–251. Ladner situates Augustine’s mathematical ideas at the center of his doctrine of the continual reforming of the soul. As manifestations of divine wisdom in the world, numbers provide a trace of divine permanence that assist the soul in making its return to its Creator. This “reformative function of number” connects Augustine’s early Platonism, his doctrines of time and memory in Confessiones, and the theology of history of De civitate dei. See further the thematic essays by Schmitt, “Mathematik und Zahlenmystik”; and Beierwaltes, “Aequalitas numerosa.” 46. Brown perceives a revolution in Augustine’s thought in the mid-390s, as early as his ordination in 391 and culminating in the decision to write Confessiones in 397, in which he turns away from an optimistic early Platonism to his characteristic views of human finitude (Augustine of Hippo, 139–150). Brown’s masterful account has been tremendously influential, but also recently nuanced by Harrison, who argues that the true fulcrum is Augustine’s conversion in 386 (Rethinking Augustine’s Early Theology, 14–19). Harrison does not address,

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however, the gradual disappearance of Neopythagorean elements from Augustine’s theology, where the evidence does suggest, per Brown, a slow shift beginning in the late 390s marked by the transition from De libero arbitrio II toward Confessiones. 47. See De ordine II.11.33—II.12.35, ed. Green, 126–127. 48. Ibid. II.14.40–41, ed. Green, 129–130. 49. There is some controversy over the sources of De ordine and hence the connection between Augustine’s views of the liberal arts and the classical tradition of ἐγκύκλιος παιδεία. The majority theory follows the Ritschl hypothesis that Varro’s lost Libri disciplinarum informed Augustine’s early works. See, e.g., the standard work by Marrou, Saint Augustin et la fin de la culture antique. In his important monograph, Dyroff concurs and further maintains that Varro mediated a Neopythagorean-Stoic source that could even be Nicomachus (“Über Form und Begriffsgehalt,” 34–46). Svoboda suggests that the source is rather Posidonius himself (L’Esthétique, 41–45). Hadot criticizes Dyroff’s analysis (Arts libéraux et philosophie, 132–135) and argues that Augustine’s views are not the epitome of a long classical tradition but rather develop Porphyry’s lost De regressu animae (ibid., 101–136). Cf. Solignac, “Réminiscences plotiniennes et porphyriennes”; and Du Roy, L’Intelligence, 130–143. Shanzer has recently defended the Varro thesis againt Hadot’s critique (“Augustine’s Disciplines,” 69–112). 50. See De ordine II.16.44, ed. Green, 131. 51. “Mouit eam quoddam miraculum et suspicari coepit se ipsam fortasse numerum esse eum ipsum, quo cuncta numerarentur, aut si id non esset, ibi tamen eum esse, quo peruenire satageret.” De ordine II.15.43, ed. Green, 130; trans. Russell, 151. 52. “illo occultissimo, quo numeramus . . . illo diuino ordine occulto,” De ordine II.15.43, ed. Green, 130–131; and ibid. II.20.54, ed. Green, 136. 53. Ibid. II.16.44, ed. Green, 131. 54. Ibid. II.18.47, ed. Green, 132–133. See further Hübner, “Artes liberales.” 55. De ordine II.16.44, ed. Green, 131. 56. “in ratione autem aut nihil esse melius et potentius numeris, aut nihil aliud quam numerus esse rationem.” De ordine II.18.48, ed. Green, 133; trans. Russell, 161. On Pythagoras see De ordine II.20.53–54, ed. Green, 136. 57. See De immortalitate animae IV.6, ed. Hörmann, 107. 58. See De quantitate animae VII.12–XIV.23, ed. Hörmann, 144–159. 59. On the aesthetics of number and harmony in De musica VI, see Du Roy, L’Intelligence, 282–297; Schmitt, “Zahl und Schönheit”; and Hübner, “Hören und Sehen.” On its possible Neopythagorean sources, see Svoboda, L’Esthétique, 67–70; and Pizzani, “Du rapport entre le De musica de S. Augustin.” 60. De musica VI.9.23–24, PL 32: 1176–77. 61. Ibid. VI.17.58, PL 32: 1192–1193. 62. Ibid. VI.4.7, PL 32: 1166.

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63. “delectatione in rationis numeros restituta ad Deum tota vita nostra convertitur, dans corpori numeros sanitatis,” ibid. VI.11.33, PL 32: 1181; trans. Taliaferro, 358. 64. De musica VI.17.56–57, PL 32: 1191–1192. 65. “Unde, quaeso, ista, nisi ab illo summo atque aeterno principatu numerorum et similitudinis et aequalitatis et ordinis veniunt?” Ibid. II.17.57, PL 32:  1192; trans. Taliaferro, 377. 66. De musica VI.11.29, PL 32: 1179; and ibid. VI.13.38, PL 32: 1184. 67. “Aequalitatem illam quam in sensibilibus numeris non reperiebamus certam et manentem, sed tamen adumbratam, et praetereuntem agnoscebamus, nusquam profecto appeteret animus nisi alicubi nota esset: hoc autem alicubi non in spatiis locorum et temporum; nam illa tument, et ista praetereunt.” De musica VI.12.34, PL 32: 1181; trans. Taliaferro, 358–359. 68. Hellgardt, Zum Problem, 190–191. 69. On number in De libero arbitrio, see Horn, “Augustins Philosophie,” 396–400; Hellgardt, Zum Problem, 179–194; Cilleruelo, “Numerus et sapientia”; and Svoboda, L’Esthétique, 91–95. Chronologically De libero arbitrio II probably follows De ordine but precedes De musica VI. As far as Pythagorean background, Du Roy suggests that while Varro and Plotinus are probable sources, the most original passages seem to be Augustine’s own (L’Intelligence, 242–256). 70. De libero arbitrio II.viii.20 (80), ed. Green, 250. 71. Ibid. II.viii.22 (84), ed. Green, 251–252. Solignac links this passage to Nicomachus’s Introduction (“Doxographies et manuels,” 136). 72. On Ecclesiastes 7:26, see De libero arbitrio II.viii.24 (95), ed. Green, 253. On Wisdom 8:1, see ibid. II.xi.30 (124), ed. Green, 258. 73. De libero arbitrio II.x.29 (118), ed. Green, 257. 74. Ibid. II.xv.39 (154), ed. Green, 264. As Horn observes: “die dort vollzogene klare Identifikation der Weisheit mit Christus setzt unausgesprochen auch die Zahl mit ihm in eine enge Verbindung” (“Augustins Philosophie,” 398). Cf. the perceptive remark by Cassirer: “Die mathematische und die religiöse Gewissheit schliessen sich bei Augustin keineswegs aus; sie stehen vielmehr in einem komplementären Verhältnis und gehen ein höchst eigenartiges Bündnis miteinander ein” (“Mathematische Mystik,” 258). On the complex relation between Verbum and sapientia in Augustine’s early theology, see Johnson, “Verbum in the early Augustine.” 75. Horn, “Augustins Philosophie,” 390. 76. “Sed peruellem scire utrum uno aliquo genere contineantur haec duo, sapientia scilicet et numerus, quia coniuncta etiam in scripturis sanctis haec posita esse commemorasti, an alterum existat ab altero aut alterum in altero consistat ueluti numerus a sapientia uel in sapientia. . . . Nam cum incommutabilem ueritatem numerorum mecum ipse considero et eius quasi cubile ac penetrale uel regionem quandam uel si quod aliud nomen aptum inueniri potest quo nominemus quasi habitaculum quoddam sedemque numerorum, longe removeor

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a corpore. Et inueniens fortasse aliquid quod cogitare possim, non tamen aliquid inueniens quod uerbis proferre sufficiam, redeo tamquam lassatus in haec nostra ut loqui possim, et ea quae ante oculos sita sunt dico sicut dici solent. Hoc mihi accidit etiam cum de sapientia, quantum ualeo, uigilantissime atque intentissime cogito.” De libero arbitrio II.xi.30 (120–123), ed. Green, 258; trans. Pontifex, 110–111. 77. De libero arbitrio II.xi.30 (123), ed. Green, 258. 78. Ibid. II.xi.32 (128), ed. Green, 259. 79. Ibid. II.xi.31 (125–126), ed. Green, 258–259. Augustine admits he is tempted to view wisdom as nobler than number, since all can count but few are wise, but then allows that this is a red herring. The wise venerate number not because it is rare but because it is ubiquitous. 80. Ibid. II.xvi.41–42 (163–164), ed. Green, 265–266. 81. Ibid. II.xvi.42 (165–167), ed. Green, 266. 82. “etsi clarum nobis esse non potest utrum in sapientia uel ex sapientia numerus an ipsa sapientia ex numero an in numero sit an utrumque nomen unius rei possit ostendi, illud certe manifestum est utrumque uerum esse et incommutabiliter uerum.” Ibid. II.xi.32 (129), ed. Green, 259; trans. Pontifex, 113 (modified). 83. For a synopsis of the vast literature on Augustine’s Platonist readings, see O’Donnell’s notes in Confessiones, II:421–424. 84. Confessiones VII.9 (13), ed. O’Donnell, 80. 85. I refer to the so-called Quaestio de ideis (De diversis quaestionibus 83, Q. 46), in which Augustine locates the Platonic forms within God. For an exposition of the text, see Gersh, Middle Platonism, 403–413. The doctrine connects Augustine with the Middle Platonists in Chapter 2 and was discussed continuously throughout the Middle Ages. But since the ideas are not then connected to numbers or mathematical forms, it only bears on our theme obliquely. See generally Grabmann, “Des hl. Augustinus Quaestio de ideis”; and De Rijk, “Quaestio de Ideis.” Solignac suggests that the doctrine is linked to Neopythagoreanism, but is unable to confirm it (“Analyse et sources de la Question De Ideis,” 308, 315). Hoenen traces its fate in thirteenth- and fourteenth-century philosophy (see “Propter dicta Augustini”). 86. See Du Roy, L’Intelligence, 450–458. Ultimately Du Roy concludes that the Incarnation was left out of Augustine’s Trinitarian philosophy because of the ongoing influence of Platonism. On this controversial claim, see Madec, La Patrie et la Voie; TeSelle, Augustine the Theologian, 146–156; and Ayres, Augustine and the Trinity, 13–41. 87. See O’Donnell in Confessiones, II:461. See further Van Bavel, Recherches sur la christologie; Rémy, Le Christ médiateur, I:56–80, 96–117 (see notes in II:20–31, 37–47); and Mallard, “Incarnation in Augustine’s Conversion.” 88. Confessiones VII.6 (8–9), ed. O’Donnell, 77–78; trans. Chadwick, 116. Cf. ibid. IV.3 (4–6), ed. O’Donnell, 34–35. For a survey of Augustinian references to

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astrology see O’Donnell in Confessiones, II:210–211. Burton notes that Augustine avoids mentioning arithmetica and geometrica in the Confessiones, opting for ­numerus instead (“Vocabulary of the Liberal Arts,” 156).  89. Confessiones VII.6 (8), ed. O’Donnell, 77.   90. Ibid. VII.9 (14), ed. O’Donnell, 80–81; ibid. VII.18 (24), ed. O’Donnell, 85.   91. “Nec tamen ob hoc mediator est, quia Verbum; maxime quippe immortale et maxime beatum Verbum longe est a mortalibus miseris; sed mediator, per quod homo,” De civitate dei IX.15:50–52, ed. Dombart and Kalb, 263. Cf. ibid. IX.15–17, ed. Dombart and Kalb, 262–266; and ibid. X.20–22, ed. Dombart and Kalb, 294–296. On this theme see further Rémy, Le Christ médiateur, I:422– 436 (see notes in II:158–61).  92. Confessiones VII.19 (25), ed. O’Donnell, 85–86.   93. Ibid. VIII.2 (4), ed. O’Donnell, 90.  94. De genesi ad litteram IV.1–2 (1–6), ed. Zycha, 93–98. The number 6 has three parts (1 + 2 + 3), each of which contributes an integral proportion (respectively, 1 /6, 1/3, and 1/2) to the composition of 6. This pattern does not recur again until the number 28 (1, 2, 4, 7, 14; or 1/28, 1/14, 1/7, 1/4, 1/2). See Agaësse and Solignac, La Genèse au sens littéral, 633–635.  95. De genesi ad litteram IV.7 (14), ed. Zycha, 103. Note that this is not a traditional arithmological association of certain powers or deities to the number, but an aesthetic judgment about specific arithmetical properties of the number (aliquot) as an appropriate symbol for a specific divine activity (creation).   96. Beierwaltes remarks that Augustine’s Neopythagoreanism is always most on display when he is exegeting this verse (“Augustins Interpretation,” 53). See further Roche, “Measure, Number and Weight”; TeSelle, Augustine the Theologian, 116–123; Peri, “Omnia mensura et numero et pondere disposuisti”; and Harrison, “Measure, Number and Weight.”  97. See De genesi ad litteram IV.3 (7), ed. Zycha, 99; cf. ibid. IV.4 (10), ed. Zycha, 101.  98. See Agaësse and Solignac, La Genèse au sens littéral, 635–639; and Horn, “Augustins Philosophie,” 407–410.  99. De genesi ad litteram IV.3–5 (7–11), ed. Zycha, 99–101. 100. Ibid. IV.3 (7), ed. Zycha, 99; cf. ibid. IV.18 (34), ed. Zycha, 116–117. Later Augustine names God the “mensor caeli et numerator siderum et pensor elementorum,” ibid. V.4 (7), ed. Zycha, 48. 101. Ibid. IV.5–7 (11–13), ed. Zycha, 101–103. 102. “Magnum est paucisque concessum excedere omnia, quae metiri possunt, ut videatur mensura sine mensura, excedere omnia, quae numerari possunt, ut videatur numerus sine numero, excedere omnia, quae pendi possunt, ut videatur pondus sine pondero. . . . At haec animorum atque mentium et mensura alia mensura cohibetur, et numerus alio numero formatur, et pondus alio pondere rapitur. Mensura autem sine mensura est, cui aequatur quod de illa est, nec alicunde ipsa est; numerus sine numero est, quo formantur omnia, nec

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formatur ipse; pondus sine pondere est, quo referuntur, ut quiescant, quorum quies purus gaudium est, nec illud iam refertur ad aliud.” Ibid. IV.3–4 (8), ed. Zycha, 99–100; trans. Taylor, 108–109 (modified). Horn compares this passage to Plotinus, Enneads V.5.4 (μέτρον οὐ μετρούμενον) and VI.6.17 (ἀσχημάτιστα ἐκεῖ καὶ πρῶτα σχήματα) (“Augustins Philosophie,” 408 n. 78); for further Neoplatonist parallels see McEvoy, “The Divine as the Measure of Being.” Augustine wrote something similar in another Genesis commentary: “Non enim animalis aliucuius corpus et membra considero, ubi non mensuras et numeros et ordinem inveniam ad unitatem concordiae pertinere. Quae omnia unde veniant non intelligo, nisi a summa mensura et numero et ordine, quae in ipsa Dei sublimitate incommutabili atque aeterna consistunt. . . . In omnibus tamen cum mensuras et numeros et ordinem vides, artificem quaere. Nec alium invenies, nisi ubi summa mensura et summus numerus, et summus ordo est, id est Deum, de quo verissime dictum est, quod omnia in mensura, et numero, et pondere disposuerit.” De genesi adversus Manichaeos XVI.26, PL 34: 185–186. 103. “et, sicut vidit, ita fecit, non praeter se ipsum videns, sed in se ipso ita enumeravit omnia, quae fecit.” De genesi ad litteram V.15 (33), ed. Zycha, 158; trans. Taylor, 166. 104. See Taylor in Augustine, Literal Meaning, I:247–249, citing Plotinus, Enneads V.1.7. 105. See De genesi ad litteram V.13 (29); ed. Zycha, 156. Cf. ibid. V.5 (12–14), ed. Zycha, 146; ibid. V.7 (22), ed. Zycha, 151; and ibid. V.11 (27), ed. Zycha, 154–55. 106. See ibid. VI.5–6 (7–10), ed. Zycha, 175–77; and ibid. VI.14 (25), ed. Zycha, 189. On human generation, see especially ibid. VI.5 (8), ed. Zycha, 176; ibid. VI.9 (16), ed. Zycha, 182; and ibid. VII.22–24 (32–35), ed. Zycha, 219–223. 107. Ibid. V.12 (28), ed. Zycha, 155–156. 108. “Sed haec aliter in verbo dei, ubi ista non facta, sed aeterna sunt, aliter in elementis mundi, ubi omnia simul facta futura sunt, aliter in rebus, quae secundum causas simul creatas, non iam simul sed suo quaeque tempore creantur:  . . . aliter in seminibus, in quibus rursus quasi primordiales causae repetuntur de rebus ductae, quae secundum causas, quas primum condidit, extiterunt,” ibid. VI.10 (17), ed. Zycha, 182; trans. Taylor, 189. 109. On this passage, see Boyer, “La théorie augustinienne”; and especially Agaësse and Solignac, La Genèse au sens littéral, 653–668. Two other good studies of rationes seminales are Thonnard, “Les raisons séminales”; and Brady, “St. Augustine’s Theory.” 110. De genesi ad litteram VI.11 (18), ed. Zycha, 183–184. 111. See Colish, Stoic Tradition, I:32, II:203–204; Hahm, Origins of Stoic Cosmology, 60–76; and especially Heinze, Die Lehre vom Logos, 107–125. 112. Horn, “Augustins Philosophie,” 404. In De musica II.17.57, PL 32: 1192, for example, God is said to create creatures ex nihilo insofar as God generates rational numbers that in turn generate sensible numbers.

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113. See De genesi ad litteram V.5 (14), ed. Zycha, 146; and ibid. IV.33 (52), ed. Zycha, 132. 114. Brady suggests that rationes seminales be conceived as secondary causes that are efficient, final, and formal, but neither material nor exemplary (“St. Augustine’s Theory,” 156–157). 115. “Eadem tribus aeternitas, eadem incommutabilitas, eadem maiestas, eadem potestas. In patre unitas, in filio aequalitas, in spiritu sancto unitatis aequalitatisque concordia, et tria haec unum omnia propter patrem, aequalia omnia propter filium, conexa omnia propter spiritum sanctum.” Augustine, De doctrina christiana I.12 (V.5), ed. Green, 16–17; trans. Green, 10. 116. See De doctrina christiana I.13–14 (VI.6), ed. Green, 16–19. 117. “Deus est monos, monadem ex se gignens, in se unum reflectens ardorem. . . . Sic quidem etiam in multis: unaquaeque unitas proprium habet numerum quia super diversum ab aliis reflectitur.” Liber XXIV Philosophorum I, ed. Hudry, 150. Until recently the Liber had been dated to the twelfth century, but as Hudry has argued, it was most likely composed by Marius Victorinus, judging from similar terms he used in Adversus Arium and other works. Hudry includes Porphyry’s Vita Pythagorae among Victorinus’s major sources (see Le Livre des XXIV Philosophes, 24–29). Turner has shown that in Adversus Arium Victorinus was influenced by an anonymous Parmenides commentary close to the milieu of Moderatus, in ways parallel to the Platonizing Sethian Gnostic treatises Zostrianos and Allogenes. See Turner, “Victorinus, Parmenides Commentaries,” 59–62, 72–75. Cf. Turner, “Platonizing Sethian Treatises”; and Turner, Sethian Gnosticism, 736–744. In this connection see also Whittaker, “Historical Background of Proclus’ Doctrine.” 118. See Courcelle, Late Latin Writers, 77 n. 140. 119. Unfortunately O’Meara’s monograph on Porphyry and Augustine does not discuss the passage in question (see Porphyry’s Philosophy from Oracles). Hadot shows that Moderatus’s Logos doctrine directly influenced Porphyry’s henology (Porphyre et Victorinus, Vol. 1, 311–312). Yet to my knowledge Hadot never identified a link between the arithmetical Trinity from De doctrina christiana and Victorinus’s knowledge of the anonymous Parmenides commentary (see Porphyre et Victorinus, vol. 1, 283–312, 475–478; and also Beierwaltes, “Trinität”). Hadot still attributed this commentary, the famous Turin palimpsest, to Porphyry; this has since been challenged and the commentary more closely studied (see the edition by Bechtle, Anonymous Commentary). For a summary of these developments see Turner, Sethian Gnosticism, 396–405. 120. As Horn remarks: “die naheliegende Erwartung, Augustinus würde in breitem Umfang das Verhältnis der Trinität zum Begriff der Zahl diskutieren, bestätigt sich nicht” (“Augustins Philosophie,” 411). On the difficulty of dating De trinitate, see Ayres, Augustine and the Trinity, 118–120. 121. See, e.g., De trinitate III.viii (16), ed. Mountain, 143; and ibid. XI.xi (18), ed. Mountain, 355. However, in Book IV he briefly returns to the Neopythagoreanism

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of De genesi ad litteram in an unusual Christological passage. Christ’s mediation was a kind of harmony of human and divine, represented by the perfect number 6, a harmony that draws us out of the many into the divine One. See De trinitate IV.ii–ix (4–12) passim, ed. Mountain, 163–178. See further Scully, “De Musica as the Guide to Understanding Augustine’s Trinitarian Numerology”; and Rémy, Le Christ médiateur, I:339–370 (see notes in II:132–146). 122. On the question of Boethius’s education at Alexandria, see Guillaumin in Boethius, Institution Arithmétique, xxii–xxv. Cf. Courcelle, Late Latin Writers, 282–295; and De Vogel, “Boethiana I.” On the separate issue of his knowledge of Porphyry, Iamblichus, Syrianus, and Proclus, see Obertello, Severino Boezio, vol. 1, 476–521. Guillaumin has aptly termed Boethius’s efforts in the Institutio neither a strict translation nor a commentary but “une sorte de paraphrase éclairée” (“La transformation d’une phrase,” 869). Robbins notes all the minor deviations in detail in Nicomachus, trans. D’Ooge et al., 132–137; cf. Obertello, Severino Boezio, vol. 1, 451–475. The best introduction in English to the mathematics of Institutio arithmetica is Guillaumin, “Boethius’s De institutione arithmetica.” On Institutio musica, see Bower, “Boethius and Nicomachus”; and Heilmann, Boethius’ Musiktheorie und das Quadrivium. 123. See Bakhouche, “Boèce et le Timée.” 124. On the life of Boethius, see Opuscula Sacra, ed. Galonnier, vol. 1, 31–119; Kaylor, “Introduction:  Times, Life, and Work of Boethius”; and Obertello, Severino Boezio, vol. 1, 3–153. General bibliographies can be found in Obertello, Severino Boezio, vol. 2; and Phillips, “Anicius Manlius Severinus Boethius.” 125. See the famous Neoplatonist hymn at Consolatio philosophiae III.9, ed. Moreschini, 79:1–80:28, but also ibid. I.5, ed. Moreschini, 18:1–20:48, and ibid. II.8, ed. Moreschini, 55:1–56:30. Cf. Gruber, Kommentar, 277–290. On the apparent absence of Christianity in the Consolatio, see the discussion by De Vogel, “Boethiana II.” On the missing quadrivium, see the interesting proposal by Fournier, “Boethius and the Consolation.” 126. See Evans, “Influence of Quadrivium Studies,” 152:  “In Boethius’ ‘theological’ treatises, a number of mathematical issues are shown to be relevant to the solution of the problem of the Trinity. . . . It is in [these] textbooks that we find traces of mathematical influence which find their way into the commentaries and beyond; in works apparently on quite different subjects, mathematical discussion occurs.” 127. IA I.1.1–4 and I.1.8–12, ed. Guillaumin, 6–10; cf. Institutio musica II.2–3, ed. Friedlein, 227–229. On the principles informing the order of the sciences in Nicomachus and Boethius as compared to others, see Guillaumin, “L’ordre des sciences du Quadriuium”; Hübner, Begriffe “Astrologie” und “Astronomie” in der Antike, 49–66; and Kühnert, “Zur Reihenfolge der artes.” 128. IA Praef. 4, ed. Guillaumin, 3. 129. See the excellent analysis in Bernard, “Zur Begründung der mathematischen Wissenschaften.” See further Navari, whose brief account of the quadrivium’s

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structure is the most lucid and poetic that I have found (see “Leitmotiv in the Mathematical Thought of Gerbert of Aurillac,” 140–142). 130. “Inter omnes priscae auctoritatis uiros qui Pythagora duce puriore mentis ratione uiguerunt, constare manifestum est haud quemquam in philosophiae disciplinis ad cumulum perfectionis euadere, nisi cui talis prudentiae nobilitas quodam quasi quadruuio uestigatur; quod recte intuentis sollertiam non latebit. Est enim sapientia rerum quae sunt suique immutabilem substantiam sortiuntur comprehensio veritatis. Esse autem illa dicimus quae nec intentione crescunt nec retractione minuuntur nec variationibus permutantur, sed in propria semper ui suae se naturae subsidiis nixa custodiunt.” IA I.1.1, ed. Guillaumin, 6; trans. Masi, Boethian Number Theory, 71 (modified per Guillaumin) = Nicomachus I.1.1. 131. The same theological and moral dimensions inform his definition of philosophy in a subsequent commentary on Porphyry: “uideatur studium sapientiae studium diuinitatis et purae mentis illius amicitia. Haec igitur sapientia cuncto equidem animarum generi meritum suae diuinitatis inponit et ad propriam naturae uim puritatemque reducit.” In Isagogen I.3, ed. Brandt, 7. 132. Guillaumin, “Le term quadrivium de Boèce,” 147–148. On the ethical dimensions of arithmetic, see IA I.19.8–9, ed. Guillaumin, 41–42; IA I.21.3–4, ed. Guillaumin, 46; IA I.32.1–2, ed. Guillaumin, 66–67; and IA II.45.1, ed. Guillaumin, 153. The references to ethics in Institutio musica are more numerous and unexceptional, given the nature of the art: “Unde fit ut, cum sint quattuor matheseos disciplinae, ceterae quidem in investigatione veritatis laborent, musica vero non modo speculationi verum etiam moralitati coniuncta sit. . . . Nulla enim magis ad animum disciplinis via quam auribus patet. Cum ergo per eas rythmi modique ad animum usque descenderint, dubitari non potest, quin aequo modo mentem atque ipsa sunt afficiant atque conforment.” Institutio musica I.1, ed. Friedlein, 179–181. 133. “Hoc igitur illud quadruuium est quo his uiandum sit quibus excellentior animus a nobiscum procreatis sensibus ad intellegentiae certiora perducitur.” IA I.1.7, ed. Guillaumin, 8; trans. Masi, 73 (modified). See Republic 527de. 134. “secundum pythagoricam disciplinam,” IA I.3.1, ed. Guillaumin 12. Cf. Cusanus’s phrase Pythagoricam inquisitionem at DI I.9 (26), 36. 135. “Quibus quattuor partibus si careat inquisitor, uerum inuenire non possit. . . . Quod haec qui spernit, id est has semitas sapientiae, ei denuntio non recte philosophandum . . . quam in his spernendis ante contempserit.” IA I.1.5, ed. Guillaumin, 7; trans. Masi, 72–73 (modified). 136. On this problem see Guillaumin, “Le Statut des mathématiques chez Boèce.” 137. “Omnia quaecumque a primaeua rerum natura constructa sunt numerorum uidentur ratione formata. Hoc enim fuit principale in animo conditoris exemplar.” IA I.2.1, ed. Guillaumin, 11; trans. Masi, 75–76 (modified) = Nicomachus I.6.1. 138. “hanc [arithmetica] ille huius mundanae molis conditor deus primam suae habuit ratiocinationis exemplar et ad hanc cuncta constituit quaecumque

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fabricante ratione per numeros adsignati ordinis inuenere concordiam,” IA I.1.8, ed. Guillaumin, 8–9; trans. Masi, 74 = Nicomachus I.4.1–2. Cf. IA II.32.1, ed. Guillaumin, 129 = Nicomachus II.19.1: “ut non sine causa hoc in omnibus rebus ab numeri specie natura rerum sumpsisse uideatur.” 139. See Pizzani, “Il quadrivium boeziao e i suoi problemi.” 140. IA I.14.2–4, ed. Guillaumin, 31–32  =  Nicomachus I.11.3. Cf. IA I.7.6, ed. Guillaumin, 16. 141. IA I.9.10, ed. Guillaumin, 19 (no parallel, but cf. Nicomachus I.8.12–13). The categories structuring the quadrivium “in propria semper ui suae se naturae subsidiis nixa custodiunt” (IA I.1.1, ed. Guillaumin, 6). Cf. the discussion of the two proportions that preserve harmony at Institutio musica I.6, ed. Friedlein, 193. On “divine constancy,” see further IA I.20.8–9, ed. Guillaumin, 45 = Nicomachus I.16.8–9; and IA I.9.9, ed. Guillaumin, 19 = Nicomachus I.8.12. 142. See IA I.2.2, ed. Guillaumin, 11  =  Nicomachus I.6.2; and IA II.30.2–4, ed. Guillaumin, 124 = Nicomachus II.17.7. 143. See IA I.3.2, ed. Guillaumin, 12  =  Nicomachus I.7.1; and IA I.2.4, ed. Guillaumin, 12 = Nicomachus I.6.4. On the three definitions of number, see Guillaumin in Institution Arithmétique, 186 n. 44. On even and odd, see IA II.27.2–II.28.4, ed. Guillaumin, 119–121 = Nicomachus II.17.1–5 passim; as well as IA II.36.1, ed. Guillaumin, 135–136 = Nicomachus II.20.3. A more expansive philosophical account of the harmonies of even and odd as arithmetical reflections of the cosmogony of the Timaeus occurs in IA II.31–32, ed. Guillaumin, 125–129 = Nicomachus II.18–19. 144. See IA I.32.1–2, ed. Guillaumin, 66–67 = Nicomachus I.23.4–6; and IA II.1.1, ed. Guillaumin, 78 = Nicomachus II.1.1. Cf. Institutio musica II.7, ed. Friedlein, 232; and ibid., II.20, ed. Friedlein, 253. On the Neopythagorean valence of radix, see Guillaumin, “Boèce traducteur de Nicomaque.” 145. See IA I.21.3, ed. Guillaumin, 46  =  Nicomachus I.17.4; and IA II.1.9, ed. Guillaumin, 81 = Nicomachus II.2.2 146. See IA II.40–42, ed. Guillaumin, 140–143  =  Nicomachus II.21–22. Cf. Guillaumin’s important clarifications at Institution Arithmétique, 215–219, nn. 103, 106, 114. 147. See IA II.4.2, ed. Guillaumin, 88 = Nicomachus II.6.1. 148. See IA II.4.4–6, ed. Guillaumin, 89–90 = Nicomachus II.6.3–4. 149. See generally Opuscula Sacra, trans. Galonnier, vol. 2, 23–123. 150. See De trinitate II, ed. Moreschini, 168:68–169:80. 151. Mathematics “investigates forms of bodies apart from matter, and therefore apart from motion” (formas corporum speculatur sine materia ac per hoc sine motu), nevertheless, these “forms, however, being connected with matter cannot be really separated from bodies” (quae formae cum in materia sint, ab his separari non possunt). De trinitate II, ed. Moreschini, 169:74–76. See further Merlan, From Platonism to Neoplatonism, 71–77.

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152. See De trinitate II, ed. Moreschini, 170:110–171:117. 153. See ibid. II, ed. Moreschini, 170:102–104, 171:118–120. Gersh is right to register the distinction of this Boethian doctrine: “De Arithmetica, De Trinitate, De Consolatione Philosophiae, and other works describe God as form, thereby departing from the fundamental tenet of Plotinus’ writings that God transcends the level of form” (Middle Platonism, 689). 154. See De trinitate III, ed. Moreschini, 171:132–134, 172:138–141. Cf. Aristotle, Physics 219b5–219b9. 155. See De trinitate VI, ed. Moreschini, 179:333–180:341. 156. See generally Opuscula Sacra, ed. Galonnier, vol. 1, 287–348. 157. Pessin raises the fascinating possibility that what Boethius calls “my hebdomads” in his preface represents an esoteric allusion to Nicomachus’s Theology (see “Hebdomads: Boethius Meets the Neopythagoreans”). The heptad in that text symbolizes the demiurgic creation of the world through the divine “limb” (ἄρθρον)—implicitly comparing the “flow” (χύμα) of number to seminal fluid (σπέρμα) (see Nicomachus of Gerasa, Theologoumena arithmeticae VII, ed. De Falco, 57; trans. Waterfield, 89). As Pessin notes, this would not only explain the Roman philosopher’s circumspection in revealing his source, but would also point toward a possible provenance of the divine fluere that resolves the philosophical quandary of De hebdomadibus, namely Nicomachean number theory. Solère argues that “hebdomad” most likely stems from Proclus, but also does not exclude Nicomachus altogether (“Bien, cercles et hebdomades,” 91–105; see esp. 103 n. 235). 158. “Ut igitur in mathematica fieri solet ceterisque etiam disciplinis, praeposui terminos regulasque quibus cuncta quae sequuntur efficiam.” Quomodo substantiae, Praefatio, ed. Moreschini, 187:14–16; trans. Stewart et  al., 39–41 (modified). 159. On forma essendi, see Quomodo substantiae, Regula II, ed. Moreschini, 187:26–28. See Schrimpf, Axiomenschrift des Boethius, 13–28; and Micaelli, “De Hebdomadibus di Boezio.” Contra Schrimpf, Hadot contends that forma essendi simply denotes the property or quality of beingness (essentialité), not a transcendent bestowal of being (see “Forma essendi”). On fluere, cf. Quomodo substantiae, ed. Moreschini, 190:88, 191:110–117, 192:134–137. Galonnier proposes that Boethius’s fluere may ultimately stem from Marius Victorinus’s anti-Arian works (see Opuscula sacra, vol. 1, 289–292, 345–348). De Libera highlights its legacy in the philosophy of Albert the Great: “Both in his problem and in his solution [he] effects a radical displacement of Aristotelian ontology by introducing the Christian-Platonic idea of a procession, a flux—fluxus, defluere—of existing things, the ‘second goods’, participating in the will of the first Good. . . . This Boethian concept of fluxus is the primary foundation of Albert’s metaphysics of ‘flux’; it has unfortunately been neglected by historians of Albert’s thought. Yet it remains of capital importance” (Métaphysique et noétique, 147–148).

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160. On arithmetic and astronomy in Macrobius, see Stahl, Roman Science, 151–169; Flamant, Macrobe et le Néo-Platonisme latin, 305–482; Gersh, Middle Platonism, 547–571; and Robbins, “Tradition of Greek Arithmology,” 97–123. On sources of Macrobian arithmology, see Robbins, “Posidonius”; and Courcelle, Late Latin Writers, 36. 161. On the meaning of the quadrivium in Martianus Capella, see Stahl, Roman Science, 170–190; Stahl, “Quadrivium of Martianus Capella”; and Stahl, Martianus Capella. 162. For an overview, see Stahl, Roman Science, 202–223. For comparisons of Boethius’s selections from Nicomachus to those of Cassiodorus and Isidore, see Karpinski in Nicomachus, trans. D’Ooge et al., 138–145; and Courcelle, Late Latin Writers, 347–351. 163. For a survey of the medieval legacy of Boethian philosophy across several domains, see Gersh, Middle Platonism, 647–651. On the Consolatio in particular, see Patch, Tradition of Boethius; Courcelle, La Consolation de Philosophie; Nauta, “Some Aspects”; and Nauta, “ ‘Magis sit Platonicus quam Aristotelicus’.” 164. See Klinkenberg, “Verfall des Quadriviums,” 6, 32; cf. Reindel, “Beginn des Quadriviums.” 165. See Englisch, Artes Liberales, 12–19, 471–478. Englisch focuses exclusively on the quadrivium, comparing Macrobius, Martianus Capella, Cassiodore, Isidore of Seville, Bede, and Hrabanus Maurus, and yet skips over the important case of Boethius. 166. Kühnert discusses the evolving conceptions of ἐγκύκλιος παιδεία from Presocratic Sophists through Varro, Augustine, and Martianus Capella (see Allgemeinbildung und Fachbildung), and Gemeinhardt provides an excellent social history of the Christian transformation of Roman educational models from the second to sixth centuries but focuses on the trivium (see Das lateinischen Christentum). See further Marrou, “Les arts libéraux dans l’antiquité classique”; and Hadot, Arts libéraux et philosophie, 263–293. 167. For the following section, see the surveys by Beaujouan, “Transformation of the Quadrivium”; North, “Quadrivium”; and Moyer, “Quadrivium and the Decline of Boethian Influence.” 168. See Beaujouan, “Transformation of the Quadrivium,” 467–470. Sometimes the new algorism and arithmology collided: see the fascinating study by Lampe, “Twelfth-Century Text on the Number Nine.” 169. See White, “Boethius in the Medieval Quadrivium”; and Navari, “Leitmotiv in the Mathematical Thought of Gerbert of Aurillac.” On early medieval arithmologies, see Großmann, “Studien zur Zahlensymbolik.” 170. See Illmer, “Arithmetik in der gelehrten Arbeitweise”; Evans, “Influence of Quadrivium Studies”; Masi, “Influence of Boethius’ De Arithmetica”; and Kibre, “Boethian De Institutione Arithmetica.” Evans has analyzed a distinctive Victorine commentary (see “Commentary on Boethius’s Arithmetica”), and Thierry of Chartres probably also wrote a commentary on Institutio arithmetica, as I will discuss in Chapter 4.

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171. See Didascalicon II.3–15, trans. Taylor, 63–71. 172. See Beaujouan, “L’enseignement du ‘quadrivium’.” On rithmomachia see further Borst, “Rithmimachie und Musiktheorie”; Folkerts, “ ‘Rithmomachia’, a Mathematical Game”; and Moyer, Philosophers’ Game. 173. On the place of musica in medieval quadrivial studies, see Münxelhaus, Pythagoras musicus; Hirtler, “Musica im Übergang”; and Speer, “Scientia Quadruvii.” 174. On the medieval legacy of Institutio musica, see Pizzani, “Fortune of the De Institutione Musica”; Bower, “Role of Boethius’ De Institutione Musica”; and Burnett, “Adelard, Music and the Quadrivium.” Heller-Roazen has argued that Boethian (Pythagorean) conceptions of harmony continued to influence early modern science and philosophy through Kepler, Leibniz, and Kant (see Fifth Hammer). 175. See Ullman, “Geometry in the Mediaeval Quadrivium”; Evans, “ ‘Sub-Euclidean’ Geometry of the Earlier Middle Ages”; Folkerts, “Importance of the Pseudo-Boethian Geometria”; and Zaitsev, “Meaning of Early Medieval Geometry.” 176. Beaujouan, “Transformation of the Quadrivium,” 471–481. See further Hübner, Begriffe “Astrologie” und “Astronomie” in der Antike. 177. Beaujouan, “Transformation of the Quadrivium,” 465. Pedersen summarizes the major scientific developments of the twelfth to fourteenth centuries (see “Du Quadrivium à la Physique”), and Newsome surveys paradigmatic transformations of the quadrivium in Nicole Oresme, Prosdocimo de’ Beldomandi and Leon Battista Alberti (see “Quadrivial Pursuits”). 178. See D’Onofrio, “Dialectic and Theology”; Gibson, “Opuscula Sacra in the Middle Ages,” 215–220; and Schrimpf, Werk des Johannes Scottus Eriugena. 179. On the hypothetical influence of Eriugena on the school of Chartres, see Lucentini, Platonismo medievale; Jeauneau, “Le renouveau érigénien du XIIe siècle”; and Kijewska, “Mathematics as a Preparation for Theology.” 180. See Duchez, “Jean Scot Érigène, premier lecteur.” 181. See McEvoy, “Biblical and Platonic Measure.” Cf. Eriugena, Periphyseon III (633B), ed. Sheldon-Williams, 58. 182. See Gersh, From Iamblichus to Eriugena, 308–312. 183. See Jeauneau, “Jean Scot et la Métaphysique des Nombres.” 184. See Eriugena, Periphyseon III (659A–661C), ed. Sheldon-Williams, 118–124. 185. See Beierwaltes, “Harmonie-Gedanke.” On a more ambitious scale, cf. Gersh, Concord in Discourse. 186. O’Meara, “Metaphysical Use of Mathematical Concepts,” 145–147; cf. Gersh, “Eriugena’s Fourfold Contemplation.”

chapt er 4   1. See Brague, Eccentric Culture, 32, 40, 122.   2. See Chenu, La théologie, 142–158. The notion of a “twelfth-century renaissance” gained currency with the eponymous book by Charles Homer Haskins in 1927.

318

 3.   4.   5.

  6.   7.

  8.   9. 10.

11.

12. 13.

14. 15.

16.

17.

