Space, Geometry and Aesthetics: Through Kant and Towards Deleuze (Renewing Philosophy)

Space, Geometry and Aesthetics Through Kant and Towards Deleuze

Peg Rawes

Space, Geometry and Aesthetics

Renewing Philosophy General Editor: Gary Banham Titles include: Peg Rawes SPACE, GEOMETRY AND AESTHETICS: Through Kant and Towards Deleuze Simon O’Sullivan ART ENCOUNTERS DELEUZE AND GUATTARI: Thought Beyond Representation Jean-Paul Martinon ON FUTURITY: Malabou, Nancy & Derrida Vasiliki Tsakiri KIERKEGAARD: Anxiety, Repetition and Contemporaneity Alberto Toscano PHILOSOPHY AND INDIVIDUATION BETWEEN KANT AND DELEUZE: The Theatre of Production Philip Walsh SKEPTICISM, MODERNITY AND CRITICAL THEORY Celine Surprenant FREUD’S MASS PSYCHOLOGY Keekok Lee PHILOSOPHY AND REVOLUTIONS IN GENETICS: Deep Science and Deep Technology Kyriaki Goudeli CHALLENGES TO GERMAN IDEALISM: Schelling, Fichte and Kant Martin Weatherston HEIDEGGER’S INTERPRETATION OF KANT: Categories, Imagination and Temporality Jill Marsden AFTER NIETZSCHE

Space, Geometry and Aesthetics Through Kant and Towards Deleuze

Peg Rawes University College London

© Peg Rawes 2008 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1T 4LP. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The author has asserted her right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act 1988. First published in 2008 by PALGRAVE MACMILLAN Houndmills, Basingstoke, Hampshire RG21 6XS and 175 Fifth Avenue, New York, N.Y. 10010 Companies and representatives throughout the world. PALGRAVE MACMILLAN is the global academic imprint of the Palgrave Macmillan division of St. Martin’s Press, LLC and of Palgrave Macmillan Ltd. Macmillan® is a registered trademark in the United States, United Kingdom and other countries. Palgrave is a registered trademark in the European Union and other countries. ISBN-13: 978–0–230–55291–3 hardback ISBN-10: 0–230–55291–9 hardback This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin. A catalogue record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Rawes, Peg. Space, geometry, and aesthetics through Kant and towards Deleuze / Peg Rawes. p. cm.—(Renewing philosophy) Includes bibliographical references and index. ISBN 0–230–55291–9 (alk. paper) 1. Space. 2. Geometry – Foundations. 3. Geometry – Philosophy. 4. Aesthetics. 5. Kant, Immanuel, 1724–1804. 6. Deleuze, Gilles, 1925–1995. I. Title. BD621.R39 2008 114—dc22 10 9 8 7 6 5 4 3 2 1 17 16 15 14 13 12 11 10 09 08 Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne

2008011821

For Clare and James

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Contents

Acknowledgements

ix

Series Editor’s Preface

x

Preface

xiii

Introduction Space Geometry Aesthetic geometric figure-subjects 1 Drawing Figures Part I: Embodied Figures Forms of pure intuition Synthetic and analytic relations Space and time External and internal differentiations of space Part II: Drawing Figures Acts of construction Reflective judgment The imagination Drawing a line Summary

1 2 3 4 9 9 11 13 15 17 19 19 20 22 29 31

2 Folding-Unfolding Discursive geometry Soul Imagination Limit and unlimit Imagination, limit and unlimit Discursivity of the elements Figure Summary

34 37 41 44 48 51 54 57 60

3

62 62 65

Passages Geometric methods after Descartes Geometric method in the Ethics vii

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Contents

Substance Passage I: attributes, modes, affects and common notions Passage II: reading the text Passage III: modes of geometric thinking Summary

68 72 82 84 89

4 Plenums The transition from synthetic to analytic geometry Corporeal magnitude Incorporeal magnitude Sufficient reason The plenum Summary

91 94 99 103 109 113 119

5 Envelopes Limit and unlimit Extensity Body-image and perception Memory The envelope Space and time Intuition Envelope I: intuition as duration Envelope II: intuitive philosophy Envelope III: natural geometry and intuitive construction Summary

121 123 126 128 132 135 136 141 143 145 146 151

6 Horizons Geometric reason and ‘Teleological-historical reflection’ Geometric sense-ideas and sense-intuitions Explication Retrieving sense-intuition from Descartes and Kant Geometric self-evidence Horizons Horizon I: intersubjectivity Horizon II: we-horizons and future-horizons Summary

154 156 161 165 169 173 175 178 180 182

Notes

184

Bibliography

202

Index

207

Acknowledgements Numerous people have helped me at various times in writing this book. I owe special thanks to Howard Caygill’s supportive advice, and Gary Banham’s encouragement of the project has been invaluable. Fiona Candlin, Uriel Orlow, Joanne Morra and Jane Rendell have generously given feedback on drafts at various stages of the project. Thanks also to Laura Allen and Mark Smout for permission to use their drawing on the front cover, and colleagues at The Bartlett School of Architecture, UCL, have supported me with financial assistance from the Architectural Research Fund. Finally, I would like to thank my family, Tom and my parents, James and Clare, especially, for their constant support.

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Series Editor’s Preface Publication in a series of books that bears the title of Renewing Philosophy marks a work as having a kind of claim that may strike some as hubristic. The notion of a series having such a title is certainly indicative of a claim that philosophy both needs and is capable of renewal. The suggestion that philosophy needs such renewal is one that I will examine first in order to show how the work here being prefaced addresses this. If philosophy needs renewal this suggests that there are some matters that philosophy has traditionally not addressed sufficiently. In this work the suggestion would have to do with the status of geometry and philosophy’s engagement in it. Both ancient and modern forms of philosophy have given geometry some attention as this work makes clear. We need only think, as the author we are here introducing directs us to, of the example of geometry in Plato’s Meno to see how geometry held a significant place in the scheme of knowledge understood as significant in the ancient world. Similarly, the status of geometry in modern philosophy is clear simply by following the trail established by such founders as Descartes. If, however, the recourse of philosophy to geometry as a body of knowledge that is somehow exemplary is clear enough as a general point the question emerges as to what it is that requires this continued recourse to geometry and in what sense the recurrence of the question concerning geometry might indicate a disquiet? If geometry seems to recur as something to which philosophy points what might be the reason for this recurrence and is the recurrence of geometry really a repetition of a question concerning a solid and well-defined single body of knowledge? Or does the reference to geometry in different philosophers point to different objects? The moment these questions are posed we begin to note that the relationship between philosophy and geometry is not as evident as it first appeared. This work, the work by Peg Rawes, that I am presenting to the reader, would then be one that would first point to something recognisable (here philosophy, there geometry) in order to make clear that what appears before us is in fact not as ‘familiar’ as it initially seemed. In defamiliarising the relation philosophy has to geometry Rawes continues a peculiar ‘tradition’: the handing-down of a philosophical problem about the nature of philosophy itself. Just as the twin references of Plato x

Series Editor’s Preface

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and Descartes made earlier indicated the prevalence of a question about geometry for philosophy so also their names are equally significant of a self-declared difficulty with the nature of philosophical inquiry. Plato’s dialogues enact a continued questioning of the very inquiry that they undertake whilst Descartes is often said to ‘inaugurate’ the modern philosophy precisely by enquiring as to what it is that we can doubt. This self-questioning inquiry is certainly peculiar but the investment of philosophy in its own questioning would seem essential to marking it off from other forms of investigation that much more readily assume their problems, prospects and solutions. Therefore, if Rawes indicates some problem with philosophical investigations of geometry then it would appear that nothing could be more of a philosophical act than such indication of a problem with philosophical inquiry. The nature of the problem posed here can be seen in two ways. First, there is the suggestion that there is a ‘hidden’ history or a doubling of philosophy’s engagement with geometry such that geometry would seem in some sense to be investigating philosophy as much as vice versa. Second, there is produced here for us a type of historical investigation that does not match usual linear assumptions concerning history. Beginning with Kant we pass ‘back’ to Proclus before moving ‘forwards’ to Spinoza. Why would this kind of movement be followed as opposed to the convention of beginning with Plato or Proclus before moving on to Spinoza and Leibniz and placing Kant nicely after all the preceding? Does not the alteration of the narrative form place in question the suggestion that we can be sure which philosopher progresses over others? The later philosophers are doubtless always aware of how much they are in the shadow of the earlier, of how easy it is simply to repeat them, to recast in only slightly varied ways what has already been. But if this problem of influence has always been attested to in philosophy then the question of how philosophical progress is to be measured surely requires some attention to narration. The attention provided in the simple way of presenting here a methodological refusal to assume the place in history of the thinkers examined would appear to still be unsettled, however, by the manner of conclusion with that exemplary teleological thinker, Husserl. If, however, even this would be open to question might it not be because Husserl after all states again and again that the right method in philosophy is the adoption of a ‘zig-zag’ path, going backwards in order to go forwards but that each step further in an inquiry constantly requires the return to starting points in order to interrogate them again. This method of Husserl’s would, if applied to this book, require a revisiting

xii Series Editor’s Preface

of the earlier thinkers in its narration in the light of the later ones indicating a reason for resisting the sense that we have, with the concluding figure, reached the end point of our inquiry. This work’s engagement with geometry frees the understanding of the philosophical concern with it from a subordination to the sciences that in each of the cases examined would be false. The indication of a different sense to this engagement, one that owes its interest rather to a reinvention of aesthetic registers is what indicates in this inquiry the emergence of something new. Just as the above observation indicated the need for a renewal so the staging of this work’s investigation indicates also the capacity for such renewal to take place. Through the prism of a re- engagement with the philosophical concern with geometry a different sense of space as something that requires a set of aesthetic responses is released and this release marks this work as performing a ‘renewal’, a renewal, above all, of philosophy. GARY BANHAM

Preface Embodied subjects This inquiry reflects my interest in multiple modes of constructing spatiotemporal existences and geometric figurations of life. Broadly, the research has a critical and constructive purpose: that is, to explore how ontological relationships between space, geometry and aesthetics can be productive and meaningful expressions of modern subjectivity. Also, although it is not the aim of this book to examine these relations in specific cross-disciplinary contexts, the discussion has a bearing for cross-disciplinary spatial, geometric and aesthetic research in a number of ways; for example, it is situated with the view that aesthetic examinations of the relationship between geometry, space and subjectivity may enable the development of architectural models of spatiotemporal relations, and contribute towards rethinking the notion of ‘figuration’ in modernist art practices. My interest in geometric subjects and geometric figures is also undertaken in light of feminist philosophy’s examination of subjectivity.1 While I would certainly not suggest that the geometric subjectivities explored here are explicitly sexed subjects, my focus on spatiotemporal forms of material and immaterial embodiment, and aesthetics, is developed with the view that these discussions are also at stake for thinking about sexed subjectivities; in particular, because of the need to examine how the triadic relationship between the sensing subject, aesthetics and space is actualised as reality for the individual, and how she/he might express it. Yet, for many feminist theorists, the abstract disembodied scientific basis of space and geometry has rendered them inherently problematic for developing theories of material and sexed experience. Contemporary ontological and feminist philosophers have observed, for example, that the perceiving subject is not granted access to these transcendental geometric spatial ideas and bodies. For some critics (particularly those who read Bergson through Deleuze’s interpretations), these issues have led to the view that spatialised thinking and experience is better replaced by privileging time.2 In addition, theories which claim that temporal ontologies ‘solve’ the issue of formal representation often rely upon accepting that an exclusive opposition between space (i.e., disembodied scientific knowledge) and time (i.e., embodied xiii

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aesthetic process) is inevitable. When directed towards the visual arts, such as architectural design and spatial practices, these discussions therefore frequently ignore how the individual operates in contemporary practices; for example, by constructing material and spatial experiences through scientific and aesthetic sense-based geometric intuition.3 However, thinkers such as, Christine Battersby, Judith Butler, Claire Colebrook, Edward Casey, Moira Gatens, Elizabeth Grosz, Kathleen Lennon, Genevieve Lloyd and John Rajchman, have also shown that ‘local’, ‘particular’ and ‘embodied’ configurations of spatiotemporal and material relations exist that enable the freely acting and sensing subject to exist.4 So it is in this spirit that I invite the reader to explore productive historical examples of embodied spatiotemporal relations through which new examinations of aesthetic geometric sexed subjects may be developed.5 In particular, by locating geometric and spatial thinking within Kant’s aesthetic Critical philosophy, I suggest that scientific knowledge and aesthetic experience are brought together in the unity of ‘geometric figure-subjects’, rather than under laws of deterministic scientific progress. Each of the geometric methods and figures examined here therefore brings geometric inventions into an aesthetic project, not merely as studies of scientific progress: for example, this discussion suggests that Proclus’ Neoplatonic commentary reconfigures Euclid’s classical mathematics; Spinoza’s and Leibniz’s post-Cartesian texts reconfigure the sixteenth-century analytical geometry of Descartes; and Bergson’s and Husserl’s texts reconfigure Riemann’s nineteenthcentury theory of topological manifolds. Geometric thinking may therefore be composed of dynamic embodiments, reconfiguring the relationship between aesthetics, space and the sensing subject. As a result, new ‘figure-subjects’ are opened up through which geometry may also enable cross-disciplinary examinations; especially for ontological research in philosophy, architectural design, art history and spatial practices in the visual arts.

Towards Deleuze This project is also informed by Deleuze’s critiques of geometry, space and aesthetics, and by his examinations of ‘minor’ ontologies in Western philosophy; evident, for example, in his analysis of the fold in Proclus’ and Leibniz’s thought in The Fold, Leibniz and the Baroque (2001 [1988]), or in his examination of Spinoza’s transcendental empiricism, in Spinoza: Practical Philosophy (1988 [1970]), and in Expressionism in

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Philosophy: Spinoza (1997 [1968]). In addition, Bergsonism (1991 [1966]), Difference and Repetition (1997 [1968]) and The Logic of Sense (1990 [1969]) also demonstrate Deleuze’s reconfiguration of ontological philosophy by promoting minor traditions of abstract transcendental thought. In light of Deleuze’s philosophy, this is therefore an examination of a ‘minor’ tradition of geometric thinking. But my examination of spatial senses of geometry in Proclus, Spinoza, Leibniz, Kant, Bergson and Husserl is also differentiated from Deleuze’s readings of spatiotemporal problems in these philosophers for a number of reasons:6 First, I analyse the concepts of space, geometry and aesthetics, rather than following Deleuze and Guattari in their critiques of these problems through concepts such as, ‘smooth space’, ‘percepts’ or ‘multiplicities’.7 Second, I examine how the ontological concepts of space, geometry and aesthetics already undergo significant reconfiguration in each of the philosophical encounters, prior to Deleuze’s radical re-invention of them as new sense-ontologies. Third, Deleuze’s engagement with these philosophers is not the major subject of my research (these relationships need to be examined in a separate study); for example, developed through an analysis of the aesthetic Kantian subject, my inquiry into the sensing subject does not, in any pure way, follow Deleuze’s theory of univocal singularities. Rather, I wish to show that, between Kant’s Critical project and Deleuze’s intensive inquiry into sense-reason in ontological philosophy, productive geometric encounters do exist; for example, the last chapter on Husserl may represent a detour from a straightforward empiricist trajectory towards Deleuze. Yet this chapter is included in order to examine how Husserl’s geometric thinking, in particular, his promotion of geometric sense-intuition (which is certainly transcendental and eidetic) may bring to light connections with the transcendental empiricist sense that is expressed by Deleuze (even if Husserl’s geometric ideas are not as strongly associated with Deleuze’s thinking as some of the ideas of the other philosopher’s ‘geometric’ ideas explored here). This discussion therefore examines inventive thinking about space, geometry and aesthetics in philosophical texts that Deleuze has already significantly reanimated for contemporary researchers. Consequently, an ‘oblique’ relationship exists between my inquiry into the ‘problems’ of geometry, spatiotemporality and aesthetics and Deleuze’s examinations of these problems in Continental philosophy; and in this respect, the book may be read as a response to Deleuze’s examinations of spatiotemporality and sense.8 To an extent, this research also reflects the shifts in engaging with Deleuze’s philosophy which has taken place over the past decade in

xvi Preface

Anglo-American Continental philosophy. The project was largely developed between 1998 and 2004, although the final chapter was written more recently when discussions about the connections between rationalism, phenomenology and Deleuze’s empirical–materialist thinking have taken on a more productive tone in Anglo-American analyses. However, when I first encountered Deleuze’s writing during my postgraduate studies, his interdisciplinary collaboration with Guattari enabled many researchers to promote abstract thinking and practices that sought to reject the conventions of contemporary Continental philosophical practice. More recently, however, discussions about the relationship between Deleuze and ontological philosophy have undergone a shift towards exploring how he enables critical reconfigurations inside and outside the discipline. This project reflects these changes insofar as it examines some of the philosophical sources in Deleuze’s thinking, re-evaluating their geometric, spatial and aesthetic scope within the discipline, as well as indicating their scope for other disciplines that explore aesthetics, the sensibility and subjectivity beyond philosophical boundaries. For crossdisciplinary readers, especially readers who engage with philosophy through the writings of Deleuze, I therefore wish to contribute towards greater understanding of some of the philosophical sources of his ideas. Overall, my hope is that this project will not be seen as a step back into philosophical discussions that limit space, aesthetics and geometry to static, regulatory structures (and which Deleuze considered to be endemic in philosophy that does not engage with the living material world). Rather, this examination is undertaken in the belief that dynamic relationships between space, geometry and aesthetics are productive for contemporary practitioners working within philosophy and beyond; in particular, for generating alternative spaces through which geometry can be discussed without it being restricted to an exclusively scientific form of truth-making or knowledge. The discussion is therefore constructed ‘towards’ Deleuze; readers will not find him at the ‘end’ of the book, rather, he is a ‘virtual’ voice in the discussion. Each chapter represents an inflection of Deleuze’s desire for invention in philosophy, and insofar as I have chosen to engage with philosophers that inform his thinking, so each is implicated with his ideas.

Notes 1. Genevieve Lloyd writes of the increased contemporary interest in historiographies of philosophy, some of which are informed by feminist philosophies.

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Rather than searching for reductive historical formations, she writes that ‘Reading strategies which may originally have been motivated by feminism are passing into broader attempts to treat the intelligent reading of history of philosophy as a conceptual resource for rethinking our present’. G. Lloyd (ed.), Feminism and History of Philosophy (Oxford: Oxford University Press, 2002), p. 3. 2. Rosi Braidotti’s, Doreen Massey’s and Elizabeth Grosz’s writings reveal some of the tension with reference to geometric space that exists in many feminist theorisations of spatiotemporal life, although each of these authors also examines productive modes of spatial thinking and life for the individual. In addition, each author’s engagement with Deleuze’s influential promotion of temporal life informs their analyses; for example, Braidotti follows Deleuze’s desire for temporal and material life, over and above the spatiomaterial biological construction of subjectivity, in Transpositions: on Nomadic Ethics (Cambridge: Polity Press, 2006); Doreen Massey’s analyses of political, material and qualitative concepts of space engage with Deleuze and Bergson. However, her analysis of Bergson’s philosophy of duration also perpetuates the opposition between space and time, partly because of reading his work through Deleuze’s emphasis on the virtual and temporal, in For Space (London: Sage, 2005), pp. 20–4; Elizabeth Grosz considers geometry to be a scientific form of knowledge. Yet, in response to Bergson and Deleuze, her analysis of biological and ontological reconfigurations of ‘sociotemporal’ life in the sexed, historical subject do not completely exclude the potential for spatiotemporal expressions of life to be reconfigured. See, for example, E. Grosz, The Nick of Time: Politics, Evolution, and the Untimely (Durham: Duke University Press, 2004), p. 193 and pp. 177–8. 3. By contrast, Brian Massumi’s analysis of the act of producing architectural drawings or diagrams recognises the different modes of spatial thinking involved in the design process. He writes: Grappling with the question of double architectural vision requires acknowledging that the diagram is a technique of existence and that design is always collective. Architecture will always benefit from the application of powers of formal analysis. But its basic medium is not geometry, or topology, or CAD, or design in general, or critique, or any other formalizable field. Its basic medium is the field of experience. B. Massumi, ‘The Diagram as Technique of Existence’, in É. Alliez and E. von Samsonow (eds), Chroma Drama. (Widerstand der Farbe, Wein: Turia + Kant, 2001), p. 175 4. These philosophers may be brought together because they generate productive ontologies for differentiated and embodied subjects. See, for example, C. Battersby, The Phenomenal Woman: Feminist Metaphysics and the Patterns of Identity (Cambridge: Polity Press, 1998); J. Butler, Bodies That Matter: On the Discursive Limits Of ‘Sex’ (New York and London: Routledge, 1993); C. Colebrook, Philosophy and Post-Structuralist Theory: From Kant to Deleuze (Edinburgh: Edinburgh University Press, 2005), and edited with I. Buchanan, Deleuze and Feminist Theory (Edinburgh: Edinburgh University Press, 2000); E. Casey, The Fate of Place: A Philosophical History (Berkeley, London: University of California Press, 1998); M. Gatens, Imaginary Bodies: Ethics, Power, and Corporeality (London and New York: Routledge, 1996); E. Grosz, Architecture from the

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5.

6.

7.

8.

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Outside: Essays on Virtual and Real Space (Cambridge, MA and London: The MIT Press, 2001), The Nick of Time: Politics, Evolution, and the Untimely (Durham: Duke University Press, 2004), Space, Time, and Perversion: Essays on the Politics of Bodies (New York and London: Routledge, 1995), Time Travels: Feminism, Nature, Power (Durham: Duke University Press, 2005); K. Lennon, The World, the Flesh and the Subject: Continental Themes in Philosophy of Mind and Body, with P. Gilbert (Edinburgh: Edinburgh University Press, 2005); G. Lloyd (ed.), Feminism and History of Philosophy (Oxford: Oxford University Press, 2002); J. Rajchman, Constructions (Cambridge, MA and London: MIT Press, 1998), The Deleuze Connections (Cambridge, MA and London: The MIT Press, 2000). Readers should note that the discussions outlined here operate at an ‘abstract’ level and are not articulated in terms of examining specific examples of material socio-political expressions of lived subjectivity. Deleuze’s attention to the spatiotemporal production of thought, in particular, his promotion of immanent and virtual forms of thought may also constitute a particularly radical form of ‘geometric’ thinking. See, for example, the essay, ‘The Method of Dramatization’ (1967). Deleuze states: Spatio-temporal dynamisms have several different properties: 1) they create particular spaces and times; 2) they provide a rule of specification for concepts, which without these dynamisms would remain unable to receive their logical articulations; 3) they determine the double aspect of differentiation, qualitative and quantitative (qualities and extensions, species and parts); 4) they entail or designate a subject, through a ‘larval’ or ‘embryonic’ subject; 5) they constitute a special theatre; 6) they express Ideas. It is through all these different aspects that spatio-temporal dynamisms figure the movement of dramatization. G. Deleuze, Desert Islands and Other Texts, 1953–1974, edited by David Lapoujade and translated by Michael Taormina. (Los Angeles and New York: Semiotext(e), 2004), p. 94 However, a detailed examination of this discussion lies outside the boundaries of this current project. See, for example, Gilles Deleuze and Félix Guattari, A Thousand Plateaus: Capitalism and Schizophrenia (1980), translated by Brian Massumi (London: Athlone Press, 1987) and What is Philosophy? (1991), translated by Hugh Tomlinson and Graham Burchell (New York and Chichester: Columbia University Press, 1994). An indirect relationship exists between this and Deleuze’s examinations into space, geometry and aesthetics. Deleuze is a significant source for this project, but his philosophy is not the ‘subject’ of my study. The term ‘oblique’ also recalls Jacques Derrida’s essay, ‘Passions: “An Oblique Offering”’ in Derrida: A Critical Reader, edited by David Wood, Oxford and Cambridge MA: Blackwell Publishers, 1992, written in response to Wood’s invitation to participate in the volume. Derrida generates an ‘oblique’ response by rejecting a neat geometric relationship to the invitation, ‘One can reject, as I have done, the word “oblique”’; yet he also notes that ‘one cannot deny the destinerrant indirection […] as soon as there is a trace. Or, if you prefer, one can only deny it’ (Wood, 1992, p. 24; my emphasis).

Introduction

This book examines an overlooked ‘tradition’ of aesthetic geometries in ontological philosophy. Developed through Kant’s aesthetic subject, I explore a trajectory of geometric thinking and geometric figurations in ancient Greek, post-Cartesian and twentieth-century Continental philosophies, through which productive understandings of space and embodied subjectivities are constructed. The discussion has a twofold purpose: first, it seeks to disrupt the ‘natural’ relationship between geometric ideas or figures and limited forms of scientific ‘truth’, and second, it promotes the importance of aesthetic sense in the production of geometric thinking. I therefore outline an alternative tradition of geometric methods and figures which resist being reduced to simple repetitions of scientific geometric thinking. Kant’s Critical philosophy challenges the negative association between geometry and reductive forms of scientific thinking. For Kant, the subject’s experience of geometry is generated out of different modes of thinking: pure intuition, sense-intuitions, technical and aesthetic activities. This book explores these relationships, and proposes that an especially aesthetic form of geometry exists in the shift from the Critique of Pure Reason (1781/1787) to the Critique of Judgment (1790). In this transition, external objective geometric knowledge becomes internalised in the form of aesthetic sense-experiences and aesthetic acts of construction. Kant’s aesthetic subject constitutes a living figure through which geometry and aesthetics are re-engaged because geometry is embodied in the reflective subject’s aesthetic powers of construction. Aesthetic geometry, then, is constitutive of the reflective subject’s sensepowers. In addition, leading out of these discussions, I examine how Kant’s aesthetic subject enables neo-Platonic and post-Cartesian expressions of geometric thinking to be retrieved, and a post-Kantian 1

2 Space, Geometry and Aesthetics

lineage of radical aesthetic geometric thinking and spatiotemporality to be traced. Six chapters explore the construction of these aesthetic geometric methods and figures in a series of ‘geometric’ texts by Kant, Proclus, Spinoza, Leibniz, Bergson and Husserl. In each case, geometry is expressed as a uniquely embodied aesthetic procedure (rather than as a disembodied scientific method). I show that each text ‘enacts’ a unique geometric method and figure by examining how each philosopher’s ideas are imbued with aesthetic sensibility and geometric sense, thereby challenging the assumption that geometry pertains only to limited mechanical forms of scientific thinking (e.g., the production of finite and objective measurements of space). Kant’s Critique of Judgment provides the first site of this geometric encounter, which is established in the relationship between technical enactments (i.e., techne) and the aesthetic subject, and is embodied in the act of drawing geometric figures. Developing from this definition of geometry as an aesthetic ‘action’ or procedure, an ontology of aesthetic geometric methods and figures can be traced from Kant’s Critical writings, back to Proclus’ commentary on Euclid’s scientific geometric method, Spinoza’s and Leibniz’s post-Cartesian philosophies, and forwards to Bergson’s metaphysics of ‘duration’ and Husserl’s ‘horizons’. Therefore, located in this ‘minor’ tradition, particularly aesthetic and embodied notions of spatiotemporal life are constructed, and six geometric figures – the reflective subject, folds, passages, plenums, envelopes and horizons – constitute unique aesthetic and geometric enactments, rather than diagrammatic objects or concepts of scientific knowledge.

Space Kant’s writing on space reveals important questions about the subject’s access to spatial forms of knowledge and aesthetic experience that remain pertinent for contemporary ontological philosophy and disciplines that engage in scientific and aesthetic modes of constructing reality (e.g., architectural design).1 Initially, in the Critique of Pure Reason, space is an inadequate form of sense-knowledge, because it is divorced from the pure objective form of geometric intuition (i.e., ‘pure reason’). Space is characterised by its sensible and formal intuitive imitation of transcendental knowledge. Moreover, when understood as a formal account of sense-intuition, space and time become even weaker sources of objective knowledge, for example, when time is characterised as general and an internal imitation

Introduction

3

of spatial intuition within the subject. Here, then, space and time produce inadequate knowledge about the external world because space is rendered as an external form of sensible intuition, and time is an internally generated sense-experience. As a result, a hierarchy is invoked between space and time, because spatial intuition’s phenomenological appearance of objective knowledge is credited with an external objectivity that is made possible by its correspondingly stable scientific form of transcendental knowledge, geometry, which mediates between pure reason and sensible mathematical ideas. Kant’s definition of space in the Critique of Pure Reason is therefore deeply at odds with contemporary desires for ontological philosophy that promotes the freely thinking, sensing (and, for many, the sexed) subject. However, in the later Critique of Judgment Kant proposes an embodied theory of space, through which the autonomous sensing subject is constructed. This theory of embodied forms of geometric and spatial intuition is expressed by the reflective subject’s aesthetic powers of production, in particular, through the sensibility and the imagination. Geometry and aesthetics are brought together through the act of drawing out spatial figures, for example, when Kant discusses how the imagination is the active aesthetic generator of geometry and space because it is analogous to the ‘technical’ tools that construct geometric figures (e.g., the compass and ruler). Geometric space is constructed out of the sensing and technical powers of the subject (rather than it being conceived as a knowledge that exists independently of the subject). Space is therefore reconnected to pure forms of reasoning through the embodied subject’s powers of geometric thinking and aesthetic judgment.

Geometry Geometry is the major scientific discipline through which a priori concepts of space are formed; for example, the geometric operations of measurement and deduction produce abstract and intelligible understandings of individual spaces and relationships between spatial objects. In contrast, in its applied forms, geometric knowledge is often taken to be the material a posteriori representation or form of its a priori conditions, for example, when architectural plans are read as material imitations of the inherently deterministic transcendental laws of geometric knowledge.2 Alternatively, in its aesthetic material forms (e.g., geometric figures in the visual arts), embodied geometric intuition is frequently perceived to lack a legitimate and productive a priori ontological value; for example, the geometric figure in ‘Formalist’ painting may be viewed as either a

4 Space, Geometry and Aesthetics

transcendental a priori idea, or an a posteriori imitation of deterministic and abstract geometric ideas, thereby placing geometric aesthetic production at odds with the realisation of ontological freedom.3 In metaphysical philosophy, geometry is also frequently taken to be an exemplary form of pure intuition and abstract ideas, thereby confirming the existence of pure, but inaccessible, transcendental knowledge. Under these terms, truth, reason and intuition are observed in its operations, yet in each case, it confirms the existence of objective forms of knowledge which exist externally to the embodied living subject; for example, Kant’s first Critique examines geometric knowledge, not geometric thinking, that is, the aesthetic activity in which the sensing subject constructs spatial relationships.4 However, if we focus on the Transcendental Aesthetic of the Critique of Pure Reason we find that Kant develops a much more radical potential trajectory for geometric sense in the form of two kinds of intuition. On the one hand, Kant retains the view that geometry is a pure transcendental intuition and, on the other hand, it is the formal appearance of the ‘sense-intuitions’ (i.e., space and time). Thus, although pure geometric intuition is separated from the sense-intuitions’ aesthetic activities, and consequently from embodied spatiotemporal experience, it is nevertheless identified as an intuition that constitutes a potential link between the sensible and transcendental realms. As a result, this connection enables a legitimate aesthetic geometric ‘sense-reason’ to be traced out of the Critique of Pure Reason and embodied in the figure of the reflective subject in the later Critique of Judgment.

Aesthetic geometric figure-subjects This discussion also explores the importance of the relationship between the act of geometric drawing and the aesthetic subject. By drawing lines of connection between the texts, through Kant’s definition of aesthetics and geometry in the Critique of Judgment, I propose a series of six aesthetic geometric figures in which geometric sense-intuition exists as aesthetic actions, events or bodies. These six figures – the reflective subject, the fold, the passage, the plenum, the envelope and the horizon – demonstrate that geometry is not merely a mathematical method of constructing space but is also an aesthetic and embodied procedure, thereby challenging the view that geometry is exclusively concerned with scientific forms of knowledge. In the Critique of Judgment Kant develops an aesthetic notion of geometry out of the relationship between internal, heterogeneous

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sense-expressions of geometric intuition, technical enactments of geometric thinking, and the world, rather than focusing on disembodied axiomatic forms of geometric intuition. As a result, a shift occurs in his inquiry from a concern with external geometric forms into an examination of internal geometric sense-intuitions which are especially strongly expressed in the aesthetic and technical activities of drawing geometric figures. Figuration is therefore re-evaluated as a geometric and aesthetic mode of production, enabling a shift from limited definitions of geometric figures, as abstract disembodied diagrams, into aesthetic acts, events or subjectivities. The geometric figure becomes understood as an aesthetic realisation of geometric thinking. In addition, it constitutes the twofold activity of geometric thinking: first, because it embodies the aesthetic activity, such as the act of drawing, and second, because it embodies the sensing subject. Thus, geometric figures represent the embodied subject’s spatiotemporal sense-intuitions, that is, his/her powers of sensible production (e.g., through psychic powers, such as imagination and memory), which are also aesthetic expressions of geometric thought. The geometric figure is therefore central to this project because unique ‘figurations’ of spatiotemporal relations are constructed in each of the methods examined here; in particular, in the shift from objective geometric objects (or elements) to embodied figurations of aesthetic activity. As a result, scientific geometry is reconnected to aesthetic conditions of production (e.g., Proclus reconfigures Euclid’s Elements into an irreducibly discursive and ‘genetic’ geometric method). Consequently, this discussion also enables rethinking the problematic relationship between the disembodied geometric figure and the embodied sensing subject. Rather, by linking these two ideas, I suggest that geometric thinking and its figure-subjects express the relationship between the subject and spatiotemporal geometric intuitions that are derived from material and embodied experience. Such geometric figure-subjects represent bridges between the highly technical and scientific geometric procedures and objects and the irreducible aesthetic spatiotemporal intuitions of the sensing subject. As a result, a priori forms of geometric intuition may give rise to mathematical and aesthetic modes of expression. Each geometric figure-subject (for example, the reflective subject, fold/unfold, passage, plenum, envelope and horizon) is an expression of a radical form of sense-intuition, and in some cases, sense-reason, because an irreducible relationship is constructed between geometric reason and embodied geometric sense. Geometric sense-intuitions, sense-ideas and sense-perceptions therefore generate

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legitimate sense-reason in the subject, and consequently, this discussion proposes that these figure-subjects may constitute particularly irreducible embodiments of geometric life. Chapter 1 shows how Kant’s Critical philosophy defines the relationship between geometry, space and aesthetics, especially with regard to the shift from an objective or pure scientific geometry to a notion of geometry that is generated by the technical imagination in acts of reflective judgment. This chapter also begins the re-engagement of Kant’s aesthetic geometry back into ontological philosophy by exploring the geometric method that Plato develops in his dialogue, the Meno (380 BC). In this text, Socrates and the slave-boy embody two aspects of aesthetic geometry: recollection and embodied intuition. On the one hand, the boy represents the pure intuition of memory and, on the other hand, Socrates’ enactment of drawing out the geometric figures anticipates Kant’s ‘technical’ form of memory in the Critique of Judgment that is embodied in the aesthetic power of the subject’s imagination. A thread of aesthetic geometry is traced between Plato’s dialogue and Kant’s Critiques, thereby retrieving an alternative definition of geometry prior to Euclid’s paradigmatic mathematical geometric method in the Elements. This establishes the conditions of aesthetic geometry as a temporal and technical enactment of memory (i.e., intuition) in which the geometric figure is both the embodied subject and the drawn line of the geometric figure, and marks an important shift from the emphasis on external disembodied geometric knowledge to the internal embodied powers of the sensing subject. Chapter 2 explores the production of Classical geometry in more detail in its analysis of Proclus’ Commentary on the First Book of Euclid’s Elements (AD 410–485). I suggest that Proclus’ examination of Euclid’s scientific method is itself a version of this forgotten geometric method, because his Commentary reveals Neoplatonic and Pythagorean definitions of geometry in the form of an ‘unfolding’ geometric method. In particular, the procedure of unfolding and its figures – the fold/unfold – constitutes a ‘genetic’ expression of the discursive nature in geometric thinking. Second, Proclus’ method expresses the Stoics’ notions of divine limit and unlimit that underpin the continuity between geometric figures. In addition, this chapter also explores the nature of the synthetic geometric method, through which discrete, differentiated geometric figures are produced. In this chapter, limit is therefore derived from a synthetic ontology that contrasts with the concept of an analytic limit and geometric method which Leibniz develops in the Monadology (1714). For Proclus, however, the geometric method represents a divine

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and discursive demonstration through which the imagination and the soul are brought into harmony with the understanding. Proclus therefore reconfigures the geometrical enactment of Plato’s Meno through its genetic potential, and through which the irreducible and divine nature of limit and unlimit are realised. In Chapter 2, the geometric method is discursive, but it is an exclusively cognitive act, disembodied from the acting subject. In Chapter 3, however, we find that the notions of limit and unlimit are reconnected with the body and its affects in Spinoza’s axiomatic text, the Ethics (1677). This text is structured following Euclid’s classical scientific definition of geometry, reinforced by Descartes’ new theorisation of analytic geometric procedures, yet it is also an intensive examination of the human emotions. Spinoza does not produce a geometric figure as such, since his purpose is to reveal the acts of the geometric method as an ethical process, rather than as a ‘technical’ procedure. Rather, I suggest that the act of reading the text constitutes a geometric figuration; that is, the geometric encounter is located in the ‘passage’ from understanding subjectivity to a ‘perfect’ understanding of God. I explore how this figure of the passage arises from the way in which the text demonstrates the changes of state in the body and its passions, thereby acting out the relationship between the divine infinity of God and the body’s irreducible limits. Therefore, in Chapter 3, Proclus’ idealised and divine notions of geometric figures become reconfigured into a series of embodied modes of subjectivity, in particular, through the powers of the emotions. Spinoza’s geometric method is shown to be a synthetic and divine method that is also embodied within the powers of the subject. Chapter 4 examines a distinctly analytical form of this overlooked geometry in Leibniz’s Monadology (1714). Leibniz’s text provides perhaps the most intensive version of the notion of limit in this discussion, because limit is not only an external geometric magnitude but is also internally and qualitatively differentiated in the Monad. Thus, the geometric figure represents not just an embodied state, but one in which the notion of limit has been rationally or intelligibly explained as an internal operation (in contrast to Spinoza’s concern with the external expressions or affects). In addition, Leibniz’s method and figures are generated from an intensification of an analytic understanding of subjectivity that, in contrast to Kant’s, Proclus’ and Spinoza’s thinking, is powerful precisely because of its emphasis on limit as an aspect of infinity. Geometric limit and unlimit are redefined, therefore, in terms of infinite divisibility, in particular, through the notion of ‘incorporeal

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magnitude’. Such an approach means that a number of geometric figures are generated, including the ‘plenum’, a geometrical figure that embodies both the internal differentiation of the Monad and its continuity with the infinite divisibility of the world. In the penultimate chapter, I show that Bergson’s text Matter and Memory (1896) rejects the notion of limit and the imagination in the reformulation of a distinct set of metaphysical relations that reflect the psychical and physical conditions of the active subject. Geometry becomes re-invigorated with its sense-perceptions in the form of memory and intuition. In contrast to the Critique of Pure Reason, however, Bergson insists that intuition is formatively concerned with the thinking, living subject. This chapter therefore shows that Bergson’s figure of the ‘envelope’ is the site through which the body acts out the intuitive spatial actions of geometry. In this chapter, the initial image of Socrates’ drawing and the boy recollecting the geometric figure is redrawn by Bergson into a topological relationship between the internal psychical activities of the subject and his/her perception of the external world. Finally, in Chapter 6, I explore how Husserl’s essay, ‘The Origin of Geometry’ (1936), written as an appendix for his unfinished book, The Crisis of European Sciences and Transcendental Phenomenology (1954), develops an especially teleological and historical form of geometric sense, and through which Husserl generates an existential and highly temporalised geometric idea, called the ‘horizon’. In addition, this chapter examines Husserl’s geometric encounters with Kant and Bergson, and proposes that, like Bergson, Husserl’s post-Kantian critique retrieves a radical kind of sense-reason. Thus, for Husserl, the horizon not only expresses an embodied transcendental geometric sense-intuition but also generates a post-Kantian ‘teleological-historical’ ontological inquiry. The Origin and Crisis therefore reveal that Husserl’s phenomenology constitutes a radical ‘science’ of geometric ‘sense-ideas’, offering an irreducible form of ‘living science’ for modern societies that challenges the dominance of symbolic methods of reasoning. Moreover, Husserl’s emphasis on the transcendental and temporal nature of sense-ideas shows that the horizon is an ‘intersubjective’ figure-subject, for example, it is ‘we-horizons’ or ‘future-horizons’, thereby shifting the discussion from the singular sensing subject into an analysis of ‘multiplicity’ embodied in geometric subjectivities and geometric sense-reason.

1 Drawing Figures

Part I: Embodied Figures In this first chapter, I show that two different kinds of geometry, space and aesthetics exist in the development from Kant’s Critique of Pure Reason (1781/1789) to the Critique of Judgment (1790). First, in the Transcendental Aesthetic of the Critique of Pure Reason, geometry is an ideal knowledge or cognition that is commensurate with the ‘higher’ level of a transcendental intuition or aesthetic. Kant states that we have access to this cognition from the sense-based world, but only by means of our sense-intuitions, space and time, which represent the phenomenal forms of transcendental intuition. Hence, in this earlier text, a divide exists between the immaterial and disembodied idea of geometry, and our embodied, but ‘inadequate’, senses of space and time. However, in the later Critique, this divide disappears, because geometry, space and aesthetics are brought together within the powers of the embodied subject. A continuity of relations is revealed between the external, transcendental and ‘pure’ geometric space and the ‘reflective subject’, bringing geometry into a heterogeneous continuum of spatiotemporal relations. Here, therefore, the a priori power of geometry is transformed into the mental, sensory and bodily actions of the subject, the imagination and aesthetic judgment: geometry, space and time become connected in the embodied, aesthetic acts of the reflective subject. So if Kant’s examination of geometry, space and aesthetics in the Critique of Judgment is retrieved, then an indeterminate, yet embodied a priori judgment in the individual subject may be reinstated. In addition, the absolute connection between deterministic spatial thinking and Kant’s concepts of geometry and space is brought under scrutiny. Thus, 9

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I suggest that Kant’s theory of reflection is valuable for both geometric and spatial reasoning, and the reflective subject is significant because it transforms the pure external space and geometry of the Critique of Pure Reason into the internal and embodied experience of space in the Critique of Judgment. In the development from the construction of geometry, space and time in the first Critique to the reflective subject’s powers of judgment in the third Critique a ‘forgotten’ relationship between geometry, space and time exists which this chapter attempts to ‘recollect’: for example, I recall Kant’s earlier essay ‘Concerning the Ultimate Ground of the Differentiation of Directions in Space’ (1768) in my exploration of the reflective subject’s ability to generate judgments about space in artistic and technical acts. In addition, I suggest that the Critique of Judgment’s ‘productive’ imagination is especially concerned with embodied modes of technical and aesthetic acts of geometric thinking, and I reveal the importance of the productive imagination, with reference to Kant’s later studies of the subjectivity in The Anthropology from a Pragmatic Point of View (1798). Finally, in the last section of the chapter, I draw out Kant’s aesthetic geometric figure in relation to Plato’s discussion about geometry, memory and intuition in his dialogue, the Meno (380 BC), suggesting that Kant’s reflective subject can be linked back to Plato’s study of the geometric method, recollection and drawing. The Meno is therefore a Classical example of aesthetic geometric thinking in two related forms: first, because it examines ‘recollection’ and memory, and second, because it presents the aesthetic act of drawing geometric figures when Plato attributes these qualities to two of the human subjects in the dialogue, the slave-boy and Socrates. On the one hand, the slave-boy embodies an intuitive understanding (or unfolding) of geometry and, on the other hand, geometry is embodied in the physical actions of Socrates, as he draws geometric figures in the sand. In this context, we might therefore suggest that geometry is both a mental and a physical activity: it is extended and unextended, logical and aesthetic, ideal and particular. The subject’s ability to construct heterogeneous and aesthetic geometric figures enables the retrieval of other embodied geometric figures in the following chapters. In addition, the reader is introduced to some of the principal constituents (i.e., ‘elements’) of aesthetic geometric methods that I analyse in the following chapters; in particular, intuition, spatiotemporal relations, unextended and extended matter, the imagination and the soul, and the importance of the Stoic ideas of limit and unlimit. Kant’s reflective subject therefore enables discussions about aesthetic and technical thinking, especially with respect to the role of geometry and the subject in spatial practices (including, the visual arts and

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architectural design). Moreover, I show that the reflective subject is itself both a recollection and a preview of an ongoing tradition of embodied geometric methods in metaphysical philosophy, when it is examined in relation to the aesthetic geometries of Proclus’ neo-Platonic philosophy, the post-Cartesian philosophies of Spinoza and Leibniz, and also continuing in the encounter with Bergson’s durational geometry and Husserl’s geometric ‘horizons’.

Forms of pure intuition In the Critique of Pure Reason the relationship between geometry and space exists in the form of intuition or a priori judgments. Geometry, Kant tells us, is an intermediary knowledge, because it is unextended intuition in the special science of mathematics. Yet intuition also exists in our extended ‘sense-intuitions’, that is, space and time, which constitute our ‘outer sense’ (i.e., space) and our ‘inner sense’ (i.e., time). So intuition is both the ‘pure’ absolute form of geometric knowledge and the spatial and temporal forms of our sensibility. As a result, geometry, space and time are a priori, irreducible to simple concepts or ideas. However, a divide between pure geometric thinking and spatiotemporal thinking emerges when Kant states that space and time are limited to the status of ‘phenomena’, because they are merely forms of appearances which are generated out of the extended world of bodies and ideas, rather than from ‘pure intuition’ itself. Geometry, on the other hand, is legitimate both when it is material bodies or ideas and when it is immaterial or unextended ideas or forms. In contrast, because space and time are always considered to be extended formal manifestations of intuition, they are prevented from having a continuous relationship with the pure realms of unextended intuition and geometry. However, as we will see below, when these sense-intuitions are generated by the reflective subject’s powers, embodied connections between the extended and unextended ideas are made possible, so that spatiotemporal relations also become legitimate a priori intuitions, and the ‘limits’ between transcendental ideas and sense-based forms are brought together in the actions of the reflective subject. In addition, the productive powers of the imagination enable the abstract science of geometry and the sense intuitions to be unified in the reflective subject. A more productive development of intuition therefore exists in the reflective subject, in which experience is examined, not as a form of reason but as aesthetic judgment. Geometric thinking and the sensibility become linked through a concept of limit in the form of the

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imagination, in which different geometric figures are generated from within the subject, not determined by an external idea of formal classification; for example, it is a notion of limit that is felt in the movement between the pleasure and displeasure of experiencing the limit of the imagination’s attempt to understand the sublime. Thus, the transcendental aesthetic (i.e., the critique of non-conceptual forms of understanding), is reconfigured to generate the aesthetic experiences of the reflective subject, and the two divided forms of intuition posited in the first Critique are brought together into an active and speculative aesthetic power. Kant’s promotion of the synthetic form of a priori idea also reveals the importance of the Stoic ideas of limit and unlimit in the aesthetic geometric method; for example, his emphasis on the ‘formal’ limits of space and time introduces a problematic restriction to the scope of his spatiotemporal relations. In addition, although he attributes productive powers to the limits of the imagination, it is nevertheless linked to a mathematical limit or division. Similarly, Kant tends to view unlimit as an excessive operation, in contrast to the legitimate ‘genetic’ or ‘immanent’ relations between the subject and the outside world. In the following chapters, however, I show that there are other more radical notions of intuition, limit and unlimit, which generate a legitimate continuity of different figures and subjects, between pure geometric intuition and embodied manifestations of geometric thinking; for example, in Leibniz and Bergson’s notions of perception. First, however, the transition from Kant’s construction of aesthetic intuition in the first Critique to its manifestation in the reflective subject reveals the shift in his geometric thinking in more detail. In the Critique of Pure Reason Kant brings together scientific geometric cognition and spatiotemporal forms of the sensibility under the same doctrine or ‘aesthetic’. In a note to the First Part of the Aesthetic (in the second edition, dating from 1787), Kant defines the term ‘aesthetic’ as the ‘critique of taste’, which he develops out of Baumgarten’s theory of aesthetics. But he also distinguishes between his and Baumgarten’s theory, stating that Baumgarten’s attempts to bring the beautiful under the premise of ‘reason’ are limited because he relies upon a posteriori evidence. Instead, Kant argues that his ‘speculative philosophy’ enables the sensibility to be a transcendental condition, which is not merely determined by a posteriori empirical evidence (Kant, 1997, p. 173).1 As a result, the sensibility exists in the extended, yet a priori forms of intuition, space and time; that is, space and time are empirical experiences but are also a priori ideas (and below, we will see that the forms of feeling pleasure and displeasure are also a priori aesthetic judgments).

