Table Of ContentMATTER, IMAGINATION AND GEOMETRY
This book considers conditions of applicability of mathematics to the study of natural
phenomena. The possibility of such an application is one of the fundamental
assumptions underlying the enormous theoretical and practical success of modem
science. Addressing problems of matter, substance, infinity, number, structure of
cognitive faculties, imagination, and of construction of mathematical objects, Dmitri
Nikulin examines mathematical (geometrical) objects in their relation to geometrical or
intelligible matter and to imagination. The author explores questions in the history of
philosophy and science, particularly in late antiquity and early modernity. The focus
is on key thinkers Plotinus and Descartes (with the occasional appearance of Plato,
Aristotle, Euclid, Proclus, Newton and others), in whom the fundamental
presuppositions of ripe antiquity and of early modernity find their definite expression.
Matter, Imagination and Geometry presents one of the first studies to compare the
relation of mathematics to physics in ancient (especially Neoplatonic) and early modem
science and philosophy.
ASHGATE NEW CRITICAL THINKING
IN PHILOSOPHY
The Ashgate New Critical Thinking in Philosophy series aims to bring high quality
research monograph publishing back into focus for authors, the international library
market, and student, academic and research readers. Headed by an international editorial
advisory board of acclaimed scholars from across the philosophical spectrum, this
new monograph series presents cutting-edge research from established as well as
exciting new authors in the field; spans the breadth of philosophy and related
disciplinary and interdisciplinary perspectives; and takes contemporary philosophical
research into new directions and debate.
Series Editorial Board:
David Cooper, University of Durham, UK
Peter Lipton, University of Cambridge, UK
Sean Sayers, University of Kent at Canterbury, UK
Simon Critchley, University of Essex, UK
Simon Glendinning, University of Reading, UK
Paul Helm, King’s College London, UK
David Lamb, University of Birmingham, UK
Stephen Mulhall, University of Oxford, UK
Greg McCulloch, University of Birmingham, UK
Ernest Sosa, Brown University, Rhode Island, USA
John Post, Vanderbilt University, Nashville, USA
Alan Goldman, University of Miami, Florida, USA
Joseph Friggieri, University of Malta, Malta
Graham Priest, University of Queensland, Brisbane, Australia
Moira Gatens, University of Sydney, Australia
Alan Musgrave, University of Otago, New Zealand
Matter, Imagination and Geometry
Ontology, natural philosophy and mathematics in
Plotinus, Proclus and Descartes
DMITR^NIKULIN
New School foS Social Research
Ashgate
© Dmitri Nikulin 2002
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or
transmitted in any form or by any means, electronic, mechanical, photocopying, recording or
otherwise without the prior permission of the publisher.
Published by
Ashgate Publishing Limited
Gower House
Croft Road
Aldershot
Hants GU11 3HR
England
Ashgate Publishing Company
131 Main Street
Burlington. VT 05401-5600 USA
Ashgate website: http://www.ashgate.com
British Library Cataloguing in Publication Data
Nikulin, Dmitri
Matter, imagination and geometry : ontology, natural
philosophy and mathematics in Plotinus, Proclus and
Descartes. - (Ashgate new critical thinking in philosophy)
1. Plotinus 2. Descartes, René, 1596-1650 3. Matter
4. Mathematics - Philosophy 5. Physics - Philosophy
6. Ontology
I. Title
117
Library of Congress Cataloging-in-Publication Data
Nikulin, D. V. (Dmitri Vladimirovich)
Matter, imagination and geometry : ontology, natural philosophy and mathematics in
Plotinus, Proclus and Descartes / Dmitri Nikulin.
p. cm. -- (Ashgate new critical thinking in philosophy)
Includes bibliographical references and index.
ISBN 0-7546-1574-X
1. Mathematical physics-Philosophy. 2. Science, Ancient. 3. Plotinus. 4. Proclus, ca.
410-485. 5. Descartes, René, 1596-1650. I. Title. II. Series.
QC20.6 .N54 2001
509--dc21
2001022642
ISBN 0 7546 1574 X
Printed and bound by Athenaeum Press, Ltd.,
Gateshead, Tyne A Wear.
C ¿ .f/z 4 £ £ £
U - 'è - o - * -
Piae memoriae aviae avique meorum
cbs tou àet ôvtoç yvcboecoç
àXXà où
tou
yiyvonévou ¿nroXXuiiévou.