Notes Some valuable recent surveys are Van Engen, “Twelfth Century”; Benson and Constable, Renaissance and Renewal; and Swanson, Twelfth-Century Renaissance (see esp. 215–229). Chenu, La théologie, 157–158. See ibid., 309–322. See Speer, “Hidden Heritage,” 166, 171. As Speer notes, Chenu’s anachronism continues to inform paradigms for medieval philosophy: see, e.g., Honnefelder, “Der zweite Anfang der Metaphysik”; and Lutz-Bachmann, “Metaphysik und Theologie.” See Evans, “Discussions of the Scientific Status of Theology”; Evans, “Boethian and Euclidean Axiomatic Method”; and Dreyer, More mathematicorum. See Speer, “ ‘Erwachen der Metaphysik’,” 34–35. Moreover, as Speer remarks elsewhere, “the thirteenth-century interest in Aristotle arises from a particular intellectual impasse of the twelfth century; it is in this sense that the roots of the reception of Aristotle lie in the twelfth century” (“Discovery of Nature,” 137). Speer, “Hidden Heritage,” 165. See Aquinas, Expositio super librum Boethii De trinitate, q. 5, art. 4, ed. Decker, 190ff. See Speer, “Hidden Heritage,” 174, 178. For this reason De Libera concludes that the Boethian tradition stands uniquely outside of Heidegger’s critique of metaphysics (Métaphysique et noétique, 103–106, 144–148). On Thierry’s life in the schools, see Häring, “Chartres and Paris Revisited,” 279–295; Ward, “Date of the Commentary”; Widmer, “Thierry von Chartres”; and Stollenwerk, “Genesiskommentar,” 3–22. Häring, “Chartres and Paris Revisited,” 285. On Abelard’s connections to Thierry, see Gregory, “Abélard et Platon”; Mews, “In Search of a Name”; and Ziomkowski, “Science, Theology and Myth,” 215–226. See Ward, “Date of the Commentary,” 236–240; and Häring, “Chartres and Paris Revisited,” 281–284. See Ward, “Date of the Commentary,” 238–239; and Burnett, “Latin and Arabic Sources,” 63–65. On Thierry’s mathematics lectures, see Mews, “In Search of a Name,” 183–187. I  will not treat Thierry’s other contributions to rhetoric and logic. On rhetoric, see Fredborg, “Commentary of Thierry of Chartres”; and Fredborg, Latin Rhetorical Commentaries. On logic, see Rodrigues, “Pluralité et particularisme ontologique”; and Gracia, “Theory of Individuation.” “Nos autem non nostra sed precipuorum super his artibus inuenta doctorum quasi in unum corpus uoluminis apta modulatione coaptauimus et triuium quadruuio ad generose nationis phylosophorum propaginem quasi maritali federe copulauimus.” MS Chartres 497, fol. 2ra, cited in Jeauneau, “Prologus

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in Eptatheucon,” 174. The prologue is translated in Gersh, Concord in Discourse, 221–222. On the contents of the Heptateucon, see Evans, “Uncompleted Heptateuch of Thierry of Chartres”; and Burnett, “Contents and Affiliation of the Scientific Manuscripts.” 18. Jeauneau, Rethinking the School of Chartres, 67. As an erudite nonspecialist, John’s quadrivial education conveys the state of the art in the mid-twelfth century. See Evans, “John of Salisbury and Boethius on Arithmetic.” 19. “Nam cum sint duo precipua phylosophandi instrumenta, intellectus eiusque interpretatio, intellectum autem quadruuium illuminet, eius uero interpretationem elegantem, rationabilem, ornatam triuium subministret, manifestum est eptateuchon totius phylosophye unicum ac singulare esse instrumentum.” MS Chartres 497, fol. 2ra, cited in Jeauneau, “Prologus in Eptatheucon,” 174. 20. “quia mathematicam solebant prius antiqui discere ut ad diuinitatis intelligentiam possent peruenire.” Commentum II.15, H 73. 21. See Ziomkowski, “Science, Theology and Myth,” 176–193. 22. Chenu, La théologie, 146. 23. See, e.g., Chenu, “Une définition Pythagoricienne”; Jeauneau, “Mathématiques et Trinité”; and Rodrigues, “Thierry de Chartres, lecteur du De trinitate.” 24. Jeauneau calls Macrobius, Boethius, Calcidius, and Martianus Capella “les quatre maîtres-piliers sur lesquels s’appuie le temple de la sagesse chartraine.” Even when they offered little more than arithmologies repeated elsewhere, they provided “excellentes occasions de pythagoriser” (“Macrobe, source de platonisme chartrain,” 283, 286). 25. Some recent surveys beyond those already cited include Stiefel, Intellectual Revolution; Otten, From Paradise to Paradigm; Bezner, Vela Veritatis; and Ellard, Sacred Cosmos. 26. See Speer, Die entdeckte Natur; and Speer, “Discovery of Nature.” 27. See Swanson, Twelfth-Century Renaissance. 28. Southern, Medieval Humanism. 29. On the controversies of the cathedral schools see Ferruolo, Origins of the University; Fichtenau, Heretics and Scholars; and Jaeger, Envy of Angels. On competing theological developments, see the fine surveys by Mews, “Philosophy and Theology, 1100–1150”; and Gemeinhardt, “Logic, Tradition, and Ecumenics.” 30. See Chenu, La théologie, 314:  “Il ne s’agit plus seulement de référer à Dieu, par voie dialectique ou symbolique, les réalités créées, dont la finalité suprême dévaluerait le contenu, le comportement, l’usage terrestres; il y a de ce monde et de l’homme une connaissance autonome, valable en son ordre, efficace en vérité de spéculation et d’action, laquelle est transférable en science théologique.” Gregory likewise places the theme of “natura autonoma” at the center of the school of Chartres (“Il Timeo e i problemi del platonismo medievale,” 54–55). Cf. Rodrigues, “La conception de la philosophie,” 134: “La philosophie serait donc cet effort de l’esprit qui, par des seuls moyens rationnnels, aspire à connaître,

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non pas l’absolu—qui, d’ailleurs, est donné d’avance en tant que vérité de foi— mais, l’expression de cet absolu qu’est le monde comme ordre et comme harmonie, c’est-à-dire, comme explicatio de l’unité divine.” 31. Gregory, “La nouvelle idée de nature,” 195–196. This conception of creation differed significantly from Augustine’s rationes seminales, as Gregory shows regarding William of Conches and Thierry (see “L’Idea della natura”; and Anima mundi, 177–183). 32. Delp aptly describes this as the twelfth century’s “fascination with the expansiveness and physicality of pagan attempts to bridge the gap between an immaterial summum bonum and the material cosmos by means of hierarchical orders of hypostases, divine and quasi-divine” (“Immanence of Ratio,” 63). Two classic surveys of the sources are Jeauneau, “L’Héritage de la philosophie antique”; and Gregory, “Platonic Inheritance.” 33. See Asclepius I, 6a–11a, ed. Scott, 295–307. 34. See McGinn, “Role of the Anima Mundi as Mediator”; and Ott, “Platonische Weltseele.” 35. Southern rightly criticized older depictions of a unified cathedral school located at Chartres (cf. e.g., Klibansky, “School of Chartres”) for an imprecise romanticism (see his essay, “Humanism and the School of Chartres,” in Medieval Humanism, 61–85). But Southern’s attempt to convert his corrections into a broader attack on the novelty or coherence of the Chartrian school of thought has provoked several rebuttals. See Dronke, “New Approaches to the School of Chartres”; Giacone, “Masters, Books and Library at Chartres”; and especially Häring, “Chartres and Paris Revisited.” Southern’s reply has two parts. His focus on the institutional state of cathedral schools and the nature of academic networks in the twelfth century is very illuminating (see “Schools of Paris”), but his attempt to dismantle “Chartrian Platonism” is vaguely argued and beside the point (see Platonism). 36. Colish, Peter Lombard, vol. 1, 254. 37. Southern criticizes Wetherbee for defining the Chartrian school too vaguely (see Platonism, 4, 15). My formulation aims to update and expand Wetherbee’s claim that a certain “poetic intuition” unifies the cultural efforts of the leading Chartrians (Platonism and Poetry, 4). I have especially benefited from the important study by McGinn, “Role of the Anima Mundi as Mediator.” 38. On Alan as student of Thierry, see Lemoine, “Alain de Lille”; and Albertson, “Achard of St. Victor,” 120–128. On Hermann’s background, see Burnett in Hermann, De essentiis, ed. Burnett, 16–25; and Burnett, “Blend of Latin and Arabic Sources,” 63–65. 39. John of Salisbury states that William of Conches, Richard the Bishop, and Gilbert of Poitiers were all students of Bernard of Chartres (see Metalogicon I.24, ed. Webb, 57–58; and ibid. II.17, ed. Webb, 94–95). Dutton dates Gilbert’s tutelage to 1112–1114 and William’s to 1115–1124, before Bernard’s death around

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1126 (in Bernard, Glosae super Platonem, 29). If Thierry hailed from Brittany but was associated with Chartres, and if the Genesis commentary was written in the 1120s or 1130s, then it is entirely conceivable that Thierry studied under Bernard in the 1120s before his death, pace Dutton (ibid., 42 n. 133), who produces little evidence to the contrary. The fact that Bernard and Thierry were not brothers, as Clerval maintained decades ago, does not mean that they had no interactions at all. Häring contends that while there is no hard evidence that Thierry studied at Chartres, the fact that he was invited by church authorities to become chancellor in 1141, and left Paris to do so, strongly suggests that he had previous connections with the school (“Chartres and Paris Revisited,” 290–292). 40. Dutton has provided substantial evidence for attributing the work to Bernard of Chartres (see “Uncovering of the Glosae Super Platonem”; and Dutton in Bernard, Glosae super Platonem, 8–21), but Dronke has raised some concerns (see “Introduction,” 14–17). 41. See Dutton in Bernard, Glosae super Platonem, 18, 48–50. 42. See Somfai, “Eleventh-Century Shift”; Gibson, “Study of the ‘Timaeus’ in the Eleventh and Twelfth Centuries”; and Ziomkowski, “Science, Theology and Myth,” 33–87. Somfai points ahead to Cusanus (but neglects to mention his Chartrian sources) when she notes that “it was not until the Renaissance that a more theoretical and theological mathematics drew attention once again to the mathematical aspects of the dialogue and the Commentary” (“Eleventh-Century Shift,” 21). 43. Dutton in Bernard, Glosae super Platonem, 6–7, 69–70. See further Ziomkowski, Manegold of Lautenbach. 44. See Speer, Die entdeckte Natur, 92, 112–113; cf. Schrimpf, “Bernhard von Chartres,” 191–192, 203–206. 45. See Speer, Die entdeckte Natur, 109–111. 46. See Dutton in Bernard, Glosae super Platonem, 76–78. 47. “Item quidam philosophi dicunt Platonem intellexisse per indiuiduam substantiam, ideas; per diuiduam, hylen; per haec duo mixta, natiuas formas. . . . Natiuae enim ideis similes sunt, quia ex earum similitudine in substantia processerunt.” Glosae super Platonem 5:67–70, ed. Dutton, 175. On the immediate context of the Timaeus, see Annala, “Function of the formae nativae.” 48. See Dutton in Bernard, Glosae super Platonem, 78; and Schrimpf, “Bernhard von Chartres,” 200–201. 49. See Speer, Die entdeckte Natur, 105–108. Cf. Calcidius, Timaeus 297–299, ed. Waszink, 299–301. 50. See Speer, Die entdeckte Natur, 120–121. 51. “Vnde seruata omnium artium fere ratione, hoc opus non rudibus, sed in quadruuio promotis elaboratum est, ut si quae quaestiones de musica et aliis oriuntur, domesticis rationibus, scilicet musicis, arithmeticis, et ceteris, sopiantur.” Glosae super Platonem 1:60–63, ed. Dutton, 141. “Et per haec omnia scientia

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quadruuii intelligitur, in quo est perfectio scientiae per numeros: arithmetica per hoc quod lineares, superficiales, cubici sunt numeri; geometria per consonantias proportionaliter notatas; musica et astronomia, in qua de musico concentu sperarum agitur.” Glosae super Platonem 5:143–147, ed. Dutton, 178–179. See further Schrimpf, “Bernhard von Chartres,” 186–187. 52. See Speer, Die entdeckte Natur, 91, 122–126; cf. Schrimpf, “Bernhard von Chartres,” 209. 53. Schrimpf, “Bernhard von Chartres,” 208. 54. Speer, Die entdeckte Natur, 293–294. 55. On this point Dutton poses a false dichotomy between Bernard’s use of either Calcidius or Boethius to explicate the formae nativae (see Dutton in Bernard, Glosae super Platonem, 90). Certainly Bernard began with Calcidius’s terms, but the Boethian influence within the Glosae predates any emphases introduced by Gilbert and passed down to John of Salisbury. Gilbert only accentuates Bernard’s own appropriation of Boethius, accelerating the Glosae’s movement away from the Calcidian framework, a shift which Dutton repeatedly notes. 56. See Dutton in Bernard, Glosae super Platonem, 64–65; and Speer, Die entdeckte Natur, 294. 57. See, e.g., Glosae super Platonem 3:26–35, ed. Dutton, 146; ibid. 4:262–275, ed. Dutton, 167; ibid. 5:127–134, ed. Dutton, 178; ibid. 5:148–169, ed. Dutton, 179; ibid. 5:362–364, ed. Dutton, 187; and ibid. 6:63–64, ed. Dutton, 191. 58. “id est ipsae formae, simulacra sunt idearum quae uerum esse habent.” Ibid. 8:191–192, ed. Dutton, 225. “non quod ideae commisceantur hyle in efficientia sensilis, sed natiuae formae, quae sunt imagines idearum.” Ibid. 8:200–202, ed. Dutton, 225–226. “Simulacra horum accipit formas, quae, uenientes in hylen, procreant haec uisibilia quae proprie dicuntur simulacra idearum.” Ibid. 8:251– 252, ed. Dutton, 227. Cf. Boethius, De trinitate II, ed. Moreschini, 170:110–171:117. 59. Dutton in Bernard, Glosae super Platonem, 87, 81. 60. Gilson, “Le platonisme de Bernard de Chartres,” 15–16. See further Marenbon, “Gilbert of Poitiers and the Porretans on Mathematics.” As Dutton indicates, Gilbert alters the concept of nativa in his own commentaries on Boethius, removing its mediating function and applying it instead within his theory of universals (Dutton in Bernard, Glosae super Platonem, 99–101). 61. See Dutton, “Uncovering of the Glosae Super Platonem,” 213–218; and Dutton in Bernard, Glosae super Platonem, 88–89, 94. 62. “Est autem forma natiua originalis exemplum et que non in mente Dei consistit, sed rebus creatis inheret. Hec Greco eloquio dicitur idos, habens se ad ideam ut exemplum ad exemplar.” Metalogicon II.17, ed. Webb, 95. “Ideas tamen, quas post Deum primas essentias ponit, negat in seipsis materie admisceri aut aliquam sortiri motum; sed ex his forme prodeunt natiue, scilicet imagines exemplarium quas natura rebus singulis concreauit.” Ibid. IV.35, ed.

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Webb, 205. In this passage John invokes Seneca’s distinction of idos/idea in Ad Lucilium 58 (18–21), ed. Gummere, 396–400; cf. Seneca, Selected Philosophical Letters, trans. Inwood, 6. See Caiazzo, “Sur la distinction Sénéchienne,” 93–97; and Nothdurft, Studien zum Einfluss Senecas, 185–186. On the transmission of Seneca’s Letters 58 and 65 in the twelfth century generally, see Reynolds, Medieval Tradition, 104–124. Caiazzo observes that Seneca’s discussion of the divine Mind combined the Calcidian Timaeus and Augustine’s Quaestio de ideis in ways that proved irresistible to Bernard of Chartres and Thierry of Chartres alike (“Sur la distinction Sénéchienne,” 104). I discuss Thierry’s use in more detail below. 63. “He autem idee, id est exemplares forme, rerum primeue omnium rationes sunt, que nec diminutionem suscipiunt nec augmentum, stabiles et perpetue; ut, etsi mundus totus corporalis pereat, nequeant interire. Rerum omnium numerus consistit in his; et, sicut in libro de Libero Arbitrio uidetur astruere Augustinus, quia he semper sunt, etiamsi temporalia perire contingat, rerum numerus nec minuitur nec augetur.” Metalogicon II.17, ed. Webb, 94. Cf. Seneca, Ad Lucilium 58 (22–24), ed. Gummere, 400–402; Augustine, De libero arbitrio II.xvii.45–46 (172–177), ed. Green, 267–268; and Boethius, IA I.1.1–2, ed. Guillaumin, 6. 64. See Dutton in Bernard, Glosae super Platonem, 73–74, 91–93. On mens in Macrobius, see Commentarii in Somnium Scipionis I.6.8–9, ed. Willis, 19–20 (cited in Chapter 2). Cf. Calcidius, Timaeus 188, ed. Waszink, 212–213; and ibid. 268–269, ed. Waszink, 273–274. 65. Dutton in Bernard, Glosae super Platonem, 94. 66. Ibid., 105. Caiazzo has since identified a twelfth-century commentary on Macrobius that makes use of Bernard’s formae nativae (see “Le glosse a Macrobio”). 67. See Speer, Die entdeckte Natur, 300–301. As Dutton has wisely remarked: “What the real text of Bernard of Chartres will do to our conception of the school of Chartres and its central concern with Platonism remains to be seen. . . . Admittedly, extravagant claims have been made about the school of Chartres, as Southern argues, in the absence of facts, but by the same token extravagant criticism has been levelled in the absence of texts” (“Uncovering of the Glosae Super Platonem,” 220). 68. Wetherbee, “Philosophy, Cosmology, and the Twelfth-Century Renaissance,” 34–36. Gregory states that Thierry’s forma essendi and Bernard’s formae nativae are two comparable solutions to “il problema platonico dei rapporti tra idee e mondo sensibile” (see “Note sul Platonismo”; cf. Parent, La doctrine de la création, 86–89). In his defense of the integrity of the school of Chartres, Häring affirms that “the remarkable surge of French learning in the early century had its beginning at Chartres. The man who inspired it was Bernard

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of Chartres”—and then compares the achievements of Gilbert, William, and Thierry (“Chartres and Paris Revisited,” 269). 69. See Dutton in Bernard, Glosae super Platonem, 97; and Nauta, “Glosa as Instrument,” 6–13. 70. See Southern, Scholastic Humanism, vol. 2, 73–77; and Southern, Platonism, 22–24. Cf. e.g., William of Conches, Glosae super Platonem 71–74, ed. Jeauneau, 144–150. 71. See Gregory, Anima mundi, 133–246; cf. McGinn, “Role of the Anima Mundi as Mediator.” New evidence evaluated by Caiazzo confirms the essentials of Gregory’s account (see “La discussione sull’anima mundi”). See, for example, William’s ambivalent definition:  “Anima ergo mundi secundum quosdam spiritus sanctus est. Divina enim voluntate et bonitate (quae spiritus sanctus est, ut praediximus) omnia vivunt quae in mundo vivunt. Alii dicunt animam mundi esse naturalem vigorem rebus insitum,” Philosophia mundi IV (13), ed. Maurach, 22–23. 72. See the study and translation of Dragmaticon philosophiae by Ronca and Curr, Dialogue on Natural Philosophy. On Dragmaticon and the Timaeus gloss, see especially Elford, “William of Conches.” Nauta has shown, however, that a nascent version of the element theory—minus the theory of minima later gleaned from Constantine the African’s Pantegni—can be traced back to William’s commentary on Consolatio (“Glosa as Instrument,” 26–31). 73. See Silverstein, “Elementatum”; and Speer, Die entdeckte Natur, 189–191. 74. “natura est uis quaedam rebus insita, similia de similibus operans.” William of Conches, Dragmaticon philosophiae I.7 (3), ed. Ronca, 30. “Natura exigit quod similia de similibus nascantur.” Dragmaticon philosophiae VI.7 (2), ed. Ronca, 204. 75. See Bernardus Silvestris, Cosmographia, ed. Dronke; Bernardus Silvestris, Cosmographia, ed. Wetherbee; and Stock, Myth and Science. Following Wetherbee, I  have adopted the Latinate name to minimize confusion with Bernard of Chartres. 76. See Stock, Myth and Science, 237–262. 77. Wetherbee, “Philosophy, Commentary, and Mythic Narrative,” 212, 225. 78. Wetherbee, Platonism and Poetry, 4. 79. Newman, God and the Goddesses, 39, 49 (cf. 35–50). See further Economou, Goddess Natura. 80. Nauta has discovered that many of the major insights of William of Conches’s Philosophia mundi (ca. 1130)  were first formulated in his commentary on Boethius’s Consolatio philosophiae III.9 (ca. 1120), namely the hymn “O qui perpetua” (see “Glosa as Instrument,” 17–39; cf. Glosae super Boetium, III m. 9, ed. Nauta, 144–179). Building on Nauta’s work, Ziomkowski shows how Thierry of Chartres recycled William’s ideas from the Boethius gloss in his Tractatus,

Notes

81.

82.

83.

84. 85.

86.

87.

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setting them within a hexaemeral context and making some modest conceptual improvements (“Science, Theology and Myth,” 176–193). As Ziomkowski recognizes, Thierry’s dependence on William has been obscured in previous scholarship awed by the putative novelty of Tractatus. Hence this clarification holds tremendous consequences, not only for the dates of Tractatus and the Boethian commentaries, but also for judgments concerning their relative centrality in Thierry’s thought as a whole (ibid., 193–199). This is not to ignore the deeply aesthetic dimension of Thierry’s theology, which displays a verbal creativity akin to Bernardus or Alan. In the words of Wetherbee, Bernard had made only a rough sketch of the formae nativae, but “in Thierry’s more original project the intuition of such continuities draws him repeatedly into an essentially poetic mode of thinking” (“Philosophy, Commentary, and Mythic Narrative,” 224–225; cf. Dronke, “Thierry of Chartres,” 361, 373). “Nomina mathematica siue principalia apud naturalem philosophum dicuntur illa que significant proprietatem mathematice i.e. abstractiue . . . Mathematica enim magis tendunt ad simplicitatem. Quanto autem aliquid simplicius, tanto ad deum spectat conpetentius.” Alan of Lille, Regulae caelestis iuris XXX (1)–XXXI (1), ed. Häring, 145. On Alan’s Regulae, see Evans, “Boethian and Euclidean Axiomatic Method”; Dreyer, More mathematicorum, 142–162; Hudry, “Métaphysique et théologie”; and Hudry in Alan of Lille, Règles de théologie, 7–89. Alan not only comments on Boethius’s De trinitate (see, e.g., Regulae caelestis iuris VIII–XII, ed. Häring, 132–135) but also repeats Thierry’s arithmetical Trinity (Regulae caelestis iuris I–IV, ed. Häring, 124–128). On the latter, see Albertson, “Achard of St. Victor,” 120–128; and Trottmann, “Unitas, aequalitas, conexio.” Ziomkowski, “Science, Theology and Myth,” 211–212. As Ott notes, Thierry’s peculiar genius was to discover resources in Augustine and Boethius for ­ ­applying number theory to the Trinity, even if both of them resisted, in their own way, just such an application (Untersuchungen, 570). See Großmann, “Studien zur Zahlensymbolik.” On this circle of Cistercian arithmologists, see Evans, Language and Logic of the Bible, 59–66; Meyer, Zahlenallegorese, 40–53; Beaujouan, “Transformation of the Quadrivium,” 482–483; and especially Lange, Les données mathématiques. On William, see Leclercq, “L’arithmétique”; on Thibaut, see Deleflie, Thibaut de Langres. See Grill, “Epistola defensionis.” Lange questions Grill’s dating of the letter and provides an alternative edition. See Lange, Les données mathématiques, x–xvii; cf. Lange in Odo of Morimond, Analetica numerorum et rerum I, ed. Lange, 183–186). “unitas Deitas est, quippe Deitas rerum omnium principium est, et ante omnia et super omnia est. Et effectrix omnium est, quia omnis natura et omne donum nature ab ipsa est.” Odo of Morimond, Analetica numerorum et rerum I, Tertia clausula (De unitate) xiv, ed. Lange, 170; cf. Tractatus 31–34, H 568–570. “Mediator

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noster non unius, sed duorum Mediator est et medium est . . . quia Deus et homo est, verbum et caro est, non tantum ex duabus, sed et in duabus naturis existens. In duabus, quia neutra transivit in alteram. Una et altera retinuit quod suum est, et ex ambabus et in ambabus una persona Mediatoris est.” Odo of Morimond, Analetica numerorum et rerum II (De sacramentis dualitatis), I.vi, ed. Lange, 14. 88. “verba dura et inusitate et non catholica satis,” Grill, “Epistola defensionis,” 200. 89. “Amen dico tibi, Petre, quia non sum locutus, Deum esse numerum. . . . Scio et certus sum, quid scripserim ego. Asserui numerum et sapientiam et verbum esse unum. Quod scripsi, scripsi. . . . Nisi forte quis trutinator sillabarum cavilletur in eo me deceptum, quod ego numerum et sapientiam esse unum, Augustinus unam quandam eandemque rem numerum dixerit et sapientiam.” Grill, “Epistola defensionis,” 201. 90. See Odo of Morimond, Analetica numerorum et rerum I, Prol. XIX (Numerus, sapientia et verbum idem esse probantur), ed. Lange, 28–29. 91. I address the controversies over dating Thierry’s commentaries on Genesis and Boethius in Chapter 5. There is another important commentary by Thierry to add to this sequence, but since it has only begun to be studied I have only occasionally referenced it in what follows. Two extant manuscripts are likely to be commentaries by Thierry of Chartres on Boethius’s Institutio arithmetica. The first, MS Bern Bürgerbibliothek Cod. 633, fols. 19ra–27ra, was discovered by Raymond Klibansky (see Fredborg, Latin Rhetorical Commentaries, 3). The Bern manuscript has been transcribed and studied by Vera Rodrigues in “Creatio numerorum: Nature et rationalité chez Thierry de Chartres,” PhD dissertation, École Practique des Hautes Études, Paris, 2006. Unfortunately, I was unable to obtain a copy of this study before publication. The second manuscript is in Stuttgart, MS Württembergische Landesbibliothek Cod. mathm. 4º 33, fols. 1ra–34ra. Extracts of the Stuttgart manuscript have been transcribed by Irene Caiazzo (see “Rinvenimento”). Caiazzo demonstrates convincingly that the author of the commentary is indeed Thierry of Chartres. She does not exaggerate when she predicts that a full edition of this commentary would “significantly change” our understanding of the Breton master (ibid., 203). 92. See Tractatus 1, H 555. Cf. Tractatus 18, H 562. Thierry points out that Augustine himself used the method of secundum physicam when comparing the Trinity to number, measure, and weight (Glosa V.17, H 296). 93. See Tractatus 2–3, H 555–57. Thierry’s Trinitarian analysis of the four causes goes back to William of Conches’s Timaeus and Consolatio commentaries, where William in turned relied on Seneca’s account of causation in Letter 65 (Caiazzo, “Sur la distinction Sénéchienne,” 108–110; cf. Nothdurft, Studien zum Einfluss Senecas, 186–190). See Glosae super Platonem 32, ed. Jeauneau, 98–99; and Glosa super Boetium III m. 9, ed. Nauta, 160. Cf. John of Salisbury, Policraticus VII.5, ed. Webb, 2:108–109. 94. See Tractatus 17, H 562.

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 95. See Tractatus 24–27, H 565–567. On Thierry’s approach to the anima mundi controversy, see Maccagnolo, “Platonismo nel XII secolo.”   96. See Tractatus 29, H 567–568.  97. Tractatus 29, H 568.  98. “Adsint igitur quatuor genera rationum que ducunt hominem ad cognitionem creatoris:  scilicet arithmetice probationes et musice et geometrice et astronomice. Quibus instrumentis in hac theologia breviter utendum est ut et artificium creatoris in rebus appareat et quod proposuimus rationabiliter ostendatur.” Tractatus 30, H 568. Riesenhuber calls the probationes “deductions” (“Arithmetic and the Metaphysics of Unity,” 52–57), and Stollenwerk praises them as “not only arithmetical, but generally ontological proofs” (“Genesiskommentar,” 52). But Jeauneau clarifies that Thierry’s sense of probationes is not a rationalist “rigorous demonstration,” but either an apologetic effort to defend Christian views, or a certainty borne of contemplation akin to Anselm of Canterbury (“Mathématiques et Trinité,” 290–292). As Peter Dronke puts it, “Thierry is offering probationes not so much in the sense of ‘proofs’ as in that of ‘examinations’: he is examining the relationships between his pairs of terms, in the hope that this will illuminate the underlying intuition, which is never itself analysed discursively” (“Thierry of Chartres,” 380–382).  99. Tractatus 33, 37, H 569–570. 100. Alison White writes regarding Thierry: “Boethius’s principles are more often assumed than quoted, and in the advanced mathematical speculations of, for example, Thierry of Chartres’ studies of the Trinity, quadrivial texts have largely been lost sight of. But this was the whole point of the quadrivium: it was but a path—however indispensable—to the ‘more certain things of intelligible knowledge’ ” (“Boethius in the Medieval Quadrivium,” 181). 101. Thierry was not the first medieval author to revive Augustine’s triad, but only the first to link it systematically with the Boethian quadrivium and thus to accentuate its arithmetical meaning. Otloh of St. Emmeram proposed a similar triad in the eleventh century (Gersh, Concord in Discourse, 116). In 1163–4, Hildegard of Bingen discussed Augustine’s triad (as eternitas, aequalitas, connexio) with bishop Eberhard of Bamberg outside of the schools. She might have read De doctrina christiana herself, but more likely she simply encountered the triad in the liturgy (see Fulton Brown, “Three-in-One,” 479–480; and Mews, “Hildegard and the Schools,” 105–107). For in the late eleventh century a new office for the feast of the Trinity was instituted at Cluny that included the antiphon: “In patre manet aeternitas in filio aequalitas in spiritu sancto aeternitatis aequalitatisque connexio.” Compiled by Stephen of Liège, the office draws on prayers of Alcuin influenced by Marius Victorinus’s henology. See further Feiss, “Office for the Feast of the Trinity”; and Hadot, “Marius Victorinus et Alcuin.” 102. Evans, “Thierry of Chartres,” 440. Evans continues:  “He showed his pupils that these newly fashionable works of Boethius were of a piece with the other

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writings which were perhaps more familiar in the classrooms of the day. It is understandable that he should look for help elsewhere in Boethius’s writings when he found himself in difficulties. But he seems to have been the first scholar to try to do so systematically. He demonstrated the unity of Boethius’ thought, crudely, perhaps, and piecemeal, but in a way which was new. . . . We shall not find Thierry comparing Boethius’ views in other treatises at every turn, but when he does do so in the course of the commentaries on the opuscula sacra it is almost always with the intention of showing how smoothly the departments of Boethius’ thought fit together” (Ibid., 441–442). Evans has also shown that medieval humanists often explained Boethius’s arithmetic in terms borrowed from the trivium and vice versa (see “Introductions to Boethius’s ‘Arithmetica’ ”). 103. Tractatus 30–31, H 568; cf. Boethius, IA II.27–28, ed. Guillaumin, 119–121. 104. Tractatus 31–33, H 568–569; cf. Quomodo substantiae, Regula II, ed. Moreschini, 187:26–28. See further Brunner, “Deus forma essendi.” Maccagnolo shows that Thierry’s use of De hebdomadibus in Tractatus and throughout his Boethius commentaries is due to the influence of Remigius of Auxerre’s gloss on Regula II (see “Secondo assioma,” 195–197). 105. Jeauneau, “Mathématiques et Trinité,” 290. 106. Tractatus 34–35, H 570. 107. “Unitas uero que multiplicata componit numeros, uel unitates ex quibus numeri constant, nichil aliud sunt quam uere unitatis participationes que creaturarum existentie sunt. . . . Quoniam autem unitas omnem numerum creat—numerus autem infinitus est—necesse est unitatem non habere finem sue potentie. Unitas igitur est omnipotens in creatione numerorum. Sed creatio numerorum rerum est creatio.” Tractatus 34, 36, H 569–570. See Brunner, “Creatio numerorum.” Jeauneau points out that Thierry’s motto is not a Platonist deformation of Christianity but rather reinforces the principle of creatio ex nihilo at the expense of Platonism (“Un représentant du Platonisme,” 82–84). 108. See Boethius, IA II.27–28, ed. Guillaumin, 119–121. It is conceivable that in formulating this argument Thierry was influenced by Liber XXIV philosophorum in the same way as Alan of Lille. “Deus est monos, monadem ex se gignens, in se unum reflectens ardorem. Haec definitio data est secundum imaginationem primae causae, prout se numerose multiplicat in se, ut fuerit multiplicans acceptus sub unitate, multiplicatus sub binario, reflexus sub ternario. Sic quidem etiam est in multis: unaquaeque unitas proprium habet numerum quia super diversum ab aliis reflectitur.” Liber XXIV philosophorum I, ed. Hudry, 150. If he was, he never cited it directly in his extant works. 109. See Tractatus 37–39, H 570–571. 110. See Tractatus 40, H 571–572; cf. Boethius, IA I.32.1–2, ed. Guillaumin, 66–67. This is also a major principle of Boethian harmonics: see Institutio musica II.7,

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ed. Friedlein, 232. Thierry repeats the doctrine in Commentum, and the author of Commentarius Victorinus follows closely. See Tractatus 39, 43, 44, H 571–574; Commentum II.36, H 79; and Commentarius Victorinus 87–88, H 499. 111. See Tractatus 41, H 572. I discuss this further below. 112. Evans, Old Arts and New Theology, 132–133. 113. Rodrigues, “Pluralité et particularisme ontologique,” 536. 114. See Tractatus 42, H 572. 115. See Tractatus 41, H 572. 116. “necesse igitur hanc equalitatem existentie rerum esse equalitatem i.e. modum quendam siue diffinitionem siue determinationem eternam rerum omnium citra quam uel ultra quam nichil esse possibile est. . . . Ibi rerum ­notiones continentur. Semper enim rei notitia in ipsius equalitate continetur. Si autem excesserit uel infra substiterit non est notitia sed falsa imaginatio dicenda.” Tractatus 41–42, H 572–573. Thierry uses other paraphrases such as “nec in[ f ]‌ra permaneat nec ultra euagetur” (Tractatus 42, 45) or even “medium inter maius et minus” (Tractatus 44). As Parent has suggested, besides Institutio arithmetica and De genesi ad litteram, it is also possible that Thierry had in mind Augustine’s De libero arbitrio, in the same manner that John of Salisbury attributed the source to Bernard (La doctrine de la création, 46). Cf. Augustine, De libero arbitrio II.xvii.45–46 (172–177), ed. Green, 267–268; and Metalogicon II.17, ed. Webb, 94. 117. See Boethius, IA I.1.1, ed. Guillaumin, 6; and IA I.21.2–3, ed. Guillaumin, 46. Thierry references these passages in the prologue to Heptateucon and in Commentum. See MS Chartres 496, fol. 2ra, ed. Jeauneau, 174; and Commentum II.2, II.7, H 68–70. 118. Thierry compares forma, modus, and mensura; mensura and pondus; and modus, mensura, and pondus. See Tractatus 43–44, H 573–74. Cf. Augustine, De genesi ad litteram IV.3 (7), ed. Zycha, 99; and ibid. IV.18 (34), ed. Zycha, 116–117. 119. See Commentum II.35, H 79, where Thierry again juxtaposes Boethian aequalitas theory and Augustine’s theology of weight, number, and measure, but more explicitly and concisely: “Hec uero eadem unitatis equalitas ab unitate gignitur per semel i.e. per integritatem et perfectionem eo scilicet quod nichil ultra nichil infra plus nichil minus est quam sit in dei sapientia. Unde Pater in Uerbo creat omnia i.e. in existendi equalitate. Sic enim creat omnia ut in eis nichil ultra sit nichil infra ut dicit creauit deus omnia in modo pondere numero et mensura. Et illud in deo inquit pondus est sine pondere, numerus sine numero et cetera.” Cf. Augustine, De genesi ad litteram IV.3–4 (8), ed. Zycha, 99–100. See also Commentum IV.29, H 103; and Glosa III.22, H 284. 120. “Istum autem modum siue unitatis equalitatem antiqui philosophi tum mentem diuinitatis tum prouidentiam tum creatoris sapientiam appellauerunt.” Tractatus 42, H 572.

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121. See Boethius, IA I.1.8, ed. Guillaumin, 8–9; and IA I.2.1, 11. These passages were never far from Thierry’s mind when commenting on De trinitate:  see Glosa I.38, H 267; and Lectiones I.51, H 149–150. 122. See Tractatus 2, H 555–556; cf. inter alia Augustine, De genesi ad litteram I.2 (6), ed. Zycha, 6–7; ibid. I.5–6 (10–12), ed. Zycha, 8–10; ibid. II.6 (11–13), ed. Zycha, 40–42; and ibid. III.20 (31), ed. Zycha, 86–87. 123. “Si autem ultra uel infra animi comprehensio extiterit inde falsitas oritur cuius nulla est substantia cum ueritas rebus omnibus sit primum esse et prima substantia. . . . Quando igitur ueritatis equalitas est huiusmodi qualem superior tractatus expressit inde manifestissime colligitur eandem ipsam equalitatem esse Uerbum deitatis. Nichil enim aliud esse Uerbum deitatis quam eterna creatoris de omnibus rebus prefinitio: quid quale quantum sit unaqueque earum uel quomodo se habeat in sua dignitate uel tempore uel loco. At huiusmodi prefinitio equalitas existentie rerum est in[ f ]‌ra quam uel ultra quam nequit aliquid consistere.” Tractatus 46, H 574–575. 124. See Tractatus 41, H 572. Cf. Lectiones V.16, H 218; Lectiones VII.6–7, H 225; and Glosa V.22–29, H 297–299. See further Albertson, “Achard of St. Victor,” 107–112. 125. Häring reasons as follows: “There are good reasons to think that Thierry never completed his tractatus [on Genesis]. If he had completed it, we could expect to find traces of the continuation in at least one of the four manuscripts that contain the work without Clarenbaldus’ letter and tractatulus” (“Creation and Creator,” 144–145). 126. “Nunc quomodo conexio equalitatis et unitatis ab utraque earum procedat explicandum est secundum disciplinas propositas.” Tractatus 47, H 575. 127. See Häring in Thierry, Commentaries on Boethius, 47.

chapt er 5   1. See McGinn, “Does the Trinity Add Up,” 239; and McGinn, “The Role of the Anima Mundi as Mediator,” 290.   2. On the history of scholarship regarding the dates and authorship of Thierry’s works, see Stollenwerk, “Genesiskommentar,” 23–37; and Maccagnolo, Rerum universitas, 4–7. Häring dated the Tractatus on Genesis to between 1130 and 1140, but prior to the Boethius commentaries; Commentum to 1135 or earlier; Lectiones to around 1140; and Glosa to 1145–50. The best overview of Häring’s arguments is his introduction to the edition (see Thierry, Commentaries on Boethius, 19– 52), but it is profitable to consult his initial analyses as well: on Commentum, see “Two Commentaries on Boethius”; on Lectiones, see “Lectures of Thierry of Chartres”; and on Glosa, see “Commentary on Boethius’ De Trinitate.” Häring follows Jansen’s early judgment that Tractatus predated Commentum, Commentarius Victorinus, and the works of Clarembald of Arras (see Jansen,

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Kommentar, 120–128). Jeauneau raises minor doubts about the attribution of Commentum, but agrees in other respects (“Notes sur l’École de Chartres,” 11–13), and Ward replies to objections by M.-Th. D’Alverny and K. M. Fredborg (“Date of the Commentary,” 266–270). Mews dates Commentum even earlier, to the “early 1120s,” but otherwise concurs with Häring, concluding that “arguments for a late date for the Commentum are not based on any firm evidence” (“In Search of a Name,” 192). Speer also concurs with Häring and critiques Dronke for opposing the Genesis commentary to the Boethian commentaries; if he speculates that Tractatus could be as late as 1143–5, given Hermann of Carinthia’s silence until then, his study otherwise assumes the priority of Tractatus (see Die entdeckte Natur, 227–228, 285–286). Ziomkowski’s thorough new analysis addresses the objections of Southern, Dronke, and Speer by reexamining the dedications by Hermann of Carinthia, Bernardus Silvestris, and Clarembald of Arras in light of Thierry’s apparent dependence on William of Conches. Accordingly, Ziomkowski dates Tractatus to the 1130s in Paris, before Thierry’s chancellorship at Chartres (“Science, Theology and Myth,” 199–215). His compelling portrait of Thierry, as his interests evolve from creation to Creator, from physics to metaphysics, and from Genesis to Boethius, is entirely consonant with what I present here (see ibid., 210–213).  3. Stollenwerk overstates the differences among Thierry’s commentaries, contending that Tractatus, Commentum, and Lectiones/Glosa were written by three different individuals (“Genesiskommentar,” 154–155). Her position is based on two comparisons. First, Stollenwerk finds the experimental trinity of perpetuals in Commentum to be incongruous with the Trinitarian meditations of Tractatus (110–111, 117–123). Second, she finds the modal theory in Lectiones and Glosa (particularly necessitas complexionis) to be doctrinally problematic (141–144). Stollenwerk’s objections show the importance of a genetic reading of Thierry’s development. To assert the continuity of Thierry’s arguments requires one to explain the continuous transitions from Tractatus to Commentum, and then from Commentum to Lectiones and Glosa.   4. Maccagnolo accepts Häring’s attribution and sequence of the three Boethian commentaries, but proposes that the Tractatus on Genesis was composed after them as their philosophical consummation (Rerum universitas, 211–215). Maccagnolo actually makes two different proposals, one more modest and one more problematic. First he argues that on the grounds of its similar content (pace Stollenwerk), Tractatus could have been written simultaneously with Commentum, that is, shortly after 1141 rather than in the 1130s (ibid., 7, 200–208). Then Maccagnolo shifts his dates later, finding in Tractatus not only traces of Commentum but even the modal theory of Lectiones and Glosa, which would push the date of the Genesis commentary toward 1150 (ibid., 209–216). Maccagnolo’s second dating requires him to name concepts in Tractatus that are analogues of each of the four modes. Hence he proposes that in Tractatus Thierry

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substituted unitas and aequalitas for the first and second modes, res for the third mode, and materia informis for the fourth mode (ibid., 213). Maccagnolo’s “hypothesis” is very difficult to accept, however, because it would mean that Thierry had assimilated the necessitas complexionis to the divine Word—a conflation that he carefully avoids, and for which he is nonetheless criticized by later readers like Clarembald and Fundamentum naturae, as we shall see. It is also inconceivable to me that Thierry would have written Tractatus late in his career as his crowning work, but then (1) left the triad unitas, aequalitas, conexio unfinished (indeed unstated as a triad); (2) neglected to mention complicatio and explicatio; and (3) resolved not to mention the four modes of being precisely during a discussion of physica and quadrivium. Following Maccagnolo, Dronke asserts that the Boethian commentaries were written “towards 1150” but also “in the decade 1135–1145” (“Thierry of Chartres,” 359 n. 10). Dronke actually cites Maccagnolo’s first dating, which only puts Tractatus and Commentum around 1141, but seems to intend the second dating and its dubious interpretations. But then Dronke goes well beyond Maccagnolo by freely importing the terms explicatio and complicatio from Lectiones into his own exposition of Tractatus, without any textual basis (see “Thierry of Chartres,” 374–384). If Dronke could not resist including the four modes and folding in Tractatus, it seems unlikely Thierry would have done so. Despite its shortcomings, the Maccagnolo-Dronke reading testifies to the real theological tensions among Thierry’s works, which both Clarembald and Fundamentum tried to resolve.   5. Gersh suggests that Thierry’s Boethian commentaries “intersect through doctrinal and textual affinities in a manner too complex to permit classification into primary and derivative” (Concord in Discourse, 115). Riesenhuber deliberately disregards the “minor differences” among Thierry’s Boethius commentaries and treats his collected works as a unified “system.” This leads him to conflate the nascent five modes of Commentum with the completed four modes of Lectiones and to compare Thierry’s “system” to those of Kant and Fichte (“Arithmetic and the Metaphysics of Unity,” 50, 69).  6. Without the nuance of Stollenwerk, Southern adopts an extreme, unwarranted skepticism regarding the coherence of Thierry’s commentaries: “they contain contradictions, confusions and feebleness which preclude the idea that the three differing versions represent a steadily developing body of teaching recorded at first hand by the master at different stages of his career. . . . Since the original form of Thierry’s commentary on Boethius’s De Trinitate is veiled in anonymity we cannot hope for complete success in tracing his original words” (Platonism, 29–30). Marenbon offers similar reasons (“Twelfth Century,” 173–174). Like Stollenwerk, Southern demonstrates the need for a genetic interpretation of Thierry in which change over time, even radical change, does not render his achievements inaccessible to the historian.

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 7. Stollenwerk repeats Clarembald of Arras’s skepticism about the trinity of perpetuals and Fundamentum’s rejection of the four modes of being (“Genesiskommentar,” 117–123, 141–144). Maccagnolo repeats Fundamentum’s attempt to identify the functions of the second mode and the divine Word (Rerum universitas, 213; cf. Maccagnolo, “Platonismo nel XII secolo,” 299), as well as Cusanus’s misprision of the second mode as aequalitas essendi (cf. DI II.7 [129]). On this point, cf. Rodrigues, “Thierry de Chartres, lecteur du De trinitate,” 662–663; and Rodrigues, “La conception de la philosophie,” 132. Gersh anticipates Clarembald’s doctrinal correction when he assimilates the four modes to Augustine’s De genesi ad litteram—a conservatizing step available to Thierry but never taken in his extant works (“Platonism,” 525–526). Riesenhuber repeats the conflation of arithmetical Trinity and four modes of being found in Septem (“Arithmetic and the Metaphysics of Unity,” 69).  8. Pace Southern, who calls Tractatus “the central document for Thierry’s thought” (Platonism, 32).  9. See Commentum II.1, H 68. 10. See Commentum II.15, H 73. 11. Pneumatic accounts of the brain go back at least to Galen (see, e.g., Verbeke, L’Évolution de la doctrine du pneuma, 206–219). Many other authors in Thierry’s milieu discussed spiritus psychology, but none in quite the same way and with the same philosophical agenda. See, inter alia, Adelard of Bath, Quaestiones naturales 18, 31, ed. and trans. Burnett, 124–127, 154–157; Hugh of St. Victor, De unione corporis et spiritus, PL 177: 287B–289A; Achard of St. Victor, De discretione animae, spiritus et mentis 41–49, ed. in Häring, “Gilbert of Poitiers,” 183–185; Ps.-Hugh of St. Victor, Liber de spiritu et anima 22, PL 40:  795BC; William of Conches, Dragmaticon philosophiae VI.18 (4–5), ed. Ronca, 240–241; not to mention similar works by William of St. Thierry and Isaac of Stella. Twelfth-century psychology combined an already rich field of biblical and patristic terms with new Latin translations of Arabic sources by Constantine the African and John of Spain. See Putscher, Pneuma, Spiritus, Geist, 45–55; Burnett, “Chapter on the Spirits in the Pantegni”; McGinn, Three Treatises, 1–100; Chenu, “Spiritus:  Le vocabulaire de l’âme”; and Sudhoff, “Lehre von den Hirnventrikeln.” These surveys of contemporary scientific and religious literature confirm Jansen’s judgment that the psychological epistemology vaunted in Commentum (and repeated in Clarembald and Septem) is not easily paralleled among twelfth-century sources. According to Jansen, Thierry’s unique version of spiritus theory combined common Galenic accounts of the brain with Hermetic theories of πνεῦμα and the Boethian anthropology of intelligentia and intellectibilitas (Kommentar, 44–67). As I will suggest in Chapter 6, however, Thierry’s distinctive spiritus theory might be related to an Arabic astronomical source, the Liber de electionibus of Zahel ben Bischr. 12. See McGinn, Golden Chain, 198–221. Cf. Boethius, In Isagogen I.3, ed. Brandt, 8–9; and Consolatio philosophiae V.5 prose, ed. Moreschini, 152ff.