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So the Transcendental Aesthetic of the Critique of Pure Reason defines the ‘elements’ of a ‘speculative philosophy’ that are generated in cognition, and which link geometry, space and time to intuition. Each element is an example of a synthetic a priori judgment, and Kant draws particular attention to the relationship between metaphysical conditions of experience and scientific forms of these intuitions when he states that mathematics is a paradigm of ‘pure’ reason that is a priori, yet it is also an empirical experience: Mathematics gives us a splendid example how far we can go with a priori cognition independently of experience. Now it is occupied, to be sure, with objects and cognitions only so far as these can be exhibited in intuition. This circumstance, however, is easily overlooked, since the intuition in question can itself be given a priori, and thus can hardly be distinguished from a mere pure concept. (CPR, p. 129) However, in a section titled, ‘Transcendental Exposition of the Concept of Space’, in the second edition, Kant makes the relationship between geometry and space more explicit when he emphasises that it is intuition, not mathematics, through which the relationship between geometry and space is primarily constituted: ‘Geometry is a science that determines the properties of space synthetically and yet a priori’. He continues: ‘What, then, must the representation of space be for such a cognition of it to be possible?’ In response, he tells us it must ‘originally be intuition’ (CPR, p. 176). In addition, Kant’s examination of theoretical ideas and empirical ideas are generated by a unique kind of judgment that is a priori but which is also determined by knowledge ‘borrowed’ from experience; synthetic a priori judgments.

Synthetic and analytic relations Theoretical and empirical knowledge are mediated by a special kind of judgment that Kant calls synthetic a priori judgments, which generate both unextended and extended ideas, such as geometry and arithmetic; for example, the synthetic quality of geometry is shown in the unextended idea (i.e., the axiom) and the extended figure (i.e., the demonstration) that determine a triangle, in each case proving that its three angles combine to form 1808 (CPR, p. 140). Geometry is therefore a heterogeneous operation, insofar as it is constituted by unextended and extended matter, and generates its

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objects in both a priori and a posteriori forms. In addition, Kant considers sciences (such as, geometry, philosophy, or the ‘critique of pure reason’) to have the capacity to produce judgments out of a particular ‘method’ of construction, which attaches ‘given concepts’ to others ‘completely foreign to them’. These attachments are called analytic or synthetic judgments and express two distinct kinds of relations or unities between the subject and predicate (CPR, p. 130); and, as we will see in the following chapters, the nature of analytic and synthetic relations is central to each of the methods. In the Critique of Pure Reason, Kant defines the judgments in the following ways. First, an analytic judgment defines an agreement between the two parts (e.g., B belongs to A), because it describes what is already contained in the constituent parts.2 Analytic judgment is therefore determined by laws of identity or similarity because B does not introduce any contradiction into the relationship. Rather, it is legitimised by virtue of its agreement with A. The principle of contradiction therefore underpins Kant’s definition of analytic judgment. (In Chapter 4, by contrast, Leibniz constructs a quite different notion of ‘analytic’ judgment in which the principle of contradiction is transformed into a relation of infinity, constituted by intensities or magnitudes that are internally differentiated. For Leibniz, analytic judgment is therefore not reducible to the self-same notion of identity that is attributed to it by Kant.) In contrast to analytic judgments, Kant states that synthetic judgments generate an external heterogeneity because independent concepts are linked through a ‘synthetic combination’ or ‘amplification’ of intuitions. Synthetic judgments therefore register non-contradictory relationships between intuitions that do not require the principle of agreement in order to be legitimate: that is, if A is not self-similar to B, the judgment does not become invalid. Instead, the operation brings the subject and predicate together, and so synthetic a priori judgments may be said to ‘amplify’ the relationship between A and B, not by resorting to an empirical demonstration but by providing an original ‘pure’, yet external, knowledge. So mathematics is a synthetic form of a priori judgment because each of its modes (such as, geometry and arithmetic) requires an external intuition in order to function. In arithmetic, for example, the introduction of an additional operation (such as, addition, subtraction or the multiplication of numbers) is required in order to produce the relation between elements. In geometry, two points are related to each other because of the external difference which is generated by the operations of direction or measurement that connects them (CPR, p. 144). But Kant also tells us

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that judgments of experience are synthetic, because experience itself is a ‘synthetic combination of intuitions’; for example, an extended body is synthetic when the external predicate of mass is taken into account (CPR, p. 142). Therefore, in each case, an external function or relationship is introduced into the relationship. Space and time are therefore synthetic a priori judgments, because each is determined by an external intuition or operation, that is, geometry or arithmetic, respectively; for example, when these judgments take the form of geometry they represent a distinct class of knowledge that is an ‘intermediary’ state between the unextended pure a priori analytic judgments and the extended a priori and synthetic judgments. Kant therefore generates a theory of formal differences and unities (e.g., in the form of different geometric figures) that are produced by external operations. As a result, although geometric thinking has a degree of variation within it, Kant’s emphasis on the external and formal attribution of difference produces a number of problems when thinking about time and space that are challenged by the other methods in this discussion, and which Kant will also reconfigure into the reflective subject’s internal operations (or powers) in the third Critique. First, however, I examine Kant’s ideas of space and time in the first Critique, in more detail.

Space and time Space, Kant tells us, is our ‘outer sense’ through which we construct our relationship with the external world, for example, in our ability to determine the magnitude of forms of appearance, or the relationship between entities. In contrast, time is our ‘inner sense’, or our ‘soul’, and registers our internal experiences (CPR, p. 157). Space and time, then, are cognitions that are derived from our experience and are distinct from concepts, capable of producing legitimate phenomenal or sensory understandings of the external world and our internal experiences. As such, space and time also enable us to comprehend different places or events simultaneously or successively, without recourse to a posteriori or empirical concepts; for example, space is ‘the condition of the possibility of appearances, not as a determination dependent upon them, and is an a priori representation that necessarily grounds outer appearances’ (CPR, p. 158). In addition, space provides the possibility for mathematics, in particular, geometry, for if it were a posteriori ‘the first principles of mathematical determination would be nothing but perceptions’ (CPR, p. 158). Spatial intuition therefore confirms the purity of geometric intuition, and indicates to the fundamental harmony established between

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scientific and metaphysical orders (an issue in Kant’s theory of experience which we will see Bergson criticising for its deep-seated emphasis on symbolic values that prevents a truly speculative philosophy of continuous change or transformation from being established). Space, then, is not a category or concept but is ‘merely the form of all appearances of outer sense, that is, the subjective condition of sensibility, under which alone outer sense is possible to us’ (CPR, p. 159). In addition, it does not account for internal sense or give us access to understanding the transcendental experience; rather, it produces ‘correlates’ of ‘things in themselves’ in the form of the appearances of external objects (CPR, p. 168). Time is also a sense-intuition: it is ‘the form of inner sense, that is, of the intuition of our self and our inner state’. But Kant also tells us that, in contrast to space’s exclusive link with external phenomena, time is ‘the a priori formal condition of all appearances in general’ (CPR, p. 163). So, despite considering time to be a pure intuition and, hence, a form of synthetic a priori judgment, Kant suggests its ‘general’ condition constitutes a relationship between internal and external sense-intuitions. However, ultimately, Kant’s insistence upon the formal limits of space and time prevents a strong genetic or discursive continuity in this relationship from being fully established; for example, in the following passage from the section, ‘Elucidation on Time’, the formal definition of limit which differentiates space and time is clearly visible: Time and space are accordingly two sources of cognition, from which different synthetic cognitions can be drawn a priori, of which especially pure mathematics in regard to the conditions of space and its relations provides a splendid example. Both taken together are, namely the pure forms of all sensible intuitions, and thereby make possible synthetic a priori propositions. But these a priori sources of cognition determine their own boundaries by that very fact ... namely that they apply to objects only so far as they are considered as appearances, but do not present things in themselves. (CPR, p. 166; my emphasis) So Kant’s examination of space and time registers a degree of difference between these synthetic a priori judgments, but at the expense of a continuous relationship between internal and external intuition, which continue to be determined by the application of external and finite limits of form. Hence, geometric space is a legitimate extended ‘image’ but is ultimately cut off from pure intuition and unextended matter (in contrast, Bergson’s notion of the ‘image’ is a highly discursive idea, in

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both its extended and unextended forms). In the end, however, a discursive relationship between pure geometry and geometric space and time is impossible in the Critique of Pure Reason because their formal limits are determined by an absolute division between the sensibility and reason. Space and time therefore represent legitimate modes of knowledge in the form of a priori intuitions through which we understand the world around us. They are products of the sensibility through which we are affected by objects external to us; the means by which we experience the world. Produced by the sensibility, these sense-intuitions are brought into harmony with the categories of understanding to become concepts of form. Thus, insofar as space and time are forms of appearance, the sensory world is linked to absolute intuition, but because Kant insists that space and time are only appearances of our experience, an exclusively formal and external limit divides geometric intuition and spatiotemporal intuition. Geometric reasoning is, however, both an autonomous intuition and an empirical form. But its extended and unextended geometric figures also represent external forms of transcendental knowledge, not the embodied experience of geometric thinking or space in the individual, so that the Critique of Pure Reason is therefore primarily concerned with determining forms of knowledge that limit geometry to a pure disembodied intuition and limit space to a restricted sensible power. However, an earlier pre-Critical essay by Kant given below suggests a more discursive relationship between geometry, space and time, in which geometric spatiotemporal relations are constituted in the subject as a series of heterogeneous embodiments.3

External and internal differentiations of space In the earlier essay, ‘Concerning the Ultimate Ground of the Differentiation of Directions in Space’ (1768), Kant challenges Leibniz’s proposition that it is magnitude, not position, which provides the means through which space is differentiated into different parts.4 Kant contests that magnitude is the ‘ground’ for determining relations in space, instead suggesting that it is ‘direction’ which provides the basis for the differentiation of space.5 Direction, he argues, ‘orientates’ the parts of space and ‘refers to the space outside the thing’, showing that ‘absolute space’ has ‘a reality of its own’. In the following paragraphs, Kant then explores how spatial three-dimensionality is derived from

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our sensible understanding of other bodies in relation to our own corporeality (CDS, pp. 365–7).6 Here, internal and external forms of space are generated as qualities of direction which operates externally and internally. In contrast, Kant argues that position does not sustain this ‘proof’ of continuity and unity, whereas, when direction is considered within the context of an embodied condition, a real set of differences can be identified; for example, our sense of direction has a relationship with the spatial construction of our bodies as left or right orientated. Kant identifies various examples of the different directions of growth that exist in nature – human hair, snail shells, the growth of beans, the direction of winds according to the lunar cycle and observations of the movements of the south seas – which point to an internal principle of direction in an individual entity, confirming that it is a universal a priori principle of nature (CDS, p. 368). Furthermore, this potential is linked to the perceptions and the aesthetic sensibility of the subject, in which ‘an immediate connection between feeling and the mechanical organisation of the human body’ are brought together. As a result, ‘clear feelings’ of difference between the sensory and mechanical attributes of the left and right sides of the body distinguish the particularity of the body, despite the apparent ‘great external similarity’ (CDS, p. 369). In this essay spatial direction is therefore an aesthetic aspect of our sense-intuition, because a continuity between internal and external space in the natural world becomes possible. In addition, Kant tells us that the differences discerned between entities or objects need to be understood in relation to ‘universal absolute space, as it is conceived by geometers’. In order to make this step, he reintroduces the concept of planes, lines and surfaces through which corporeal bodies can be understood to be similar or different. Thus, apparently incongruent aspects of bodies or unrelated bodies can be made to appear similar depending upon their relationship to each other on a single plane; for example, despite the impossibility of the surface of one being transferable onto the other, the left and right hand can be viewed as ‘similar and equal and yet incongruent’ (CDS, p. 370). In this earlier essay, therefore, geometric intuition connects a series of synthetic states of nature and differentiated extended bodies, suggesting a more discursive or genetic relationship between entities. In addition, direction produces both the internal spatial specificity of an embodied entity and its relation with the general order. Space, then, can be similar and incongruent, whether it is derived from both an

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extended body or from the unextended principles of absolute space, and ‘inner differences’ are founded upon the different positions within an entity; for example, the different positions of the right and left hands, which are also related to absolute space. Yet, in absolute space their ‘true differences’ are determined by the specific constitution of the body. So absolute space is not constructed externally of our sensory perception, but is a ‘fundamental concept’, which grounds any outer perception which we might have (CDS, p. 371). Thus, space is both extended as part of the corporeal body and unextended in the form of absolute space. Here, Kant’s theory of space is less recognisable as an exclusively external, synthetic a priori cognition; rather, its determination as a continuity of relations between the internal incongruence and the external congruence of a body suggests a notion of space that is constituted by distinct ‘figures’ in a discursive order. Kant’s pre-Critical examination of direction suggests a notion of spatial ‘intuition’ in which the subject produces space both in absolute and particular terms; a line of inquiry which is broken in the first Critique. However, this link is retrieved in the powers of the reflective subject in the Critique of Judgment, especially because space is understood in relation to the thinking body; that is, it is confirmed as an internally produced intuition which is not reducible to an externally derived form. In this earlier essay, therefore, space is not exclusively defined as the external formal appearance of pure intuition, because Kant emphasises the irreducible difference of the subject in both external and internal space, once again offering the scope for a more discursive set of relations.7

Part II: Drawing Figures Acts of construction In the Critique of Judgment Kant’s formulation of the relationship between aesthetic judgment, space and geometry removes the exclusive divide between pure transcendental intuition and the sensibility. Consequently, geometry and space are brought together through the sensibility’s powers of spatial construction in the form of the reflective subject. Also, unlike the rules of aesthetics that are harmonised with the conceptual categories of the understanding in the Critique of Pure Reason, here Kant constructs a legitimate relationship between aesthetic judgments, reason and the productive imagination.

20 Space, Geometry and Aesthetics

Thus, in the development from the first to the third Critique, the relationship between experience and space undergoes a dramatic shift in which external or transcendental understandings of spatial relations become internalised into the powers of the sensing and reflective subject. Space and time are transformed from limited transcendental forms into indeterminate cognitions or judgments that are constituted by the thinking subject: cognizing geometry and space become embodied aesthetic judgments so that disembodied reason is transformed into a perceiving and embodied sense-reason. In addition, in the context of discussions about embodied reasoning, which operates in the production of aesthetic experiences (e.g., the act of drawing or the inhabitation of an architectural space), the division between external, abstract or disembodied space, versus the local and embodied spatial judgments, represents an important shift for those who seek to challenge the perception that geometry and space are always, exclusively, deterministic. Kant’s discussion about the production of embodied spatial and aesthetic judgments, as legitimate forms of reason, therefore offers a valuable route through which to show the connections and differences between the construction of pure space and embodied spatial experience. Kant calls this form of sense-reason ‘reflective’ or ‘indeterminate’ judgment, through which the subject produces aesthetic judgments of the world. Here, the subject’s reflective powers (such as, the powers of sensing, perceiving, imagining, and so on) are productive judgments, underpinning the differences between the Critique of Pure Reason, in which space and geometry are external ideas, and the Critique of Judgment, in which they are embodied expressions of the thinking individual’s internal powers. 8

Reflective judgment Kant describes the power of reflective judgment in §V of the First Introduction to the Critique of Judgment when he refers to its ability to determine a general concept out of a particular example. He gives two descriptions of its powers: first, it is ‘the ability to reflect, in terms of a certain principle, on a given presentation so as to [make] a concept possible’. Second, it is ‘the ability to determine an underlying concept by means of a given empirical presentation’ (CoJ, p. 399). Previously, Kant distinguishes between reflective judgment’s powers of production and those of reason and understanding, when he writes that reason’s powers of judgment generate ideas (i.e., ‘the ability to determine

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the particular through the universal’), and the understanding generates concepts (i.e., ‘the ability to cognize the universal [i.e., rules]’). In contrast, he states that the reflective subject generates judgments through ‘the ability to subsume the particular under the universal’ (CoJ, p. 391). As a result, reflective judgment is defined, not through conceptual frameworks but through the subject’s power to produce a relationship between itself and a particular experience in the world. In contrast to the sensibility’s limited powers in the first Critique, here its powers of reflection enable the subject to independently produce a ‘transcendental schematism’ so that the particular example is subsumed to the general rule, but without recourse to a conceptual schematism (CoJ, pp. 400–1). Furthermore, because Kant emphasises the reflective subject’s ability to generate judgments, the process of this activity becomes primary, highlighting the link between reflective thinking and aesthetic judgments. Reflective judgment therefore represents a special kind of aesthetic activity, a relationship in which the particular is subsumed to the indeterminate, yet embodied, judgment. Kant goes on to explain how the act of reflecting on nature is an ‘artistic’, not merely a ‘schematic’ or logical action. Reflective judgment, he writes, is a ‘subjective relation’ that enables aesthetic judgment to be conceived as a subjective, indeterminate, yet technical (i.e., techne), power derived from nature (CoJ, p. 402). Moreover, in the final paragraphs of this section, Kant tells us that the powers of reflective judgment reveal a ‘purposiveness’ in nature, that is, an autonomous ‘lawfulness’ (CoJ, p. 406). Reflective judgment therefore produces a kind of agency, which is neither ‘merely subjective’, and thereby restricted to the sensibility of the individual, nor exclusively cognitive. In addition, when nature is conceived in this way, its powers of synthetic and ‘purposive’ autonomy are reflected in the subject, and in whom neither is determined by a conceptual or a purely cognitive understanding. Kant therefore defines it as an ‘AESTHETIC judgment of reflection’ (CoJ, p. 409). Thus, the relationship between the subject and the world is brought about in the act of making aesthetic judgments in the First Introduction of the Critique, so that the procedures which might previously have been defined as deterministic knowledge become understood as aesthetic acts. As a result, scientific or technical procedures, such as geometry, are understood as ‘technical’ forms of aesthetic activity, and spatial and geometric judgments are defined in terms of their ability to express the powers of the sensing reflective subject, and his

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or her experiences in the world, not as formal concepts or general ideas. The power to think reflectively is presented as an aesthetic activity which is generated, for example, through the power of the emotions or feelings, and thus Kant argues that aesthetic judgment operates through the dynamic potential for agreement or disagreement between the subject and the world.9 In addition, reflective judgment’s ability to generate relations in the form of an ‘aesthetic of feeling’ is central to its powers. This is especially evident in the productive imagination’s ability to make connections between a judgment and its objects which, although limited in the scope of products or ideas that are available to it (in contrast to the understanding), nevertheless enables the possibility of an artistic or technical (i.e., an aesthetic) understanding of geometry. But this discussion also suggests that reflective judgment represents a particularly spatial form of aesthetic judgment, which is generated in the shift between the concept of geometry in the Critique of Pure Reason and the concept of space in the Critique of Judgment. Furthermore, this transition develops from an abstract concept of space into a kind of spatial judgment that is embodied in the individual subject. Space and geometry are reconfigured into productive internal judgments, contributing to the aesthetic power of reflective thinking in the subject. Geometry, for example, shifts from being an intellectual branch of mathematical reasoning into a series of judgments formed by feelings; for example, in the individual’s struggle to understand nature in terms of magnitudes and the sublime.10 The relationship between the subject, space and geometry therefore constitutes a special kind of aesthetic sensibility in the form of the reflective subject. The work of the ‘productive’ imagination in the Critique of Judgment is central to this shift, in particular, because of the ways in which its powers enable aesthetic and embodied relationships to be constructed between the reflective subject and geometric figures. In turn, the productive imagination’s powers are also central to reflective judgment’s technical activities.

The imagination Two sections in the Critique of Judgment explore the imagination’s activities, in particular, highlighting its role as a power of reflective judgment. In §26, ‘On Estimating the Magnitude of Natural Things, as We Must for the Idea of the Sublime’, Kant explores the role of the

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agitated imagination, which generates the reflective subject in the form of an ‘aggregate’ of feelings of pleasure and displeasure. In a second example, from the Comment to the First Introduction, Kant gives a technical explanation of reflective judgment, showing how the imagination represents a technical analogy of geometric instruments. As a result, the productive imagination is involved in constructing embodied judgments of space and geometry in the process of reflective thinking.11 The imagination’s ability to produce aesthetic judgments is a primary constituent of the Kant’s analysis of the aesthetic of taste in Part One of the Critique. In Book II, A, ‘On the Mathematically Sublime’, Kant also examines how the imagination constitutes a link between the subject and the transcendental sublime, by drawing attention to its attempts to comprehend magnitudes (CoJ, pp. 103–6). This is an aesthetic operation in two ways: first, the imagination performs mathematical or geometric operations in its attempt to cognize the sublime by subsuming it to formal ‘limits’ or magnitudes, thereby generating an internal ‘agitated’ and indeterminate limit in the subject (CoJ, p. 112). Second, Kant examines how the mental agitation that results from the individual’s attempts to understand the sublime, and their inevitable failure to do so, produces a particularly intensive form of feeling in the displeasure which ‘arises from the imagination’s inadequacy’, and yet also in the pleasure which is felt by the affirmation of reason’s purposiveness (CoJ, p. 115). So, as an aesthetic activity, the ‘mental agitation’ which the imagination undergoes is central to the formation of aesthetic judgments. Once again, reflective judgment is derived from a dynamic (and, in the production of the sublime, a conflicting) relationship between reason and an embodied faculty or power. In this particular section of the Critique, the imagination therefore produces aesthetic indeterminacy (CoJ, p. 116). As a result, the imagination is attributed a significant role in the production of reflective judgment because it provides the content of aesthetic experience, unlike the restrictions which are placed on it in the Critique of Pure Reason.12 Instead, Kant tells us that the imagination is autonomous insofar as it is ‘productive and spontaneous (as the originator of chosen forms of possible intuitions)’ (CoJ, p. 91; my emphasis). However, Kant is ultimately cautious about the extent of this freedom, stating that it is restricted to aesthetic judgments, and a harmony is only permissible with ideas because, in order for its products to be fully harmonised under the ‘teleological judgment’, its perceptions and judgments must be brought under the conceptual categories of the understanding.

24 Space, Geometry and Aesthetics

Nevertheless, in the context of highlighting the extent to which reflective thinking synthesises the powers of the sensibility with the powers of reason in the embodied subject, the imagination is important for enabling space and geometry to become embodied (i.e., sensed or felt) intuitions, rather than merely cognitive ideas. This potential is located in the mental agitation that constructs spatial and geometric ideas, highlighting the embodied effort of judging and the attending feelings of pleasure and displeasure. Thus, in the formation of aesthetic judgments, the imagination has a particularly productive nature through which space and geometry, as sense-intuitions, are brought into harmony with pure intuition. In this context, the activities of the imagination represent a form of reflective judgment in the production of concepts of nature as art. It is an expression of the embodied subject’s powers of production, so that geometry becomes expressed, not as an objective or cognitive knowledge but as a technical activity brought about through the powers of the imagination. The imagination therefore modifies the relationship between cognitive limit and sensation into an embodied series of enactments that belong to the ‘freely acting individual’. As a result, geometric boundaries become expressions of embodied thinking, shifting from the objectivity of the mathematical geometric figure into the powers of the reflective subject. In turn, geometric drawing becomes conceived artistically, not as a deterministic representation of concepts. This production of embodied geometric thinking is also evident in an examination of the imagination’s powers in the ‘Comment to the First Introduction’. Here, Kant argues that reflective judgment’s power of subsumption links the productive imagination together with a ‘technical’ geometry in the discipline of critical philosophy, enabling theoretical forms of geometry to be subsumed to practical or applied geometries (CoJ, p. 388). Theory and practice are therefore necessary parts of a Critical philosophy (but as shown here, they are not always determined by underlying concepts), so that Kant enables a harmony to exist between the power of the imagination to generate practical geometries and the power of reason to generate theoretical geometries. In turn, Kant develops this idea into a discussion about the construction of geometric figures in which he argues that particular geometric figures produced by the imagination are ‘special parts’ (i.e., scholia) of an ‘absolute’ geometry, thereby further highlighting how the imagination generates legitimate content in reflective thinking. But it is in the supplementary note to this passage in which the aesthetic power of the

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productive and ‘geometric’ imagination is promoted most strongly, when he writes: This pure and, precisely because of that purity, sublime, science of geometry seems to comprise some of its dignity if it confesses that on its elementary level it needs instruments to construct its concepts, even if only two: compass and ruler. These constructions alone are called geometric, while those of higher geometry are called mechanical, because to construct the concepts of higher geometry we need more complex machines. Yet even when we call compass [Zirkel] and ruler [Lineal] (circinus et regular) instruments, we mean not the actual instruments, which could never produce those figures [circle and (straight) line] with mathematical precision, but only the simplest ways [these figures can] be exhibited by our a priori imagination, [a power] that no instrument can equal. (CoJ, p. 388)13 In this passage, the pure science of geometry is transformed into practical action, which results from the imagination’s ability to produce sensible forms of experience, independently of the understanding. Here, therefore, geometry is transformed from a pure theoretical reason (idea) into a series of actions, instruments or enactments which constitute the different forms of ‘practical’ or applied geometric methods. Importantly, it is the imagination’s powers that provide the conduit for this passage, from the ‘pure’ geometric relations to the technical acts of artistic production, so that each is brought into harmony with the other, and which establishes heterogeneity in geometric thinking. Practical geometry therefore demonstrates the possibility of a theoretical object, but Kant also tells us that ‘All other propositions of performance, with whatever science they may be affiliated, we might call technical rather than practical ... . For they belong to the art of bringing about something that we want to exist [sein]’ (CoJ, pp. 389–90). Thus, geometric demonstrations, diagrams or figures, are understood as different forms of artistic ‘techne’. Rather than being viewed as determinate procedures, they are ‘proposition[s] of performance’, acts or ‘presentation[s] of forms’ (CoJ, p. 388). Once again, the scientific nature of geometry becomes connected to the way in which we judge nature – that is, it is given a value as an analogy to art. So, in contrast to scientific understandings of space and geometry in which the ‘technical’ procedure is separated from the ‘aesthetic’ act, this discussion suggests that Kant’s reflective judgment sustains the link. As a result, discussions about geometric ‘techniques’ may be

26 Space, Geometry and Aesthetics

rethought because they are expressions of reflective thinking, not constructions of ‘ideal’ scientific truths.14 Geometric spaces therefore become embodied in the powers of reasoning, aesthetic judgment and the actions of the imagination, and are transformed into the mental, sensory and bodily actions of the subject. Moreover, these powers represent embodied forms of indeterminate geometric and spatiotemporal intuition. This potential is evident, in particular, in the role attributed to the imagination, which is conceived as an intensive and indeterminate operation. Here, the sensibility is given access to ‘pure’ intuition, not through a formal series of appearances, but through the feelings of pleasure and displeasure. In addition, the imagination is an aspect of the embodied subject that is examined in terms of its powers of enactment. Therefore, geometric procedures, instruments and figures are not objective or cognitive knowledge, but technical and aesthetic enactments constructed by the productive imagination’s powers. The reflective subject is both the aesthetic experience of the sensing individual and the technical construction of geometric figures, thereby challenging the suggestion that geometric figures are always products of deterministic scientific procedures. Rather, in the figure of the reflective subject, aesthetic and technical expressions of geometry are brought together into a reflexive, thinking and perceiving unity. In the final section of the chapter, this revitalised notion of geometric unity is also developed out of the spatiotemporal experiences of geometric drawing and recollection. Kant’s investigations into the productive imagination also continue beyond the third Critique; for example, it is explored in some depth in the later text, the Anthropology from a Pragmatic Point of View (1798). Here, the imagination is productive insofar as it can create images (i.e., perceptions, notions or projections) that are in harmony with a ‘higher level’ of cognition. Thus, the Anthropology studies the structure of sensibility, the senses and the imagination in the individual, in a way that we will see is implied in the imagination’s production of reflective judgment in the Critique of Judgment. In both of these texts, then, the imagination may be considered a ‘productive’ faculty in its own right. A short discussion of the Anthropology highlights some of its capacities in more detail, which will also be useful for considering the role of imagination in the other geometric methods, especially those of Proclus and Spinoza. In the first book of the Anthropology, ‘On the Cognitive Faculty of Self’, Kant tells us that the imagination is a mode of the sensibility or the ‘faculty of intuitive ideas’. In particular, it is the form of the

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sensibility, that is, intuition ‘without the presence of the object’ (APP, §15, p. 40).15 The imagination is therefore able to produce images or notions of space and time that are derived from an internally generated sense, independently of external or empirical objects. The imagination exists in two forms, productive and reproductive; when it is a priori and synthetic, it is productive. Kant writes that it is the ‘faculty of the original representation of the object (exhibito originaria), which consequently precedes experience’ (APP, §28, p. 56). Thus, the sensibility, as a cognition through which forms of appearances might be generated in the Critique of Pure Reason, is developed into a faculty belonging to the embodied subject. In contrast, when the imagination produces images from previously gained ‘empirical perceptions’, Kant tells us its powers are reproductive (APP, §28, p. 56). Although, ultimately, Kant considers the knowledge generated by the imagination to be inadequate (because it produces analogous forms of sense experience, such as space and time), it is not merely confined to an empirical order of objects; for example, when he notes that the imagination makes pictorial sensations in space, such as ‘corporeal forms’. Geometric ‘sense’ derived from the imagination may therefore be a faculty that is of a ‘higher level’ than thinking, when Kant explains: All this is based upon the fact that the imagination, which supplies the content of understanding, that is, content to its concepts for the sake of knowledge, seems to give a reality to its invented notions because of the analogy between them and real perceptions. (APP, §28, p. 58) But, since the imagination is always determined by the ‘rules of sensibility’ (i.e., it does not generate concepts or ideas), it ‘provides the material whose association is achieved without consciousness of the rule, consonant with the understanding but not derived from it’ (APP, §31, p. 67). In addition, the imagination’s capacity for producing images from perception also indicates to its limitations, because, in contrast to the understanding’s ability to produce clear judgments, the imagination cannot produce analytic or ‘pure’ reason; rather its powers are unruly and excessive. As we will see in the Critique of Judgment, however, it is this agitation or excessiveness that generates Kant’s aesthetic. The imagination and its limits construct a link to the transcendental notion of the sublime within Kant’s Critical philosophy, in which the limit between sensible and intelligible realms becomes understood as a

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limit-operation in the imagination and the feelings of pleasure and displeasure that it produces: first, because the formal, mathematical and external limit is subsumed to an internal, ‘agitated’ and indeterminate limit in the efforts of the imagination to cognise the magnitude of the sublime. Second, the reflective subject’s feelings of pleasure and displeasure construct an intensive notion of limit, rather than an exclusive prohibition of the sensibility from the transcendental realm, so that the individual is him/herself constituted by an aesthetic limitlessness or irreducibility generated by the mathematical magnitude of limit and sensation. The imagination modifies this relationship between limit and sensation into an embodied series of enactments that belong to the ‘freely acting individual’, such as the feelings of pleasure and displeasure. As a result, limit is an embodied state and can be said to reflect the shift from the objective reality of the mathematical geometric figure into the subjective reality of the reflective subject. However, for Kant, the transcendental relationship between the subject and geometry is still demarcated by the unknowable sublime so that the ‘limitlessness’ of the subject is registered as an excessive presentation, rather than as the eruption of an immanent power that constitutes the individual. The geometric method and its figure are therefore aspects of the aesthetic powers of the reflective subject insofar as they are enactments of the imagination; however, they also remain attendant to the absolute divisions between the powers of the sensibility and pure reason or God, and results in limiting the subject’s powers versus the limitless powers of the sublime (although Kant’s pre-Critical texts recognise this more complex and heterogeneous notion of geometric enactments, insofar as the geometric figure is distinguished by its external and internal spatial relations16). Later, the chapters explore philosophers who develop strong immanent and genetic continuums of geometric relations between the subject and nature, in which the subject’s irreducible aesthetic and intuitive powers generate truly autonomous modes of sense-reason.17 For the moment, however, Kant’s examinations into aesthetics, geometry and space in the Critique of Pure Reason and the Critique of Judgment are brought together in the aesthetic subject. Plato also examines the enactment of physical and psychic geometric intuition and geometric drawing, which are precursors to Kant’s technical acts of construction in the subject. For Kant, the production of geometric figures is engendered in the technical and aesthetic enactments of the imagination. In the Meno, however, Plato focuses, not on the role of the imagination but on memory or recollection in the production of

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the geometric drawing. Kant’s technical enactment therefore retrieves a kind of memory that Plato reveals in the activities of drawing and recollection.

Drawing a line18 By taking ‘constructive’ aspects of Kant’s aesthetic geometry, a link exists between the embodied notion of limit in the productive imagination, the technical status of the geometric figure, and the act of drawing geometric figures in Plato’s dialogue, the Meno (380 BC). In addition, this dialogue echoes the Phaedo’s proposition that the mathematical diagram is the site for the recollection of memories: ‘If you take a person to a diagram ... then you can show most clearly that learning is recollection.’19 Principally, the Meno has been promoted as an examination of the nature of virtue and whether it is learnt or ‘recollected’. During the course of the dialogue, however, Plato demonstrates the nature of virtue by using geometric examples to explore the Stoic principles of limit and unlimit, and the principles of the one and the many. Socrates, for example, explains that virtue is both particular to each person and exists as a ‘single virtue’ that ‘permeates each of them’, developing the point with the analogy of ‘shape’ (Plato, 1989, 74b–75d, pp. 357–8). He continues, stating that the concept of limit produces a definition of shape; shape is defined as ‘the limit of a solid’ (Plato, 1989, 76a, p. 359). For Plato, limit is therefore equated with an identifiable boundary or end, which supports the notion of the geometric figure as a ‘bounded figure’: shape, he tells us, is limit. But Plato’s identification of shape with limit is problematic, for where limit might produce an ‘intensive’ relationship with infinity (i.e., unlimit or limitlessness) Plato tends to affirm the exactness of formal limits. In the next chapter, by contrast, Proclus emphasises the discursive nature of shape and limit, not the formation of determinate boundaries. Geometric figures, then, are used to explain extended ideas about the qualities of virtue, such as its magnitude and limit. But Plato also considers the act of drawing to constitute a distinct notion of geometric method, which transforms geometry from a mathematical knowledge into a sensible enactment, that is, as intuitive acts. This is presented in two forms: first, the boy’s intuitive recollection of geometry and second, Socrates’ act of drawing figures in the sand: ‘Socrates begins to draw figures in the sand at this feet [...]’ (Plato, 1989, 82b, p. 365). In these lines geometric demonstration is therefore linked to an aesthetic and

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reflective set of judgments in the figures of Socrates and the boy and in the following dialogue between Socrates and Meno, the relationship between geometric intuition and recollection is also evident: Socrates: What do you think, Meno? Has he answered with any opinions that were not his own? Meno: No, they were all his. Socrates: Yet he did not know, as we agreed a few minutes ago. Meno: True. Socrates: But these opinions were somewhere in him, were they not? Meno: Yes. Socrates: So a man who does not know has in himself true opinions on a subject without having knowledge. Meno: It would appear so. Socrates: At present these opinions, being newly aroused, have a dreamlike quality. But if the same questions are put to him on many occasions and in different ways, you can see that in the end he will have a knowledge on the subject as accurate as anybody’s. Meno: Probably. Socrates: This knowledge will not come from teaching but from questioning. He will recover it for himself. Meno: Yes. Socrates: And the spontaneous recovery of knowledge that is in him is recollection, isn’t it? (Plato, 1989, 85c–d, p. 370) A logical process of intuitive geometric reasoning unfolds, beginning with the recognition that the ideas belong to the boy, which are distinct from understanding or pure reason. Rather, these ideas are indistinct, having a ‘dreamlike quality’ and are made clear, not through learning, but by questioning, which Plato calls ‘recollection’. Memory is therefore considered to be inherent in the enactments of the geometric method. So, whilst the dialogue demonstrates a series of mathematical operations, it is also an enactment of geometric method in Socrates’ actions and the boy’s recollections. In addition, Plato reveals how the intuitive basis of knowledge in geometry is embodied in the activities of the boy’s soul, also enabling him to suggest that geometry is a discursive and immaterial procedure; for example, when Socrates says that the nature of the soul is an active and inquisitive form of memory: Thus the soul, since it is immortal and has been born many times, and has seen all things both here and in the other world, has learned

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everything that is. So we need not be surprised if it can recall the knowledge of virtue or anything else which, as we see, it once possessed. All nature is akin, and the soul has learned everything, so that when a man has recalled a single piece of knowledge ... there is no reason why he should not find out all the rest ... for seeking and learning are in fact nothing but recollection. (Plato, 1989, 81b, p. 364) Socrates’ enactment of geometric figures also demonstrates the relationship between geometry and the aesthetic actions of the body. Logical geometric knowledge becomes not only discursive in the operations of the soul and memory but it is also the discursive act of drawing out geometric figures, which demonstrates a shift from geometry as an externally derived and axiomatic order of knowledge to an internal and aesthetic procedure. Thus, we find that Plato’s examination provides an aesthetic and intuitive geometric method from which Kant’s aesthetic subject might be drawn and provides the possibility for Kant’s thinking to be considered as a re-enactment of Plato’s metaphysics, although there are, of course, differences; for example, Kant’s theory of the imagination in the technical, aesthetic geometric act is a counterpoint to Plato’s analysis of intuition. Plato, on the other hand, considers the soul to be the discursive site of memory or intuition through which the geometric method is embodied within the subject. Furthermore, these connections between Kant and Plato raises the potential for examining the ‘origins’ of aesthetic geometries, a question that Husserl also explores, and is discussed in the final chapter.

Summary A line is drawn between the Meno and the Critique of Judgment in which geometry is expressed as an aesthetic ‘act’ of drawing and construction, and indicates an overlooked geometric method and figuration. In particular, we find that a relationship between the ‘pure’ science of geometry and the ‘sensible’ act of drawing geometric figures is manifested in the boy’s intuitive grasp of space, Socrates’ drawings in the sand, and the production of geometric figures in the Critique of Judgment, and which reveals an aesthetic reflective judgment. Between these two encounters we find that the absolute geometric method becomes embodied into the aesthetic powers of the reflective subject, and the geometric method is therefore presented both in the body of the

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reflective subject and in the geometric diagram or figure so that an ‘aesthetic origin’ of geometry is instantiated. Kant’s Critical philosophy suggests a shift from an external to an internal aesthetic geometry in the first and third Critiques. First, Kant’s synthetic a priori judgment, although radical in positing the particular and a priori difference of individual states, remains a problematic notion of difference because it is determined by the ‘external’ limit. In keeping with the classical notion of ‘synthetic’ division in mathematics, Kant sustains the exclusive, formal autonomy of an individual. In addition, his critique of the forms of knowledge and their related faculties (i.e., reason, understanding, intuition and imagination) in the Critique of Pure Reason is a major innovation of a neo-Platonic thought. It is, however, contested by the other methods examined in this discussion, which prioritise the importance of the sensibility (for example, in Leibniz and Spinoza’s examinations of non-cognitive knowledge, such as perception and memory). Although the powers of the imagination are promoted, the imagination is still taken to represent a scientific order, that is, division or limit, rather than in its extended perceptual capacities. The possibility of internal or embodied geometric difference is, however, evident in texts such as, ‘Concerning the Ultimate Ground of the Differentiation of Directions in Space’. But Kant does not sustain this possibility in the Critique of Pure Reason in the development of a ‘pure’ geometric reason and so geometry remains determined by an absolute division between reason and the sensibility. In the Critique of Judgment, however, the reflective subject retrieves a notion of geometric enactment in which its ‘rules of construction’ generate a speculative and discontinuous unity. In particular, the ‘mathematical’ principles of limit are engendered in the faculty of the imagination to form an intensive limit of feeling and sensation. In addition, the imagination enables the reflective judgment to constitute a ‘technical’ or artistic notion of geometric method and figuration, and which is reflected in the intuitive recollection and performed enactments of geometry in Plato’s Meno. Ultimately, however, Kant underestimates the scope of the sense perception and non-cognitive activities of the body in the aesthetic subject. But, in the following chapters, each philosopher generates a method that is both geometric and sensory. Kant’s Critical philosophy therefore constitutes a key geometric and aesthetic encounter or re-enactment. The first encounter is in the Critique of Pure Reason’s examination of intuition. In its secondary form it is re-enacted in the Critique of Judgment through the powers of the reflective

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subject, in particular, through the technical powers of drawing and construction that the imagination provides. Finally, this geometric aesthetic is itself retrieved from metaphysical philosophy in the figures of Socrates and the slave-boy in Plato’s Meno, providing an additional ‘enactment’ of the aesthetic that re-engages Kant’s project with earlier geometric methods. Two figures of a geometric memory – intuition and recollection – are transformed into a technical form of enactment in Kant’s reflective judgment. In the next chapter Proclus’ Classical aesthetic geometry, which is derived from its Platonic and Pythagorean sources, demonstrates a particularly discursive form of geometric enactment.

2 Folding-Unfolding

In Kant’s theory of aesthetic judgment, the reflective subject’s aesthetic activities are embodied in the geometric figure. In Proclus’ text, A Commentary on the First Book of Euclid’s Elements, the aesthetic geometric method and its figures are derived from Stoic ideas of divine and ‘intelligible’ powers, but are not generated by embodied faculties. However, Proclus’ (AD 410–485) contribution to the study is valuable because it demonstrates the extent to which these Stoic concepts generate a relationship between geometry and aesthetics that is genetic, serial and continuous throughout the divine and sensible realms. A continuous or ‘discursive’ relationship between the pure figures of the Gods and the sensible figures (or natural forms) is therefore established. In addition, Proclus’ text provides an early precursor in the shift from external to internal geometric methods because it shows how external and axiomatic mathematical ‘elements’ are reconfigured into a series of singular and immanent figures. Finally, a series of key metaphysical principles are present in this text, which recur throughout the chapters: in particular, synthetic and analytic geometric figures, the operations of the imagination, the soul and the notions of ‘limit’ and ‘unlimit’. Thus, Proclus’ Classical work is an important counterpoint to the view that Euclid’s Elements (c.300 BC) is exclusively derived from rational and scientific principles; instead, his Commentary demonstrates that Euclid’s geometric text is also an aesthetic form of geometry. In the Commentary on Euclid’s Elements, Proclus generates a special unfolding and folding procedure by the discursive movement between the divine and sensible powers of the soul, the understanding and the imagination, which produces a series of intermediate figures.1 Out of this discussion, I reveal Proclus’ explicit examination of a procedure of ‘unfolding’ and its implied figure of the ‘fold’, by exploring the structure 34

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of the text and its geometric ‘elements’, which are the constituents of an early form of aesthetic geometry. As a result, a more complex philosophical understanding of a neo-Platonic geometric method is generated in which the geometric method and its figures construct a double movement of folding and unfolding through a series of finite and infinite figurations. However, before focusing on the text in detail, a brief account of the context in which Proclus was writing in fifth-century Greece shows that mathematics, and its derivative procedures (including geometry), are informed by neo-Platonic and Pythagorean metaphysics. In this respect, Proclus’ Commentary situates Euclid’s technical (i.e., a mathematical) analysis of geometry firmly in the context of aesthetic and philosophical ideas. For Proclus, the value of the Elements is twofold: first, its subject matter demonstrates the power of ‘the cosmic figures’ in Plato’s Timaeus, which makes it an ‘elementary exposition’ of metaphysical import. Second, its explication of these Platonic origins represents ‘a method of perfecting’ the geometric method in both its scientific and metaphysical potentials. The Elements therefore represents a paradigm of philosophical and mathematical inquiry, both in its technical examination of geometry and in the philosophical ideas through which its ‘figures’ are produced (CEE, p. 58).2 However, Proclus does not completely transform Euclid from a mathematician into a metaphysician. Instead, by emphasising the relationship between Euclid’s mathematical propositions and the Platonic and Pythagorean principles from which it is generated, he explains an overlooked philosophical context through which he reconsiders the formation of geometry in the Elements, constituting an important text through which the philosophical value of geometric principles after Plato is re-evaluated.3 Beginning with an examination of the statement that mathematics is ‘imaginative and discursive thinking’ the Commentary represents a striking ‘anticipation’ of Kant’s geometric method and imagination in the third Critique. Both Proclus and Kant assign the imagination to a position of mediation between the intelligible and sensible realms, and each attributes the productive nature of imagination to its powers of division that link it to the Stoic notions of limit and unlimit; that is, not exclusively to its powers of imitating insensible figures. Within the Commentary, the imagination’s discursive powers therefore constitute a form of ‘unfolding’ and ‘folding’ geometric movement through a series of metaphysical orders, and through which the divine, insensible principle of unlimit is immanent in particular, sensible limits (i.e., geometric figures are not produced by recourse to a theory of imitation of ideal forms).4

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A more discursive relationship between the transcendental realms of nous (intellect or intuition) and diavoia (understanding) also comes to the fore, because the activities of folding and unfolding enable both the notion of limit and unlimit to be present simultaneously. Moreover, the ‘genetic’ potential in geometric figures becomes dramatically enhanced in the shift from a principle of mathematical certainty that is a finite, ‘bounded’ and ‘contained’ synthetic identity, into a synthetic and infinite unity. The relationship between geometric figures and the notion of a singular ‘unity’ is therefore constituted, not through the emphasis on different classifications of finite identities but as a result of the discursive relationship between different figures. Thus, we will see that Proclus’ affirmation of Pythagorean principles generates synthetic principles of construction, within a continuity of different figures. Proclus’ theory of geometry is explicit about its discursive form of an aesthetic unfolding; however, within his discussion the notion of folding remains implicit (in contrast to the explicit fold that Leibniz generates in Chapter 4). Hence, not only is the figure of the fold present but it is also implicit; for example, in Proclus’ emphasis on the discursive movement of the understanding, and in its implicit double, the relationship between the imagination and the soul. In this context, the imagination (in the form of recollection) is an implicit ‘enfolding’ or countermovement to the explicit discursive ‘unfolding’ of the understanding.5 In addition, the procedure and figure of the fold/unfold remain firmly linked to the divine principles of discursivity that originate from the ideas of the ‘One’ and the ‘Many’. As a result, the scope of the imagination to produce mathematical objects remains an intellectual ‘operation’, not a power that is generated by a fully embodied and thinking subject. Thus, unlike the autonomous and embodied powers of Spinoza’s and Leibniz’s ‘infinite substance’, Proclus’ Pythagorean ideas do not enable speculative thinking to be generated from an autonomous order of sensible, living things, because geometry, substance and the subject are determined by an ideal order of divine elements. In addition, Proclus only examines the nature of matter insofar as it is a derivative of the higher levels of thought. Although the sensible realm is a positive product of the imagination’s ability to unfold the intellect, understanding and the soul, sense opinion and matter are considered to be ‘contaminated’ by it, and consequently, intellect embodied in matter is always considered less significant than divine ideas. Within these metaphysical structures of unfolding and folding, the axiomatic organisation of the text shows how Proclus’ analysis is deeply embedded in Pythagorean principles of serial progression and notions

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of divisibility and multiplicity, for example, in the principles of the One (i.e., limit and divisibility) and the Many (i.e., unlimit, indivisibility and multiplicity), which represent the highest, unknowable and transcendental realities, but which are also immanent in the lower realms and particular entities. The geometric elements – that is, the axiom, postulate, proposition, problem, theorem, hypothesis and definition – are both transcendent and sensible operations, rather than merely sensible ‘abstractions’ of higher forms of idea. As a result, the geometric figures of folds and unfolds manifest these relationships, and provide evidence of Proclus’ examination of the Elements in which geometry becomes a special site of mediation between the intelligible and the sensible realms without recourse to the ‘Divided Line’ that separates the intellect from the senses.6 In addition, a series of relations are proposed in the ‘common notion’ of the figure, which links the highest and the lowest realms. Therefore, diaonetic (or discursive) thinking demonstrates the potential for the intellect to move between ideal and particular ‘figures’ by folding and unfolding the ‘simple’ axiom or point into complex and ‘combined’ geometric figures. In addition, this discursivity is an activity of both the understanding and the irreducible soul’s activities, for example, when the imagination produces ‘enmattered images’, representing an important anticipation of Kant’s theory of the imagination in the third Critique and the aesthetic geometries of Spinoza, Leibniz, Bergson and Husserl.