TroTé t i kcci
EÙonoXôyriTOv. IçTy yàp àei
tou ôvtoç
ri yeconETpia1! yvcbaiç ècmv
Plato. RP 527b 5-8
Contents
Introduction ix
PART I: MATTER 1
1.1 The notion of matter 2
1.1.1 On the notion of matter in Aristotle 2
1.1.2 Plotinus on matter 6
1.1.3 Matter and extension in Descartes 12
1.2 Substance: ontology vs. henology 19
1.2.1 The One and the multitude in Plotinus 20
1.2.2 Distinction between the One and matter 26
1.2.3 Substance and attribute in Descartes 28
1.3 Infinity 34
1.3.1 Infinity in Aristotle and Plotinus 34
1.3.2 Infinity as perfection: Infini and ind€ftni 39
1.3.3 Paradox and infinity 51
1.3.4 Indefinite divisibility of physical and mathematical
entities in Descartes 54
1.4 Substance and essence 58
1.4.1 Otherness in substance 58
1.4.2 Essence and existence: Descartes’ essentialism 63
PART II: INTELLIGIBLE MATTER AND GEOMETRY 69
2.1 Geometry, arithmetic and physics in antiquity 70
2.1.1 Foundations of arithmetic in Plotinus 72
2.1.2 Constitution and the structure of number in Plotinus 81
2.1.3 Number and magnitude in ancient mathematics: The point 91
2.1.4 The in(de)finite in mathematical reasoning 95
2.2 Geometry, metaphysics and method in Descartes 103
2.2.1 Geometry in its relation to physics according to Descartes 113
2.2.2 Number and magnitude in Descartes 122
2.3 Intermediary 127
2.3.1 Geometrical objects as intermediary:
Proclus vs. Descartes 128
2.4 The notion of intelligible matter 132
2.4.1 Intelligible matter in Plotinus 135
2.4.2 Intelligible matter in Proclus 141
viii Matter, Imagination and Geometry
PART III: REASON, IMAGINATION AND CONSTRUCTION 145
3.1 Reason and the structure of cognitive faculties 146
3.1.1 Intellect-noys in Plotinus 146
3.1.2 Discursive thinking-dianoia 150
3.1.3 Life 152
3.1.4 Mind and its ideas: Descartes 157
3.2 Imagination in ancient philosophy 171
3.2.1 Aristotle and Plato on imagination 171
3.2.2 Imagination in Plotinus 175
3.2.3 Main features of imagination in Neoplatonism: Porphyry,
Syrianus, Proclus 179
3.2.4 Imagination and intelligible matter 183
3.3 Imagination in Descartes 187
3.3.1 Mind, imagination and the infinite 192
3.3.2 Imagination and mathematics according to Descartes 204
3.4 Imagination and kinematic construction in geometry 210
3.4.1 Construction and the verum factum principle: Cartesian
reconstruction of the world 210
3.4.2 Construction in geometry: Kinematic constructibility in
Descartes 223
3.4.3 Imagination and geometry: Imagination as constructive 230
3.4.4 Movement in the intellect according to Plotinus 239
3.4.5 Motion and construction in Proclus: Production of
a geometrical figure by movement in imagination 245
Conclusion 255
Bibliography 261
Index 283
Introduction ix
Introduction
One of the definite and most important assumptions underlying the enormous
theoretical and practical success of modem science is, as is commonly recognized,
the mathematization and mechanization of the physical world, based on the
possibility of applying mathematics to the study of physical reality, of nature.1
Early modem scientists constantly return to this theme: Galilei's famous statement
in the “Assayer” asserts that science is written in the book of nature in the language
of mathematics, the only language that allows one to read this magnificent book.2
For Kepler, the world—primarily the celestial cosmos, but also the terrestrial
one—in its structure and regularity of movements, follows a divine harmonic
pattern, exemplified and known primarily through mathematical objects, which are
further expressible in proportions and equations of numbers and in regular
geometrical solids.3 In the French Preface to the “Principles of Philosophy”,
Descartes presents a simile of the tree of knowledge where mechanics, as well as
other sciences, appear as branches on the trunk of physics, itself rooted in
metaphysics; if, therefore, geometry is also to be recognized as a science, the
principles that underlie geometry and mechanics must be taken to be the same.4
Newton begins his “Principia” with the claim that physics, considered as
1 Cp. Koyré 1968, 12-14; Maier 1938, passim; Duhem 1954, 107 sqq.; Kuhn 1961, 161-193;
Burtt 1964, 97-107; Funkenstein 1986, 29-30; Gloy 1996,98-117 et al. On various senses of
the notion of applicability see: Steiner 1998, 16 sqq.
2 “Philosophy [i.e. science in general] is written in this grand book, the universe, which
stands continuously 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
geometrical figures without which it is humanly impossible to understand a single word of
it; without them we would go around as in a dark labyrinth.” - Galilei, Assayer (Galilei
1957,237-238).
3 See: Mysterium Cosmographicum and Harmonice Mundi, -J. Kepler. Gesammelte Werke,
vols. I (1938) and VI (1940). Cp. Koyré 1973, 127 sqq.
4 Descartes. Princ., Pref., AT IXB 14-15; Princ. IV 203, AT VIIIA 326. Cp. Gabbey 1993,
311-323. On the historical background of the attempt of Galileo, Beeckman and Descartes to
use mathematical description for the study of free fall, see: Gaukroger 1978, 192 sqq.;
Schuster 1977, 53 sqq. As Gaukroger argues, unlike Galilei who tries to represent physical
objects as mathematical ones, Descartes “wants both to ‘mathematize’ physics and to
‘physicalize’ mathematics in one and the same operation. He does not simply want to use
mathematics in physics, he wants to unify mathematics and physics in certain critical
respect” - Gaukroger 1980, 97-98.