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13. See Commentum II.3–6, H 68–70. 14. See Commentum II.8–9, H 70–71. 15. See Commentum II.14–15, H 72–73. 16. “Ad huius rei intelligentiam pauca quidem premittenda sunt sed admodum necessaria. De eterno igitur et perpetuo ea quidem in quibus philosophi cum sanctis scripturis concordant quantum humana fert ratio attingamus. Illa enim in quibus philosophia a sacra dissidet scriptura utpote a uia ueritatis exorbitantia reicimus.” Commentum II.18, H 74. The conclusion of the excursus is unambiguous: “Littere continuatio erit hec,” Commentum II.50, H 84. 17. See Commentum II.22, H 75; cf. Lectiones II.48, H 170. 18. See Commentum II.28–29, H 77. In fact Calcidius transmits the dualism of Numenius, who distinguished God as unica singularitas from primordial matter as duitas indeterminata; cf. Calcidius, Timaeus 295, ed. Waszink, 297–298. 19. See Commentum II.29–30, H 77. 20. See Commentum II.37–38, H 80. 21. See Commentum II.33–34, H 78–79; cf. Commentarius Victorinus 95, H 501. Thierry’s doctrine of the Son as Square works from a prophetic fragment known as the Spanish Sibyl that circulated shortly before the Second Crusade (see Albertson, “Achard of St. Victor,” 114–115). The text of the Sibyl in MSS Munich (clm) 5254 and 9516 concerns German nobles travelling from Constantinople to Jerusalem (see Giesebrecht, Geschichte, 502–506; and Dronke, “Hermes and the Sibyls”). Thierry’s commentary on Institutio arithmetica in the Stuttgart MS also cites the Spanish Sibyl (fol. 18vb) and also compares God to a square (fol. 27vb): “Vis ergo quadrati in ipsa forma est. Ex vi namque quadrati forma ipsa quoque essendi est aequalitas . . . . Prima enim forma essendi ex vi unitatis est immutabilitas. Ex hoc enim deus immutabilis est, qui semper unus est, non nisi uno modo habere se potest. Rursus cum forma sit aequalitas essendi, dico quod divinitas ex vi quadrati forma vel causa est. Ex vi namque quadrati aequalitas et forma est existendi aequalitas, ut iam sepe dictum est” (Caiazzo, “Rinvenimento,” 195–198). This link to Boethian arithmetic discovered by Caiazzo confirms Rodrigues’s contention that Thierry’s discussion of the tetragonus is indebted to his study of Institutio arithmetica (“Thierry de Chartres, lecteur du De trinitate,” 658–659). Cf. Boethius, IA I.27.7, ed. Guillaumin, 56; and IA II.11.1, 98. Gersh, on the other hand, argues that Thierry fundamentally misconstrues Boethian arithmetic in Commentum (“Platonism,” 514–516). Of course it may be that Thierry misunderstood Boethius in a creative and productive way. 22. “Ab hac igitur sancta et summa Trinitate descendit quedam perpetuorum trinitas.” Commentum II.39, H 80. Thierry defines perpetuitas later in Commentum: “i.e. perpetuitatem que ab eternitate differt eo quod perpetuum fine caret cum habeat principium. Eternum uero et fine caret et principio.” Commentum IV.44, H 107. On the utter novelty of Thierry’s “trinity of perpetuals,” see Gersh, “Platonism,” 522–523; cf. Stollenwerk, “Genesiskommentar,” 117–127.

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23. See Commentum II.39–42, H 81–82. 24. See Commentum II.43–44, H 82. 25. “Secundum haec ergo cum ex summo deo mens, ex mente anima fit, anima vero et condat et vita compleat omnia quae sequuntur, cunctaque hic unus fulgor illuminet et in universis appareat, ut in multis speculis per ordinem positis vultus unus, cumque omnia continuis successionibus se sequantur degenerantia per ordinem ad imum meandi:  invenietur pressius intuenti a summo deo usque ad ultimam rerum faecem una mutuis se vinculis religans et nusquam interrupta conexio.” Macrobius, Commentarii in Somnium Scipionis I.15.15, ed. Willis, 58. Macrobius identifies this image, originally Plotinian (see Enneads I.1.8), as the Homeric catena aurea. 26. “Sicut enim facies una in diuersis renitens speculis una quidem in se est sed pro speculorum diuersitate hec una illa uero altera esse putatur ita quoque, si comparare liceat, forma quidem diuina in omnibus quodam modo relucet nec est nisi una quantum in se est rerum omnium forma si harum que forme putantur diuerse illam puram ueramque simplicitatem consideres.” Commentum II.48, H 83. 27. In Lectiones and Glosa, Thierry mentions his dependence on the distinction between fate and providence in Consolatio. See Lectiones II.6, H 156; Lectiones II.32, H 165; and Glosa II.23, H 273. 28. “Providentia namque cuncta pariter quamvis diversa quamvis infinita complectitur, fatum vero singula digerit in motum locis, formis ac temporibus distributa, ut haec temporalis ordinis explicatio, in divinae mentis adunata prospectum, providentia sit, eadem vero adunatio, digesta atque explicata temporibus, fatum vocetur. . . . Ordo enim quidam cuncta complectitur, ut quod ab adsignata ordinis ratione decesserit, hoc licet in alium, tamen ordinem relabatur, ne quid in regno providentiae liceat temeritati.” Boethius, Consolatio philosophiae IV.6.10, IV.6.53, ed. Moreschini, 122:34–41, 128:186–129:189 (my emphases). For possible Neoplatonic sources, see the commentary by Gruber, Kommentar, 356–363. The conceit of the moving sphere in the same chapter also opposes explicare and implicare: as the circle’s center is to its circumference, so are eternity to time, intellect to reason, and the simplicity of providence to the motion of fate (Consolatio philosophiae IV.6.15–17, ed. Moreschini, 123:62–124:78). On these passages as sources for Thierry of Chartres, see McTighe, “Eternity and Time”; cf. McTighe, “Neglected Feature of Neoplatonic Metaphysics,” 32–34. 29. See Boethius, Consolatio philosophiae V.6.14, ed. Moreschini, 157:54–57; cf. Boethius, De trinitate IV, ed. Moreschini, 176:243–248. 30. “Quas igitur formas per pluralitatis diuersitatem possibilitatis i.e. materie mutabilitas explicat, easdem quodam modo in unum forma diuina conplicat et ad unius forme simplicitatem inexplicabili modo reuocat.” Commentum II.49, H 84. 31. See Commentum II.66, H 88; and Commentum IV.42–43, H 107.

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32. “estimo quod quinque modis rerum consideratur uniuersitas. Est enim rerum uniuersitas in deo, est in spiritu creato, est in numeris, est in materia, est etiam rerum uniuersitas in actu ita uidelicet quod deus est omnia, spiritus creatus omnia, natura quoque omnia, materia iterum omnia, actualia quoque nemo dubitat esse omnia.” Commentum IV.7, H 97. On possible sources of Thierry’s fivefold modal theory in Commentum, see Gersh, “Platonism,” 525–529; and Stollenwerk, “Genesiskommentar,” 124–128. 33. See Commentum IV.8–11, H 97–98. 34. “Sed quoniam que de rerum universitate proposita sunt maioris indigent inquisitionis quam tractatus presens exigat dimittantur hec interim quorum explanationem uoluminis integri series expectat. Ilud uero presentis speculationis est quod predicationes secundum diuersas universitatis considerationes uariantur cum uocabula tamen omnia propter naturas et secundum rationis motum reperta sint.” Commentum IV.10, H 98. 35. Both commentaries refer to Gilbert of Poitiers’s censure in 1148, and Thierry retired to a Cistercian monastery in 1155 (see Häring in Thierry, Commentaries on Boethius, 33). Häring characterizes Commentum and Glosa as commentaries by Thierry, but Lectiones as a “reportatio” of Thierry’s lectures by an able scribe (see “Commentary on Boethius’ De Trinitate,” 260–261; “Lectures of Thierry of Chartres,” 116; and “Two Commentaries on Boethius,” 66–67). Among the student treatises, by contrast, Commentarius Victorinus is more of a compilation of Thierry’s ideas than an original composition, but nevertheless a commentary constrained by the text of De trinitate, while the Tractatus de Trinitate is a freestanding treatise (see Häring, “Two Commentaries on Boethius,” 75). 36. Boethius, De trinitate II, ed. Moreschini, 168:68–169:80. 37. “Est item rerum uniuersitas et eadem omnino: non enim rerum uniuersitates plures esse possunt . . . . Est itaque rerum uniuersitas subiecta theologie, mathematice et phisice sed alio modo quam theologie. Et item eadem uniuersitas subiecta phisice et alio modo: scilicet ut in actu est.” Lectiones II.6, H 156; cf. Lectiones II.13–15, H 158–159. 38. Dronke, “Thierry of Chartres,” 368. 39. See Lectiones Prol. 4, H 126. 40. See Lectiones Prol. 5, H 126; and Lectiones Prol. 21, H 130. Cf. Lectiones I.1–30 passim, H 133–141. By contrast, Thierry first mentions rationes theologicae in Commentum II.1, H 68. 41. See Lectiones V.16, H 218; and Lectiones VII.5–7, H 224–225. Here Thierry adds a new layer to the arithmetical Trinity by praising the triad as the eternal foundation of the quadrivium itself:  numerus (i.e., arithmetic), proportio (harmonics), and proportionalitas (geometry). See Lectiones VII.7, H 225; cf. Boethius, IA II.40.1–3, ed. Guillaumin, 140; and Boethius, Institutio musica II.7, ed. Friedlein, 232.

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42. In Lectiones II, Thierry explains the three disciplines in terms of folding (§§1–6), then the four modes in terms of the three disciplines (§§7–13), and then obversely the three disciplines in terms of the four modes (§§14–34). Thereafter he turns to problems of unity and plurality of form (§§35–67). 43. See Lectiones II.6, H 156; and Lectiones II.32, H 165. Cf. Commentum II.49, H 84. 44. See Lectiones II.4–6, H 155–156. 45. “Et ea quidem uniuersitas est in necessitate absoluta in simplicitate et unione quadam omnium rerum que deus est. Est etiam in necessitate conplexionis in quodam ordine et progressione:  inmutabiliter tamen. Est in possibilitate absoluta: in possibilitate tamen sine actu omni. Est etiam in determinata possibilitate:  possibiliter et actu.” Lectiones II.9, H 157. Compare Thierry’s alternative fourfold schema, much less convincing, in Abbreviatio monacensis De trinitate II.31, H 345. The first mode is indiuidua substantia, the second indiuidua naturam, the third possibilitas determinata, and the fourth diuidua substantia. This schema builds on Calcidius’s exposition of Aristotle’s account of individuated substance in Physics 192a3–192a34. See Calcidius, Timaeus 286ff., ed. Waszink, 289ff. Plotinus proposed a similar quaternity in Enneads IV.2.2: the One (eternal); soul (indivisibly divided); embedded forms (divisibly undivided); and body (divisibly divided). See Merlan, From Platonism to Neoplatonism, 33. 46. “Absoluta enim necessitas rerum omnium conplicatio est in simplicitate. Necessitas conplexionis earum rerum explicatio in quodam ordine. Qui ordo a phisicis fatum dicitur. Absoluta autem possibilitas est eiusdem uniuersitatis rerum complicatio in possibilitate tantum de qua ueniunt ad actum. Et uocatur a phisicis primordialis materia siue caos. Determinata uero possibilitas est explicatio possibilitatis absolute in actu cum possibilitate. Sic eadem rerum uniuersitas quatuor modis est.” Lectiones II.10, H 157–158. 47. “Considerat enim theologia necessitatem que unitas est et simplicitas. Mathematica considerat necessitatem conplexionis que est explicatio simplicitatis. Mathematica enim formas rerum in ueritate sua considerat. Phisica uero considerat determinatam possibilitatem et absolutam. Absoluta autem necessitas et absoluta possibilitas in his extrema sunt. Reliqui uero modi uelud media. Quod alibi melius explicatur.” Lectiones II.11, H 158. The “elsewhere” to which he refers is Commentum II.28, H 77. Thierry experiments with a few other characterizations of the disciplines. If theology is the science of simplicity, mathematics is the science of abstractions (see Lectiones II.18, H 160). Theology grasps the single, indivisible substance; mathematics the indivisible natures; and physics studies divisible substances (see Lectiones II.31, H 165). 48. On Lectiones, Speer comments that “auffallig ist das enge Explikationsverhältnis zwischen Theologie und Mathematik, das in der Notwendigkeitsstruktur der Seinswirklichkeit gründet. Denn auch auf Seiten der modi existendi besteht zwischen der ewigen Einfachheit der absoluten Notwendigkeit

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und der Notwendigkeit der Verknüpfung oder Zusammenfassung ein Explikationsverhältnis” (“Erwachen der Metaphysik,” 32). 49. See Lectiones II.12, H 158; and Lectiones II.30, H 164. 50. As Rodrigues writes, in his modal theory Thierry “aurait su assimiler et développer, dans une synthèse complexe et riche de multiples influences, les ambiguïtés et les oscillations qui, entre platonisme et aristotélisme, étaient déjà celles de Boèce” (“Pluralité et particularisme ontologique,” 536). Dronke likewise praises Thierry’s “exceptional gift for synthesis, bringing together the most disparate ranges of literature and thought” (“Thierry of Chartres,” 370). 51. See Gregory, “Il Timeo e i problemi del platonismo medievale,” 112–122. 52. McTighe suggests Plotinus as a predecessor of Thierry’s doctrine of folding (see McTighe, “Meaning of the Couple”; and McTighe, “Neglected Feature of Neoplatonic Metaphysics”). But he finds better evidence of explicatio in Plotinus than of complicatio, and some of the henological doctrines he attributes to Plotinus originated in Eudorus and Moderatus. On pre-Chartrian instances of “folding,” see also Counet, “Les complications de l’histoire de la philosophie”; and Moritz, Explizite Komplikationen, 226–228. Maccagnolo suggests that Remigius of Auxerre’s gloss on De hebdomadibus informed Thierry’s folding and modal theory (see “Secondo assioma,” 197–199). 53. See Caiazzo, “Rinvenimento,” 201. Caiazzo notes that Thierry’s commentary on Institutio arithmetica must have been composed while the Breton master was still working intensively on the liberal arts represented in Heptateucon (that is, in the 1120s and 1130s) as opposed to his later turn to the Boethian opuscula (in the 1130s and 1140s). The Stuttgart commentary contains the arithmetical Trinity as in Tractatus, the Spanish Sibyl as in Commentum, but also the reciprocal folding that appears in full form in Lectiones. Caiazzo therefore dates the commentary between Tractatus and Commentum (see “Rinvenimento,” 186–188). But one can also easily imagine that this commentary marked the moment between Commentum and Lectiones when Thierry first discovered the power of folding in the first place, that is, around 1140. 54. “quia sicut unitas est complicatio numeri, et numerus explicatio unitatis, ita quoque punctum est complicatio omnis magnitudinis, et magnitudo est explicatio puncti.” MS Württembergische Landesbibliothek Cod. mathm. 4º 33, fol. 21vb, in Caiazzo, “Rinvenimento,” 202. Cf. Boethius, IA II.4.6, ed. Guillaumin, 90 (my emphasis): “Ex hoc igitur principio, id est ex unitate, prima omnium longitudo succrescit quae a binarii numeri principio in cunctos sese numeros explicat, quoniam primum interuallum linea est.” 55. See Glosa V.17–20, H 296–297. 56. See Glosa II.1–11, H 268–270. 57. See Glosa II.24–28, H 273–274. 58. See Glosa II.14–15, H 271–272.

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59. See Glosa II.18, II.22, H 272–273. Cf. Calcidius, Timaeus 123, ed. Waszink, 167; and ibid. 268–269, ed. Waszink, 273–274. 60. For Rodrigues it is “plus exactement dans la conception du mode de la rerum universitas qu’est la necessitas complexionis, que réside de notre point de vue le nerf de la position de Thierry” (“Pluralité et particularisme ontologique,” 514). 61. See Glosa II.21, H 273. 62. “Hec igitur uniuersitas quam in quandam simplicitatem in se complicauit absoluta necessitas explicatur in formarum atque in imaginum ueritates quas ideas dicimus. Easque disponit ordine quodam in seriem causarum quam sic necesse est. Nam res eam sequuntur cum eius alicui se subiecit cause. Hec uero determinata necessitas uel necessitas complexionis eo quod cum aliquam eius materiam incurrimus causarum reliquarum seriatam conexionem uitare non possumus. . . . Quod si nullam eius causam attigerimus ei causarum conexioni minime subiacemus.” Glosa II.20–21, H 273; cf. Calcidius, Timaeus 144, ed. Waszink, 182–183. Cf. Dronke’s translation in “Thierry of Chartres,” 369. 63. “Quam alii legem naturalem alii naturam alii mundi animam alii iusticiam naturalem alii ymarmenem nuncupauerunt. At uero alii eam dixere fatum alii parchas alii intelligentiam dei.” Glosa II.21, H 273. Cf. the parallel list in Septem: “Hic igitur motus a Mercurio natura dicitur, a Platone anima mundi, a quibusdam fatum, a theologis divina dispositio apellatur” (961D–962A). The Stoic concept of fate as εἱμαρμένη is closely related to the Logos (see Aall, Logos, I:130–133; and Heinze, Lehre vom Logos, 125–129). The term appears in Asclepius and later in Martianus Capella in association with divine necessity:  “Quam εἱμαρμένην nuncupamus, o Asclepi, ea est necessitas effectrix rerum omnium quae geruntur semper sibi concatenatis necessitatis nexibus vinctae. Haec itaque est aut effectrix rerum aut deus summus, aut ab ipso deo qui secundus effectus est deus, et omnium caelestium terrenarumque rerum firmata divinis legibus disciplina. Haec itaque εἱμαρμένη et necessitas ambae sibi invicem individuo conexae sunt glutino,” Asclepius III, 39, ed. Scott, 362; cf. Martianus Capella, De nuptiis Philologiae et Mercurii I.64, ed. Willis, 19. Clarembald of Arras seems to have referenced such texts in his discussions of the second mode: see Tractatus super librum Boetii De Trinitate II.44, ed. Häring, 124–25; Expositio super librum Boetii De Hebdomadibus III.19, ed. Häring, 201; and Tractatulus super librum Genesis 23, ed. Häring, 236. 64. “Numerus enim, ut habet Arithmetice prologus, principale exemplar extitit in mente conditoris.” Glosa I.38, H 267. 65. “Nam nullus omnino subintrat numerus in deo nisi numerus qui est sine numero de quo nunc non loquimur.” Glosa III.22, H 284. Cf. Commentum II.35, H 79, cited in the last chapter; as well as Commentum IV.29, H 103: “In deo namque est qualitas sine qualitate quantitas sine quantitate numerus sine numero.”

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66. According to Speer, the modal theory of Lectiones is not only the culmination of Thierry’s thought but also grounds the vision of autonomous nature that Tractatus shares with Bernard of Chartres, William of Conches, and Adelard of Bath:  “im Mittelpunkt der Frage nach dem Gesamtbezug der Ursachenordnung und ihrer Begründung in einer Prinzipienreflexion wird der Notwendigkeitsgedanke stehen, der schließlich zu einer Ontologie der universitas rerum führt” (Die entdeckte Natur, 232; cf. 239, 264). 67. Maccagnolo contends that Thierry’s four modes provide the overarching “metaphysics” that unifies his contributions to theology, mathematics, and physics (Rerum universitas, 213–216, 250). As Speer rightly notes, this is both imprecise and anachronistic, since Maccagnolo’s understanding of metaphysics, aligned as it is with Chenu’s faulty paradigm of “éveil métaphysique,” belongs more to the thirteenth century than the twelfth (Die entdeckte Natur, 286–288). 68. Dronke suggests that “interweaving” (Verflechtung), “connection” (Zussamenhang), “construction” (Gefüge), “constitution” (Beschaffenheit), and “composition” (Verfassung) are all possible translations, but he decides on “necessity of make-up” (“Thierry of Chartres,” 369 n. 39). Speer uses “die Notwendigkeit der Verknüpfung oder Zusammenfassung” (Die entdeckte Natur, 257). Consulting only Glosa, Wetherbee proposes “necessary continuity” in the sense of a “mediating movement” (“Philosophy, Commentary, and Mythic Narrative,” 223–224). When the same term appears in Cusanus, Bond translates it as “necessity of connection,” Hopkins as “connecting necessity,” Honecker as “Notwendigkeit des Kausalzusammenhanges,” and Wilpert-Senger as “die Notwendigkeit der Verknüpfung.” McTighe rejects “necessité relative” (Gandillac) and “necessity of combination” (Häring) and proposes instead “necessity of composition (see “Thierry of Chartres and Nicholas of Cusa’s Epistemology,” 175 n. 21). The problem already troubled Hermann Löb in 1907, who criticized the imprecise translations of Johann Uebinger (“begreifender Notwendigkeit”) and Franz Anton Scharpff (“das alles umschliessende Band”) (Bedeutung der Mathematik, 38 n. 58). 69. See Abbreviatio monacensis De Trinitate II.9–23 passim, H 339–343; and Abbreviatio monacensis De Hebdomadibus II.40, H 412. 70. “Sicut enim aequalitas est complexio inaequalitatis, et ab ea inaequalitas omnis descendit,” MS Württembergische Landesbibliothek Cod. mathm. 4º 33, fol. 31rb; see Caiazzo, “Rinvenimento,” 200. 71. See Lectiones II.9–10, H 157; Glosa II.20, H 273; and Abbreviatio monacensis De Hebdomadibus II.40–42, H 412. In light of such passages, my proposed translation of the second mode as “necessity of enfolding” is admittedly somewhat confusing. For the ordo that the second mode establishes is not itself an enfolding but an unfolding, that is, the eternally originary unfolding of the divine enfolding. But Thierry’s idea is a complex one, and I see no better alternative at present. Nicholas of Cusa wrestled with this very problem in his dialogue De mente, as we shall see.

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72. Thierry does remark in his Genesis commentary that some philosophers identified aequalitas with the Macrobian mens. But then, interested more by the Augustinian sapientia, he never returns to the problem of the divine Mind. See Tractatus 42, H 572; and Lectiones II.53, H 172. 73. “Sole inquit forme que in mente diuina sunt merito forme dicuntur.” Commentum II.64, H 88. 74. See Commentum II.63–66, H 87–88; cf. Commentum II.49, H 83–84. On Thierry’s initial discussion of Calcidius and Boethius on plural exemplars in Commentum, see Gersh, “Platonism,” 518–520. 75. Thierry’s forma formarum is closely related to the mens divina in the Boethian commentaries. Boethius had named God forma essendi in De hebdomadibus, and in De trinitate had distinguished God as the one true Form. Häring notes that Augustine called God the forma omnium formatorum in Sermon 117 (see Thierry, Commentaries on Boethius, 410). Thierry combines both ideas by naming God the forma formarum. See Abbreviatio monacensis De Hebdomadibus II.26–30, H 409–410; Lectiones II.38–43, H 167–168; and Glosa II.28–36, H 274–276. Notably in Tractatus Thierry never uses forma formarum, but does reference forma essendi from De hebdomadibus (see Tractatus 45, H 574). On the two terms in Thierry of Chartres, see Brunner, “Deus forma essendi”; and Counet, Mathématiques et dialectique, 153–165. 76. See Lectiones II.40, H 167. 77. “Mens etenim diuina generat et concipit intra se formas i.e. naturas rerum que a philosophis uocantur ydee. Unde diuinitas nichil aliud est quam ipsa mens diuina que est generatiua ydearum. Concipit enim et tenet eas intra se et ab ipsa ueniunt in possibilitatem,” Lectiones II.43, H 168; see further Lectiones II.44–46, H 168–170. As Caiazzo points out, Thierry’s discussions of ideas in the divine mind often lean on Seneca’s Letter 65—another link with Bernard of Chartres (“Sur la distinction Sénéchienne,” 108–110). Beyond this passage, see also Lectiones II.65–66, H 176; and Glosa II.30, H 275. 78. See Abbreviatio monacensis De Hebdomadibus II.25, H 409:  “Forma enim essendi omnium rerum que deus est conceptiua est omnium formarum. Mens enim diuina omnes formas omnium rerum intra se concipit conplectitur continet in simplicitate quadam.” 79. “Ex his enim formis quae praeter materiam sunt, istae formae venerunt quae sunt in materia et corpus efficiunt. Nam ceteras quae in corporibus sunt abutimur formas vocantes, dum imagines sint,” Boethius, De trinitate II, ed. Moreschini 171:113–116. 80. “Praeter materiam sunt in ueritate sua:  scilicet in necessitate conplexionis. Formis dixit pluraliter quia sunt ibi in necessitate conplexionis plura rerum exemplaria que omnia sunt unum exemplar in mente diuina. Secundum quod Plato dicit in Parmenide Calcidio testante quod unum est exemplar omnium rerum et plura exemplaria in quo nulla diuersitas nulla ex diuersitate

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contrarietas sicut in Platone dicitur.” Lectiones II.66, H 176. See by contrast how Thierry navigates the same passage in Boethius (“ex his enim formis”) without the forma formarum or the four modes of being in Commentum II.63– 66, H 87–88. 81. See Calcidius, Timaeus 272, ed. Waszink, 276–277; cf. Gersh, Middle Platonism, 460–467, esp. 466 n. 203. Caiazzo speculates that Thierry’s doctrine of ideas in Commentum II.63–66, H 87–88, might derive from Bernard’s formae nativae (“Le glosse a Macrobio,” 223). Cf. Lectiones II.64, H 176; and Glosa II.49, H 279. 82. See, e.g., Lectiones II.53, H 172, where Thierry appears to identify mens divina with divina sapientia. 83. See Glosa II.28–36, H 274–276. The smoking gun is the awkward circumlocution Thierry inserts just at the moment when we would expect to find necessitas complexionis: “Possibilitas enim determinat et ad actum ducit in causarum quandam seriem concipiendo ydeas et in actualia hec: conectendo ipsas materie.” Glosa II.36, H 276 (my emphasis). By contrast, the reference to the second mode is retained in Abbreviatio monacensis De trinitate II.66, H 354–355, even though the four modes of being are adjusted in other minor ways throughout that text. 84. According to Gersh, there are eight dominant “philosophemes” in Latin Neoplatonism that structure Christian discourse on the hierarchy of being. The repetition of paradigmatic source texts in Boethius, Augustine, and Macrobius partially explains the apparent similarities between Thierry of Chartres and Nicholas of Cusa (see “First Principles of Latin Neoplatonism,” 136–137). 85. Brunner remarks that in Thierry’s theology the simultaneity of numerical series promotes the spatializing of creation, while the discontinuity of number defines the “singularity” of each creature (“Creatio numerorum,” 725). 86. As Gregory observes, one can already see how the diverse concepts governing Thierry’s overlapping theologies of mediation—aequalitas, forma essendi, and the divine Verbum—might generate tensions in his thought (see Anima mundi, 73–97).

chapt er 6   1. See Ricklin, “Plato im zwölften Jahrhundert,” 160–163.   2. See Reynolds, “Essence, Power and Presence of God.” Hugh of St. Victor may have innovated the triad before Abelard: see Poirel, Livre de la nature, 345–420; and Mews, “World as Text.”  3. On the history of the theory of appropriated Trinitarian names, see Ott, Untersuchungen, 254–266, 581–594; and Hödl, Wirklichkeit und Wirksamkeit, 5–14, 28–59.   4. See Albertson, “Achard of St. Victor.”

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  5. See, e.g., Tractatus de Trinitate 12–19, H 306–308; and Commentarius Victorinus 81–88, H 498–499. I  discuss these treatises in greater detail in Albertson, “Achard of St. Victor,” 112–116. They appear after the text of the sole exemplar of Thierry’s Lectiones in MS Paris BN Lat 14489, fols. 62–66 (Tractatus de Trinitate), and fols. 67–95v (Commentarius Victorinus, formerly called In titulo and attributed to Ps.-Bede). On Commentarius Victorinus, see further Bertola, “Il ‘De Trinitate’ Dello Pseudo Beda”; and Evans, “Influence of Quadrivium Studies,” 158–163. Häring notes several reasons for dating the two works after Thierry’s Glosa, whether they were written by Thierry or by one of his students:  “Especially the manner of handling the ‘mathematical’ explanation of the Trinity, based on the Augustinian dictum cited above, offers impressive evidence to the effect that both works belong to the school of Thierry of Chartres” (“Short Treatise on the Trinity,” 128). Häring would later argue, however, that Commentarius Victorinus could well have been written by Thierry himself, noting “very striking points of contact” with the anonymous Tractatus de Trinitate (Häring in Thierry, Commentaries on Boethius, 40–45).   6. See, e.g., Tractatus de Trinitate 26–28, H 309–310; and Commentarius Victorinus 99–110 passim, H 501–503.   7. On Achard, see Achard, L’Unité de Dieu, ed. Martineau; Châtillon, Théologie, spiritualité et métaphysique; and Ilkhani, La philosophie de la création.   8. See Albertson, “Achard of St. Victor.”  9. On fourteenth-century quantification, see, inter alia, Maier, “Concept of the Function”; Maier, “Achievements of Late Scholastic Natural Philosophy”; Murdoch, “Mathesis in philosophiam scholasticam introducta”; Murdoch, “Rationes Mathematice”; and Sylla, “Medieval Quantifications of Qualities.” In a broader scope, other valuable studies of medieval “mathematization” include Lindberg, “On the Applicability of Mathematics”; Crosby, Measure of Reality; Roche, Mathematics of Measurement; and Kaye, Economy and Nature in the Fourteenth Century. 10. See Wood, “Calculating Grace”; Sylla, “Autonomous and Handmaiden Science”; Davenport, Measure of a Different Greatness; and above all, Murdoch, “From Social into Intellectual Factors.” 11. See Aertsen et al., Nach der Verurteilung von 1277. For a concise overview of the historiographical problems surrounding fourteenth-century philosophy (and “nominalism”), see Courtenay, Changing Approaches. 12. See Trapp, “Augustinian Theology”; Courtenay, Schools and Scholars, 307–324; and Oberman, “Fourteenth-Century Religious Thought.” 13. On medieval Proclianism, see Kristeller, “Proclus as a Reader”; and Imbach, “Le (néo-) Platonisme médiéval.” On the Albertist school, see Hoenen, “Thomismus, Skotismus und Albertismus”; and Kaluza, “Les débuts de l’albertisme tardif.” 14. On Albert, see Mahoney, “Albert the Great on Christ and Hierarchy.” On Bonaventure, see Dettloff, “ ‘Christus tenens medium in omnibus’ ”; and

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Gerken, Theologie des Wortes. On Eckhart, see McGinn, Mystical Thought of Meister Eckhart, 72–90. Bonaventure viewed number as a special vestigium of God and even made use of arithmology (see Welte, “Zahl als göttliche Spur”; and Bernath, “Mensura fidei”). 15. See Jeauneau, “Note sur l’École de Chartres,” 22–23; and Häring in Thierry, Commentaries on Boethius, 46–52. In Sermon II, Helinand (1162–1237) seems familiar with Thierry’s Commentum. See “In natali Domini I,” PL 212: 486A–498C. He expands on Thierry’s doctrine of aequalitas essendi (491C) and on his image of the Word as figura of the Father (490B–490D). Although Helinand discusses unitas and aequalitas at length, he never treats the arithmetical Trinity per se (see, e.g., 490C). 16. See Häring in Thierry, Commentaries on Boethius, 25, 52. Cf. Southern, Platonism, 33–34; and Ziomkowski, “Science, Theology and Myth,” 202. 17. See early analyses by Flatten, “Die ‘materia primordialis’ ”; Parent, La doctrine de la création; Jeauneau, “Simples notes”; and Crombie, Augustine to Galileo, 13–18. More recent studies include Zimmerman, “Kosmogonie des Thierry von Chartres”; Lemoine, “Le nombre dans l’École de Chartres”; and Otten, “Nature and Scripture.” 18. See, e.g., Beierwaltes, “Einheit und Gleichheit”; and Trottmann, “Unitas, aequalitas, conexio.” 19. See Duhem, “Thierry de Chartres et Nicolas de Cues”; and Chenu, “Une définition Pythagoricienne.” 20. See Jaeger, Envy of Angels; Fichtenau, Heretics and Scholars, 267–280; and Ferruolo, Origins of the University, 47–92. 21. See PL 199: 945D–964D. The sole known manuscript is in the British Museum, London, MS Harley 3969, beginning at fol. 206. What appears to be the last page is missing, breaking off the author’s conclusion in mid-sentence: “Unde notandum, quia triplex est veritatis causarum seu principiorum inquisitio et triplex cognitio. Prima inquisitio est mathematica, quae contemplatur [. . .]” (Septem 964D). 22. The Latin Asclepius survives in six manuscripts, all from the twelfth century (Lapidge, “Stoic Inheritance,” 103). 23. See Lucentini, “L’Asclepius Hermetico nel Secolo XII.” Lucentini mentions Septem only briefly as a “fragment” by John of Salisbury and notes that John mentions “Hermes Trismegistus” once by name in Policraticus. But the references to “Hermes” in Septem denote De sex rerum principiis: see “Liber Hermetis,” ed. Silverstein, 238; and Gregory, “Il Timeo e i problemi del platonismo medievale,” 141–142. Lucentini has edited another twelfth-century Hermetic text, possibly by Alan of Lille, in “Glosae super Trismegistum.” 24. Schaarschmidt considers Septem “einer bewussten, wenn auch sehr ungeschickten Nachahmung” of John of Salisbury, reminiscent of the pseudonymous De quinque septenis transmitted as a work of Hugo of St. Victor (Johannes

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Saresbariensis, 278–281). Even if some of Schaarschmidt’s evidence for dating is inaccurate, recent scholarship on Septem has concurred with his conclusions. Nederman concludes that Septem “bears no tangible relation to any of John’s other writings and rightly deserves to be excised from his corpus” (John of Salisbury, 80). Cf. “Liber Hermetis,” ed. Silverstein, 238; Dronke, Fabula, 35; and Wetherbee in Bernardus Silvestris, Cosmographia, 11. 25. See Meyer, Zahlenallegorese, 38–40. 26. See, e.g., Septem 960A. Robilliard argues that the magister could not be Thierry himself. Although Septem exhibits close knowledge of the Commentum, he says, it displays too great a “flexibility of vocabulary.” Robilliard grants that the a­ uthor of Septem copies from Hugh of St. Victor’s commentary on the Celestial Hierarchy, but insists that the master in question is Richard of St. Victor (see “Hugues de Saint-Victor”). Häring notes that Septem was once attributed to Robert of Courçon and that the title occurs in a work of Isaac of Stella (Häring in Thierry, Commentaries on Boethius, 81). Nevertheless: “it seems that the writer of the De septem septenis is the first witness to Thierry’s Commentum” (Häring, “Chartres and Paris,” 290). 27. See Septem 947C–948B. Septem’s statements regarding Pythagoras and quadrivium are probably related to similar references in Hugh of St. Victor’s Didascalicon (I.7, III.2) and in the Liber Hermetis edited by Silverstein. As Silverstein explains, Hugh cites a work titled Matentetrade, attributed to Pythagoras, concerning the ethical aspects of the quadrivium. Hugh’s title may itself be a misapprehension of a passage in Remigius of Auxerre’s commentary on Martianus Capella that refers to Pythagoras “qui non tacuit mathen tetraden,” that is, to his teaching (μάθησις) about the tetrad. A  marginal gloss in Liber Hermetis calls the same work the “liber de doctrina quadruuii.” See “Liber Hermetis,” ed. Silverstein, 224–225; and Hugh, Didascalicon, trans. Taylor, 20, 189–190. 28. Arithmetic teaches one “to divide multitudes,” music “to return dissonance to harmony,” geometry “to make equality out of inequality,” and astronomy to read signs of future portents. See Septem 948C. 29. See Septem 948D–949A. 30. See Septem 949BC. The PL text should be corrected at 949B: musico for unifico and musicalem for unificalem. 31. “Ista septem disciplinae, sapienter consideranti, septem sunt viae animae in vitae et morum honestate. . . . Sic igitur animus erudientis per has septem trivii et quadrivii viae eloquentiam et sapientiam adipiscitur, et anima cujuslibet in iisdem viis et iisdem modis, quibus dictum est, sui reformationem et accessum ad Deum consequitur. Per has igitur animae vias artificiosas percipiuntur universae viae Domini, scilicet misericordia et veritas. . . . De qua Dominus: Ego sum via, veritas et vita; via in exemplo, veritas in promisso, vita in praemio, vel via ad se, veritas ex se, vita in se.” Septem 948D, 949D–950E.

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32. The author lists imaginatio, ratio, and intellectus, as well as four miscellaneous powers (animus, mens, opinio, and memoria) to fill out his complement of seven. See Septem 951BC. 33. Septem 951D–952A. See further McGinn, Three Treatises, 55. 34. Compare Septem 952D–954A (“In libro vero Electionis invenitur scriptum de modo actionum eius sensualium . . . sine omni partium compositione comprehendit”) and Commentum II.4–7, H 69–70. Septem refers to his source as “Liber Electionis.” Silverstein has identified this work as the Liber eleccionis or Liber de electionibus by Zahel ben Bischr (Sahl Ibn-Bišr), an early medieval Arabic astronomical handbook originally edited by Octavianus Scotus (Venice, 1519) (see “Liber Hermetis,” ed. Silverstein, 228, 236). The argument of the Liber cited in Septem, however, is notably more complete than the corresponding passage in Commentum. Most likely Septem is citing verbatim from a source that Thierry paraphrased during his lectures on Boethius (much as Clarembald fills out Thierry’s allusions to De genesi ad litteram, as we shall see). If so, this would represent Thierry’s most extensive use of an Arabic source. 35. See Septem 953D–954A:  “In media vero parte capitis, in cellula quae dicitur intellectualis anima, cum simplex tum immutabilis, ad suam quoque immutabilitatem se recipit, et rerum formas extra materiam in sua immutabilitate considerat. Haec vis animae disciplina vocatur, quia per disciplinam et doctrinam ad hanc formarum considerationem venitur. In eadem vero cellula, qua anima formas rerum in sua simplicitate considerat, se ipsa utens pro instrumento, ita scilicet ut formam circuli non solum a materia abstrahat, verum etiam et sine omni partium compositione intelligat. Haec vis intelligentia dicitur quae solius Dei est, et praeter hoc. Differt autem a disciplina, quia haec immutabiliter rerum formas ut ex partibus compositas considerat; intelligentia vero sine omni partium compositione comprehendit.” Cf. Septem 951D–952A, where the author makes the same point before citing “Liber Electionis.” 36. See Septem 957C–960A; cf. Ps.-Hugh of St. Victor, Liber de spiritu et anima 38 (PL 40: 808D–809A). Septem defines the three kinds of contemplation in terms that mirror the Aristotelian-Boethian division of the sciences in Book II of De trinitate:

Revelatio

Emissio

Inspiratio

Cum materia et forma Sensibilis Mundana [Physica]

Sine materia cum forma Intelligibilis Humana [Mathematica]

Sine materia et forma Intellectibilis Divina [Theologia]

37. See Asclepius I, 12b–14a, ed. Scott, 308–311. 38. “Propterea hoc primum principium et caetera omnes quaerunt studiosius in trivio, inquirunt perspicacius in quadrivio, perquirunt subtilius in theologiae

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et philosophiae scrutinio. Haec omnes quaerunt, sed non inveniunt, quoniam modus inquisitionis et ignorantia veritatis eis obsistunt.” Septem 964C; cf. Septem 960BC, 962D. 39. “Secundum diversos auctores, septem sunt rerum principia quae dicuntur causae primordiales; secundum theologos unum est principium, Deus creator omnium; secundum physicos, tria sunt principia: materia, forma, et spiritus creatus, id est natura; secundum Mercurium quatuor:  lex astrorum, natura, mundus et mundi machina.” Septem 960C. The author’s schema identifies some principles with others, and suggests each is the mode of one cosmic whole. Combining the overlapping accounts yields this overview (see 962CD):

Auctores

Principium rerum

[Quomodo omnia in principio]

I. Theologi II. Physici

(1) Deus, Aeternitas (2) Materia (3) Forma (4) Spiritus creatus [(4) Natura] (5) Lex astrorum (6) Mundus (7) Machina mundi

Immutabiliter Apte et possibiliter Incorporaliter et visibiliter Actualiter [Naturaliter] Ordinabiliter Motabiliter Proportionaliter et concorditer

III. Mercurius

40. See Septem 962AB. 41. The principles of De sex rerum principiis are causa, ratio, natura, mundus, machina mundi, and tempus et temporalia. See De sex rerum principiis, Pars Prima (8–40), ed. Silverstein, 248–251. On the author’s sources, see Delp, “Immanence of Ratio,” 66–72. 42. Septem 960D–961B. The author’s citations from Hermes, the Sibyl, the Gospel of John, and “Augustine” in this passage stem from the fifth-century bishop Quodvultdeus, Adversus quinque haereses, III.1–21, ed. Braun, 264–268. 43. “Parmenides quoque dicit:  Deus est cui esse quidlibet quod est esse omne id quod est. Item idem: Deus est unitas: ab unitate gignitur unitatis aequalitas. Connexio vero ab unitate et unitatis aequalitate procedit. Hinc igitur Augustinus:  Omni recte intuenti perspicuum est, quare a sanctae Scripturae doctoribus Patri assignatur unitas, Filio aequalitas, Spiritui sancto connexio; et licet ab unitate gignitur aequalitas, ab utroque connexio procedat: unum tamen et idem sunt. Haec est illa trium unitas: quam solam adorandam esse docuit Pythagoras. . . . Opinor ideo cum qui illam veram unitatem considerare desiderat, mathematica consideratione praetermissa, necesse est ad intelligentiae simplicitatem animus sese erigat.” Septem 961BC. It is difficult to determine possible sources of this passage among the known texts of Thierry’s circle. The

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best conjecture seems to be Commentarius Victorinus, which includes the first sentence quoted by Parmenides, but does not attribute the triad to him:  “Et secundum theologicam affirmationis data est illa descriptio de deo a Parmenide philosopho quam utinam dedisset aliquis sanctorum: deus inquit est cui quodlibet esse quod est est esse omne id quod est.” Commentarius Victorinus 99, H 502. But Septem’s summary of the arithmetical Trinity resembles not only Commentarius Victorinus 83–85, H 498–499, but also Commentum II.38, H 80. For other possible sources of the Parmenides reference, see Häring, “Creation and Creator,” 163. 44. “Ab hac ergo summa et aeterna trinitate descendit quaedam perpetuorum trinitas. Ab unitate namque descendit materia, ab unitatis aequalitate forma, a connexione utriusque spiritus creatus, id est natura.” Septem 961C. Cf. Commentum II.39, H 80: “Ab hac igitur sancta et summa Trinitate descendit quedam perpetuorum trinitas.” See further Dronke, Fabula, 178–179. 45. See Commentum II.18–29, H 74–77, especially II.22–23 and II.28. 46. “Haec sunt tria principia, a primo principio descendentia. Unde magister meus dicit, primum principium aeternitas, quae quia immutabilis est, dicitur necessitas, secundum principium: materia quae quia apta est recipere omnes formas, dicitur possibilitas; tertium principium forma, quae quia materiam in alicujus statum terminat, dicitur finalitas; quartum principium spiritus creatus, qui, quia motus est rerum universalis, dicitur actualitas.” Septem 961CD. Cf. Glosae super Trismegistum 36, ed. Lucentini, 235:  “Nota quod aliter dicuntur omnia esse in anima, aliter in primordiali materia, aliter in mundiali machina, aliter in divina sapientia: omnia enim in anima dicuntur esse intelligibiliter, in primordiali materia materialiter, in mundo essentialiter, in divina sapientia causaliter.” 47. See Septem 961D–962D; cf. Glosa II.12–23, H 271–273. 48. Among Thierry’s extant texts, the three perpetuals, the nascent four modes, references to Pythagoras by name, and the spiritus theory of mind appear only in Commentum. Thierry’s interests in Hermes Trismegistus and in Augustine’s arithmetical Trinity are more pronounced in Commentum than in Lectiones or Glosa. See, for instance, the juxtaposition of John 1:4, a saying of Hermes, the nascent four modes, and the universal spiritus creatus in Commentum IV.7–9, H 97–98. 49. On Clarembald’s life, see Häring in Clarembald, Life and Works, 4–23; and Fortin, Clarembald of Arras, 15–30. 50. On Clarembald’s Boethius commentaries, see Jansen, Kommentar; Fortin, Clarembald of Arras; and Fortin and George, Boethian Commentaries. On his Genesis commentary, see Martello, Fisica della Creazione. 51. Clarembald, Epistola ad Odonem 2–3, 7–8, ed. Häring, 63–65. On Clarembald’s critique of Abelard and Gilbert of Poitiers, see Häring in Clarembald, Life and Works, 38–45; Jansen, Kommentar, 18–22; and Fortin, Clarembald, 44–48. Occasionally Thierry also weighed in against Gilbert; see, e.g., Lectiones II.56,

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H 173. Gibson notes that several of the conservatives reacting against Gilbert of Poitiers in the 1150s stemmed from Thierry’s circle (see “Opuscula Sacra,” 223). 52. Jansen, Kommentar, 148; cf. Häring in Clarembald, Life and Works, 51. 53. Häring in Clarembald, Life and Works, 52. As Dronke observes:  “Clarembald repeats a number of Thierry’s detailed insights regarding the natural unfolding of the universe, but does not fully grasp Thierry’s underlying principles of natural explanation. Instead, he mingles sentences drawn from Augustine, seemingly unaware of the vast difference in orientation between the two writers on Genesis, so that [Clarembald’s] tractatulus, though assembling some valuable materials, uses them for little more than a banal essay in Christian apologetics” (“Thierry of Chartres,” 381; cf. Southern, Platonism, 32). 54. I  assume Häring’s analysis that dates Clarembald’s Tractatus on De trinitate ­before the Tractatulus on Genesis (in Clarembald, Life and Works, 21–23). 55. As Ziomkowski points out, the fact that Clarembald found it necessary in his own Genesis commentary to connect Thierry’s four modes to Thierry’s hexaemeron—something Thierry never did—is further evidence that Thierry’s Tractatus predated his Boethian commentaries (“Science, Theology and Myth,” 380–382). 56. See in Clarembald’s Tractatus the Epistola ad Odonem 1–2, ed. Häring, 63; as well as his Introductio 18–19, ed. Häring, 72–73. 57. See Tractatus II.34–40, ed. Häring, 120–123; cf. Commentum II.30–II.38, H 77–80. Of the nine paragraphs on the arithmetical Trinity in Clarembald’s Tractatus, six concern aequalitas. 58. “Ab hac autem Necessitate absoluta necessitas descendit complexionis sive concatenationis cum ea, quae in absoluta necessitate complicata sunt ab aeterno, in temporum continuatione quasi concatenatis et sese complectentibus causis amministrantur.” Tractatus II.44, ed. Häring, 124–125. Clarembald repeats this definition of necessitas complexionis on two more occasions: see Tractatulus 23, ed. Häring, 236; and De Hebdomadibus III.19, ed. Häring, 201. 59. See Tractatus II.45, ed. Häring, 124; cf. Glosa II.20–21, H 273. On concatenatio see further Häring, “Creation and Creator of the World,” 143; and Asclepius III, 39, ed. Scott, 362 (cited above). 60. See Commentum II.36, II.39, II.43, H 79–82; Lectiones II.4, H 155; and Lectiones II.25, H 163. Clarembald knows this pattern well: see Tractatulus 25, ed. Häring, 237. To be fair, Thierry does state on one occasion that “inter hec autem quasi inter extrema sunt forme rerum et actualia. Forme namque rerum a deo quasi a primo descendunt principio” (Commentum II.28, H 77, my emphasis). 61. See, e.g., Lectiones II.9–10, H 157–158; and Glosa II.20, H 273. One exception might be Abbreviatio monacensis De Hebdomadibus II.39–42, H 412, but even here the notion of the second mode as descent is left implicit. 62. On the date of Clarembald’s De hebdomadibus commentary, see Häring in Clarembald, Life and Works, 19. Schrimpf provides a thorough evaluation (see Axiomenschrift, 88–118).