Discursive geometry Proclus follows the Platonic belief that mathematics is discursive. He writes that mathematics’ methods involve ‘diaonetic and imaginative thinking’ and, in this respect, it represents a demonstration of the faculty of understanding (CEE, p. 15). Mathematics’ powers situate it at a special level in the order of knowledge, in which the intelligible and sensory worlds are brought together through the act of discursive thought (diavoia) to form a distinct kind of knowledge. But it is the aesthetic value of these powers that Proclus emphasises most strongly to develop a method in which the relations between its objects are as important as its forms. In the following sections an examination of these powers – that is, the nous, understanding, soul and the imagination – reveal the extent to which Proclus’ interpretation of geometry constitutes an aesthetic series of unfolding and folding movements between these metaphysical operations.

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The discursive nature of mathematics is expressed at the beginning of the Commentary. Confirming Plato’s classification of mathematical knowledge, Proclus emphasises the discursive powers of mathematical demonstration, telling us that, by moving from one fact to another in the construction of their respective objects, geometry and arithmetic are deductive procedures that generate clear and precise descriptions of the world, mediating between the realms of pure ‘intelligence’ (nous) and the imperfect sense-perception or ‘opinions’ (doxa): Mathematical being necessarily belongs neither among the first nor among the last and least simple of the kinds of being, but occupies the middle ground between the partless realities – simple, incomposite, and indivisible – and divisible things are characterised by every variety of composition and differentiation . . . . But the discursiveness of [the mathematical] procedure, its dealing with its subjects as extended, and its setting up of different prior principles for different objects – these give to mathematical being a rank below that indivisible nature that is completely grounded in itself. (CEE, p. 3; my emphasis) Geometric procedures and their figures are attributed with a special kind of autonomy which mediates between the unknowable and extended realms and (as a result of this intermediary nature), geometry enables the powers of understanding and imagination to be harmonised with the self-determined activities of the soul. Geometry therefore lies between the imperfection of sensible, empirical entities and the perfection of insensible, immaterial forms when Proclus writes: ‘the intermediate status of mathematical genera and species’ lies ‘between absolutely indivisible realities and the divisible things that come to be in the world of matter’ (CEE, p. 4).7 Thus, a neo-Platonic order is confirmed in which there are four orders of reality that move in descending order, as follows: 1. 2. 3. 4.

the partless unity of the One (union); the Ideal Forms of Being (nous); the logoi of Mathematics (diavoia); the sensible entities of Becoming(sens).8

But we are also able to observe the neo-Platonic value attributed to the nous in this schema. For Proclus, the nous represents the ideal, insensible forms from which all sensible ideas are generated and all perceptions are images of these ‘first patterns of all things’ (CEE, p. 13). Its content is indivisible and non-discursive so that it is an ‘all-at-once-grasping of totality’ (CEE, p. xx). But it also provides the soul with its content and is,

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in this sense, an ‘external’ source of ideas for the soul. (As we will see in Chapter 5, such a definition also appears to foreground Bergson’s concept of intuition in which the soul is conceived as a ‘psychic’ activity that ‘grasps’ reality as an intuition and as a discursive activity. But Bergson’s wish to distance himself from Platonic metaphysics, especially the problematic status attributed to perception and matter also represents a crucial difference between his notion of ‘totality’ that is grounded in a ‘superior empiricism’ and Proclus’ notion of a divine totality.) Also, because any mathematical procedure lies between the simple indivisible forms and divisible nature, Proclus considers it to be commensurate with the understanding: ‘a faculty higher in rank than opinion, but inferior to intellect’ (CEE, p. 10). However, he distinguishes between the nous and the understanding on the basis that the understanding’s discursive and deductive activities unravel the unintelligible, indivisible and pure intellect into intelligible and divisible forms, in a manner of ‘unfolding’: Though second in rank to intellect and the highest knowledge, understanding is more perfect, more exact and purer than opinion. For it traverses and unfolds the measureless content of Nous by making articulate its concentrated intellectual insight, and then gathers together again the things it has distinguished and refers them back to Nous. (CEE, p. 3) So the understanding makes intelligible the unintelligible ‘mathematical ideas’ in the nous, and also those of the soul, to generate the ‘substantial and self-moving’ varieties of mathematics, such as geometry and arithmetic. As the explicator of pure intellect it therefore mediates between the nous’ ‘originating principles’ of ‘partless ideas’ and its own products, the sensible mathematical bodies (CEE, p. 15). Furthermore, this constant explicatory or discursive movement distinguishes mathematical understanding from the nous, which is non-discursive, unified, ideal and constant. But Proclus also considers its unfolding movement to be analogous to a kind of life-giving activity, that is, a genetic production of ideas. (The relationship between notions of life and ‘activity’ is central to this discussion; it underpins the power of each geometric method and figure.) As a result, the discursivity of mathematics, the understanding and the soul represent a particularly creative series of activities:9 By contrast mathematics, though beginning with reminders from the outside world, ends with the ideas that it has within; it is awakened

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to activity by lower realities, but its destination is the higher being of forms. Its activity is not motionless, like that of the intellect, but because its motion is not change of place or quality, as is that of the sense, but a life-giving activity, it unfolds and traverses the immaterial cosmos of ideas, now moving from first principles to conclusions, now proceeding in the opposite direction, now advancing from what it already knows to what it seeks to know, and again referring its results back to the principles that are prior in knowledge. Moreover, it is not, like Nous, above inquiry because filled from itself, nor is it satisfied, like perception, with matters other than itself; rather it advances through inquiry to discovery and moves from imperfection to perfection. (CEE, p. 16; my emphasis) In addition, this activity connects Proclus back to Plato’s discursive examination of geometric questions in the Meno, because in each, mathematics is in a constant, double movement of inquiry, operating between the higher activities of the soul and the lower levels of sensible ideas: first, in the form of the understanding, mathematics’ deductive powers unfold (i.e., reproduce) the indivisible first principles as the extended and sensible forms. Second, in the form of the soul, mathematics is a more ‘creative’ and reflective form of production in which the sensible forms are brought together (i.e., ‘enfolded’) under the general form of the ‘manifold’ of ideas. Proclus continues this discussion by emphasising the soul’s dynamic movement through its differentiating powers of production: And its powers are manifestly of two sorts. Some develop its principles to plurality and open up the multiform paths of speculation, while others assemble the results of these many excursions and refer them back to their native hypotheses . . . . Consequently it is only natural, I think, that the cognitive powers operating in the general science that deals with these objects should appear as twofold, some aiming at the unification and collection of the manifold for us, others at dividing the simple into the diverse, the more general into the particular, and the primary ideas into secondary and remoter consequences of the principles. (CEE, p. 16; my emphasis) Extending in two directions, mathematics’ procedures unfold from the purest immaterial idea downwards to the natural and sensory world of matter and, by implication, in an ‘enfolding’ movement, upwards from its empirical applications; for example, in mechanical or optical

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applications, or in the ‘unitary and immaterial insights’ that comprise its universality (CEE, p. 17). Forms of applied mathematics are therefore conceived as speculative activities, which are not merely derived from sense-perception. Instead, the content of the nous and soul, which are unfolded by the understanding, constitute immaterial ideas and affirm the a priori status of mathematical knowledge. (This potential is also evident in Descartes’ investigations into rational and analytic definitions of the a priori which inform Spinoza and Leibniz’s geometric methods.) Proclus therefore disagrees with the claim that ‘mathematical forms’ are abstractions ‘from material things’ or ‘common’ notions, which are derived from sensible entities. Instead, he considers the soul and nous to be the origins of mathematical ideas (CEE, p. 13). Nevertheless, mathematics is not completely divorced from the sensible realm, because Proclus also strongly affirms the powers of the imagination through which mathematics is connected to the production of images or projections in the sensible realm.

Soul The relationships between the intellectual powers, in particular, the relationship between the soul and the imagination, are central to the formation of this aesthetic geometry. Proclus gives a detailed outline of the soul in the first Prologue. Autonomous and ‘self moving’, the soul is modelled upon Plato’s notion of the ‘world soul’ in the Timaeus, where the soul and mathematics are brought together in a series of divine mathematical figures. Proclus writes: Plato constructs the soul out of all the mathematical forms, divides her according to numbers, binds her together with proportions and harmonious ratios, deposits in her the primal principles of figure, the straight line and the circle, and sets the circles in her moving in intelligent fashion. All mathematicals are thus present in the soul from the first. (CEE, p. 14)10 So the soul’s indivisibility closely resembles the limitless nature of the nous. But, as shown above, its importance lies particularly in its value as a ‘higher’ realm of discursivity from which geometric objects and figures are unfolded by the understanding. In a revealing passage on the origin of ideas in the soul, Proclus explains that the soul draws her concepts both from herself and from Nous, that she is herself the company of the forms, which received their

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constitution from the intelligible patterns but enter spontaneously upon the stage of being. The soul therefore was never a writingtablet bare of inscriptions; she is a tablet that has always been inscribed and is always writing itself and being written on by Nous. For soul is also Nous, unfolding herself by virtue of the Nous that presides over her, and having become its likeness and external replica. Consequently if Nous is everything after the fashion of intellect, so is soul everything after the fashion of soul; if Nous is exemplar, soul is copy; if Nous is everything, soul is everything discursively. (CEE, p. 14; my emphasis) The soul’s discursivity therefore attributes the status of ‘recollection’ or memory to mathematics, a countermovement to the ‘forward’ movement of deduction through which the understanding operates. As a result, the soul’s expression of the content of the nous occurs in multiple forms, because it is both the activity of unfolding that progresses from the nous to the soul, as well as the progression from the soul to the understanding. In addition, the analogy of the soul, as a continual site of inscription, also attributes discursivity to an aesthetic image and, in turn, this recalls the geometric encounter in the Meno, in particular, the boy’s act of recollecting geometric figures and Socrates’ ‘inscription’ of figures in the sand. The soul’s unlimited powers of discursivity and recollection are therefore activities of construction (i.e., recollection and inscription) so that mathematical procedures and their figures are themselves attributed with the potential for limitless acts of unfolding and folding. Furthermore, the powers of recollection and writing, which are brought together in the discursive act of inscription, mean that geometry is once again attributed with the technical (i.e., aesthetic) expression of nature and the soul. Moreover, these powers are amplified by the role of the productive imagination in diaonetic thinking. In addition, Proclus emphasises the soul’s productive nature when he calls it the generatrix through which the discursive element is produced: We must therefore posit the soul as the generatrix of mathematical forms and ideas. And if we say that the soul produces them by having their patterns in her own essence and that these offspring are the projections of forms previously existing in her, we shall be in agreement with Plato and shall have found the truth with regard to mathematical being. (CEE, p. 11)

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The geometric figure is therefore both a ‘projection’ and an ‘offspring’ of the soul, attributing two distinct ideas of production to the operations of the soul. On the one hand, mathematical forms constitute imitations or images of the ‘original patterns’, and on the other hand, they are considered to be ‘genetic’, because they embody the Pythagorean notions of continuity and the plenitude of forms. The nature of the geometric figure, as a projection of the imagination, is significant. But in this particular passage, the soul’s discursivity, is also shown to be analogous to biological associations of ‘life’ and reproduction, thereby augmenting an important sense-based potential in mathematics’ discursive powers. Thus, in contrast to the understanding, which unfolds the ideas given to it by the nous, the soul generates its own ideas as well as receiving them ‘from elsewhere’ (CEE, pp. 13–14). In addition, its origination from Plato’s ‘world soul’, which expresses the nature or the cosmos, is also made evident in Proclus’ attention to its ‘life-giving’ qualities. As a result, the notion of soul carries within it the idea of ‘plenitude’ that is also important to each of the geometric methods in which the soul is discussed, in particular, for Spinoza and Leibniz’s concepts of substance. Here, however, the soul that Proclus promotes is primarily considered to be an elevated theological state and is not explicitly embodied as the soul of the thinking subject; rather, for Proclus, mathematical learning provides a route through which individuals can strive to reach the higher realms of existence. Proclus’ confirmation of the hierarchy in the Platonic order is central to his argument that discursive learning and recollection are directed towards the ‘discovery of pure nous’ and the possibility of achieving ‘the blessed life’ (CEE, p. 38). Discursivity, learning and recollection therefore have an ethical significance which is brought about in acts of geometric thinking, and it is in the activities of the soul that the immaterial, intelligible ideas of the nous are unfolded and become most closely associated with a ‘psychic’ power. However, as Mueller notes in his ‘Foreword’ to the Commentary, Proclus does not seek to explain this activity in terms of a fully embodied ‘psychic’ operation (in the manner that is developed in Spinoza’s Ethics). But the text does suggest a ‘transitional psychic activity’ in two forms: first in mathematics’ powers of discursivity, and second, in the soul’s powers of recollection (CEE: p. xx). Proclus’ geometric procedure of unfolding therefore has some similarity to Spinoza’s Ethics, insofar as it is directed towards a ‘theological’ pursuit of knowledge. In contrast to Proclus, however, Spinoza will develop this ‘psychic’ movement as an embodied knowledge in the form of a ‘passage’ from the emotions of the individual through to a divine love of God.

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Imagination Unlike Kant’s aesthetic figure, the ‘reflective subject’, Proclus does not consider the imagination, space or time, to be fully embodied in the thinking subject. For Proclus, the imagination represents a productive faculty of thinking which has the power to embody geometric figures. Furthermore, by implication, the sense-perceptions of space and time are always implicitly located in its empirical images; for example, in the operations of mechanics, optics or astronomy, space and time are implicit orders of perception. As Morrow also notes, the imagination is the key innovation in Proclus’ adaptation of Platonic theory that ‘anticipates’ Kant’s schematism of the imagination and understanding (CEE, p. lix). It is the second kind of dianoetic thinking, which generates the unfolding geometric relationship between the nous, soul and understanding. In addition, its unique contribution is derived from its relationship to material forms of geometric figures, so that its activities are always determined by the excessive or ‘boundless’ unlimit of extended matter, rather than by the orderly nature of divine limit. The imagination is therefore an embodied faculty of cognition, providing an original connection between its position ‘in the body’ and the production of images from the external ‘undivided centre of life’. Proclus explains: By contrast the imagination, occupying the central position in the scale of knowing, is moved by itself to put forth what it knows, but because it is not outside the body, when it draws its objects out of the undivided center of life, it expresses them in the medium of division, extension and figure. (CEE, p. 42; my emphasis) Moreover, Proclus rejects Aristotle’s classification of the imagination as a form of ‘passive’ nous. Instead, he promotes the imagination’s power in its production of the multiplicity of extended beings that comprise mathematics, geometry, ‘nature’ and life. It is granted a special relationship with extension, in which its powers of production are underpinned by division and indivisibility to give it a unique role in unfolding the geometric method from insensible to sensible form. Proclus writes: ‘For imagination, both by virtue of its formative activity and because it has existence with and in the body, always produces individual pictures that have divisible extension and shape, and everything that it knows has this kind of existence’ (CEE, p. 41; my emphasis). Thus, although the imagination is restricted insofar as it produces images or beings which

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are extended ideas, rather than immaterial ideas, it is still a necessary faculty of cognition towards fulfilling the dianoetic potential of geometry; for example, the imagination gives shape to thought when it provides the intelligible matter of a specific geometric figure. It is therefore a kind of mathematical embodiment, which is inherently linked to the potential for divisibility and indivisibility. Derived from the Pythagorean notion of unlimit, these Stoic ideas provide the imagination with an irreducible power of geometric figuration, further underscoring its importance for mathematical thinking.11 But Proclus also reminds us that the imagination does not produce pure ideas of the intellect (as do the understanding and soul). Instead, its images or projections are always secondary to the ideas produced in the understanding and those ‘in nature’ because ‘the idea in the understanding is undivided, so also is the idea in nature’. So, for example, an ideal circle of the understanding is undivided, without magnitude or extension, yet ‘the circle in imagination is divisible, formed, extended – not one only, but one and many, and not a form only, but a form in instances’ (CEE, p. 43). However, Proclus also adds that the ‘abstract image’ of the circle in the imagination provides a more adequate abstraction of the ‘sensible’ circle in nature, which ‘is inferior in precision, infected with straightness and falls short of the purity of immaterial circles’ (CEE, p. 43). Hence, the imagination is an important mathematical faculty because its ability to abstract images or projections from sense-objects constitutes a step towards the divine ideas of pure intellect (despite its inability to generate ideas in the same manner as the soul and nous). Furthermore, these abstractions of the sensory world are evidence of its original powers of production, because each figure or projection is determined by its relationship to an extended body and, by implication, the thinking subject. So the act of shaping matter produces extended figures in the imagination when Proclus writes: ‘it is in imagination that the constructions, sectionings, superpositions, comparisons, additions, and subtractions take place, whereas the contents of our understanding all stand fixed without any generation or change’ (CEE, p. 64). In addition, the imagination’s production of ‘intelligible matter’ is the basis of diversity in nature, life and the sensory world; for example, it provides a ‘common element’ between different magnitudes of a figure or number, such as, a series of concentric circles, which are connected by the ‘immaterial substratum’ of the image of the circle, yet each one is distinguished by having a different magnitude (CEE, pp. 42–3). Once again, the relationship between the acts of shaping and nature therefore establishes an aesthetic

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relationship, which anticipates Kant’s examination of the ‘technical’ relationship between nature and art in the Critique of Judgment. In addition, the imagination has the capacity to move in two directions; on the one hand, it represents an ‘unfolding’ movement that generates particular, extended images or projections in the sensory world, derived from the immaterial ideas of the nous. On the other hand, it moves in the opposite direction, which is an ‘enfolding’ that produces general abstractions of the sense-world. Proclus explains the relationship between the imagination and the understanding as follows: For the understanding contains the ideas but, being unable to see them when they are wrapped up, unfolds and exposes them and presents them to the imagination sitting in the vestibule; and in the imagination, or with its aid, it explicates its knowledge of them, happy in their separation from sensible things and finding in the matter of imagination a medium apt for receiving its forms. (CEE, p. 44) Insensible and sensible matter are therefore embedded in these activities, so that Proclus continues to examine the relationship between the imagination and the understanding in the figure of a ‘screen’ onto which the understanding ‘projects’ its ideas. In this context, the imagination is not just a ‘passive nous’ but it is also productive, insofar as it provides the means through which the ‘partless’ ideas of the understanding are ‘inscribed’ into extended forms. Projections or diagrams therefore establish an active connection between the two faculties (CEE, p. 45).12 However, projections are also generated by the imagination and soul. In this instance, the imagination’s images or pictures are passive because they are inscriptions of the soul’s activity; yet they also demonstrate the imagination’s powers of construction. In addition, in the analogy of the imagination as a screen, Proclus suggests that the reception of figures from the soul is a more ‘reflective’ and inward kind of movement, which also generates a correlation between the imagination’s operations with the soul’s operations: Therefore, just as nature stands creatively above the visible figures, so the soul, exercising her capacity to know, projects on the imagination, as a mirror, the ideas of the figures; and the imagination, receiving in pictorial form these impressions of the ideas within the soul, by their means affords the soul an opportunity to turn inward from the pictures and attend to herself. (CEE, p. 113)

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The analogy of the mirror and the figure in which the soul looks ‘outside herself’ at the figures of the imagination continues in Proclus’ discussion of how the soul is ‘struck by the beauty’ of the reflections of herself; although the soul ultimately rejects these reflections in favour of its ‘own beauty’. Nevertheless, the imagination and the soul are brought together in an act of recollection (i.e., an enfolding) through which the soul distinguishes its original figures from the secondary figures of the imagination. In addition, Proclus attributes autonomous movement to the soul and the imagination when he writes that they are both ‘self-moving’. So, although the soul’s abilities to recollect or reflect are ultimately more active than the projection and reception of figures onto the screen of the imagination, this is a significant discussion in the Commentary, because it is highly prescient of the harmony between the imagination and the ‘reflective subject’ in Kant’s third Critique. The following chapter also explores Spinoza’s analysis of the imagination’s powers, in conjunction with the emotions and embodied ‘common notions’, which are generated in the journey towards a state of ‘blessedness’ in the subject. For Spinoza, however, these imagined and projected images are also considered ‘adequate’ figures or ‘common notions’. Leibniz’s exploration of ‘fictional’ or ‘approximate’ figures that constitute a ‘sufficient reason’ also resonates with Proclus’ affirmation of the imagination’s role in the production of extended mathematical ideas. In addition, Proclus’ emphasis on the activities of the imagination reminds us of Kant’s attention to the dynamic nature of the productive imagination in the Critique of Judgment, when he tells us that it is directed towards an activity of life because, like the soul, its movements are self-generated, in contrast to the ‘contents’ of the understanding which are, on their own, static and constant. Thus, whilst the Commentary anticipates Kant’s construction of the synthetic a priori in which mathematics mediates between the intelligible and the sensible realms, a hierarchy between the two levels is more strongly demarcated in Proclus’ thinking, because of the underlying movement towards the divine and intellectual realm. In contrast, Kant’s construction of the sensible realities of space and time in the Critique of Pure Reason provides a more developed analysis of transcendental knowledge, even if it is accessible, ultimately, only through appearances. In addition, Kant’s discussion of the imagination’s role in the production of pleasure and displeasure in the Critique of Judgment provides a more developed psychological description of the ‘double’ movement of unfolding and folding in Proclus’ text, and

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therefore reveals the importance of the subject for an aesthetic analysis of the imagination, space and time, which Proclus does not recognise. But, as the next two sections show, despite the lack of a fully embodied psychical movement in Proclus’ discussion, the basis of his thinking in the Pythagorean notions of limit and unlimit, provides an especially powerful genetic and aesthetic connection between the sensible and insensible realms.

Limit and unlimit The imagination’s powers are both discursive and reflective, yet Proclus’ mathematics is profoundly Pythagorean in nature, because the operations of limit and unlimit generate the aesthetic nature of the geometric method by establishing it in an irreducible and immanent continuum of unfolding and enfolding movement. Limit and unlimit therefore amplify neo-Platonic interpretations of metaphysics and science. Under Proclus’ guidance, Euclid’s geometry therefore demonstrates the Pythagorean metaphysics of limit and unlimit, in the form of dianoetic thinking. As a science, geometry examines and expounds ‘only that indivisible nature which is appropriate to his first principles’, whilst as philosophy it examines ‘everything that is in anyway divisible as well as the nature of the indivisibles that are sovereign over them’ (CEE, p. 76). As a result, geometry’s scientific discursivity is amplified by the metaphysical powers of the limit and unlimit operations. In addition, the discursivity of the Commentary is itself comprised of disruptive ‘interruptions’ (CEE, p. li) of Pythagorean argument (and these excessive interruptions or disruptive ‘asides’ will also be an important feature of the ‘scholia’ in Spinoza’s Ethics). Geometric discursivity is therefore determined by the Pythagorean principles of the One, the Many, the Limit, Unlimit and Mixture, an ontology in which the divine principles are manifested in the sensible world, so that figures, such as ‘Number’ and sensible beings, are constituted by a divine irreducibility (Guthrie, 1987, p. 21). In contrast, Platonic metaphysics proposes that the sensible world is related to the indivisible realm through the mediation of another level – that is, representation or form – so that the ‘contemplation’ of the immanent divine order can be known only through appearances or ‘phenomena’: It was different for Plato. He adopts the Pythagorean notion that number is the principle of order in the cosmos and life, but number

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as such to him is not yet a theion [divinity]. It points at a purely intelligible Number which is a ‘Form’ [eidos] – no immanent principle of order within the objects, but a transcendent Example. This is the basic difference between the Pythagorean doctrine of number and Plato’s Theory of Forms. Plato’s philosophy is a metaphysic of the transcendent; the Pythagorean philosophy is a metaphysic of the immanent order. (de Vogel, 1996, p. 35) Mathematics in Pythagorean thought therefore constitutes an especially direct demonstration of the divine laws; for example, in the operation of ratio when the regularity of limit can be expressed in the ‘even’ ratio of 2n/n (such as, 2/1, 4/2, 6/3 […]), and the irregularity of unlimit is expressed in the ‘uneven’ ratio of n11/n (such as, 2/1, 3/2, 4/3 […]) (CEE, pp. xxv–xxvi). Proclus’ promotion of limit and unlimit as ratios also confirms his neo-Platonic inheritance of the Timaeus’ theory of the divine, in which the world’s soul is expressed as a series of ratios (Plato, 1989, 36a–36b, pp. 1165–6). In addition, it reminds us of the mathematical classification that Plato constructs in the Republic which is determined by Pythagorean principles, producing divisions of discrete or continuous things, which are also controlled by either multiplicity (plethos) or magnitude (megethos). As a result, any classification is determined by unlimit; for example, multiplicity cannot be limited to a maximum number (poson) and magnitude cannot be limited to a minimum quantity (pelican; CEE, p. xxvii). Moreover, Proclus underlines the importance of unlimit in the Pythagorean order by distinguishing geometry from arithmetic because magnitude provides the grounds for its ‘irrationality’ and ‘irreducibility’ (CEE, p. 5). (These discussions of ratio, magnitude and multiplicity are also important precursors for Leibniz’s investigations into the principles of an analytic geometric method.) In chapter II of the first Prologue, Proclus focuses particularly on the importance of limit and unlimit for generating discursive drives which produce a totality of realities. Derived from ‘the indescribable and utterly incomprehensible causation of the One’, they are ‘all-pervading principles that generate everything from themselves’ (CEE, p. 4). So, although Proclus’ concept of the single, original One is consistent with the problems of a ‘formless’ and unknowable ‘infinity’, his argument insists upon the discursive powers of the Pythagorean principles in the production of immaterial and material realities. The inexpressibility of the One is omitted in favour of the discursivity that mathematics produces, for example, in the discursivity between the geometric axioms

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and elements, which demonstrate the original divine irreducibility of the One (CEE, p. xviii). Limit and unlimit therefore constitute a causal progression of order. Principles ‘proceed’ from them and ‘go forth’ into the divisions of the nous, soul, understanding and mathematics; for example, the stable existence of the ideal forms is determined by limit, yet their ‘variety, generative fertility, and their divine otherness and progression’ are drawn from unlimit. Mathematical objects are the limit and unlimit’s ‘offspring’, demonstrating their ‘co-operation’ with each other and representing intermediary states that ‘proceed’ towards infinity as a series of identifiable ratios under the control of limit (CEE, p. 5). But of all these ‘intelligibles’ or ‘higher realities’, unlimit is ‘the first creative cause and generative power of all things’ (CEE, p. 73).13 Thus, Proclus constructs a genetic ‘plenitude’ or development in diaonetic thinking, and an emphasis is placed on the dynamic discursivity which grounds mathematics, bringing it into agreement with Plato’s argument in the Republic that mathematics is the highest form of dialectic methods (Plato Republic: 543e, cited in CEE, p. 35). A geometric unfolding of the Platonic order (from the One, through to the nous, soul, understanding and finally in sensible things) is therefore determined by the constitution of identity as limit. However, the plenitude of the Pythagorean unlimit that is immanent in all realms also prevents a divide being instantiated between the transcendental and the sensible realms that Plato’s Divided Line constructs in the Republic. In addition, magnitude provides an important power in limit and unlimit for generating the irreducible and extended geometric continuum, when Proclus tells us that magnitudes constitute ‘infinite’ divisibility because they are ‘divisible without end’, yet each is ‘bounded’ from one another, providing another form of ratio (CEE, p. 5). Magnitude accounts for the divisibility of extended geometric objects, yet it also keeps intact the irreducibility of immaterial geometric ideas (CEE, p. 40). Magnitude therefore constitutes both limit and unlimit in extended matter, but not for unextended ideas in the understanding, such as the universal idea of the ‘simple and unextended’ circle. Here, Proclus tells us, that magnitude and shape are not produced, ‘for such objects in the understanding are ideas devoid of matter’ (CEE, p. 43). Therefore, magnitude is inherently connected to extended entities and is an important manifestation of limit and unlimit in the imagination’s production of the geometric figure. Thus, geometry and arithmetic provide a discursive unfolding of the divine Pythagorean principles. In addition, the notions of limit and

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unlimit constitute an irreducible series of original and discursive operations, in contrast to the non-discursivity of the ideal forms in the nous. However, the divine powers tend to be defined in relation to the external powers of the geometric figure, and so explanations about how limit and unlimit might become embodied ‘transitional psychic activities’ are not developed. Nevertheless, if we suggest that the faculty of the imagination embodies these powers, despite their ‘impure’ classification as ‘sensible’ (rather than divine) knowledge, then geometry and its figures may also be said to embody an unlimited and material discursivity.

Imagination, limit and unlimit In his analysis of Euclid’s first definition, ‘A point is what has no parts’, Proclus undertakes a lengthy discussion about the metaphysical natures of limit and unlimit, and their relationship to the imagination (CEE, pp. 70–8). He distinguishes between the idea of limit and the idea of what is delimited. Limit produces different kinds of figures: ‘immaterial things’, ‘forms that require matter’ and ‘objects that appear in the imagination’. So, in immaterial things, limit is the indivisible unity of the thing in itself, whereas, in imaginary and material objects, it is a boundary. Perfection, Proclus states, is found in the simple and primary entities rather than composite substances. Thus, in the immaterial things of the nous and soul, the conditions of limit constitute the things in themselves; that is, indivisibility, uniformity and unity are attributable to the perfection of their internal causes. By contrast, extended forms are determined by external causes in which the notion of unity is ‘imported’. Therefore, a boundary or limit is prior to matter in the shapes and objects of the imagination and sensible objects; for example, we may think of three-dimensional objects by attributing ‘planes’ to provide a limit or ‘containment’ to the figure. As a result, two kinds of ‘forms’ are possible from this relationship between limit and matter: first, forms separated into idea and matter (such as mathematical ideas) which have their own agency, so that unity arises from ‘boundaries existing in themselves’; second, forms that are inseparable from matter, and are constituted by limit as ‘parts’, ‘filled’ with matter. So the inherent ‘boundlessness’ of matter represents a contamination of the potential for an autonomous unit or limit to be established, and results in ideas forgoing ‘their native simplicity for alien combinations and extensions’ (CEE, p. 71).

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Proclus’ study of the metaphysical potentials in limit and unlimit also leads him to distinguish between the ‘offspring’ of limit and unlimit, which exist in the ‘point’ generated by the nous or the soul. He suggests that the point constitutes an autonomous limit, ‘completely without parts’ and yet also ‘secretly contains the potentiality of the unlimit’. Significantly, by insisting upon the more radical Pythagorean concept of limit in which matter is constitutive of limit as difference within the monad (rather than an external application of classification or form), it is possible for Proclus to state that the particular, extended, geometric figure is an example of ‘self-sufficient’ limit: ‘The point, then being a limit, preserves its character when things participate in it’ (CEE, p. 72). The axiomatic point therefore outwardly expresses limit, whilst it also inwardly ‘secretly’ possesses unlimit, and from which it derives its potential for discursive and indivisible plenitude in the world and the cosmos (CEE, p. 75). This paradox of the partless limit also introduces an important operation in geometric thinking, which Proclus goes on to consider in relation to the imagination’s powers, because he emphasises the simultaneous divisibility and indivisibility of the point which is carried through the imagination from the nous and soul (and this paradox also anticipates the power of ‘vice-diction’ in Leibniz’s discussions). Once received by the imagination, the point is shaped and divided into extended matter and, because it has the ‘double character of indivisibility and divisibility’, it is understood to be both divided and undivided in ‘intervals’ (CEE, p. 78). In addition, the Pythagorean definition of the geometric point underscores this condition; the point is ‘a unit that has position’, which arises from it being produced ‘in the bosom of the imagination [so that it] is therefore enmattered’ (CEE, p. 77). Proclus explains that if a unit is determined independently of position it is Number or arithmetic, whereas if the point is determined by position it is a geometric figure. The relationship between the point, imagination and embodiment therefore represents a shift from an abstract or intellectual concept of (pure) Number into a concept that is inherently concerned with extension and limit: ‘By contrast the point is projected in imagination and comes to be, as it were, in a place and embodied in intelligible matter’ (CEE, p. 78). The faculty of the imagination is therefore central to the relationship between limit and unlimit in the geometric method and the point’s status as a kind of ‘interval’. (This is an important development in which to briefly note that space and time are implied as limits of the imagination. The discussion therefore bears a resemblance

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to the discussions about space and time’s irreducible unity in Leibniz’s Monad and Bergson’s discussion of ‘perception’. In addition, a succession of ‘impressions’ in time and the simultaneous occupation of space are implied, which Proclus argues prevents a collapse into pure divisibility. Finally, in the following paragraphs, he writes that indivisibility is also a characteristic of time; a discussion, however, that is too large to be expanded here, in detail.14) But in answer to the question, ‘how is the indivisible point possible if the imagination is determined by limit, shaping and division?’ Proclus writes that ‘the imagination in its activity is not divisible only, neither is it indivisible’ (CEE, p. 77). The imagination is neither exclusively divisible nor indivisible; rather, it moves ‘from the undivided to the divided, from the unformed to what is formed’. He continues that if the imagination was divisible ‘it would be unable to preserve in itself the various impressions of the objects that come to it, since the later ones would obscure those that preceded them – just as no body can at the same time in the same place have a series of shapes, for the earlier ones are erased by the later’. Alternatively, if it were only indivisible, the imagination would ‘view everything as undivided’, as do the understanding and the soul, and could not ‘exercise form-giving functions’ (CEE, p. 77). An irreducible indivisibility defines the point, Proclus concludes. It is ‘the being’ of the point and, as a result, because it is derived from the point, the line is also determined by partlessness. Limit, therefore, attributes extension in the form of the point or ‘interval’ when Proclus writes that ‘Possessing this double character of divisibility and divisibility, the imagination contains the point in undivided and intervals in divided fashion’ (CEE, p. 78). So an infinite partlessness forms the basis for Proclus’ explanation of Definition II in the Elements: ‘A line is length without breadth’ (CEE, p. 79). Employing the Pythagorean principles of the divine monad, dyad, triad and tetrad, he examines the discursivity between one geometric principle and another. Geometric limits are shown to express a confluence of divine states; for example, the point is equated with the monad because it is not just ‘a limit only’ but it is also ‘twofold’, because it is neither ‘wholeness nor parts’. In addition, he observes that a ‘forthgoing’ dialectic between the attributes of the monad and dyad is produced in the definition because the line has parts and is a unity; that is, the line is infinitely divisible since it is an extended entity (monad) and because it is ‘extended oneness and generates duality’ it also demonstrates the properties of the dyad (CEE, p. 80).

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Ultimately, however, limit and unlimit are divine states, existing independently of matter and are therefore ‘intelligible’, not sensible, ‘agents’ of extension. In the following chapters, these definitions will become increasingly ‘embodied’ in Spinoza’s and Leibniz’s theories of discussions of infinite ‘substance’. Here, however, limit and unlimit produce the discursivity of the geometric objects, elements and figures.

Discursivity of the elements The axiomatic structure of Proclus’ Commentary, its elements and figures, prepares the way for the following chapters on Spinoza and Leibniz in which the axiomatic structure of each text contributes to the aesthetic form of the geometric method (for example, Spinoza employs it to emphasise of his ‘affirmation’ of an indivisible God and Leibniz uses axiomatic statements in order to underpin an infinitely divisible limit). Proclus writes that mathematical figures are produced in the nous as immaterial forms. But the plenitude of an inherent indivisibility in the dianoetic method means that they also mediate between the divine and the material worlds. Thus, ‘numbers, points, lines, planes, and all their derivatives’ mediate between the insensible and sensible objects ‘since they are independent of matter’. Yet they also have attributes of extension, insofar as they can be divisible into parts; that is, they have a certain kind of ‘mathematical matter’ (CEE, p. lviii). Geometric objects are therefore immanent to their method, each determined by the limit and unlimit so that ‘unfolding’ is itself a kind of geometric figure, an expression of the dialectic of the limit and unlimit operations. Furthermore, dianoetic or ‘imaginative and discursive thinking’ is ‘triadic’ because it is comprised of a ‘mixture’ of the three orders of knowledge and their respective cognitive powers: the nous or intuition, the understanding or discourse, and sense or opinion, which are brought together to constitute ‘a texture of all these strands’ (CEE, p. 29). Thus, an inherent continuity underpins the term ‘element’. Proclus explains: We call ‘elements’ those theorems whose understanding leads to the knowledge of the rest and by which the difficulties in them are resolved. As in written language there are certain primal elements, simple and indivisible, to which we give the name … and out of which every word is constructed, and every sentence, so also in geometry as a whole there are certain primary theorems that have the rank of starting-points for the theorems that follow, being implicated in

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them all and providing demonstrations for many conjunctions of qualities; and these we call ‘elements’. (CEE, pp. 59–60) Proclus’ promotion of unlimit as the first ‘creative cause’ of both simple and complex elements also underpins his analysis of the potential discontinuity in the discursive geometric method; for example, he writes that the discontinuous potential of the figure is evident in Book One of the Elements in the form of problems and theorems that provide explanations ‘interwoven’ into the ends of sections (CEE, p. 67). In addition, he tells us that problems and theorems represent two different modes of a proposition; problems are ‘the construction of figures, the division of them into sections, subtractions from and additions to them, and in general the characters that result from such procedures’; whereas, theorems are ‘concerned with demonstrating inherent properties belonging to each figure’ (CEE, p. 63; my emphasis). Thus, a proposition is a figure defined by two different modes of operation; construction or demonstration. Furthermore, he distinguishes between theorems and problems, because theorems are concerned with the general, whereas problems are concerned with the particular: ‘In general, then, all cases in which the property is universal, that is, coextensive with the whole of the matter, must be called theorems; but whenever the character is not universal, that is, does not belong to the whole genus of the subject, then it must be called a problem’ (CEE, p. 65). In addition, a range of analytic or synthetic figures are generated because propositions are constituted by different operations; for example, theorems are analytic because they contain ‘only a given attribute, not its antithesis’, and problems are synthetic because they ‘admit the possibility of antithetical predicates in its matter – the attribute sought as well as its opposite’ (CEE, p. 65). So problems are figures of amplification, because their subject matter is comprised of different elements, whilst also generating particular kinds of propositions (and this potential for geometry to be synthetic and analytic continues into Spinoza’s and Leibniz’s divergent solutions). Proclus continues his analysis of the diversity of elements, concluding that axioms and postulates are distinct from each other, in a fashion similar to the differences that distinguish theorems and problems, because both axioms and theorems ‘take for granted things that are immediately evident to our knowledge and easily grasped by our untaught understandings’; for example, they demonstrate that a straight line is the shortest distance between two points. Thus, axioms are ‘clear knowledge without demonstration’, just as theorems are ‘knowing from

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demonstration’, in each case, displaying characteristics of completion and a unitary identity (CEE, pp. 140–1). However, postulates and problems undertake a more speculative form of deduction, assuming that a figure can be posited as a simple idea without demonstration; for example, a spiral or an equilateral triangle can be easily be assumed without demonstration, but in the act of drawing, ‘complex motions’ are revealed (CEE, pp. 140–2). Postulates and problems therefore require an additional act of construction in order for them to be realised, because each is determined by the act of figuration, and each is aided by the construction or ‘exhibition of a character’. So, although postulates are given the same general character as axioms, insofar as they are ‘undemonstrated starting points’, they differ because they require an additional construction to be completed. Proclus tells us that postulates are specific to geometry, whereas axioms are generic (i.e., universal) to all the sciences of quantity and magnitude. He concludes that there are three ways of distinguishing between axioms and postulates: (a) postulates ‘produce’ geometry, whereas axioms ‘know’ geometry; (b) postulates are the particular ‘subject matter of geometry’ (e.g., drawing is required to prove ‘that all right angles are equal’); and (c) postulates demonstrate proof, whereas axioms are beyond demonstration (CEE, pp. 142–3). As a result, postulates and problems provide an explicit discursive link to the sense world because they are more complex, and require the demonstration of unextended ideas by the act of drawing extended geometric figures. Once again, like Plato’s Meno and Kant’s study of the technical actions of the imagination, Proclus’ emphasis on the synthetic act of geometric thinking reveals that an aesthetic act of construction is required. The importance of multiplicity in the geometric method is also evident in Proclus’ examination of two lesser known elements, the ‘lemma’ and the ‘porism’. First, he writes that a lemma designates ‘any proposition invoked for the purpose of establishing another’ and requires a particular ‘mental aptitude’ which directs two methods of explication, analysis and diaeresis (division), or the ‘reduction to impossibility’. Diaeresis is a kind of ‘lemma’ because it ‘does not directly show the thing itself that is wanted but by refuting its contradictory [nature] indirectly establishes its truth’ (CEE, p. 166). Thus, a lemma is an intermediate or partial figure, which is produced as a result of the difference existing between two other elements. Second, Proclus defines the ‘porism’ as a particular kind of problem that designates the liberation of ‘some other theorem’, and is ‘an incidental gain resulting from the scientific demonstration’. It is therefore

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a kind of by-product, ‘bonus’ or discovery that lies between problems and theorems and is explained in two forms: first, it is ‘a theorem whose establishment is an incidental result of the proof of another theorem, a lucky find’; second, it may be ‘problems whose solution requires discovery, not merely construction or simple theory’ (CEE, p. 166). Therefore, a porism is a figure produced by speculation or hypothesis, rather than by a deductive and deterministic construction. In addition, like the lemma, it is an interval or figure of speculative inquiry. Proclus writes: But to find the centre of a given circle, or the greatest common measure of two given commensurable magnitudes, and the like – these lie in a sense between problems and theorems. For in these inquiries there is no construction of the things sought, but a finding of them. Nor is the procedure purely theoretical; for it is necessary to bring what is sought into view and exhibit it before the eyes. (CEE, p. 236) So, even though Proclus endorses the scientific value of geometry, based upon ‘self evident’ and ‘determinate first principles’, he repeatedly emphasises the complexity within and between elements, in particular, when examining the eruptions of difference that are produced in the aesthetic acts of drawing and thinking (CEE, p. 61). As a result, we find that the construction of the elements is constituted by multiple discursive operations, which are enfolded in and unfold through the aesthetic geometric method. A final discussion about the structure of the geometric figure shows how Proclus further distinguishes between these differences within a discursive continuum.

Figure Section XIV of the Definitions analyses Euclid’s term ‘figure’. Here, Proclus states that a ‘figure is that which is contained by any boundary or boundaries’ (CEE, p. 109). It is ‘something that results from change, arising from an effect produced in things that are struck, or divided, or decreased, or added to, or altered in form, or affected in any one of various other ways’. Demonstrating the figure’s discursivity, Proclus outlines the ascending hierarchy of figures in six levels: 1. sensible figures of ‘art’; 2. ‘nature’s craftsmanship’;

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3. 4. 5. 6.

‘heavenly bodies’ or ‘intelligible forms’; ‘figures of souls’; ‘intelligible figures’; and finally ‘unknowable gods’.

Figure is therefore a notion that is ‘derived from the first causes’ of the gods and, although there is an increasing descent from the perfection of the gods to the imperfect material figures, Proclus rejects the neo-Platonic suggestion that sensible figures are incomplete versions of the higher forms, stating that they too ‘contain the primary cause of their products’. Nor does he accept the suggestion that the immaterial figures of souls or intelligible figures ‘lack reality’ (CEE, pp. 111–12). Rather, immaterial and material figures have a certain kind of selfsufficient agency, similar to the reflective agency of the soul. Each figure contains ‘self-moving ideas’ of ‘other things’ that are external to it, which ‘unfold’ internally in the figure to ‘bring back all things to themselves and enclose them’. Thus, at each level, the figure has the ability to apprehend itself depending upon the magnitude of its powers; for example, gods have knowledge of the universe, souls have ‘immaterial thought and spontaneous knowledge’ and figures in nature ‘create appearances’ (CEE, p. 112). Having made these distinctions, Proclus explains the relationship between movement and the figure by means of a Pythagorean principle: ‘Clearly, then, the self-moved figure is a priori to what is moved by another; the partless is prior to the self-moved; and prior to the partless is the figure which is identical with unity’ (CEE, p. 113). The figure therefore becomes multiple in its potential forms, structured through ‘movement’ in which the divine exists in all states. But although the scope of this movement is restricted to the hierarchical order of perfection, and in which the senseobject will always be less autonomous than the divine, the figure is not reduced to a merely formal categorisation of ‘static’ identity. Furthermore, in this analysis of the nature of ‘unity’ in the geometric figure, Proclus reminds us that the figure produced in the imagination is extended and bounded, exhibiting the ‘twofold progression of the limit and the unlimited’ (CEE, p. 114). He then considers the validity of Euclid’s definition of figure suggesting that, although it is contained, the figure is nevertheless considered as a ‘whole’, rather than a separation of matter and boundary. So, according to Proclus’ examination, the self-sufficiency of the figure and ‘the powers it contains’ are affirmed (and this emphasis on the ‘autonomy’, ‘powers’ and ‘self-sufficiency’ of a heterogeneous series of figures will also be important for Spinoza and Leibniz).

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Geometric figures therefore constitute a multiplicity of limits or boundaries through which a figure is conceptualised by its own ‘selfsufficient’ singularity, rather than by being measured as an imitation of a primary ‘genus’. This discursive distinction is also promoted in the emphasis on the figure, as a series of successive ‘intervals’ or singularities in the unfolding method. The geometric figure represents a qualitative series of differentials that are ‘irreducible’ to the reductive notion of a single essence or limit so that the ‘primary cause’ of the figure becomes, not ‘incomplete processes’ but a generative power or movement, folding and unfolding between, and internally within, each figure. Finally, in a series of definitions of the figure (which we will see recur in the writings of Spinoza and Leibniz, especially), Proclus summarises the Pythagorean origins of the idea of figure and suggests that its mulitiplicity is determined by five conditions. First, the notion of figure is descended from the limit and unlimit; that is, it is a ‘mixture’ of the two and is therefore inherently irreducible. Second, it has a unity that is constituted by ‘different forms’; for example, the different parts of a circle or rectilinear figure. Third, it ‘has the potency of thoroughgoing plurality’, exhibiting an infinity of shapes and magnitudes in an unceasing ‘unfolding’; for example, just as the notion of ‘one’ is implied in the idea of the geometric figure, circular lines may also be ‘implicated’ in straight lines, and vice-versa. (Although this notion of continuity is appealing, it is also problematic because it is exclusively synthetic. Leibniz, on the other hand, produces a more successful notion of analytic continuity.) Fourth, figure is commensurate with the successive development of complexity, and the ‘inexhaustible’ discursivity of arithmetic and number. Fifth, the Pythagorean figure has a ‘secondary’ and harmonious internal order of unity, which can be divided into similar extended or unextended parts, for example, the division of a triangle or square into smaller versions (CEE, pp. 115–16). Thus, although there is a transcendental hierarchy of perfection towards the divine limit and unlimit, the concept of boundary or limit is nevertheless immanent to each figure’s autonomy. Extended geometric objects are aesthetic demonstrations of the method’s discursive movement, providing the possibility for a continuous unfolding or folding expression of different states within a continuum. Therefore, the geometric figure of the ‘fold’ is an unlimited limit operation. Nevertheless, its internal differentiation remains underdeveloped in Proclus’ account, because it is determined externally by the divine principles of discursivity, limit and unlimit.

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Summary Proclus constructs an aesthetic and discursive continuity from Euclid’s geometric method and figures in the Commentary. However, the transformative principles of unfolding and folding are problematic, insofar as they are determined by the external powers of limit and unlimit which constitute the ‘synthetic’ nature of the geometric figures and method, and so we are left with the problem that the geometric figures are derived from synthetic and external differences; for example, difference is derived from the external, synthetic powers of limit, unlimit and mixture, which restrict the extent to which Proclus can distinguish the internal changes that generate geometric figures. Thus, although each figure is a discrete, differentiated limit within an irreducible or unlimited continuum, they are nevertheless determined by the ‘external’ powers of limit and unlimit. This important issue will return as a key discussion in Chapter 5 about Leibniz’s analytic method, which reveals an intelligible transformation between geometric figures, in the form of an internal and intensive limit or ratio. So, insofar as geometric figures of unfolding and folding are determined by the divine and original causes of the limit and unlimit, they may represent discontinuous unities, and Pythagorean principles generate the discursivity of geometric figures and elements, counteracting the precedence of a representational and formal order. In addition, the theological, metaphysical and aesthetic powers of these principles suggest a series of discontinuous unities. However, although the discursive movement still upholds the transcendence of the nous, world, soul or reason in the form of the mystical powers of the limit and unlimit, the actual internal changes between its figures are not defined in analytic terms. Sensible beings are therefore imbued with a kind of irreducibility or infinity, but an explanation which relies upon a ‘mystical’ solution, rather than intelligible psychological explanations. Thus, the claim that multiplicity exists is still at odds with a clear understanding of the difference between and within particular embodied geometries. In addition, although the self-sufficiency of figures is suggested, their internal discursivity is not as clearly defined in terms of embodied or human ‘psychic activities’. The importance of an internal folding and unfolding is hinted at, weakening the emphasis on external reason, but the imagination and the soul are primarily characterised as ‘logical functions’ rather than embodied ‘physio-psychic’ processes generated by the subject.