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63. “In necessitate enim absoluta ab aeterno omnia simplicitate quadam complicata constiterunt et in ea omnia erant quoad ipsa, ut Iohannes summus theologorum testatur:  Quod factum est, in ipso vita erat. Cumque in ipso vita esset, descendentia per necessitatem complexionis ad inmarmenen iam se in possibilitate definita manifeste depromunt ac fato subsunt.” De Hebdomadibus III.19, ed. Häring, 201. Cf. Commentum IV.8, H 97:  “Quod autem deus sit omnia testatur Iohannes Apostolus dicens omnia inquit in ipso uita erant i.e. sapientia.” This passage follows immediately after Thierry’s abortive fivefold modal system. Thierry’s appeal to John 1:4, repeated by Clarembald here, is an allusion to De genesi ad litteram V.13 (29–30), ed. Zycha, 156–157, where Augustine is constructing his distinction between rationes divinae and rationes seminales. 64. Häring notes that Clarembald “could not simply shake off the effects of Thierry’s lectures and made it a special point to reconcile ‘most views of the philosophers’ [viz. of Thierry] with the Christian truth” (“Creator and Creation,” 180). 65. See Tractatulus 1–7, ed. Häring, 227–228; cf. Häring’s comments in Clarembald, Life and Works, 21. 66. See Tractatulus 8, ed. Häring, 229; and ibid. 45, ed. Häring, 247. 67. See ibid. 11–16, ed. Häring, 230–233. 68. Häring, “Creator and Creation,” 180. 69. Clarembald’s four sections cover primordial matter (§§18–25), seminal reasons (§§26–29), the motion of time (§§30), and the four modes (§§31–32). See, e.g., Augustine, De genesi ad litteram VI.10 (17), ed. Zycha, 182–183. 70. Tractatulus 17, ed. Häring, 233. 71. See Clarembald’s discussion of perpetuitas at Tractatulus 46–49, ed. Häring, 248. By contrast, Martello interprets Clarembald’s three inchoatives as a gloss on Thierry’s four modes (see Fisica della Creazione, 309–313, 319–321). 72. See Augustine, De genesi ad litteram VI.11 (18–19), ed. Zycha, 183–185. 73. “Et haec sunt tria inchoativa quorum Creatorem Filium Dei credimus et asseveramus. . . . Filius Dei Creator universorum est tam inchoativorum quam perfectivorum. Creator inchoativorum est secundum hoc quod est Principium. Perfectivorum vero secundum hoc quod est Verbum.” Tractatulus 17–18, ed. Häring, 233. 74. See, e.g., Augustine, De genesi ad litteram I.1–6 (1–12), ed. Zycha, 1–10. Cf. Tractatulus 24, ed. Häring, 237; and Tractatulus 30, ed. Häring, 240. 75. On the second inchoative, see Tractatulus 27–29, ed. Häring, 238–239; cf. e.g., Augustine, De genesi ad litteram VI.10 (17), ed. Zycha, 182–183. On the third inchoative, see Tractatulus 30, ed. Häring, 239. 76. Lapidge suggests that Clarembald’s emphasis on seminales rationes pushes Thierry’s modal theory toward the Stoic doctrine of causation as fate (see “Stoic Inheritance,” 110–112). 77. See Tractatulus 21–23, ed. Häring, 235–236.

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78. “Haec sunt itaque quattuor rerum universitates quas Augustinus aliis nominibus significans: Omnia, inquit, per ipsum facta sunt in Verbo, in materia, in seminalibus rationibus, in opere. Verbum autem absolutam Necessitatem, hoc est divinam sapientiam vocat. Seminales rationes vocat vires occultas insertas materiae secundum quas per necessitam conplexionis alia ex aliis temporibus consuete producuntur. Materiam vero vocat possibilitatem absolutam. Opus vero possibilitatem definitam.” Ibid. 23, ed. Häring, 236. (I omit Häring’s quotation marks.) 79. “in unitate Deum i.e. Necessitatem absolutam, in binario materiam i.e. absolutam possibilitatem ratione alteritatis constitueret, in ternario qui primus omnium numerorum medio termino connectitur necessitatem conplexionis, in quaternario qui primus opere et actu tetragonus est materiam quattuor elementorum formis vestitam hoc est possibilitatem definitam intelligeret.” Ibid. 24, ed. Häring, 236. Cf. Commentum II.28, H 77; and Calcidius, Timaeus 295, ed. Waszink, 297–298. Martello proposes Eriugena as a possible source for Clarembald’s “Pythagorean” doctrine, but this seems unnecessary in light of Thierry and Calcidius (see Fisica della Creazione, 321–326). 80. See Tractatulus 21–22, ed. Häring, 235–236; cf. Tractatus II.46–47, ed. Häring, 125–126. 81. “Divina namque sapientia in materia operatur. Et sicut omnia naturalia in definita possibilitate actu et natura subsistunt ita eadem in divina sapientia sive providentia per quandam simplicitatem conplicata. Nichil in ipsa sunt nisi quod ipsa divina sapientia est. Quod etiam Iohannes evangelista, summus theologus, testatur dicens: Omnia in ipso vita erant.” Tractatulus 22, ed. Häring, 235–236. 82. See Tractatulus 23, ed. Häring, 236; cf. Tractatus II.44, ed. Häring, 124–125. 83. One liminal case is Achard of St. Victor, who discussed Thierry’s arithmetical Trinity at length in De unitate dei et pluralitate creaturarum. There are hints that he also engaged reciprocal folding in his concept of rationes explicatrices, but this portion of his book has been lost. See, e.g., Achard of St. Victor, De unitate dei et pluralitate creaturarum I.39–42, ed. Martineau, 108–112. 84. See Hoenen, “ ‘Ista prius inaudita’ ”; and Albertson, “Learned Thief.” 85. The following pages summarize the argument of Albertson, “Late Medieval Reaction.” 86. If on the basis of future manuscript discoveries Fundamentum naturae is somehow proven to be an early work of Cusanus himself, all of these observations will remain valid. It seems inconceivable, future discovery or not, that De docta ignorantia precedes Fundamentum, for reasons that Hoenen has explained well (see “ ‘Ista prius inaudita’ ”). Even if Nicholas is the author of Fundamentum and De docta ignorantia, then his earlier, indubitably negative reading of Chartrian sources in Fundamentum would still be just as valuable as an initial reception of Thierry’s theology. Indeed, given the self-contradiction that the positive reception of Thierry’s ideas in De docta ignorantia would in that

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case represent for Nicholas as an author, a careful study of Fundamentum would become even more important, not less. The key here is to see that whoever its author may be, Fundamentum was written to refute Thierry’s theology.   87. See F 4r, 448. The four sections correspond to DI II.7–10. The introductory part appears at the end of DI II.7; the subsequent three roughly correspond to DI II.8–10, with extensive additions by Cusanus, as I will discuss in Chapter 7.   88. “Est enim modus essendi, qui absoluta necessitas dicitur, uti deus est forma formarum, ens entium, rerum ratio sive quiditas, et in hoc essendi modo omnia in deo sunt ipsa necessitas absoluta. Alius modus est, ut res sunt in necessitate complexionis, in qua sunt formae rerum in se verae cum distinctione et ordine naturae, sicut in mente. Alius modus essendi est, ut res sunt in possibilitate determinata actu hoc vel illud. Et quartus ultimus modus essendi est, ut res possunt esse, et est possibilitas absoluta.” F 4r, 448.   89. “Tres ultimi modi essendi sunt in una universitate,” F 4r, 448.  90. “Quos tres modos ultimos aliquantulum discutiamus a possibilitate inchoantes. . . . Et tantum de possibilitate seu materia universi.” F 4r–6v, 448–458.   91. “Anima sive forma universi seu mundi non est elementum nec ex elemento. . . . Et haec de anima mundi seu forma universi sufficiant.” F 6v–8r, 460–468.   92. “Motum, per quem est conexio formae et materiae, spiritum quendam esse. . . . Et hoc de spiritu conexionis seu virtute universi sufficiant.” F 8v–9v, 468–476. The manuscript ends here.  93. Thierry attributes the plurality and mutability of the universe to its fourth mode: see Glosa II.17, H 272.  94. See Vescovini, “Temi ermetico-neoplatonici”; and Catana, Concept of Contraction, 103–134.   95. See F 5v, 456.   96. “Ita quidem motus gradatim de universo in particulare descendit et ibi contrahitur ordine temporali aut naturali.” F 8v, 470.  97. Ibid.   98. See F 9r, 472.  99. “Et nulla potest esse creatura, quae non sit ex contractione diminuta, ab isto opere divino per infinitum cadens. Solus deus est absolutus, omnia alia contracta. Nec cadit eo modo medium inter absolutum et contractum, ut illi Platonici imaginati sunt, qui animam mundi mentem putarunt post deum et ante contractionem mundi.” F 8r, 468. 100. “Cadunt autem differentiae et graduationes, ut unum actu magis sit, aliud magis potentia, absque hoc quod deveniatur ad maximum et minimum simpliciter, quoniam maximus et minimus actus coincidunt cum maxima et minima potentia ut sunt maximum absolute dictum.” F 5v, 456. 101. See, e.g., Metaphysics 990b1–991b10. 102. See Physics 195b30–198b8. 103. See F 5v, 456.

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104. “Quare possibilitas absoluta in deo est deus, extra vero non est possibilis.” F 5r, 454. 105. See F 7r, 460. 106. “Unde necessitas complexionis non est, ut posuerunt Platonici, scilicet mens minor gignente, sed est verbum et filius aequalis patri in divinis, et dicitur logos seu ratio, quoniam est ratio omnium.” F 7v–8r, 466. 107. “Unum enim infinitum exemplar tantum est sufficiens et necessarium, in quo sunt omnia ut ordinata in ordine, omnes quantumcumque distinctas rerum rationes adaequatissime complicans,” F 7v, 466. 108. See F 7r–8r, 466–468. 109. “Philosophi quidem de verbo divino et maximo absoluto sufficienter instructi non erant. Ideo mentem et animam ac necessitatem in quadam explicatione necessitatis absolutae sine contractione considerarunt.” F 8r, 468. 110. See F 7v, 466; and F 8r, 468. In Fundamentum’s third part, the author also uses complicatio to define the divine rest from which all motion proceeds: “Non est ergo aliquis motus simpliciter maximus, quia ille cum quiete coincidit. Quare non est motus aliquis absolutus, quoniam absolutus motus est quies et deus. Et illa quies complicat omnes motus” (F 9r, 472). 111. See F 9r, 472. 112. McTighe’s analysis of DI II.8 (139) and DI II.9 (148) reveals the same rejection of the fourth and second modes (see “Contingentia and Alteritas,” 60–61). Yet without the perspective added by the Fundamentum treatise, McTighe mistakenly considers this rejection inherent to reciprocal folding itself, whether in Boethius, Thierry, or Nicholas—a surmise belied by the example of Thierry himself (see “Neglected Feature of Neoplatonic Metaphysics,” 30–34; cf. Lectiones II.66, H 176). 113. See Septem 960C–960D. 114. “de quibus nisi sciatur quid teneri debeatur multae haereses proveniunt quem­ admodum ex ignorantia creationis rerum haeresis Euticiana et Nestoriana orta sunt.” Tractatulus 11, ed. Häring, 230. 115. “Haec omnes quaerunt, sed non inveniunt, quoniam modus inquisitionis et ignorantia veritatis eis obsistunt.” Septem 964D; cf. Septem 962D. 116. See Lectiones II.3–15 passim, H 155–159.

chapt er 7   1. See Watanabe, Nicholas of Cusa:  A  Companion, 284–289; and Meuthen, Nicholas of Cusa, 54–56.   2. See Bond, “Nicholas of Cusa from Constantinople to ‘Learned Ignorance’,” 143.   3. See Meuthen, Nicholas of Cusa, 80–83; cf. Cassirer, Individuum und Kosmos, 64. To get a sense of Cusanus’s reformist activities (as opposed to ideas), see Sullivan, “Nicholas of Cusa as Reformer”; and Pavlac, “Reform.”

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  4. See Meuthen, Nicholas of Cusa, 26–29, 33–34; and Aris, “Leser im Buch.”   5. Watts provides a fascinating tour through the cardinal’s library (Nicolaus Cusanus, 13–24).   6. See “Epistola auctoris,” DI III (263), 98–100. Regarding citations to De docta ignorantia, the revised edition and translation by Paul Wilpert and Hans Gerhard Senger, Nikolaus von Kues. Philosophisch-Theologische Werke, vol. 1, supersedes the original Heidelberg Cusanus edition upon which it is based (and which I otherwise use exclusively). This 2002 volume incorporates all three books of De docta ignorantia published individually between 1994 and 1999, such that each book is paginated separately. I cite accordingly by book, paragraph number, and page number to the respective portion of the Wilpert-Senger 2002 edition.   7. Honecker calculates that between February 1438 and February 1440, Nicholas had only three separate stretches of leisure to compose De docta ignorantia, totalling perhaps eight months (see “Entstehungszeit der ‘Docta ignorantia’ ”; cf. Bond, “Nicholas of Cusa from Constantinople to ‘Learned Ignorance’,” 151–154). Honecker speculates that Cusanus acquired a Ps.-Dionysian text to read on the sea voyage back to Italy. But of course De concordantia catholica is already influenced by Ps.-Dionysius, and Nicholas did not yet read Greek. Klibansky reaches similar conclusions (“Geschichte der Überlieferung,” 216–219). Even after crediting the revelation, Bond concludes: “The weight of official responsibilities might have deprived Cusanus of intense theological engagement had not the illumination on shipboard provided a vision of the divine mysteries that demanded telling. Yet neither Cusanus’ recorded activities nor his correspondence provide evidence of the development or process of composition of De docta ignorantia” (“Nicholas of Cusa from Constantinople to ‘Learned Ignorance’,” 162). Boyle considers Nicholas’s testimony a rhetorical topos (see “Cusanus at Sea”).  8. Senger, Philosophie des Nikolaus von Kues, 7.   9. On Pythagorean trends in the Renaissance, see Heninger, “Some Renaissance Versions of the Pythagorean Tetrad”; Celenza, Piety and Pythagoras in Renaissance Florence; Celenza, “Pythagoras in the Renaissance”; and Joost-Gaugier, Pythagoras and Renaissance Europe, 19–31, 65–92. 10. See Monfasani, “Nicholas of Cusa, the Byzantines, and the Greek Language,” 222–223. 11. On the mathematical sources informing Nicholas’s education, see Stuloff, “Mathematische Tradition in Byzanz”; Hofmann, “Mutmassungen über das früheste mathematische Wissen”; and Folkerts, “Die Quellen und die Bedeutung.” On Cusanus’s early scientific efforts before 1440, see Senger, Philosophie des Nikolaus von Kues, 106–154; and Böhlandt, Verborgene Zahl, 41–82. 12. See Vescovini, “Cusanus und das wissenschaftliche Studium in Padua.” 13. See Müller’s excellent account of the Wissensnetwerk of Toscanelli, Alberti, and Cusanus in Perspektivität und Unendlichkeit, 15–33.

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14. On Beldomandi, see Senger, Philosophie des Nikolaus von Kues, 150–153; and Newsome, “Quadrivial Pursuits,” 142–208. 15. See Rose, Italian Renaissance of Mathematics, 29–30; and Watanabe, Nicholas of Cusa: A Companion, 238–239. Yet Regiomontanus was sufficiently interested in Cusanus’s approach that he began an unfinished work from 1464 with a tribute to docta ignorantia (see Zinner, Regiomontanus, 76–78). On Nicholas’s encounters with contemporary mathematicians, see Hofmann, “Über Regiomontans und Buteons Stellungnahme”; Nagel, Nicolaus Cusanus und die Entstehung der exakten Wissenschaften, 86–96; Nicolle, Mathématiques et Métaphysique, 250–262; and especially De Bernart, Cusano e i matematici. Nicolle has argued that Cusanus’s theological conceptions of the infinite, unity (the arithmetical Trinity), and proportionality (ratio, similitudo, aequalitas) represent metaphysical obstacles that ultimately frustrate his geometrical efforts (Mathématiques et Métaphysique, 188–248). Hofmann nevertheless considers Nicholas’s achievements impressive, given his scant mathematical education and the fact that few others in his century attempted the quadrature of the circle (“Sinn und Bedeutung,” 396). 16. See Haubst, “Der junge Cusanus war im Jahre 1428”; and Colomer, “Zu dem Aufsatz von Rudolf Haubst.” 17. See Haubst, “Albert, wie Cusanus ihn Sah”; Haubst, “Fortleben Alberts des Grossen”; Hoenen, “Academics and Intellectual Life”; and Hoenen, “Heymeric van de Velde (†1460).” 18. See Albertson, “In Search of Unity.” 19. See Meier, “Figura ad oculum demonstrata”; and Zorach, Passionate Triangle, 73–80. 20. On Cusanus’s debt to Heymeric, see Colomer, Nikolaus von Kues und Raimund Llull, 5–46; Colomer, “Nikolaus von Kues und Heimeric van den Velde”; Hoenen, “Tradition and Renewal”; and now especially Hamann, Siegel der Ewigkeit, 230–262. 21. On Proclus, see e.g., Klibansky, Proklos-Fund; and Beierwaltes, “ ‘Centrum tocius vite’.” On Eriugena, see e.g., Riccati, “Processio” et “explicatio”; Beierwaltes, “Eriugena und Cusanus”; and the several studies in Kijewska, et al., Eriugena—Cusanus. 22. See DI I.9 (26), 36. 23. See “Epistola auctoris,” DI III (264), 100. Cf. DI I.2 (5–7), 10–12; and DI III.1 (182), 2–4. 24. See Hoenen, “ ‘Ista prius inaudita’,” 401–402. 25. See Albertson, “Learned Thief,” 378–389. 26. Ibid., 374–375. 27. Schnarr’s much-cited monograph on the four modes of being addresses De docta ignorantia and De coniecturis but neglects De mente and De ludo globi altogether (see Modi essendi). In their place, Schnarr examines unrelated instances

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of the term “modi essendi” in De venatione sapientia XVII (49), ed. Klibansky and Senger, 46; ibid. XXVI (78), 75; and ibid. XXXIX (116), 107–108. Other usages unconnected to the Chartrian legacy include De beryllo 22, ed. Senger and Bormann, 25–26; and De principio 39, ed. Bormann and Riemann, 56. Schnarr could not perceive that the four modes in De docta ignorantia were framed by Fundamentum through the three perpetuals (Modi essendi, 24–39). Subsequently he characterized the Cusan modal theory in De docta ignorantia as “ternary,” in contrast to the “quaternary” modes of De coniecturis (ibid., 167). Thomas commits the same mistake when, following Schnarr, he judges the fourfold epistemological modes in De coniecturis more highly than the apparently threefold ontological modes in De docta ignorantia—underscoring the hazards of reading DI II.7–10 without accounting for Thierry’s differentiated reception history (see Teilhabegedanke, 63–67). While attempting to establish that fourteenth-century scholastic physics informs De docta ignorantia, Powrie attributes ideas to Cusanus that may have originated with Fundamentum instead (see “Importance of Fourteenth-Century Natural Philosophy,” 46–51). Other older studies likewise require updating, e.g., Hirschberger, “Platonbild”; McTighe, “Meaning of the Couple”; McTighe, “Neglected Feature of Neoplatonic Metaphysics”; Herold, Menschliche Perspektive, 16–25; De Gandillac, “Explicatio-Complicatio”; Winkler, “Amphibolien des cusanischen All-Einheitsdenkens”; Leinkauf, “Bestimmung des Einzelseienden”; and even Dupré, Passage to Modernity, 58–60. 28. See, e.g., Rusconi, “Commentator Boethii,” 272; cf. Albertson, “Late Medieval Reaction.” 29. Takashima compares the doctrine of anima mundi in Thierry’s Tractatus and DI II.10 without appreciating the hermeneutical obstacles now posed by Fundamentum (see “Nicolaus Cusanus und der Einfluss der Schule von Chartres,” 102–103). Counet understands that the four modes in De docta ignorantia are continued in De coniecturis and De mente, and that they originate with Thierry of Chartres. He also detects “les approfondissements et les changements” worked by Cusanus on Thierry’s ideas in DI II.8 and II.9 (Mathématiques et dialectique, 173–176). But without the guidance provided by the Fundamentum treatise, Counet’s interpretations sometimes run into problems. First, he interprets DI II.8 (136) as Nicholas identifying the fourth mode with God (cf. Albertson, “Late Medieval Reaction,” 65–73; and McTighe, “Contingentia and Alteritas,” 60–61). On this basis Counet asserts that Cusanus wished to assimilate Thierry’s four modes of being to Eriugena’s four modalities of nature (Mathématiques et dialectique, 177–78). Second, Counet reads DI II.9 (145, 149) as Nicholas’s sincere rejection of the world-soul. According to Counet this antirealism opened up the possibility of what Rombach calls a functionalist ontology (Mathématiques et dialectique, 178–184). Yet Counet fails to consider the later occasions when Cusanus shows deep interest in the world-soul (ibid., 185). See, e.g., DM XIII (145–146), 198–199; and LG I (40–43), 45–49.

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30. Moritz briefly but trenchantly engages the Hoenen theory, listing three objections before proceeding with his own interpretation of Book II (see Explizite Komplikationen, 234–235). This marks the most substantive critical response to Hoenen since Pauli and Senger. Rusconi has more frequently addressed the Hoenen controversy but her analysis is problematic. She attempts to revive Pauli’s suggestion that a common source stands behind Cusanus and Fundamentum alike (see Albertson, “Learned Thief,” 361). According to Rusconi, this source is none other than Thierry of Chartres (“Commentator Boethii,” 268; cf. “El uso simbólico,” 46). This proposal has two major problems. First, a common source is meaningful in this context only to the extent that it resembles the derivative texts closely and thus illuminates their provenance (Boethius is also a common source). As Pauli and Hoenen surely appreciated, DI II.7–10 and Fundamentum resemble each other far more than they do Thierry’s commentaries (see further Albertson, “Learned Thief,” 377–378). Second, Fundamentum does not simply transmit Thierry’s doctrines but contradicts them (see Albertson, “Late Medieval Reaction”). Consequently Rusconi’s interpretations of DI II.7–10 run up against several difficulties. Initially she holds that when Nicholas appropriated Thierry’s four modes in DI II.7–10, his opposition of absolute and contracted being did not contradict the Platonists’ modal theory but defended and qualified it (“Natürliche und künstliche Formen,” 259). After investigating it further, Rusconi discovers that DI II.7–10 in fact discards Thierry’s fourth and second modes (see “Commentator Boethii,” 289). Ultimately Rusconi holds open the possibility that Cusanus did not use Thierry’s commentaries at all, but rather “irgend­eine Zusammenfassung dieser Kommentare, die wir nicht mehr zur Verfügung haben” (ibid., 290). But this is just what the Fundamentum treatise represents. 31. Hirschberger explains a parallel problem regarding putative citations of Plato (“Platonbild,” 114–115). 32. See Septem 961C. 33. Cusanus begins his discussion of the arithmetical Trinity as follows: “Pythagoras autem, vir suo aevo auctoritate irrefragabili clarissimus, unitatem illam trinam astruebat. . . . Et haec est illa trina unitas, quam Pythagoras, omnium philosophorum primus, Italiae et Graeciae decus, docuit adorandam. Sed adhuc aliqua de generatione aequalitatis ab unitate subiungamus expressius.” DI I.7 (18, 21), 26–30. Cf. Septem 961C: “Haec est illa trium unitas: quam solam adorandam esse docuit Pythagoras.” Then several pages later Cusanus concludes:  “iuxta Pythagoricam inquisitionem trinitatis in unitate et unitatis in trinitate semper adorandae. . . . Nunc inquiramus, quid sibi velit Martianus, quando ait philosophiam ad huius trinitatis notitiam ascendere volentem circulos et sphaeras evomuisse.” DI I.9–10 (26–27), 34–36. Cf. Septem 961C: “De qua Marcianus dicit, quod cum eam adorare vellet philosophia, evomuit circulos et sphaeras.” 34. See DI I.10 (28–29), 36–40; and DI I.12 (33), 44–46. Cf. Septem 961C: “Opinor ideo cum qui illam veram unitatem considerare desiderat, mathematica

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consideratione praetermissa, necesse est ad intelligentiae simplicitatem animus sese erigat.” 35. See DI II.7 (128–130), 50–54. 36. The triad appears in Commentum II.37–38, H 80; Lectiones V.16–17, H 218; Lectiones VII.5–7, H 224–225; Glosa V.17–29, H 296–299; Tractatus de Trinitate 12–19, H 306–307; Commentarius Victorinus 81–85, H 498–499; and Septem 961B. 37. See DI I.7 (18–19), 26–28. Cf. Tractatus 30–31, H 568; and Tractatus 39–40, H 571– 572. Duhem first discovered Nicholas’s dependence on Tractatus (see “Thierry de Chartres”). Reinhardt provides a more detailed confirmation, but overlooks the centrality of Septem and Commentarius Victorinus (see “Gemeinsamkeiten und Unterschiede”). Cusanus surely encountered Augustine’s triad earlier through more common scholastic sources. As Haubst has discovered, around 1428 Nicholas studied excerpts from Aquinas’s Summa theologiae that compared similar triads (see Haubst, “Thomas- und Proklos-Exzerpte,” 23–24; and Vansteenberghe, “Quelques lectures,” 282). But this evidence only brings into relief the very different meaning given to Augustine’s triad by Thierry of Chartres (and followed by Nicholas in De docta ignorantia), which runs against the grain of the standard “appropriation” interpretation of Peter Lombard, Thomas Aquinas, and most others (see McGinn, “Unitrinum Seu Triunum,” 94, 102; and Albertson, “Achard of St. Victor”). 38. See DI I.7 (20–21), 28–30; cf. Tractatus 37–38, H 570–571. It would be unusual in the mid-twelfth century to call the Father the causa of the Spirit. 39. “Unitas dicitur quasi ὠντας ab ὠν [sic] Graeco, quod Latine ens dicitur. Et est unitas quasi entitas. Deus namque ipsa est rerum entitas.” DI I.8 (22), 30. See the comments on Greek orthography in the Heidelberg edition of De docta ignorantia I.8 (22), ed. Hoffmann and Klibansky, 17. The closest version in Thierry’s writings comes from Commentum II.22, H 75: “Hec [necessitas] autem a Platone eternitas, ab aliis unitas quasi onitas ab on Greco i.e. entitas, ab omnibus autem usitato uocabulo appellatur deus.” Cf. Lectiones II.48, H 170; Lectiones VII.5, H 224; and Glosa V.18, H 297. 40. See DI I.8 (22), 30–32. Forma essendi recurs throughout Tractatus, Commentum, Lectiones, and Glosa. But Cusanus’s formulation points toward Commentum: “Aequalitas vero essendi est, quod in re neque plus neque minus est, nihil ultra, nihil infra.” DI I.8 (22), 30–32. “In rebus enim nichil ultra est quam dei sapientia comprehendat, nichil infra nichil plus nichil minus. Dei ergo sapientia essendi est equalitas. . . . Hec uero eadem unitatis equalitas ab unitate gignitur per semel i.e. per integritatem et perfectionem eo scilicet quod nichil ultra nichil infra nichil plus nichil minus est quam sit in dei sapientia. Unde Pater in Uerbo creat omnia i.e. in existendi equalitate. Sic enim creat omnia ut in eis nichil ultra sit nichil infra,” Commentum II.31, II.35, H 78–79. Cf. Tractatus 42–46, H 573–575; Lectiones VII.6, H 225; and Glosa V.19–20, H

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297. Cusanus’s later repetition of the arithmetical Trinity apparently uses the same passage in Commentum: see DI I.24 (80–81), 102. 41. On “hoc, id, idem,” see DI I.9 (25), 34; cf. Commentarius Victorinus 128–131, H 507–508. On Parmenides, see DI I.23 (70–71), 92–94; cf. Commentarius Victorinus 99, H 502, as well as Septem 961B. 42. See Commentarius Victorinus 97–112, H 501–504. The anonymous author probably learned this from Thierry’s commentary on Boethius’s tractate Contra Eutychen. See Abbreviatio Monacensis Contra Eutychen III.62–63, in Thierry, Commentaries on Boethius, ed. Häring, 465; and Fragmentum Londinense: Contra Eutychen III.62–63, in ibid., ed. Häring, 246. 43. See DI I.26 (87–88), 110–112. 44. Note that Commentarius Victorinus survives with Lectiones in the same codex, Paris BN Lat 14489. Cusanus makes use of the former in De docta ignorantia but apparently neglects the latter; for instance, he never uses Lectiones to correct Fundamentum’s tendentious reading of the four modes. So this may not have been the volume he used. Yet one must still contend with the tantalizing detail that the codex was purchased by Prior John Lamasse (d. 1458) for the library of Saint Victor around the same time that Cusanus was composing his 1440 work (see Häring in Thierry, Commentaries on Boethius, 27–28, 38–40). 45. These findings concur with Haubst’s early conclusions. Haubst names Tractatus, Commentum, and Commentarius Victorinus as the major sources of the arithmetical Trinity in Book I (Bild des Einen, 234–235). For the trinity of perpetuals in Book II, he names Tractatus, Commentum, Glosa, and Septem (ibid., 99); as I will show, the apparent reference to Glosa is better explained through Fundamentum. Haubst has since conjectured: “Insgesamt bestätigt all das die Vermutung, daß Nikolaus eine Kompilation oder Exzerpten-Sammlung aus den Boethius-Kommentaren Thierrys, die er eifrig benutzte, vorlag. Vielleicht war es auch ein Sammelkodex. Doch den Namen des Verfassers verriet dann auch dieser nicht” (Streifzüge, 174). 46. Gilson, History of Christian Philosophy, 540. Likewise, after noting the “unzeitgemässe Auftreten” of Cusanus’s affinity with Boethius, Wilpert remarks: “Dass dieser Versuch [the Christian Neoplatonism of Boethius and Ps.-Dionysius] im neunten und zwölften Jahrhundert weitergeführt . . . ist verständlich. Kann aber ein Denker des fünfzehnten Jahrhunderts meinen, mit diesem Rückgriff auf eine ehrwürdige, aber seit drei Jahrhunderten unterbrochene Tradition die Probleme seiner Zeit zu lösen?” (“Philosophiegeschichtliche Stellung,” 391). 47. “Unde ait commentator Boethii De Trinitate, vir facile omnium, quos legerim, ingenio clarissimus,” Apologia doctae ignorantiae 35, ed. Klibansky, 24. See further Albertson, “Learned Thief,” 378–389. 48. “Quare in illis sapientes exempla indagandarum rerum per intellectum sollerter quaesiverunt, et nemo antiquorum, qui magnus habitus est, res difficiles alia similitudine quam mathematica aggressus est, ita ut Boethius, ille

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Romanorum litteratissimus, assereret neminem divinorum scientiam, qui penitus in mathematicis exercitio careret, attingere posse. Nonne Pythagoras, primus et nomine et re philosophus, omnem veritatis inquisitionem in numeris posuit? Quem Platonici et nostri etiam primi in tantum secuti sunt, ut Augustinus noster et post ipsum Boethius affirmarent indubie numerum creandum rerum ‘in animo conditoris principale exemplar’ fuisse. . . . Aurelius etiam Augustinus Platonicus, quando de quantitate animae et eiusdem immortalitate et ceteris altissimis investigavit, ad mathematica [sic] pro adiutorio convolavit. Ista via Boethio nostro adeo placere visa est, ut constanter assereret omnem veritatis doctrinam in multitudine et magnitudine comprehendi.” DI I.11 (31–32), 42–44; trans. Bond, 101 (modified). Albert the Great referenced the same quote from Boethius during his exposition of Pythagorean number theory in his Metaphysics commentary (see Metaphysica I, Tract. 4, Cap. 2, ed. Geyer, 50). 49. Cusanus repeats this genealogy in the late work, De venatione sapientiae XXI (59), ed. Klibansky and Senger, 56–57. On the centrality of Augustine and Boethius as Cusan sources, see Haubst, Bild des Einen, 206–211; Santinello, Pensiero di Nicolò Cusano, 54–62; and Santinello, “Mittelalterliche Quellen.” 50. Horn praises Cusanus’s grasp of the Pythagorean roots of Platonic number theory in this passage and in related loci from De mente and De ludo globi (see “Cusanus über Platon”). 51. “Sed sicut ex formula uerborum haberi potest uolens Augustinus quoquo modo insinuare quod ineffabile erat et incomprehensibile confugit ad mathematicam.” Commentarius Victorinus 81, H 498. 52. Like other late medieval schools, the Albertists favored by Heymeric and Nicholas claimed ancient philosophical heroes as their partisans. While the nominalists preferred Heraclitus and Epicurus, and the Scotists preferred Plato, the Thomists and Albertists praised Boethius above all (see Kaluza, Querelles doctrinales à Paris, 15–17). 53. See, e.g., DI I.1 (3), 6–8. 54. Compare Offerman’s outline of Books I  and II in Christus—Wahrheit des Denkens, 59–60, 109–110, respectively. 55. Clearly the parallels I have drawn between the actual composition of Book I and the hypothetical integration of the Fundamentum treatise within Book II are not dispositive. But it does lend a great deal more respectability to Hoenen’s contention that the cardinal inserts Chartrian source material in the middle of the second book of De docta ignorantia without attribution and then proceeds to develop his own ideas on that basis. Since this is demonstrably what occurs in Book I, it is difficult to object prima facie to a comparable occurrence in Book II. 56. See Haubst’s perceptive observation about Apologia 35:  “Trotzdem wird man eher damit rechnen müssen, daß Cusanus einen uns unbekannten Kommentar benutzte. Das hohe Lob, das er diesem zollt, läßt sogar daran denken, daß er

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in diesem auch das Gedankengut Thierrys und Ps.-Bedas, die er sonst nie erwähnt, vereint vorfand” (Bild des Einen, 249 n. 133). 57. On the arithmetical Trinity in DI I.7–10, see Haubst, Bild des Einen, 231–254; Haubst, Streifzüge, 275–289; Santinello, Pensiero di Nicolò Cusano, 132–140; Beierwaltes, “Einheit und Gleichheit”; and McGinn, “Unitrinum Seu Triunum.” Counet reads the triad as the ground of coincidentia oppositorum, such that unity, equality, and connection signify the three moments of identity of opposites, equalization of inequal opposites, and reciprocal immanence of opposites (see Mathématiques et dialectique, 91–92, 208–209, 415–416). On geometrical figures in DI I.11–23, see Haubst, Bild des Einen, 262–284; Counet, Mathématiques et dialectique, 191–198; Bergmans, “Nicholas of Cusa’s Vanishing Geometrical Figures”; Yamaki, “Bedeutung geometrischer Symbole”; Nicolle, “How to Look at Cusanus’ Geometrical Figures”; and Böhlandt, Verborgene Zahl, 92–103. On cosmological speculations in DI II.11–12, see Meurers, “Nikolaus von Kues und die Entwicklung des astronomischen Weltbildes”; Brient, Immanence of the Infinite, 204–218; Counet, Mathématiques et dialectique, 224–246; and especially Krafft, “Kosmologische Weltbild.” 58. See DI I.1 (2–4), 6–8; DI I.24 (77, 80), 98–102; and DI I.26 (86–88), 108–112. 59. On Cusanus’s efforts to explain Fundamentum’s concepts of maximum and contractio, see Albertson, “Learned Thief,” 367–372. 60. “Cadunt autem differentiae et graduationes, ut unum actu magis sit, aliud magis potentia, absque hoc quod deveniatur ad maximum et minimum simpliciter, quoniam maximus et minimus actus coincidunt cum maxima et minima potentia ut sunt maximum absolute dictum.” F 5v, 456 = DI II.8 (137), 60–62. There is a textual problem in the last clause, where one would expect a subjunctive verb. The Prague manuscript substitutes et for ut: see De docta ignorantia II.8 (137), ed. Hoffmann and Klibansky, 88:22. 61. See F 9r, 472. 62. “Et quoniam aequalitatem reperimus gradualem, . . . patet non posse aut duo aut plura adeo similia et aequalia reperiri, quin adhuc in infinitum similiora esse possint.” DI I.3 (9), 14; trans. Bond, 90. 63. Bocken, “Zahl als Grundlage,” 219. On the theme of incommensurability in Cusanus, see further Hirschberger, “Prinzip der Inkommensurabilität”; Müller, Perspektivität und Unendlichkeit, 94–103; and Powrie, “Importance of Fourteenth-Century Natural Philosophy.” 64. See DI I.3-4 (10–11), 14–16. 65. See DI I.4 (11), 16; cf. a more developed version of this idea at DI II.1 (95–96), 8–10. 66. Nicolle suggests that this theological concept of Equality engendered confusion in Cusanus’s later geometrical works on the quadrature of the circle. By conflating quantitative equivalence with pure qualitative identity, it encouraged Nicholas to read every geometrical difference as an inequality (see “Égalité et identité,” 158–160).

362 67. 68. 69. 70. 71. 72.

Notes

See DI I.11 (30), 40. See DI I.16 (46), 62. See DI I.17 (49), 66. See DI I.24 (80), 102. See DI II.1 (91-92), 4–6. “Ascende hic quomodo praecissima maxima harmonia est proportio in aequalitate, quam vivus homo audire non potest in carne, quoniam ad se attraheret rationem animae nostrae,” DI II.1 (93), 6; trans. Bond, 129. 73. See DI II.1 (94), 8. 74. See DI II.1 (91), 4. 75. “Est autem deus arithmetica, geometria atque musica simul et astronomia usus in mundi creatione, quibus artibus etiam et nos utimur, dum proportiones rerum et elementorum atque motuum investigamus. Per arithmeticam enim ipsa coadunavit; per geometriam figuravit, ut ex hoc consequerentur firmitatem et stabilitatem atque mobilitatem secundum condiciones suas; per musica proportionavit taliter, ut non plus terrae sit in terra quam aquae in aqua et aëris in aëre et ignis in igne, ut nullum elementorum in aliud sit penitus resolubile.” DI II.13 (175), 108; trans. Bond, 166. The ancient theme of God as geometer appears already in Cassiodorus, who imagined that “the Holy Trinity employs geometry [geometrizat] when it grants various species and forms to its creatures which it has even now caused to exist” (Zaitsev, “Meaning of Early Medieval Geometry,” 540). As McEvoy has shown, Robert Grosseteste (d. 1253)  described God as Numerator and Mensurator primus whose mind contains the infinities that ground the geometrical space of creation (see Philosophy of Robert Grosseteste, 168–180). McEvoy is right to emphasize both the rarity of such mathematical theologoumena in medieval Latin theology and their profound importance for the history of early modern science (ibid., 213–215). 76. “Admirabili itaque ordine elementa constituta sunt per deum, qui ‘omnia in numero, pondere et mensura’ creavit. Numerus pertinet ad arithmeticam, pondus ad musicam, mensura ad geometriam. . . . Quis non admiraretur hunc opificem, qui etiam tali siquidem arte in sphaeris et stellis ac regionalibus astrorum usus est, ut sine omni praecisione cum omnium diversitate sit omnium concordantia,” DI II.13 (176, 178), 110–112; trans. Bond, 166–167. On music in these passages, see Hüschen, “Nikolaus von Kues und sein Musikdenken.” 77. See Tractatus 30, H 568. 78. Führer, “Evolution of the Quadrivial Modes,” 334. On the quadrivium in Cusanus, see Schulze, Zahl, Proportion, Analogie, 93–121; and Schulze, Harmonik und Theologie, 27–32, 69–76. In opposition to my interpretation, Vengeon envisions a Cusan “humanism of the quadrivium” celebrating the Promethean power of the human spirit (Nicolas de Cues, 95–106). Führer detects an inchoate “quadrivial cosmology” in De docta ignorantia that gives way to a fully developed “quadrivial psychology” and “quadrivial theology” in De mente, but this

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seems too stark of a contrast given the Boethian and Chartrian basis shared by both works.  79. Breidert makes a similar remark: “Gott ist der Magister, der Mensch der Schüler; nur in uneigentlichem Sinne erschafft der Mensch die artes. Indem der menschliche Geist die mathematicalia erschafft, expliziert er nur das ursprünglich in ihm Eingefaltete” (“Mathematische und symbolische Erkenntnis,” 125). Or Führer again: “the human mind is quadrivial in its essential constitution and is therefore able to participate in the divine mind, which is also quadrivial” (“Evolution of the Quadrivial Modes,” 331). 80. See DI II.1 (97), 12. While Cusanus discusses infinity as both a mathematical and theological concept, one must take care, as Breidert has warned, not to read the status of the infinite in seventeenth-century mathematics back into Cusanus (see “Mathematische und symbolische Erkenntnis,” 122–123). Enders has provided a thorough overview of the literature in “Unendlichkeit und All-Einheit.” See further Alvarez-Gómez, Verborgene Gegenwart des Unendlichen; Harries, Infinity and Perspective; Hösle, “Platonism and Anti-Platonism”; Brient, Immanence of the Infinite; and Vengeon, Nicolas de Cues, 17–52. 81. On complicatio and explicatio in Cusanus, see Riccati, “Processio” et “explicatio,” 110–122; De Gandillac, “Explicatio-Complicatio”; and Counet, Mathématiques et dialectique, 80–84. The most sustained analysis is Moritz, Explizite Komplikationen, 27–90. But he dismisses the “Neoplatonic henology” of Thierry of Chartres, Ps.-Dionysius, or Proclus as possible sources of the “holism” of Cusanus’s reciprocal folding, crediting Aristotle, Duns Scotus, and Meister Eckhart instead (see Explizite Komplikationen, 172–239). 82. See, respectively, DI I.22 (67–69), 88–90; DI I.24 (75–79), 96–102; and DI III.11 (244), 74–76. 83. See DI II.2 (100–103), 16–20. It is no accident that in his questions Nicholas relies upon Thierry’s hallmarks:  forma essendi in DI II.2 (102), and forma formarum in DI II.2 (103). 84. “Deus ergo est omnia complicans in hoc, quod omnia in eo. Est omnia explicans in hoc, quod ipse in omnibus.” DI II.3 (107), 24; trans. Bond, 135. Cf. DI II.3 (111), 28: “licet etiam scias deum omnium rerum complicationem et explicationem, et—ut est complicatio—omnia in ipso esse ipse, et—ut est explicatio— ipsum in omnibus esse id quod sunt sicut veritas in imagine.” 85. “Excedit autem mentem nostram modus complicationis et explicationis.” DI II.3 (109), 26; trans. Bond, 136. 86. See DI II.3 (105–106), 22–24. In light of the Stuttgart commmentary on the Institutio arithmetica recently discovered by Caiazzo, Cusanus’s attempt to connect the Boethian quadrivium directly to Thierry’s complicatio and explicatio reflects the cardinal’s similar orientation and emphasis. 87. “Sicut igitur ex nostra mente per hoc, quod circa unum commune multa singulariter intelligimus, numerus exoritur, ita rerum pluralitas ex divina mente,

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in qua sunt plura sine pluralitate quia in unitate complicante.” DI II.3 (108), 24; trans. Bond, 135 (modified). This curious passage stands in tension with the exclusion of plural exemplars found in DI II.9 (148), 74 = F 7v, 466. Cusanus revisits the idea in DC I.2 (7), 12, where, however, he changes mens divina to deus mens infinita. It also bears a strong resemblance to Thierry’s discussion of mens divina and plural forms in Lectiones II.66, H 176, discussed above. 88. See DI II.3 (107), 24. 89. On the connection between singularitas and contractio in Cusanus’s account of individuation, see Leinkauf, “Bestimmung des Einzelseienden,” 185–195. 90. “unde quando recte consideratur de contractione, omnia sunt clara.” DI II.4 (114), 32. 91. See DI II.4 (114–115), 32–34. See further Albertson, “Learned Thief,” 365–366. 92. “Et ita intelligi poterit, quomodo deus, qui est unitas simplicissima, exsistendo in uno universo est quasi ex consequenti mediante universo in omnibus, et pluralitas rerum mediante uno universo in deo.” DI II.4 (116), 34–36; cf. DI II.5 (117), 36–38. Goris has shown that the notion of a “mediating universe” is originally Eckhartian (see “Mediante universo”), and Benz emphasizes the Neoplatonist background of emanatio simplex (see Individualität und Subjektivität, 138–144). Wilpert and Senger trace the Anaxagoran doctrine of DI II.5 (117–121) to Eckhart as well (see Philosophisch-Theologische Werke, II:121 n. 52). Hence we can compare DI I.19–20 and DI II.4–5 as two parallel attempts to approach Fundamentum by way of Eckhart (see further Albertson, “Learned Thief,” 379–383). 93. For the former, see DI II.5 (122), 42; for the latter, see DI II.6 (123–126), 42–48. As we shall see in the next chapter, DI II.6 effectively operates as the prefatory interpretation of Fundamentum for De coniecturis, just as DI II.7 is for De docta ignorantia. 94. “Non potest enim contractio esse sine contrahibili, contrahente et nexu, qui per communem actum utriusque perficitur.” DI II.7 (128), 50. Haubst shows how Cusanus used similar Llullian grammatical triads in his early sermons and then foregrounded them in De docta ignorantia (see Bild des Einen, 72–83; cf. Colomer, Nikolaus von Kues und Raimund Llull, 85–113). As Nicholas worked with his difficult new source, it is not surprising that he tried to paraphrase contractio into more familiar Llullian categories. 95. “Possibilitas igitur ab aeterna unitate descendit.” DI II.7 (128), 50. 96. “Ipsum autem contrahens cum terminet possibilitas contrahibilis, ab aequalitate unitatis descendit.” DI II.7 (129), 50. 97. See DI II.7 (130), 52. 98. “et illa ab unitate gignente in divinis descendit sicut alteritas ab unitate. . . . Nihil enim praecedere videtur posse. Quomodo enim quid esset, si non potuisset esse? Possibilitas igitur ab aeterna unitate descendit.” DI II.7 (128), 50. The antecedent of illa is contrahibilitas (the contractible substrate), which Cusanus immediately defines as quaedam possibilitas.

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  99. “Unde cum contrahens sit adaequans possibilitatem ad contracte istud vel aliud essendum, recte ab aequalitate essendi, quae est verbum in divinis, descendere dicitur. Et quoniam ipsum verbum, quod est ratio et idea atque absoluta rerum necessitas, possibilitatem per ipsum tale contrahens necessitat et constringit, hinc ipsum contrahens quidam formam aut animam mundi et possibilitatem materiam vocaverunt, alii fatum in substantia, alii, ut Platonici, necessitatem complexionis, quoniam a necessitate absoluta descendit, ut sit quasi quaedam contracta necessitas et forma contracta, in qua sint omnes formae in veritate. De quo infra dicetur.” DI II.7 (129), 52; trans. Bond, 145–146 (modified). When Cusanus lists alternative names for the second mode, he is repeating Fundamentum’s litany of cognates that will be cited in DI II.9 (142), 64–66 = F 7r, 460. But that list is itself a modified version of Thierry’s names for the second mode in Glosa II.21, H 273. Cusanus’s dependence on Fundamentum for his introductory remarks in DI II.7 suggests that he either did not have access to Glosa or gave priority to Fundamentum. It appears possible, however, that he did use Glosa in DI II.8 (133) and DI II.10 (151); see below. 100. Although there is little evidence of influence, Cusanus’s impulse to view the second mode as a mediation of the first and third modes matches the instincts of Clarembald of Arras in his Boethian commentary: “descendentia per necessitatem complexionis ad inmarmenen iam se in possibilitate definita manifeste depromunt ac fato subsunt” (De Hebdomadibus III.19, ed. Häring, 201). 101. See DI II.9 (142–148), 66–72. After witnessing his zeal to correlate aequalitas with Fundamentum’s ideas in DI II.7 and (as we shall see) in DI III.3, it seems incredible that Cusanus would have suddenly gone silent on the matter in DI II.9, where the status of the second mode’s divinity is directly addressed. This is further evidence in favor of Hoenen’s hypothesis that Nicholas was not the original author of that passage. 102. See Leinkauf, “Bestimmung des Einzelseienden,” 193–195, 199–200. Leinkauf is one of the few who have perceived how unusual and yet how important is this postulate of “die einschränkend-angleichende Tätigkeit des göttlichen Wortes” or simply “der göttlichen contractio adaequans” (ibid., 199–200). As Leinkauf demonstrates, the combination of contractio and aequalitas (DI II.7) into Nicholas’s doctrine of singularitas (DI III.1) is fundamental for the future development of his thought. Without it there is no Christology and no dialectical similitudo in the late works, which in turn provides the conceptual basis for the divine names non-aliud, idem, and possest (ibid., 184, 201–203). Bredow has investigated the doctrine’s prominence in De venatione sapientiae, where Cusanus expressly roots the singularity of creatures in the singularity of God (see “Gedanke der singularitas”). “Unde sicut singularissimus deus est maxime implurificabilis. . . . Gaudet igitur unumquodque de sua singularitate, quae tanta in ipso est, quod non est plurificabilis. . . . In hoc enim omnia se gaudent similitudinem dei participare.” De venatione sapientiae XXII (65), ed. Klibansky

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and Senger, 63–64. On the concept of singularitas, see further Meinhardt, “Christliche Impuls”; Bredow, “Participatio Singularitatis”; Benz, Individualität und Subjektivität, 200–210; and Beierwaltes, “Nicolaus Cusanus:  Innovation durch Einsicht,” 363–366. 103. On Cusanus’s relation to Plato and especially the mediating Platonic ideas, see Schnarr, “Das Wort Idea”; and Hirschberger, “Platonbild.” Meinhardt speaks aptly of a Cusan “Fundamentalplatonismus” that exceeds particular doctrines (“Christliche Impuls,” 107–109). Senger has analyzed the tension between Plato and Aristotle in Cusan thought from 1450 forward (see “Aristotelismus vs. Platonismus”). 104. Many of these are references to his doctrine of docta ignorantia or to other chapters of his work intended to give the impression that the Fundamentum material in DI II.7–10 fits seamlessly within the rest of De docta ignorantia. See, e.g., Hoenen, “ ‘Ista prius inaudita’,” 402, 410, 421. 105. The interpolations correspond roughly to DI II.8 (132–135), 54–58; DI II.9 (142–148), 64–74; and DI II.10 (151), 78–80. Cf. Hoenen, “ ‘Ista prius inaudita’,” 453–71. 106. See DI II.8 (132-34), 54–58. Cusanus’s account of carentia, aptitudo, and informitas seems to rely upon Glosa II.18, H 72: “Quare ipsa est materia primordialis quam alii ylem alii siluam alii cahos alii infernum quidam aptitudinem atquae carentiam dixerunt que a deo creata est.” Cf. also Calcidius, Timaeus 268, ed. Waszink, 273; and ibid. 283, ed. Waszink, 286; as well as Clarembald of Arras, Tractatus I.22, ed. Häring, 94. This account of matter, deeply linked to the Timaeus exegesis of the school of Chartres, is repeated chiefly in De mente and De ludo globi (see Thiel, “Rezeption des platonischen Timaios”). 107. See DI II.9 (142), 64–66 = F 7r, 460. 108. Führer attempts to make sense of such references across all of the cardinal’s works. His difficulties with certain passages of De docta ignorantia highlight the philosophical tensions introduced by Fundamentum (see “Cusanus Platonicus,” 347–352). 109. See DI II.9 (148), 72–74. 110. Fundamentum uses variants of complicatio six times and explicatio only once. See F 5r, 452 (complicata); F 7v, 466 (complicans); F 8r, 468 (complicans, explicatione); F 8v, 470 (complicatio); F 9r, 472 (complicante, complicat). In his interpolation in II.9 alone, Cusanus uses explicatio ten times and complicatio three times: see DI II.9 (143–147), 66–72. Given this sophisticated, balanced handling of Thierry’s conception of folding as reciprocal, most scholars have assumed that Nicholas had access to Lectiones. As we have seen, reciprocal folding is prominently featured only in Lectiones, and there in connection with the four modes of being. It is not transmitted in Tractatus, Commentum, the student treatises or Septem, and appears in Glosa and Clarembald less conspicuously than in Lectiones. Yet this prompts the question of why Nicholas

Notes

111. 112.

113.

114.

367

would have allowed Fundamentum to stand uncorrected by Lectiones, if indeed he had the latter in his possession. If Nicholas had read Lectiones carefully enough to glean the confident understanding of Thierry’s folding displayed in DI II.3–4, how could he fail to notice the contradiction between Lectiones and Fundamentum? One must weigh this problem against the possibility that Cusanus found everything he needed from the examples of complicatio and explicatio in Fundamentum itself, slight as they are, and thus may not have required Lectiones in order to compose Book II. Another possibility is that Cusanus used (1) a separate work by Thierry, now lost to us, which (2) explores reciprocal folding in detail, and yet (3) not in connection with the four modes of being. Notably, Caiazzo’s Stuttgart commentary on the Institutio arithmetica fulfills all three criteria. See DI II.9 (142, 143, 147), 66–72. “Et ordine naturae perfectum prius est imperfecto. . . . Et aliud est perfectum in genere, extra quod nihil est sui ipsius determinati generis, sed ordine temporis, hoc est via generationis seu originis, imperfectius prius est perfecto,” F 9v, 474; cf. Aristotle, De caelo 269a19–269a21 and Metaphysics 1021b13–1022a4. Hoenen demonstrates parallels with the florilegium Auctoritates Aristotelis (“ ‘Ista prius inaudita’,” 442–443). Cusanus adverts to the same distinction again in DI I.20 (59), 78; and DI III.3 (202), 24. See DI II.9 (148), 72–74. After observing the contradictions and duplications in Cusanus’s anima mundi doctrine in DI II.9, Schwarz surmises that the cardinal mixed together different sources in this passage and that these must have “contaminated” each other (Problem der Seinsvermittlung, 102). “Multi Christianorum illi viae Platonicae acquieverunt.” DI II.9 (146), 70. The valence of this important statement hangs upon the sense of acquieverunt. Depending on one’s translation, Cusanus aligns himself more or less with his source’s condemnation of “the Platonists.” Bond’s translation is “acquiesced,” suggesting that Christians conceded the truth of the via Platonica slowly or reluctantly (Nicholas of Cusa, 152). This aligns Cusanus with Fundamentum’s forthcoming attack on the “Platonist” position in DI II.9 (148–150), where the cardinal ceased his interpolation and cited Fundamentum’s critique of the second mode. Likewise Hopkins translates it as “consented” (Nicholas of Cusa on Learned Ignorance, 109). Taking into account the Fundamentum treatise, however, the cardinal’s editorial relationship to his source need not be limited to affirmation. Wilpert’s German translation suggests “were satisfied with,” a more accurately ambiguous reading:  “Viele unter den christlichen Denkern fanden in diesem platonischen Weg ihr Genüge” (Philosophisch-Theologische Werke, vol. 1, 71); cf. Haubst’s translation: “sich mit dieser via Platonica zufrieden gaben” (Bild des Einen, 118). Another possible translation of ad-quiescere is “to find rest in,” i.e., to settle on, to find sufficient, to accept. Given Cusanus’s evident desire to defend Platonism and how avidly he read Christian Platonists

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like Ps.-Dionysius, Eriugena and Eckhart, this last reading has many points in its favor. 115. Haubst searches for signs of Nicholas’s shifting allegiances (see, e.g., Bild des Einen, 118). Schnarr is surprised that “ausführlich und mit deutlich spürbarer Sympathie legt her die Meinung der Platoniker dar” (Modi essendi, 32) and so must clarify when Cusanus is ventriloquizing Peripatetic voices and when Platonic (ibid., 26–36). Watts identifies the contradiction between Nicholas embracing folding in DI II.3 but then rejecting “hierarchical descent” in DI II.9 (see Nicolaus Cusanus, 67–71). Particularly instructive is the case of Benz, who feels compelled to divide DI II.9 into two separate parts. He calls the first part, DI II.9 (141–147), an arm’s length “report” on the Platonist anima mundi doctrine (see Individualität und Subjektivität, 175–181). The second part, DI II.9 (148–150), represents Nicholas’s intervention in the controversy (see ibid., 182–188). Benz marvels at Cusanus’s irenic objectivity while explaining the doctrine in the first part that he will apparently reject in the second, and accordingly attempts to minimize Nicholas’s disagreement with the Platonists by qualifying his refutation of the second mode. Since he tacitly accepted the doctrine elsewhere in Book II, the cardinal must have intended only the friendly “clarification” that certain “accents” in the anima mundi doctrine should not overshadow the Christian Logos (ibid., 185, 188). Given Benz’s exceptionally careful reading, it is striking that he cannot take Cusanus’s rejection of the second mode at face value. It is even more startling to realize that Benz’s proposed division accords perfectly with the Fundamentum hypothesis. What Benz isolates as the curious second part of DI II.9 is precisely the extract of Fundamentum (F 7r–8r, 466–468 = DI II.9 [148–150]) that Cusanus introduces in Benz’s “report” (DI II.9 [141–147]). What Benz (quite rightly) cannot accept as entirely Cusan is not Cusan at all; and what strikes Benz as a remarkably evenhanded “report” was in fact Nicholas’s attempt to provide context for the passage from Fundamentum that he was about to cite. 116. In this passage, Cusanus follows the treatise verbatim with one exception:



F 8r, 468

DI II.9 (150), 76

Nec cadit eo modo medium inter absolutum et contractum, ut illi Platonici imaginati sunt, qui animam mundi mentem putarunt post deum et ante contractionem mundi.

Nec cadit eo modo medium inter absolutum et contractum, ut illi imaginati sunt, qui animam mundi mentem putarunt post deum et ante contractionem mundi.

The cardinal’s careful paraphrase of this very sentence earlier in the same chapter (“putabant has rationes distinctas . . . post deum et ante res esse”:  DI II.9 [146], 70) suggests that the omission was not accidental.

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117. If Cusanus did misunderstand Fundamentum so radically, this makes it very difficult to maintain that the treatise was his own work, or even a work of his youth. A clear proof that someone is using a source not their own is evidence that they have misinterpreted that source. 118. “Unde necessitas complexionis non est, ut posuerunt Platonici, scilicet mens minor gignente, sed est verbum et filius aequalis patri in divinis, et dicitur logos seu ratio, quoniam est ratio omnium.” F 7v–8r, 466. 119. In this light a passing remark in Cusanus’s Christmas sermon from 1438 takes on new significance. There he appeals to Augustine’s views in Confessiones and De civitate dei concerning the “Platonists” who dimly grasped the divine Word:  “Tamen ‘non enuntiavit Plato hoc Verbum tamquam personam in divinis, sed ut rationem idealem rerum, per quam Deus omnia condidit, quae appropriatur Filio. Nam aliqui Philosophi sub deo ponebant aliam substantiam, quam vocaverunt supremam intelligentiam sive intellectum’, per quam deum cuncta creare ponebant,” Sermo XIX (6), ed. Haubst and Bodewig, 295. 120. See DI II.7 (129), 50–52. Haubst finds that he has to correct Cusanus when the cardinal assigns functions to necessitas complexionis that should have been given to the divine Word (see Bild des Einen, 117 n. 52). 121. “Die Problemstellung, der Cusanus mit seiner besonders in De docta Ignorantia systematisch ausgearbeiteten Christologie begegnet, zeichnet sich schon in den frühen Predigten ab und klingt an in bestimmten für sein Denken charakteristischen Einstellungen und Begriffen bzw. Denkmustern” (Dahm, Soteriologie, 71). Dahm lists four examples of such themes from the Christology of Book III that he finds anticipated in pre-1440 sermons: (1) a theology of creation ordered teleologically to the Incarnation; (2) the separation of finite and infinite beyond analogy; (3) the Christological interpretation of procession and return; and (4) the human being as cosmic medium (ibid., 72–76). See further Hundersmarck and Izbicki, “Nicholas of Cusa’s Early Sermons”; and McGinn, “Maximum Contractum et Absolutum,” 164–168. 122. See De concordantia catholica I.2 (9–12), ed. Kallen, 33–36. 123. Ibid. I.1 (5–8), ed. Kallen, 30–33. 124. “Sic verbum supernum lapidem imaginor, cuius virtus penetrat cuncta usque ad ultimum, non quod gradatim deficiat virtus infinita, sed ut finitis et terminatis creaturis insit mirabilis conexionis ordo.” Ibid. I.2 (10), ed. Kallen, 35; trans. Sigmund, 9.  See further Miroy, Tracing Nicholas of Cusa’s Early Development, 79–93. 125. See Sermo I (3), ed. Haubst, 4. 126. See Sermones XVI–XVII, ed. Haubst and Bodewig, 261–278. These were given on Christmas Day between 1432 and 1435, but the exact years remain unknown. On Cusanus’s knowledge of patristic and medieval Christological distinctions, see Haubst, Christologie, 109–138.

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127. See Sermo XX, ed. Haubst and Bodewig, 291–317. On this sermon see Casarella, “His Name is Jesus.” 128. “Eckhart presents a functional rather than an ontological Christology. . . . [T]‌he Incarnation is, so to say, between the procession of the divine persons and the production of creatures. The Incarnation shares in both realms, and is, therefore, the expression of the eternal divine relationships and the exemplar of the created world” (Schneider, “Functional Christology of Meister Eckhart,” 293, 301). On Christ as “center” (medium) in Bonaventure, see Cousins, Bonaventure and the Coincidence of Opposites, 131–159. Reinhardt has examined the same theme in De visione dei (see “Christus, die ‘absolute Mitte’ ”). Wilpert and Senger, taking DI II.9 (149) as authentically Cusan, trace that Logos doctrine to the Neoplatonist Christologies of Augustine, Eriugena, and Eckhart (see Philosophisch-Theologische Werke, II:131 n. 129). It seems that Cusanus heard the same Neoplatonist possibilities in Fundamentum’s words. 129. See Hoenen, “Tradition and Renewal”; Colomer, Nikolaus von Kues und Raimund Llull, 104–113; and Reinhardt, “Christus, die ‘absolute Mitte’,” 219– 220. As McTighe observes, “Cusa brings to a head many of the species of Platonism that were operative in the period from the 12th to the 14th century and many that were latent” (“Meaning of the Couple,” 207–208). 130. See F 8r, 468 = DI II.9 (150), 76–78. 131. See F 8v, 470 = DI II.10 (152), 80. 132. On the centrality of Book III’s Christology for Cusan theology generally, see Metzke, “Nicolaus von Cues,” 231–234; Bond, “Nicholas of Cusa and the Reconstruction of Theology”; Schönborn, “ ‘De docta ignorantia’,” 138–142; and Offerman, Christus—Wahrheit des Denkens, 4–22. Unfortunately Book III has been entirely neglected in discussions of Hoenen’s Fundamentum hypothesis, beginning with Hoenen himself (see Albertson, “Learned Thief,” 373–375). But in fact it is pivotal. If that anonymous treatise served as the primary instigation (per Hoenen, the “Vorlage”) for the composition of De docta ignorantia, and if the new theological vision of De docta ignorantia is consummated in the Christology that reconciles the absolute and the contracted, then one must be able to explain how Fundamentum is connected to, or even inspires, that Christology. Hence the more Hoenen’s hypothesis can illuminate the Christology of Book III, the more credible his theory becomes. 133. See Schönborn, “ ‘De docta ignorantia’,” 155. 134. See DI III.2 (194), 16. 135. See DI III.Prol. (181), 2. 136. Haubst analyzes Cusanus’s argument in DI III.1–3 in painstaking detail (Christologie, 143–167), focusing especially on DI III.3 (see ibid., 158–160, 166– 167). But even Haubst relies exclusively on the general definition of the hypostatic union between absolute maximum and contracted maximum in DI III.2 (191–192), 14. It is not until later in the book that Haubst briefly mentions

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Cusanus’s specific definition in DI III.3 (202), 24 (see ibid., 176). The same problem afflicts Meinhardt, “Christliche Impuls”; and Benz, Individualität und Subjektivität, 218–231. 137. “quasi universalis rerum omnium contractio aequalitati omnia essendi hypostatice et personaliter unita,” DI III.3 (202), 24; trans. Bond, 177. 138. See, inter alia, Haubst, Christologie; Bond, “Nicholas of Cusa and the Reconstruction of Theology”; Schönborn, “ ‘De docta ignorantia’ ”; Offermann, Christus—Wahrheit des Denkens; Roth, Suchende Vernunft, 48–92; and McGinn, “Maximum Contractum et Absolutum.” 139. See, e.g., DI I.2 (7), 12; and DI II.5 (122), 42. 140. For the final section of Fundamentum, see F 8v–9r, 470–472. Cusanus’s brief interpolation in DI II.10 (151), 78–80, abruptly changes the subject to cosmic fate, alluding to Calcidius, Timaeus 144, ed. Waszink, 182–183. Nicholas may have been inspired by Septem’s equation of “motus universalis et naturalis,” “anima mundi,” and “fatum” (951D–952A). But note that among Thierry’s commentaries only Glosa (not Lectiones) characterizes the second mode in terms of causal necessity and fatum (see Glosa II.20–21, H 273). Nicholas will return to these texts at LG I (40), 45–46; and LG I (57), 64. 141. Schwarz perceives the deep connection between DI II.8–10 and DI III.1–4 forged through the substitution of Verbum for necessitas complexionis in DI II.9. As he puts it, the Mittlerfunktion of the cosmos rejected in Book II is restored in Book III as the Mittlerfunktion of the Incarnation (see Problem der Seinsvermittlung, 89–94, 159–162). Meurers has to compare the same passa­ ges to define the maximum and minimum (see “Nikolaus von Kues und die Entwicklung des astronomischen Weltbildes,” 398–407). 142. See F 8v–9r, 470–472  =  DI II.10 (153–154), 80–84. Lohr suggests that such ideas in De docta ignorantia can be traced back to Llull (see “Lull’s Theory of the Continuous and the Discrete”). 143. “Nam dum omnia moveantur singulariter, ut sint hoc quod sunt meliori modo, et nullum sicut aliud aequaliter, tamen motum cuiuslibet quodlibet suo motu contrahit et participat mediate aut immediate,” F 9r, 472 = DI II.10 (154), 82. As Bredow and Dahm have noticed, the sense of the phrase meliori modo quo is somewhat opaque in the context of De docta ignorantia as a whole, and it rarely appears in later Cusan writings (see Bredow, “Der Sinn der Formel ‘Meliori modo quo’ ”; and Dahm, Soteriologie, 95–99). This would make sense if the phrase did indeed originate in Fundamentum. 144. See DI III.1 (182–186), 2–8. 145. “ut nihil sit in universo, quod non gaudeat quadam singularitate quae in nullo alio reperibilis est, ita quod nullum omnia in omnibus vincat aut diversa aequaliter, sicut cum nullo ullo umquam tempore minus eo fuerit et alio maius, hunc transitum facit in quadam singularitate, ut numquam aequalitatem praecisam attingat,” DI III.1 (188), 10; trans. Bond, 172 (modified).

372

Notes

146. Leinkauf, “Bestimmung des Einzelseienden,” 186–187. 147. See DI III.1 (185), 6. 148. “Quapropter concluditur species ad instar numeri esse ordinatim progredientis . . . ut sive sursum numeremus sive deorsum, ab unitate absoluta, quae Deus est, ut ab omnium principio, initium sumamus,” DI III.1 (187–188), 8; trans. Bond, 171. 149. Leinkauf aptly describes the “analogische Identität” and “asymmetrische Komplementarität” between singular creatures (the numerically one) and divine Equality (the transcendent One), “eine nicht nur formale oder logische Ähnlichkeit, sondern . . . im Konzept der singularitas auf eine—bei allem Unterschied—ontologisch, im Wesen der Einheit gründende Gemeinsamkeit von göttlicher und kreatürlichen Existenz” (“Bestimmung des Einzelseienden,” 189–190, 182). 150. “Est ergo hic spiritus per totum universum et singulas eius partes diffusus et contractus, qui natura dicitur. Unde natura est quasi complicatio omnium, quae per motum fiunt.” F 8v, 470 = DI II.10 (153), 80. “Et ad hoc moventur, ut in se aut in specie conserverentur per naturalem conexionem diversorum sexuum, qui in natura complicante motum sunt uniti et divisive contracti in individuis.” F 9r, 472 = DI II.10 (154), 82–84. 151. “Hoc tale maximum contractum supra omnem naturam contractionis illius terminus finalis exsistens, in se complicans omnem eius perfectionem, cum quocumque dato supra omnem proportionem summam teneret aequalitatem, ut nulli maior et nulli minor esset, omnium perfectiones in sua plenitudine complicans.” DI III.2 (191), 14; trans. Bond, 173. 152. See DI III.2 (192–94), 14–16. 153. “Qui motus est medium conexionis potentiae et actus,” F 8v, 468–470 = DI II.10 (152), 80. 154. “Quapropter natura media, quae est medium conexionis inferioris et superioris, est solum illa, quae ad maximum convenienter elevabilis est potentia maximi infiniti Dei. Nam cum ipsa intra se complicet omnes naturas, ut supremum inferioris et infimum superioris,” DI III.3 (197), 20. 155. “per ipsum, qui est maximum contractum, a maximo absoluto omnia in esse contractionis prodirent et in absolutum per medium eiusdem redirent, tamquam per principium emanationis et per finem reductionis.” DI III.3 (199), 20–22; trans. Bond, 176. 156. Cusanus returns once more in DI III.3 to the final section of Fundamentum after stating his Christological synthesis. He ends the chapter with a caveat that borrows from the Aristotelian distinction between ordo naturae and ordo temporis used throughout DI II.9 (see above). Cusanus recognizes that his Christology entails a paradox. All beings, including human beings, are created through the hypostatic union. But this means that the Incarnation of the Word into humanity precedes the creation of humanity. The cardinal attempts to mitigate the problem by suggesting that this order is not temporal sequence

Notes

373

but natural priority: “Hic autem ordo non temporaliter considerari debet . . . sed natura et ordine perfectionis supra omne tempus,” DI III.3 (202), 24. On the eternal predestination of the Incarnation, see McGinn “Maximum Contractum et Absolutum”; and Albertson, “ ‘That He Might Fill All Things’.” 157. “ut sic primo sit deus creator, secundo deus et homo creata humanitate supreme in unitatem sui assumpta, quasi universalis rerum contractio aequalitati omnia essendi hypostatice ac personaliter unita, ut sic per deum absolutissimum mediante contractione universali, quae humanitas est, tertio loco omnia in esse contractum prodeant, ut sic hoc ipsum, quod sunt, esse possint ordine et modo meliori.” DI III.3 (202), 24; trans. Bond, 177 (my emphasis). 158. See DI III.4 (204), 28. First, God as aequalitas is not constrained by partial degrees. Second, all creatures exist according to degrees of difference. Third, God therefore abides in creatures through Jesus, the aequalitas essendi whose humanity “universally enfolds” (universaliter complicanti) all things. Even if the term contractio is lacking in this version of the argument, it is strongly implied by the terms gradus and gradatim and by the function of universal enfolding. 159. It is remarkable how rarely Nicholas’s correlation of aequalitas and contractio in this passage is examined, even in the most detailed studies of the Christology of Book III: cf. Schönborn, “ ‘De docta ignorantia’,” 143–154; Offermann, Christus— Wahrheit des Denkens, 142–167; Counet, Mathématiques et dialectique, 366–392; and Roth, Suchende Vernunft, 48–92 (but see the fine discussion of contractio at 75–79). Even Leinkauf does not address this passage (see “Bestimmung des Einzelseienden,” 195–200). A notable exception is Dahm, Soteriologie, 111–113. 160. At one point the contradiction is so precise as to appear deliberate. “Nec cadit eo modo medium inter absolutum et contractum,” DI II.9 (150), 76  =  F 8r, 468. “Et ita humanitas Iesu est ut medium inter pure absolutum et pure contractum.” DI III.7 (225), 52. 161. Jacobi has proposed a parallel but more abstract and ahistorical reading of the Christology of Book III by building on Rombach’s notion of Funktionalismus (see Substanz, System, Struktur, 206–228). If the henology of Book I expresses a substantialist ontology of identity (i.e., theology), repeating ancient Platonism, then Book II gives voice to the new Cusan functionalist ontology of pure relation (i.e., cosmology). According to Jacobi these two motives are then somehow reconciled in Book III (see Methode der cusanischen Philosophie, 131–142, 192–200), although in fact he turns instead to De dato patris luminum (ibid., 143–173). Counet attempts to link Jacobi’s reading to the hypostatic union of Book III (see Mathématiques et dialectique, 372–398).

Chapt er 8   1. “Die Spekulation des Cusanus wird zum Kampfplatz, auf dem sich nun Gedankenelemente, die in der mittelalterlichen Philosophie unterschiedlos

374

Notes

ineinander übergehen, begegnen, auf dem sie sich erkennen und aneinander messen.” Cassirer, Individuum und Kosmos, 17.  2. Nicolle suggests that Cusanus’s peculiar use of mathematics alters Neoplatonism in two ways, both transforming its customary dualism into a graduated continuum of being and inverting the Platonic hierarchy of mathematics and dialectic (see “Quelques sources,” 57)—in other words, a monistic henology.   3. See McGinn, “Maximum Contractum et Absolutum”; and Albertson, “ ‘That He Might Fill All Things’.”   4. See DI III.Prol. (181), 2.   5. On the frequency of productive misunderstandings, see Dodds, “Parmenides of Plato,” 134 (on Plotinus, 140).   6. Jeauneau, “Mathématiques et Trinité,” 295.   7. Wilpert contends that Cusanus was still making revisions of De coniecturis in 1445 while he was completing his first two mathematical works, De geometricis transmutationibus and De arithmeticis complementis (see “Kontinuum oder Quantensprung,” 112). See further Koch, Ars coniecturalis; and Koch, “Über eine aus nächsten Umgebung.”   8. Haubst has noted that the references to De coniecturis in De docta ignorantia all pertain to ideas from Book II. As he indicates, this suggests that it was not, surprisingly, the Pythagorean and mathematical topics filling Book I that excited Cusanus enough to pen another lengthy work, but rather something in Book II (see “Das Neue in De docta ignorantia,” 47).   9. See DI II.1 (95), 10; DI II.6 (123, 126), 44–46; DI II.8 (140), 64; DI II.9 (150), 78; DI II.10 (155), 84; and DI III.3 (187–188), 8–10. 10. See DI II.1 (95), 10; DI II.6 (126), 46; and DI III.3 (187–188), 8–10. 11. See DI II.6 (123–124), 42–44; cf. DC I.4 (13–15), 18–20. 12. See DI II.5–6 (122–123), 42–44. 13. The sermon was previously dated to 1439, on the assumption that it must have preceded De docta ignorantia, before new manuscript evidence fixed its date in December 1440. On the centrality of sermons for determining the rhythms of Cusan development, see Euler, “Entwicklungsgeschichtliche Etappen.” 14. See Sermo XXII (10–11), ed. Haubst and Bodewig, 338–339. 15. “Videmus in quolibet, quod est res una, discreta et conexa; haec enim reperiuntur in essentia omnis esse. Et ipsa unitas dicit indivisionem, discretionem et conexionem. Si igitur reperimus in omni esse participante esse ista, et videmus unitatem contractam, quae a rebus participatur, non esse nisi sit trina.” Sermo XXII (17), ed. Haubst and Bodewig, 342–343. The triad of indivisio, discretio, and conexio also appears at DI I.10 (28), 38. 16. See DI I.11 (31–32), 42–44. 17. “Sed sufficit nunc nobis scire, quod si principium universorum intueri volumus, tunc videmus omnem intelligentiam rationalem claudi multitudine et

Notes

18.

19. 20. 21.

22.

23.

24.

25. 26.

27. 28. 29. 30.

375

magnitudine; nihil enim extra multitudinem et magnitudinem ratio deprehendit.” Sermo XXII (19), ed. Haubst and Bodewig, 344. See Sermo XXII (19), ed. Haubst and Bodewig, 345. As we have only recently learned from the Stuttgart MS, Thierry grounded reciprocal folding upon the originary explicationes of quadrivial dimensions, that is, unity unfolding into multitude and points unfolding into magnitude (see Caiazzo, “Rinvenimento,” 202–203). Cusanus’s efforts to understand Thierry in light of the quadrivium apparently led him to formulate similar ideas. See, e.g., DC I.1 (6), 10; DC I.8 (35), 40–41; DM IV (75), 115; DM IX (122), 176; and TC 9:33–45, 43–45. See Sermo XXII (20–22), ed. Haubst and Bodewig, 345–346. See ibid. (23), ed. Haubst and Bodewig, 347. “Et Verbum non potest minor Deo esse, cum sit una aeternitas et infinitas.” Ibid. (23), ed. Haubst and Bodewig, 348. Cf. F 8r, 466 = DI II.9 (149), 76: “Unde necessitas complexionis non est, . . . scilicet mens minor gignente, sed est verbum et filius aequalis patri in divinis, et dicitus logos seu ratio, quoniam est ratio omnium.” “omnia in arte infinita sunt aequalitas, et simplicissima ratio infinita est omnium ratio. Omnes enim diversitates in unitate infinitae rationis complicantur.” Sermo XXII (24), ed. Haubst and Bodewig, 348. Cf. F 7v, 466 = DI II.9 (148), 74: “Unum enim infinitum exemplar . . . omnes quantumcumque distinctas rerum rationes adaequatissime complicans, ita quod ipsa infinita ratio est.” “in qua recipitur, secundum magis et minus et similis conceptui, sed numquam aequalis praecise, ita formae rerum sunt imagines artis divinae, etc.” Sermo XXII (29), ed. Haubst and Bodewig, 350. Cf. F 7v–8r, 466  =  DI II.9 (149), 76: “ratio simplicissima non recipiens in se una nec magis nec minus . . . quoniam non est nisi una infinita forma formarum, cuius omnes formae sunt imagines.” “universum, si consideratur in pura unitate, tunc est Deus.” Sermo XXII (28), ed. Haubst and Bodewig, 349. Cf. F 6r, 456 = DI II.8 (140), 64: “si mundum consideramus ut ipsa est, tunc est ut in deo.” See Sermo XXII (32, 33, 36), ed. Haubst and Bodewig, 351–353. “Christus Dominus in hoc, quod supra omnem creaturam maximitati absolutae coniunctus est, quia eo maior dari nequit, in quo potentia infinita in se completa et perfecta est, tunc Deus est et ipsa ars seu forma infinita omnium, quae sunt.” Ibid. (35), ed. Haubst and Bodewig, 352. See ibid. (37), ed. Haubst and Bodewig, 353–354. On ars in De docta ignorantia, see e.g., DI II.1 (91), 4; DI II.1 (94), 8; DI II.13 (175), 108; and DI II.13 (178), 112. See Sermo XXIII (14–15), ed. Haubst and Bodewig, 366–367. See ibid. (16–17), ed. Haubst and Bodewig, 368. Cf. DI II.3 (106), 24; but also Commentarius Victorinus 87–88, H 499. See Boethius, IA II.40.1–3, ed. Guillaumin, 140; and Boethius, Institutio musica II.7, ed. Friedlein, 232.