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Finally, although the ‘synthetic’ order admits the intermediary status of the geometric figure, such as the mixture, it is not considered in terms of a series of internal differences. Each figure’s singularity is a finite identity limit versus the infinitude of an internal and analytic continuum in which the figure’s unity is constructed through synthetic and analytic differences. In the following chapters, however, we will find Spinoza, Leibniz, Bergson and Husserl propose aesthetic geometries in which the method and figure become fully commensurate with the internal and autonomous irreducibility of the thinking subject.

3 Passages

Spinoza’s geometric method is ‘expressed’ in the axiomatic structure of his text, the Ethics (1677).1 Here, in contrast to Proclus’ explicit analysis of an unfolding procedure, Spinoza does not use a geometric figure to define his geometric method; and the figure of the ‘passage’ that I propose is not a term which will therefore be found located in the text itself. Rather, I suggest that it is through modes of engaging with the text (e.g., in the acts of reading or thinking) that the aesthetic geometry of Spinoza’s thought is revealed. Passage is therefore a kind of ‘comportment’ or ethic, which is produced in the reader by the text. Furthermore, the text is an expression of an aesthetic geometry because it brings together the axiomatic scientific method and the aesthetic experience of reading, which produces an ethical subject or reader. As a result, a ‘forgotten’ aesthetic geometry is generated in the reader’s different modes of engaging with the text, rather than a ‘drawn’ or ‘technical’ geometric figure generated explicitly in its argument, as is evident in Proclus’ and Kant’s writing. Before analysing the Ethics, however, a short ‘scholia’ on the relationship between Descartes’ and Spinoza’s analyses of analytic and synthetic scientific methods and the modal ideas of mind and body, situates Spinoza’s geometric method in relation to Descartes’ Cartesian philosophy and geometric thinking.2

Geometric methods after Descartes Descartes’ writings on scientific method include the Discourse on Method (1637) and Principles of Philosophy (1644), and Spinoza’s commentaries on Descartes’ Cartesian principles are critical, yet respectful, examinations of his ideas. In the ‘Commentary on Descartes Principles’ (1663), for 62

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example, Spinoza agrees with Descartes about the value of proper scientific comportment which provides ‘clear and distinct’ ideas. Both philosophers consider analytic and synthetic forms of science to be the best way in which philosophical understandings about the perfection of God can be reached. Under these terms, geometry is conceived as a scientific discipline, first and foremost, which provides evidence of clear and distinct ideas about God’s perfection. In an introduction to Spinoza’s Commentary, the seventeenth-century philosopher, Meyer, also draws attention to the importance of analytic and synthetic methods in Spinoza’s and Descartes’ work; for example, referring to Descartes’ definition of analytic method in his Reply to the Second Objections (1642), Meyer writes ‘[that] which shows the true way by which the thing was discovered, methodologically, and as it were a priori’. In addition, he observes that Descartes’ definition of synthetic methods is [that] which uses a long series of definitions, postulates, axioms and theorems, and problems, so that if a reader denies one of the consequences, the presentation shows him that it is contained immediately in the antecedents, and so forces his assent from him, no matter how stubborn and contrary he may be. (Curley, 1985, p. 226) Meyer also notes that Descartes’ contribution to the development of the axiomatic method lies in his innovative analytic thinking, which forms the basis of his impact on developments in the physical sciences and philosophy. Descartes’ intensive examination of analytic and synthetic methods therefore informs Spinoza’s geometric thinking. However, although Meyer considers Descartes’ legacy to be primarily developed out of the analytic method, this discussion emphasises the extent to which Spinoza’s method is also connected to Descartes’ examination of synthetic thinking, in particular, with respect to Spinoza’s use of ‘scholia’. Spinoza’s Commentary on Descartes, for example, is organised according to Descartes’ analytic method as it is laid out in the Principles but, like the Ethics, the Commentary also demonstrates Spinoza’s own deliberate investigations into synthetic procedures. So, although Spinoza’s Prolegomenon in the Commentary states that Descartes’ analytic method ‘brought to light solid foundations for the sciences, and finally, by what means he freed himself from all doubts’, Spinoza also highlights the importance of synthetic objects and figures which are produced in mathematical methods. He writes

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that Descartes’ clear and distinct ideas are significant, because they do not represent a series of discrete conclusions, but ‘should all be seen in a single act of contemplation, as in a picture’ (Curley, 1985, p. 231; my emphasis). Forms of scientific objects and figures therefore have an aesthetic significance for Spinoza, in addition to their analytic value. Spinoza’s attention to the aesthetic manner in which the geometric method produces its forms is also evident in his emphasis on the different modes of thinking that constitute Descartes’ thinking subject, yet also points to an important difference between the two philosophers. In the Prolegomenon, and repeated throughout the Commentary, Spinoza draws attention to the way in which Descartes’ thinking is constituted by different modes of thought, some of which provide clear understanding and some less, depending upon their cause. He writes: So when he said, I think, all these modes of thinking were understood, viz. Doubting, understanding, affirming, denying, willing, not willing, imagining and sensing. But here the chief things to be noted – because they will be very useful later, when we deal with the distinction between the mind and body – are (i) that these modes of thinking are understood clearly and distinctly without the rest, concerning which there is still doubt, and (ii) that the clear and distinct concept we have of them is made obscure and confused, if we wish to ascribe to them anything concerning which we still doubt. (Curley, 1985, pp. 234–5) Later, in the Corollory to Descartes’ fourth Proposition, Spinoza explains that knowledge of our body is less clear than knowledge of our mind, writing: ‘Hence it is evident that the mind, or thinking thing, is better known than the body [ . . . ]’ (Curley, 1985, p. 242). Therefore, although Spinoza recognises that the body is distinct from the mind, because it does not produce clear understandings of the subject, nevertheless, he considers it to be a necessary mode of man’s existence. In addition, he develops Descartes’ analysis of the ‘different degrees of reality, or being’, in order to emphasise how infinite modes of substance are not matters of ‘accident’ but are unique states of being. Commenting on Axiom 4, he writes: This axiom comes to be known just from the contemplation of our ideas, of whose existence we are certain, because they are modes of thinking. For we know how much reality or perfection the idea of substance affirms of a

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substance, and much the idea of mode affirms of a mode. Hence we necessarily find that the idea of substance contains more objective reality than that of some accident. (Curley, 1985, p. 243) Descartes’ examinations into the differentiation of the subject are therefore central to the development of Spinoza’s geometric method, informing his analytic and synthetic studies into axiomatic and geometric modes of expressing the thinking subject. However, by highlighting Spinoza’s comments on the ‘modes’ of thinking in Descartes’ method, we can also see how Spinoza develops the union between the mind and body as particular modes of an infinite substance; rather than relegating the body to an unthinking or ‘accidental’ form of substance. Therefore, Descartes’ and Spinoza’s definitions of substance and the potential for synthetic union between the mind and the body radically diverge (see also, Curley’s Preface to the Commentary in which he states that the main difference lies in Descartes’ belief that the mind is a distinct kind of substance; Curley, 1985, p. 221).3 The following sections show how Spinoza’s commitment to an infinite modal expression of thinking and feeling embodies a uniquely ethical and aesthetic geometry within this discussion.

Geometric method in the Ethics Spinoza’s Ethics (1677) provides a particularly ‘human’ expression of the geometric method. It is both a demonstration of God’s existence and a ‘practical’ guide about how to achieve this understanding.4 The axiomatic method is therefore presented as a meaningful procedure through which the individual explores metaphysical, theological and psychological steps towards well-being or ‘perfection’. In addition, on the one hand, it is a paradigmatic scientific procedure, and on the other hand, it is a route through which understandings of the senses (i.e., the emotions) may enable a joyful life. At the very least, the Ethics is significant because it affirms a science of the emotions, constituting an early form of ‘psychology’ that brings together the scientific form of geometric thinking and the mutability of emotional life. Thus, unlike Proclus, who is silent on the details of the senses and their union with the intellect, Spinoza proposes that the axiomatic formation of geometric thinking is a means through which the aesthetic unity between the immaterial ‘mind’ and the material ‘body’ is produced. Spinoza’s method is therefore a kind of anticipation of Kant’s aesthetics because, in each case, the sensibility is a legitimate subject of study within an a priori and

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scientific method. Second, in each method, the subject’s experiences of pleasure and pain constitute the passage towards a transcendental notion of subjectivity. (Also, the experience of reading the Ethics is a demanding undertaking, since the reader is expected to engage with a text that is technically intense scientific and aesthetic experience.5) In the Chapter 2, the geometric method was a discursive and synthetic unfolding derived from the mystical powers of limit and unlimit. Geometric figures were transcendental and multiple, driven by Proclus’ Pythagorean commitment to the hierarchy between unextended intellect and extended matter. Under these terms, infinity was the primary value; however, it remained immaterial and unknowable. As a result, extended things – for example, the body, senses and extended Nature – were reduced to impure versions of pure intellect, and situated into a continuum of figures that did not define the internal union between their immaterial and material constituents (i.e., the relationship between the body, soul and mind); for example, the fold was an implied geometric figure generated between the operations of the soul, understanding and the imagination, and so it remained limited to an ideal because its potential as an embodied subject was not explicit. Thus, the figure of the fold was always an implied action in the discursive ‘unfolding’ and its various actions, although recognisable intermediary figures were grounded in an ideal notion of synthetic development, but were not developed into a clear union of unextended and extended substance. In addition, these ‘general’ ideas of the soul, understanding or imagination did not fully account for the transition from the immaterial realm into a conduct of living; for example, the discursivity of the soul and its relationship to the imagination was explained through an analogy of representations. Therefore, ultimately, reason derived from the emotions was denied and Proclus’ method never made the transition from the abstract idea to the corporeal figure explicit. For Spinoza, however, the relationship between the body and mind is central to his metaphysics, constructing a notion of unity that is established in inherently embodied human conditions, the emotions. In addition, infinity is not restricted to the immaterial realms but is the basis for the single indivisible substance (i.e., God-as-Nature). ‘Real’ expressions of a transcendental infinity in multiple and extended beings are central to Spinoza’s thinking, as Feldman explores in his introduction to the Ethics; for example, he asks: ‘how does extended substance come about from unextended substance?’ and, ‘how do the two interact if they are self-sufficient?’ (Ethics, 1992, p. 9).6

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As a result, these questions lead on to two persistent levels of metaphysical inquiry in the text: first, Spinoza’s defence of substance as existence (not as extension), so that God is connected to substance as Nature, rather than representing existence as an abstract idea that is separated from the material world. Second, Spinoza examines embodiment in terms of a series of modes of existence in which this multiplicitous substance is underpinned by the modality of the axiomatic procedure and the scholiatic episodes. The geometric method therefore generates intrinsic differences between each distinct expression and mode. In addition, substance’s inherent infinity is further complicated by its two ‘parallel’ and indivisible attributes, the mind and body, which are revealed in the ‘logical’ unfolding of the axiomatic method, and in the intuitive and ‘corporeal’ scholia that accompany it. So, in contrast to Proclus, for whom discursivity is an intellectual route to knowledge, geometric discursivity in Spinoza is reconfigured into an embodied passage via the sensibility’s operations. The Ethics provides an examination of the unity between the unextended mind and extended body that are derived from a multiplicitous substance, representing a passage through the emotions. Here, the union between the immaterial and the material is not just derived from the faculty of the imagination but also arises out of the unruly, excessive ‘affects’ of sense-perception, so that God’s ‘immanence’ is also expressed in the subject. Proclus’ concept of unfolding is therefore transformed into a method in which the geometric figure becomes embodied in the body-in-process, a lived body or subject. Geometric ‘truths’ are reached by ‘stepping-stones’ in a subject’s life, and may take the form of an intellectual or theological journey, but are especially closely linked to psychological journeys.7 I begin by examining the notion of substance in relation to the dialectic of limit and unlimit, in order to demonstrate the text’s metaphysical structure and geometric ‘discursivity’, as it is embodied in God, Nature and man (and also briefly point to the differences between Spinoza’s and Leibniz’s strategies). I then examine the key ‘elements’ – that is, the attributes, modes, affects and common notions – which constitute this infinite substance to explore how it unfolds in a step by step way, which Spinoza describes as ordine geometrico demonstrata (E, p. 7).8 In addition, this discussion examines the infinite perfection of God which Spinoza implies is expressed in the deductive procedure of the geometric method and points to the importance of ‘practical’ demonstrations; that is, Spinoza’s study of how a ‘perfect’ unity between body and mind might be realised. I then turn to a brief examination of

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the development of the imagination and memory, showing that Spinoza makes an important contribution in the shift from Proclus’ cognitive faculty, to the ‘psychical’ operations of Kant’s imagination, Bergson’s examination of memory and Husserl’s sense-ideas. Finally, having established the ‘elements’ of the geometric method, I explore the text’s passage through axiomatic and scholiatic forms, showing how geometric examples are used to provide particularly ‘embodied’ forms of geometric thinking in the ‘scholia’. I therefore suggest that the text represents the heterogeneous, aesthetic and internal structures of the geometric figure of ‘passage’.

Substance As outlined in the earlier ‘scholia’ on Descartes, Spinoza’s break from the Cartesian tradition is most strongly expressed in his notion of a univocal, yet infinite, substance. Challenging Descartes’ postulation that extension and thought are two distinct substances, Spinoza proposes that an indivisible substance immanently manifests the infinity of God in nature and in man. The Ethics is therefore a continuation of the Stoic principles of indivisibility and divisibility, brought forward in seventeenth-century debates about finity and infinity, substance and ‘atomism’ (and informed by theoretical debates about atomism, bodies and motion in the natural sciences and physics, aesthetics and metaphysics).9 However, this discussion focuses on Spinoza’s theories of substance, attributes, modes and affects, which embody the transcendental notions of limit and unlimit, to argue that Spinoza’s ‘elements’ represent a new set of embodied geometric and ‘aesthetic’ qualities. As this and the following chapter show, the relationship between geometric thinking and embodiment take on a new dynamic form after Descartes; in particular, because Spinoza and Leibniz tackle the problem of division in Cartesian theories of substance in critiques of the dialectical relationship between limit and unlimit, and its significance for corporeal embodiment. For Spinoza, the solution is posited in a ‘univocal’ substance in which the synthetic notions of limit and unlimit are reconstructed in the union between the emotions and the corporeal subject. Leibniz, however, takes a route in the opposite direction, so that limit and unlimit are analytically differentiated corporealities. Nevertheless, a number of important similarities between the two methods are evident. First, the geometric method is a process that constructs internal and external relations, which are generated from within the individual subject.

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Therefore, for Spinoza, the modalities of human emotions (the affects) enable the subject to understand the external environment and objects, whereas, for Leibniz, the infinite divisibility of the Monad (i.e., the subject) produces a continuum between its interior and exterior. Second, as a result, formal difference is constituted internally and is expressed externally (i.e., it is not derived from an external source). Third, Spinoza’s affects and Leibniz’s magnitudes generate discursive geometries that are characterised by intensity, rather than logical or mechanical relations of agreement, and are therefore important precedents to Kant’s aesthetic and the reflective subject in the Critique of Judgment. Therefore the dialectic between divisibility and indivisibility is the primary generator for Spinoza’s axiomatic discussion of substance, which he examines in the first passages of the text. In Part I, ‘Concerning God’, Spinoza explores how these ‘divine’ causes provide the basis for a geometric method that unifies God’s immaterial infinity in Nature with extended or embodied modes of being. In Definition 1, Spinoza writes that God ‘exists’ absolutely and is ‘self-caused’. In the following five Definitions and Explication, God’s absolute infinite existence is distinguished from the ‘limited’ finitude of things, thought or bodies. Therefore, although God is unlimited, bodies and things are limited, because one thought limits another, and one body limits another body. Different modes of substance therefore cannot affect another mode. Consequently, a significant shift takes place in which limit is determined by the nature of substance; rather than by the entity or thing in which it is expressed. In Definition 3, Spinoza is explicit about this potential of substance ‘By substance I mean that which is in itself and is conceived through itself; that is, that the conception of which does not require the conception of another thing from which it has to be formed’ (E, p. 31). Thus, in contrast to the neo-Platonic belief that the higher realms of intellect, reason and the gods, constitute transcendental knowledge which is inaccessible to man, Spinoza’s theory of substance provides the ground through which unextended (i.e., thought) and extended matter are derived from a single infinite ‘substance’. Substance is constructed under three metaphysical principles: infinity (and finitude), existence and immanence. In particular, its restricted definition as ‘extension’ or matter is replaced by its affiliation with existence that ‘belongs to the nature of substance’ (E, I, Prop. 7, p. 34). Extended matter is one mode in which substance is expressed, rather than it being conceived as the sole defining principle by which it is understood. Instead, multiplicity in substance is the primary cause of extended and unextended beings, and is also evidence of God’s infinite powers, for

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example, when Spinoza writes: ‘Absolutely infinite substance is indivisible’ (E, I, Prop. 13, p. 39). God, thought and bodies therefore share the qualities of substance, because each is a ‘self-caused’ existence in itself, and so Spinoza distinguishes between the infinite substance of God and the limited but autonomous divisions of substance as thought and body. Later, in Part III, Spinoza emphasises the importance of existence, not in relation to the divine but in the concept of life or duration in the individual entity. Here, the relationship between existence and substance is given its biological duration in the form of the ‘conatus’, the entity’s power to strive to exist (which is also crucial to Bergson’s ‘progressive’ philosophy). Spinoza explains: Therefore, the power of any thing, or the conatus with which it acts or endeavours to act, alone or in conjunction with other things, that is . . . , the power or conatus by which it endeavours to persist in its own being, is nothing but the given, or actual, essence of the thing. (E, III, Prop. 7, Proof, p. 108) The conatus is specific to the finite mode of substance and points to a secondary mode of temporality that is brought about in the attribute of body: ‘Each thing, in so far as it is in itself, endeavours to persist in its own being’ (E, III, Prop. 6, p. 108). God’s existence, on the other hand, is not limited, it is infinite. The subject’s conatus, then, is limited by the duration of the body: ‘The conatus with which each thing endeavours to persist in its own being is nothing but the actual essence of the thing itself’ (E, III, Prop. 7, p. 108). The subject and God-as-Nature are also differentiated because of the durational ‘life-force’ that determines man’s essence: ‘The being of substance does not pertain to the essence of man; i.e. substance does not constitute the form (forma) of man’ (E, II, Prop. 10, p. 69). So substance is distinct from the extended form of man (whilst it is also immanent to mind and body) because it is the principle of existence. Once again, substance’s essence is confirmed to be existence, not extended matter or form, and extension is not determined by the limitations of form, but by the limit of a durational existence; that is, the endeavour to exist. In addition, extended bodies are not excluded from transcendental values of existence, but are legitimate modes of substance’s infinite potential, so that extended entities also display the paradox of division and infinity. Moreover, Spinoza explores how

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infinite substance is the primary cause of his triumvirate order of ‘existence’ in the attributes, modes and affects. Substance is infinite and indivisible in its multiple forms of existence; in God and Nature it is infinite and indivisible; however, in its extended forms of existence it has discrete and finite limits. Spinoza explores this paradox in our ability to understand water as both divided and indivisible. He observes how our imagination divides water into parts, but in our ‘intellectual’ understanding of it as a substance, it is indivisible. Therefore, like Proclus, Spinoza also considers the imagination to be a key operation in the production of limit, in contrast to the indivisibility of a pure intellect. Hence, God is both an Omnipotent immaterial power and Nature, and Spinoza promotes this immanence further by highlighting its ‘creative’ or productive activity when he states that ‘God is the immanent, not the transitive, cause of all things’ (E, I, Prop. 18, p. 46). Alternatively, in the Scholium of Proposition 29 he draws attention to the definition of Nature as a creative cause, distinguishing between its active sense as producer or ‘nature naturing’ (Natura naturans) and its passive sense as product or ‘nature natured’ (Natura naturata). Nature is therefore explicitly defined in terms of its powers of production and the modes in which these powers are manifest (E, p. 52). God-as-Nature is the immanent cause, expressed in the infinite attributes and modes of existence.10 In addition, this immanence generates the singular forms of extended substance, for example, a general idea of nature or man, as well as particular embodiments of substance, such as an emotion, or individual creature or person. Substance therefore embodies the metaphysical notions of limit and unlimit, which make up the continuum of metaphysical relations and orders. Like Proclus, Spinoza’s ontology affirms the notion of an absolute, infinite God/One. However, these methods are also distinct because Spinoza’s dialectic does not commit to the hierarchy of unextended limit and unlimit vis-à-vis extended limit. Rather, by constituting God, Nature and man as ‘self-caused’ existences, limit and unlimit become embodied in the notion of the human (him/herself). Extended bodies therefore manifest this existence, rather than representing impure derivations of pure transcendental and immaterial sources. Consequently, Spinoza’s concept of extension is not reduced to an opposing ‘materialist’ power (although it may be defined as a ‘materialist’ metaphysics11) but is determined by the infinite nature of existence. Spinoza’s ‘univocal’ substance is therefore imbued with the paradoxical relationship

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between limit and unlimit, divisibility and indivisibility, in unextended and extended bodies.

Passage I: attributes, modes, affects and common notions Spinoza constructs an ‘embodied reason’ in the form of the triad of interdependent operations (i.e., the attributes, modes and common notions) to produce a highly complex union of the different orders of human nature. In geometric terms, these operations constitute: first, the strands that unfold from the unity of indivisible substance; and second, the ‘elements’ of the method that express the different figurations of unity in a series of increasingly defined stages, developing from the attributes of the mind and body, to the ‘modes’ or ‘affects’ of emotions; and third, the unity of the ‘common notions’. As a result, there is a shift from the logical deduction of a geometric method into one that is both embodied (i.e., it is intimately linked to the living person) and is attributed with ‘psychic’ powers of transformation (i.e., the emotions). Thus, Spinoza’s axiomatic explication of univocal substance provides an embodied and aesthetic expression of geometry. Attributes Substance is first expressed in two attributes, the intellect and the body; each attribute intrinsically embodying both the common and the particular forms of existence. Thus, the complexity of indivisible substance is made clear and distinct because of its division (i.e., its limits) into attributes, modes and common notions; however, because division or limit is also intrinsic to existence, the intellect or body do not collapse into merely formal distinctions. Spinoza tells us that attributes express the essence of God or Nature: ‘By attribute I mean that which the intellect perceives of substance as constituting its essence’ (E, I, Def. 4, p. 31). Prior to modes and affects they express the infinity of substance in its extended and unextended states, and Part II examines their structure in detail, highlighting the extent to which they are aspects of God’s infinite existence; for example: ‘Thought is an attribute of God; i.e. God is a thinking thing’, and ‘Extension is an attribute of God; i.e. God is an extended thing’ (E, II, Props 1 and 2, p. 64). Mind and body are therefore distinct, yet unlimited, attributes that constitute God’s powers and confirm that existence is fundamental to substance in all its forms of expression. Spinoza gives a number of explanations: ‘The more reality or being a thing has, the

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more attributes it has’ (E, I, Prop. 9, p. 36); and, ‘nothing in Nature is clearer than that each entity must be conceived under some attribute, and the more reality of being it has, the more are its attributes which express necessity, or eternity, and infinity’ (E, I, Prop. 10, Schol., p. 36). Nevertheless, although each attribute is autonomous from one another, they are brought together under a commonality of ‘ideas’. Substance, in the form of the intellect, is continuous with substance in the form of the body so that the ‘idea of the mind is united to the mind in the same way as the mind is united to the body’ (E, II, Prop. 21, p. 80). Hence, Spinoza maintains the notion of agreement between different extended and unextended attributes, because they are derived from an indivisible unity (the univocal substance or idea). In addition, by insisting on a common agreement between two distinct attributes, the discrete singularities of the mind and body are brought into a synthetic union in which neither is conflated with the other, since each expresses a specific quality of substance. Modes Spinoza also distinguishes between the general expression of God and the particular capacities of these attributes, which are expressed in the modes or affects: ‘the affections of substance; that is, that which is something else and is conceived through something else’ (E, I, Def. 5, p. 31). Modes provide a tertiary level of distinction to substance and, once again, the principles of finity and infinity determine their particular infinity: ‘From the necessity of the divine nature there must follow infinite things in infinite ways (modis) (i.e., everything that can come within the scope of infinite [intellect]’ (E, I, Prop. 16, p. 43). So, the triadic, causal structure of substance is reconfirmed: developing from the primary, indivisible God or Substance into the second level of the attributes of mind and body, and finally, to the third level of definite modes: Every mode which exists necessarily and as infinite must have necessarily followed either from the absolute nature of some attribute of God or from some attribute modified by a modification which exists necessarily and as infinite. (E, I, Prop. 23, p. 48) Modes are finite, for example, as the body and the mind they are two distinct limits of substance’s existence. They are also self-caused or autonomous: ‘By “body” I understand a mode that expresses in a

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definite and determinate way God’s essence in so far as he is considered as an extended thing’ (E, II, Def. 1, p. 63). So the attributes of extension and thought are general ‘essences’ of God and Nature, however, modes also constitute distinct kinds of ideas (reason, imagination, love, desire or hate, and so on) and bodies (animals, plants or the individual man or woman, and so on). In addition, modes express the unlimited agreement in which Nature and God are unified into a multiplicity of modal types. In one of the most important examples of this agreement Spinoza tells us that the idea of the mind cannot be said to exist unless the parallel state of the body also exists. He writes: ‘Modes of thinking such as love, desire or whatever emotions are designated by name, do not occur unless there is in the same individual the idea of the thing loved, desired, etc.’ (E, II, Ax. 3, p. 64). Thus, there is an increasing definition of limit or finitude in the sequence of causal relations. Modes sustain the ‘connection’ between the different versions, but they also have a greater definition of autonomy than attributes and, hence, definition of their different powers. Like Proclus (and for Leibniz and Bergson), there is an increased continuity of change that is generated, especially in the different kinds of agreement which exist between different modes of ideas and things: for example, the discursivity of the geometric method is evident in the modal continuity of one idea of a circle passing into another idea, or in the passage from an extended form of a circle transforming into another (E, p. 67). Therefore, although the synthetic ‘parallel’ between thought and body is kept intact – that is, thinking cannot become extension – particular differences between modes may be less clearly exclusivey demarcated as either, thought or extension. Below, we will see that emotions or affects produce the most ambiguous kind of unity, which is expressed as both thought and body, in particular, as adequate ideas or common notions, in which divisions between mind and body are most disrupted because they become embodied modes of reason. So geometric discursivity, or the step by step deduction of the geometric method, is augmented by the ‘agreement’ or commonality that inheres in the multiple modes of expressing univocal substance. Second, an immanent and genetic discursivity is inscribed into the axiomatic procedure of the Ethics, which is also underlined by the creative production of God-as-Nature (i.e., Natura Naturans). However, a clear understanding of the agreement between the mind and body still needs to be outlined, and much of Parts II and III explore the union in detail. First, Spinoza states that each is ‘the object of the

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idea’ of each other, confirming the unity of the affections and the body (E, II, Props 12 and 13, p. 71). In the Scholium to Proposition 13, for example, he states that ‘the human Mind is united to the Body’ pausing, however, to note that this can only be proved if there is ‘adequate knowledge of the Nature of the body’ (E, p. 72). This leads him to define the particular nature of the human body in order to also cast light on the particular nature of the human mind. The mind, he suggests, is understood through a principle of proportion (i.e., ratio); for example, the ‘proportionate’ activity of the body that is reflected in the mind of its accompanying body.12 Developing here is the notion of ‘ratio’ (i.e., reason) in which a unity is produced out of the relationship between two independent aspects (and which is also considered to be a crucial aspect of Leibniz’s method). In addition, in Proposition 19, II, Spinoza defines the particular way that the union is constituted (i.e., the way in which we can understand the body), even more closely. He writes that it is in ‘the ideas’ and the ‘perceptions’ of the affections ‘by which the body is affected’ (E, p. 80). Thus, the body is understood through forms of appearances, as ideas and perceptions of ‘affections’ (rather than arising from its nature as an extended thing), so that it reflects the human mind’s knowledge of itself ‘except in so far as it perceives ideas of affections of the body’ (E, pp. 80–1). In this respect, Spinoza produces an intermediary form of knowledge not dissimilar to Kant’s theory of forms and appearances. For Spinoza, however, the emphasis on ‘expression’ provides a more immanent mode of synthesis than Kant allows, because the emotions are the route through which a divine perfection can be attained. Kant, by contrast, considers the emotions to be limited to the sensible realm. Nevertheless, this confluence between Spinoza and Kant is apparent in the aesthetic potential of the emotions, insofar as they represent the movement (the passage) between pleasure and pain. In addition, Spinoza’s conception of the imagination bears a strong resemblance to Kant’s productive imagination in the third Critique. But the embodied powers of the emotions are, however, only connected to the unity of the individual’s mind and body, and do not mean that the individual has knowledge of other external bodies, unless it is through the ‘ideas of affections of its own body’ (E, II, Prop. 25, pp. 82–3). As we will see in the following section, these affects define man’s modal autonomy and internal ‘sufficiency’, representing a kind of ‘sufficient reason’ through which the internal and external nature of the individual is brought into being. Furthermore, like Leibniz’s

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‘sufficient reason’, the affects are considered in terms of their intensity, introducing an important shift into the powers of demonstration and construction in the geometric method. But, in contrast to Leibniz’s concept of the Monad, Spinoza’s unity relies on a synthetic harmony, whereas, Leibniz’s ‘sufficient reason’ is generated from an analytic continuum or unity. Affects Affects, or the emotions, represent the entirely unique mode of existence in the human subject. In addition, they are evidence of the unlimited power of the agreements and commonalities that may exist between the attributes and modes. Crucial to the possibility of a harmonious embodied subject, the affects represent the level in which the potential for harmony between the perceiving and extended subject is explored through its capacities for happiness, sadness, passion, agency, activity and passivity. Parts III and IV conduct an intense explication of the emotions, which are the special expressive powers that humans embody. Here, then, the transcendental ‘plenitude’ of unlimit is expanded in terms of the ‘physiopsychic’ human condition. The emotions represent a set of ‘transitive’ powers that are continuously and internally produced by the subject, yet they are expressed in its external modes, constituting the subject’s autonomous actions and the potential for realising a ‘joyful’ life. Part III presents Spinoza’s extended analysis of the emotions as affects, in which he examines their production and duration in the subject, especially through their activity in adequate ideas, and passivity in inadequate ideas. He analyses these relations in detail, stating that adequate ideas are the embodiment of the active state, whereas inadequate ideas are the embodiment of the passive state. So, in each case, the cause and effect are interdependent because each reflects the other; for example, the inadequate idea produces the passive state, not because the body leads the mind to passivity due to its limited nature but because the relationship does not share a commonality. Hence, the body is not an external impurity or obstruction towards an active notion of being (i.e., becoming), but Spinoza does require that it be harmonised into a unity with the emotions in the realisation of perfection. Perfection is therefore the underlying reason, through which active and passive emotions are brought together in the form of the conatus, or the subject’s drive for existence. In themselves, however, the affections do not constitute adequate ideas; rather, they are ‘confused’ ideas of the body and external entities

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(E, II, Prop. 28, p. 83). Propositions 29–31 underline the inadequacy of this confused, fragmentary and discontinuous kind of ‘knowledge’. In the following propositions, however, Spinoza explains that ideas and affects do attain a unity and truth when they are constituted in God (E, II, Prop. 32. p. 85). Adequacy is that which is ‘common and proper’ between things, so that affects and ideas can be conceived as adequate once they are attributed with a commonality and a definition of limit in relation to ‘the whole’ (E, II, Props 38–9, pp. 87–8). In addition, Spinoza’s analysis of activity and passivity leads him to explore their manifestation as pleasure and pain, and to promote the temporality of the passage from one state to another; for example, in the transformation from an inadequate into an adequate idea. Thus, the duration of the conatus is partly defined through the movement between active and passive emotions. ‘Adequate’ knowledge, for example, can be produced from inadequate passions and confusion: ‘A passive emotion ceases to be a passive emotion as soon as we form a clear and distinct idea of it’. Or in the statement: ‘There is no affection of the body of which we cannot form a clear and distinct conception’ (E, V, Prop. 3, p. 204). The next section shows how important this active harmony is for establishing the unified and autonomous subject. Common notions In the final development of the ‘modalities’ Spinoza tells us that adequate ideas constitute the common notions, which are ‘those things that can lead us as it were by the hand to the knowledge of the human mind and its utmost blessedness’ (E, p. 63).13 Common notions are the clear, distinct and embodied ideas through which we can come to understand the perfection of God and, hence, the axiomatic method (i.e., geometry) is shown to be a step by step agreement between mind and body, which expresses the ‘perfection’ in the human subject. Common notions therefore represent the irreducible unity of the human figure that reflects the perfection of God, as an idea and a reality, in the agreement between an emotion and the body.14 So Spinoza inaugurates an agreement between man and God, not merely because of the existential essence of substance but through the different distinctions of limit and unlimit. In addition, as we pass through his metaphysical levels, the definitions of unlimit become more clearly demarcated, to the extent that the emotions are embodied as kinds of ‘reason’ derived from unlimit, but brought into proportion (i.e., ratio) with the body towards a joyful existence. As a result, the notion of agreement is not just determined by synthesis but is also

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achieved by increasing or decreasing intensity (i.e., magnitude). Agreement becomes defined in terms of ‘agreeing more’ (i.e., active emotion or happiness), or ‘agreeing less’ (passive emotions or sadness).15 As a result, ‘harmony’ is determined by degrees of intensity in the emotions (which agitates a kind of agreement that is also important for Leibniz) and represents a shift, from a purely synthetic agreement, into an analytic agreement. Common notions therefore represent the most ‘unified’ form of geometric figure in the text, in which the power of the subject is generated out of an agreement between its immaterial and material modes of existence. In addition, Spinoza insists that modes, or affections, are the primary means through which the knowledge of an infinite unity is generated, and which embody the ‘transitive’ powers of change and duration. A shift takes place, from an emphasis on the external activities or powers of the cognitive faculties (especially, the hierarchy between the understanding and the imagination in producing the content for the affects) to an emphasis on the ways in which the human figure is constituted by active or passive powers and the potential for harmony or unity to exist. Thus, the notion of an internal activity of each ‘sufficient’ subject is promoted in favour of the external ‘independence’ of the understanding that Proclus requires. The transformation, from the production of a synthetic unity by means of a ‘mixture’ of states to ‘affective’ modes that constitute embodied common notions also places greater emphasis on the aesthetic of the ‘sensibility’ that Kant promotes. Common notions are therefore important, not merely as quantifiable differences between states (such as the external physical differences between a man and horse) but because they are modulations of embodied differences: that is, they are adequate and transient ‘reasons’ or figures of passage. Thus, their geometric value is not logical but derived from the sensibility, because their existence is determined by the continuous change in the emotions. As a result, the relationship between God (i.e., the knowledge or ‘love’ of God) and the mode of expression (i.e., the method) is also continuously under transition. Thus, for Spinoza, common notions become ‘real’ kinds of embodied knowledge, granted a kind of ‘reason’ that registers a range of adequate or inadequate states, so that he can also propose that ‘love’ or ‘desire’ represent embodied expressions of an infinite substance. Second, they represent a form of aesthetic unity, in which the particular expressions of the indivisible substance are brought into harmony with God and Nature to constitute a series of indivisible, yet

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embodied unities. As the most complex and unified level of geometric figures – that is, they are an adequate idea or unity – the common notions’ sufficiency represents a real union between body and mind. That is, they are the figures through which Spinoza explains the problem of union between unextended and extended matter. Thus, we also see that there is a shift from an unfolding understanding, which is an extensive movement ‘down’ from the immaterial idea to the extended body, to its reverse, an intensive ‘enfolding’ in which the unity of God is confirmed in the particular powers (affects) that the common notions embody. Moreover, this ‘enfolding’ also represents a precursor to Kant’s reflective judgment in which knowledge of the universal is developed out of the particular; for example, Spinoza states: ‘The more we understand particular things, the more we understand God’ (E, V, Prop. 24, p. 214). In contrast to Kant’s schema of the faculties of the understanding and imagination in the Critique of Pure Reason, however, Spinoza posits the emotions as the ‘powers’ through which knowledge of God can be developed. Thus, we find that Spinoza’s concern with the emotional experience of pleasure and pain represents an aesthetic dynamic of unity and sufficiency. Another link between Spinoza, Kant and Proclus is found in his discussion about the duration of inadequate or limited ideas, when he explores the formation of ideas by the imagination; for example, his suggestion that its powers of division link it to inadequate ideas. Like Proclus, therefore, Spinoza considers the imagination to be an embodied limit-operation, emphasising that its powers are determined by its corporeality. In the lengthy Scholium of Proposition 15, I, he explains: I reply that we conceive quantity in two ways, to wit, abstractly, or superficially – in other words, as represented in the imagination – or as substance, which we do only through the intellect. If therefore we consider quantity as it is presented in the imagination – and this is what we more frequently and readily do – we find it to be finite, divisible, and made up of parts. But if we consider it intellectually and conceive it in so far as it is substance – and this is very difficult – then it will be found to be infinite, one, and indivisible, as we have already sufficiently proved. (E, p. 42) In Part II, the imagination’s powers and products are examined in more detail, when it is described as ‘a conception of the mind’ that produces ‘images of things that we imagine’ (E, p. 97). The imagination and its products, images or words are ‘constituted solely by corporeal

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motions, far removed from the concept of thought’ (E, II, Prop. 49, Schol., p. 97). As has been shown above, however, ideas are the source of ‘commonality’ between the mind and the body, which constitute understandings of the mind and body in a unity. The imagination is therefore important as one mode of producing ideas and perceptions, although, in this context, its powers are ultimately restricted to extended matter. But Spinoza also considers the images of absent objects and ideas, in which the imagination produces memories of the ‘affections’ that the body perceives, so that it is the common ground between the images of the mind and the perceptions of the body (E, II, Prop. 17, Coroll. and Prop. 18, Schol., pp. 77–9). Here, imagination is conceived as memory, because it is ‘simply a linking of ideas involving the nature of things outside the human body, a linking which occurs in the mind parallel to the order and linking of the affections of the human body’ (E, p. 79). As a result, the imagination produces a continuous link between the internal perceptions and ideas of the body and its exterior world (and, as we will see later, this continuous passage of perceptions and images from the interior to the exterior, is a key aspect of Bergson’s discussions about matter and memory). The geometric method expressed in the seventeenth-century Ethics therefore lifts the restrictions from embodied perception that limit it in the neo-Platonic method, in particular, in its relationship to internal operations, which become explicit, not implicit. Instead, Spinoza proposes a series of modes in which the geometric figure is irreducible from its material, bodily expressions, in particular, in its evolution from a multiplicitous substance. Extension becomes a productive condition of the geometric figure, in particular, in the form of the thinking and emotional subject, so that unity is defined as forms of agreement between distinct kinds of existence, which are also ‘common’ elements of the single, originary substance. So how does the textual organisation and subject matter in the Ethics constitute a geometric method? Up to this point in the discussion it is the construction of a univocal, yet modal, substance that can be observed in an extensive explication. But I also draw attention to the importance of intensive developments; for example, the common notions embody a shift from an ‘unfolding’ and its reverse movement, an ‘enfolding’, in which the general is produced out of the particular. Coupled with this relationship, the extensive movement is succeeded by an ‘intensive’ movement between the emotions; thus, Spinoza’s geometric method may be considered as extensive and intensive, unfolded

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and enfolded.16 These reverse movements of intensive enfolding produce an important shift in the development of form that is generated from the particular, rather than the general, and in which the emotions or affects are intrinsic ‘origins’ in the production of union, harmony or concord between the mind and the intellect. The geometric method is therefore imbued with a new potential in the corporeal irreducibility of the emotional subject (not the transcendental immateriality of God). In addition, the emphasis on these intensive powers (i.e., the actions of the emotions) results in diverse movements. Thus, the ‘form’ of the elements enables the dialectic of limit and unlimit to produce diverse commonalities of the method, because form is generated internally, rather than being attributed by an external idea, classification or identity. Modes, whether they are active or passive emotions, are sufficient in themselves, insofar as they are intermediate states on the way to a state of ‘blessedness’, but their form also most strongly reflects the enfolding and unfolding movement between the general, external states of unity and the particular, inner unity of the subject. In addition, the common notions are akin to the ‘all-in-one totalities’ of intuition, yet are derived from the sense perceptions and body, versus the step by step procedure of the discursive geometric method, which is derived from ‘intellectual’ origins. Thus, Spinoza’s detailed axiomatic explanation of the common notions is underscored by the irreducibility of the emotions, which are always in duration and at different ‘speeds’.17 In addition, the emotions always ‘go forth’; for example, we may experience the same emotions, but the sequence of their transition and their duration is always different, so that the extended body is always a durational experience. We can therefore propose that geometric discursivity in the Ethics is characterised by the following qualities: a. The relationship between substance, attributes and modes develops in an increasingly intensified dialectic of limit and unlimit; that is, the affections represent the most distinct and intensive forms of division; b. Emotions may transform into another, because they are determined by the plenitude of unlimit. As a result, emotional modes are the passages through which to get to the common notions, because they represent embodied ‘stepping stones’ to the common notions; c. Common notions represent the highest state of geometric ‘figure’ in which pure intellect becomes accessible to the individual, for example, in the adequate ideas that we construct. They are produced out of the union between the mind and body, rather than being attributed

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by pure intellect from outside, and they are therefore akin to intuition or the soul; d. Limit becomes internalised (i.e., it becomes a psycho-physiological discussion) and so the individual is constituted by the internal limits or ‘thresholds’ between the emotions and body. The following sections examine how passages through the text are presented to the reader, first by discussing the overall structure of the text, and then by exploring the transitive nature of axioms, propositions and, especially, the scholia, showing that the figure of the passage exists in multiple forms in the Ethics.

Passage II: reading the text The experience of reading the Ethics requires a relationship of ‘agreement’ to be developed between the reader and the text as we progress through its various parts. In this respect, geometry becomes an ‘affective’ method: first, as it lays out the nature of the emotions; and second, because it is intended to lead the reader towards the understanding of God and ‘a joyous life’. Thus, the text brings together geometry and the sensibility in the conjunction between feelings and the scientific geometric method. It is a rigorous and ‘affective’ document that employs scientific and aesthetic methods, and through which the reader is asked to engage or become ‘affected’. In addition, the axiomatic method propels the reader through the text so that the experience of reading is not just developed out of geometric order, proposition, analysis and argument but is also realised in the subject, constituting a ‘practical’ philosophy; that is, the experience of reading the Ethics is an aesthetic experience of geometric method. In the last section of the chapter an analysis of the axiomatic and scholiatic structure of the geometric method underpins this potential in the text, but first I outline the passage of metaphysical argument in the text, through the five parts, in order to underline the ‘ethical’ development of the geometric method. Part I examines the fundamental structure of the world, in its most essential substance, that is, God. God is the only form in which substance can be equated with absolute certainty with extension: God is the ultimate self-determining expression of infinite substance. Also, for Spinoza, the ‘truth’ of a multiplicitous substance is derived from the existence of God but is infused with nature and physics. The Ethics therefore begins by examining the internal constitution of substance in a series of definitions that express the elements and

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their operations, in turn, revealing the multiplicity of each of these constituents within Spinoza’s aesthetic and geometric thinking. In Part II, the propositions and proofs develop an analysis of the structure of the subject through which God’s transcendental capacities are expressed in the fundamental attributes, thought and extension. In order for this journey to be achieved, Spinoza shows how these attributes may join together in agreement to promote as great a sense of joy as possible. In addition, this emphasis on the human powers (i.e., the emotions or affects) is taken up in the two subsequent parts of the Ethics, which are intensive studies of them in practice. The emotions are therefore active agents towards a state of ‘blessedness’ in the individual reader. Part III analyses the modes of the body and mind, that is, its affects, examining the relationship between inadequate ideas and adequate ideas. Spinoza tells us that adequate ideas are the active states between ideas and the body, in which each reflects the power of the other. Inadequate ideas, in contrast, are passive because a confused relationship between the body and mind remains limited by contradiction, so that the body and mind are unaffected by each other, and the movement towards an active state of happiness is obstructed. Happiness (i.e., absolute affirmation) is therefore a form of self-knowledge (selfcause), which is determined by the movement between the ‘powerknowledges’ of the mind and body that ‘bind’ its energies towards perfection. In a detailed study of the emotions and the subject’s ability to achieve an adequate and ethical life, Part IV examines perfection and how this state might be realised in the individual. Here, the embodied emotions are assessed not only in relation to the qualitative effects of good and bad but Spinoza also emphasises how these qualities themselves are expressions of a substantive unity; ‘perfection and imperfection are in reality only modes of thinking’ (E, IV, Preface, p. 153). So when someone changes states from perfection to a lesser perfection, it is not a change of essence, ‘but that we conceive his power of activity, in so far as this is understood through his nature, to be increased or diminished’ (E, p. 154). Spinoza’s emphasis is not therefore just based upon the moral qualities that come from being ‘guided solely by reason’ but also promotes a reason that is derived from the emotions (E, p. 192). Rather, it is an argument for ‘the right way of living’ through a ‘practical’ examination of the emotions (E, p. 195). Nevertheless, it is important to recognise that, for Spinoza, the most perfect mode of happiness is ‘blessedness’, which ‘is nothing other than

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that self-contentment that arises from the intuitive knowledge of God’ (E, IV, Appendix, p. 196). The common notions therefore represent the unity of desire and reason to form ‘intuitive knowledge’ out of the agreement between the mind and body. Embodied reason is therefore the mode that is examined in the final Part of the Ethics. But it also amplifies ‘the method, or way, leading to freedom’, because the geometric method is a continuously changing and embodied activity in the individual, not a totalising set of truths (E, p. 201). Part V therefore represents Spinoza’s most explicit challenge to Descartes’ ‘occultist’ confusion of the mind–body relationship. Arising from his insistence upon the distinction of two kinds of substance, he argues that Descartes’ notion of the pineal gland fails to account for a properly embodied state, because it gives no clear explanation of the ‘union of mind and body’ (E, p. 202). Spinoza, on the other hand, resolves this confusion by returning to the primary examination of the mind and body union, demonstrating how blessedness can be realised, and affirms that the subject has the same qualitative value (or reality) as the divine; for example, he states that passivity is transformed into activity once it becomes an adequate idea (E, V, Prop. 3, p. 204). So, in Part V, rather than reason and emotion exclusively opposing one another, Spinoza argues that they are expressions of the same unified substance and are more ‘truthful’ realities through which to conduct life. Emotions become intimately tied to reason, rather than being rejected as confused or inadequate ideas. This is undertaken in an analysis of the realities – ideas, images and God – that affect the subject; although the limited endurance of the body still remains a fundamental restriction to the scope of human power and hence, perfection, and constitutes an irrevocable distinction between the nature of the method (the power to love) and its goal (God). Nevertheless, the geometric method might therefore be considered the most ‘perfect’ human way to strive for blessedness, because it is the most divine expression or manifestation of God; it is the ‘third kind of knowledge’ – intuition – through which the subject can be led to the greatest contentment (E, p. 215). In addition, the final Part is also a summary of the aesthetic unity that the method produces, insofar as embodied reason may represent a ‘sensibility’ or ‘aesthetic’. Thus, the text and its objects become passages through which perfection is made possible.