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31. See Sermo XXIII (6–9), ed. Haubst and Bodewig, 361–363. 32. See ibid. (18–22), ed. Haubst and Bodewig, 368–371. 33. “Vides igitur manifeste, quod non solum nomen Dei est innominabile, sed nec etiam nomen praecisum cuiusque rei. Et relucet nomen Dei in omnibus nominibus ut in imagine. Et sicut Dei Verbum est Nomen infinitum, ita et Sermo infinitus; et omnes linguae et sermones sunt explicatio illius.” Ibid. (33), ed. Haubst and Bodewig, 376. On the ineffability of God, see Theruvathu, “Die Unaussagbarkeit (ineffabilitas) Gottes.” 34. See Sermo XXIII (33–34), ed. Haubst and Bodewig, 376. 35. See ibid. (36–40), ed. Haubst and Bodewig, 377–380. 36. See Koch, Ars coniecturalis; and Koch, “Sinn des zweiten Hauptwerkes.”  37. Bocken, L’Art de la collection, 19. Haubst has also described De coniecturis as an intensification of the line of thought pursued in DI II.7–9 (see “Zusammenfassende theologische Erwägungen,” 196–197). 38. See Bocken, L’Art de la collection, 78–81. 39. Schwarz, Problem der Seinsvermittlung, 163. 40. “Quapropter unitas mentis in se omnem complicat multitudinem eiusque aequalitas omnem magnitudinem, sicut et conexio compositionem.” DC I.1 (6), 10. Cusanus returns to the arithmetical Trinity throughout the work’s final chapter, DC II.17 (171–184), 173–183 passim. Note the Boethian or quadrivial sense of multitudo rather than the Proclian sense found in the annotated excerpts that Haubst discovered, discussed below (see “Thomas- und Proklos-Exzerpte,” 27, 31). Similarly Bormann shows that alteritas in De coniecturis is more Chartrian than Platonic, Plotinian, or Proclian (see “Zur Lehre des Nikolaus von Kues,” 135–136). See further Riccati, “Processio” et “explicatio”, 69–74, 118–121. 41. See DC I.1 (6), 9. Later Nicholas proposes that unity is the principle of multitude, and trinity the principle of magnitude: see DC I.8 (35), 40–41. See further Rusconi, “Grandeur et multiplicité.” 42. See DC I.2 (7–9), 11–14; cf. Boethius, IA I.2.1, ed. Guillaumin, 11. 43. Although I  use “quadrivium” to speak about multitude and magnitude in De coniecturis, I  should note that Cusanus never lists the four mathematical arts in De coniecturis as he had in DI II.1 and II.13. It is true that the cardinal mentions some of the liberal arts in passing (see DC II.2 [82–86], 79–83), but nothing approaches the Nicomachean system preserved in Institutio arithmetica. Böhlandt points out that De coniecturis nevertheless contains arithmetic (the numerical progressions), geometrical figures, and even nascent harmonics and astronomy (see Verborgene Zahl, 107–187). 44. See DC I.4 (12–16), 18–21. For a fine overview of the four unities, see Oide, “Über die Grundlagen.” 45. See DC I.4 (12), 18. 46. See Koch, Ars coniecturalis, 10–12. Koch emphasizes that Cusanus himself critiqued his own methods in De docta ignorantia. “Nam in ante expositis De docta

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ignorantia memor sum de deo me intellectualiter saepe locutum per contradictoriorum copulationem in unitate simplici. Iam autem in proxime praemissis divinaliter intentum explicavi.” DC I.6 (24), 30–31 (my emphasis). Nicholas’s remark, however, corresponds directly with his treatment of ratio and intelligentia in De coniecturis, which I have discussed above. 47. See Koch, Ars coniecturalis, 15–16. Bocken notes on the contrary that De docta ignorantia is not self-evidently Seinsmetaphysik and that De coniecturis becomes significant only insofar as it goes beyond generic Neoplatonist Einheitsmetaphysik (see Bocken, L’Art de la collection, 45–47). Stallmach and Flasch have also found Koch’s distinction inadequate (see Stallmach, Ineinsfall der Gegensätze, 85–119; and Flasch, Metaphysik des Einen, 236). One might also ask why the arithmetical model of the Trinity in DI I.7–10 is not already an exemplary instance of Einheitsmetaphysik. 48. See, e.g., Velthoven, Gottesschau und menschliche Kreativität, 52; Oide, “Über die Grundlagen”; as well as Bormann’s notae 1 and 38 in his edition of De coniecturis. Thomas attempts to extend Koch’s theory by proposing that the driving force behind De coniecturis was Cusanus’s deep engagement with Proclus’s doctrine of participation (see Teilhabegedanke, 14–21). As Thomas recognizes, however, reciprocal folding is among the most essential doctrines for Cusan participation; yet this concept, pace Thomas, stems directly from Thierry of Chartres, not generally from “Neoplatonism” or even the “School of Chartres” (ibid., 57–61). 49. To make his case Koch draws two sharp distinctions: between the four modes of being in DI.7–10 and those in DC II.19, and between the four unities in DI II.6 and those in DC I.4. If these were not enforced, one might well conclude that the four unities in DC I.4 simply develop the four unities of DI II.6, which in turn appear inspired by the four modes of being in DI II.7–10, and hence that the sources of De docta ignorantia are merely reworked in a different way in De coniecturis. Regarding the four modes, Koch performs a rather acrobatic reading of Figures U and P, such that the modes in DC II.19 add up to ten intermediate transitions, whereas the modes in DI II.7–10 add up to twelve. From this evidence alone Koch concludes that even if the two passages use the “same words,” they have “nothing” to do with each other (see Ars coniecturalis, 33 n. 60). Regarding the four unities, Koch conjectures that while in DI II.6 Nicholas was clearly preoccupied with the debate over universals, in DC I.4 he must have been inspired by a mystery source (on which see below). But as Bocken rightly suggests, even if Cusanus’s new formulae in De coniecturis betray some Proclian influence, the major outlines of his new program are indubitably rooted in De docta ignorantia II.6, II.8, and II.9 (see L’Art de la collection, 20). 50. Koch, Ars coniecturalis, 35. 51. See Haubst, “Thomas- und Proklos-Exzerpte,” 48–49. Haubst argues that the Proclus excerpts did not influence De docta ignorantia, noting that Ps.-Dionysius

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and especially Thierry of Chartres were the central sources of that work (ibid., 33–34, 44–45). Nicholas may have acquired some sample fragments of Ambrogio Traversari’s Proclus translations as early as the winter of 1437–8, but the first meaningful trace of Proclian influence arrives in De coniecturis (ibid., 41, 48). 52. After citing Haubst, Koch concedes:  “Trotzdem dürfte damit die Frage nach den Quellen der Lehre von den vier Einheiten noch nicht völlig beantwortet sein” (“Nikolaus von Kues und Meister Eckhart,” 165). The missing element, Koch explains, is the principle relating higher and lower unities which is so central to Cusanus’s argument in De coniecturis, and “als Quelle für diesen Gedanken reichen die zitierten Sätze aus der Theologia Platonis nicht aus” (ibid., 167). Citing Wackerzapp, Koch concludes that Eckhart stated this principle frequently, albeit in terms that differ significantly from De coniecturis. 53. Watts has identified several passages in more traditional sources that Nicholas could have used to construct his four unities, and which might also have informed Thierry’s four modes: Martianus Capella, Calcidius, Augustine’s De musica, and Ps.-Boethius’s Ars geometrica (see texts in Nicolaus Cusanus, 97). 54. See Bocken, L’Art de la collection, 45–47. 55. See Haubst, Streifzüge, 217; and Haubst, “Das Neue in De docta ignorantia,” 46–51. 56. Consider for instance how Cusanus used his knowledge of Eckhart to approach difficult Chartrian ideas in De docta ignorantia and indeed already in Sermons XXII and XXIII. On Thierry’s unitas and aequalitas in DI I.3 and I.5, see Wackerzapp, Einfluss Meister Eckharts, 23–29; on Fundamentum’s concepts in DI II.4, see ibid., 109–111; on the two sermons, see ibid., 115–118. Wackerzapp’s claims of Eckhartian influence on DI II.6 are far less convincing (see ibid., 49–54). 57. See DC I.2 (8), 13–14. 58. See DC II.8–12 (112–133), 108–129. Cf. Bormann’s table (ibid., 206–207); and Burkert, Lore and Science, 51–52. 59. On Figure P (figura pyramidis), see DC I.9 (41–43), 45–47; and DC I.10 (49–51), 51–53. Cf. F 5r, 454. 60. “quasi in quattuor quibusdam simplicibus distinctis essendi modis.” DC I.11 (59), 60. 61. On Figure U (figura universi), see DC I.13 (65), 63–64. 62. See DC I.12 (61), 61; and DC I.13 (66), 65. 63. “Sic enim quattuor essendi modos in denarium resolvi videt, qui universus est numerus.” DC II.10 (119), 115. See also DC I.11 (58), 59. On the τετρακτύς see Heninger, “Some Renaissance Versions of the Pythagorean Tetrad”; and Burkert, Lore and Science, 71–73. 64. For instance, on the hexad (senarius): Number is the simultaneous procession from and return to unity, and in doing so it passes from God through the intelligible and rational “worlds” into the sensible world, its final destination and

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return point. Since this circuit adds up to six stages—two sets of four unities make eight, but the beginning and end points of the circle overlap—the hexad represents numerical progressions into sameness (just as the heptad represents difference). See DC II.7 (106–107), 102–104; cf. by contrast the material in Robbins, “Tradition of Greek Arithmology.” Schulze has tried to construct a consistent Cusan arithmology out of such statements in De coniecturis and elsewhere (see Zahl, Proportion, Analogie, 81–90). 65. See DC I.2 (7–9), 12–14; cf. Boethius, IA I.2.1, ed. Guillaumin, 11. 66. See, e.g., DC II.13 (137), 133–135; DC I.9 (38–40), 43–45; and DC I.2 (9), 14. On number theory in Cusanus see Haubst, Bild des Einen, 212–231, 246–254; Velthoven, Gottesschau und menschliche Kreativität, 153–159, 166–184; Schulze, Zahl, Proportion, Analogie, 67–92; and Bocken, “Zahl als Grundlage der Bedeutung.” The most thorough study is Eisenkopf, Zahl und Erkenntnis. 67. See DC I.4 (13), 18–19. Nicholas repeats this reasoning at DC I.11 (59), 60; and DC II.9 (117), 113. Cf. Commentum II.28, H 77. 68. See DC I.4 (15), 20; DC I.8 (35), 41; and DC I.12 (63), 62. Cf. Commentum IV.7, H 97; Lectiones II.6–9, H 156–57; Glosa II.13, H 271; and Glosa II.23, H 273. 69. See DC I.11 (59), 60; and DC II.1 (71–72), 72. 70. See DC I.4 (13–16), 18–21. For Thierry, the first and fourth modes are enfoldings, and the second and third modes are unfoldings. 71. “ut admiranda in invicem progressione divina atque absoluta unitate gradatim in intelligentia et ratione descendente et contracta sensibili per rationem in intelligentiam ascendente mens omnia distinguat pariterque conectat.” DC I.4 (16), 21; cf. F 8v, 470 = DI II.10 (152), 80. 72. “Hosque essendi modos trium regionum ad invicem continuari coniecturatur, ut unum sit universum.” DC II.9 (119), 114; cf. F 4r, 448 = DI II.7 (131). 73. See DC I.1 (5–6), 7–10; and DC II.1 (71–72), 72. 74. See DC II.9 (117–118), 113–114. Schnarr comments on this passage in Modi essendi, 62–67. 75. “Es ist nicht klar, warum Cusanus für die Notwendigkeit der Verknüpfung eine neue Formulierung gebraucht.” Rusconi, “Commentator Boethii,” 265. 76. See Lectiones II.9–11, H 157–158; Lectiones II.18, H 160; Lectiones II.34, H 166; and Glosa II.26–27, H 274. 77. See, e.g., DC I.7 (27), 34: “Anima numerus intelligentiae, quam quadrate explicat, non incongrue concipitur, sicut ipsa intelligentia numerus est unitatis supersimplicis; unitas enim intelligentiae numeratur in anima, dum multipliciter contrahitur.” 78. “Et quia unitas absoluta est prima et unitas universi ab ista, erit unitas universi secunda unitas, quae in quadam pluralitate consistit. Et quoniam, ut in De coniecturis ostendetur, secunda unitas est denaria, decem uniens praedicamenta, erit universum unum explicans primam absolutam unitatem simplicem denaria contractione.” DI II.6 (123), 42–44.

380 79. 80. 81. 82. 83. 84. 85.

Notes

See DC I.4 (13), 18–19; DC I.6 (22), 28–29; and DC I.11 (60), 61. See DC I.10 (53), 54; and DC I.12 (61), 61. See Counet, Mathématiques et dialectique, 320–321, 332–341. See DC I.6 (24), 31. See DC I.7 (27), 34. See DC I.6 (23–25), 29–33; and DC I.8 (35), 40–41. “Quapropter haec est radix omnium rationabilium assertionum, scilicet non esse oppositorum coincidentiam attingibilem. Hinc omnis numerus aut par aut impar, hinc ordo numeri, hinc progressio, hinc proportio.” DC II.1 (76), 75; trans. Hopkins, 200. 86. See DC II.1 (77), 76. Cusanus reasons that the foundation of every mathematical demonstration is the assumption that if what is demonstrated were not true, then the coincidence of opposites would obtain. Since reason cannot attain that coincidence, the given demonstration holds. 87. Nicholas makes this point repeatedly. See DC I.5 (21), 26–28; DC I.6 (25), 32–33; DC I.10 (52), 53–54; and DC II.1 (75), 73–74. 88. See DC I.10 (53), 54. 89. “Intelligentia autem, vocabulorum rationalium ineptitudinem advertens, hos abicit terminos,” DC I.8 (35), 40. “Intellectus autem debilitatem rationis advertens has abicit coniecturas,” DC I.10 (52), 53. 90. See DC I.6 (25), 32–33; and DC I.7 (28), 35. 91. “cum ibi omnem numerum rationis in unitatem simplicissimam absolutum conspicias,” DC II.1 (75), 74. 92. See Schwarz, Problem der Seinsvermittlung, 295–297. Schwarz also reads De docta ignorantia as the staging ground for a controversy only fully prosecuted and resolved in De coniecturis and De mente. But where I see the tension arising between mediations of Logos and Arithmos, Schwarz locates the fault line between transcendental idealism (the priority of mens over res) and objective idealism (the priority of res over mens), the reconciliation of which reconfigures the medieval problematic of Platonic mediation into early modern idealism. Another instance is Metzke’s opposition of Christology and ontology in Cusan thought (see Metzke, “Nicolaus von Cues”; and Jacobi, Methode der cusanischen Philosophie, 111–114). 93. On the development of Cusan Christology in the 1440s, see Dahm, Soteriologie, 151–204. On the related theology of creation in those years, see Wolter, Apparitio Dei, 159–203. 94. Nagel, Nicolaus Cusanus—mathematicus theologus, 15. 95. “Oportet intellegentem in omnibus ita ad complicationem sicut explicationem advertere, ne erret.” Sermo XXIII (6), ed. Haubst and Bodewig, 361. 96. See “Epistola ad Rodericum de Trevino,” in Nicholas of Cusa, Writings on Church and Reform, trans. Izbicki, 430–449. Cf. also De genesi IV (168), ed. Wilpert, 120; and De dato patris luminum IV (110), ed. Wilpert, 81.

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  97. In March and April 1444, see Sermo XXX (8–12), ed. Haubst and Schnarr, 45– 49; Sermo XXXII (4), ed. Haubst and Schnarr, 54–55; and Sermo XXXIII (4–5), ed. Haubst and Schnarr, 58–60.   98. In May and June 1444, see Sermo XXXVII (5–6), ed. Haubst and Schnarr, 75– 77; and Sermo XXXVIII (13–16), ed. Haubst and Schnarr, 111–115.   99. See Sermo XLV (5–6), ed. Haubst and Schnarr, 189–190; Sermo XLVII (10–11), ed. Haubst and Schnarr, 199; and Sermo XLVIII (24–31), ed. Haubst and Schnarr, 210–212, which informs Dialogus de deo abscondito and De quaerendo deum. 100. Stallmach suggests that De quaerendo deum continued the epistemology of De coniecturis (see “Geist als Einheit und Andersheit”). 101. See further Albertson, “ ‘That He Might Fill All Things’.” 102. On theosis in Cusanus, see Hudson, Becoming God. 103. See De filiatione dei II (60), ed. Wilpert, 45; ibid. IV (76), ed. Wilpert, 56; and ibid. VI (85–87), ed. Wilpert, 61–62. 104. See De filiatione dei III (70), ed. Wilpert, 51–52; and ibid. VI (85), ed. Wilpert, 61. Nicolle notes that “resolutio” is also a mathematical term (“Quelques sources,” 58). 105. See De filiatione dei III (65–68), ed. Wilpert, 48–50. 106. See De dato patris luminum IV (111), ed. Wilpert, 81–82. 107. See De dato patris luminum II (97), ed. Wilpert, 72; and ibid. III (106), ed. Wilpert, 78. In other words, the aporiae of creation (DI II.2) that Nicholas formerly resolved through reciprocal folding (DI II.3) are now reconciled in terms of the Incarnation. 108. See De dato patris luminum II (103), ed. Wilpert, 77. 109. See De filiatione dei IV (72), ed. Wilpert, 52–54; cf. ibid. VI (88), ed. Wilpert, 62–63. The concept of “innumerable monad” is probably Proclian. 110. “Et hoc ipsum est creatoris creare, quod est rationis rationcinari seu numerare.” De dato patris luminum III (105), ed. Wilpert, 78; trans. Hopkins, 122. 111. On the history of the geometrical problem and its significance, see Mueller, “Aristotle and the Quadrature of the Circle.” For an overview of Cusanus’s geometrical writings, see Hofmann, “Sinn und Bedeutung”; and Nagel, Nicolaus Cusanus und die Entstehung der exakten Wissenschaften, 61–82. Nicholas’s mathematical works can be read in German or French translations, both with very fine introductions worth consulting. See Die mathematischen Schriften, trans. Hofmann; and Les Écrits mathématiques, trans. Nicolle. 112. “Inter quae cum nulla rationalis proportio cadat, oportet in quadam coincidentia extremorum hoc latere secretum. Quae cum in maximo sit, ut alibi traditur, et maximum sit circulus, qui ignoratur, in minimo, qui est triangulus, inquiri ipsam debere ibi ostenditur.” De geometricis transmutationibus 3, in Nicholas of Cusa, Scripta mathematica, ed. Folkerts, 5. 113. For a scientific evaluation of the cardinal’s efforts, see Nicolle, Mathématiques et métaphysique, 66–89; Böhlandt, Verborgene Zahl, 188–207; and Müller, Perspektivität und Unendlichkeit, 47–73.

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114. See De genesi II (155–156), ed. Wilpert 112–113; and ibid. II (158–159), ed. Wilpert, 114–15. Reinhardt has rightly emphasized this aspect of the treatise (see “L’interprétation philosophique de la création”). 115. On the immediate context of De filiatione dei, see Schwaetzer, “Viva similitudo.” 116. See De genesi I (147), ed. Wilpert, 107–108; and ibid. II (154), ed. Wilpert, 111–112. On idem absolutum as a divine name, see Beierwaltes, “Identität und Differenz,” 117–120; and Yamaki, “Cusanische Entdeckung des Begriffs ‘idem absolutum’.” 117. “Potest igitur creatio seu genesis dici ipsa assimilatio entitatis absolutae, quia ipsa, quia idem, identificando vocat nihil aut non-ens ad se. Hinc sancti creaturam dei dixerunt similitudinem ac imaginem.” De genesi I (149), ed. Wilpert, 109. This concept will prove crucial in De mente (1450). 118. See De genesi I (150–151), ed. Wilpert, 109–110. 119. Mandrella, “Wissenschaftstheoretische Primat,” 199. Mandrella contends that passages from De mente and De possest suggest that Cusanus was familiar with Aquinas’s commentary on Boethius’s De trinitate II. This may well be, but if Cusanus was intensively studying Thierry of Chartres’s commentaries on De trinitate—and indeed on points where Thomas questioned the Boethian division of science (cf. Speer, “Hidden Heritage,” 165)—then it is equally important to track his evolving relationship with those Chartrian sources.

chapt er 9   1. See Inthorn and Reder, Philosophie und Mathematik, 3–5.  2. Inthorn and Reder describe four possible ways to integrate the two disciplines:  (1)  the priority of the dialectical (Cusan mathematics as metaphors within theology); (2)  the priority of the binary (Cusan theology as axiomatic proofs more geometrico); and (3)  disjunction (the failure of the Cusan experiment). None of these suffice, they rightly note. Hence they propose (4) a higher synthesis of the binary and dialectical. But the “basis” of this synthesis turns out to be vaguely “human rationality” (see Philosophie und Mathematik, 26–29).   3. See Nickel, “Nikolaus von Kues: Möglichkeit mathematischer Theologie,” 20–25.  4. Müller uses the term in a different sense. Instead of addressing the disciplinary gap between mathematics and theology, Müller describes what he calls a “dual mathematics” in Cusanus, the gap between arithmetical geometry (deductive mathematics) and speculative geometry (inductive mathematics) (see Perspektivität und Unendlichkeit, 45–46, 76–77). This builds on Bocken’s analysis of the “problematic relationship of arithmetic and geometry” in De docta ignorantia and De coniecturis (see “Zahl als Grundlage,” 210–219).   5. See Velthoven, Gottesschau und menschliche Kreativität, 15–29.   6. See ibid., 115; cf. Haubst, “Thomas- und Proklos-Exzerpte,” 26–39. Velthoven claims that Cusanus found his notion of mens as complicatio in the Proclus fragments and then inferred explicatio on his own. One searches in vain

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for any trace of complicatio or explicatio in these Proclus texts. On one occasion Velthoven notes Cusanus’s dependence on Thierry’s reciprocal folding (Gottesschau und menschliche Kreativität, 94–95), but this fails to inform his analysis (see ibid., 112–116). Velthoven also credits Proclus for the pivotal transition from the second to first mode in DM VII (106–107) (see ibid., 123–124), on which see further below.  7. Velthoven, Gottesschau und menschliche Kreativität, 143.  8. Folkerts, the editor of the critical edition of Cusanus’s mathematical works, repeats the common view that the treatment of number in De mente puts an end to Pythagorean confusions:  “Cusanus teilt nicht die pythagoreische Auffassung von der Identität zwischen der mathematischen Ordnung und der Naturordnung. Da die Mathematik dem Seinsbereich des menschlichen Geistes angehört, ist sie nicht gleichbedeutend mit der Struktur der Welt, sondern wird zum Instrument des Menschen, um diese Struktur zu erfassen” (“Quellen und die Bedeutung,” 293). Yet Folkerts is only lightly paraphrasing Nagel, who in turn is citing the same page from Velthoven (see Nicolaus Cusanus und die Entstehung der exakten Wissenschaften, 59). See further instances in Müller, Perspektivität und Unendlichkeit, 40–43; Eisenkopf, Zahl und Erkenntnis, 71–73; and Yamaki, “Bedeutung geometrischer Symbole,” 297–300. Bocken (“Zahl as Grundlage,” 208–209) and Watts (Nicolaus Cusanus, 179) are rightly more cautious.   9. In support of his view, Velthoven cites the famous passage in De beryllo where Cusanus endorses Aristotle’s rejection of Pythagoreanism (Gottesschau und menschliche Kreativität, 142–143; cf. De beryllo 56, ed. Senger and Bormann, 63–64). (Note that Velthoven’s citations follow the older Baur edition.) He adduces Haubst’s comment on the passage:  “[Cusanus] kennt den Irrtum der Pythagoreer, welche die Zahlen und überhaupt das Mathematische substantialisierten und dann in diesen begrifflichen Schöpfungen unserer ratio die Prinzipien der sinnenfälligen Welt erblickten” (Bild des Einen, 205). There are three problems here. (1) The De beryllo passage selected by Velthoven contradicts every other occasion of Cusanus mentioning Pythagoras or Pythagoreanism in De docta ignorantia, De mente, and De ludo globi, where he uniformly sings their praises; see further Horn, “Cusanus über Platon.” To grant the exceptional case priority over the most common is hermeneutically dubious. (2) On the page after Velthoven’s citation, Haubst notes that the Boethian trope of number as the first exemplar in God’s mind appears continuously from De docta ignorantia to De coniecturis to De mente. “Auf diese Weise findet Nikolaus von Kues aber den Zugang zu dem christlichen, näherhin dem boethianischen Pythagoreismus, welcher der Zahl eine grundlegende Proportion in der göttlichen Schöpfungsidee zumißt” (Haubst, Bild des Einen, 206). A  few pages later:  “Cusanus sieht sich also selbst in der Linie der pythagoreischen Tradition” (ibid., 209–210). (3)  Velthoven defines Pythagoreanism by citing

384

10. 11.

12. 13. 14. 15. 16. 17. 18.

19. 20.

Notes Aristotle’s judgment against them in Metaphysics (see Gottesschau und menschliche Kreativität, 136). This confuses Cusanus’s apprehension of Pythagoreanism via Aristotle with the historian’s burden first to judge what Pythagoreans actually taught in the fourth century bce, and then to compare them to Cusanus’s Neopythagoreanism. To observe that Cusanus repeated Aristotle’s questionable account is not to demonstrate that Cusan philosophy is intrinsically ­anti-Pythagorean. Instead one must evaluate the sources that historically transmitted such ideas to Cusanus, namely, Boethius and Thierry of Chartres. See Velthoven, Gottesschau und menschliche Kreativität, 167–69. On Boethius, see e.g., DI I.11 (32), 42; DC I.2 (9), 14; and DM VI (94), 140. Some of Velthoven’s objections are rather superficial, such as his dismissal of Boethian number theory, or his insistence that if mathematicals are projections of the human mind, they are disqualified from naming God (see Gottesschau und menschliche Kreativität, 167–69, 179–180). Velthoven too quickly defers to Karl Jasper’s answer to the question “Ist der Weltschöpfer ein Mathematiker?” (ibid., 180–181). Jaspers simply asserts that in passages like DI II.13 (175), Cusanus “begreift vielmehr, daß die gesamte mathematische Welt eine Hervorbringung des menschlichen Geistes sei, der selber mit diesem seinem hervorbringenden Vermögen von Gott geschaffen wurde. Daher ist mathematisches Denken menschliches Denken. Der Mathematiker ist nicht Gott, sondern der Mensch” (Nikolaus Cusanus, 87–88). Velthoven weighs some passing remarks by Apel as a representative of the opposing view (one is tempted to say a straw man; cf. Apel, “Das ‘Verstehen,’ ” 146–48), but then concurs with Jaspers, whom he cites frequently throughout his book. See Velthoven, Gottesschau und menschliche Kreativität, 169. See respectively ibid., 177–178; and ibid., 171–173. Ibid., 166; cf. Jaspers, Nikolaus Cusanus, 87. Stallmach makes some helpful comments on this point, contra Jaspers (see “Cusanische Erkenntnisauffassung,” 50–52). Flasch, “Nikolaus von Kues: Idee der Koinzidenz,” 251. See ibid., 251–255. Cranz offers a valuable alternative to Flasch’s theory of development (see “Late Works of Nicholas of Cusa”; and “Development in Cusanus?”). For Cranz, Cusanus underwent three transitions that illustrate the difference between his middle (1440–1453) and late (1454–1464) periods:  (1)  from the search for God in darkness toward the immediate vision of God in light; (2)  from the ontic univocity of God and creatures toward the progressive weakening of ontology; and (3)  from situating himself within Greek philosophical traditions toward radically reinterpreting them. Nothing in my argument disagrees with Cranz’s account. See Flasch, Nikolaus von Kues: Geschichte einer Entwicklung, 103–115; cf. Cassirer, Individuum und Kosmos, 10–11. See Flasch, Nikolaus von Kues: Geschichte einer Entwicklung, 99–102, 121, 141–142.

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21. See ibid., 130. This requires Flasch to interpret Nicholas’s discussions of the divine Word in Sermon XXII as closer to De aequalitate and De principio in the late 1450s than to Book III of De docta ignorantia just months before. Similarly, when Cusanus uses mathematical concepts from Boethius and Thierry in the sermon, Flasch states that these passages are not about number or quadrivium, but actually about conceptual thinking and the principles of philosophical analysis (see ibid., 133–134). Flasch does grant, however, that Thierry’s doctrines are the elements from De docta ignorantia that most endure in later works (see ibid., 116–117), and that they save Cusanus from pantheism (see Metaphysik des Einen, 256–260). 22. See Flasch, Nikolaus von Kues:  Geschichte einer Entwicklung, 135–142. With his various triads, Cusanus “sucht die Trinität philosophisch argumentierend zu beweisen” (ibid., 98). Likewise the cardinal’s doctrine of Incarnation is essentially “einer Art Perfektionslogik”:  “Sein Hauptinteresse gilt dem kosmologischen Sinn und damit der vernünftig aufweisbaren Notwendigkeit der Vereinigung von Gottheit und Menschheit” (ibid., 137). Cf. Flasch, Metaphysik des Einen, 324–326. 23. See Flasch, Nikolaus von Kues: Geschichte einer Entwicklung, 152–159. 24. See ibid., 168–169, 174–175. 25. See ibid., 273. 26. See ibid., 292–293. 27. See ibid., 289, 403. Flasch rues the priority given to De docta ignorantia in Cusanus studies and speculates what could have happened had De mente provided the point of departure instead (see ibid., 273–275). For Flasch the cardinal’s negative theology frustrated the potential of his philosophical henology until it was corrected around 1450 (see Metaphysik des Einen, 318–329). Hoye rightly objects: “Flasch ist zwar zuzustimmen, daß die negative Theologie nicht das Letze sei, aber seine Interpretation geht insofern fehl, als sie den Grund für diese Eingrenzung der negativen Theologie in der Einheitsphilosophie zu finden meint” (Mystische Theologie, 59). Hoye ably lays out the distorting effects that this presupposition exerts on Flasch’s exposition of Cusan texts (ibid., 58–65; cf. Hoye’s insightful review of Nikolaus von Kues: Geschichte einer Entwicklung). Ironically, Flasch’s historical account of the genesis of negative theology from the Parmenides, Plotinus, and Proclus is insufficiently genetic (see Nikolaus von Kues:  Geschichte einer Entwicklung, 403–404). As I have shown, between Plato and Plotinus the development of negative theology passed directly through Neopythagoreanism. So when Cusanus combines mathematical concepts, Neoplatonist henology, and negative theology together in De docta ignorantia, these are by no means inherently opposed elements that need sorting out in the 1450s. 28. Flasch is sensitive to the unusual valences of mens in Cusanus’s account, which as he notes is difficult to locate in terms of historical sources (see Nikolaus von Kues: Geschichte einer Entwicklung, 290–302). Flasch tries to link two early passages on mens as imago dei to Proclus and Plotinus rather than to Boethian

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or Chartrian traditions. On DM I  (57), see Nikolaus von Kues:  Geschichte einer Entwicklung, 277; on DM III (73), see ibid., 279. He considers other comparable sources for the mens doctrine, including Averroes, Albert the Great, Proclus, Liber de causis, and Eckhart, before swearing off further attempts:  “Von einer Herleitung des Cusanus aus einer einzelnen ‘Quelle’ kann nicht die Rede sein; nur sei vor Rückübertragungen modernen Konzepte gewarnt” (ibid., 291–292). When he finally mentions Thierry as the source of reciprocal folding, it is only (but now rightly) to sharpen the distinction between explicatio in De mente and in De docta ignorantia (ibid., 294). Flasch’s abortive search is missing a vital clue. The crucial separation of mens from the function of explicatio, as I explain below, first originates as a critique of Thierry’s thought in F 8r, 468 = DI II.9 (150), 78. 29. See Flasch, Nikolaus von Kues: Geschichte einer Entwicklung, 312–317. 30. See ibid., 389–410. 31. See Meuthen, Nicholas of Cusa, 86; and Duclow, “Life and Works,” 37–38. 32. See De sapientia I (5–6), ed. Steiger, 8–12; and De staticis experimentis 162, ed. Baur, 222. 33. See Müller, Perspektivität und Unendlichkeit, 155–168. 34. Bormann is one of the few scholars to have scrutinized the continuities between the four modes in De docta ignorantia, the ascent and descent through four unities in De coniecturis, and the ascent through the four modes in De mente (see “Koordinierung des Erkenntnisstufen,” 67–70; note the recorded discussion at 82). 35. To his credit Haubst briefly suggests Commentum and Glosa as sources for De mente VII (see Bild des Einen, 190 n. 118). But the first to confirm the discovery in detail was Thomas P. McTighe, whose important article carries broader consequences than is often recognized, particularly in light of the Fundamentum ­treatise: “The De mente’s account of the levels of knowledge is in important respects derived from the description of the levels of knowledge given in Thierry of Chartres’s Commentum. The two accounts resemble each other in such remarkable detail as to preclude either the supposition of fortuitous agreement or of a derivation from some unknown common source. The unavoidable conclusion, I shall argue, is that Nicholas had the Commentum before him when he was composing his own De mente. There is also some evidence, though this is less certain, that he also utilized the Lectiones and the Glosa” (McTighe, “Thierry of Chartres and Nicholas of Cusa’s Epistemology,” 170). McTighe faults Baur’s edition not only for failing to identify the Chartrian origins of De mente’s epistemology but for giving credit to Proclus instead (ibid., 169, 173). 36. Rusconi has critiqued McTighe for claiming that Cusanus used Commentum, stating that in fact Nicholas used Commentum, Lectiones, and Glosa (see “Commentator Boethii,” 251, 289). But this is precisely what McTighe has argued, and Rusconi offers no new evidence beyond the parallels provided by him (ibid., 287–288). Rusconi proposes instead that the four modes of Lectiones appear in De docta ignorantia II in Thierry’s ontological sense, and then in De mente VII–VIII in a new epistemological sense (ibid., 289). But this theory faces several challenges. (1)  Fundamentum does not reproduce the modal theory of Lectiones in De docta

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ignorantia II, but rather contradicts it. (2) There are several reasons to prefer Glosa to Lectiones as Nicholas’s source for De mente VII–VIII (see below). (3) The characterizations of necessitas complexionis in De docta ignorantia and in De mente are dramatically different. In 1440 the second mode is rejected as an inadequate substitute for the divine Word; in 1450, it is prized as the apex of rationality. 37. See De sapientia I (22–23), ed. Steiger, 45–50. 38. Compare the following sequence: F 8r, 466 = DI II.9 (149), 76

De sapientia I (23), ed. Steiger, 48

sed est verbum et filius aequalis patris in divinis, et dicitur logos seu ratio, quoniam est ratio omnium . . . quoniam non est nisi una infinita forma formarum cuius omnes formae sunt imagines.

Sapientia igitur, quae est ipsa essendi aequalitas, verbum seu ratio rerum est. Est enim ut infinita intellectualis forma, forma enim dat formatum esse rei.

F 7v, 466 = DI II.9 (148), 74

De sapientia I (23), ed. Steiger, 48–49

Unum enim infinitum exemplar tantum est sufficiens et necessarium, in quo sunt omnia ut ordinata in ordine, omnes quantumcumque distinctas rerum rationes adaequatissime complicans, ita quod ipsa infinita ratio est verissima ratio circuli, et non maior nec minor nec diversa aut alia.

Unde infinita forma est actualitas omnium formabilium formarum ac omnium talium praecisissima aequalitas. Sicut enim infinitus circulus, si foret, omnium figurarum figurabilium verum exemplar foret et cuiuslibet figurae essendi aequalitas . . . et omnium mensura adaequatissima licet simplicissima figura, sic infinita sapientia est simplicitas omnes formas complicans et omnium adaequatissima mensura.

F 7v, 466 = DI II.9 (149), 74

De sapientia I (24), ed. Steiger, 50

Visis rerum diversitatibus admirandum est, quomodo unica simplicissima ratio omnium etiam sit diversa singulorum.

O quam admiranda est illa forma, cuius infinitatem simplicissimam omnes formabiles formae nequeunt explicare!

F 7v, 466 = DI II.9 (149), 74

De sapientia I (25), ed. Steiger, 51

In hoc enim, quod videmus diversitatem rationum omnium rerum verissime esse, tunc in hoc, quod hoc est verissimum, apprehendimus una rationem verissimam, quae est ipsa veritas maxima.

Sic vides unicam et simplicissimam dei sapientiam, quia est infinita, esse omnium formarum formabilium verissimum exemplar.

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39. See De sapientia II (38–42), ed. Steiger, 70–75; cf. F 5r, 454 and F 7r, 466. 40. See DM I (57), 90. 41. See DM IX (122–125), 176–178. Cusanus also proposes that what Plato calls anima mundi and what Aristotle calls natura both designate necessitas complexionis: see DM XIII (145–146), 198–199. 42. See DM II (58–66), 92–103; cf. DM III (71), 107. 43. See DM II (67), 103–104. Cf. De sapientia I (23), 48–50; and F 7r–7v, 466. 44. See DM II–III (68–70), 104–107. 45. “Et ita mens est imago complicationis divinae prima omnes imagines complicationis sua simplicitate et virtute complicantis. Sicut enim deus est complicationum complicatio, sic mens, quae est dei imago, est imago complicationis complicationum.” DM IV (74), 113; cf. DM III (72), 108–110. The point is also emphatically repeated in an important passage on the second mode (see below), DM IX (122), 176. The term complicatio complicationum recurs in only one other book in the Cusan corpus, and there with specific reference to the quadrivium: see LG II (86), 105; and LG II (92), 115. On the mind’s creativity as an image of God, see Velthoven, Gottesschau und menschliche Kreativität, 123–128; and Stallmach, Ineinsfall der Gegensätze, 37–58. 46. “Attende aliam esse imaginem, aliam explicationem. Nam aequalitas est unitatis imago. . . . Et non est aequalitas unitatis explicatio, sed pluralitas. Complicationis igitur unitatis aequalitas est imago, non explicatio.” DM IV (74), 113. 47. See F 8r, 468 = DI II.9 (150), 78. 48. On assimilation, see DM IV (75), 115; cf. De genesi I (149–152), ed. Wilpert, 108– 111. Cusanus is fond of describing the mediating position of mens between God and world with the phrase “post mentem”; cf. his appropriation of the phrase already in DI II.9 (146), 70. “Unde quantum omnes res post simplicem mentem de mente participant, tantum de dei imagine, ut mens sit per se dei imago et omnia post mentem non nisi per mentem.” DM III (73), 112. “Proprie ita est, quoniam omnia, quae post mentem sunt, non sunt dei imago nisi inquantum in ipsis mens ipsa relucet.” DM IV (76), 116. But this echoes the very idiom used by Fundamentum to define the false mediation of the Platonist mens: “Nec cadit eo modo medium inter absolutum et contractum, ut illi Platonici imaginati sunt, qui animam mundi mentem putarunt post deum et ante contractionem mundi.” F 8r, 468. Cusanus also applies Fundamentum’s distinction of temporal and natural priority (previously borrowed for his Christology in De docta ignorantia) to define mens: see DM V (81), 123. 49. Flasch suggests that the destiny of mens in De mente has a prehistory in De docta ignorantia and De coniecturis (see Nikolaus von Kues: Geschichte einer Entwicklung, 112, 149–150). 50. “What is unique to the De mente is the theme of the mind’s use of its own immutability and its own simplicity as instruments by which it achieves knowledge. What, then, is the provenance of this epistemological feature? It is the same as

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that of the metaphysics which undertrusses the epistemology—the commentaries of Thierry. But, whereas Cusa’s metaphysics of complicatio-explicatio and the four modi essendi is borrowed from Thierry’s Lectiones (and, possibly, also the Glosa), his epistemology derives from the Commentum” (McTighe, “Thierry of Chartres and Nicholas of Cusa’s Epistemology,” 171). Neither McTighe nor Rusconi discusses Septem’s critical role in De mente VII–VIII. 51. There are only seven extant sources that repeat Thierry’s modal theory, and only Glosa has the distinctive characteristics that appear in Cusanus’s version in De mente. The seven possible sources are Lectiones, Glosa, the Munich Abbreviatio of the Lectiones, Septem, Clarembald’s Tractatus and Tractatulus, and Fundamentum. Septem and Fundamentum use unusual or partial formulae that do not connect the four modes with mathematics. The Abbreviatio most often calls the second mode necessitas connexionis, not necessitas complexionis (see Abbreviatio monacensis De trinitate II.9, H 339; and ibid. II.21–23, H 342–343). Neither of Clarembald’s accounts of the four modes attends to the disciplinary consequences of the doctrine, as emphasized in De mente (see Tractatus II.43–47, ed. Häring, 124–126; Tractatulus 21–24, ed. Häring, 235–237). These criteria leave Lectiones and Glosa as the remaining possibilities, and there are five arguments in favor of Glosa. (1) The initial description of the four modes in DM VII (97) matches Glosa II.12–23, H 271–273, better than Lectiones II.9–14, H 157–159. In Lectiones (and in Fundamentum) the four modes are subdivided as modes of necessity or possibility; in Glosa and De mente they are distinguished as absolute or determined. (2)  The structure of the four modes is explained in terms of simple unity and alterity in DM VII (105) and in Glosa II.12–23, whereas in Lectiones Thierry explained them in terms of reciprocal folding. (3) Unlike Lectiones, only Glosa II.21, H 273, links the second mode to anima mundi and intelligentia, much as Fundamentum had done, and as evidently concerned Cusanus in De mente. (4) In Lectiones, Thierry foregrounded the disciplines as they relate to the four modes; but in Glosa Thierry instead began with the faculties of knowledge, as he had in Commentum, but now in relation to the four modes. The interests of Cusanus in De mente run parallel to Glosa on this point, not Lectiones—hence the need for the spiritus theory of mind. (5) The image of the “circle in the mind” from DM VII (103), 155, most resembles Thierry’s version in Glosa II.7, H 270, not Lectiones II.20, H 161, contra Steiger’s notes. For these reasons we can be confident that the new text Cusanus drew on in 1450 was Thierry’s Glosa, even if we remain unsure how he accessed the text. 52. See Commentum II.3–6, H 68–70; and Septem 952C–954A. Notably Cusanus has the Orator and Philosopher attribute the spiritus theory to certain physici— the name Septem had used to refer to Thierry. See DM VIII (112), 165. 53. See Commentum II.8–14, H 70–72; and Lectiones II.11–13, H 158. 54. See Glosa II.1–10, H 268–270; and Glosa II.24–27, H 273–274.

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55. See Septem 952D–953D. Cusanus briefly mentions the same spiritus epistemology in DC II.14 (142), 142; and LG II (101), 125–127. On similar ideas in De quaerendo deum, see Oide, “Einfluss der galenischen Pneumatheorie.” 56. “mentem esse, ex qua omnium rerum terminus et mensura. Mentem quidem a mensurando dici conicio.” DM I  (57), 90. According to Stadler, not only is mensura central to Cusan epistemology, but furthermore “die gänzliche Einbindung des Maßbegriffes in die Erkenntnisproblematik kann als cusanisches Spezificum gegenüber anderen Traditionen der Verwendung des Maßbegriffes . . . gelten” (“Zum Begriff der mensuratio,” 118). 57. See DM IX (123–124), 177. The image of God using compasses to create the world was common since the eleventh century, but Nicholas applies the metaphor to the human mind (see Friedman, “Architect’s Compass in Creation Miniatures”). 58. See DM IX (116), 171; and DM VII (97–98), 146–148. 59. See DM VII (97), 147. 60. See DM IX (125), 178. 61. See DM VII (102–107), 153–161. 62. See DM X (127–128), 179–181. 63. See DM IX (116), 171–172. 64. See Commentum II.15, H 73; Lectiones II.30, H 164; Glosa II.9–10, H 270; and especially (contra Steiger’s notes) Glosa II.27, H 274. 65. See DM VIII (111), 164–165. On the meaning of disciplina in De mente and its roots in Boethian traditions, see Rusconi, “Intellectu qui est disciplina.” 66. See DM X (126), 179. 67. See DM VII (107), 161. 68. See DM VII (102), 153–154; and DM VII (104), 156. 69. See DM VII (103), 155. Cf. Velthoven’s interpretation of this passage in Gottesschau und menschliche Kreativität, 152. 70. See DM VII (105), 157. 71. “talis profecto supra determinatam complexionis necessitatem videret omnia, quae vidit in varietate, absque illa in absoluta necessitate simplicissime, sine numero et magnitudine ac omni alteritate. Utitur autem hoc altissimo modo mens se ipsa, ut ipsa est dei imago, et deus, qui est omnia, in ea relucet, scilicet quando ut viva imago dei ad exemplar suum se omni conatu assimilando convertit.” DM VII (106), 158–159; trans. Hopkins, 560 (modified). Velthoven incorrectly claims that Proclus is Nicholas’s source for this passage, describing it not as an ascent from the second to first mode of being, but as mens “in sich gehen, um Auge in Auge mit sich selbst zu stehen,” a Socratic-Augustinian “Reflexion auf sich selbst” (see Gottesschau und menschliche Kreativität, 123–124). 72. See DM IX (122), 176. 73. See DM XV (156), 212. 74. “omnes complicationes sunt imagines complicationis simplicitatis infinitae et non explicationes eius, sed imagines, et sunt in necessitate complexionis. Et

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mens prima imago complicationis simplicitatis infinitae vim harum complicationum sua vi complectens est locus seu regio necessitatis complexionis,” DM IX (122), 176 (my trans.). On the epistemological function of the enfolding of mens, see Velthoven, Gottesschau und menschliche Kreativität, 100–112. 75. See DM IV (74), 112–114, discussed above. 76. See DM IX (123), 177. 77. “Sunt autem omnia sensibilia in quadam continua instabilitate propter possibilitatem materialem in ipsis abundantem. Abstractiora autem istis, ubi de rebus consideratio habetur, non ut appendiciis materialibus, sine quibus imaginari nequeunt, penitus careant neque penitus possibilitati fluctuanti subsint, firmissima videmus atque nobis certissima, ut sunt ipsa mathematicalia.” DI I.11 (31), 42; trans. Bond, 100–101. 78. See DI I.12 (33), 44–46. 79. See DM VII (107), 161. On this typically Cusan model of “double ascent,” see Velthoven, Gottesschau und menschliche Kreativität, 185–196; cf. Müller, Perspektivität und Unendlichkeit, 103–105. 80. On the complicated question of mathematical abstraction in De mente, see Velthoven, Gottesschau und menschliche Kreativität, 144–153; and Kremer, “Erkennen bei Nikolaus von Kues,” 51–57. Velthoven and Kremer rightly contrast the active unfolding of mathematicals in De mente with the passive abstraction of mathematicals in De docta ignorantia. But the crucial ascent through the four modes is not a transition from passive to active but from an abstractive logic to a quite different visual logic. Bormann considers this passage to be the key to harmonizing abstraction and production in De mente (see “Koordinierung der Erkenntnisstufen,” 67–68). 81. See DM VI (88), 132–133. 82. See DM VI (91), 135–136. Cf. Philolaus, Fragments 4 and 5, in Huffman, Philolaus, 172–193; and Boethius, IA II.32.3, ed. Guillaumin, 128. 83. See DM VII (97), 145–146; and DM VI (92), 136. On Cusanus’s treatment of the soul as self-moving number in De mente and De ludo globi, see Eisenkopf, Zahl und Erkenntnis, 133–157. 84. See DM VI (94–95), 140–141. Cf. Boethius, IA I.2.1, ed. Guillaumin, 11; and Glosa I.38, H 267. 85. “Nescio, an Pythagoricus vel alius sim. Hoc scio, quod nullius auctoritas me ducit, etiamsi me movere tentet.” DM VI (88), 133. 86. “sed symbolice ac rationabiliter locuti sunt de numero, qui ex divina mente procedit, cuius mathematicus est imago. Sicut enim mens nostra se habet ad infinitam aeternam mentem, ita numerus nostrae mentis ad numerum illum. Et damus illi numero nomen nostrum sicut menti illi nomen mentis nostrae,” DM VI (88), 133; trans. Hopkins, 551 (modified). Nicholas states an incomplete version of the same idea in DI II.3 (108), 24–26. 87. “Nam sola mens numerat; sublata mente numerus discretus non est.” DM VI (93), 138.