Passage III: modes of geometric thinking In this final section Spinoza’s rigorous and intense geometric method is generated out of the scientific axiomatic elements and figures and the

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‘intuitive’ asides of the scholia, to form a ‘productive’ mode of geometric thinking (or passage). In this respect, Spinoza affirms the a priori definition of ‘clear and distinct’ ideas in a discussion about adequate and inadequate ideas in the axiomatic method. The geometric method therefore enables Spinoza to propose embodied, not disembodied, ‘truths’. Furthermore, Spinoza develops Descartes’ axiomatic method (that uses sceptical doubt) into an affirmative practice. So in Part I the axiomatic method affirms the infinitely expressed substance, God-as-Nature; in Part II it explicates the human attributes, thought and extension; in Part III the human powers of expression are revealed in the analysis of the emotions or affects; in Part IV the relationship between the intellect and the emotions is examined for its ‘rightness’; and finally, in Part V, the active self-knowledge (i.e., the agency) of the subject is explored in relation to a divine, yet embodied, immanence. Axioms The axiomatic method enables Spinoza to bring together the embodied subject and the ‘scientific’ figure. As a result, a relationship between God, nature and the human is generated in which the embodied geometric figure is a ‘vital’ human expression of God. If we consider the text’s axioms, we can see how they represent different figures of this relationship: for example, in Part I, the axioms represent the figure of God; in Parts II, III and IV they constitute the figure of man; and finally, in Part V, they represent the figure of the ‘reflective subject’, that is, the internalisation of God in the subject. In this respect, the geometric method enables Spinoza to ‘invent’ a series of intrinsic geometric figures – especially in the common notions – which represent ‘real’ truths because they reflect God. The axiomatic method therefore generates an increasingly embodied series of geometric figures which demonstrate the power of God in divine, scientific, emotional, moral, adequate and inadequate ideas. Looking more closely at the axioms of Part I, for example, we can see how the scientific axiomatic figure is employed, in order to posit a series of embodied, irreducible and aesthetic figures in the following ways: 1. Figures are infinite modes of singularity: All beings exist inherently in themselves or in another being, so that being is conceived ‘through another thing’, and when this is not so, ‘through itself’. Thus, beings are predicated upon a finitude that is also expressive of an underlying infinity (Axioms 1 and 2).

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2. Figures express different realities at the same time: More than one reality exists at any one time. Any reality (being/entity/form) is an expression of a preceding cause or reality (Axiom 3). 3. The reality of the figure is based upon its (self-)knowledge: The greater the knowledge of the realities, that is, its effect and cause, the greater the scope of the effect. The scope of the being is dependent upon the knowledge of the cause (Axiom 4). 4. Figures are individual, yet are common and in agreement with other figures: When there is nothing in common between things they cannot involve/cause/affect another (Axiom 5). Agreement provides the ground, that is, the ‘truth’ of an idea. Ideas, when true, are concrete expressions of other realities, that is, they are made real in conjunction with other expressions of them. They are, therefore, a kind of internal expression of an external set of relations (Axiom 6). 5. Figures are finite and infinite: Limit and unlimit provide the possibility of realities (Axiom 7). Developing from these axioms, we can suggest that a series of operations or figures are generated, which embody both infinite expressions and increasingly singular differences, such as modes or affects. Alternatively, if we look at the propositions we see that these also generate diverse modes of reality – substance, attribute or mode – which express infinity in the form of logical and discrete statements. Once again, a ‘genetic’ relationship between singular geometric figures also reveals an intrinsic connection between one another. Therefore, Spinoza’s axiomatic method is an analytic demonstration of synthetic powers of production (creation), in its examination of the two-way relationship between the divine and the embodied subject. As a result, this passage through ‘creation-production-demonstration’ suggests a highly reflexive method of delivery, in which the geometric method is immanent to the subject, rather than limited to a representational or transcendental structure that is separate from its content. In addition, while this ‘genetic’ discursivity may remind us of Proclus’ geometric thinking, the emphasis on reflexivity between the figure’s operations and the subject’s actions will also be significant aspects of Bergson’s and Husserl’s thinking in the final two chapters. The geometrical method is therefore a ‘volition’ or ‘a mode of thinking’ that affirms the ‘conception’ of an idea; for example, in the Proof to Proposition 49, II, the geometric figure affirms the idea of a triangle because it demonstrates that a triangle must involve the idea that ‘its three angles are equal to two right angles’ (E, p. 96). As a result, Spinoza

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suggests that affirmative thinking belongs to the ‘essence’ of a thing, and hence the geometric figure affirms both ideas and Nature. In addition, Spinoza explores the geometric ‘manner’ in the ‘Preface’ to Part III when he states that it is an expression of ‘the universal laws and rules of Nature’, illustrating this point in the statement that emotions ‘follow from the same necessity and force of Nature as all other particular things’. Emotions are therefore attributed ‘definite causes’ and have specific ‘properties’ that can be examined in the same manner as God and ‘the mind’. In this respect, Spinoza considers ‘human actions and appetites just as if it were an investigation into lines, planes, or bodies’ (E, p. 103). The geometric method is therefore removed from its abstract and lofty generalisations, and has the scope to embody particular realities, which also generates a passage between the text and the reader; that is, an intensively expressed procedure through which the text becomes immanent in the reader and vice-versa. Scholia This reflexive power is most strongly expressed in the scholia, which constitute a dramatic shift in the ‘comportment’ of the method, moving away from the logical delivery of scientific procedures into a series of rhetorical ‘interruptions’ or interlocutions.18 Thus, the Ethics is not merely a scientific demonstration of agreed truths but becomes a passage of expressions or ‘voices’ which agree, disrupt or expand the deductive scientific method in a manner that is reminiscent of the movement between the Pythagorean and Platonic discourses in Proclus’ text. In particular, the scholia embody the passions, affects or emotions so that the axiomatic argument is further amplified. As such, they are figures, singularities or common notions that expand and intensify the method, and confirm its value towards the production of multiplicity rather than the uniformity of ‘One’; for example, Scholium to Proposition 15, Part I, is one of the most forceful arguments about divine extended matter, but it also uses a scientific example of geometry to make its case (E, pp. 41–2). Spinoza therefore revisits the geometric method (i.e., the figures and modes) in order to emphasise the necessity of the adequate ‘truth’ of an infinite God; and he also returns (a second enfolding, perhaps) to the mathematical principles of geometry, thereby emphasising the embodied aesthetic in the scholiatic figure. Thus, these scientific examples enable Spinoza to propose a discursive set of relations, and to demonstrate the divine in the extended figures. In addition, they also constitute ‘scholiatic’ elements, modes or embodiments of geometric figures. Geometric matter is not therefore merely

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constrained to scientific reality but is also embodied in the ‘common notions’, especially in the scholiatic episodes in which discussions about embodiment are most prominent. Occurring principally in Parts I and II these examples are used to distinguish the relationship between substance, limit and extension in the production of the common notions; for example, the first scholium of the text is an extended proof that ‘substance is necessarily infinite’, in contrast to the confused or imagined notions that trees talk or that man is derived from stones (E, I, Schol. 2, Prop. 8, p. 34). Here, therefore, Spinoza provides a geometric example to demonstrate the ‘eternal truth’ of infinite substance when he writes: No definition involves or expresses a fixed number of individuals, since it expresses nothing but the nature of the thing defined. For example, the definition of a triangle expresses nothing other than simply the nature of a triangle, and not a fixed number of triangles. (E, p. 35) This geometric definition of a triangle is not used to make a quantitative distinction (i.e., it does not refer to its limits, as a form or quantity). Rather, it expresses the intrinsic quality of a triangle without limit. The triangle therefore demonstrates God’s infinity, whilst also embodying a ‘sufficient reason’ or mode of knowledge. It affirms ‘corporeal’ or ‘real’ Nature, so that its value as a mathematical figure is not separated from the natural order of Nature, but is instead, an additional expression of it. Thus, the epistemological division between mathematical knowledge (which is limited to the production of ideal or abstract forms) and the sensible realm are brought into ‘commonality’ in the emphasis on embodied and divine realities. This genetic discursivity is also evident in Spinoza’s explanation about the order of ideas and things in the Scholium to Proposition 7, II: Consequently, thinking substance and extended substance are one and the same substance, comprehended now under this attribute, now under that . . . . For example, a circle existing in Nature and the idea of the existing circle – which is also in God – are one and the same thing, explicated through different attributes. (E, p. 67) In addition, this scholium amplifies the statement that the ‘order and connection of ideas is the same as the order and connection of things’

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(E, p. 66). God is the cause of both an unextended idea of the circle and a drawing of a circle, so that the form of the circle is produced from a continuous series of unextended and extended figures (E, p. 67). Thus, the geometric figure is a ‘formal being’ that expresses the ‘order of the whole of Nature’ or of God, and the geometric procedure is one mode of the immanent divine not just an order of representation.

Summary Spinoza enables the geometric figure to be an expression of the subject in its various modes of intellectual, spiritual and emotional embodiment. In addition, these configurations constitute an embodied geometric passage or aesthetic experience. In particular, not only do Spinoza’s common notions express an aesthetic geometry in the passage from one mode to another (such as, the shift from the scientific geometric method into the aesthetic response) but they also suggest a progressive procedure and a practical demonstration.19 The geometric method is not limited to a single mode of representation, but extends into a range of embodied geometric figures, which reveal passages that exist between mathematical, philosophical and aesthetic relations. The Ethics is therefore an aesthetic passage in the journey from the divine to the concrete, staged in five parts to provide a ‘practical’ ethics, and in which the geometric method expresses a complex metaphysical process. Each part develops a path through which distinct modes of metaphysical realities are expressed in nature, God, man, intellect, body and the emotions. In addition, this passage emphasises the importance of modes of enactment or comportment in the method (i.e., it asks, ‘how does the method work?’ not, ‘what object is produced?’). As a result, not only does Spinoza enable the geometric method to express a productive philosophy but it is also a method through which ‘truths’ are reconnected to internal powers belonging to the subject. The geometric method is a rigorous procedure through which a unified, yet complex, and irreducible substance is generated. As a result, its value shifts, from a concern with idealistic truths that are pregiven and inaccessible to an embodied expression of God in specific human qualities. An internally generated reason therefore determines its activities. Rather than producing limited identities or representations, the process is infused with an immanent notion of nature and God, so that the geometric method demonstrates embodied reason (and therefore also reflects Spinoza’s life, which was frequently interspersed with disruptions, risks and confrontations).

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Spinoza’s passages therefore mediate between the thoroughly synthetic geometric method posited by Proclus and Leibniz’s analytic method, explored in the following chapter. In particular, their various modes embody an ambiguous moment in the development of the geometric method, in which the synthetic divisions between mind and body become less distinct as a result of the focus on the intensive movement of the emotions. In addition, we find that Spinoza’s concept of the idea (especially, the adequate idea) presents a complex version of ‘reason’ (i.e., an indivisible ratio between the body and mind) that is derived from the emotions. Thus, a critical shift in the development of the geometric method is established, because Spinoza’s project is also brought into close proximity with Kant’s reflective subject and, as will be shown in the last two chapters, it bears a strong resemblance to Bergson’s ‘intuitive’ and Husserl’s ‘intersubjective’ geometric methods and figures.

4 Plenums

In Leibniz, we find a philosopher whose writings reveal an especially intensive examination of geometry in relation to the principles of division and infinity. In addition, Leibniz’s mathematical theory of geometry, Calculus, is evidence of how he reconfigures principles of quantity into a continuum of differential magnitudes or figures. This chapter suggests that, alongside these analytical mathematical geometric figures, Leibniz also develops an aesthetic geometric method and figures, which are infinitely divisible and embody a qualitative notion of magnitude. Leibniz’s late text, the Monadology (1714), is constituted by two kinds of magnitude which register the inherent infinite divisibility of the aesthetic geometric figure (e.g., the figure of the plenum): first, as a corporeal magnitude, through which an intensive extension is constructed, and second, as an incorporeal magnitude, comprised of the unextended ‘forces’ of perception and appetite. The discussion is also developed with reference to Leibniz’s earlier essays and letters in which the development of these principles can be observed, suggesting that his philosophical and scientific writings are a continuous investigation into the nature of limit and unlimit in mathematical and aesthetic geometric thinking. As a result, Leibniz’s geometric method mediates between Spinoza’s predominantly synthetic method, Kant’s reflective subject, Bergson’s geometric duration and Husserl’s intersubjective subjects, especially because he examines the internal constitution of limit and unlimit in the form of intensive magnitudes within the subject (or Monad). Leibniz’s method is therefore unique amongst the other geometries examined here, particularly because he promotes an analytical understanding of the subject. Nevertheless, there are other notable threads that run between Leibniz, Spinoza, Kant, Bergson and Husserl’s texts, most prominent of 91

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which is the issue of unity between body and mind; for example, Leibniz’s notion of substance and perception has several traits that are similar to Bergson’s ideas of ‘matter and memory’. But Leibniz’s method is distinct from Bergson’s in one important aspect, the value of reason. Reason, for Leibniz, represents a necessary harmony with the divine, which Bergson considers an artificial ‘symbolism’ that limits the irreducible unity to a pregiven value and, hence, restricts the freedom of the individual to an intellectual form of ratio. Bergson’s solution to the issue will place an even stronger emphasis on the scope of the internal ‘transcendental’ powers of the subject in the form of the psychic activity of memory, suggesting that pure and external knowledge is relegated to an obsolete symbolism. However, Leibniz’s analysis of the infinitely divisible subject presents an inventive and rigorous predecessor to the enquiries of Kant, Bergson and Husserl into the transcendental subject; for example, in the next chapter, we will see that Bergson’s metaphysics of duration is also an engaged critique of geometry in his predecessors’, especially, Kant’s, Leibniz’s and Spinoza’s understandings of science. In this chapter Leibniz’s development of an internal differentiation of substance is therefore a particularly important precursor to Bergson’s reconfigurations of matter and memory. Recalling the last chapter, we saw that Spinoza’s geometric method constructed geometric difference through indivisible substance and finite modes to affirm geometric infinity and the irreducible, yet autonomous, subject. But, for Leibniz, magnitude is a special geometric ‘limit-function’, in which the notion of divisibility and indivisibility are reconstructed to form a continuously changing series of irreducible and aesthetic figures. Geometry, as a science of magnitudes, can therefore be described as the construction of bodies that are brought about through the division of bodies into parts. According to this scientific definition, Euclid’s explication of the point, line, plane or surface represents a series of geometric figures or abstract notions that are constructed through the principle of division into finite bodies. Leibniz, however, amplifies the principle of division or magnitude into a radically new form that is an aesthetic principle of unity, and which further augments the infinite divisibility of the scientific form. Magnitude is understood, not merely as the scientific operation that generates discrete and finite divisibility but as a distinctly aesthetic geometric operation. In addition, Leibniz constructs his discrete, yet infinite, figures through a particularly analytic understanding of limit and unlimit; for example, as an analytic mathematical procedure, magnitude produces intermediate states between figures or limits, such as the different

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calibrations between the curve and the straight line, or the curve and the circle; geometry is therefore redefined as an analytic procedure of infinite differentiation. By focusing on the notion of an aesthetic geometric magnitude in Leibniz’s Monadology, this chapter suggests that two forms of the operation are expressed: first, magnitude and substance (i.e., extensity); and second, magnitude in the form of perception and appetite. Extended magnitude (i.e., the body) and unextended magnitude (i.e., the psyche or mind) are therefore unique versions of the ‘limit operation’ in this discussion, which construct an analytic and aesthetic geometric method and figures, and are characterised by internal intensities or magnitudes. Once having examined the constituents of Leibniz’s aesthetic magnitude, the discussion develops to consider his principle of ‘sufficient reason’ in which a qualitative notion of ratio (or reason) is produced. Finally, I explore the formulation of this aesthetic magnitude in the geometric figure of the ‘plenum’ (i.e., the Monad or soul). The Monadology is therefore a demonstration of the geometric method in which the notion of limit is transformed into internal and intensive magnitudes. Leibniz intensifies the operation of division in the geometric method so that the finite geometric identities of the whole and part become a continuous plenitude of irreducible singularities; Monads or souls therefore constitute intensive magnitudes. In addition, the chapter suggests that the Monad constitutes the ‘plenum’, a geometric figure in which a qualitative notion of internal space is generated, because of the emphasis on a continuum of material and immaterial relations that are both internal and external. The plenum therefore represents a kind of topological figure, through which the relationship between the internal structures are continuous with the external form, rather than a finite limit between the interior and exterior, which constructs the discrete autonomy of a geometric figure (and topological geometry will again be important in the following chapter, especially in Bergson’s figure, the ‘envelope’, which bears a strong resemblance to the plenum). The structural organisation of the text also represents a kind of ‘plenum’, because it is a space in which the geometric principles of division and identity (i.e., limit, whole and part) are reconfigured and its internal differentiation is promoted; for example, the form of the text reveals both the continuity between its geometric elements and the division of these ideas into discrete axiomatic sections. Hence, the principle of intensive magnitudes informs the construction of a text that is made up of highly condensed sections, yet these elements also form an extensive plenum, and are reminiscent of the double movements of

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unfolding and enfolding in Proclus’ writing. Geometric thinking is therefore reconfigured, both as a metaphysical argument and in the form of the text itself. Simple, indivisible and qualitative internal orders of differentiation are promoted in favour of the scientific geometric methods that only generate external and formal difference. In particular, we will see how Leibniz’s concern with the powers of perception and appetite and his principle of ‘sufficient reason’ generate this intensive and aesthetic magnitude. Rejecting the dualism that is derived from Cartesian philosophy, the text and its figures – the Monad and plenum – are constructed as a result of Leibniz’s resistance to the opposition between the mind and the body; a resistance that was previously expressed in his earlier writings on the mechanistic opposition between solid and fluid states.1 Instead, we find that Leibniz considers mathematics, especially geometry, to be a valuable intermediate operation through which its figures are not limited to a reductive, representational identity but are understood to represent the inherent difference or ratio between two individuated states or magnitudes. So geometry and its figures are particular differentials, ratios or ‘reasons’, when ‘reason’ designates a particular idea, rather than general ‘truths’. (See, especially, discussions about ‘little perceptions’ and sufficient reason below.) First, however, a more detailed discussion about the development of a post-Cartesian analytic geometric method shows the extent to which Leibniz provides a particularly original position in this discussion and points to some of the connections that link his concepts with the other methods explored here, especially in the shift from the discrete and synthetic figures in a continuum, which characterise Proclus’ and Spinoza’s methods, to a continuum of discrete, yet analytic, figures.

The transition from synthetic to analytic geometry A brief account of the shift from the neo-Platonic synthetic geometric method to a post-Cartesian analytic method clarifies the differences between Leibniz’s method and those of Spinoza and Proclus. Leibniz’s geometric method resists a synthetic order of difference, which generates the identities of its figures out of a series of external limit operations. In the second chapter, for example, we saw that Proclus’ definition of the element or limit was intensive and genetic, insofar as it produced an infinite form of geometric thinking; however, it was still defined by the notion of a synthetic order of difference, attributable to the a priori oppositions of limit and unlimit. Leibniz, however, registers the invention of infinity as an analytic differential, that is, as an intensive

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limit- operation; and the basis of his analytic theory of infinity therefore reflects the rethinking of atomism in the seventeenth century in which atomism loses its synthetic definition (e.g., the idea of the indivisible ‘seed’ of the soul, which is surrounded by matter) to become an analytic discussion of infinite differentiation (Arthur, 2001, p. xlviii). So, for Leibniz, the inheritance of the Pythagorean principles of limit and unlimit enables him to generate unusually ‘analytic’ conclusions about the limit, atom and axiom; and through which the structure, function and operations of limit and unlimit are radically reformulated, not in order to produce exclusively discrete identities or the unchanging external differentiation of limit in the curve and the circle, but to express infinite internal degrees of difference. Thus, seventeenth-century developments in mathematics modify the belief in synthetic and external magnitude that underlies geometry in Classical metaphysics, so that it becomes an internalised series of differential changes. Robert Latta’s commentary on Leibniz’s method succinctly describes this important shift in understanding, explaining the ‘transition’, as follows: Early in the seventeenth century a considerable advance was made in the science of Mathematics, mainly through the work of Kepler, Cavalieri and Descartes. The Geometry of the Greeks was synthetic or synoptic. It dealt with the ideal figures as discrete wholes, not taking into consideration the possibility of them being analysed into elements, of which they are combinations or functions. Thus the relations of the figures to one another are considered external. Each is what it is: no one is regarded as having in it the possibility of passing into another. A rectilineal figure is one thing; a curvilinear figure is another. The barriers between them are insurmountable, at least by the methods of exact or demonstrative science. Thus a curve is still a curve, however small may be its curvature. A polygon is still a polygon, however numerous may be its sides. And the kinds of curves are each independent of the others. An ellipse is still an ellipse however distant one focus may be from the other. Kepler’s introduction of the notion and name of infinity into Geometry was the beginning of a great change in mathematical models. The geometrical figures of the Greeks were all finite, and therefore capable of representation to the eye, or, in the other words, capable of being pictured . . . . Kepler, in order to attain a greater exactness in the statement of mathematical relations, suggested that finite (or definite) figures might be regarded as consisting of an infinite (or indefinite) number of elements. (Latta, 1985, pp. 75–6)

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An ‘intensive notion of substance’ is therefore generated when the mathematical principle of division is made more substantial, so that an infinite progression between the curve and straight line is emphasised, rather than the ‘purity’ of the finite difference in synthetic geometry. As a result, this also shifts the long-standing opposition between quality and quantity, so that these two modes of difference are brought into a single concept of magnitude that is qualitatively differentiated by degrees within a continuum (i.e., it is not a ‘mixture’, as the Stoics propose). As we have seen in the previous chapter, Spinoza’s thinking is also informed by seventeenth-century debates about infinity in relation to the problematic Cartesian dualism of the mind and body. The notion of an indivisible, yet multiplicitous, substance provides Spinoza with his solution to the problem, whereby internal difference exists in finite modes or affects. However, for Spinoza, any unity or ‘figure’ is synthetic because, although it is indivisible as an irreducible union of modes, it is also determined by a finite notion of limit in each singularity. In contrast, Leibniz proposes multiple and infinite substances that generate infinitely divisible unities or figures, and so his notion of limit is intensive, because it is generated out of an infinitely analytic divisibility. Second Spinoza’s concept of substance is predominantly extensive and modal, that is, its powers generate discrete external modes of existence. Leibniz, on the other hand, produces a notion of extended bodies or figures that are intensive and multiple. Each method therefore resists the Cartesian premise of mechanised substance that reduces limit to the finite divisions of the whole and the part. Leibniz undertakes this challenge through an increased emphasis on an analytical understanding of ‘infinity’ in metaphysics and mathematics (i.e., in his theory of Calculus).2 Thus, Leibniz’s theory of Calculus generates a mathematical version of the intensive limit-operation that enables, for example, analytic geometric knowledge to be applied to the development of the physical sciences. (In contrast, Spinoza invents synthetic geometric ideas that represent an early form of psychology, because they are formed by the sensibility.) Thus, both philosophers adopt a critical position towards the assumed Cartesian split between mind and body, and Leibniz’s magnification of division or limit enables geometric intensity to be generated from a scientific procedure that also expresses a particularly intensive substance. In addition, Leibniz’s intensive substance provides a fascinating ‘parallel’ to Spinoza’s conceptualisation of extensive substance, especially because each philosopher is sceptical of the divisible body, leading them to affirm the inherent immateriality in

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substance, and which therefore also reflects neo-Platonic discussions about the soul. Thus, Spinoza proposes the ‘common notions’ as a form of this irreducible unity and Leibniz proposes the Monad (an Entelechy or soul). In each case the soul is the site of a complex, irreducible substance that is inherently related to matter, yet independent from it, and expresses a continuity between extended and unextended matter. So infinity is immanent in both geometric methods: in the Ethics it is understood as an indivisible condition of unity in a univocal substance, whilst for Leibniz it is an infinitely divisible operation or logic; that is, it produces unity in which the concept of infinity is augmented by an intensive analysis of infinitely divisible substances. As an analytic operation, then, infinity becomes an active principle through which intensive substances are generated, and the connection between the soul and the body is more clearly comprehensible, as a continuum of aesthetic and differential magnitudes. By contrast, in Spinzoa’s modal differentiation, the details of the relationship between the different affects or emotions (i.e., the causes) are defined as sensible powers, not in logical terms. Spinoza attributes causality to the powers of mind and body, in which a creative God is immanent, so that modes register a genetic evolution of differentiation, but are emphasised as attributes of this infinite God, therefore accounting less for the incremental changes that take place between the internal and external states, or the shift from the sensory to the divine. Thus, because Spinoza posits a divine indivisibility as his premise for extended infinity (which also underpins the principles of movement or causation), the clarity of explanation that an analytic method brings to a discussion of infinity is overlooked. For Leibniz, however, cause is explained as distinctly differentiated internal forces (such as perception and ‘appetition’, which are immanent in the individual Monad) and he promotes the multiplicity of these substances, so that God’s powers, although omnipresent, are not the first order of expression.3 Thus, Leibniz’s geometric principles of extended and unextended ideas and bodies are characterised by an analytical infinity (i.e., magnitude), rather than by an indivisible modal substance, thereby demonstrating the extent to which his geometry transforms the production of synthetic or absolute truths and bodies, into analytical and intensive magnitudes.4 Leibniz’s analytic method is also distinct from Kant’s notion of geometry in the Critique of Pure Reason.5 The first chapter distinguished between Kant’s concept of analytic and synthetic agreement in the first Critique; for Kant, analytic identity is an internal agreement between two related elements, and synthetic identity is comprised of an external

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agreement between two independent elements. Qualitative difference is therefore generated by the heterogeneous mixture derived from a synthetic operation, whilst analytic operations are considered homogeneous because no external, that is independent, difference is introduced into the relationship. Leibniz, however, promotes analytic difference as the means through which to generate internal, heterogeneous difference in quantity (i.e., magnitude), therefore rejecting the claims that analytic agreements exclusively generate ‘self-same’ and determinate unities. In the Critique of Pure Reason, therefore, synthetic and analytic agreements are distinct from the Pythagorean notions of limit and unlimit. However, in the Critique of Judgment, infinite and differentiated magnitudes are present, especially in the relationship between limit and the imagination in the production of the sublime (and Leibniz’s study of ‘fictional figures’ also anticipates this relationship). In addition, there is commensurability between Leibniz’s geometric figures and the reflective subject, in particular, in each philosopher’s emphasis on the discrete, yet continuously changing, autonomy of the subject. Nevertheless, Kant’s argument upholds the value of synthetic limit and the imagination’s powers of synthetic production so that the power in an analytic definition of limit remains under-explored in his thinking. For Leibniz, however, the irreducibility of an analytic notion of magnitude (i.e., the inherent ‘ratio’ between limit and unlimit) is precisely where the power of his method lies in the construction of geometric figures.6 As a result, a priori definitions of quantitative magnitude that persist in the Cartesian and neo-Platonic methods are reconfigured by Leibniz’s analytic geometric method into infinitely divisible, qualitative difference. In addition, the division between extension and unextended ideas is broken, insofar as limit is an operation that produces unlimited relations of magnitude. Each of the previously necessary geometric relations or ratios (e.g., whole/part, divisible/indivisible or quality/ quantity) are rethought to constitute continuously changing multiplicities or geometric figures. Internal difference is brought to the fore, in the irreversible shift from quantitative and homogenous (i.e., undifferentiated) magnitude, into qualitatively differentiated and heterogeneous magnitude, which radically distinguishes Leibniz’s geometric method from his predecessors. Later in the chapter, I consider Leibniz’s qualitative differentiation of substance in the notion of incorporeal magnitude which is produced by perception and appetite. First, however, the structure qualitative difference produced in corporeal magnitude is examined, in order to understand the corporeal nature of his aesthetic geometric method and figures.

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Corporeal magnitude Leibniz’s research into magnitude can be traced back to a discussion about ‘incommensurable’ figures in the essay, ‘On the Nature of Corporeal Things’ (1671), where he describes magnitude as ‘the multiplicity of parts’ (Arthur, 2001, p. 345). He also writes that magnitude is ‘the constitution of a thing by the recognition of which it can be regarded as a whole’, in an essay written from 1676 called, ‘On the Secrets of the Sublime, or On the Supreme Being’ (Arthur, 2001, p. lxxii). The Monad is therefore a geometric figure which arises out of a longstanding investigation into the principles of magnitude and limit. Moreover, the Monadology’s exploration of these geometric principles strongly reflects Leibniz’s engagement in geometric thinking, which can be drawn from Euclid’s Elements, through Proclus’ study of unlimited limit in the axiomatic method, and Descartes’ analytic method. In §3, for example, Leibniz calls Monads ‘the Elements of things’, positing an explicit relationship between the substance of the Monad and Euclid’s term ‘element’ (Monadology, 1973, p. 251).7 In particular, if we recall Euclid’s first definition in the Elements – a ‘point is that which has no part’ – we find the paradox of limit and divisibility present in both the notion of the point and the Monad. But Leibniz’s notion of limit also represents an analytic version of division because, rather than defining atoms, points or elements in terms of a synthetic relationship of limit and unlimit, the Monad is an irreducible, analytic magnitude; for example, in the initial sections of the text we saw that the notions of magnitude, infinity and substance produce a highly complex kind of entity or geometric figure, in which division or limit are dramatically redefined. In §1 Leibniz calls the Monad a ‘simple’ substance ‘without parts’ and in the following sections he develops its instantiation in relation to magnitude (M, p. 251). It is expressed in the notion of ‘aggregate’ in §2; and in §3 it is ‘Atoms of Nature’ that are ‘neither extension, nor form, nor divisibility’; then in §§4–5 it is ‘indissoluble’ or without beginning. Magnitude, as a principle of geometric construction, is therefore integral to the first axiomatic statements of the Monadology. The operation of divisibility is made complex so that the pregiven divisions of a unity (such as division into the whole and part) are disrupted because an amplified or multiple kind of limit is introduced. Division becomes a multiplicitous operation, registering limit and unlimit together, rather than its separation into either a limit that demarcates a divisible magnitude (e.g., the finite figure of a circle, and its division into two parts or semi-circles) or unlimit which lapses into

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‘indivisible’ infinity. Thus, both limit and unlimit are constituted through a procedure of infinite divisibility, and from which geometric figures are imbued with different magnitudes or powers of unlimit and plenitude. Leibniz’s critique of substance also engages with seventeenth-century mathematical and Cartesian debates about atomism and geometric division, for example, the contemporary view that the atom is neither a simple geometric point nor a concrete, unchanging entity.8 In this respect, the Monad contributes to anti-Cartesian discussions about the potential for unity between the mind and body in an infinite entity; that is, the expression of an infinite substance or extensity through which the Monad is conceived as indivisible and corporeal. In addition, Leibniz’s contribution highlights the limitations in theories constituted by the synthetic division of the whole, part, soul, matter and mind, because Leibniz’s simple ‘element’ or atom is not an abstract notion but an active ‘substance’. Limit therefore becomes immanent to concepts of life in multiple ‘simple substances’. But neither does Leibniz’s theory restrict the Monad to a finite extended corporeality, since that would limit it once again to a determinate quantity or magnitude. Later, in §§40–8, Leibniz explores how God lends another essential expression of qualitative infinity to the Monad, establishing a connection between the Monad’s limit and all other realities. In §40, Leibniz states that God is infinite; he is ‘a pure sequence of possible being’ and contains ‘as much reality as possible’, and in §43, God is the source of existence and essences, that is, ‘the source of whatever there is real in the possible’ (M, pp. 259–60). However, God’s infinity is also determined, not just by the magnitude of everything actual and possible (i.e., the ‘immensum’) but in his perfection, as §41 explains: God is absolutely perfect, perfection being understood as the magnitude of positive reality in the strict sense, when the limitations or the bounds of those things which have them are removed. There where there are no limits, that is to say, in God, perfection is absolutely infinite. (M, pp. 259–60)9 However, ‘created things’ are determined by their natural limits; for example, in the ‘natural inertia’ of bodies versus the unlimited perfection of God (M, §42, p. 260). Extended beings therefore embody a limitthreshold of magnitude, which is determined by their own internal capacities, whereas God has no limit-threshold to his perfection. This examination of the magnitude in God’s infinite perfection continues in

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§§45–8. For example, §47 states that God is the ‘ultimate unity or the original simple substance’ from which all other realities are derived (M, p. 261). Like the Ethics, therefore, the Monadology confirms God to be the primary cause of infinity or unlimit; but Leibniz’s geometric method also emphasises the corporeality of the infinite and intensive extension (perhaps to a greater extent than Spinoza), insofar as the notion of the divine is not the primary discussion of the text. Instead, it studies the Monad’s internal magnitude (for Spinoza, however, God is the first principle of infinity that is examined in the Ethics; also see note 3 in this chapter). Later in the chapter, I show that this distinction also underpins Leibniz’s principle of ‘sufficient reason’, in which internal ratios are the cause through which man and God are brought into harmony. The Monad therefore represents a notion of substance that is immaterial and material, intensive, yet also, extensive. In addition, its relations are determined, not by the production of either a single divisibility (limit) or an indivisible unity, but as a result of degrees of infinite divisibility. Thus, divisibility and indivisibility come under the principle of magnitude, because they are qualitative degrees of difference and intensity, rather than representing two opposing kinds of quantity; and below I argue that this differentiation produces the qualities of incompossibility or ‘vice-diction’ in sufficient reason. In this respect, magnitude is a continuum, generated through an internally differentiated limit. The premise that finite divisions lie between the whole and part is transformed into a series of infinite evolutions in which the Monad is a geometric figure generated not only from a progression of discrete elements, such as the axiom, but also from embodied psychic relations. Magnitude, then, is consistent with an aesthetic passage from the unextended idea into the extended figure (rather than being a purely mathematical determination), because simple and discrete corporealities are shown to be inherently incommensurable, and therefore refute the imposition of finite beginnings or ends that are commonly associated with quantitative magnitude. So magnitude is inherently derived from the plenitude of ‘extensity’, and Leibniz tells us that there are multiple substances, rather than one infinite substance, such as Spinoza’s univocal substance. The concept of existence is therefore expressed in relation to infinite substances that are divisible into infinite parts. Substance is considered to be infinite, not because of its formal limits but as a result of its relationship to the soul or memory (i.e., an intensive extensity), so that ‘wholeness’ becomes untenable because its pregiven limits – for example, beginnings and

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ends – are unthinkable (this emphasis on the immaterial forces of the Monad also recalls Spinoza’s idea of the conatus and prefigures Bergson’s discussion of duration). Once again, the notion of determinate, divisible wholes or parts becomes highly problematic, and it is a discussion to which Leibniz returns throughout the Monadology; for example, in §8 he writes of the continuous change in the plenum that cannot be reduced to a division of whole and parts. Thus, in the notion of multiple ‘simple substances’, such as the Monad or the ‘plenum’, an infinite notion of division and difference is posited, which also reinforces the double operation of limit and infinity. In addition, infinity is prioritised in the sections which examine how simple substances form ‘aggregates’ or ‘composites’, rather than finite wholes (M, §2, p. 251). Magnitude is the key operation in the formation of conglomerates or aggregates, because it is concerned with continuous differentiation, rather than being limited to either the infinitely small (e.g., ‘infinitessimals’) or the largest quantity (e.g., the ‘immensum’). Instead, the notion of limit becomes a kind of ‘approximation’ or ‘accident’, rather than a determined or pregiven ‘end’. First, however, in a short summary of the discussion so far, we find that the scientific and quantifiable notion of geometric magnitude (represented by the whole and the part, finite limit and indivisible infinity) is untenable in the following ways: 1. It is aligned with a mechanical division of Cartesian substance that overlooks the possibility of an immaterial extensity; 2. Division into whole identities does not admit the provisionality of ‘ratio’ or ‘sufficiency’ in the operation of infinite divisibility. Wholeness is therefore an inadequate notion of identity, because magnitude is only partially explained, rather than registering its value, as an intensive operation in infinite continuums; and 3. The part cannot be a smaller or finite imitation of the whole. The complex geometric multiplicity of the Monad’s magnitude (i.e., that which defines its unity) is therefore founded upon; a. differential limit that cannot be reduced to a finite part or whole; and b. infinite divisibility in extended matter and unextended thought. In addition, because the Monad is promoted in the shift from an abstract principle (i.e., the point) into a metaphysical substance, Leibniz augments a connective principle of infinite change by bringing together a highly defined mathematical concept and an ontological inquiry, so that the atom and element become ‘concrete’ realities. The Monad

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therefore constitutes the geometric principles of limit and unlimit in God, nature and life. As a result, the persistent belief that finite division determines analytic geometric methods is transformed into an aesthetic and qualitative discussion. In the following section, the immaterial magnitudes which produce embodied geometric thinking are explored in more detail.

Incorporeal magnitude Magnitude produces substance (extension), however, Leibniz also produces a theory of magnitude which is derived from qualitative and incorporeal differences (forces) in his analysis of perception and appetite. As a result, this ‘incorporeal magnitude’ connects Leibniz’s theory to Spinoza’s affects and Bergson’s notion of memory; for example, Bergson constructs an aesthetic geometry in relation to the psychic activities of the individual which, on first sight, might appear to be closer to Spinoza’s theory of psychic activities than Leibniz’s theory of ‘internal activities’, because Spinoza’s emotions represent more developed modes of psychic definition and activity than Leibniz’s ‘logical’ conditions of perception and appetite.10 However, I show below that this is too sharp a distinction, because Leibniz’s theories also bear a strong resemblance to Bergson’s theories of ‘matter and memory’, in particular, when Leibniz argues that the psychic activities of the Monad prevent it from merely being an inert, yet infinitely, divisible, thing. So although Bergson’s ideas are clearly more developed as psychic activities, nevertheless, they display similar traits of intensive corporeal magnitude as Leibniz’s writings on perception and appetite.11 Thus, we can suggest that Leibniz’s discussion of perception, appetition and the soul are important constituents of the Monad’s geometric aesthetic unity and agency. Perception Perception is the incorporeal principle of change, which determines the Monad’s unity, in particular, because it is the condition through which the soul and body are brought into continuity. It is therefore a kind of ‘magnitude’, an intensive limit or ratio; for example, in §14 Leibniz defines it as ‘The passing condition which involves and represents a multiplicity in the unity, or in the simple substance, is nothing else than what is called Perception’ (M, §14, p. 253). Leibniz’s examination of perception’s multiplicitous nature shows that it generates all perceptions, even those of which we are not aware

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(e.g., dreams). This, Leibniz explains, is apperception or consciousness. Perception is therefore different psychic levels of activity through which different intensities of harmony and order are produced; so that, in the production of its singularity, perception provides the Monad with the capacity for greater or weaker intensities of ‘attention’, which results in different degrees of harmony between it and the world. (Once again, Leibniz’s discussion bears a striking resemblance to the notion of perception and the ‘virtual’ in Bergson’s writings; and in the following section, we also see that Leibniz’s figures of ‘small perceptions’ are continuous with Bergson’s concept of memory.) Leibniz reinforces his argument with reference to the ‘mistakes’ of those Cartesians who consider consciousness to be the defining attribute of existence, not the founding principle of existence, which includes a real virtual level of existence (M, §14, p. 253). Perception, for Leibniz, therefore resists the Cartesian principle of exclusion between the soul and the body. Furthermore, as a challenge to Cartesian claims that the soul and the body are separate from each other, Leibniz accuses them of having ‘adopted the Scholastic error that souls can exist entirely separated from the bodies, and have even confirmed ill-balanced minds in the belief that souls are mortal’ (M, p. 253). By contrast, the Monad’s powers of perception and, by implication, the soul, exist as a series of embodied intensities. Thus, Leibniz considers souls to be ‘indestructible’, because of the relationship between the different intensities of activity (or awareness) that constitute substance and the mind. Appetition Having demonstrated that perception is consciousness in general, Leibniz examines the production of particular degrees of awareness and directed thought in the principle of ‘appetition’, again emphasising the embodied nature of perception, rather than its value as a disembodied cognitive operation. Appetition is the ‘internal principle’ that ‘brings about the change or the passing from one perception to another’ (M, §15, p. 253). This ‘desire’ (l’appetit) strives for ‘the whole of perception’ but does not attain it; however, in doing so, appetition produces ‘new’ perceptions. In §17, Leibniz explains that perception and its appetites are not reducible to symbolic explanations, such as ‘mechanical causes’ or the ‘figures and motions’ that they produce. Here, we therefore see Leibniz’s development towards the principle of sufficient reason, which will further demonstrate that the mechanics of causal change, such as sequential order and oppositional truths, are inadequate explanations to

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describe either, the infinity of the Monad, or its internal structures (M, §17, p. 254). Leibniz elaborates, writing that perception and its products are not reducible to a mechanical diagram of magnitudes, parts and wholes, visible in the analogy of the internal workings of a mill. Perception, he tells us, is ‘sought’ in the simple substance (rather than the ‘composite’ substance or the ‘machine’) through which the ‘internal activities’ of perceptions and their changing appetites resist reduction to a mechanical or composite set of elements. Perception therefore represents an internally differentiated force that also reflects Spinoza’s discussion about modes and affects, insofar as both inherently express the ‘life’ of the unity and are qualitatively differentiated into distinct thoughts, ideas or images. In the following passages, Leibniz considers the qualitative nature of perception, as an infinitely divisible magnitude, in the form of ‘little perceptions’. In §21 he describes the different qualities of perception in which different states can be observed, and in this form such perceptions are considered to be ‘weak . . . in which nothing stands out distinctively’; for example, the act of ‘spinning around’ which causes the ‘power of perception’ to be weakened (M, p. 251). Alternatively, Leibniz writes that these ‘little thoughts’ are akin to states of ‘undirected’ or ‘approximate’ perception, such as unconsciousness or dreams, in contrast to the suggestion that the loss of consciousness in sleep results in a nonthinking substance (M, §§21–3, pp. 255–6). Perception, then, endures continuously in the Monad, and this continuous passage through different states of awareness means that any given perception is always a concatenation of its past and futural states: ‘Every present state of a simple substance is a natural consequence of its preceding state, in such a way that its present is big with its future’ (M, §22, p. 256). So in §23, Leibniz is able to demonstrate the ‘virtuality’ of perception when the present perception is understood to be part of a previous perception (which is also strongly echoed in Bergson’s discussion of the contraction and expansion of memory). Leibniz writes: ‘for one perception can come in a natural way only from another perception, just as a motion can come in a natural way only from a motion’ (M, §23, p. 256). In addition, this emergence of perceptions from preceding perceptions is reminiscent of both the enfolding and unfolding of images that we observed in the relationship between the imagination and soul in Proclus’ Commentary, and the duration of Spinoza’s conatus in the embodied subject. Alternatively, in §25, Leibniz tells us that the soul’s perception ‘represents that which goes on in the sense-organs’, reinforcing the

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relationship between the unextended capacities of the soul and the qualitatively different perceptions that are generated in the sensing body. A ‘concatenation’ also exists between Leibniz’s idea of perceptions and Spinoza’s theory of affects because each philosopher posits an intensive, psychophysical relationship between unextended and extended materialities. These ‘small perceptions’ or memory therefore represent different states of embodied perception in which memory is an internal kind of reason, when Leibniz writes: ‘The memory furnishes a sort of consecutiveness which imitates reason but is to be distinguished from it’ (M, §26, p. 256). Furthermore, like Bergson, Leibniz also reflects on the natural sciences and the how memory is expressed in an animal’s perceptions; for example, suggesting that animals ‘are led by the representation of their memory to expect that which was associated in the preceding perception, and they come to have feelings like those which they had before’ (M, §26, p. 256). Thus, in the analogy of a dog remembering the pain that comes from being struck by a stick, Leibniz suggests that ‘reason’ arises out of representations (i.e., images, ‘reasons’ or ideas) generated by the memory, which is a continuous unity of different perceptions. Here, the strength of a perception of an image or ‘picture’ is therefore dependent upon the magnitude derived ‘from the number of the previous perceptions’ (M, §27, p. 257). Perception, then, is the operation through which action is explained, countering the Cartesian belief that bodily actions are the effects of external mechanical causes, because it is embodied in the autonomous and internal changes in memory and appetite. Soul Having revealed how the corporeal forces embody the Monad, as a series of intensive limits, Leibniz devotes the following sections to a detailed explanation of the primary incorporeal unity, that is, the soul. On the one hand, the soul designates the Monad’s nature as a ‘simple substance’, and on the other hand, it is the principle of sufficient reason (ratio) because it produces unity out of an internally generated, divine reason. But we have also seen that the Monad is an embodied perceiving entity, and so Leibniz connects the continuously perceiving substance with the incorporeality of the soul in the statement: ‘[If we] designate as soul everything which has perceptions and desires in the general sense that I have just explained, all simple substances or created Monads could be called souls’ (M, §19, p. 255). Thus, for Leibniz, the Monad is a particular unity of perception and appetite (rather than a general class of entity) which he defines in

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greater detail, arguing that whilst it is true that a Monad is a perceiving entity in general, it is the Monad’s capacity for memory, or perceiving as a duration, that distinguishes its definition as a soul from a simple Monad (or Entelechy). In addition, we will see that the Monad is a corporeal kind of infinity called sufficient reason, in contrast to the problematic ‘potentiality’ of infinity that is produced in a ‘pure reason’ of infinity.12 ‘Soul’, then, is really only applicable to those Monads in which perception is more distinct than a general ‘feeling’. An Entelechy or Monad designates perception, whereas a soul embodies perception which is produced in part by memory; that is, ‘the term Soul [refers to] those whose perception is more distinct and is accompanied by memory’ (M, §19, p. 255). Thus, in the following §20, the soul is considered to endure and, as a result, registers the possibility of different states of consciousness, such as dreams or fainting. The soul enables consciousness to pass through one state into another; that is, the loss of consciousness in dreams, sleep or fantasy does not designate the loss of existence or soul but is evidence of different kinds of intensity and perception that exist in the Monad (M, §20, p. 255). In addition, Leibniz considers the soul to be the infinite unity through which a special kind of reason is constituted. In §§29–30 he explains that the mind is the ‘rational soul’ that distinguishes us from ‘lower’ Monads by ‘the knowledge of eternal and necessary truths’ (M, p. 257). So this rational soul or mind constitutes the faculty of reason that connects us to the ‘necessary’ laws and ‘abstractions’ of nature and enables us to perform Reflective Acts. Moreover, it underpins our understanding of God as the principle of perfection. Thus, the mind is defined as a particular kind of perceptive reason that brings us into a natural order of infinity and perfection, and its reflective acts provide the basis for the formation of the self-conscious subject (i.e., the ‘I’). For Leibniz, then, the soul is the principle of sufficient reason, because it generates an aesthetic magnitude that is derived out of the harmony between the internal activities of perception, desire and memory, and the external laws of God and nature.13 Located between the theories of Spinoza, Kant, Bergson and Husserl, Leibniz’s autonomous thinking subject therefore provides an important figure of continuity within the different geometric methods. Thus, the Monad is constituted by self-generated reason or internal action, rather than being determined by external causes. Interiority and the internal activities of the Monad become primary concerns, when Leibniz defines the notion of limit in the Monad in terms of its ‘internal

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activities’, intensifying the aesthetic geometric unity of the Monad so that it is neither reducible to an external notion of form nor determined by external laws of mechanical cause and effect (M, §18, p. 255). Instead, the intensive, corporeal and internal qualities of the Monad are generated through an internal continuum of magnitude. This marks an important shift in the scope of the geometric method on two counts: first, geometric thinking is dramatically recast by the introduction of intensive, internal relations that produce differentiated figures and limit, thereby undermining the prominence given to formal procedures which generate exterior space; and second, a concrete, yet analytical, explanation is given for the previously mystical notion of limit (as indivisibility) that was observed in Proclus’ method. As a result, limit is brought into an internal and embodied series of relations, rather than remaining an abstract principle of production; for example, §7 contains the famous description of this autonomy, which states that the Monad is completely independent from all external causes or affects: There is also no way of explaining how a Monad can be altered or changed in its inner being by any other created thing, since there is no possibility of transposition within it, nor can we conceive of any internal movement which can be produced, directed, increased or diminished there within the substance, such as can take place in the case of composites where a change can occur among the parts. The Monads have no windows through which anything may come in or go out. (M, §7, p. 251) The Monad is therefore constructed by an internal imperative. In addition, it transforms the subject from mechanical and external causes and effects into an aesthetic principle of life. In the following sentences of §7 Leibniz reveals the integrity of internal difference in the Monad in his concept of the ‘attribute’, and which is in contrast to Spinoza’s modal notion. He writes: ‘The Attributes are not liable to detach themselves and make an excursion outside the substance, as could sensible species of the Schoolmen. In the same way neither substance nor attribute can enter from without into a Monad’ (M, p. 251). So, unlike Spinoza’s finite modes, Leibniz promotes the infinite magnitude of internal attributes. Externality, however, is also a real state, since it is in the external relations between Monads and a natural order to which Spinoza’s modes and Leibniz’s ‘sufficient’ reason correspond. The Monad is, then, both extensive and intensive: a singularity or an irreducibly discrete entity with its own agency; for example, §12 introduces the concept of the

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Monad as ‘manifold’, expressing its capacity for extensive change in a differentiated and intensive substance. Leibniz explains that the manifold ‘constitutes, so to speak, the specific nature and the variety of the simple substance’ (M, §12, p. 253); and he continues this explanation in §13: This manifoldness must involve a multiplicity in the unity or in that which is simple. For since every natural change takes place by degrees, there must be something which changes and something which remains unchanged, and consequently there must be in the simple substance a plurality of conditions and relations, even though it has no parts. (M, §13, p. 253) Change, then, is not just a single, consistent measure of intensity but is as varied as the multiple states of difference existing within the manifold, rather than being derived from an external force, or dividable into units or parts. The manifold or continuum therefore displays the characteristics of intension and extension that have been observed in Proclus’ and Spinoza’s theories and which will also constitute the continuity of duration in Bergson’s philosophy.14 A further correspondence with Spinoza’s geometric ideas is evident in Leibniz’s examination of the perceptual limits of the body, as a form of intense magnitude, and Spinoza’s concepts of active and passive, adequate and inadequate ideas. Thus, the Monad’s capacity for action is also brought under the condition of intensive magnitude because when a Monad is active it has ‘distinct perceptions’ and when it is passive ‘it has confused perceptions’ (M, §49, pp. 261–2). Active and passive expressions of its forces, ‘endeavours’ or conatus, therefore reveal its ‘perfection’, because they transmit the Monad’s internal order of magnitude into external actions of magnitude or limit (§52). In addition, we can see that these discussions of active and passive forces have geometric value towards the production of fictional geometric figures; for example, the inaccuracy of small perceptions, such as dreams or dizziness, produces infinite figures that are similar to the imperceptible states of change, which are registered in the calibration from a curve to a straight line; that is, ‘small perceptions’ are analogous to Leibniz’s mathematical invention of ‘approximate’ or ‘indiscernible’ figures in Calculus.

Sufficient reason In the principle of sufficient reason, Leibniz invents a theory of logic or ratio that produces the aesthetic geometric unity of the Monad or soul.

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Underpinning the aesthetic premise of the Monad, sufficient reason operates by means of an analytic logic to generate the internal, autonomous and reflective ‘I’ of the Monad, especially in the shift from a relationship of contradiction to ‘incompossibility’ or ‘vice-diction’. Thus, it constructs qualitative reason in which ‘truth’ is equated with the idea of the ‘best’ or most ‘fitting’ relationship in a continuous series of possibilities. Such a notion of reason means that the predicate and its agreement (such as, the opposites of mind and body, or internal and external relations) are brought together, not as quantitative magnitude but as a qualitative ratio of different relations. Sufficient reason is first expressed in §18, following Leibniz’s definitions of Perception and Appetition, when he tells us that the perfection of the Monads is to be understood in terms of their ‘sufficiency’: All simple substances or created Monads may be called Entelechies, because they have in themselves a certain perfection . . . . There is in them a sufficiency . . . which makes them the source of their internal activities, and renders them, so to speak, incorporeal Automatons. (M, §18, pp. 254–5; my emphasis) Monads are perfect insofar as they have a sufficient source of internal relations that comprise their ‘incorporeality’. Sufficiency is equated with the composition of the Monad, as an immaterial substance, so that in the following sections (§§19–28), Leibniz explains sufficiency in terms of a continuum of perceptions, duration and memory, which constitute the sensuous and perceiving subject. Then, in §29, he turns to the particular knowledge of self and God – that is, self-consciousness – which he calls, the ‘Rational Soul or the Mind’; and through which we are able to construct the special unity of thinking substance, the ‘reflective I’ (M, p. 257). Leibniz continues to distinguish the principles through which we produce reason, drawing the distinction between the internal structures of sufficient reason and ‘reason’ that is gained through the principle of contradiction (M, p. 258). These two principles are explained in §§31–2 when he states that contradiction enables us to produce the notions, ‘truth’ and ‘false’, whereas sufficient reason provides us with a contingent notion of truth, based upon the fact that the existence of any truth is itself a sign of its own sufficient reason. Truth, under the principle of sufficiency, therefore becomes a contingent or embodied form of ‘reason’ that exists in a ‘fact’. In §33, Leibniz continues, writing that there are two kinds of truth, which are produced by reason – Reasoning and

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Fact. Reasoning is necessary, he states, whereas facts are contingent, allowing the existence of the contradictory facts in the same statement. In addition, he notes that reasoning can be divided into ‘simpler ideas and simpler truths’ until primary truths are given; for example, in the progression of mathematical or geometric proofs. Then, in §34, Leibniz examines the geometric method with respect to an analytical understanding of reason, rather than as a synthetic principle of reason which is determined by the external division of contradiction; for example, the analytic relationship between the axiom and the problem is not defined by external difference, rather, in the analytic method the principle of change is from within the subject. Alternatively, Leibniz writes that mathematics ‘resolves’ these contingent speculations into axioms, definitions or postulates. Speculations are not false, therefore, because they do not ‘contradict’ the primary principles of axioms. Similarly, axioms do not exclude differences, because they contain internal differences, such as the hypothesis of the speculative problem. In §35 Leibniz explains: There are finally simple ideas of which no definition can be given. There are also the Axioms and Postulates or, in a word, the primary principles which cannot be proved and, indeed, have no need of proof. These are identical propositions whose opposites involve express contradictions. (M, p. 258) Thus, if we take Leibniz’s critique of the finite axiom, together with his emphasis on ‘infinite divisibility’, the geometric ‘element’ (such as the axiom or proposition) is not a certainty but a principle that can hold a ‘mixture’ of differences or contradictions within itself; that is, geometric elements and their products are comprised of ‘incompossibilites’ of contradiction (or, as Deleuze writes, ‘vice-diction’).15 Geometry therefore expresses a continuously changing continuum of internal ‘sufficient’ reason, not systematic and finite ‘laws’ or truths. Furthermore, sufficient reason is constructed from the principles of sufficiency that reconfigure intensive magnitudes into ‘fictions’, ‘agglomerations’ or ‘approximations’ of truth. Arthur’s study of the continuum in Leibniz’s writings briefly refers to the ‘fictional’ quality of Leibniz’s geometric figures, which show ‘the connection of the doctrine of petites perceptions with the analysis of geometric figures as fictional entities approximated arbitrarily closely by polygons’ (Arthur, 2001, p. xxvi). Three consequences arise from this observation: first, more than one geometric figure is produced out of a continuum: second, each

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figure is connected to ‘virtual’ or imperceptible, yet embodied, conditions of thought and matter (i.e., the petite perceptions or memory); and third, fictional figures confirm geometric principles of ‘sufficiency’, rather than finite perfection. Thus, the concept of sufficient reason upholds Leibniz’s insistence on the ‘substantial form’ of the Monad or soul that has the capacity to produce reason through its own activities. Nevertheless, this internal and individuated reason does not exclude a relationship with the ‘perfect’ and infinite reason of God, which is never merely ‘sufficient’. In this respect, sufficient reason is always in relation to an external principle of sufficiency and infinity, rather than releasing the Monad’s sufficient reason to an unregulated agency or will. Sufficient reason is therefore not just necessary for the internal harmony of the Monad but is also required for discerning the relationship between the individual entity and the external world, God or nature. In this respect, the internal sufficiency of an entity is always contingent upon the infinitude (plenitude) of the world, so that the ‘truth’ of the external world becomes infinitely and immanently enfolded (implicatio) within it. As §36 states: But there must be also a sufficient reason for contingent truths or truths of fact; that is to say, for the sequence of the things which extend throughout the universe of created beings, where the analysis into more particular reasons can be continued into greatest detail without limit because of the immense variety of the things in nature and because of the infinite division of bodies. There is an infinity of figures and of movements, present and past, which enter into the efficient cause of my present writing, and in its final cause there are an infinity of slight tendencies and dispositions of my soul, present and past. (M, §36, p. 259) Such a concept of harmony extends between God-as-reason and embodied reason, representing two different kinds of perfection, one infinite and one sufficient (M, §37, p. 259). Thus, in §38, sufficient reason is described as ‘sufficient’ substance, that is, a ‘substantial’ reason or God, and Leibniz explains God’s sufficiency in the following passage from §39: ‘Now, since this substance is a sufficient reason for all the above mentioned details, which are linked together throughout, there is but one God, and this God is sufficient’ (M, p. 259). Here, then, Leibniz reinforces the harmony of a metaphysical order in which sufficient reason (the embodied, yet Rational Mind) is a more appropriate

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form of reason, in contrast to the disembodied, pure reason that is exclusive of matter or the body. Instead, sufficiency is originated in the body and its perceptive powers are confirmed in its harmony with the infinite substance, God. Thus, according to this argument, reason (ratio) is produced by a thinking substance, rather than merely representing a product of idealised intellect. Sufficient reason is therefore an aesthetic geometric principle, insofar as it demonstrates the relationship (i.e., the ratio), between internal and external relations, or discrete and universal infinites, so that geometric figures express a continuum of different aesthetic ‘reasons’ (or ratio). As a result, reason becomes understood as a continuum of magnitudes, a kind of limit-operation from within the geometric figure or body, rather than merely imitating an external agency, law or ‘Reason’. Thus, sufficient reason demonstrates the powers of the soul, memory, perception and appetition, perfection and sufficiency, which produce an infinitely divisible series of ‘ratios’, ideas or concepts, rather than a logic in which perfection designates the finite idea or body. Furthermore, Leibniz distinguishes between reason and ‘sufficient reason’ in his examination of the Monad’s autonomous, incorporeal structure, and in so doing reason becomes understood as a differential principle – a ratio – that is not reducible to finite representations. The following section shows how the intensive and extensive qualities of this aesthetic are expressed in the geometric figure of the plenum.

The plenum Leibniz is the philosopher whose method can be said to most clearly generate a continuum of differentiated figures in this discussion. Arthur has pointed to this continuum (or plenitude) of figures in Leibniz’s writings that include, the ‘net’ and the ‘fold’; for example, in an extract titled ‘On the Origin of Things from Forms’ (1676), Leibniz explains the figure of the net in the following passage: But this universal space is an entity by aggregation, and is continuously variable; in other words, it is a composite of spaces empty and full, like a net, and this net continuously receives another form, and thus changes; but what persists through this change is the immensum itself. But the immensum itself is God insofar as he is thought to be everywhere, i.e. insofar as he contains that perfection or absolute affirmative form which is attributed to things when they are said to be somewhere. (Arthur, 2001, p. 121)

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According to this earlier essay, infinite change is therefore inherent in the continuous development from one figure to another, and is also reflected in the infinite immensity of God and the world. In addition, the figure of the net suggests a metaphor that constantly receives and exchanges states in both a spatial and temporal spectrum.16 Thus, the infinite figure is constructed in relation to a genetic order of infinity, which represents the plenitude of the world. These infinite spatiotemporal relations are also present in other essays on the continuum that date from this period, especially in his radical invention of the geometric figure which is constituted by imaginary or fictional qualities; for example, Leibniz calls the polygon a ‘fictitious entity’ that is a kind of ‘ideal limit to a sequence of polygons’ (Arthur, 2001, p. lvi). Arthur also observes Leibniz’s argument that, in order for a body to have unity ‘in space and self-identity and continuity through time’, an ‘immaterial’ principle or ‘something imaginary’ must be involved (Arthur, 2001, p. lxii). So here, the imagination is introduced in order to explain figures that approximate to a given moment in time, since ‘magnitude, shape, and motion all “involve something imaginary”’ (Arthur, 2001, p. lxii). Thus, immaterial or imaginary figures represent the provisional assignation of place, time or movement to an imperceptible difference. In this respect, they are ‘sufficient’ geometric identities, shapes or forms, which approximate to an infinitely continuous unity. ‘Substantial form’ is therefore an infinitely divisible spatiotemporal unity that is brought about by immaterial operations, such as the imagination (Arthur, 2001, p. lxii). Leibniz’s earlier writings underpin the importance of the imagination in the construction of geometric figures; although, the imagination is not discussed explicitly in the Monadology. But, as we saw in the previous section, the sensibility is represented by perception and appetition, and in the following discussion I show that the figure of the plenum embodies spatiotemporal relationships that resonate with Bergson’s theory of matter and memory and Husserl’s horizons. The plenum is first mentioned in the Monadology in §8 which lays out the nature of its qualities. It is a ‘completely filled space’, that is, a space that contradicts the existence of the vacuum since it is constituted by matter.17 Leibniz writes: ‘For instance, if we imagine a plenum or completely filled space, where each part receives only the equivalent of its previous motion, one state of things would not be distinguishable from one another’ (M, §8, p. 252). However, rather than constructing the plenum as a divisible figure of equivalent finite parts, Leibniz reconciles the contradiction between the ‘immaterial’ void and the materiality of the spatial figure into a series of material relations, that is, a continuum.

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In addition, this passage includes a discussion about different qualities which affirm an intensive magnitude – that is, the infinite multiple singularity – and are generated out of its internal and differential powers, so that qualities of one Monad distinguish it from another. Thus, the plenum’s ‘paradoxical nature’ is an expression of the multiplicity in ‘vice-diction’ or sufficient reason, and it is a geometric figure that is immanently related to an intensive substance and an intensive extensity, but not derived from the external movement of bodies in space. Furthermore, it is also a figure that is infinite twice over: first, it is inherently ‘fictional’, because it is an unassignable infinity; and, second, because of its relationship to material plenitude. Moreover, this logic of immaterial and material continuity is carried through the various geometric figures named in the text (i.e., the envelope, fold and the plenum) suggesting that Leibniz’s text embodies a continuum of figures out of an ‘analytic’ contraction and expansion of geometric states. In addition, its geometric figures (i.e., the Monad or plenum) represent both the contraction (implicatio) of all other elements in the text into one idea, and the expansion (explicatio) of all these ideas into a unity of incompossible elements (these movements will also be retrieved again in Bergson’s theory of the contraction and expansion of memory, and in Husserl’s theory of horizons). In addition, the plenum is present in Leibniz’s earlier writings on the continuum and closely relates to his examination of extended bodies and movement; for example, in the essay ‘On Matter, Motion, Minima, and the Continuum’ 1675, he writes: Now, I conceive everything to be a plenum, i.e. to be matter with various motions, for if some whole infinite mass were understood to be moving with a certain universal motion, this motion could be considered nonexistent. Therefore, supposing the plenitude of things – in other words, supposing there is no part of space that does not contain matter moving with a motion different from an infinity of others – I show that the same quantity of motion is conserved. (Arthur, 2001, p. 33) So, in this examination of the heterogeneity of moving bodies Leibniz also makes a discussion of internal difference in the geometric figure possible, because the notion of a homogenous, external movement is considered to be redundant in distinguishing differences between bodies. Latta provides an insightful discussion about the internal forces of movement that constitute the plenum, and suggests that the

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continuum is comprised of the interrelated forces of appetition and perception and the external forces of movement in the world: The conception of continuity, however, by implying a plenum, escapes the contradictions that are involved in the idea of the void. But it still has to be shown how change is possible within a plenum, or how change can take place without disturbing the continuity of the infinite series of Monads. Any change within a plenum affects every part of it . . . . If, however, the universe be a quantitative plenum, it is impossible to understand how any change could originate within it. It must receive its motion from outside, and must thus be regarded as finite, which again is inconsistent with its reality as a plenum. Leibniz overcomes this difficulty by regarding the universe, not as an infinite mass occupying all that there is to occupy, but as a continuity or infinite gradation of qualitative differences, each containing within itself the principle of its own changes. He substitutes for an extensive plenum of mass an intensive continuum of force or life. (Latta, 1985 p. 40; my emphasis) Latta’s explanation reflects the principle of sufficient reason in the harmonised ratio that connects the internal activities of the Monad and nature. But his argument also highlights the extent to which the plenum is an intensive infinite unity, and is a spatiotemporal figure that is constituted out of an intensive matter, not a finite space filled with matter. Interestingly, Latta also continues his examination in a discussion about ‘passage’ and the ‘pre-Cartesian’ notion of influxus physicus or ‘the actual passage of elements from the one substance to the other’, in order to explain the relationship between the soul and body. Here, then, we have a concept of passage that reflects the extensive passage of Spinoza’s affects, but discussed in the context of a method that is intensive (Latta, 1985, p. 42). In the context of the Monadology, the plenum is therefore an important figure of sufficient reason, designating both the internal forces of the Monad and the external plenitude of the nature, and expresses the relationship between the world, soul, mind and the body in seventeenthcentury theories of plenitude.18 So it is an aesthetic geometric figure that is internally differentiated as a result of its material and immaterial forces of activity, preventing extended matter from being reduced to a mechanical series of parts, or determined by motion that is generated by an external source. Instead, change is brought about by the internal forces of movement, such as perception and appetition. The plenum is therefore produced in relation to an intensive and qualitative series of

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magnitudes or immaterial and material forces, thus marking an important moment of development towards a truly differential figure. In addition, it is an aesthetic geometric figure derived from a ‘discursive’ plenitude (as seen in Proclus’ method), but reconfigures the general principle of genetic discursivity because it is situated within the discrete infinity of the Monad: thus, it provides a ‘natural’ or genetic continuity of relations with others in the universe. In this respect, the plenum can be considered to be both, the figure of the world, and the discrete singularity of the Monad in a continuous spatiotemporal infinity, in which ‘every body responds to all that happens in the universe, so that he who saw all, could read in each one what is happening everywhere’ (M, §61, pp. 264–5). Extensive and intensive, on the one hand, are the plenitude of relations between discrete singularities and, on the other hand, the intensive enfolding of the Monad or the soul ‘reads’ itself, but ‘only what is there represented distinctly’. So, in contrast to the infinitely connected space of the plenum, the soul ‘cannot all at once open up all its folds, because they extend to infinity’ (M, §61, p. 265). Thus, the plenum embodies the discursive movements between figures and the principles of reflection and memory. In the following section, the immanence of the world is further interiorised when Leibniz considers the plenum in relation to the body and the actions of the Monad (i.e., a living being). Thus, in §62, the plenum, having been assigned a limit, as a Monad or a soul, now becomes understood as the ‘universe’, once again confirming its infinite unity through internal and external sufficient reason. Furthermore, if we recall Plato’s notion of the ‘world soul’, the Monad is a representation of the universe. But it is also the embodied soul, because Leibniz writes that it is ‘more distinctly the body which specially pertains to it, and of which it constitutes the entelechy. And as the body expresses all the universe through the interconnection of all matter in the plenum, the soul also represents the whole universe in representing this body, which belongs to it in a particular way’ (M, §62, p. 265; my emphasis). This self-conscious subject is also examined in §63, when Leibniz defines the Monad as ‘a living being’, writing: ‘The body belonging to a Monad, which is its entelechy or soul, constitutes together with the entelechy what may be called a living being, and with a soul what is called an animal’ (M, p. 265). Thus, the relationship between the geometric figure of the plenum, matter and the soul means that the plenum is both a singularity (the Monad) and the divisible infinity of the world and God. It is both a differential geometric figure and the corporeal subject, so that the notion of figure itself becomes a continuum between analytic geometry and

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the aesthetic forces of consciousness, constituting an aesthetic and geometric aggregate of ‘sufficient reason’. So Leibniz’s theory of difference is founded upon a principle of identity in which matter and space are non-similar at any point in time; embodied in the plenum, identity cannot remain the same over a period of time. Instead, identity may be said to unify a body undergoing constant and continuous change. Thus, identity becomes a differential operation in an aesthetic geometry and, as a result, the ‘perfection’ of a geometric figure (e.g., the ‘perfect’ circle) exists in the difference or the potential for change between one magnitude and another within a continuum of relations, not as an equivalent truth for a sensible figure. The aesthetic geometric method is therefore, not the production of equivalents but the production of differences between one Monad’s duration and in its individuation from another. The concepts of identity, figure or form are derived from the principle of a continuous series of changes, marking a shift from the production of discrete mathematical identities into the aesthetic ‘soul’ or thinking subject. Thus, identity is not merely an abstract and rational approximation but is embodied in the individuating powers of perception and appetition that Leibniz emphasises between the fictional geometric figure and in the specificity of the thinking Monad, both of which are irreducible to quantifiable magnitude. So, the plenum is not constructed out of a split between two externally recognisable forms but through the degrees of difference that exist within a body or state; and so the relationship between geometry, nature and limit is transformed by the intensification of an infinite and aesthetic geometric magnitude. As a result, the concepts of space and time are dramatically liberated from a hierarchy in which space is generated at the expense of temporality, into a relationship in which time is immanent. In this new configuration, the continuum between space and time is generated in the shift from a quantitative spatial understanding to a qualitative temporality, and in the emphasis on the infinite, yet continuous, incommensurability that exists between the two modes of perception. Space and time therefore become understood as ‘relations’, which will be explored further in Bergson’s discussions of heterogeneous space in the next chapter. However, as expressions of sufficient reason (i.e., in harmony with God), space and time are constituted by a scientific ‘symbolism’ that Bergson will reject in his pursuit of a ‘progressive’ philosophy. Nevertheless, through Leibniz’s analytic and aesthetic geometric method and figures, space and time are intensive magnitudes, rather than opposing finite operations, because they are always constructed in

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relation to other states, and although divisible, they are also expressions of infinity.

Summary Leibniz’s aesthetic geometric method demonstrates the scope of an analytic procedure that emphasises an infinite divisibility, and which generates a series of aesthetic figures that are aggregates of both immaterial and embodied intensities. In particular, this chapter has explored this procedure in relation to a qualitative notion of magnitude that not only encompasses the discrete limit of division but also affirms the necessity of aesthetic or qualitative difference in the geometric figure. Thus, the Monad or plenum’s magnitude is an intensive ‘ratio’ between the incorporeal and corporeal qualities, which generate a uniquely embodied geometric figure. Highly reflective of the Stoic and Cartesian principles of geometry, division and magnitude, Leibniz’s aesthetic magnitude enables multiplicity to be generated, rather than a reconstitution of the finite One (Form), or the formless and timeless infinity of the ‘Many’; for example, in contrast to Proclus, the geometric figure (or soul) is an intensive extensity (rather than a mystical symbol or supernatural power) that is constructed from the indivisible embodied forces of perception and appetite. Magnitude therefore becomes embodied and internal, in contrast to the discursive, yet general, magnitude of Proclus’ unfolding, so that Leibniz’s aesthetic geometry mediates between the symbolic powers that Proclus upholds and the intuitive and embodied ‘life’ that Bergson and Husserl posit. In addition, the plenum corresponds with the genetic plenitude of this intensive geometric procedure: first, because it is determined by a discursive division that is intensive and infinite; and second, because it is internally and externally differentiated by the multiplicity of extended substances, and by the unextended soul’s activities of ‘perception and appetite’. As a result, the geometric figure represents an infinite unity of corporeality and incorporeality in a third order of magnitude, that is, sufficient reason or embodied reason (ratio), in the reflective subject. Furthermore, this emphasis on the production of ‘incompossibles’ or figures that are internally differentiated by forces and limits, highlights the extent to which Leibniz’s aesthetic figure, or ‘reflective I’, constitutes an important precedent to Kant’s development of the aesthetic subject in the Critique of Judgment; Kant’s ‘aesthetic judgment’ is therefore reminiscent of the ‘incompossibility’ in Leibniz’s ‘aesthetic’ reason.

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The Monadology therefore represents a unique metaphysics in which a radical version of the geometric principle of magnitude generates qualitative difference in a series of infinite figures, and in the immaterial ‘forces’ of perception and appetition. As a result, Leibniz’s method is an important mediator between Spinoza’s and Bergson’s methods in which the autonomy of the geometric figure passes from a principle of an internal, yet finite, limit (i.e., the modes) into a series of internal and infinite continuities (i.e., duration). In addition, like Spinoza’s method, the continuously divisible forms of the plenum and the Monad are reflected in the formal structure of the text insofar as they are constructed out of a series of discrete statements that constitute both entities in themselves and are expressions of a greater plenitude. However, unlike his predecessors, Leibniz proposes not just one geometric figure but a series of evolving forms – the net, envelope, plenum and the fold – that suggest a shift towards a more ‘topological’ notion of geometry in which the relationship between the internal and external conditions become more intensive, not as limit but in the embodied and intuitive actions of the individual.19 In the next chapter, an especially topological and aesthetic geometric method will be revealed in Bergson’s writings on matter and memory.

5 Envelopes

Bergson’s philosophy emphasises the importance of intuition in the geometric method, and reflects my earlier discussions about extensity, intensity, memory, the soul and the body. But his rigorous critique of scientific geometric thinking and metaphysics also suggests a radical departure from these methods, which is embodied in the notion of the ‘living act’. In this chapter, the geometric method is therefore pushed to one of its most intensive limits in the constitution of the aesthetic and intuitive body. In earlier chapters I suggested that Spinoza’s notion of the extensive body and Leibniz’s intensive body provided innovative geometric solutions to the metaphysical problem of the division between extended and unextended matter. For Bergson, however, the discussion about division is developed even further to include the limits of the geometric method, as it is defined by metaphysics. Bergson rejects the a priori ground that materialist and idealist thinking require which, he argues, results in the exclusive divisions between matter and intellect that are derived from the traditions of a symbolic metaphysics (especially a neoPlatonic or Kantian metaphysics). Bergson’s method not only promotes new concepts within the boundaries of metaphysics but also seeks to reconfigure spatiotemporal relations into a new understanding of ‘psychic’ realities within a ‘progressive’ philosophy. However, whilst being highly critical of the limits that form philosophical thinking, Bergson’s engagement with a history of metaphysical ideas is also inclusive; for example, Leibniz’s notion of infinity and perception correspond with Bergson’s ideas about infinity and perception. Yet, this continuity is also provisional because of Bergson’s insistence upon the ‘psychic’ duration of ‘active’ life, which opposes Leibniz’s logical or ‘symbolic’ concept of infinity. Also, although Bergson rejects the ‘ready-made’ aspect of 121

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geometry – that is, as an a priori diagram – the ‘attention’ he pays to extensity reveals how the geometric method can be conceived as an immanent aspect of the living body.1 The discussion below examines this argument, drawing attention to those aspects of Spinoza and Leibniz’s theories that provide important precedents to Bergson’s theories of memory, extensity, intensity and intuition. Bergson’s intensive revision of geometric relations constitutes much of the ‘energetic’ impetus of his book, Matter and Memory (1896). Here, geometric and spatiotemporal relations are rethought through the production of a ‘real’, that is, an independent, ontology of duration. Yet geometric relations are also necessary constituents of his ‘progressive philosophy’; for example, space and time are two minor figures of discussion that are reconfigured into the notions of matter and memory, perception, intuition and especially duration, ‘releasing’ them from their exclusivity as scientific concepts to become ‘intermediate’ aesthetic forms of matter and memory. The initial sections of this chapter outline these new constituents in the formation of the ‘body-image’, the activities of perception and the two modes of memory, suggesting that their dynamic relationship produces the aesthetic figure of the envelope. Thus, Matter and Memory produces both a highly intensive critique of the geometric method and a revision of spatiotemporal relations in the construction of an aesthetic of duration. Second, the chapter considers Matter and Memory in relation to two later texts, ‘Introduction to Metaphysics’ (1903) and Creative Evolution (1907), in which the geometric method becomes more strongly defined as an aesthetic or ‘natural’ geometry, so that the aesthetic unity of the living body and its acts (which are derived either from memory or from the habits of the body) are emphasised in the concept of ‘extensity’. In addition, I explore Bergson’s notions of space and time, in relation to his critique of ‘pre-modern’ science and the problems of the Cartesian scientific method. Geometry, space and time, together with perception and duration, are therefore defined in terms of metaphysics of life and intuition. ‘Introduction to Metaphysics’, for example, presents a crucial moment in the construction of intuition, because it demonstrates the importance of intuition as an aesthetic consideration that has been ‘forgotten’ by philosophy and science. In this essay Bergson argues that philosophy and ‘pre-modern’ science have been misled by the insistence on relative truths and symbolic knowledge, at the expense of concrete reality and progressive philosophy. Bergson’s philosophy therefore produces a radical notion of natural geometry or ‘intuition’ in which geometry is infused with living expressions of space and time.

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This chapter also suggests that Bergson’s development of an ontology of time is enabled partly as a result of his sophisticated understanding of geometry, which informs his reconstruction of the relationship between space, time and intuition. In particular, Bergson’s understanding of geometric methods in philosophy retrieves Spinoza and Leibniz’s concepts of extensity to inform the topological potential of duration. Thus, duration produces topological relations between philosophy and the subject that dramatically reconfigure the nature of science, philosophy and life; it is a topological geometric method, through which unique notions of unity are proposed that are lived, rather than pregiven, symbolic harmonies.2 Bergson therefore conducts an intensive reconstruction of the metaphysical relations of space and time, the self and the world to retrieve forgotten relations that constitute an ‘absolute’, but not symbolic, nature. In addition, questions about the production and the structure of metaphysical relations are framed through an aesthetic and geometric reconfiguration of the relations between matter/memory, whole/part, limit/body, quality/quantity, and Bergson shows how (if we think beyond the form of these concepts that are inherited from a ‘limited’ metaphysics) unique and liberating expressions of life can be re-established.

Limit and unlimit As the preceding chapters have shown, the dialectic between the concepts of limit and unlimit, finitude and infinity, are crucial aspects in the ‘union’ between extended and unextended matter; for example, in Proclus’ Commentary, unlimit constitutes the divine infinity of the geometric figure; for Spinoza, an indivisible God is immanent in the modes of the subject and for Leibniz, the infinite divisibility of limit produces an intensive, yet autonomous, being. Bergson also examines the ‘tension’ between limit and unlimit in order to construct the irreducible notion of the ‘living act’, suggesting a strong correspondence to Spinoza’s theories of substance, in particular, because duration (i.e., pure memory) represents the ‘virtual’, and unlimited infinity in matter (i.e., perception) constitutes the extensive limit of the living organism.3 This emphasis on an extensive indivisibility resembles Spinoza’s concept of extensity or indivisible matter in which the virtual or absolute is attributed to God and is internalised in the form of the ‘conatus’. Also, like Spinoza, Bergson considers extended matter in terms of a synthetic organisation of magnitude and rejects the analytic basis of difference in the aesthetic geometric method.

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Bergson’s rejection of reason also indicates his affiliation with Spinoza insofar as geometry, space and time are intermediary aspects of a discursive soul or intuition – that is, pure duration – so that the intuitive body provides the basis for a different kind of reason. But in this rejection of ‘reason’ does Bergson also reject the principle of relations or ratio? Or does his study of ‘relations’ in fact correspond with Leibniz’s investigations into ratio? As we will see below, ratio is evident in the ‘relationship’ between matter (i.e., perception) and memory that is produced in the intuitive, aesthetic geometry or topology. So, perhaps we can say that Bergson’s procedure constitutes a ‘natural’ or intuitive sense-reason since ratio (or magnitude) is generated by aesthetic psycho-physical activities, which are reminiscent of both Spinoza’s theological notion of harmony between God and the emotions and Leibniz’s incorporeal magnitude. But how does Bergson relate to Leibniz’s notion of sufficient reason? In the discussion about scientific method below, Bergson’s rejection of the Cartesian analytic method will lead him to reject this form of ratio, because it represents the symbolic harmony between a mathematical procedure and God. Yet, we will also see that Leibniz’s study of the powers of perception and memory in a ‘psychic’ topology is continued in Bergson’s thinking, especially because Bergson’s notions of perception, aggregate and the continuous intensity of the life in action have similarities to Leibniz’s analytic notion of the intensive Monad. Bergson therefore not only appears to accept Leibniz’s claim that the psychic forces are both qualitative and internal differences but he also considers Leibniz’s analytic method to be a scientific reduction. In this discussion, however, Leibniz has been considered to affirm an aesthetic and ‘transcendental’ matter through his intensive notion of perception that produces a continuously changing process. In addition, I have argued that Leibniz’s notion of the ‘sufficiency’ of perception in the Monad also bears a strong resemblance to Bergson’s notion of duration, in particular, because each is internal and infinitely extensive. I therefore suggest that there are more similarities between Bergson and Leibniz, than Bergson’s opposition to ‘symbolic’ relations might first admit. Finally, we will also see that the seventeenth-century investigations into the ‘unity’ of the subject, such as ‘sufficient reason’ or ‘common notions’, have similar ‘aesthetic’ characteristics to those found in Bergson’s duration, further underlining the ‘progressive’ potential of seventeenthcentury understandings of extensity. Consequently, we may propose that actions of the body and of memory constitute Bergson’s notion of an intensive extensity, and display principles of both synthetic and analytic geometry derived from Spinoza and Leibniz.

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Matter and Memory therefore constitutes a radical form of aesthetic geometric method and aesthetic figure or intuition, in which the living subject embodies the heterogeneity of extended and unextended memory and duration. In addition, we find a reprise of the intensive dialectic between limit and unlimit in Bergson’s rethinking of reason in Matter and Memory. On the one hand, the dialectic generates a text that seeks to intensify the limits of metaphysics and, on the other hand, it proposes a highly complex figure that is developed out of the relations between the body and its image, internal and external space, the part and the whole and, especially, matter and memory. The book’s highly critical engagement with the limits of philosophy also constitutes an intensification of the concept of limit, in which duration (i.e., time) provides an intensive challenge to the conditions of geometry and metaphysics, and contrasts with Kant’s view in the Critique of Pure Reason that time is a ‘repetition’ of space. Rather, by revisiting (or recollecting) seventeenth-century concepts of extensity Bergson rejects the Critique’s proposal that the ‘formal’ intuitions of space and time cannot be related to the ‘pure reason’ of geometry. Matter and Memory therefore reveals the interiority of an aesthetic geometry through a highly intensive examination of the metaphysical conditions that produce space and time. Geometry becomes radicalised into an intensive ‘tension’ of its internal and external limits (i.e., the division between external space and internal time) in an account of a ‘forgotten’ geometry. Thus, the notion of limit is both a fundamental aspect of Bergson’s ontology and provides the means for a formidable critical analysis. In this respect, Matter and Memory’s critique of metaphysics operates on three different levels of tension between limit and unlimit: 1. A unique kind of dualism between unextended and extended matter is produced that challenges the ‘symbolic’ and ‘parallel’ metaphysical relations upon which Spinoza and Leibniz depend (see the Introduction and chapter I, especially, for a defence of a radicalised dualism). 2. A revision of the relations between philosophy and science, in particular, by reconsidering the tensions between quality and quantity, and the whole and part (see the Conclusion, especially). 3. The proposition of intuitive or ‘psychic’ relations between matter and memory by a series of changes of degree (rather than the transformation of extended matter into unextended matter or vice-versa); that is, matter and memory are neither identical nor equivalent (see chapters I, II, III and IV, especially).

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Extensity For Bergson, limit is not sufficiently defined by a mathematical or metaphysical explanation of magnitude; in contrast to the rationalism of Leibniz and Spinoza, because in each case, ratio or difference is constituted by a symbolic value. In addition, Bergson does not accept that the imagination’s powers of division represent a satisfactory account of limit, because it is also determined by division.4 Instead, he seeks to define a notion of infinite limit that is produced by the ‘psychic’ powers of the subject, which enables him to rethink the dualistic explanations of unextended and extended matter. Limit is therefore informed by the tension between a series of psychic or material states that construct the subject, and in turn Bergson argues that these revisions to mind–body relations challenge the ‘incomprehensibility’ which exists in metaphysical accounts of ‘real’ movement and change in the living subject. For Bergson, such incomprehension is evidence of a relationship that is determined by symbolic limits and division, whereas explanations of the ‘natural’ psychic ‘life’ enable clear understandings of the individual to be established without recourse to the pregiven harmony required by rational explanations. Thus, Bergson promotes the psychic duration of the subject as the basis for the ‘real’ tension between limit and unlimit. However, like Spinoza and Leibniz, Bergson’s solution to the symbolic restrictions of scientific thought lies in the concept of extensity, through which the division between perception (mind) and matter (body) is removed, and the relationship between inextensive and extensive matter is rethought. So we find that the text explores two methods through which extensity is produced: first, it is an investigation of extensity as perception, that is, the nature of its extension in space in order to produce an understanding of ‘action’, and second, Bergson explores how extensity is ‘subtilized’ or ‘dissolved’ into the ‘affective sensations’ towards the production of inextensive matter or pure memory (Matter and Memory, 1991, p. 245);5 for example, he writes: ‘That which is given, that which is real, is something intermediate between divided extension and pure inextension. It is what we have termed the extensive’ (MM, p. 245; my emphasis). So extended matter is pure perception not only derived from our consciousness but also affective to it. Absolutely distinct from the soul, matter is nevertheless imbued with duration and action in itself. Extended matter is therefore, not a duplicate of intuition or memory, but an aspect of perception or the living body that has its own inherent

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extensity, when Bergson writes: ‘we eliminate all virtuality, all hidden power, from matter and establish the phenomena of the spirit as an independent reality. But to do this we must leave to matter those qualities which materialists and spiritualists alike strip from it’ (MM, p. 72). In contrast, as we will see below, he criticises materialist and idealist thinking for having confused extensity, so that for materialists, matter is a ‘representation of the spirit’, and for idealists, it is ‘the accidental garb of space’ (MM, p. 72). Thus, Bergson defines matter in relation to the psychic activities of the body; for example, promoting the psychic distinctions between our responses and movements that are generated by our nervous system, yet also noting that both matter and perception are necessary to our notion of life. In this sense, the body reflects our perception of the exterior world; for example, we respond to external stimulations through a set of ‘mechanical, physical and chemical reactions’ (MM, p. 28). ‘Living matter’ represents zones of ‘indetermination’ or centers of real action through which conscious perception is produced. In addition, Bergson suggests that the living body perceives and acts as a result of varying intensities of stimulation and activities, which demonstrate a reflexive relationship between perception and the actions of the organism (MM, p. 31). Perception and the actions of the body that arise from it constitute a continuity between space and time. Bergson explains that ‘perception is master of space in the exact measure in which action is master of time’ (MM, p. 32). In the first instance, extensity is therefore a relationship between space and time that is produced out of the perceiving and acting body; the subject is an indeterminate ‘unity’ of the mental perceptions and the actions of the body, and perception is distinguished as either, internally or externally produced extensity, because the body represents indeterminate centres of action or ‘variable’ relations between the organism and its external environment; that is, ‘when perception is internal it is called memory, and when it is external it is matter’ (MM, pp. 33–4). However, memory that is derived from perception is always related to extension since it is only pure memory that is unextended, whereas, perception is always comprised of ‘duration’. Bergson explains how the reconfiguration of perception and memory enables a removal of the division between extension and inextension: But, just because we have pushed dualism to an extreme, our analysis has perhaps dissociated its contradictory elements. The theory of pure perception, on the one hand, of pure memory, on the other

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hand, may thus prepare the way for a reconciliation between the unextended and the extended, between quality and quantity. To take pure perception first. When we make the cerebral state the beginning of an action, and in no sense the condition of a perception, we place the perceived images of things outside the image of our body, and thus replace perception within the things themselves. But then, our perception being a part of things, things participate in the nature of our perception. Material extensity is not, cannot any longer be, that composite extensity which is considered in geometry; it indeed resembles rather the undivided extension of our own representation. That is to say, the analysis of pure perception allows us to foreshadow in the idea of extension the possible approach to each other of the extended and unextended. (MM, pp. 181–2; my emphasis) The extended and perceiving body is an infinite limit: first, because it is an infinitely variable, reflective ‘centre of real action’, and second, because it is internally and externally generated. Extended matter, or a spatiotemporal body, therefore becomes understood as a fulcrum of action so that geometry is not restricted to a limit-boundary of an extended figure but is considered to be embodied into the discursive and aesthetic actions of the body, which are generated from the internal memory and external perceptions. Moreover, as one of the most intensive limits that Bergson constructs, extended matter and the presentation of duration provide a highly complex concretion of the tension between matter and memory or space and time: If matter, so far as extended in space is to be defined (as we believe it must) as a present which is always beginning again, inversely, our present is the very materiality of our existence, that is to say, a system of sensations and movements and nothing else. (MM, p. 139) In the next section, we will see this concrete tension expressed in the invention of the ‘body image’, which enables the figure of the reflective subject in the Critique of Judgment to be reconfigured into a more radical form.

Body-image and perception Bergson intensifies his analysis of the perceiving body in a reconceptualisation of the notion of ‘image’, further developing his critique of the divisions that symbolic limits of idealist and materialist metaphysics

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generate. An intensive and aesthetic notion of the body is generated by promoting the body as a specific kind of image, through which a transformative and reflexive relationship between the world and the subject are brought together.6 In chapter I Bergson proposes an alternative dualism in the relationship between the image and body, developing the concept of body image to provide a material site of the relationship between internal aspects of the body (i.e., the mind) and the external world (matter). As a result, this theory of image reflects the production of extended realities that are determined by relations; for example: ‘All these images act and react upon one another in all their elementary parts according to constant laws which I call the laws of nature’ (MM, p. 17). Thus, images designate ‘psychic’ relations, and reflect our different levels of engagement or ‘attention to life’. Such a reflective notion highlights the importance which Bergson places on rethinking ‘life’ as a continuously changing psychic act, in contrast to the symbolic claims made by rational and speculative philosophies (MM, p. 14). Thus, within this aggregate of image relations, the body is a unique kind of image, its internal qualities produced by the affections and its external qualities derived from perception. The subject and its relations with the world are therefore created through intensive mental processes: the subject is in effect, an aggregate of images. Bersgon explains: ‘All seems to take place as if, in this aggregate of images which I call the universe, nothing really new could happen except through the medium of certain particular images, the type of which is furnished me by my body’ (MM, p. 18). In addition, this relationship is one of constant activity, determined by giving to, and receiving movement from, the ‘external world’: My body is, then, in the aggregate of the material world, an image which acts like other images, receiving and giving back movement, with, perhaps, this difference only, that my body appears to choose, within certain limits, the manner in which it shall restore what it receives. (MM, p. 19) But Bergson’s notion of the perceiving image is also reminiscent of Kant’s reflective subject in the third Critique, introducing a confluence that Bergson keeps hidden behind his fierce critique of materialism and idealism in Kant’s metaphysical thinking. Bersgon’s challenge is developed out of his analysis of Kant’s first Critique, in which he concludes that materialism produces insufficient explanations of the relations between mental phenomena and consciousness, and Kant’s idealism

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also resulting in the body being understood as a perception of the subject’s consciousness. He argues that Kant considers mind, perception and memory to be ‘operations of pure knowledge’ that make either ineffective duplications of an external reality or inert and disinterested notions of mental production, and ‘always [neglect] the relation of perception with action and of memory with conduct’ (MM, p. 227). Thus, Bergson argues that Kant fails to properly account for the relationship between sense and understanding in the Critique of Pure Reason because his idealism and realism are determined by the exclusion of a ‘real’ materiality, which prevents the ‘reciprocal influence’ of a more radically dualistic substance. Bergson’s evaluation of Kant’s Critical philosophy therefore rests firmly on its limited idealism and realism, which lead to the misrepresentation of the powers of perception and the body. As an idealist philosophy, Bergson suggests it fails to recognise the intermediate links between different sensations by categorising them under the understanding; and as a realist philosophy, it allows ‘no conceivable relation’ between the ‘thing in itself’ and the ‘sensuous manifold’, so that in each case a homogenous space is constructed as a ‘barrier’ between the mind and external objects, and perception is determined towards pure knowledge, not action (MM, p. 231). But, as has been suggested in Chapter 1, this disagreement cannot be applied to the third Critique in which Kant constructs aesthetic judgments, not cognitive ideas or forms; for example, images are both forms and sensations, which are determined by embodied expressions of space and time. Thus, aesthetic judgment represents a more sympathetic predecessor to Bersgon’s notion of the ‘body-image’, insofar as it is also concerned with the ‘psychic’ and physical activities of the individual. In this respect, aesthetic judgment is therefore more closely aligned to the notion of ‘perception’ than to cognitive thinking. So although Bergson appears not to recognise this potential in his resistance to Kant’s ‘speculative’ philosophy, his notion of the body-image, which is a discrete, yet irreducible, unity does bear some similarity to Kant’s reflective subject. Nevertheless, Bergson’s desire to disrupt the overriding emphasis on the cognitive ideas that determine materialist and idealist thinking represents a forceful critique of the problematic limitations of ‘images’ in the first Critique, which can only be forms of sensations, rather than sensations in themselves. In the following paragraphs, Bergson’s emphasis on the construction of this body-image demonstrates the extent to which the psychic relations of perception and memory are actively embodied in the individual, rather than existing as purely ‘speculative’ ideas.

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Perception Perception, Bergson tells us, is understood as an activity of the living body in which the body is a perceptive centre that is also part of a moving plane, which vibrates as part of an ‘aggregate’ of other images. Bergson explains this activity, writing: ‘since there is no material image which does not owe its qualities, its determinations, in short, its existence, to the place which it occupies in the totality of the universe’ (MM, p. 228). So perception is an aspect of powers of selecting and editing. Bergson writes: ‘Perception, therefore, consists in detaching, from the totality of objects, the possible action of my body upon them. Perception appears, then, as only a choice’ (MM, p. 229). Continuing this discussion, he writes that images are not finite. Instead, they ‘outrun perception on every side’, and it is the work of science and metaphysics to reconstitute these images, in order to ‘restore’ the relationship between the part and the whole. Hence, as a reprise of Leibniz’s perception, the scope of perception is also changed in Bergson’s unity, from being a constituent of the ‘appearance of reality’, to a relational state in the aggregate of the ‘body-image’, because it indicates ‘in the aggregate of things, that which interests my possible action upon them’ (MM, p. 230). Thus, the body-image is not constructed through perceptions of space in which the perceiving individual is situated into a pregiven ‘anterior’ spatiotemporal order. Rather, the perceiving body is related to a ‘homogenous’ notion of space insofar as our actions are embodied in the form of ‘concrete extensities’ (MM, p. 231). Bergson emphasises the immanent relationship that produces both perception and the body when he writes: To sum up: if we suppose an extended continuum, and, in this continuum, the center of real action which is represented by our body, its activity will appear to illuminate all those parts of matter with which at each successive moment it can deal [ . . . ]. Everything will happen as if we allowed to filter through us that action of external things which is real, in order to arrest and retain that which is virtual: this virtual action of things upon our body and of our body upon things is our perception itself. (MM, p. 232) Our perception of the world is therefore an expression of our relationship to it; that is, our perception ‘is a part of things’ and the consequence of this order of expression is that ‘things participate in the nature of our perception’, which also provokes a radical rethinking of the relationship between matter and geometry, because the division

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between the intellect and nature is removed (MM, p. 182). But this reconfiguration also prevents a return to the pregiven harmony of symbolic correspondences; instead, Bergson insists that perception and the body-image enact a highly reflexive engagement with the external world. Furthermore, it is not only matter or pure perception that reconfigures the notion of the subject, because memory also reconfigures the relationship between quality and quantity (MM, p. 182). Thus, in Bergson’s retrieval of an irreducible unity (forgotten by homogenous geometry, which denies the immanent and reflexive senses of the subject in the world), perception and memory form an intensive limit or variable relation that expresses the ‘indeterminacy’ of the body as a centre of action. He writes: ‘From this indetermination’, we ‘have been able to infer the necessity of a perception, that is to say, a variable relation between the living being and the more-or-less distant influence of the objects which interest it’ (MM, p. 33). This ‘variable relation’ he concludes, is ‘consciousness’, given by memory to perception, because ‘there is no perception which is not full of memories’ (MM, p. 33). In the following section, I examine the structure of memory and its forms as habit and duration, in more detail.