392

Notes

 88. “Unde numerus substantiarum separatarum non plus est nobis numerus quam non-numerus, quia adeo a nobis est innumerabilis,” DM XII (143), 196; trans. Hopkins, 579.   89. See DM VI (92), 136–138.   90. “Unde sicut quoad deum rerum pluralitas est a mente divina, ita quoad nos rerum pluralitas est a nostra mente.” DM VI (93), 138. Cf. DM XII (143), 196: “Nam quamvis nos sublata varietate materiae non capiamus multiplicationem numeri, propter hoc tamen non desinit rerum pluralitas, quae est divinae mentis numerus.” This doctrine has a remarkable parallel in Achard of St. Victor’s notion of divine pluralitas (see De unitate dei et pluralitate creaturarum I.5–10, ed. Martineau, 72–78).   91. See DM VI (93–94), 138–140.   92. See DM VI (93), 138–139.   93. See DM VI (95), 141–142. The formulation of the arithmetical Trinity in this passage most closely resembles Commentum II.22, H 75, and Glosa V.18, H 297, in accordance with my discussion of Lectiones above.   94. “Mensurat etiam symbolice comparationis modo, ut quando utitur numero et figuris geometricis et ad similitudinem talium se transfert. Unde subtiliter intuenti mens est viva et incontracta infinitae aequalitatis similitudo.” DM IX (125), 178; trans. Hopkins, 569.   95. “ille videt omnem creaturam numerum divinae mentis aufugere non posse.” DM XII (144), 197; trans. Hopkins, 580.   96. “Ex quo habes inter mentem divinam et res non mediare numerum, qui habeat actuale esse, sed numerus rerum res sunt.” DM VI (96), 145; trans. Hopkins, 555 (modified). Cf. Velthoven’s interpretation of this passage in Gottesschau und menschliche Kreativität, 175–176.   97. See F 8r, 468 = DI II.9 (150), 76–78.   98. See DM IX (125), 178. By contrast, Fundamentum proposed the triad of form, matter, and connection as a “similitudo contracta” of the Trinity (see F 9r, 472).   99. See DM X (126–128), 178–181; cf. Boethius, IA I.1.1–5, ed. Guillaumin, 6–7. See also Nicholas’s meditations on quadrivial categories in terms of reciprocal folding in DM IX (116–121), 171–175; cf. Boethius, IA II.4.4–9, ed. Guillaumin, 89–91. 100. See Meuthen, Nicholas of Cusa, 87–96; Watanabe, Nicholas of Cusa: A Companion, 29–33; and Duclow, “Life and Works,” 38–40. 101. See Böhlandt, Verborgene Zahl, 236–240. 102. Nicholas never considered De theologicis complementis to be a separate volume from the geometrical work to which it was appended. He expressly forbade them from being copied separately (see TC 1:10–12, 4–5; given the length of chapters in De theologicis complementis, I provide line numbers as well for convenience). Judging from the copy he personally corrected and sent to Heymeric, they stand together as one book; the separate title was added later by a fifteenth-century hand (see Van de Vyver, “Brüsseler Handschriften,” 325–326).

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103. “in quo transtuli mathematicas figuras ad theologicalem infinitatem.” Letter 5 (14 September 1453), ed. in Vansteenberghe, Autour de la docte ignorance, 116. Cf. the similar statement at TC 3:75–80, 19. 104. See Imbach, “ ‘Centheologicon’,” 475. See further Hamann, Siegel der Ewigkeit, 50–59; and Nagel, Nicolaus Cusanus—mathematicus theologus, 5–6. 105. See Letter 5 (14 September 1453), ed. in Vansteenberghe, Autour de la docte ignorance, 113–117. 106. “Was Cusanus in De mente über das Urbild-Abbild-Verhältnis von Gott und Welt gesagt hat und über unseren Geist, der als naheres Bild Gottes die Dinge abbildlich in sich begreift, findet besonders in Complementum theologicum seine mathematische Veranschaulichung.” Haubst, Bild des Einen, 185. 107. See TC 2:77–85, 12–13; TC 7:1–25, 33–35; TC 11:1–15, 54–55; and TC 13:1–60, 75–80. 108. On infinite circles, see TC 3:37–63, 16–18; TC 6:1–26, 29–31; and TC 9:1–14, 40–41. On tangent circles, see TC 8:1–43, 37–40. 109. See TC 2:52–77, 9–12; TC 3:75–83, 19–20; TC 4:42–56, 23–25; and TC 5:21–27, 27. 110. The literature on De theologicis complementis, particularly its rich Trinitarian theology, is surprisingly slight. To his credit, Velthoven more than anyone gives the treatise due weight alongside De mente (see Gottesschau und menschliche Kreativität, 131–196 passim). Flasch understands the Complement’s centrality for his argument, even if his interpretation is somewhat idiosyncratic (see Nikolaus von Kues: Geschichte einer Entwicklung, 389–410). Rusconi has explained most of the important geometrical figures in the treatise (see “Visio und mensura”; cf. “El uso simbolíco,” 215–233). Bormann-Kranz’s Untersuchungen provides a useful chapter-by-chapter orientation. 111. “Si sic est in mathematicis, sic erit verius in theologicis.” TC 3:19–20, 14. Flasch interprets this sentence to mean that mathematics provides a sufficient, independent and certain basis for philosophizing before the leap into doctrinal theology—as if the cardinal wished to set mathematics and theology on their separate ways (see Nikolaus von Kues: Geschichte einer Entwicklung, 398). He is right that the Complement continues in the humanist tone of De mente (“Hymnen auf die Souveränität unseres Geistes”:  ibid., 394). But it surely runs against the grain of Nicholas’s project to conclude that in the fusion of mathematics and theology, mathematical rationality has the upper hand: “Die Cusanische Theologisierung der Mathematik, die schon seinen Zeitgenossen als ein Bruch mit der Mathematik vorkam, war zugleich eine Mathematisierung der Theologie” (ibid., 401). On this passage in Flasch, see Hoye, Mystische Theologie, 133–138. 112. See TC 5:33–48, 28–29. 113. “Creator igitur, dum omnia creat, ad se ipsum conversus omnia creat, quia ipse est infinitas illa, quae est essendi aequalitas.” TC 5:50–52, 29; trans. Hopkins, 756. 114. See TC 6:38–42, 32–33.

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115. See TC 9:36–45, 44–45. 116. See TC 6:27, 31. On fecunditas, see further Haubst, Bild des Einen, 166–171. 117. “Sic creator ipse ad se ipsum et infinitam fecunditatem respiciendo creat fecundam essentiam creaturae, in qua est principium complicativum virtutis, quod est centrum seu entitas creaturae, quae complicat in se virtutem suam.” TC 6:26–30, 31–32; trans. Hopkins, 756–757 (modified). 118. See TC 12:24–33, 63–64. 119. “Deus igitur mediante se ipso operatur quidquid vult etiam unum in aliud transmutando, et non est opus, quod deus habeat varios angulos ad varias transmutationes seu varia instrumenta, sicut oportet geometrum habere, sed unico angulo infinito omnia transfert. Et quia angulus ille est deus, estque voluntas dei deus. Et ita angulus ille simpliciter maximus est dei voluntas. Ideo deus sola voluntare omnia transfert et immutat.” TC 12:34–40, 64–65; trans. Hopkins, 768. Cf. TC 12:6–8, 62. 120. See TC 12:47–54, 65–66. 121. “Sic vides numerum incomprehensibilem et infinitum et innumerabilem, qui est maximus pariter et minimus, quem nulla ratio attingit nisi in umbra et caligine, quia est improportionalis ad omnem numerum numerabilem, et quomodo deus, qui dicitur numerus omnium rerum, ita est numerum sine quantitate discreta, sicut est magnus sine quantitate continua, et est idem angulus infinitus, qui est numerus infinitus, ut ipse simplicissimus simplicissime omnia et singula numeret, mensuret et transmutet.” TC 12:54–62, 66–67; trans. Hopkins, 768–769. 122. Cusanus’s remarks on number in De theologicis complementis build upon what he had written in De coniecturis and De mente. Cf. e.g., TC 9:33–36, 43–44; and DM VI (88–96), 132–145. 123. See TC 10:50–69, 52–54. 124. Here Flasch protests that the Complement’s references to vision need not be mystical, since visio and intuitio can be technical terms in mathematics (Nikolaus von Kues: Geschichte einer Entwicklung, 400). 125. See TC 2:28–70, 7–11. 126. “mentali visu intuear, quomodo in speculo mathematico verum illud . . . reluceat non modo remota similitudine, sed fulgida quadam propinquitate.” TC 1:7–10, 4; trans. Hopkins, 747. 127. “Coincidunt in deo mensurare et mensurari, quia est mensura et mensuratum.” TC 14:19–20, 82; trans. Hopkins, 772. 128. “Se igitur intuendo intuetur simul et omnia creata et nequaquam differenter se et  alia, et videndo creata simul et se videt. Creata enim, quia creata, non videntur perfecte, nisi creator videatur . . . et sic videre se est videri a se et videre creaturas est videri in creaturis. Eodem modo, si quaeritur de creatione; creatio enim in deo est visio.” TC 14:13–23, 81–82; trans. Hopkins, 772 (modified). 129. See DI I.3–4 (10–11), 14–16.

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130. “Ad circularem igitur capacitatem continue accedit, quam sua virtute numquam attingit, sed de gratia creatoris rapitur de angulari capacitate in circularem,” TC 9:68–70, 46–47; trans. Hopkins, 762 (my emphasis). 131. “Et haec est similitudo figuralis conveniens, qua duci poteris, ut videas differentiam esse inter eas mentes, quae assecutae sunt perfectionem capacitatis mentalis per raptum in mundum intelligibilem, et inter eas, quae venantur capacitatem in sensibili mundo sub sensibilibus particularibus signis, sicut de polygoniis et circulo mathematice experimur.” TC 9:74–79, 47–48 (my trans. and emphasis). 132. For an overview of ecstasy in Christian mysticism, see the older but still useful Baumgartner, “Extase,” 2072–2171. 133. McGinn shows that while Cusanus used the term without fanfare in De docta ignorantia (see, e.g., DI III.11 [246], 76–78), he once warned against deceptive ecstatic experiences in a letter to Caspar Aindorffer from 1452 (see Harvest of Mysticism, 437–456; cf. Letter 4 [22 September 1452], ed. in Vansteenberghe, Autour de la docte ignorance, 112). Dahm observes that Nicholas referred to raptus frequently in his sermons in the 1440s (see Soteriologie, 180–185). Nicholas also speaks of an incomprehensible vision occurring in a sudden rapture (“quodam incomprehensibili intuitu quasi via momentanei raptus fiat”: Apologia doctae ignorantiae 16, ed. Klibansky, 12), a mental rapture (“in raptu quodam mentali”:  De visione dei XVI [70], ed. Riemann, 57), or a rapture from image to exemplar (“ex similitudine in exemplar rapitur”: Letter to Niccolò Albergati 5, in Nicholas of Cusa, Cusanus-Texte IV, ed. Bredow, 28). Flasch is quick to cite the 1452 letter as evidence of the cardinal’s disavowal of De docta ignorantia, yet somehow overlooks the reappearance of raptus (after 1450) in De theologicis complementis (see Nikolaus von Kues: Geschichte einer Entwicklung, 440, 443). 134. See DM VI (96), 145. 135. Nagel writes that “so wie er einst die theologische Figuren aus De docta ignorantia mathematische nutzbar machte, verwandelt er nun in dieser Schrift [De theologicis complementis] die mathematischen Figuren in theologische” (Nicolaus Cusanus—mathematicus theologus, 23). See also Yamaki, “Bedeutung geometrischer Symbole,” 300–305; and Flasch, Nikolaus von Kues: Geschichte einer Entwicklung, 393–394. 136. Against this view, some see the Complement as a return to the abstractive ascent of De docta ignorantia I. See, e.g., Rusconi, “Visio und mensura,” 189; and Flasch, Nikolaus von Kues: Geschichte einer Entwicklung, 393. 137. See De beryllo 20, ed. Senger and Bormann, 24–25. 138. See respectively De beryllo 52–53, ed. Senger and Bormann, 59–60; and ibid. 58–63, ed. Senger and Bormann, 66–73. 139. See ibid. 35, ed. Senger and Bormann, 39–40. 140. See ibid. 36–37, ed. Senger and Bormann, 40–43. “Solum autem notes non esse necessarium universalem esse creatum intellectum aut universalem mundi

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animam propter participationem, quae Platonem movit. Se ad omnem essendi modum sufficit habunde primum principium unitrinum, licet sit absolutum et superexaltatum, cum non sit principium contractum ut natura, quae ex necessitate operatur, sed sit principium ipsius naturae et ita supernaturale, liberum quo voluntate creat omnia.” Ibid. 37, ed. Senger and Bormann, 42–43. Cf. F 8r, 468 = DI II.9 (150), 76. 141. See De beryllo 39, ed. Senger and Bormann, 45–46. Fundamentum proposes a similar triad of forma absoluta, possibilitas absoluta, and conexio absoluta (see F 9r–9v, 472). 142. See Fundamentum’s lengthy account of the veteres on the fourth mode (F 4r–5r, 448–452) and the author’s list of historical cognates for the second mode (F 7r, 460). 143. See De beryllo 56, ed. Senger and Bormann, 64. 144. See Velthoven, Gottesschau und menschliche Kreativität, 143. 145. On De mathematica perfectione and Aurea propositio in mathematicis, see Böhlandt, “Vollendung und Anfang”; cf. Böhlandt, Verborgene Zahl, 282–304. 146. “Circa unitrinum igitur principium et rerum ab eo effluxum versabitur altissima sapientis speculatio.” Aurea propositio in mathematicis 7, in Nicholas of Cusa, Scripta mathematica, ed. Folkerts, 226. See further Böhlandt, Verborgene Zahl, 301–304.

Chapt er 10   1. See Meuthen, Nicholas of Cusa, 133–138; and Duclow, “Life and Works,” 42–47.  2. Three of the fields comprise the arithmetical Trinity, though now viewed through the lens of Proclus and Ps.-Dionysius: see De venatione sapientiae XXI (59)–XXVI (79), ed. Klibansky and Senger, 56–76.   3. See LG II (109), 136.   4. Euler notes that during these four years Cusanus invested all of his intellectual efforts into sermons, producing roughly forty per year. These sermons, which regularly lasted over an hour when delivered, account for all of his theological writings in this period and over half of his total sermons (see “Proclamation of Christ,” 89–90).   5. See the sermons cited in Euler, “Proclamation of Christ,” 92–99.   6. “Christus igitur si cognoscitur, omnia in ipso cognoscuntur. Christus si habetur, omnia in ipso habentur.” Sermo CCLXXX (12), ed. Riemann, 587; trans. in Euler, “Proclamation of Christ,” 91.  7. On these works, see Cranz, “De aequalitate and the De principio”; Nicolle, “Égalité, identité et répétition de l’Un”; Pasqua, “L’Un sans l’être”; and Flasch, Nikolaus von Kues: Geschichte einer Entwicklung, 482–516.  8. See De aequalitate 7–10, ed. Senger, 10–15; and ibid. 14–17, ed. Senger, 19–23.

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  9. See ibid. 23–26, ed. Senger, 30–35; and ibid. 34, ed. Senger, 44–46. See further Schwaetzer, Aequalitas. 10. See the coordination of multitudo, dualitas, and numerus in De principio 28–32, ed. Bormann and Riemann, 40–44; or the discussion of the monad as primal enfolding of numbers in ibid. 39, ed. Bormann and Riemann, 54–57. 11. See DM VII (107), 161. 12. Casarella has explained the historical significance of Cusanus’s turn to possibility in comparison with Aristotle, Aquinas, and Heidegger (see “Nicholas of Cusa and the Power of the Possible”). 13. “Non enim unitas quae de deo dicitur est mathematica, sed est vera et viva omnia complicans. Nec trinitas est mathematica, sed vivaciter correlativa. Unitrina enim vita est, sine qua non est laetitia sempiterna et perfectio suprema.” De possest 50, ed. Steiger, 61–62. Of course this should not be read as a rejection of Thierry’s arithmetical Trinity, which he continued to use in Cribratio Alkorani (1461), De li non-aliud (1462), De venatione sapientiae (1463), and De ludo globi (1463). Haubst’s judgment that De possest marks the final “Entalgebrasierung der Trinitätsvorstellung” does not consider the context of this work as the site where Cusanus articulated his new orientation to physics (Bild des Einen, 250– 254). Cusanus was simply repeating the commonplace (found also in Boethius and Thierry) that the Trinity is not reducible to numbers but rather their living source. Flasch too states that after De possest Nicholas distanced himself from mathematical theology and its “Transposition geometrischer Argumente” (see Nikolaus von Kues: Geschichte einer Entwicklung, 540). But this forgets the case of De ludo globi, as I explain below. All that Cusanus intended here was what he had already stated in a sermon from 1455: the truest meaning of number is not quantitative number but the self-subsisting (i.e., divine) number from which all mathematical concepts originate. “Quod si quis diceret numerum esse quantitatem, dico non me sic dicere animam vivum numerum quasi mathematicum seu quantum, sed substantialem, a quo effluit conceptus mathematicalis in similitudine. Nam numerus mathematicalis quemdam numerum in se subsistentem praesupponit, qui de mathematicali iudicium profert.” Sermo CCII (4), ed. Donati and Mandrella, 446. 14. “Coeterna ergo sunt absoluta potentia et actus et utriusque nexus.” De possest 6, ed. Steiger, 7–8; cf. ibid. 8, ed. Steiger, 8–9. Cusanus later calls the triad posse, esse, nexus (see ibid. 48–52, ed. Steiger, 59–63). 15. See ibid. 14, ed. Steiger, 17–18. 16. “quia possest est possest, quae est unica ratio omnium modorum essendi.” Ibid. 28, ed. Steiger, 34; trans. Hopkins, 95. 17. “quoniam maximus et minimus actus coincidunt cum maxima et minima potentia ut sunt maximum absolute dictum.” F 5v, 456. 18. “Sed cum Christiani dicant aliam esse personam ipsius absoluti posse, quam nominamus patrem omnipotentem, et aliam ipsius esse, quam quia est ipsius

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posse nominamus filium patris, et  aliam utriusque nexum, quam spiritum vocamus, cum naturalis amor sit nexus spiritalis patris et filii,” De possest 48, ed. Steiger, 59. Cf. F 9r, 472: “Sicut ergo omnis possibilitas est in absoluta, quae est deus aeternus, et omnis forma et actus in absoluta forma, quae est verbum patris et filius in divinis, ita omnis motus conexionis et proportio ac harmonia uniens est in absoluta conexione divini spiritus, ut sit unum omnium principium deus.” 19. See De possest 6–7, ed. Steiger, 7–8; cf. F 6r, 458 and F 8r, 468. 20. See De possest 8–10, ed. Steiger, 9–13; cf. F 9r, 472. Although he introduces several modifications, Cusanus found the image originally in Republic 436de. 21. On God as entitas or forma formarum, see De possest 12–14, ed. Steiger, 14–18; cf. Commentum II.20–22, H 74–75. On the Son as figura, see De possest 58, ed. Steiger, 69; cf. Commentum II.32–34, H 78–79. On forma essendi and other such examples, see further Rusconi, “Cusanus und Thierry von Chartres.” 22. Nicolle has suggested, however, that even the figures in De docta ignorantia are designed to become mobile in the mind’s eye (see “How to Look at Cusanus’ Geometrical Figures?”). On motion and rest in De possest, see further Eisenkopf, “Thinking between quies and motus.” On the centrality of space and motion in Renaissance philosophy, see Cassirer, Individuum und Kosmos, 192–201. 23. See De possest 18–19, ed. Steiger, 23–25. 24. “Maximus ergo motus esset simul et minimum et nullus.” Ibid. 19, ed. Steiger, 24; trans. Hopkins, 85. Compare Cusanus’s prior theological statement that “nullus enim motus est in fine seu id quod esse potest nisi qui deo convenit, qui est motus maximus pariter et minimus seu quietissimus” (ibid. 10, ed. Steiger, 13). 25. “Non est ergo aliquis motus simpliciter maximus, quia ille cum quiete coincidit. Quare non est motus aliquis absolutus, quoniam absolutus motus est quies et deus.” F 9r, 472 = DI II.10 (155), 84. 26. See De possest 23, ed. Steiger, 28–29. 27. See Maier, “Significance of the Theory of Impetus”; and Murdoch and Scylla, “Science of Motion.” 28. See, e.g., De possest 72, ed. Steiger, 84–85. 29. See ibid. 69–71, ed. Steiger, 81–83. Although this theme appears in earlier works like De quaerendo deum, it surely takes on a new meaning after the developments of the 1450s. 30. On physics, see ibid. 62, ed. Steiger, 73–74. On theology, see ibid. 63–64, ed. Steiger, 74–76. 31. “Et vocatur speculatio illa mathesis seu disciplina.” Ibid. 63, ed. Steiger, 75. See Commentum II.15, H 73; and Glosa II.7–9, H 269–270. On this point see Rusconi, “Cusanus und Thierry von Chartres.” 32. See De possest 43, ed. Steiger, 52–53. 33. “nihil certi habemus in nostra scientia nisi nostram mathematicam, et illa est aenigma ad venationem operum dei.” Ibid. 44, ed. Steiger, 54; trans. Hopkins, 113 (modified).

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34. “Quare secundum mathematicae perfectam comprehensionem ad theologiam aenigma propinquius fieri posse arbitror.” Ibid. 61, ed. Steiger, 73; trans. Hopkins, 137 (modified). Note that this theme of mathematical aenigmata and natural theology occurs already in DI I.11 (30), 40. On the Cusan concept of aenigmata, see Velthoven, Gottesschau und menschliche Kreativität, 184–196; D’Amico, “Rolle der geometrischen Figur”; Rusconi, “Nature of Mirror”; and Rusconi, “El uso simbólico,” 203–254. 35. See De possest 44, ed. Steiger, 55. 36. Thurner has aptly described the deep connection between the sensible nature of Cusan aenigmata and the centrality of the Incarnation for Cusan thought as a whole (see “Theologische Unendlichkeitsspekulation,” 82–86). 37. Perhaps Nicholas was inspired (like Thierry) by Boethius’s image of rotating spheres that coordinate time and eternity (see Consolatio IV.6.15–17, ed. Moreschini, 123:62–124:78). Boethius even draws attention to the labyrinthine “play” of the circle: “Ludisne, inquam, me inextricabilem labyrinthum rationibus texens, quae nunc quidem qua egrediaris introeas, nunc vero quo introieris egrediare, an mirabilem quendam divinae simplicitatis orbem complicas?” Ibid. III.12.30, ed. Moreschini, 94:77–95:81. 38. See LG I (1), 3. 39. The early Paris and Basel editions of De ludo globi misidentified the center point as the number 1, although according to the text it should clearly be 10. See LG I  (50), 56; cf. Miller, “Nicholas of Cusa’s De ludo globi,” 138–139. The center, the divine King, holds the highest score for the players of the game. Pace Miller, Cusanus had good reason not simply to depict God once again as the divine One, but rather more provocatively and trenchantly, given the history of Neopythagoreanism, to depict Christ as the divine decad. 40. See LG I (5), 6–7. 41. See LG I  (6–7), 7–8. Cusanus calls such pebbles “impediments and suffocations” of the ball’s motion, bringing to mind the seeds falling on stony ground in the Parable of the Sower (see Matthew 13:7). 42. See LG I  (51), 57; cf. Nicholas’s example of tossing peas to illustrate contingency and difference in LG II (81), 100. For McTighe, De ludo globi represents Cusanus’s deepest engagement with creaturely difference (see “Contingentia and Alteritas,” 62–65). 43. See LG I (4), 5. 44. See LG I (5), 7; and LG I (59), 66. 45. “Cessat igitur impetu qui impressus est ei deficiente. Sed si globus ille foret perfecte rotundus, ut praedictum est, quia illi globo rotundus motus esset naturalis ac nequaquam violentus, numquam cessaret.” LG I (23), 27. Remarkably, the unceasing regular motion that Cusanus thus envisioned anticipated the seventeenth-century theory of inertia, as Maier has shown: “[Galileo] war nicht der erste, der auf das Phänomen [inertia] aufmerksam geworden ist:  dieses

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Verdienst scheint Nicolaus Cusanus zu gebühren (in De ludo globi, Opera I, Basel 1625, S.123). Aber er versucht keine mechanische Erklärung, sondern beschränkt sich auf eine neu-platonisch gefärbte philosophische Begründung” (Vorläufer Galileis, 152 n. 39). 46. One major exception to note in De possest is the doctrine of Christ as the universal manifestation (ostensio) of being: see De possest 31, ed. Steiger, 36–37. 47. “Iste, inquam, ludus significat motum animae nostrae de suo regno ad regnum vitae, in quo est quies et felicitas aeterna. In cuius centro rex noster et dator vitae Christus Iesus praesidet. Qui, cum similis nobis esset, personae suae globum sic movit, ut in medio vitae quiescat, nobis exemplum relinquens, ut quemadmodum fecit faciamus et globus noster suum sequatur, licet impossibile sit quod alius globus in eodem centro vitae, in quo globus Christi quiescit, quietem attingat.” LG I (51), 56–57; trans. Watts, 79. 48. See LG II (68–75), 81–90. 49. “Est autem circularis et centralis motus, qui vita est viventium. Quanto autem circulus centro est propinquior, tanto citius circumvolvi potest. Igitur, qui sic est circulus quod et centrum, in nunc instanti circumvolvi potest. Erit igitur motus infinitus. Centrum autem punctus fixus est. Erit igitur motus maximus seu infinitus et pariter minimus, ubi idem est centrum et circumferentia, et vocamus ipsum vitam viventium in sua fixa aeternitate omnem possibilem vitae motum complicantem.” LG II (69), 82–83; trans. Watts, 93. 50. “Ibi enim idem est centrum vitae creatoris et circumferentia creaturae. Christus enim deus et homo est, creator et creatura. Quare omnium beatarum creaturarum ipse est centrum. . . . [O]‌mnes beati per circumferentia circulorum figurati in circumferentia Christi, quae est similis creatae naturae, quiescunt et finem attingunt propter circumferentiae naturae creatae cum increata natura hypostaticam unionem, qua nulla maior esse potest. . . . [I]pse est unicus mediator, per quem accessus haberi potest ad viventem vitam.” LG II (75), 88–90; trans. Watts, 95–97. 51. See DI II.11 (157), 88. 52. See LG II (110–114), 137–140. The passage is rarely studied in depth, but see Senger, “Globus intellectualis,” 110–114. 53. See LG II (115–116), 140–142. 54. “Neque aliud est imago quam nomen suprascriptum. Sic dicebat Christus: ‘Cuius est imago et superscriptio’ eius? Responderunt: ‘Caesaris’. Facies igitur et nomen et figura substantiae et filius monetarii idem sunt. Filius igitur est ‘imago’ viva et ‘figura substantiae’ et ‘splendor’ patris, ‘per quem’ pater monetarius facit seu monetat sive signat omnia. Et cum sine signo tali non sit moneta, id unum, quod in omni moneta signatur, est exemplar unicum et formalis causa omnium monetarum.” LG II (116–117), 143; trans. Hopkins, 1245 (modified). 55. See Tractatus 41, H 572; Commentum II.32, H 78; and Commentarius Victorinus 95, H 501.

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56. Roth has discovered a fragment written in Nicholas’s hand on a blank page in Cod. Cus. 50, which might record the cardinal’s initial conception of the coinage metaphor, not long before De ludo globi was composed. The fragment establishes that mathematical theology was central to the original idea. Cusanus writes that when God numbers a creature (numerare dei) it is equivalent to God forming the creature (essenciare seu formare) (see Roth, “Geld und Begriffskunst”). That said, Cusanus made a similar remark in De dato patris luminum III (105), ed. Wilpert, 78. In a 1455 sermon he had already compared the intellect to a living gold coin that can evaluate any other currency by unfolding or enfolding: see Sermo CCII (3), ed. Donati and Mandrella, 445–446. 57. See LG II (120), 148. 58. See LG II (115), 141. 59. Like Haubst (see “Zusammenfassende theologische Erwägungen,” 194–195), Böhlandt has suggested that De ludo globi represents a rewriting of De coniecturis and draws attention to three similarities:  the fourfold unities resembling the fourfold analysis of coins; the prominence of Boethian arithmetic and the decad; and the aim of constructing a total world-picture in one figura. But Böhlandt also observes major differences: the prominent Christology of De ludo globi, and the cardinal’s new desire to reveal his sources, in contrast to the premium placed on “Originalitätssicherung” in De coniecturis (Verborgene Zahl, 175–187, 131). 60. “habent enim aliquae scientiae instrumentum et ludos, arithmetrica [sic] rhythmimachiam musica monochordum.” LG I (2), 4; trans. Hopkins, 1182. 61. On the history of rithmomachia, see Breidert, “Rhythmomachie und Globusspiel”; Borst, Mittelalterliche Zahlenkampfspiel; Borst, “Rithmimachie und Musiktheorie”; Folkerts, “Rithmomachia”; and Moyer, Philosophers’ Game. Breidert clarifies the close linguistic link between ῥυθμός and ἀριθμός (see “Rhythmomachie und Globusspiel,” 159–161). 62. See LG I (31), 35–36. 63. See, respectively, LG II (82), 100–102; LG II (83–88), 102–109; and LG II (118), 144–146. Flasch uses the same metaphor of Cusan development as a complex “spiralförmige Bewegung,” albeit with different intent (see Nikolaus von Kues: Geschichte einer Entwicklung, 393). 64. See LG II (104–106), 129–133. 65. See LG II (90), 111–113. 66. See LG II (91–96), 113–121. 67. See Senger’s “Praefatio” to his edition of De ludo globi, xxxii. Gandillac calls the dialogue a “très curieux texte, aussi archaïsant que novateur” (“Explicatio-Complicatio,” 84). Flasch finds it “zugleich didaktisch und innovativ” (Nikolaus von Kues: Geschichte einer Entwicklung, 602). 68. Bredow compares Nicholas’s interest in games to the necessity of myth in Plato’s dialogues (see “Über das Globusspiel”). For Watts it exemplifies the cardinal’s

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embrace of Renaissance “metaphors of culture” in his later works (see Nicolaus Cusanus, 191–207). In Haug’s analysis of the dialogue’s “Renaissance poetics,” the category of play as the productive craft of thinking is the perfect complement to the cardinal’s new epistemological constructivism since 1450, and a welcome alternative to tired medieval allegory (see “Das Kugelspiel des Nicolaus Cusanus”). Senger considers the mystical and scientific aspects of ludus but stresses the ethical dimensions in particular (see “Globus intellectualis,” 94–107). Thurner interprets the ludus globi as a concrete aenigma of the second order, which represents philosophical speculation to itself reflexively: its infinity, ease, and delight (see “Theologische Unendlichkeitsspekulation,” 99–121). Other ludic interpretations include Blumenberg, Legitimacy of the Modern Age, 535– 536; Heinz-Mohr, Globusspiel des Nikolaus von Kues; Schär, “Spiel und Denken”; Gandillac, “Symbolismes ludiques”; and Kijewska, “De ludo globi.” 69. Watts praises the anthropocentrism of Book I, but then strains to find such “humanist” ideas in Book II (see Nicolaus Cusanus, 204). Bond expertly reads the dialogue as a Christocentric spiritual ascent, yet cannot integrate its Neopythagorean ruminations (see “Journey of the Soul to God”). 70. Eisenkopf has helpfully analyzed the text into four levels of meaning: Pythagorean, temporal, epistemological, and metaphysical (see “Mensch, Bewegung und Zeit im Globusspiel,” 59). The moving ball, static gameboard, and its center are three spaces that assume metaphorical meaning in each of the four levels. Eisenkopf is right, in my view, to seek sophisticated and even plural orders of meaning within the dialogue. At the same time, her abstract categories do not stem from Nicholas’s actual sources, such as Asclepius, Thierry of Chartres, or even past Cusan works. 71. “Fuit autem propositum meum hunc ludum noviter inventum . . . in ordinem proposito utilem redigere.” LG I  (50), 55; trans. Hopkins, 1206. Cf. LG I  (7), 8:  “Haec omnia considerare necesse est, ut deveniamus ex istis ad speculationem philosophicam, quam venari propositum.” 72. The decadic figura universi in De coniecturis is three major circles containing two further iterations of triplet circles:  see DC I.13 (65), 64. As Yamaki has pointed out, we can interpret the nine concentric circles of the gameboard in De ludo globi as a modification of Figure U that first draws its circles around the Christ-decad and then sets them in motion (see “Bedeutung geometrischer Symbole,” 307). In De venatione sapientiae, Cusanus refers to a certain “libellus de figura mundi” that he wrote in Rome regarding singularitas: see De venatione sapientiae XXII (67), ed. Klibansky and Senger, 65. While at first this sounds quite like De ludo globi, Klibansky and Senger have listed several reasons (pace Bredow) why this the phrase probably refers to a lost work instead (see ibid., 155–156). The concept can nevertheless shed light on De ludo globi, not least because Nicholas calls the work a figuratio that reveals the forma mundi. See, e.g., Bredow’s still fruitful discussions of rotunditas in “Figura mundi.”

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73. Thus Oberrauch:  “Die Spielkugel stellt ein räumliches Modell dar und ihre spezifische Körperlichkeit ist konstitutiv für das Spiel. Die Bewegung der Kugel ist als ein Prozess im Raum aufzufassen. . . . Der Raum bildet so die Bedingungsvoraussetzung für Vergleiche” (Aspekte der Operationalität, 98–99). Note that unlike De docta ignorantia, De coniecturis, and De theologicis complementis (if the latter is properly joined with its geometrical complement), De ludo globi did not include any figures in the edition prepared by Cusanus. Yet all three early editions of the dialogue (Krakow 1495, Paris 1514, Basel 1565; see the reproductions in Senger’s edition) felt compelled to add circular figures to the text, as if the dialogue possessed an inherent momentum toward figuration. 74. See Asclepius I, 10, ed. Scott, 305. See further Allers, “Microcosmos”; and Thurner, “Explikation der Welt.” 75. “triplex est mundus: parvus qui homo, maximus qui est deus, magnus qui universum dicitur,” LG I (42), 47; trans. Hopkins, 1201. 76. See LG I (42), 47; and LG I (44), 49. 77. “ideo unius perfecti universi plures particulares et discreti homines speciem gestant et imaginem, ut stabilis unitas magni universi in tam varia pluralitate multorum parvorum fluidorum mundorum sibi invicem succedentium perfectius explicetur.” LG I (42), 48; trans. Hopkins, 1202. 78. Cusanus not only possessed Asclepius in his library in Kues but made extensive annotations that survive in MS Brussels, Bibliothèque Royale Albert 1er, 10054–10056 (see Arfé, “Annotations of Nicolaus Cusanus and Giovanni Andrea Bussi”). Arfé analyzes the cardinal’s marginalia inscribed during his years in Rome between 1458 and 1464. The annotations indicate that Cusanus’s primary interest was the text’s coordination of God, world, and humanity. The human being and the world are equally images of God, and the sensible world imitates God’s eternity, being contained eternally in God—precisely the themes of De ludo globi. For an overview of Hermetic themes in other Cusan works see Arfé, “Ermete Trismegisto e Nicola Cusano.” 79. On the double movement of the game’s spherical space, see Senger, “Globus intellectualis,” 93–94. 80. Schär captures this special effect perfectly:  “Am Spiel bleibend, führt sie ständig auch davon weg, indem sie Dinge ins Spiel bringt, die außerhalb stehen. . . . Spielend versenkt sich das Denken ins Bild und in die merkwürdige Erfahrung, die es vermittelt und läßt sich so den Anstoß zu seinen Überlegungen geben” (“Spiel und Denken,” 411, 413–414). 81. Senger glosses this “triadische Denkstruktur” as a threefold globus materialis (world), globus intellectualis (soul), and globus intelligibilis (God), which together generate “einem globalen Gedankenspiel” and “eine globale Theorie menschlicher Welterfahrung und Welterkenntnis” (“Globus intellectualis,” 92–93). 82. Unfortunately I  cannot treat Cusanus’s profound account of rotunditas here. It is discussed at length in Bredow, “Figura mundi”; Blumenberg,

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Pseudoplatonismen; Oberrauch, Aspekte der Operationalität, 105–156; and especially Senger, “Metaphysischer Atomismus.” 83. As Blumenberg has observed about De ludo globi, it is one thing for Cusanus to isolate a pure geometrical idea such as rotunditas beyond physical determinations, it is quite another to then apply this rarified concept to actual cosmological questions (see Pseudoplatonismen, 11). 84. See DI I.23 (70–73), 92–96. The image goes back to Alan of Lille, Regulae caelestis iuris VII, ed. Häring, 131–132; and before him to Liber XXIV philosophorum II, ed. Hudry, 152. Several essays on this famous metaphor are worth consulting, above all Mahnke, Unendliche Sphäre, 144–176. See further Harries, “Infinite Sphere”; Keefer, “World Turned Inside Out”; Nicolle, “L’Infinitisation de l’espace”; and Brient, Immanence of the Infinite, 147–242. In reference to De ludo globi, see Butterworth, “Form and Significance of the Sphere”; and Senger, “Globus intellectualis,” 88–93. 85. See DI I.21 (63–66), 84–88; and DI II.11 (156–61), 84–92. 86. See DI III.8 (232), 60. 87. See LG I (14–15), 15–16 ; cf. F 6r, 456–458 = DI II.8 (140), 64. 88. See LG I (40), 45–46. Cf. F 7r, 460 = DI II.9 (142), 64; Septem 961D–962A; and Glosa II.20–21, H 273. Flasch is right to marvel at the cardinal’s about-face on the world-soul between De docta ignorantia II.9 and De ludo globi (see Nikolaus von Kues: Geschichte einer Entwicklung, 596–598). 89. See LG I (55–59), 61–66; cf. DI II.10 (151), 78–80. 90. See LG II (62), 75; and LG II (64), 76. 91. “non est nisi una vera et praecisa ac sufficientissima forma omnia formans,” LG II (121), 149. Cf. F 7r–8r, 466 = DI II.9 (148–149), 74–76. 92. See LG II (109), 136; cf. the entire passage, LG II (82–109), 100–136. 93. Flasch comments:  “Cusanus verweist auffallend oft auf seine eigenen früheren Werke, zustimmend, nicht kritisierend. Er fordert seine Besucher zu deren Lektüre auf. Das Buch vom Spiel mit den Kugeln ist eine anschauliche Hinführung zu anderen Werken, zu De docta ignorantia und zu De mente insbesondere” (Nikolaus von Kues: Geschichte einer Entwicklung, 582). 94. See LG I (45–47), 50–53. 95. “[i]‌omnia sunt in deo et ibi sunt veritas, quae nec est plus nec minus. Sed ibi sunt complicite et inevolute sicut circulus in puncto. [ii] Omnia sunt in motu. Sed ibi sunt, ut evolvuntur, sicut cum punctus unius pedis circini super alio evolvitur. Tunc enim punctus ille explicat circulum prius complicatum. [iv] Omnia in posse fieri sicut circulus in materia, quae in circulum duci potest. [iii] Et omnia sunt in possibilitate determinata sicut circulus actu descriptus.” LG I (49), 54–55. 96. “Satis summarie haec resumpsisti, quae nescio quomodo—et extra propositum—in sermonem pervenerunt. Igitur revertamur nunc ad ludum nostrum et intentum brevissime pandam.” LG I (49), 55. On the “propositum” see LG I (7), 8; and LG I (50), 55.

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  97. “Nonne sic vides alium essendi modum monetae in arte omnipotentis monetarii? Alium in monetabili materia? Alium in motu et instrumentis ut monetatur? Alium ut est actu monetata? Et hi omnes modi circa esse ipsius monetae consistunt.” LG II (116), 142.   98. “Deinde est alius modus, qui circa illos essendi modos versatur, scilicet ut est ratione discernente monetam.” LG II (116), 142.   99. See LG II (118), 144–146. 100. “Eo modo intellectus videt negative infinitam actualitatem seu deum et infinitatem possibilitatem seu materiam. Media affirmative videt in intelligibili et rationali virtute. Modos igitur essendi, ut sunt intelligibiles, intellectus intra se ut vivum speculum contemplatur.” LG II (119), 147. 101. “Inter illos essendi extremos modos sunt duo,” LG II (118), 145.