Memory In chapter II, Bergson examines how body-limit constitutes the limits of memory, when he explains that memory provides the means through which to bring together mind and matter, as a ‘place of passage’, rather than as a receptacle for storing images. He writes: ‘It is then the place of passage of movements received and thrown back, a hyphen, a connecting link between the things which act upon me and the things upon which I act – the seat, in a word, of the sensori-motor phenomena’ (MM, pp. 151–2). This definition therefore draws attention to the absolute difference between sensations and memory. Memory is not of the body, but as it passes into sensations, it becomes lived by the body. As a result, the body is the intensive limit (or relation) between its sensations and memory, and matter and memory therefore become radical revisions of the relationship between space-time and perception in which the living body is the intensive limit that links image to sensation (MM, p. 182). The distinction between perception and memory is also partly evident in the difference between ‘external perception’, which is the ‘pure’ form of perception defined independently of its relationship to memory, and

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perception, which is constructed by memory. However, Bergson points out that pure perception is always inadequate and requires memory in perception to provide a means of determining ‘with more precision the point of contact between consciousness and things, between body and spirit’ (MM, p. 65). So perception becomes understood as a ‘concrete’ state that describes the tension between the internal consciousness of the subject and external matter, constituting a kind of envelope through which homogenous movement becomes heterogeneous change. In addition, this ‘concrete perception’ or body-envelope is ‘the living synthesis of pure perception and pure memory’, which has different rhythms of duration and internal ‘tension’ (MM, p. 246). Furthermore, this envelope (or tension) provides the means through which ‘to overcome the opposition between quality and quantity [in] the idea of extension, that [lies] between the unextended and extended. Extension and tension admit of degrees, multiple but always determined’ (MM, p. 247). However, up to this point in the discussion, the notion of ‘pure perception’ remains disengaged from the body and from its subjectivity, so that Bergson sets out to demonstrate how this ‘consciousness’ is not just a geometric principle of concretion but is a rethinking that will involve him in ‘[restoring] to the body its extensity and to perception its duration’. Consciousness is therefore reconnected with ‘its two subjective elements, affectivity and memory’ (MM, p. 233). Bergson explains these ‘elements’ in the following pages; affection, he writes, represents the internal ‘senses’ of the body that enter into our perception so that the body’s surface constitutes the ‘common limit’ between our affection and other external bodies to produce both sensations (i.e., feelings) and images (i.e., other objects). The body surface is therefore a double site for internal and external relations (MM, p. 233). Alternatively, in an earlier discussion he explains that ‘[a]ffection is, then, that part or aspect of the inside of our body which we mix with the image of external bodies; it is what we must first of all subtract from perception to get the image in its purity’ (MM, p. 58). However, in contrast to perception and sensation being considered distinct, because they are different degrees of the same order, ‘pure perceptions’ or ‘images’ are the limits from which sensations are produced by the body. Sensations and images are therefore considered to be true relations of one another, so that sensations ‘will then appear as the impurity which is introduced into [the image] being that part of our own body which we project into all others’ (MM, pp. 234–5). As a result, memory and affection are strongly reminiscent of the powers of the embodied soul, explored in the preceding chapters.

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Memory is the key metaphysical innovation through which Bergson produces a dramatic reconfiguration of the relationship between the material and spiritual realities of the subject, and is expressed in two forms: duration and habit. Duration is the principle which confirms the reality of ‘life’ as a necessity for metaphysics and science. It affirms the intuitive basis of life, not as a division between representations and the sense-perceptions of space and time (or a repetition of a higher knowledge) but as an intuition that is inherently creative and active; for example, in the introduction, Bergson writes that Matter and Memory is an affirmation of ‘the reality of the spirit and the reality of matter, and tries to determine the relation of the one to the other by the study of a definite example, that of memory’ (MM, p. 9). Memory is the relation of ‘the spirit’, which informs any cerebral state, and produces a unique set of inextensive images, in contrast to those generated by pure perception. Memory does not therefore produce extensive images when Bergson writes: ‘Memory, inseparable in practice from perception, imports the past into the present, contracts into a single intuition many moments of duration, and thus by a twofold operation compels us, de facto, to perceive matter in ourselves, whereas we, de jure, perceive matter within matter’ (MM, p. 73). Instead, memory gives intuition to matter through which we might perceive ‘matter in ourselves’, rather than ‘matter within matter’. Underlining the distinction between matter and memory, Bergson insists that matter cannot in itself be intuition, because perception is a ‘choice’ not an intuition; that is, perception or the selection of images arises from a more visible ‘discernment which foreshadows spirit’ (MM, p. 235). Memory, however, is related to the consciousness through which intuition is generated so that the material universe may then be considered ‘a kind of consciousness’ of relations and action between parts. Moreover, in order to ‘touch the reality of spirit’ a continuity between the present and past is required in the form of memory so that matter is abandoned for spirit. Memory is therefore a ‘theoretic consequence and the experimental verification of our theory of pure perception’ (MM, p. 235). Bergson identifies two forms of memory that are ‘actualised’: first, those realised through our habits and, second, duration or true memory. He explains: Habit rather than memory, [ . . . ] acts our past experience but does not call up its image. The other is the true memory. Coextensive with consciousness, it retains and ranges alongside of each other all our

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states in the order in which they occur [ . . . ]. Truly moving in the past and not, like the first, in an ever renewed present. (MM, p. 151) Thus, these ‘cerebral states’ are neither the cause nor duplicates of perception; for example, perceived objects are present in pure perception in which ‘the perceived object is a present object, a body which modifies our own’, whereas memory is concerned with absent objects or images insofar as ‘a remembrance is the representation of an absent object’. But Bergson also tells us that in order for an image of an absent object to be generated, the sufficiency of the body must be even greater (MM, p. 236). Therefore, both memory and perception constitute the ‘sufficiency’ of the body through which images are constructed, once again, recalling the irreducibility of Leibniz’s Monad. In the section below, I explore this notion of the ‘sufficient’ body in more detail in relation to the envelope which, I suggest, constitutes a particularly aesthetic geometric figure in Matter and Memory.

The envelope The body is a centre of action and has the ability to generate ‘new action’ that represents an intensive aggregation of ‘limits’. This constant tension, between the production of internal sensations and external images in the ‘body-limit’, can be characterised as a topological surface-limit or event in which interiority and exteriority are an intensive limit; it is, ‘merely the distinction between my body and other bodies’, rather than being separated by an irreducible difference between the interiority of the body and the external world. Bergson explains: The distinction between the inside and the outside will then be only a distinction between the part and the whole. There is, first of all, the aggregate of images; and, then, in this aggregate, there are ‘centers of action’, from which the interesting images appear to be reflected: thus perceptions are born and actions made ready. (MM, p. 47) Thus, the body surface is constituted by ‘the common limit of the external and the internal [and] is the only portion of space which is both perceived and felt’, a conduit for the transmission of the virtual into real action (MM, p. 57). This topological and aesthetic continuity generates the body, not as a mathematical point in space but as a ‘privileged image’ in which ‘its virtual actions are complicated by, and impregnated with, real actions, or, in other words, that there is no perception

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without affection’. Bergson continues that ‘affection is, then, that part or aspect of the inside of our body which we mix with the image of external bodies; it is what we must first of all subtract from perception to get the image in its purity’ (MM, p. 58).7 In particular, chapter IV reveals the complexity of this topology, which is comprised of the extreme dualism between the memory and perception that reconfigures the relationship between the body and soul (i.e., intuition). In addition, this dualism opposes the symbolic parallelism of idealist and materialist metaphysics in which the mental and physical realms are taken as ‘duplicates’ of each other because, for Bergson, the body is neither a pure site of creating perceptions nor a site of storage of ‘recollections or images’, but becomes ‘an instrument of action, and of action only’ (MM, pp. 225–6). Furthermore, because of this emphasis on the body as an instrument of duration, we also see a reprise of Kant’s reflective subject and its ‘technical’ powers of memory in the act of constructing geometric figures; and in the following section, I suggest that Kant and Bergson are also connected by Bergson’s construction of the relations between matter and memory, and his critique of space and time. In addition, I go on to suggest that Bergson’s retrieval of an embodied intuition constitutes an enfolding, back to Kant and the embodied intuition of the Meno, and an unfolding forward towards Husserl’s geometric sense-horizons.

Space and time This chapter proposes that one of the most dramatic reconfigurations in Matter and Memory is the relationship between space and time, because Bergson’s critique of pregiven, symbolic relations in metaphysics enables new concepts of space and time to be generated, therefore removing the restriction of time to the status of a formal imitation of space. By producing a unique and qualitatively different notion of time, Bergson also enables space to be rethought, constituting a relation that is not determined by equivalence, and is liberated from the pregiven harmony of space-time. Qualitative distinctions of space and aesthetic geometric relations are affirmed in the notion of duration, which also revises the aesthetic ‘relations’ between space and time. The nature of space is therefore radicalised in relation to both time and geometry. Furthermore, Bergson’s intensive critique means that ‘intermediary’ geometries exist, challenging the assumption that modern geometries can only be repetitions of an absolute reason. Instead, his analysis suggests that there are a series of intermediary geometries generated between

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the homogenous and diagrammatic ‘intuitions’ of space and time and the ‘true’ aesthetic of ‘duration’. As discussed earlier in the chapter, this is partly achieved by limiting space and time to minor figures in Matter and Memory and constructing them from an aesthetic discussion of extensity, so that they are expressed independently of the limited planes of ‘ready-made’ science and metaphysics.8 But what value do space and time have? First, we see that they are necessarily homogenous inasmuch as they provide a site for actualising the virtual duration of memory. Space and time are therefore necessary aspects of extensity when Bergson tells us that they are concrete perceptions that are situated on the plane of the measured and physical ‘diagram’. In this respect, they are constituents of a diagrammatic concretisation of life or ‘fulcrums of action’. He writes: Homogenous space and time are then neither properties of things nor essential conditions of our faculty of knowing them: they express, in an abstract form, the double work of solidification and of division which we effect on the moving continuity of the real in order to obtain there a fulcrum for our action, in order to fix within it starting points for our operation, in short, to introduce into it real change. They are the diagrammatic design of our eventual action upon matter. (MM, p. 211; my emphasis) Space and time are therefore natural effects of our intuition, and are required in order for real change to be actualised, because they provide bridges between the virtual and the physical act and operate as discursive functions (in contrast to Kant’s homogenous and restricted intuitions). In Bergson’s writing we therefore also see that space and time constitute a recollection of the unfolding of the unextended soul into the extended images of the imagination in Proclus’ Commentary, so they may be understood as aspects of the unfolding of inextensive matter into extensive matter. Bergson’s critique of ‘homogeneous’ geometry therefore reveals the operations of space and time to be ‘actualisations’ of extensity. But this analysis also enables him to identify the forgotten relations through which a new concept of a heterogeneous and aesthetic geometry is produced; that is, the natural geometry of the body, and its relationship to the aggregates of matter and memory, becomes the primary site for reconfiguring the relationship between intuition and geometry. In addition, Bergson’s acknowledgement that the extended forms of geometry are necessary ‘fulcrums’, through which

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space and time can be actualised from their virtual states of pure memory, also constitutes the ‘double movement’ of the different states of reality (i.e., the movement from the virtual to the actual and vice-versa), enabling an aesthetic geometry to be embodied in the individual. As a result, this reflects not only the problem of geometry’s universalising tendencies but also enables geometry to be reinvigorated with a qualitative specificity that has been forgotten in post-Kantian metaphysics. Thus, in his examination of extensive matter, Bergson insists upon how matter can be actualised without it being handmaiden to an ‘amorphous and inert space’, thereby positing a relationship between matter and space that is productive and ‘active’; that is, each is constructed in the act. In addition, he suggests that the diagram mediates between the liberating procedure of extensive action (or extensity) and the limitations of homogenous space when he writes: ‘It might, then, be possible, in a certain measure, to transcend space without stepping out from extensity; and here we should really have a return to the immediate, since we do indeed perceive extensity, whereas space is merely conceived – being a kind of mental diagram’ (MM, p. 187). This leads Bergson to assess the concept of space and its objects that are constructed in a symbolic scientific method, especially in the Cartesian method, and results in his rejection of the imagination, as an operation of this symbolic limit; for example, he considers the nature of mathematical movement in the image of a hand moving from one point to another, writing that, without the limit of the imagination introducing moments of division or ‘halt’, the movement is ‘one’ or a unified ‘passage’, so that movement becomes understood as the passage of a body in space (MM, p. 189). Without the imposition of the imagination, ‘real’ movement is therefore generated, rather than the illusion of fixed points in space. In addition, science’s representation of actions as external and symbolic geometric properties (such as, the definition of the point and line along which a hand is seen to move) delimits movement to a representational equivalent, rather than attributing it with a ‘real’ condition of extensity. Duration, in contrast, resists all symbolic representations of its changes of state because it is irreducible to being measured or divided into instants, however infinite they might be (MM, p. 190). As a result, this discussion recalls Leibniz’s theory of the approximate or ‘fictional’ geometric figure and infinite divisibility. But as noted above, Leibniz’s account is ultimately problematic for Bergson, since it upholds the symbolic harmony of God. In contrast, Bergson suggests that space can be an irreducible aspect of ‘extensity’ and pure movement

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(rather than a concept that designates either scientific or imagined representations of division): Concrete extensity, that is to say, the diversity of sensible qualities, is not within space; rather is it space that we thrust into extensity. Space is not a ground on which real motion is posited; rather it is real motion that deposits space beneath itself. But our imagination, which is preoccupied above all by the convenience of expression and the exigencies of material life, prefers to invert the natural order of the terms. (MM, p. 217; my emphasis) Duration is constructed as the fundamental principle of the living body in the movement from the mental state to the idea, from the idea to the image and from the image to sensation and action. As movement generated from unextended matter, the soul or image does not therefore involve a dislocation from extension; instead, the soul (or the virtual) remains part of the continuum and is expressed, not in ideal space, but in ‘pure’ time. Bergson explains this ‘passage’ between the unextended to the actualised or extended space-matter, as follows: if there is a gradual passage from the idea to the image and from the image to the sensation; if, in the measure in which it evolves toward actuality, that is to say, toward action, the mental state draws nearer to extension; if, finally, this extension once attained remains undivided and therefore is not out of harmony with the unity of the soul; we can understand that spirit can rest upon matter and, consequently, unite with it in the act of pure perception, yet nevertheless be radically distinct from it. It is distinct from matter in that it is, even, memory, that is to say, a synthesis of past and present with a view to the future . . . . We were right, then when we said, at the beginning of this book, that the distinction between body and mind must be established in terms not of space but of time. (MM, p. 220; my emphasis) In chapter II, Bergson highlights the extent to which he rejects ideal space in his criticism of the Cartesian dependency upon mechanical relations; for example, suggesting that Cartesians are confused about the structure of movement between the parts and the whole so that a relativity between the terms is introduced, which collapses into concepts of universal movement. Bergson critiques Descartes’ scientific method, suggesting that he produces a confused metaphysic because

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his physical understandings of movement are curtailed by a symbolic set of geometric relations. Descartes conflates his methods, Bergson argues, handling ‘motion as a physicist after having defined it as a geometer’ so that it is limited to a symbolic expression of relations. He continues: ‘For the geometer all movement is relative: which signifies only, in our view, that none of our mathematical symbols can express the fact that it is the moving body which is in motion rather than the axes or the points to which it is referred’ (MM, p. 194). Once again, movement and space are reduced to absolute states and Bergson’s critique of ‘movement’ and motion in this chapter reveals the limitations of an undivided and homogenous principle of space that classical metaphysics and scientific geometry generate. Thus, although the classical traditions of space and time are useful, insofar as they provide ‘fulcrums’ of reality, they are nevertheless always related to a symbolic relationship rather than a continuous notion of ‘life’. However, when movement is considered within the modern and qualitative sciences (e.g., as in Riemann’s topology), Bergson suggests that there is a significant shift from ‘the abstract study of motion’, to an examination of ‘the concrete changes occurring in the universe’, which properly defines internal movement. He writes that movement ‘whatever its inner nature, becomes an indisputable reality’ (MM, p. 193). Thus, modern qualitative science and philosophy break with the symbolic traditions of science and philosophy to enable heterogeneous spacetime and ‘real movement’: A moving continuity is given to us, in which everything changes and yet remains: why then do we dissociate the two terms, permanence and change, and then represent permanence by bodies and change by homogeneous movements in space? This is no teaching of immediate intuition; but neither is it a demand of science, for the object of science is, on the contrary, to rediscover the natural articulations of a universe we have carved artificially. (MM, p. 197) In this respect, Matter and Memory affirms that science and metaphysics constitute qualitative time and movement because they promote dynamic internal relations that challenge the problematic ‘ready-made’ harmony which underpin symbolic forms of philosophy and science. In addition, we find that these relations of heterogeneous change are determined by the reality of an intuition or aesthetic geometry that also reflects the scope of duration in the modern sciences. This, I suggest, constitutes a reprise of a ‘natural geometry’.

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Intuition Intuition is dramatically rethought in Bergson’s philosophy of geometry. As we saw in Chapter 1, Kant’s Critique of Pure Reason restricts intuition to either a version of ‘pure’ reason or knowledge or a form of the sensibility. For Bergson, however, intuition is not knowledge but the actions of the living and irreducible subject, reinstated in the unity between the body and memory and liberated from its position as, either a cognitive or a material entity. Intuition, for Bergson, is therefore formed out of actual, concrete living, rather than its value as a symbolic knowledge that is situated into a schema of different modes of understanding the world. In addition, it increasingly develops into a fundamental concept of duration or life in his writing, further distinguishing him from the symbolic limits that determine the intuitions of space and time in Kant’s Critique of Pure Reason. Thus, within this reconfiguration of relationships, geometry is not a closed system that is relegated exclusively to the symbolic artifice of mathematics and metaphysics, but is reconnected to intuition. Intuitive geometry is therefore the aesthetic principle – that is, the envelope – that constitutes the reflective and living subject. In Chapter 1, the relationship between geometry, intuition and the body the Meno is implied in Socrates’ drawings of geometric figures and the boy’s ability to answer questions about their construction. In this chapter, Bergson amplifies these possibilities by showing how the intuitive acts of constructing geometry are expressed: first, in the physical movements of the body (i.e., as physical activities); and second, in the ‘recollection’ of geometric principles (i.e., as mental activities) that constitute an embodied geometry, memory or absolute intuition. Thus, Bersgon underlines the activity of geometric construction, not as ‘reason’ but as ‘living’ or ‘natural intuition’, because the act of drawing geometric figures is derived from the bodily perceptions of ‘habit’ or memory. So intuition is not cognitive thinking, but action in which space and time are brought together as extensity and duration; and in a manner which also recalls Spinoza’s study of the living subject’s active comportment. In contrast, Kant’s notion of intuition in the Critique of Pure Reason is a cognitive idea of image-perceptions that are not sensuous, but forms of thought, and although Kant develops his geometric thinking in the Critique of Judgment, which retrieves the act of construction, it is nevertheless distinct from Bergson’s critique, because the event is mediated through the faculty of the imagination, rather than being immanent with the sensuous manifold of perception and memory, that is, the body.

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Bergson’s metaphysics therefore involves a retrieval of geometric thinking in which intuition, as the actions of the body, embodies the discursive activities of perception and memory; it is a return to the body as the site of discursivity, undoing the restrictive harmony that ties intuition to the non-discursive intellect or faculty of the nous. Instead, discursivity is embodied both in the body and in the mind, and reveals a way back (an enfolding) to Proclus, Spinoza and Leibniz’s notions of extensive and unextensive discursivity. In addition, intuition is integral to the scope of the aesthetic and thinking subject to be a fully temporalised and autonomous unity. Intuition remains a transcendental concept, but rather than accepting its formulation as an incomprehensible and non-discursive level of cognition (i.e., the inexplicable ‘all-in-one’ grasping of an idea), Bergson reveals its discursive interiors in the aesthetic subject and in the psychic powers of memory (the soul) and perception. Thus, intuition is not brought back to Kant’s understanding in which it is the sense-based form of an absolute intuition (i.e., space and time) and the absolute, yet inexpressible intuition. Instead, it is a ‘union’ of discursive and nondiscursive activities that constitute the living subject in which the aesthetic geometric method and its figures are aspects of this discursivity, not as anterior diagrams but as aesthetic expressions of intuition’s ‘natural’ orders. So, in Matter and Memory, intuition unifies duration, memory and ‘life’ in a unique kind of subject, in contrast to the restricted notion of intuition that Kant produces in the Critique of Pure Reason. For Kant, the sense-intuitions are merely empirical powers of presentation and are limited to a symbolic harmony with the transcendental ‘thing-in-itself’ (noumenon); thus, as a constituent of unification or harmonious construction, embodied intuition is always a limited and symbolic function. However, Bergson seeks to reveal the ‘true’ nature of intuition in relation to the subject in his critique of spatial and temporal intuitions so that, rather than producing formal appearances, intuition is an affirmation of corporeal acts of construction in life, and provides an intensive inquiry into the relationship between concepts of ‘life’ and the subject. (In the essay, ‘Philosophical Intuition’, written in 1933, Bergson writes that his ‘modern’ ancestors, Spinoza and Leibniz, also desired these metaphysical relations, but were ultimately beholden to the symbolism embedded in rationalist metaphysics and science9). Bergson’s notion of intuition and its operations are not, however, harmonised with pregiven metaphysical principles, so that the notions of limit, unlimit, quality and quantity are also intensively cross-examined towards

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revealing (not symbolic equivalents) but intuitive powers that are fundamental constituents of duration and the body as a centre of action. This embodied intuition is even more explicitly promoted in the subsequent texts, ‘Introduction to Metaphysics’, and Creative Evolution. In the former, intuition is proposed as a progressive philosophy and in the latter text, it becomes a ‘natural’ geometry that is intuitively produced in the act of drawing geometric figures; for example, in ‘Introduction to Metaphysics’ Bergson examines the cause of ‘forgetting’ intuition in philosophy that results in philosophy’s inability to generate a real relationship with its ‘origins’ and its failure to express the inherent ‘extensity’ of its nature. Moreover, Bergson argues that philosophy takes its own limited methods to be real truths, for example, when he writes: ‘Relative is symbolic knowledge through pre-existing concepts, which goes from the fixed to the moving, but not so intuitive knowledge which establishes itself in the moving reality and adopts life itself of things’ (‘Introduction to Metaphysics’, 1903, p. 276).10 In the following sections this discussion is developed to show that three forms of intuition are identifiable in Matter and Memory, ‘Introduction to Metaphysics’ and Creative Evolution: first, as duration (action); second, as philosophy; and third, as ‘natural geometry’ (construction).

Envelope I: intuition as duration In Matter and Memory intuition expresses a topological relationship between the body and duration. Intuition becomes a form of geometry that is generated out of Bergson’s insistence upon the body, as a centre of action, so that life is also resituated into the centre of philosophical thinking. Action and life are therefore fundamental concepts within Bergson’s philosophy, through which a new kind of intuition is invented in the form of duration. Thus, the unification of life and intuition are brought together in the notion of the ‘act’, and the body’s actions in turn reveal the inherent relationship between intuition and spatialised experience. In turn, duration always has a relationship to space, but not so that its reality, as duration, is lost. For Bergson, then, the body must always be an active body – a body in action or event – for the link between the body and space to be realised in duration and as extensity. Thus, although Matter and Memory is concerned with releasing time from the dominant perceptions of space and geometry, it nevertheless provides an innovative notion of geometry that is generated from the immanent relationships which connect the body, geometry and

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intuition together in the form of duration. Intuition is therefore the continuum in which geometry is an intermediate ‘plane’, and is the connection between the body, as a site of intuitive action, and geometry, insofar as it is an ideal reality of space. Furthermore, this embodied geometric intuition represents ‘life’ in which the actions of the body unify space and time and the duration of the body; for example, Bergson writes: if our belief in a more or less homogeneous substratum of sensible qualities has any ground, this can only be found in an act which makes us seize or divine, in quality itself, something which goes beyond sensation, as if this sensation itself were pregnant with details suspected yet unperceived. Its objectivity . . . must then consist . . . precisely in the immense multiplicity of the movements which it executes, so to speak, within itself as a chrysalis. Motionless on the surface, in its very depth it lives and vibrates. (MM, p. 204) Previously, Bergson writes that the living body and its actions also embody the axis of homogeneous space and time, as well as the axis of multiplicitous duration, and so we can therefore suggest that intuition is constituted in the continuously transforming, topological unification of these internal and external relations, when he writes that ‘Pure intuition, external or internal, is that of an undivided continuity (MM, p. 183; my emphasis). Here we see how Bergson revitalises the concept of intuition, yet also upholds the necessity of spatiotemporal relations, in his challenge to the ‘impotence of speculative reason as Kant has demonstrated it’, which perpetuates the divide between the noumenal and sense perception (MM, pp. 184–5). Rather, Bergson’s solution lies in his alternative, ‘third’ unique intuition, that is, duration. He explains: It seemed to us that a third course lay open. This is to replace ourselves in pure duration, of which the flow is continuous and in which we pass insensibly from one state to another: a continuity which is really lived, but artificially decomposed for the greater convenience of customary knowledge. (MM, p. 186) Thus, the connection between geometric space and duration is posited when Bergson suggests that geometric space is an ideal towards which we move, but never achieve. Geometry is not therefore banned, but is necessary for extensity which is constitutive of both duration and the intermediary spatiotemporal conditions which exist between geometric

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ideals and embodied, that is, lived, durational space. Moreover, in the section below, I suggest that Bergson’s study of ‘proper’ metaphysical relations directly informs his theory of intuition and its potential for retrieving lost connections between duration and embodied geometric thinking.

Envelope II: intuitive philosophy In the essay, ‘Introduction to Metaphysics’, Bergson makes the case for ‘intuitive thinking’ that will transform the scope of philosophy into a ‘progressive philosophy’. Intuitive thinking, he suggests, has the power to disrupt the logical artifice of symbolic metaphysical and scientific thought by introducing ‘life’. He writes: ‘But the simple act which has set analysis in motion and which hides behind analysis, emanates from a faculty quite different from that of analysing. This is by very definition intuition’ (IM, p. 281). Intuition, then, is a ‘simple act’ in which a particular kind of ‘philosophising’ or thinking is generated that is not reducible to analytic methods. So, although the discipline of philosophy requires logical and analytic reasoning, Bergson suggests that intuition is able to reverse this procedure, so that a progressive philosophy is formed. He writes: ‘our mind is able to follow the reverse procedure. It can be installed in the mobile reality, adopt its ceaselessly changing direction, in short, grasp it intuitively’ (IM, p. 275). Continuing this argument, he states that intuition introduces a ‘violent’ rupture in the dominant procedure, and enables the generation of ‘fluid concepts, capable of following reality in all its windings and of adopting the very movement of the inner life of things’ (IM, p. 275). So the essay’s critique of intuition demonstrates the scope of a unique unity, which both augments the purposes of modern philosophy and science and intensifies their internal structures and external relations to each other. Thus, he is not suggesting that intuition is reinstated in order to undermine the work of philosophy or science, rather that it should be properly accounted for by each method: Science and metaphysics then meet in intuition. A truly intuitive philosophy would realise the union so greatly desired, of metaphysics and science . . . . Its result would be to re-establish the continuity between the intuitions which the various sciences have obtained at intervals in the course of their history, and which they have obtained only by strokes of genius. (IM, pp. 276–7)

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Furthermore, Bergson argues that intuition is intimately connected to the existence of different realities, when he suggests that modern science and metaphysics are determined by ‘understanding’, which is given to fixing, dividing and reconstructing to produce ‘stability either in relations or in things’. As a result, science’s ‘relational concepts’ and ‘concepts of things’ in metaphysics continue to forget the existence of a relationship between understanding and the underlying ‘intuition of reality’ (IM, p. 278). But this is also a positive critique, because Bergson shows how each tradition – although failing to acknowledge its relationship to life and intuition – does in fact contain ‘real’ concepts of intuition within; for example, he agrees that ‘modern’ science has introduced a ‘proper’ concept of movement and modern philosophy displays a latent preoccupation with ‘life’ as duration (IM, p. 277). However, for Bergson, these ideas are ultimately underdeveloped because they are determined by the limited homogenous spatiotemporal metaphors, movements or structures in each discipline – for example, the ‘tunnelling’ of metaphysics or the construction of bridges by scientists – which continue to ‘forget’ the aesthetic ‘moving river of things’ that ‘passes between these two works of art without touching them’ (IM, p. 278). In addition, this disciplinary ‘blindness’ is made more acute by Kant’s intensification of the symbolic operations of science and metaphysics, so that each discipline is even more autonomous from other ‘external’ realities; and here he criticises Kant’s misunderstanding of ‘intellectual intuition’ that is motivated towards the relative symbolism of science and the artificial symbolism of metaphysics. For Bergson, therefore, the retrieval of extensive intuition is absolutely necessary, in order to correct modern philosophical and geometric thinking, and in the next section I explore this ‘recovered intuition’ in his discussion about ‘natural geometry’ (IM, p. 279).

Envelope III: natural geometry and intuitive construction In Creative Evolution the importance of intuition for geometry, duration, space and time becomes more explicit in Bergson’s concept of ‘natural geometry’ and in his study of the act of drawing the geometric figure. Natural geometry – that is, the envelope – therefore embodies the body and the geometric figure in the intuitive act, so that geometry is conceived as a particular form of extensity, which is generated through intuition’s active extension of space.11 But, Bergson also suggests that intuition is a logical extension of a ‘natural geometry’ so that the act of constructing philosophy is also

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reconfigured from a purely intellectual endeavour into an embodied activity because geometric principles are derived from the body. Below, I explore how Bergson ‘recollects’ this embodied intuition in his examination of the act of drawing geometric figures, which challenges the perception that geometry can only be a handmaiden to logical manufacture and reasoning. As a result, we find that the activities of drawing and recollection do not become reduced to a series of logical demonstrations but are embodiments of aesthetic and geometric duration. In addition, natural geometry is a reprise of Bergson’s discussion about the intuitive body as a ‘centre of action’ in Matter and Memory; for example, he writes: ‘Besides consciousness and science, there is life. Beneath the principles of speculation, so carefully analysed by philosophers, there are tendencies of which the study has been neglected, and which are to be explained simply by the necessity of living, that is, of acting [ . . . ]’ (MM, p. 198). The embodied geometric act is therefore expressed in intuition or life; and, in a discussion about the concrete construction of intuitive geometry, Bergson explores its potential in Creative Evolution (a discussion which also recalls Socrates’ act of drawing in the Meno). In the context of philosophy, the act of drawing has a special relation to geometry because the aesthetic act of drawing figures is also inherently tied to the propensity for deductive thinking. But Bergson reminds us that the mind also has the propensity for intuitive thought in the form of duration, which can ‘violently’ reverse this logical progression of space and time. In this respect, we see the necessity of a psychological power in Bergson’s argument, because it is not enough for him to propose a different way of thinking as knowledge. Rather, Bergson’s desire is to find a way that reflects the specificity of the living organism, which resists the tendency to attribute the same methods of construction (e.g., deductive and inductive thought) to both organic organisms and inorganic matter. Bergson reminds us that geometry is a pure knowledge, which also exists in an intuitive state in the act of drawing. This relationship between geometry and our psychic powers is outlined in the first paragraph of a section entitled ‘Geometry and Deduction’, in which he argues that the powers of geometry are both, a step by step discursive extensity and an intuitive construction of embodied spatiotemporal unities (figures): All the operations of our intellect tend to geometry, as to the goal where they find their perfect fulfilment. But, as geometry is necessarily prior to them (since these operations have not as their end to

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construct space and cannot do otherwise than take it as given), it is evident that it is a latent geometry, immanent in our idea of space, which is the mainspring of our intellect and the cause of its working. We shall be convinced of this if we consider the two essential functions of intellect, the faculty of deduction and that of induction. (CE, 1964, p. 222)12 Here, Bergson’s discussion of geometry and intuitive acts in Matter and Memory are developed into a study of the ‘intuitive’ relationship between the subject’s powers of embodied spatial deduction and induction. But this later text also emphasises the power of intuition as an act of construction, rather than as a mode of knowledge, so that the notion of geometry is understood to be a productive sensibility, that is, an aesthetic; for example, when he explains that deduction is ‘the same movement by which I trace a figure in space engenders its properties: they are visible and tangible in the movement itself; I feel, I see in space the relation of the definition to its consequences, of the premises to the conclusion’ (CE, pp. 222–3). Deduction therefore generates ideas as part of an ongoing or infinite process which is not only a priori but also ‘participates’ in the ‘imperfection’ of the constructive act (CE, p. 223). In contrast, induction enables the construction of a unity, either as an image or as an action, for example: But when I trace roughly in the sand the base of a triangle, as I begin to form the two angles at the base, I know positively, and understand absolutely, that if these two angles are equal the sides will be equal also, the figure being then able to be turned over on itself without there being any change whatever. I know it before I have learnt geometry. Thus, prior to the science of geometry, there is a natural geometry whose clearness and evidence surpass the clearness and evidence of other deductions. (CE, p. 223) The geometric figure is therefore defined by an infinite duration and discursivity that are the aesthetic conditions of its construction, and which Bergson insists are vital because ‘[y]ou cannot represent this space to yourself without introducing, in the same act, a virtual geometry which will, of itself, degrade itself into logic’. Thus, he continues, natural geometry embodies an intellectual ‘letting go’ or detension, so that a ‘pure spatial intuition’ or unity is also produced (CE, p. 224). So we find that geometry and other space-acts are not mutually exclusive from each other. Instead, they are tied by the tension between the

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genetic development of spatial embodiments (e.g., in the act of drawing), and the absolute duration of the body. As we shall see below this ‘double movement’ is one of the most radical manifestations of relations that Bergson promotes in his ‘progressive philosophy’. Earlier, I showed that Bergson explores the tension of ‘double movements’ in chapter II of Matter and Memory, in order to produce more complex, heterogeneous notions of space and time. Bergson’s double movement is therefore a multiple kind of intuition; for example, it is used to express the spatiotemporal series and ‘diagram’ of actual perceptions and the virtual. Alternatively, it describes the perception of external objects, which arise from the movement between perceiving the objects as independent of consciousness, and our states of consciousness, which are independent of ‘objective reality’ (MM, p. 143). In addition, space and time are brought into tension between the necessity of space to ‘preserve’ reality and the necessity of time to ‘devour’ reality. Also, later in the chapter, I suggest that this double movement expresses the production of ‘general ideas’ in the movement between the ‘plane of action’ and the ‘plane of pure memory’, when Bergson states that ‘the general idea escapes us as soon as we try to fix it at either of the two extremities. It consists in the double current which goes from the one to the other – always ready either to crystallise into uttered words or to evaporate into memories’ (MM, p. 162). Finally, extremes of memory – that is, action and dream – are also examined in relation to the term; for example, in the subject’s capacity to produce infinite ‘possible states of memory’, which comprise the ‘different planes’ of continuity between action and dream (MM, pp. 168–70). Once again, this double movement therefore registers the continuum of intermediate states that are produced between the body and mind, in which the mind travels unceasingly over the interval comprised between its two extreme limits, the plane of action the plane of dream . . . but the action is not able to become real unless it succeeds in encasing itself in the actual situation, that is to say, in that particular assemblage of circumstances which is due to the particular position of the body in time and space. (MM, p. 172) Thus, geometry cannot be considered only as a pregiven artifice of our idealist natures nor is it merely a material reduction of these ideas in the natural world. Rather, it is embodied in the psychic and lived world of duration and intuition – the vital elan – as an ideal state that is never achieved. As a result, geometry is released from its problematic

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role of accountability to readymade laws and is given back its relationship to immanent, intuitive acts and psychic powers, and in which the infinite continuum or process is not a logic, but life. As such, the intuitive acting body is a general discursive principle in Matter and Memory (i.e., it is an axiomatic text), which challenges metaphysical principles of a priori geometric production, yet the particular construction of geometric intuition remains implicit (in contrast to memory). In Creative Evolution, however, Bergson develops a more explicit discussion about the nature of an embodied, intuitive geometric method as his promotion of a philosophy of ‘life’ becomes more prominent; for example, mind, matter and space are explained as ‘evolutionary’ relations in the later texts. As a result, the subject’s thinking and acting are shown to have a greater affinity with spatialisation, so that we might suggest that intuition is an envelope between our habits and space; for example, Bergson writes: Thus, the space of our geometry and the spatiality of things are mutually engendered by the reciprocal action and reaction of two terms which are essentially the same, but which move each in the direction inverse of the other. Neither is space so foreign to our nature as we imagine, nor is matter as completely extended in space as our senses and intellect represent it. (CE, p. 213–4) By drawing from this discussion in ‘Creative Evolution’, it is evident that Bergson’s earlier theories of space in Matter and Memory demonstrate its power as an intuitive and highly radical notion of a durational matter or geometric figure – that is, an envelope – not just as an expression of a logical geometric diagram. In contrast to Kant’s view that mathematical geometry is independent of other forms of knowledge, Bergson considers it to be inherently related to our perceptions and our relationship to the external world. The Critique of Pure Reason shows us how, within the pregiven tradition of metaphysics and science to which Kant is tied, geometry is a highly constructed artifice of their symbolic systems. Bergson, however, resists this symbolism and instead posits an alternative relationship between geometry and other kinds of relationships with the world. Moreover, I have also shown that Bergson is sympathetic to the ‘modernity’ of Spinoza and Leibniz’s methods, especially their development of physics. But he also criticises them for upholding the symbolic determinations of Cartesian science and metaphysics, which perpetuate the negation of time and the dominance of a homogenous space, for example, when he

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suggests that seventeenth-century metaphysics repeats the limitations of ‘ancient metaphysics’ and science, so that their pursuit of a proper scientific notion of movement in modern physics – and hence the metaphysical exploration of matter and the soul – is restricted (CE, p. 167). So, according to Bergson, Spinoza and Leibniz fail to progress beyond the pregiven metaphysical harmonies of unintelligible powers – that is, God and the soul – which results in understandings of nature that rely upon unexplained relations and exclusions. As a result, his critique of the construction of space and geometry constitutes geometric thinking in one of its most original and most intensive forms, and is an impressive demonstration of an interrogation into the internal and modern ‘origins’ of geometry and philosophy. Furthermore, geometry and its interiors are made intensive, not only because Bergson demonstrates the power of rethinking the interiority of science and metaphysics but also because his thinking represents a highly reflexive practice. Geometry is not just a product of pure reason or intellect, rather it is also a product of our intuition, soul and other corporeal powers of sensereason (recalling Spinoza’s and Leibniz’s theories, especially), and underpins Bergson’s rejection of geometry as a limited product of symbolic reason. Moreover, the symbolic correspondence between matter and geometry is removed, because matter is not restricted to the harmonising effect of a logical geometry or cleaved from a metaphysics of ‘life’.

Summary Bergson’s writing is important in this discussion because of his affirmative, yet critical, engagement in the history of philosophy. In particular, his examination of Kant’s intuition in the Critique of Pure Reason is one of the most intense sites of engagement through which he develops key insights into new kinds of thinking that Kant overlooks. Moreover, Bergson radicalises the geometric method so that it becomes an intuitive act, which provides an alternative notion of construction; and, despite the inherent difficulties of Kant’s intuition, both he and Bergson share a concern about the need to rethink the scope of geometric methods and their role in philosophy. But it is only Bergson who releases time from space and in so doing enables a radically different notion of space to be expressed. For Bergson, then, we find that the construction and the actualisation of time represent intuitive acts and are evidence of a radically intuitive geometric method. In each case, because Kant reduces intuition to a form of ‘absolute’ knowledge any link between its different kinds is disallowed. In the Critique of Pure Reason, construction will

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always fall into two concepts – that is, different knowledges – that are dependent upon an irreducible abyss of unintelligibility defining their limits and are therefore at odds with the evolution of a complex thinking organism. Nevertheless, the potential for intuition to be a critical and creative active form of reason fascinates Bergson; and one might also suggest that it represents a highly creative relation in Bergson’s metaphysics and, in turn, establishes a hidden topological relation between his metaphysics and Kant’s. Intuition, then, is vital to Bergson’s ‘progressive’ notions of space and time, and he brings the revision of geometric ‘relations’ in philosophy to the forefront of this discussion in a highly critical examination of the symbolic theories of geometry and metaphysics that have perpetually forgotten the meaningful relations between philosophy, science and life. For Bergson, these symbolic systems mean that no real material or relationship can be realised, and spatiotemporal relations remain limited to representations or ‘signs’ of intellectual reason, rather than manifesting proper expressions of ‘lived’ experience. Hence, he argues that space and time need to be liberated from the closed systems of Platonic or Cartesian symbolism and their respective sciences, and explored in terms of ‘psychic’ necessity. Therefore, space is not objective; rather space and geometry embody concrete subjects and events in life. In this respect, space and geometry are attributed with a genetic condition of change that is intimately tied to duration, which is also highly reminiscent of Spinoza’s study of spatiotemporal relations and extensity. However, although Bergson applauds Spinoza’s project for reviving the intuitive status of geometry, he is critical of Spinoza’s reliance upon a relativist metaphysics that collapses a potentially liberated expression of the world and life back into a harmony with understanding. In contrast, Bergson constructs space as an intuition that manifests habit (memory), and the tendency of the organism to orientate and express itself through physical spatialisations, which promote a ‘natural’ geometry that is properly intuitive, constructive and reflective of the limits of the body as a centre of action and duration. In addition, in Creative Evolution, Bergson suggests that geometry is reflexive of the living subject, in contrast to symbolic geometry that is produced by an ancient metaphysics and underlying ‘pre-modern’ science. Geometry is therefore interiorised into the actions of the subject or envelope (rather than externalised or repressed), representing a forgotten geometry, which Bergson suggests can be realised once a new metaphysics of ‘intensive quality’ is retrieved. Geometry is not merely consigned to being a

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product of a closed metaphysics or science, but is brought into a ‘lived’ philosophy (and, although we have seen that Kant’s reflective judgment proposes this shift, it remains ‘speculative’). In this penultimate chapter, the relations between geometry, space and time are central to the production of a truly progressive philosophy. In the last chapter, Husserl’s geometric thinking generates multiple ‘horizon-figures’, through which geometric figures are embodied in relation to other subjects and historical forms of sense-reason, enabling a continuously changing aesthetic and historical tradition of geometric thinking.

6 Horizons

This final chapter examines Husserl’s essay, The Origin of Geometry, in which he analyses a special kind of geometric sense-idea. Calling for a ‘reactivation’ of a Historie of geometric thinking, he explores a special temporal horizon of existence between living subjects. Husserl’s twentieth-century analysis of this teleological-history of horizons therefore retrieves the relationship between geometry, sense and the subject that has been developed in the preceding chapters. The Origin is also developed as part of his last unfinished philosophical work, The Crisis of European Sciences and Transcendental Phenomenology: An Introduction to Phenomenological Thought (1954) in which he posits a new kind of apodictic knowledge for living in the world. Although the Origin was first published in 1939, Husserl completed the manuscript in 1936, as an appendix to the Crisis, which presents a ‘spiritual’ or historically informed transcendental philosophy of objective knowledge. As a result, the Crisis’ emphasis on a phenomenological basis of apodictic knowledge is central to understanding how the Origin’s historical and teleological examination of geometric thinking is constructed, and provides important clarification about Husserl’s concept of sense in the Origin. Moreover, Husserl’s personal subjectivity underpins the historical formation of his critique; as an Austrian-born Jewish philosopher living in Germany, and writing at the height of Nationalist Socialism in Europe during the turbulent decades of early twentieth century, his critique of sciences still has historical resonance for discussions about the role of science in contemporary societies.1 For Husserl, geometry’s exemplary status as an apodictic form of knowledge (i.e., irrefutable reason) reveals its virtue for generating clarity in thought. Yet this potential for clear thinking also highlights how the 154

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modern sciences have become separated from their ‘extra-scientific’ historical and cultural values. He shows how geometry is, in fact, exemplary of a special ‘spiritual’ class of ontological entities (a special category of life), which cannot be sufficiently explained by the opposing metaphysical orders developed by empirical (materialist) and objective (idealist) philosophies. Instead, he proposes that geometry is a ‘living science’, which is constituted by an internal genetic or living ‘tradition’ (CES, p. 356).2 Calling this special ‘spiritual’ or ‘teleological’ progression and recollection of geometric ideas, ‘Historie’, the Origin recovers an existential reason in geometric thinking. In addition, this complex geometric discussion provides an important illustration of the new kind of phenomenology that Husserl develops in the Crisis, in which historical imperatives are reinstated into ontological philosophy (CES, pp. 354–5). Husserl defines this new philosophical science of sense, ‘transcendental phenomenology’, calling it the ‘novel universal science of subjectivity as pregiving the world’ (CES, §38, p. 147). In addition, this ‘teleologicalhistorical’ analysis of geometry shows that a new kind of philosophical inquiry is required for modern life in which spiritual forms of existence directly inform apodictic knowledge. Geometric ideas are significant transcendental forms of this ‘sense-history’, through which the symbolic value of magnitude placed on geometry by the modern sciences is challenged and its powers of reasoning in speculative philosophy are reactivated. Thus, the Origin and Crisis oppose the inadequate ‘readymade’ and symbolic geometric reasoning of the modern scientific disciplines. Husserl’s geometric horizons also construct a geometric encounter with Kant’s transcendental and aesthetic thinking. Like the philosophers who precede him in this study, Husserl’s geometric method and geometric ideas constitute legitimate forms of sense-reason and are necessary states of the living world. However, unlike Kant, Spinoza, Leibniz or Bergson, Husserl does not explicitly promote a theory of perceptual or aesthetic geometry in his critique of modern scientific method; for example, he is sceptical of aesthetics’ ability to produce objective ‘senseideas’, and this prevents him from endorsing empirical or materialist theories of ‘perception’ and the sensibility (in particular, those developed by Hume, Berkeley and Locke). Instead, his focus on the ‘problemhorizon of reason’ (CES, p. 378) is firmly directed towards the need for a new method of apodictic clarity and he identifies transcendental phenomenology as the philosophical project through which this can take place.

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However, this discussion suggests that a correspondence does exist between Husserl and Kant’s Critical and aesthetic geometric thinking. The connection is threefold: first because of the importance of critique that informs Kant’s aesthetic project and Husserl’s phenomenological project. Second, Husserl’s belief that geometric thinking is a special example of temporal historical sense, not a conceptual form of scientific thinking, connects him to Kant’s technical and aesthetic geometric subject. Third, Husserl’s emphasis on ‘intersubjective’ horizons bears a strong resemblance to Kant’s theory of reflective and aesthetic judgment, because of his focus on transcendental and synthetic a priori ideas. Also, Husserl’s opposition to aesthetics and his critique of the sensibility is developed out of an analysis of the Critique of Pure Reason, not the Critique of Judgment. However, if Husserl’s theory of a ‘reactivated science’ is examined in relation to Kant’s technical aesthetics, in particular, his ‘aesthetic subject’, then Husserl’s phenomenological geometric ideas also express aesthetic qualities. By suggesting that Husserl’s sense-ideas correlate with Kant’s theory of aesthetic judgments, transcendental phenomenology and aesthetics are brought together in ideas about life and existence.3 Moreover, Husserl’s teleological-history of geometric thinking and his attention to an original form of temporality are also evidence of a particularly aesthetic form of geometric sense-reason.