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Index

Abelard, Peter, 95, 105–106, 117–118, 140, 142, 149, 151, 160 abstraction, mathematical, 28, 49, 72, 103, 107 in Greek Neoplatonism, 62–63, 66 Nicholas of Cusa and, 219, 224, 237–238, 250–252, 260, 391 Thierry of Chartres and, 132, 136 Achard of St. Victor, 141, 176, 351, 392 Adelard of Bath, 96, 100, 106, 108, 111, 131 aenigma, 226, 260, 262, 267 aesthetics, 61, 69, 100, 104–106, 237, 273, 276 akousmatics (ἀκούσματα), 24, 42, 50, 173 Alan of Lille, 15, 96, 100, 107–108, 141, 239, 404 Albert the Great (Albertus Magnus), 87, 143, 172, 178, 193, 315, 360 Alberti, Leon Battista, 171 Albertism, 2, 143, 172 Alcinous (Albinus), 45, 47, 49, 59, 253, 298 Alcuin of York, 88, 108, 327 Alexander Polyhistor, 41 alterity (alteritas) in Boethius, 85, 112 in Nicholas of Cusa, 208, 211, 216, 236–237 in Thierry of Chartres, 124, 151 anagogy, 25, 30–34, 67–68, 81–83, 146, 240, 275–276 Anaxagoras, 28

Anselm of Canterbury, 193, 287, 327 Antiochus of Ascalon, 40, 42, 56 apophasis (ἀπόφασις) henological origins of, 43, 49–51, 58, 63, 183, 226–227, 253, 385 in Nicholas of Cusa and equality and quadrivium, 181–185, 194–197, 199, 251, 253 as Ps.-Dionysian mystical theology, 141, 162, 176, 178–179, 205, 218 Apuleius, 51, 69 Archimedes, 243 Archytas of Tarentum, 23–34, 42, 51–53, 60–61, 80, 138, 208, 253, 255 Aristotle, 9–10, 52, 94, 123, 140, 178 and the debate on universals, 231, 251 on the disciplines, 39, 63, 85 and Fundamentum naturae, 158–160, 256 against the Pythagoreans, 23–24, 28, 30, 37–39, 239, 251–252 arithmetic (ἀριθμητική). See also quadrivium, four sciences of in Augustine, 70–71 in Boethius, 82–85 in Iamblichus, 63–67 as mediation (See mediation(s), mathematical) in Nicholas of Cusa, 200, 214, 334 in Nicomachus of Gerasa, 50–58 in Plato, 33–35 in Thierry of Chartres, 112–113, 129

468

Index

arithmology, 6, 24–26, 46, 48 medieval traditions of, 86–87, 89, 97, 108 in Nicholas of Cusa, 211, 254 in Nicomachus of Gerasa, 57–58 “S” tradition of, 41–44 Arithmos. See Logos and Arithmos art (ars), divine, 204, 272 ascent. See also descent in Augustine, 70, 73–74 in Nicholas of Cusa, 213, 216, 274, 391 and four modes, 236–240, 242, 390 and mathematical abstraction, 238, 250, 259, 395 as theosis or filiation, 219, 276 in Plato, 30, 33 Asclepius, 99, 106, 145, 147, 197, 268, 339. See also Hermeticism Asclepius of Tralles, 51 assimilation, 220, 388 astronomy (σφαιρική), 88. See also quadrivium, four sciences of Athenagoras, 49 Augustine of Hippo and Augustinian traditions, 61–62, 142–143 and Bernard of Chartres, 101, 104 on Christology, 71, 73–75, 79–80, 308 and Clarembald of Arras, 150–156 on creation, 75–78 and John Scotus Eriugena, 88–89 as Neopythagorean, 14, 69–73 and Nicholas of Cusa, 176–178, 211, 240, 247 and Odo of Morimond, 108–109 and Thierry of Chartres, 109–118, 126, 133–138 on Trinity, 78–80, 141 Averroes (Ibn Rushd), 386 Avicenna (Ibn Sīnā), 94, 302 Bacon, Roger, 87 Badiou, Alain, 2, 284

Basil the Great, 169 Basilides, 49 Beldomandi, Prosdocimo de’, 172 Bernard of Chartres, 15, 100–107, 200, 276 and Thierry of Chartres, 107–119, 126, 129, 131–135, 320–321 Bernardus Silvestris, 95–96, 100, 105–108, 145 Berthold of Moosburg, 142 Bessarion, Basilios, 169, 171 Blumenberg, Hans, 2, 11 Boethius, Anicius Manlius Severinus on the disciplines, 39, 84–85, 119–121, 127, 259 on the faculties of the soul, 122, 146 as medieval source, 93–96, 99, 101–109 in Nicholas of Cusa, 171–173, 177–178, 182, 203, 211, 225–226, 233–235, 237, 240–241, 243, 255, 259–260, 266, 274 in Thierry of Chartres, 107, 111–114, 122, 126, 129, 133–134 as translator of quadrivium, 10, 27, 51, 61, 82–84, 87–88, 111, 203, 205, 235 as transmitter of Neopythagoreanism, 9, 13–14, 80–81, 83, 86, 97, 117 Bonaventure, 87, 143, 193, 197, 217, 246, 344 Bracciolini, Poggio, 171 Bradwardine, Thomas, 1, 87, 142 Brixen (Bressanone), 170, 243, 257, 260 Bruni, Leonardo, 171 Bruno, Giordano, 1, 11 Buridan, Jean, 142 Byzantium, 173 Calcidius, 41, 50, 56, 61, 97, 100–101, 103, 123, 134–135, 154, 334 calculation (λογιστικά), 31 Carolingians, 87–88, 94 Cassiodorus, 87, 362

Index Cassirer, Ernst, 4–8, 11, 19, 223 categories, Aristotelian, 52–53, 83 cathedral schools, 27, 95–96, 98, 101, 126, 145, 149, 320 causation as Aristotelian four causes, 109–110, 117, 160, 326 natural or necessary, 98, 101–102, 104–105, 116, 131, 133, 137, 165 and necessitas complexionis, 131, 150 in Platonism, 49, 64, 101 as seminal causes, 78, 110, 153–154 center, 70, 211, 246, 305 Christ as, 193, 197, 261–264, 268–269, 370, 399 and circumference, 33, 245, 263–264, 274–275, 335 certainty, mathematical, 7, 29, 68, 133, 236, 327 Cesarini, Julian, 170, 173 Christology. See also Incarnation Augustinian, 71, 73–75, 79–80, 308 medieval Platonist, 11, 86–87, 89, 143, 152, 312, 370 in Nicholas of Cusa, 16, 20, 279 and Cusan development, 202–207, 216–220, 223, 226, 229, 250–252 in De docta ignorantia III, 174–175, 185, 190–201, 253–254, 369–370, 372–373 in late works, 256–257, 260–265, 267–269, 272–276 as (Neo-)Chalcedonian, 193–194, 197, 219, 263, 275 church, 3, 142, 169, 173, 193, 201, 218 Cicero, Marcus Tullius, 40, 42 circles, 70, 84, 125, 182, 211, 245–246, 259, 261, 263–264, 269, 271–272, 274 Cistercians, 108, 122, 143 Clarembald of Arras, 95, 144, 149–158, 162–165, 177, 188, 199, 333, 339

469

Clement of Alexandria, 41, 46–47, 49, 58, 60, 71, 87, 253, 297–298 coincidence of opposites (coincidentia oppositorum), 191, 216, 220, 224, 226, 258–259, 380 Cologne, 143, 170, 172–173, 193 Commentarius Victorinus (Ps.-Bede, In titulo), 141, 165, 176–179, 181, 233 conjecture, 206–207, 213 connection (conexio, nexus). See also Trinity, arithmetical in Augustine, 77, 79–80 in Nicholas of Cusa, 176, 186, 197, 203, 258 in Thierry of Chartres, 117–118, 124, 132–133 Constantine the African, 106, 324, 333 contemplation, 1, 93, 123, 142, 146, 179, 237, 248, 264 contraction (contractio) in De coniecturis, 208–210, 213, 216 in De docta ignorantia, 174, 178–179, 181–182, 185, 187, 189, 192, 194–200, 254 in Fundamentum naturae, 159–161 Copernicus, Nicolaus, 2 cosmogony, 33, 46, 48, 55, 102, 105–110, 119, 246 cosmology, 8, 11, 99, 104, 159, 174, 182, 276, 278 cosmos. See universe Council of Basel, 169–170, 172 Creator, 46–47, 55, 147–148, 153, 159–160 in Augustine, 75–77 in Nicholas of Cusa, 178, 183–184, 219, 246, 249–250, 259, 263–264, 269 in Thierry of Chartres, 98, 105, 111–115, 136–138 Cusanus. See Nicholas of Cusa

470

Index

decad, 24, 35, 41, 44 in Iamblichus, 62–64, 67–68 in Nicholas of Cusa, 201, 208, 211, 214–215, 219, 254, 261, 266, 274, 402 in Nicomachus of Gerasa, 51–52, 55, 57–58, 83 De communi mathematica scientia, 65, 303 degrees (gradus), continuum of, 159–160, 181–182, 187, 195–196, 199–200, 213, 263, 269, 373 demiurge, 33, 37, 41, 43–44, 55, 66, 83, 99, 101, 118, 251, 315 Democritus, 28 Descartes, René, 2–13, 103, 223, 239, 278, 286 descent. See also ascent in Moderatus, 44, 48, 57 in Nicholas of Cusa, 186–187, 192, 194–197, 208, 211, 213–215, 219, 386 in Thierry of Chartres and his critics, 124–125, 148–152, 154–155, 159, 163 De sex rerum principiis, 145, 347 dialectic, 30–32, 36–37, 53, 65–67, 70 Dietrich of Freiburg, 142 Dietrich von Xanten, 244 Ps.-Dionysius the Areopagite, 2, 43, 88–89, 170–172, 177, 180, 192–194, 201, 204–205, 217, 223, 250, 305, 354 and medieval Christology, 143, 193, 197, 254 and medieval Proclian traditions, 61, 68, 93–94, 142, 173 and negative theology, 49, 141, 162, 176, 178–179, 196, 218, 253 disciplines, 42, 70, 146–147 and disciplina, 7, 85, 122, 233, 235, 260

threefold Aristotelian division of, 39, 42, 53, 58, 63, 84 in Nicholas of Cusa, 234–235, 238–239 in Thierry of Chartres, 101–102, 107, 119, 121, 127–129, 136–137, 150–151, 337 discourse, 10, 19–20, 60–62, 73, 80–81, 86, 89, 93, 95, 110, 112, 117, 137, 142, 162, 205, 243, 247, 255, 278–279 Dominicus Gundissalinus, 96 dualism, 42–43, 50, 60, 101, 103, 295, 374. See also monism dyad, 35, 38, 41–44, 46, 48, 54–55, 57, 64, 69, 79, 154, 211 Eberhard of Bamberg, 327 Eckhart von Hochheim, Johannes (Meister Eckhart), 2, 13, 15, 43, 170, 177, 180, 185, 210, 217, 286–287, 363 theology of Word of, 11, 143, 193, 197, 370 and Fundamentum naturae, 185, 364, 378 ecstasy (raptus), 239, 245, 248–250, 254, 395 Eichstätt treatise. See Fundamentum naturae elements, four, 28, 33, 104–106, 109–110, 127, 132, 183 emanation, 45, 59, 68, 138, 143, 185, 197, 206, 276 enfolding (complicatio). See also folding, reciprocal and necessity of enfolding in Nicholas of Cusa and contractio, 196–197 by divine Word or Christ, 198, 204–205, 218, 246–247, 263 as geometrical, 246, 251, 274

Index by God as Absolute, 184–185, 230–232, 237, 388 by human mind, 207–208, 227, 231–232 in Thierry of Chartres and his critics, 125–136, 150, 152, 154, 161 epistemology, 4, 27, 29, 31–33, 64, 67, 122–123, 146, 205–209. See also mathematization of epistemology equality (ἰσότης, aequalitas). See also Trinity, arithmetical in Augustine, 70–71, 79–80 of being (aequalitas essendi or existentiae), 114–115, 176, 187–188, 194, 196, 198, 200, 207, 246 and inequality, 43, 84, 113, 151, 182–184, 204–205, 361 in Nicholas of Cusa as negative precision, 181–183, 253–254, 262 as basis of Incarnation, 191–192, 194–200, 205, 257 as Platonic and Pythagorean doctrine, 43–44, 47, 53, 57, 66, 84, 87 in Thierry of Chartres, 113–115, 124 in Thierry’s critics, 150, 155–156, 159, 162, 165 Eriugena, John Scotus, 2, 39, 88–89, 94, 172, 351 ethics, 24, 27, 33, 63–64, 76, 82, 84, 102, 146, 262, 295, 313, 402 Euclid, 6, 9, 12, 49, 65–66, 82, 86–88, 290, 301 Eudorus of Alexandria, 38, 40–51, 54, 56–58, 64, 79, 86, 99, 254, 285, 292, 295 Eugenius IV, 169, 218 Eurytus, 23, 25, 290 even and odd, 35, 42, 84, 211, 216, 240, 314

471

exemplar(s) and images, 63, 122, 134, 236–238, 241, 245, 265, 273–275, 395 in mind of God, 88, 99, 103–104, 114, 131, 134, 178, 211 plurality of, 134–135, 139, 161–162, 189–191, 213, 231, 270–271 Word as, 143, 228, 230–231 extension (διάστημα), 36, 44, 48, 82, 269, 298 face, divine, 125, 264–265, 272–273, 276 fate, 99, 128, 131, 132, 152, 165, 187, 189, 270–271, 371. See also providence as εἱμαρμένη, 131, 152, 339 vs. providence, 125 fecundity (fecunditas), 83, 246 Ficino, Marsilio, 12, 171, 173 figures, geometrical and divine Son as figura, 124, 265, 276 and constructing polygons, 182, 219, 246, 249 as the divine Mind, 37–38, 56, 67, 246–248 as dynamic, 259–260, 269 and figurae mundi, 267, 402 and figures P and U, 211 as instruments of contemplation in De docta ignorantia, 176, 179–180, 194, 237–238 in De ludo globi, 259–263, 269, 274 in De theologicis complementis, 243, 245, 249–250 and the mathesis narrative, 5–7 as projections, 65–67 filiation (filiatio), 218–219, 239, 276 Flasch, Kurt, 18–20, 223, 226–228, 250, 384–386, 393–395, 397, 401, 404

472

Index

flow (χύμα, ῥύσις, flux), 54–55, 64, 68, 84–85, 89, 135, 196, 241, 315 folding, reciprocal. See also enfolding and unfolding contested sources of, 125–126, 129, 224, 338, 363, 375 and four modes of being, 127–129, 161 and four unities, 201, 208–210, 212, 214, 216 as geometrical space, 136–138, 237, 245–246, 261 in Nicholas of Cusa access to Chartrian doctrine of, 213, 215, 234–237, 276, 366–367 new interpretations of, 184–185, 189, 218, 231, 254–255 and three disciplines, 128–129, 136–137, 215, 337 form of being (forma essendi), 85, 112–113, 171, 176, 181, 323, 341, 363 divine, 85, 103, 125–126, 133–134, 230, 265, 271 of forms (forma formarum), 130, 134–135, 158, 161, 341, 363 and matter, 39, 42–44, 57, 101–105, 123, 133–134, 197, 237, 251, 260 native (formae nativae), 101–106, 129, 135, 199 Fulbert of Chartres, 101 fullness (πλήρωμα), 48–49 Fundamentum naturae in the context of Thierry’s reception history, 156–157, 162–165 as problem for Cusanus studies, 16–17, 174–175, 351–352, 356–357, 368–370, 386–387 against Thierry’s modal theory, 158–162 Galen, 146, 333 Galilei, Galileo, 5–9, 11, 13–14, 278, 284–285, 399

game of sphere(s) (ludus globi), 261–263, 266–269 genealogy, 9–10, 18, 278 genetic method, 18–20, 120, 227, 288, 385 Geoffrey of Auxerre, 108 geometry (γεωμετρία), 5–6, 29, 37, 53, 67–68, 70, 88–89. See also quadrivium, four sciences of as method (more geometrico), 239, 382 in Nicholas of Cusa, 172, 222, 250, 256, 261–264, 269, 274–275 geometrical works, 172, 218–220, 243–244, 251–252 God as geometer, 245–250, 255, 362 theologia geometrica, 244–245, 252, 248 in Thierry of Chartres, 124, 138–140 Gerbert of Aurillac, 87, 101 Gilbert of Poitiers, 100, 103–108, 134–135, 149–151, 285, 336, 349 Giles of Rome, 159 Gnosticism, 45, 48–50, 75, 311 Gospel of John, 71, 141, 147, 151, 165, 193, 257 Gregory of Nyssa, 50 Gregory of Rimini, 142 Grosseteste, Robert, 1, 362 Gundissalinus, Dominicus, 39, 96 harmonics. See music harmony cosmic, 24–25, 32–35, 44, 80, 86, 146, 182–183, 296, 317 divine, 79, 89, 117, 138, 182–183, 241, 287, 312 ethical, 34, 146 number as, 28–29, 45, 57, 84, 240, 314 Haubst, Rudolf, 2, 13, 16, 210, 359, 370, 383

Index Heidegger, Martin, 2, 5, 8–11, 277, 318, 397 Helinand de Froidmont, 143 henads, 25, 64, 68, 303–304 henology, 25–26, 290 Academic and Middle Platonist foundations of, 36, 38, 41–49, 292, 294 Neopythagorean, 51, 54–58, 62–63, 81–84, 253 in Nicholas of Cusa ambivalence toward, 200, 220, 248 in De coniecturis, 206–207, 209–210, 216, 257 Trinitarian, 79, 112–113, 132, 138 Heraclitus, 28, 147, 163 heresy, 48, 95, 141, 163, 172 Hermann of Carinthia, 95, 100, 106, 108, 145 Hermeticism, 50, 98–99, 106, 130–131, 145–149, 176, 205, 268, 273–274. See also Asclepius Heymeric de Campo, 172, 193, 243–244 Hilary of Poitiers, 176 Hildegard of Bingen, 96, 107–108, 327 Hincmar of Reims, 108 Hippolytus of Rome, 48 Hoenen, Maarten J. F. M., 16–17, 157, 163, 174–175, 180, 188, 351, 357, 360, 365, 370 Hugh of St. Victor, 87, 108, 122, 144, 149–150, 345 Husserl, Edmund, 5–11, 15, 277 Hussites, 172 Iamblichus of Chalcis, 6, 16, 62–64, 69, 136, 211, 299, 303 as critiqued by Proclus, 61, 65, 67, 80, 99, 138 as Nicomachean Neopythagorean, 30, 36–37, 51, 68, 117, 142, 253–254, 279

473

and Syrianus, 65, 71, 85 ignorance learned (docta ignorantia), 171, 179, 183–184, 205–207, 224 of the philosophers, 147, 152, 161, 163 image (imago) and exemplar, 63, 122, 134, 236–238, 241, 245, 265, 273–275, 395 geometrical (See figures, geometrical) as imago dei, 231–232, 236–238, 240–242, 245, 385–386 number as, 31, 33–34, 47, 63–64, 66, 70, 85, 122, 211, 237, 240–241, 245 as sign, 265, 272–273 imagination (φαντασία, imaginatio), 67, 122, 129, 147 immutability, 103–104, 122–123, 128, 236, 239 impetus, 124, 259, 262 Incarnation. See also Christology in Augustine, 71, 74–75 in Nicholas of Cusa, 11, 16 in De docta ignorantia III, 174, 193, 198, 200–201, 204, 217–220, 253, 372–373 in late works, 256–257, 264–265, 268–269, 273–276 inchoatives (inchoativa), 78, 153–155, 163–164 inertia, 259, 399 infinity as category for Cusanus studies, 18, 227, 363 of degrees of difference, 159–160 divine, 142, 176, 237, 246 and function of number, 32, 66 and geometrical figures, 238, 243, 245, 247 as infinite sphere, 269, 404 as infinite velocity, 2, 259, 263–264 negative vs. privative, 184

474

Index

intelligence or intellect (intellectus, intelligentia), 68, 72, 85 vs. intellectibility (intellectibilitas), 122, 129, 235, 239, 333 in Nicholas of Cusa, 182, 203–204, 206, 208–209, 215–219, 234–235, 239, 251, 260, 264–265, 272, 274 in Thierry of Chartres, 96, 129, 146–148 Irenaeus of Lyons, 48 Isaac of Stella, 333, 345 Isidore of Seville, 87, 108, 316 Islam, 10, 140, 302

Kant, Immanuel, 2, 5, 8, 223, 225, 279, 283, 285, 303, 317, 332 Klibansky, Raymond, 4–5, 13–16, 302, 326 Koch, Josef, 206, 209–211, 226, 290, 377 Koyré, Alexandre, 6–9, 284

Leibniz, Gottfried Wilhelm, 1-2, 15 Liber de causis, 142, 159, 172, 201, 302 Limit (πέρας, terminus) and Unlimited (ἄπειρον), 32, 35, 42–43, 48, 76–77, 84 in Nicholas of Cusa, 196–197, 253 in Philolaus, 28, 32, 49, 65–66, 138, 253 Llull, Ramon, 13, 170, 172, 193, 364, 371 logic, 50, 62, 96, 103, 111 Logos (λόγος). See also Christology and Incarnation all-cutting (λόγος τομεύς), 47, 253 and Christian theology of Son, 71, 73–74, 152, 161, 191, 194, 251, 254, 257 as divine Mind, 50, 54–58, 99, 226 of mathematics, 29, 36, 52 Stoic doctrine of, 28, 46–47, 56, 78 unitary (ἑνιαῖος λόγος), 44, 48, 54 Logos and Arithmos. See also Logos (λόγος) and mediation(s) in harmony in Augustine, 71–73 in Nicholas of Cusa, 199–200, 214, 228, 239, 242, 251, 254, 273, 276, 279 in Nicomachus of Gerasa, 58 in Thierry of Chartres, 117, 121 in tension in late antique Neoplatonism, 56, 61, 68–69, 74, 80, 86, 102–103, 278, 301 in Nicholas of Cusa, 211, 217, 380 among Thierry’s critics, 135, 143, 152, 156, 162 Lombard, Peter, 141, 193, 358 love, 79, 99, 107, 124, 186, 205

Lactantius, 82 language, 6, 69, 73, 79, 205, 260 Lebenswelt, 5, 277

Macrobius, 86–87 on catena aurea, 125–126 on divine ideas, 56

Jacob of Cremona, 243 Jaspers, Karl, 225, 384 Jerome, 79, 82, 240 Jesus. See Christology Johannes de Muris, 87 John Chrysostom, 169 John, Gospel of. See Gospel of John John Philoponus, 51 John of Salisbury, 96, 100, 103, 145 John Scotus Eriugena. See Eriugena, John Scotus John of Spain, 333 Jordanus Nemorarius, 87 Judaism, 10, 40–41, 45–46, 50–51, 94, 140 Justin Martyr, 41, 47

Index on mens divina, 98–99, 104–105, 113–114, 133–135, 161 and “S” arithmology, 41, 97, 211, 254 magnitude or quantity (πηλίκον, μέγεθος, ποσότης, magnitudo, integritas, quantitas), 44–45, 48–49, 54, 84, 246, 260. See also quadrivium, foundations of in multitude and magnitude Mahnke, Dietrich, 15, 223, 288 Manegold of Lautenbach, 101 Marcellus of Ancyra, 48 Marius Victorinus, 40, 74, 79, 311, 315, 327 Martianus Capella, 41, 82, 86, 88–89, 96–97, 339, 345 mathematicals (μαθήματα), 24, 29, 31, 36 and abstraction (See abstraction, mathematical) and forms, 35, 37–39 in God (See figures as the divine Mind and number as first exemplar in divine Mind) mediating, 25–26, 31–34, 68, 84, 123, 171, 225, 242 (see also mediation(s), mathematical mathematics. See also quadrivium and akousmatics (See akousmata) ancient Greek, 29–30, 49–52, 65–67 and certainty (See certainty, mathematical) divine (God as mathematician), 6–7, 83, 183–184, 207–208, 225–226, 238, 242, 245–250, 255, 362 in the Middle Ages (See quadrivium) in Nicholas of Cusa’s works, 171–172, 218–220, 243–244, 251–252 and physics and theology (See disciplines)

475

mathematization. See also abstraction, mathematical of epistemology, 3–4, 11, 19, 223–225, 257 as mathesis universalis Cartesian, 7–11, 13, 15, 103, 223, 277–279 Chartrian, 102, 119 Neopythagorean, 56, 58, 82, 87, 89 of method (more geometrico), 11–13, 63, 67–68, 85, 215, 239, 382 of nature, 5–6, 8, 10, 102–103, 200, 222, 277–279 mathesis (μάθησις), 1, 31, 87–88, 122, 238, 254, 260, 290, 345. See also mathematization as mathesis universalis and mathesis narrative mathesis narrative, 8–13, 19, 223, 277–279, 284 matter and dualism, 50, 60, 101, 103, 334 as dyadic, 42–44, 48, 57 prime (ὕλη, silva), 36, 101, 110, 126, 128, 130, 151, 153–155, 157–159, 163–164, 189 and separable form, 39, 103–104, 123–125, 133–134, 148, 237–238, 260 maximum Christ as absolute and contracted, 173, 194, 197 as contracted, 159–160, 186, 196–197 as divine absolute, 161, 178–179, 186, 191, 197 as equality, 181–182, 185, 197 vs. minimum, 189, 191, 220, 230, 259 Maximus Confessor, 88 means arithmetical and geometrical, 84 between extremes, 114, 123, 128, 146, 211–212, 273

476

Index

measure. See also Wisdom 11:21 as geometrical measurement, 3, 5, 7, 88 God as, 47, 55–56, 76, 114, 230, 237–242, 245–249, 251–252 and mens, 234–235, 241 as self-measure, 208, 228, 231, 233–235, 237–242, 257 mediation(s) in competition, 46–47, 57–58, 138–139, 156, 161–163, 217–219, 278 harmonization of, 199–200, 239, 242, 253–254, 260, 273, 275–276 mathematical Augustinian, 71–73, 108–109 autonomy of, 97–100, 105–107 Chartrian, 102–103, 116–117, 125–126, 131, 135, 137, 165 Cusan, 202, 205, 211, 214–215, 238 Neopythagorean, 52–53, 57–58, 61–64, 82 Platonic, 25–26, 30–35, 38–39 non-mathematical Christological, 74, 77–78, 143–144, 156, 159–163, 193–194 Proclian, 67–68 religious sense of, 45–48 Meister Eckhart. See Eckhart von Hochheim, Johannes metaphysics, 8, 32, 94, 119, 142 microcosmos, 99, 197, 268 Middle Platonism, 9, 24, 26, 36, 40–41, 44–45, 51–52, 55–56, 58, 60, 73, 97, 101, 292, 308 mind (νοῦς, mens) as divine mediator in Fundamentum naturae, 159–161 in Nicholas of Cusa, 189–191, 231–232 of God in Greek Platonism, 38, 54–58, 66, 68, 304

in Macrobius, 99, 104, 114–115, 133–135 in Nicholas of Cusa, 185, 240–242 as human faculty in Nicholas of Cusa, 232–235, 237, 241, 247 in Plato, 31 mirror, 125, 219, 248, 260 modal theory. See modes of being Moderatus of Gades, 40–45, 48–51 as founder of monistic henology, 58, 60, 64, 86, 142, 201, 255 on number as emanative flow, 54, 59, 68, 84, 138, 196, 254 as source of arithmetical triad, 43–44, 57, 66, 79, 117, 240, 311 modernity, 2–5, 8–12, 20, 102, 222–223, 242, 277–279 modes of being, four (modi essendi), 127–138. See also necessity and possibility as anagogical ascent, 236–242 as five modes, 126, 129, 332, 350 as fourfold sign-image, 271–273, 276 as four unities, 202, 209–217 monad (μονάς) and divine One, 25, 36, 43, 45–47, 56, 219 and dyad, 38, 41–42, 44, 54–55, 57, 69, 154 and generation of matter, 48, 54 and generation of number, 45, 48, 52, 54, 64–65, 68, 138, 305, 381 God as, 44, 49, 54, 57, 79, 89 and point, 36–37, 49, 246 monism, 45, 48, 50, 58, 60, 62, 295, 299. See also dualism and henology, 38, 42 and Nicholas of Cusa, 207, 374 monochord, 88, 266 motion cosmic-temporal, 33–35, 96, 99, 125, 153, 182, 293

Index emanation as, 44, 48, 68, 143, 148 between form and matter, 67, 124, 137, 156, 163–164, 186 helical, 262, 305 mathematization of, 7, 9, 116, 142 vs. rest, 56, 66, 181, 184, 259, 263, 269, 276 as structuring the quadrivium, 39, 52, 76–77, 85, 183, 258, 260–261, 264 and velocity, 259, 263–264 multiplication, 54, 113, 124, 146 multiplicity (πλῆθος), 32, 36–37, 45, 48, 65–66, 298 multitude or number (ποσόν, πλῆθος, multitudo, discretio), 85, 218, 220. See also quadrivium, foundations of in multitude and magnitude music (μουσική), 34, 42, 50, 69, 88–89, 241. See also quadrivium, four sciences of mysticism, 1–2, 11, 61, 174, 179, 237, 245, 249, 255, 259, 261, 288 and mathematics, 15, 236, 239, 248, 254 and mystical experience (See ecstasy) as number mysticism (See arithmology) as Ps.-Dionysian mystical theology, 20, 171–172, 223, 227, 238, 250 names, divine, 1, 49, 141, 180, 182, 184, 204–205, 218, 230, 246, 276, 365 nature (natura), 6–8, 98–107, 116–117, 126, 129, 131, 137, 143, 147–148, 157–158, 196, 270 necessity. See also certainty, mathematical and fate absolute (necessitas absoluta: first mode of being)

477

in Nicholas of Cusa, 185, 187–189, 192, 213, 235, 237 in Thierry of Chartres, 127–131, 133, 151–152, 154, 158–161, 164 causal (See causation) of enfolding (necessitas complexionis: second mode of being), 130–135, 236–238, 340 in Clarembald of Arras, 150–156 as critiqued by Fundamentum naturae, 158–163 in Nicholas of Cusa, 187–192, 199, 213–216, 229–238, 240, 242–243, 250–251, 254, 257–258, 261, 270–273 and contraction, 187–188, 192, 196 and divine Word, 161–165, 191–192, 198, 202, 273, 333, 369, 371 in Thierry of Chartres, 127–138 Neo-Kantianism, 3–4, 6, 8–9, 11–13, 285 Neoplatonism and Christianity, 10, 27, 50, 60–61, 74, 93, 142–143, 193, 279, 342 origins and constitution of, 41, 45, 62–63, 65, 86, 138, 293 as source of Nicholas of Cusa, 13, 172–173, 192–193, 206, 210, 374, 377 Neopythagoreanism Christian, 10, 12, 19, 48, 61–62, 80–81, 89 in Augustine, 71, 73–75, 109, 117 in Nicholas of Cusa, 13, 27, 173, 178, 206–208, 216–217, 226–229, 232, 243, 255, 275–278, 383–384 in Thierry of Chartres, 97, 112, 137–140, 142, 156 origins and constitution of, 36–37, 41–42, 45, 51, 58, 60–62, 64, 66, 69, 285

478

Index

New Academy, 40–42, 51 Nicene orthodoxy, 71, 80, 113, 156 Nicholas V, 228, 243 Nicholas of Cusa life of, 169–173, 193, 202–203, 217–218, 228–229, 243–244, 255–257 state of scholarship on, 2–5, 10–17, 171, 173, 206, 209–211, 222–228, 249–250, 252, 266–267 Nicomachus of Gerasa, 14, 19, 27, 30, 36–37, 44–45, 48–69, 71–72, 75, 81, 84–86, 93, 99, 113, 117, 137, 142, 162, 196, 214, 243, 285, 296–297, 299, 301, 303, 306, 315–316 and arithmology, 41, 83, 97, 211 and mathematical theology, 26, 51, 57–58 and number as divine exemplar, 55, 114, 131, 138, 178, 240 as positive henology, 58–59, 253 and quadrivium, 52–53, 56, 66, 77, 80, 82–83, 104, 112, 138, 202–203, 208, 253–255, 274, 276, 376 Nigidius Figulus, 42, 295 nominalism, 11, 116, 142, 172, 279 number (ἀριθμός) definitions of, 28, 32, 45, 52, 54, 84–85, 239–240, 298–299, 301 as divine icon, 238–239, 250, 254, 260, 275 as first exemplar in divine Mind, 55–56, 83, 114, 131, 138, 178, 211, 225, 240, 255, 383 God as Number without, 76, 79–80, 115, 131, 240, 242, 247, 249, 254 levels and kinds of, 57–58, 63–64, 70, 89, 208, 211, 214, 248, 254

as multitude (See multitude or number) as serial order, 36–37, 55, 83, 130, 133, 136, 159, 196, 200, 219 Numenius of Apamea, 43, 50, 60, 101, 299, 334 Odo of Morimond, 108–109 Old Academy, 9, 26, 28, 32, 36, 40–42, 51, 53, 58, 99, 137 One (τὸ ἕν), 31, 36. See also henology and unity as divine, 36–37, 41, 43–46, 54–57, 94, 112, 132, 138, 148, 196, 226, 275, 399 and many, 32, 75, 125, 184, 192, 206, 220 ontology, 94, 129, 290, 315 and entitas, 123, 176, 398 and interpretations of Nicholas of Cusa, 11, 373, 380, 384 opposites, Pythagorean (συστοιχίαι), 42, 211 optics, 7, 42, 88, 172 Oresme, Nicole, 1, 142, 317 Origen of Alexandria, 40, 47, 50, 60, 193 Padua, 171, 173 Paris, 9, 14, 23, 87, 95, 97, 140, 142–145, 149, 172, 177, 321 Parmenides, 28, 29, 46, 147–148, 176 perpetuals, trinity of in Clarembald of Arras, 150–156 in Fundamentum naturae, 156–158, 163–164 in Nicholas of Cusa, 176, 179, 186, 192 in Septem, 147–148 in Thierry of Chartres, 124–126 Peter of Traves, 108 Peuerbach, Georg von, 172, 243

Index Philip of Opus, 35 Philo of Alexandria, 40–41, 44, 46–50, 57–58, 71–72, 97, 99, 162, 211, 254 Philolaus of Croton, 16, 19, 23–24, 26–30, 32–34, 42–43, 51–53, 58, 60–61, 137–138, 196, 208, 240, 253, 255 physici, 128, 147–149, 157–158, 163, 180, 347 physics Aristotelian, 123–124, 158–160 in Bernard and Thierry of Chartres, 101–103, 110–111, 116–117, 131 fourteenth-century, 9, 11, 116, 142 in Nicholas of Cusa, 258–265, 274–275 Pythagorean, 24, 63–64 and exegesis secundum physicam, 109–110, 152, 158, 164 and the threefold Aristotelian division, 39, 84–85, 119, 122, 126–129, 234 in William of Conches, 105–107, 109, 132 Piccolomini, Aeneas Sylvius (Pius II), 170, 256 Pico della Mirandola, Giovanni, 171, 173 Plato, 10, 14, 19, 23–39 vs. Aristotle in Middle Ages, 103, 158, 171, 201, 231, 251–252 and Neopythagoreanism, 43, 46–47, 51, 53, 55–56, 58, 65 and twelfth-century Platonism, 95, 97, 100, 105, 123, 132, 134, 151 Plethon, Georgios Gemistos, 171 Plotinus and medieval Christian theology, 45, 60–61, 68, 74, 77–78, 93, 239 and Neopythagoreanism, 38, 42–44, 49–50, 62–63, 68, 201, 298–299

479

as source of Nicholas of Cusa, 206, 227, 338, 385 plurality (pluralitas) as difference, 85, 220, 352 of exemplars, 133–135, 185, 190, 270–271 and folding, 125, 127, 130, 231, 254 and number, 28, 32, 68, 83 Trinity as, 241, 392 Plutarch, 40, 45, 55, 169, 301 point (punctum), 67, 84, 203, 235, 259, 375 polygons. See figures, geometrical Porphyry, 40, 43–44, 50–51, 60, 62, 68, 74, 79, 85, 86, 122, 306, 311 Posidonius, 56, 306 possibility absolute (possibilitas absoluta: fourth mode of being), 128, 130, 154, 158, 352, 396 as critiqued by Fundamentum naturae, 159–160 in Nicholas of Cusa, 181, 187, 236, 257–259, 261 vs. actuality, 123, 125–129, 134, 148 and contraction, 186–187 determined (possibilitas determinata: third mode of being), 128, 130, 133, 152, 154, 158, 197 divine, 258–260, 270 precision, 5, 115–116, 182–183, 199, 204, 216, 236–237, 260, 262. See also equality and measure Presocratics, 23–28, 51, 65, 97, 287 procession and return, 45, 59, 67–68, 138, 143, 192–193, 216, 305, 315, 369, 378 Proclus, 6, 14, 16, 35, 40–43, 89, 93, 138, 173, 178, 206, 315, 381 and early modern philosophy, 9–10, 12, 68, 223, 285–286

480

Index

Proclus (Cont.) and medieval Christian theology, 13, 61–62, 68–69, 142–143, 172, 200–201, 302 and Neopythagoreanism, 59–69, 80, 82, 85–86, 98–99, 208 as source of Nicholas of Cusa, 12–13, 19, 169–170, 172, 210, 220, 224, 226–228, 257, 288, 363, 376–378, 382–383, 385–386, 396 on spatialized mediation, 45, 59, 67–68, 130 projection (προβολαί), 64, 66–67, 208, 223, 225, 228, 303–304, 384 proportion (ἀναλογία, proportio), 29, 31, 33, 35–36, 71, 84, 182–184, 216, 220, 241, 336, 355 providence, 71, 76, 80–81, 99, 114, 125, 129–130, 146, 154, 164, 184. See also fate Pythagoras, 1, 38, 63, 70, 82, 123, 165, 189, 345 the historical, 24–25, 173, 255, 289–290, 292, 295 Nicholas of Cusa in praise of, 171, 173, 176, 252, 256, 273, 383–384 as sage, 46, 148, 177–178, 204, 266 quadrature of the circle, 1, 172, 219–220, 243–245, 252, 355, 361, 381 quadrivium. See also arithmetic, astronomy, geometry, and music bond (δεσμός) or commonality (κοινωνία) among, 29, 31, 35–36, 52, 66 and Christian theology in Augustine, 69–73 in Nicholas of Cusa, 27, 176, 179, 182–184, 202–208, 211–212, 228, 233, 235, 243, 246–251,

253, 255–257, 266, 269, 273–276 in Thierry of Chartres, 27, 94–95, 104, 111–119, 136–139 as ethical training, 31, 34, 82–83, 146 foundations of in multitude and magnitude in Nicholas of Cusa, 178, 184, 203–205, 207–208, 215–216, 233, 235, 239, 246, 255, 260, 274–276, 376 in Nicomachus of Gerasa and his readers, 52–55, 63, 66, 76–77, 82–83 in Thierry of Chartres, 114–115, 117, 137, 375 four sciences of, 7, 10–11, 301 in Nicholas of Cusa, 172, 182–183, 203–205, 235, 274–276 in Nicomachus of Gerasa and his readers, 52–53, 56–58, 80–83 in Presocratics and Plato, 26, 31–38, 42 in Thierry of Chartres, 111, 146–147, 336 in medieval schools, 86–89, 95–96, 98, 101, 142, 145–147, 172–173, 266 quantity. See magnitude or quantity Qur’an, 169 rapture. See ecstasy reason(s) eternal (rationes aeternae), 75, 77–78 seminal (λόγοι σπερματικοί, rationes seminales), 56, 78–79, 99, 111, 153–155, 163, 165 Regiomontanus (Johannes Müller von Königsberg), 172, 243, 355 remaining (μονή), 45, 54, 67, 84 Remigius of Auxerre, 328, 338, 345 Reuchlin, Johannes, 173

Index rhythm, 31–35, 69, 183, 401 Richard of St. Victor, 141, 345 rithmomachia, 88, 266, 273 Robert of Courçon, 345 Rombach, Heinrich, 11, 356, 373 Rome, 39–40, 42, 48, 228–229, 243, 256, 402–403 roundness (rotunditas), 255, 268–270, 402–404 Salutati, Coluccio, 171 salvation, 45, 64, 70, 142 Same (idem), 33–34, 44, 220 Sánchez de Arévalo, Rodrigo, 218 School of Chartres, 15–16, 98–100, 102, 104, 140 Seneca, 56, 103–104, 323, 326, 341 senses, bodily (sensus), 31, 34, 53, 67, 70, 72, 77, 82, 105, 122, 129, 208, 215–216, 260 Septem (De septem septenis), 145–149 in reception history of Thierry of Chartres, 156–157, 162–165 as source of Nicholas of Cusa in De docta ignorantia, 175–179, 181, 186, 190–192 in Idiota de mente, 229, 232–235, 239 Sibyl, 147, 334, 338, 347 Sigismund of Bavaria, 256 Simon of Tournai, 141 simplicity (simplicitas), 125, 127–129, 131, 152, 154, 214–215, 236–239, 245, 248, 337 Simplicius, 44 singularity (singularitas), 159, 185, 195–196, 200, 254, 334, 342, 365, 402 Socrates, 29, 31–32, 40, 53, 204, 229, 390 soul. See also world-soul immortal, 10, 24, 178, 239, 262

481

as knowledge, 33, 66, 122, 234, 304 as mediator, 44, 99, 116, 125, 146, 148 as microcosmos, 268 moral purification of, 34, 70, 76, 82 as self-moving number, 38, 64, 240, 269 space of Dialogus de ludo globi, 263–264, 267–269, 274 divine, 136–137, 249, 251, 258, 264 of folding, 125, 130, 184, 250, 254, 261 as geometrical, 7, 36–37, 65, 67, 259, 275–276 as mediation, 68, 80, 138 of necessitas complexionis, 237, 248 Speusippus of Athens, 35–38, 40–43, 48, 51, 56, 62–63, 68, 99, 239, 295 sphere(s) 113, 125 game of, 261–264, 269, 271, 274 God as infinite, 269, 404 music of the, 42, 86 Spinoza, Baruch, 2, 239 spiritus psychology, 122, 124, 146, 156, 158, 161, 233–234, 254, 270, 333 spreading (ἐκτείνειν, πλατύνειν), 48–49, 59, 138 squares, 113, 124, 201, 209, 211, 220, 334 squaring the circle. See quadrature of the circle Stephen of Liège, 327 Stoicism, 10, 40, 46–47, 56, 63, 78, 93, 102, 130–131 Syrianus, 40, 64–66, 69, 71, 85, 208, 304, 312 Tertullian, 41 tetrad (τετρακτύς), 34, 49, 211, 289, 345, 378 theologia geometrica, 244–245, 248–249, 251, 256, 258, 260, 264, 267, 273

482

Index

theology, mathematical, 2, 44, 62, 68, 94, 103, 109 in Augustine, 69–71, 75–78 definition of, 58 in Nicholas of Cusa, 1, 11–12, 20, 199–202, 205, 211–212, 217, 223–227, 229, 243, 245, 247, 250–251, 253–255, 261, 276, 279 in Nicomachus of Gerasa, 51, 56–61 in Thierry of Chartres, 127, 137–138, 140 Theon of Smyrna, 41, 44, 47, 50, 86, 254 theophany, 211, 214, 219, 250, 265, 273 Theophrastus, 23 theosis (θέωσις), 218, 239 theurgy, 42, 61–64, 253 Thibaut of Langres, 108 Thierry of Chartres life of, 95–96, 100 medieval reception of, 17, 162–165, 175–176, 180, 192, 200, 351 state of scholarship on, 14–15, 96–97, 120–121, 143–145 Thomas Aquinas, 39, 94, 117, 358, 382, 397 time, 33–34, 74, 77–78, 125, 153–154, 184, 219, 259, 264 Toscanelli, Paolo, 171–172, 219, 243 Tractatus de Trinitate, 141, 165, 336, 343 transcendence, divine, 46, 57, 158–161, 164–165, 181, 190, 231, 278 Traversari, Ambrogio, 378 triads of Abelard, 140–141 Neopythagorean, 54–55, 57, 67 of Eudorus, 42–43, 99, 138 of Moderatus, 43–45, 48, 79, 138 of Nicholas of Cusa absolute potential, absolute actuality, connection, 258–259

contractible substrate, contracting agent, connection, 186 indivision, distinction, connection, 203, 205 multitude, inequality, division, 204–205 multitude, magnitude, connection, 207–208 of Philolaus, 28, 138 of Thierry of Chartres of matter, form, and motion (See perpetuals, trinity of) of unity, equality, and connection (See Trinity, arithmetical) triangles, 1, 54, 220, 246, 289 Trinity appropriated vs. proper names, 141–142 arithmetical, 14, 147–148, 150–151, 162 in Augustine, 79–80 in Nicholas of Cusa, 176–179, 182–183, 203–208, 238, 241, 254–255, 271 in Thierry of Chartres, 111–118, 121, 136–144, 327 of perpetuals (See perpetuals, trinity of) trivium, 96, 111, 145–147, 316, 328 truth divine, 24, 42, 161, 182, 209, 248 ethical, 146–147 mathematical, 7, 25, 34, 70, 72–73, 82, 111, 114, 134, 177 in necessitas complexionis, 128, 131, 134, 161 unfolding (explicatio). See also folding, reciprocal in Boethius, 84, 125, 375 in Nicholas of Cusa, 189, 192, 205, 215, 231–232, 237, 242, 268 in Proclus, 66–68, 304

Index in Thierry of Chartres, 125, 128, 133, 137 union, hypostatic, 193–194, 198, 200, 264, 274–275 unities, four, 207–217 unity (ἑνότης, unitas), 43, 79, 108, 112–113, 117, 123–124, 130, 176, 214. See also Trinity, arithmetical universals, problem of, 103, 231, 322, 377 universe as contracted, 159, 173, 181–188, 195–198 as divinely created, 33, 36, 55, 78, 99, 192, 220, 268 and the four modes (as universitas rerum), 127, 131, 158, 160, 164 as mathematically ordered, 6, 14, 33–34, 44, 55, 57, 67, 102, 113, 131, 158, 188, 196, 201, 211, 278 as mundus triplex, 268–269, 274 Unlimited. See Limit and Unlimited Valentinus, 47–49, 59–60, 71, 138 Valla, Lorenzo, 96 value (valor), monetary, 264–265, 272–273 Varro, Marcus Terentius, 20, 40–42, 69–70, 82, 86, 306 Velthoven, Theo van, 224–227, 383–384 Vincent of Beauvais, 143 virtue. See ethics William of Auberive, 108 William of Conches, 96–97, 100, 104–111, 116–118, 131–133, 160, 180

483

William of Ockham, 142 Wilpert, Paul, 16 wisdom (σοφία, sapientia) as abstraction, 46, 48, 107 as divine Mind, 6, 71–74, 78, 109–110, 114–115, 117, 135, 152, 154, 156, 204, 230, 256, 342 as part of the Trinity, 108–109, 140 of the philosophers, 43, 52, 70, 72, 82–83, 99, 146, 163, 165, 266 Wisdom 11:21 (numerus, pondus, mensura), 72, 75, 77, 80, 88, 108–109, 114, 183, 228–229, 265 Word, divine (Verbum). See Logos, Christology and Incarnation world. See universe world-soul (anima mundi) and the divine Word, 46, 57, 200, 214, 232 and the hierarchy of mediators, 37–38, 44, 48, 99, 101, 131–132, 165, 199, 251 and the Holy Spirit, 104–105, 107, 110, 116–118, 151, 160 as necessitas complexionis, 158–161, 189–191, 198, 213, 231, 270–271, 274, 356, 367–368, 404 as source of mathematical order, 33–35, 66, 99, 116, 137, 239 Xenocrates of Chalcedon, 35–38, 40, 42–43, 51, 56, 99, 240 Zahel ben Bischr, 333, 346 zodiac, 259