Geometric reason and ‘Teleological-historical reflection’ Husserl examines geometry because it is a paradigm of apodictic knowledge, and it is therefore an excellent example of pure objective reason that is sought by the modern sciences. He defines geometry as ‘all disciplines that deal with shapes existing mathematically in pure spacetime’ (CES, p. 354). He clarifies this definition further in the final paragraphs of the Origin, writing that ‘Geometry and the sciences most closely related to it have to do with space-time and the shapes, figures, also shapes of motion, alterations of deformation, etc., that are possible within space-time, particularly as measurable magnitudes’ (CES, p. 375). Therefore, for Husserl, geometry and its associated methods are inherently concerned with the production of autonomous a priori forms of objective knowledge. However, having identified geometry’s potential for generating objective knowledge, Husserl also outlines the purpose of his ‘reactivation’ of geometry in the first pages of the essay; that is, to reveal its ‘submerged’ histories, in particular, to recover an a priori tradition of forgotten geometric sense-ideas and a historically constituted

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‘living science’ (CES, p. 354). This approach enables Husserl to reinstate a ‘spiritual’ form of reason into scientific thinking; for example, he critiques Galileo’s innovations in modern scientific method which bring greater abstract clarity to geometry, but conceals the ‘true’ project of a living reason and, consequently, the potential of a ‘living science’. By contrast, Husserl proposes a dynamic and living form of reason that releases geometry from its value as an objective, disembodied form of knowledge in modern science, so that its methods and objects are reactivated; for example, in Part I of the Crisis, he describes the need to reconnect reason and ‘Existenz’ by means of a ‘teleological-historical’ method of reflection (CES, p. xxiv). Existential reason, he writes, is concerned with the question or ‘problem’ that underpins all metaphysical questions about God, man, immortality and ideas of existence: Reason is the explicit theme in the disciplines concerning knowledge (i.e. of true and genuine, rational knowledge), of true and genuine valuation (genuine values as values of reason), of ethical action (truly good acting, acting from practical reason); here reason is the title for ‘absolute’, ‘eternal’, ‘supertemporal’, ‘unconditionally’ valid ideas and ideals. (CES, p. 9) For Husserl, reason is inherently connected to hidden notions of existence or life which dominant rational scientific procedures conceal through their production of disembodied knowledge and methods. He defines this special ‘total’ form of existence as, ‘Existenz’, and proposes that the modern crisis in the European sciences is due to a lack of clarity about how this ‘total’ concept of ‘Existenz’ constitutes modern European ‘cultural life’ (CES, p. 12). In contrast to scientific reason, this living form of geometric reason cannot be determined merely by recording changes in scientific or historical facts. Husserl therefore argues that philosophical and scientific inquiry needs to be redirected towards the ‘deepest essential interrelation between reason and what is in general, the enigma of all enigmas’ (CES, p. 13). Transcendental philosophy, he argues, reinvigorates these metaphysical questions. It retrieves this ‘latent reason’ in ‘the understanding of its own possibilities and thus to bring to insight the possibility of metaphysics as a true possibility’ (CES, p. 15). In the Origin, Husserl develops this argument by examining the ‘most original sense in which geometry once arose, was present as the tradition of millennia, is still present for us, and is still being worked on in a lively forward development’ (CES, p. 354; my emphasis). Geometry is not the

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simple eidetic repetition of pregiven mathematical ideas of abstract space and measurement. Nor is the essay a claim for the objective historical study of ancient geometric personae, facts, events, methods or innovations, for example, when he states: ‘We must focus our gaze not merely upon the ready-made, handed-down geometry and upon the manner of being which its meaning had in [Galileo’s] thinking’ (CES, p. 353). Instead, it is an inquiry into ‘depth-problems’, especially, the ‘sense in which [geometry] appeared in history for the first time’ (my emphasis). Husserl examines these lost historical and psychic activities in geometric thinking, in order to retrieve the ‘original’ meanings in geometry: ‘which continued to be valid with this very same meaning – continued and at the same time was developed further, remaining simply “geometry” in all its new forms’ (CES, p. 353). Geometry is therefore constructed by a powerful a priori sense-idea of existence, that is, Historie or ‘spiritual’ tradition.4 In addition, Husserl’s Historie is a critical form of philosophical reactivation, in response to the lack of sense-reason in modern European scientific methods. Returning to this argument in the final paragraphs of the text, Husserl underlines the significance of historical purposiveness for geometric thinking, emphasising the need for an internal history of geometry, not the continuation of disembodied and idealised mathematical knowledge (CES, p. 378). He criticises the ‘historically factual aspect of mathematics’, which is determined by a symbolic and externally situated ‘romanticism’ that ‘overlooks’ the ‘genuine problem, the internal-historical problem’ (CES, p. 378; my emphasis). Even more emphatically, he states that it has a teleological imperative, defining Historie as ‘nothing other than the vital movement of the coexistence and the interweaving of original formations and sedimentations of meaning’ (CES, p. 371). Historie therefore constitutes a teleological examination of geometry; that is, an inquiry into a new purpose in geometry that is not yet accounted for by modern scientific procedures. Moreover, if factual history is to have any original value, then it must be developed out of a universal a priori ‘internal history’, which ‘necessarily leads further to the indicated highest question of a universal teleology of reason’ (CES, p. 378). Husserl’s inquiry is therefore driven by teleological and historical imperatives. The Origin examines the production of geometry as a teleological-historical science and, with the Crisis, it sets out to reinstate historical reason in philosophy and in geometry.5 Each reveals an irrefutable reason that originates from a fundamental idea of ‘sense’ (i.e., ‘primally establishing’ functions) that will reactivate the historical value of scientific knowledge (CES, p. 354).

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In the first pages of the Crisis, Husserl writes that this approach ‘is the only way to decide whether the telos which was inborn in European humanity at the birth of Greek philosophy . . . was not rather the first breakthrough to what is essential to humanity as such, its entelechy’ (CES, p. 15). Then, in a following section he also observes that it is a form of ‘historical reflection’, writing: ‘What is clearly necessary . . . is that we reflect back, in a thorough historical and critical fashion, in order to provide before all decisions, for a radical self-understanding: we must inquire back into what was originally and always sought in philosophy’ (CES, p. 17). In addition, when Husserl calls this approach a ‘historical meditation’ (CES, p. 353), the Origin reflects his reassessment of Descartes’ Cartesian analysis of the relationship between the thinking ego (cogito) and reason in the Crisis. As a result, the Origin’s value as a ‘historical meditation’ on geometric thinking examines philosophical innovation through which the ‘submerged original beginnings of geometry’ is retrieved and through which its ‘proper’ scientific value is reinstated (CES, p. 354). He writes: Clearly, then, geometry must have arisen out of a first acquisition, out of first creative activities . . . it is not only a mobile forward process from one set of acquisitions to another but a continuous synthesis in which all acquisitions maintain their validity, . . . Geometry necessarily has this mobility and has a horizon of geometrical future in precisely this style; this is its meaning for every geometer who has the consciousness (the constant implicit knowledge) of existing within a forward development understood as the progress of knowledge being built into the horizon. (CES, p. 355) Thus, Husserl’s teleological project is not a simple forward-moving scientific form of reasoning but is directed towards retrieving lost methods and forms of apodictic sense-reason which, he suggests, existed in Greek civilization and were available to Descartes. Geometric thinking is therefore both a historically specific scientific method and an intuitive activity that is constructed by the mental activities of the scientific ego. In previous chapters we have seen versions of this energetic geometry in the dynamic agitation of Kant’s reflective judgment, the unfolding ‘transitional psychic activity’ of the Proclus’ nous (CEE, p. xx), and Bergson’s durational memory. Like Husserl’s idea of teleological-history, each of these genetic geometric methods is imbued with ‘life-giving activities’ (CEE, p. 16). In each, an enfolding, retroactive movement or

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recollection is drawn out simultaneously with the forward unfolding movement of deductive geometric thinking and ‘natural’ geometric life. Husserl also emphasises the importance of this dual movement of geometric purpose, for example, describing history as an unusual ‘thematic direction’, and arguing that the historically constituted depth-problems of geometry do not provide clear objects or ends and ‘can naturally not yet be seen at the beginning’ (CES, p. 354). Therefore, for Husserl, this inquiry into the ‘genesis’ of geometric origins or beginnings is not concerned with the simple eidetic imitation of discrete geometric objects but is focused on the dynamic, vital and energetic transmission of a historical geometric sense (CES, p. 370). However, he is also careful to show that, despite this internal genetic repetition, reactivating geometric sense is not a repetition of self-same abstract geometric ideals but represents a passage of geometric sense-tradition that is constituted in specific historical horizons. Thus, Husserl’s geometric horizons are also inherently concerned with a purposive (i.e., a teleological) set of questions. Moreover, below we will see that his attention to the transmission of geometric ideas and actions in the discursive forms of language and writing is further evidence that this genetic spirit in geometric thinking and its actions is established through a teleological analysis. Teleological-history therefore resists the abstraction of scientific geometric reasoning into absolute disembodied ideas, by constituting geometric thinking within a sense-based tradition that generates apodictic knowledge, but is not restricted to self-same imitations of geometric ideals. Instead, Husserl argues that teleological-history is imbued with psychic forms of sense that produce the potential for ‘ideality’ in geometric actions and figures because it is a ‘spiritual’ form of scientific method, thereby enabling ‘the whole living science’ of geometry to be reactivated (CES, p. 356). He explains: ‘tradition is precisely tradition, having arisen within our human space through human activity, i.e., spiritually’ (CES, p. 355). Geometric tradition is therefore generated out of this psychic kind of existence:

We understand our geometry, available to us through tradition . . . to be a total acquisition of spiritual accomplishments which grows through the continued work of new spiritual acts into new acquisitions. We know of its handed-down, earlier forms, as those from which it has arisen; but with every form the reference to an earlier one is repeated. (CES, p. 355)

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Also, by examining the different kinds of activity in geometric science, Husserl observes that they embody a natural ‘progressive’ activity. He suggests that geometric existence (i.e., the totality of geometric methods and ideas) is both the synthesis of a ‘set of acquisitions’ that form a ‘totality’ and the retention of this totality of knowledge in any singular geometric ‘acquisition’, event or invention (CES, p. 355). Geometry is therefore formed (in part) by this dynamic relationship of retention and progression that repeatedly flows through its tradition, from its origins towards individual geometric events that construct it, and in its reverse movements. However, the Origin’s main purpose is to show that this potential activity (i.e., the potential for apodictic sense-intuition) exists as a dynamic spiritual geometric tradition in itself. Geometry is not just a projective science that is determined by an objective or mechanistic implementation of geometric methods by scientific personae. Instead, Husserl shows how pure geometric ideas are not merely reducible to the production of objective scientific discourse but are in themselves original states of pure existence (idealities) that generate the movement of deductive and inductive geometric thinking. Like Proclus’ analysis of the dynamic movement of the Stoic ideas of limit and unlimit, Husserl also retrieves a transcendental kind of activity or existence for geometry, which he defines as a spiritual tradition or Historie. However, for Husserl, this spirit is not an immaterial notion of divine limitlessness: rather, it is embodied in the movement of a teleological-historical analysis. The following sections explore the importance of this original sense-intuition for geometric thinking in more detail.

Geometric sense-ideas and sense-intuitions In Part II of the Crisis, Husserl explores the ‘origins’ of ‘true’ apodictic knowledge in more detail. He examines the production of axiomatic knowledge in Platonic mathematics (e.g., Euclid’s Elements) through which geometric objects are idealised, independently of their natural empirical occurrences (CES, pp. 21–3). Even in this early period of Western geometric science, Husserl concludes that its activities are directed towards ‘finite tasks’. However, it is in the work of the modern rationalist scientists, in particular, in Galileo’s mathematical account of nature (that is, the formation of modern ‘natural science’) that Husserl considers the pursuit of objectivity to be most strongly instantiated because Galileo is the first to conduct a full ‘mathematisation of nature’ so that ‘nature itself is idealized’ (CES, p. 23). However, Husserl also observes that Galileo’s

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historical position is on the cusp of ideas about ‘prescientific’ and ‘everyday sense-experiences’ which are not accounted for in his systematic mathematisation of nature. Surely, he suggests, these prescientific sensebased experiences must also include ‘pure geometry’ and sense-intuitions. In addition, he argues that these sense-ideas could not have been as invisible to Galileo’s understanding of reason, as they are for mathematicians working in twentieth-century Europe (CES, p. 24).6 Husserl explores the value of these ‘pregiven’ [vorgegeben] ‘spatiotemporal shapes’ which Galileo inherited from the Greeks, and draws attention to the slippage in meaning that exists in between idealised theories of geometry and empirical geometric experiences: So familiar to us is the shift between a priori theory and empirical inquiry in everyday life that we usually tend not to separate the space and the spatial shapes geometry talks about from the space and spatial shapes of experiential actuality, as if they were one and the same. (CES, p. 24) He observes that we need to be precise about what we understand ideal intuitive geometry to be – being careful to understand that it is not the general, fluctuating, approximate or imagined shapes that we intuitively experience in day to day life (CES, p. 25). By contrast, idealised geometric thinking involves the development of praxes or technologies that produce exactness (ideality) in geometric ideas or ‘limit-shapes’ (e.g., the production of the line, triangle and circle), and through which geometry becomes a systematic set of praxes, such as measuring or surveying (CES, pp. 26–7). However, Husserl also shows that the pregiven geometry which exists in Galileo’s innovations and any other subsequent mathematical development of ‘exactness’ in geometric ideas (e.g., in geometric measurements and technical geometry) is not divorced from the intuited pregiven ‘plena’ of material qualities. Rather, each geometric ideality is a legitimate and necessary form of geometric intuition or origin of geometry; for example, he examines the natural phenomena of ‘material plena’ that constitute the specific intuited spatiotemporal experiences of geometry in everyday life. These are ‘ “specific sense-qualities” – which concretely fill out the spatiotemporal shape-aspects of the world of bodies’. Material plena, however, ‘cannot, in their own gradations, be directly treated as are the shapes themselves’. Nevertheless, he adds, ‘these qualities, and everything that makes up the concreteness of the sensibly intuited world, must count as manifestations of an “objective”

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world’ (CES, p. 33). Galilean geometry is therefore important for Husserl, because it instantiates and brings to the fore, the crisis (or cleavage) in modern scientific thinking between these two special, yet interrelated, kinds of sense or ‘meaning-content’ that exist in geometry (CES, p. 41). Consequently, geometry’s sense-meaning, its techniques of exactness towards establishing apodictic knowledge and its sensible spatiotemporal qualities, constitute the material plena. In addition, although he does not specifically refer to the Stoic ideas of limit and limitlessness, this discussion bears a close affinity to the discussion of plenitude and infinity in Leibniz’s discussion. Husserl therefore not only develops a definition of ‘the total sense of natural-scientific method’ (CES, p. 43), but he also locates the crisis in the modern sciences in the problematic ‘substitution’ of ‘mathematical idealities’ for ‘the only real world, the one that is actually given through perception, that is ever experienced and experienceable – our everyday life-world’ (CES, p. 49). Once again referring to Galileo’s innovations, Husserl observes that even at the beginnings of modern science, Galileo’s geometric method and ideas were ‘already empty of meaning’, already composed of an inadequate rational intuition that does not properly refer back to the life-world of experience. Leading him to discuss the development of modern scientific methods that transform the ancient art of surveying, which was not concerned with measurement in its original form, into a technique for constructing exact geometric ideas, he observes that these new sciences submerged the Classical forms of surveying which generated an immanent relationship between the ‘immediately intuited world’ and the ‘vitally practiced’ geometric activity (CES, p. 49). He criticises Galileo for overlooking this vital potential in geometry: ‘He did not reflect closely on all this: on how the free, imaginative variation of this world and its shapes results only in possible empirically intuitable shapes and not in exact shapes’ (CES, p. 49). Thus, he concludes that Galileo’s emphasis on a self-sufficient rational geometric intuition ignored the value of this relationship, thereby substituting mathematical idealized nature in favour of ‘prescientifically intuited nature’ (CES, p. 50). Also, later in the discussion, Husserl returns to this critique of Galileo, because he simultaneously generates a divorce between embodied subjective experiences of intuition and pure reason by concealing their actual relationship under the disembodied technical procedures: ‘Galileo abstracts from subjects as persons leading a personal life; he abstracts from all that is in any way spiritual, from all cultural properties which are attached to things in human praxis’ (CES, p. 60).

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Instead, for Husserl, this pregiven scientific existence, ‘the horizon of all meaningful induction’, retains the link between ‘actually experiencing intuition’ (i.e., forms of space-time) and ‘all the bodily [korperlich] shapes incorporated in it’. He continues, emphasising this immanent relationship between our active corporeal experience of space, time and sense-intuition, writing: ‘it is in this world that we ourselves live, in accord with our bodily [leiblich], personal way of being. But here we find nothing of geometrical idealities, not geometrical space or mathematical time with all their shapes’ (CES, p. 50). Husserl’s account of geometry in the Crisis is therefore crucial to understanding the value of the Origin because it provides insights into the lost historical value of sense-intuition in modern geometry’s apodictic and spatiotemporal forms. For Husserl, Galileo’s significant discovery of the autonomy of mathematical reason also signals the historical beginnings of an active concealment of the actual ‘pregiven’ sense-intuition or sense-reason that exists in the everyday geometric experience of the natural world. This is where he identifies the cleavage of reason is first established in modern rational science (and where it also represents a challenge to metaphysics; CES, pp. 52–3). Thus, Husserl’s teleological-historical project is concerned with the desire to reconnect these different kinds of intuition, rather than leaving the split between them intact, as has previously been allowed by modern scientific methods, and by metaphysical philosophy. This renewed ‘living’ geometric science is also generated in Husserl’s discussion of praxis, which he reconnects to teleological-historical production, releasing it from its role in generating self-same ‘exactness’ in geometric ideas. In contrast, Husserl argues that ‘historical praxis’ and ‘technical praxis’ emphasise the importance of the activities that generate geometric bodies, inventions and events. Praxis is therefore defined through a teleological understanding of geometric activity, for example, in an examination of its preceding and forthcoming manifestations, but also because praxis brings to the fore the potential for practical innovations that generate new geometric states, conditions, ‘tendencies’ or applications. Husserl, for example, discusses the practical development (praxis) of geometric methods, through which certain ‘preferred’ shapes and ‘improvements’ are made, such as, the attention towards ‘surfaces’ that are ‘more or less “smooth”, more or less perfect surfaces’ (CES, p. 376). The praxis of shaping and measuring geometric ideas indicates to tendencies of progression or a ‘new sort of construction’. Thus, praxis generates ‘products’ ‘arising out of an idealizing, spiritual act, one of

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“pure” thinking, which has its materials in the designated general pregivens of the factual humanity and human surrounding world and creates “ideal objects” out of them’ (CES, p. 376). Yet he warns against promoting the ‘thinking activity of a scientist’ from being ‘time-bound’ to specific local a posteriori values. Rather, the empirical and concrete changes in method that are brought about in praxis must be examined as part of an original historical meaning ‘which necessarily was able to give and did give to the whole becoming of geometry its persisting truth-meaning’ (CES, p. 377). Thus, Husserl enables sense or temporality to inform geometric thinking thereby establishing a deeper ontological meaning in its activities. Praxis, for Husserl, is evidence of the teleological sense of geometric actions (i.e., thinking), and underlines his inquiry into a sense-based form of ideal reason (and, in this respect, his emphasis on praxis suggests that embodied and aesthetic techniques of geometric thinking may also be linked back to the importance placed on geometric actions in Kant’s aesthetic judgment and Bergson’s natural intuition). Thus, Husserl’s notion of geometric sense refers to a process of clarification and of spatiotemporal experience, but not one explicitly directed towards a discussion of perception. Rather, geometric sense embodies the production of meaning as a specific form of mathematical intuition or reason (i.e., the ideality or exactness of a geometric idea), and the intuited experience of spatiotemporal qualities of geometric bodies. Sense, for Husserl, establishes the precise ‘contentmeaning’ of geometry as an apodictic value, not as a mental faculty that merely constructs the individual’s subjective or sensible experiences. Husserl’s geometric sense therefore comes into proximity with Kant’s synthetic a priori, because it constitutes the autonomous (i.e., an objective) synthetic existence of geometric intuition. As a result, we might also suggest that Husserl’s project is concerned with aesthetics, if we agree that he retrieves an actual sense-reason (i.e., pure intuition) in geometric thinking, and insofar as he develops a theory of geometry that is not merely a disembodied intellectual technical activity but it is an activity concerned with the production of pure intuition in geometric ideas and in the living subject’s everyday spatiotemporal experiences.7

Explication Husserl also conducts a critique of deductive and logical progress in mathematics, examining the nature of geometric thinking and its

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production of geometric sense, whilst simultaneously rediscovering the ‘true’ metaphysical existence of geometric ideas. For Husserl, logic is not a merely disembodied atemporal state of self-same repetition but is reconnected to its ‘original’ eidetic existence. In contrast, modern science has evacuated eidetic thought of its original transcendental power or truth-status. Husserl’s project is necessary because mathematics which has ‘been emptied of meaning, could generally propagate itself, constantly being added to logically’ (CES, p. 368). Against this tendency to logical propagation, Husserl proposes that his ‘regressive inquiry had to be reawakened. More than this: the true sense of such an inquiry had to be discovered’ (CES, p. 368). He criticises the seductive quality of scientific geometric operations in an analysis of ‘evacuated’ forms of geometric science, which reduce geometric meaning to limited forms of ideas, because they progress from the ‘first oral cooperation of the beginning geometers’ to the need for an ‘exact fixing’ of geometric descriptions in the axiomatic method and, consequently, apply geometric praxis to these ideas in practical forms. Such reductive geometric methods, Husserl concludes, has resulted in the loss of geometric ‘original self-evidence’, and in the development of mathematics as a ‘self-propagating’ form of logical inquiry (CES, p. 368). Examining the discursive nature of mathematical sciences, he attributes geometry’s ‘immense proliferation’ to the fact that each individual innovation is interdependent with the totality of the science, ‘no building block within the mental structure is self-sufficient’ (CES, p. 363). He suggests that there is an inherent paradox in geometric science because of the retention of its ‘living reactivability’ – that is, the retention of its idealities – and yet its propensity to generate new objects that exceed its preceding states (CES, p. 363). In the following pages, Husserl asks how geometric idealities are both thematically constant (i.e., exist in their own right), yet allow for the production of new knowledge. He examines the specific discursive sense-structure in deductive geometry, explication, explaining that it is the temporalised method which produces an unusual double action in geometric thinking. Geometry, he writes, operates through a ‘peculiar ‘logical’ activity which is tied specifically to language’ (CES, p. 364). He distinguishes between two explicatory movements. First, explication is a ‘passively emerging sentence (e.g., in memory), or one heard and passively understood, is at first merely received with a passive egoparticipation, taken up as valid; and in this form it is already our meaning.’ (CES, p. 364; my emphasis.) Thus, in its passive form it is the reception of

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already-given objective knowledge. Yet, in its active form, Husserl argues that it embodies an intentionality that expresses the individual ‘elements of meaning’ in distinction to generally perceived meanings: ‘What was a passive meaning-pattern has now become one constructed through active production’. Explication therefore activates original, general or indeterminate forms of geometric meaning. This, Husserl tells us is an example of ‘self-evidence’ in which ‘the structure arising out of it is in the mode of having been originally produced’ (CES, p. 364). This power to produce geometric sense-ideas is therefore attributed to the underlying actions or operations of geometric thinking, rather than in the geometric ‘elements’ (i.e., the axioms, propositions, and so on). Instead, Husserl argues that the logical sentences or judgments that are expressed in geometric explication, carry in them the ‘peculiar’ discursive potential for new judgments or new activities to be actualised, generating a spatiotemporal account of geometric sense-ideas which are produced within the discursive structures of geometric thinking. Ideal geometric objects are ‘capable of being passed on’ through the activity of explication (CES, p. 364). Thus, explication registers a temporality of meaning in which geometric sense is already given. Examining in more detail how linguistic structures, such as ‘sentences’, enable the reactivation of original ‘self evident’ geometric idealities, Husserl observes that sentences are ‘reproductive transformations of an original meaning produced out of an actual original activity; that is, in themselves they refer to such a genesis’ (CES, p. 365). He concludes that sentence structures therefore enable the ‘genuine reactivation in full originality’ of the ‘primal self-evidences’, establishing a kind of discursive ‘infinitization’ in the production of geometric meaning. So geometry is not just a passive ‘handing-down’ of pregiven meaning. Rather, its sentence structures are ‘a lively, productively advancing formation of meaning, which always has the documented, as a sediment of earlier production, at its disposal’ (CES, p. 365). Yet the development of this ‘reactive’ geometric continuity also requires a consistency to exist in the production of logical sentence structures, so that the actual recovery of geometry’s original meaning is ensured. Husserl then examines the ‘ontic’ forms of this ‘living geometric tradition’; that is, the sensible manifestations of geometric deduction in the form of the axiomatic elements, definitions, propositions or drawn figures. He distinguishes between these operations and the ‘superstructure’ of geometric thinking when he observes that these are, in fact, rigorous methods of expressing ‘ready-made concepts and sentences’

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(CES, p. 366); for example, in the act of drawing geometric figures, we ‘substitute’ the ‘actual production of the primal idealities’ with ‘sensible intuitions’ in the form of drawn figures. In each case, geometry progresses by virtue of its practical discursive potential, rather than by the actual activation of its original idealities (CES, p. 366). Here, however, Husserl also reminds us of the innate power of the deductive sciences to construct incorrect apodictic forms of objectivity, whereby their methods appear to operate on the basis of original truth-structures, but in fact extract the primal geometric matter and therefore prevent ‘genuine’ reactivation of its meaning. He warns that although individuated ‘cultural structures’ of geometric thinking, such as propositions, claim ‘to be sedimentations of a truth-meaning’, they do not actually constitute the original geometric meaning (CES, p. 367). Instead, for Husserl, the validity of a deductive science, like geometry, must be tested by its potential to actually reactivate truth-meaning that cannot be assumed to exist in the purely progressive powers of an objective scientific method. Therefore, deductive geometric operations do not constitute geometric truths in their own right; rather, Husserl argues that a pure autonomous form of geometric sense exists in advance of these operations. In order to close the gap between these inadequate objective geometric methods and the actual living tradition of geometry, Husserl attempts to retrieve the missing historical ‘problems or investigations’ that will reinstate the ‘original truth-meaning’ of geometry (CES, p. 368). He identifies the potential for geometric explication to be more than a logical reiteration of itself, identifying how explication is in itself an activity through which the historical truth-value of geometry can be constructed. This actuality lies in explication’s powers of ‘making it explicit’. Explication’s powers of generating ‘self-evidence’, Husserl writes, constitutes ‘nothing other than historical disclosure’ (CES, p. 370). Thus, explication is reconnected back to a lost tradition of geometric existence that is not merely disembodied logical deduction. This historical form of explication implies a continuity of pasts which imply one another, each in itself being a past cultural present. And this whole continuity is a unity of traditionalization up to the present, which is our present as [a process of] traditionalizing itself in flowing-static vitality. (CES, p. 371) Once again, the temporalisation of the geometric method into a discursive existence is generated through the activities of explication

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(explicato) and implication (implicato). The geometric method of explication therefore constitutes the ‘inner structure of meaning’ through which the historicity of geometry is constructed, rather than the production of disembodied objective apodictic knowledge that is divorced from its living tradition (CES, p. 371). Thus, for Husserl, logic constitutes a specific geometric (i.e., a technical) form of embodiment. It is therefore also a reminder of the agitation that constitutes the discursive movement of geometry in Proclus’ and Kant’s geometric thinking. Yet, Husserl identifies it in order to re-invest logical structures with an embodied sense-value, even more strongly. Husserl therefore constructs a metaphysics of historical existence (i.e., a phenomenology) of ideal geometric thinking by examining the different states of intuitive activity that construct it, beginning from the transcendental intuition of the geometric object which endures within the totality of the science, to an examination of this intuition within each specific activity (or praxis) of constructing geometric bodies and objects. The horizons of living science, language and history are not constructed as externally determined and fixed sense-structures. Rather, for Husserl, they are oscillating event-horizons or activities through which geometric sense is constructed.

Retrieving sense-intuition from Descartes and Kant In preparation to his discussion of transcendental phenomenology in the Crisis, Husserl undertakes an important historical analysis of sense-reason (i.e., sense-intuition) in Cartesian and post-Cartesian philosophy. This ‘clarification of the unifying sense of the modern philosophical movement’ takes him back to Descartes, who ‘inaugurates’ the tradition in his rational philosophical and mathematical enquiries (CES, p. 73). Husserl argues that Descartes’ analytic method represents the first historical expression of a radical ‘critique of knowledge’ (CES, p. 76). Yet he also points out that Descartes fails to develop this project to its fullest potential, especially because by focusing on the validity of the cogito, Descartes overlooks the importance of the ‘living body’, thereby generating a critique that disembodies pure reason and pure intuition from sensible forms of intuition (CES, p. 79). Nevertheless, Husserl observes that Descartes is an important threshold to the ‘pre-given’ senses of the world because he exists at the boundary between premodern and modern scientific methods. Descartes is significant

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because he accounts for both the ‘validity of all previous sciences – even mathematics’ and for the validity of the pre- and extrascientific life-world, i.e., the world of sense-experience constantly pre-given as taken for granted unquestioningly and all the life of thought which is nourished by it – the unscientific and finally even the scientific. (CES, p. 76) However, like Galileo, Descartes ultimately ignores the actual pre-existing value of sense-reason and ‘pure soul’ in his pursuit of the ego (cogito) (CES, pp. 79–80). Yet, Husserl acknowledges that Descartes simultaneously succeeds in establishing a significant ‘new telos’ for the ego (CES, p. 81). Nevertheless, Husserl considers this new theory of ‘intentionality’ (that is, the production of reason through the experiences of thinking, feeling, willing and so on), to also be an underdeveloped expression of pure sense-reason (CES, pp. 82–3). Husserl continues to explore the limitations of rationalist examinations of geometry that exclude the extra-psychic dimension of the ‘living world’; that is, apodictic knowledge which is situated in the world. In the following section, for example, he identifies Kant with Descartes’ ‘egological’ project, in Kant’s claim that the mathematical reasoning of his ‘critique of reason’ produces actual objective metaphysical truths (CES, p. 81). Husserl, however, is sceptical of both Descartes’ and Kant’s belief in the totality of their methods, accusing each of making actual pregiven truth or reason inaccessible. But he also continues his critique of the relation between metaphysics and scientific method with a discussion of Locke, Berkeley and Hume in the following sections (§§21–4), whom he accuses of further perpetuating the decline of intuited sense-reason for logical philosophy, through their misadventures into theories of immanent sense-data (CES, pp. 83–8). Nevertheless, in §24 he identifies the importance of Locke’s, Berkeley’s and Hume’s empirical theories of the world, applauding each for taking the sensibility (or for Hume, the ‘entire soul’) to be immanent with ‘impressions’ and ‘ideas’ (CES, p. 89). Husserl understands that, despite their inconsistencies of understanding the relationship between the ego-experience and sense-data, these critiques of ‘dogmatic objectivism’ nevertheless reveal ‘a completely new way of assessing the objectivity of the world [ . . . ] and, correlatively, that of the objective sciences’ (CES, p. 90). Ultimately, Husserl is therefore highly critical of a definition of immanence which excludes apodictic objective knowledge entirely.

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While recognising Kant’s debt to Hume’s ‘shaking of objectivism’ (CES, p. 91), Husserl writes that Kant’s significance for philosophy actually lies in his invention of ‘a new sort of transcendental subjectivism’ (CES, p. 91). In the following section, Husserl defines what he means by ‘transcendental’. It is the motif of inquiring back into the ultimate source of all the formations of knowledge, the motif of the knower’s reflecting upon himself and his knowing life in which all the scientific structures are valid for him occur purposefully, are stored up as acquisitions, and have become and continue to become freely available. . . . Rather, it is a concept acquired by pondering the coherent history of the entire philosophical modern period: the concept of its task which is demonstrable only in this way, lying within it as the driving force of its development, striving forward from vague dynamis towards its energeia. (CES, pp. 97–8) Thus, transcendental knowledge is inherently concerned with a double movement of retention and projection, and Husserl concludes that Kant’s transcendental critique is therefore in opposition to the postCartesian rationalism of Leibniz but not Hume’s empiricism. This is because Kant invents a ‘two-fold’ account of how pure reason functions and ‘shows itself’. First, it is the objective and natural intuitive self-exposition of reason in the mathematical sciences and forms of sense-intuitions. Second, it is the subjective and ‘sensibly intuited world of objects’ (CES, p. 94). Husserl links Kant’s and Hume’s projects together because, like Hume’s empiricism, Kant’s transcendental science does not, in itself, ‘explain the origin, the “cause” of the factual manifolds of sense-data’. However, in so doing, Kant submerges the infinite indetermination that Leibniz releases in his inquiry into the limitlessness of manifold (i.e., the sense-plena) (CES, p. 95). In the following section, Husserl also shows that Kant’s project is also developed upon an ‘unquestioned ground of presuppositions which codetermine the meaning of his questions’ (CES, p. 104). Husserl argues that Kant, following in Descartes’ footsteps, succeeds in developing a transcendental theory of the senses, in the form of the sensibility or sense-intuitions. However, he concludes that this is a perceptual form of knowledge that forgets the living body as a constituent of the thinking individual’s engagement with the world, and he distinguishes between the physical body (Körper) and the living body (Leib), arguing that it is only the living body in which the self is actually fully developed

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(CES, p. 107). He describes the importance of this intuitive reasoning corporeality: The sensibility . . . proceeds in consciousness not as a mere series of body-appearances, as if these in themselves, through themselves alone and in their coalescences, were appearances of bodies; rather they are such in consciousness only in combination with the kinaesthetically functioning living body [Leiblichkeit], the ego functioning here in a peculiar sort of activity and habituality. (CES, pp. 106–7) Clarifying what he means by ‘living ego-body’, Husserl suggests that it is not merely the experience of how the world relates via the physical body, rather, it is through the kinaesthetic activities of seeing, hearing, touching that the ego or consciousness directs our attention towards ourselves (the acting subject) and towards others (‘living-together’) (CES, p. 108). Thus, for Husserl, consciousness is ‘in constant motion’ in the living subject (CES, p. 109). In contrast, he argues that Kant’s Critique of Reason never quite situates his theory of feeling in the world. It is situated in the subject, but the subject himself is never reconnected into an actual historical reality. Kant’s project also retrieves the sense-intuitions, yet they remain technical faculties or sense-perceptions, rather than embodied into the cultural or teleological-historical traditions. Husserl’s figure of historical geometric sense-intuition therefore presents a more resolved version of Kant’s synthetic a priori in the Critique of Pure Reason because of his emphasis on the ‘life-world’ of the subject, and his analysis of intersubjectivity.8 For each, however, the transcendental and synthetic nature of geometric intuition is central, and in this respect a correspondence between the aesthetic judgment of Kant and the phenomenological sense-idea for Husserl is apparent, despite Husserl’s rejection of aesthetics as a method for the reactivation of reason in modern philosophy. The Crisis and the Origin do not therefore provide a return to the ‘psychological empiricism’ of Locke and Hume, nor do they promote the sensibility, but they do imply a potential relationship between Kant’s aesthetic judgment and Husserl’s geometric sense-idea, which Husserl himself overlooks. Kant’s attempt to construct ‘apodictic’ reason in the form of the reflective judgment and aesthetic subject reveals an interesting link between his theory of embodied geometric sense in the technical agitation of the sensibility, and Husserl’s promotion of geometric sense-reason. In the next section I show how Husserl develops an autonomous subjectivity in geometric thinking, which is also

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evidence of its unique spiritual quality, and consequently, its potential for developing an intersubjective theory of geometric sense-ideas.

Geometric self-evidence Husserl describes how science is an original kind of existence. However, its ‘self-evidence’ or ‘originaliter’ is brought into actuality through the mental thoughts of the individual scientist. In addition, Husserl carefully distinguishes between the primary living yet immaterial geometric idea and human existence, when he writes that ‘geometrical existence is not psychic existence’ (CES, p. 356). Rather, geometry exists independently of the internal cognitions of an individual and the technical praxis of geometric science, because it has an ‘objective’ existence. Husserl continues: ‘Indeed, it has, from its primal establishment, an existence which is peculiarly supertemporal’ that is accessible to everyone (CES, p. 356). Thus, he suggests that geometry is a particular ‘class of spiritual products of the cultural world’ which have an ‘“ideal” objectivity’ (CES, p. 356). These ideal objects (ideale Gegenständlichkeit) are not merely embodied as spatiotemporal forms of sense in the language structures of scientific discourse (e.g., as ‘sensible utterances’). Rather, they exist as ‘two-leveled repetitions’. Geometric objects are simultaneously self-individuated (self-evident) geometric entities, presented as general geometric identities (i.e., circles, triangles, and so on), and also exist in ‘sensibly embodying repetitions’ or intuitions (CES, p. 357). Husserl therefore invests geometric objects with a transcendental nature that is not reducible to individual sense-phenomena or sense-meanings but which constitutes the movement between the totality of geometric ideas and particular embodied expressions of geometry. He distinguishes these ideal objects further by examining the difference between linguistic structures, such as ideal geometric words or theories, versus the actual ideal existence of geometric ‘truth’, which he considers to be a distinct class of ‘idealities’, ‘geometrical objects’ or ‘states of affairs’ (CES, p. 357). But this then leads him to pose the question about how these transcendental idealities are generated from the ‘primary intrapersonal origin’ of the individual. He asks how are ideal geometric truths constructed out of the relationship between the general objective geometric activities and individuated states of geometric activity, and identifies language as the principal structure for this double expression. Language, he concludes, provides both the ‘linguistic embodiment’ and the ‘intersubjective’ structure through which the ideal geometric object is activated in the world (CES, p. 358).

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Later in the essay, Husserl returns to this paradox, when he examines the relationship between the internal psychic activity (i.e., ‘intrapsychic’ geometric ideas) of the individual and the autonomous geometric object which is ‘anything but a real psychic object, even though it has arisen psychically’ (CES, p. 359). He identifies how geometry’s ‘vivid self-evidence’ is brought into actuality through its movement towards and recollection of its original ideality. However, he observes that this dynamic repetitive movement of deduction (explicato) and recollection (implicato) does not itself establish a shared external objectivity with others. Rather, the geometric idea (i.e., the sense-idea) at this stage still exists only in an intrapsychic form of the original ‘self-evident’ geometric object not as a truly autonomous geometric idea (or sense). Thus, it is only when there is ‘reciprocal linguistic understanding’ that the connection between geometry’s original production and the local individuated manifestation of it can be ‘actively understood by others’ (CES, p. 360). Here, therefore, Husserl’s interest in the discursive structures that construct geometric thinking highlights how geometry is constituted in the intersubjective relations between two or more subjects. Intrapsychic states of geometric objects represent the internal spiritual sense of geometry, but they do not account for the intersubjective, external sense of geometry which is distributed through embodied language-structures. So, in contrast to each of the preceding geometric methods, Husserl explicitly enquires into the dynamic relationship between the intrapsychic and intersubjective construction of geometric thinking between subjects, not just for the single subject. Geometry shifts from being embodied in the multiple aesthetic sense-perceptions of the thinking subject into the discursive sense-ideas of geometric subjectivities. In the following sections, I explore how Husserl’s figure of the horizon constitutes a sense-structure or sense-history that is original in this discussion because of its attention to the formation of intersubjective geometric methods and figures, not singular figures. Thus, Husserl’s original geometric sense-idea (spirit or Historie) transcends its embodied meaning in language. It is also a ‘super’ sense that endures in the discursive activities of geometric unfolding and enfolding, which constitute the ‘thematic’ reactivation of geometric ideas. Husserl therefore invests the a priori production of objective truth with extrascientific, that is, transcendental existence. Consequently, his discussions about geometric objectivity also resonate with the preceding examinations of geometric intuition, because in each case intuition (i.e., existence, life or activity) constructs geometric objects and states of

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affairs in absolute pure forms of ideality, yet also constructs geometric thought as embodied spatiotemporal sense-ideas. Husserl’s geometric sense is particularly closely aligned with Kant’s and Bergson’s theories of transcendental sense-intuition because, for each, geometric thinking is both the activity of pure intuition and thought. As a result, his geometric idealities can therefore be described as problem-horizons of sense-reason.

Horizons Husserl’s theory of the horizon represents a teleological geometric idea through which he generates a unique temporality in geometric thinking. Horizons constitute the internal temporal structure of the spirit of geometry. However, unlike the preceding examinations of geometric figures, the horizon operates more precisely as an idea of thought, rather than as a figure of perception. This emphasis on the discursive power of intuitive geometric ideas (rather than geometric bodies or techniques) enables Husserl to explain how geometric ideas are intersubjective, transmissible as temporalisations between subjects, rather than spatialised actualisations of corporeal subjects. Thus, for Husserl, the horizon is more correctly defined as a sense-idea (than a figure-subject, as is the case for Spinoza, Leibniz and Bergson). Nevertheless, like Bergson’s durational geometry, Husserl’s project is also significant because the horizon is constructed in opposition to the limited objective (i.e., symbolic) geometry of the modern scientific method. Insufficient in the form of scientific logical operations, the horizon embodies a unique historical sense (or intuition), and retains an extrasensory meaning of temporality. Husserl first uses the term horizon in the Origin when he examines the relationship between language as a ‘function of man within human civilization’ and ‘the world as a horizon of human existence’ (CES, p. 358). He continues, stating that this ‘horizon of civilization’ constructs our consciousness of ‘living wakefully in the world’. We are conscious of the world as the horizon of our experience. The world is a horizon of our life, as a horizon of ‘things’ (real objects), of our actual and possible interests and activities. Always standing out against the world-horizon is the horizon of our fellow men, whether there are any of them present or not. . . . we are conscious of the open horizon of our fellow men . . . We are thereby coconscious of the men on our external horizon in each case as ‘others’; in each case ‘I’ am

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conscious of them as ‘my’ others, as those with whom I can enter in to actual and potential, immediate and mediate relations of empathy. (CES, p. 358; my emphasis) Horizons constitute the specific events, objects and actions of culture and the underlying concept of human existence or life which generates geometric meaning-structures, for example, the ‘common language’ that connects subjects together in a society. Horizons therefore embody the immediate concrete actualisations of geometric thought, the reciprocal relations between the subject’s internal geometric thoughts and other subjects’ (a world-horizon), and the ideas of those who are not immediate to ourselves. Horizons are the original geometric structures (i.e., Historie) through which internal intrapsychic geometric ideas become external intersubjective geometric ideas (CES, p. 359). Husserl identifies six different horizons that generate this discursive flow and retention of geometric life: the ‘horizon of our life’, the ‘horizon of things’, the ‘world-horizon’, the ‘horizon of our fellow men’, the ‘open horizon’, the ‘external horizon’ (CES, p. 358). Geometric horizons are therefore relational spatiotemporal ideas that are genetic and discursive. They are ‘relational unities’, which are explicit and implicit of the world that we actively or implicitly express in our language-structures. In addition, each horizon is a unique state of the ‘we-horizon’ of civilization, ‘a community of those who can reciprocally express themselves’ (CES, p. 359). Here, therefore, a shift occurs in Husserl’s geometric thinking, from an emphasis on the internal production of geometric figures to the activation of geometric sense-meaning by means of intersubjective relations with others. Thus, geometric ideas are embedded within material forms of human relations, not just as internal psychic acts of production and intuition, so that geometric sense is essential for a living world of human interaction. Returning to this discussion later in the essay, Husserl evaluates the horizon’s potential for generating a new concept of history which is not limited to an uncritical forward-moving progression of factual events. Through an account of the relationship between an internal state of geometric intuition and its external intuitive state, he shows that geometric thinking is also a uniquely temporal event-horizon that constructs relations between living historical subjects. He writes: We are constantly, vitally conscious of this [historical] horizon, and specifically as a temporal horizon implied in our given present

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horizon. . . . We stand, then, within the historical horizon in which everything is historical, even though we may know very little about it in a definite way. (CES, p. 369) In addition, the processes of geometric thinking, especially explication, disclose ‘the horizon of its history within itself’ (CES, p. 371). Horizons therefore constitute distinct geometric ideas through which Husserl is able to retrieve the original temporality of geometric thinking. Furthermore, he reinstates the relationship between past, present and future events by highlighting the continuous teleological activities of explication and recollection. History [Geschichte] is therefore understood to be ‘the universal horizon of questioning’ because it carries within it the indeterminate beginnings of geometric thinking (CES, p. 373). A present and future indeterminacy in history is established in its already-given, but unknown, origins that constitute the ‘openly endless horizon of unknown actualities’. Consequently, this ‘presupposed’, already-existing ‘horizon-certainty’ is not something learned, not knowledge which was once actual and has merely sunk back to become part of the background; the horizon-certainty had to be there in order to be capable of being laid out thematically; it is already presupposed in order that we can seek to know what we do not know. (CES, p. 374) So geometric horizons are constituted by ‘not-knowing’. In particular, the endless future-horizon ‘exists in advance for us as world, as the horizon of all questions of the present and thus also questions which are specifically historical’ (CES, p. 374). This ‘flowing, vital horizon’ of history is constituted by ongoing ‘past presents’ and Husserl analyses how these implied pasts are made present as actual apodictic ideas in the ‘horizon-exposition’ of the ‘life-world’ (CES, p. 375). Finally, in the closing paragraphs of the essay, Husserl identifies the horizon with reason, calling it ‘the problem-horizon of reason’ (CES, p. 378). Thus, geometry is a unique sense-idea through which reason and sense are retrieved from objective and logical thought (i.e., symbolic thinking). In addition, geometry’s relationship to spatiotemporal sense-intuitions, which exist independently of objective scientific methods, enables a unique way to establish sense-reason. In this respect, Husserl’s geometric horizons constitute an original geometric ‘ratio’ (reason) that describes the relationship between geometric sense and historical reason. By exploring Husserl’s concept of horizon a tradition of geometric sense is revealed

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that informs the past and future construction of scientific forms of geometric thinking, because it operates from the intuitive and spiritual senses that comprise Historie, the ‘living present’. These ‘many-sided problem-horizons’ therefore construct the Origin’s inquiry into the ‘meaning structure “geometry”’ (CES, p. 378). Hence, the horizon is a geometric figure of ‘the whole living science’ (CES, p. 356).

Horizon I: intersubjectivity The horizon also registers how geometric sense-reason is generated between internal psychic geometric sense-ideas, and geometric senseideas that are generated in the intersubjective horizons of society. Husserl’s discussion shifts the production of the geometric idea from an analysis of exclusively internal scientific operations to an analysis of the external cultural states of existence through which it is transmitted; especially in his examination of the relationship between sense and language. Thus, Husserl reconfigures the value of internal and external relations in geometric methods, particularly in his examination of historical horizons developed in his figure of ‘intersubjectivity’. Husserl views language to be a structure that allows everything to be expressible because it mediates between internal and external subjects. Language provides the geometer with the structure through which he can ‘express his internal structure’ (CES, p. 359). This leads Husserl to ask how it is possible for something which is psychic and original for an individual subject to also be objective at the same time. He asks: ‘how does the intrapsychically constituted structure arrive at an intersubjective being of its own as an ideal object’ (CES, p. 359). Husserl explores the relationship between the internal psychic ideas of the individual subject and the external psychic, cultural and empirical ideas shared in a community. Once again, he posits a necessary movement between the two states that generates a specific kind of ‘self-evidence’ in the geometric idea itself, which is extralinguistic (rather than eliding internal geometric ideas in the act of prioritising the external formation of geometric origins, or vice-versa). In addition, geometric ideas are not just disembodied transcendental states but are embodied in the activities of the individual subject or geometer. Husserl explains the process of how self-evident geometric origins are experienced by the individual subject. He describes the process of recollection and actual production that form the ‘flowingly fading consciousness’ and the ‘reawakening’ of the origin within the subject that generates an increasing ‘clarity’ through which the ‘past experiencing

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[Erleben] is lived through a quasi-new and quasi-active way’ (CES, pp. 359–60). However, despite this ‘co-established’ repetition and production of origins in the subject at this stage, geometric ideas still remain internal to the subject. It is only in the form of ‘reciprocal linguistic understanding’ that the original meaning of geometry ‘can be actively understood by the others’ (CES, p. 360). In the process of this shared repetition between people, geometry becomes ‘an object of consciousness’, which is not determined by ‘likeness’ but is nevertheless a ‘structure’ that is ‘common to all’. Yet this intersubjective languagestructure does not, in itself, reproduce the actual persisting existence of ideal geometric origins, which exist or ‘continue-to-be’ even when culture does not see them; for example, written forms of communication enable the passive and associative retention and reactivation of these ideals (CES, p. 360). In themselves, however, written signs or linguistic sounds do not properly reactivate the original geometric ‘self-evidence’. In addition, Husserl observes a kind of retroactive passive association (i.e., passive intuition) which we develop through our induction into language – for example, it is embedded in our acquisition of language – but he continues, rejecting this passive kind of retention or intuitive activity. Instead, he promotes geometric explication that is concerned with truth-making (not passive repetition) which reactivates sense-ideas and is generated by the reciprocal activity of the reader or ‘other’ (CES, p. 361). Geometric ideas are therefore associated with the sensing subject, whereas geometric Historie has an extrasubjective value. In addition, his analysis of geometric ideality contains the potential for being transmitted or communicated within itself, thereby associating geometry with temporal existence in its own right. Temporal geometric existence is generated by the movement between the retention (sedimentation) and explication (formation) of Historie. Geometric sense-ideas are therefore constituted through the original reproducibility (i.e., by the repeatability of its existence) not through factual accounts of scientific innovation or change. Geometric thinking’s propensity towards eidetic repetition therefore embodies a kind of genetic existence. Moreover, geometric sense-ideas embody a ‘supertemporal’ form of existence, not just because of the powers of reproducibility that exist in scientific operations but also because of the inherent temporality of sense-reason. Husserl’s emphasis on the activity of the geometric method and figure therefore brings his project in close proximity with the earlier discussions of aesthetic geometric methods. Once again, we are also reminded

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of Kant’s reflective subject, in which the technical imagination constructs an agitated geometric relation between the single subject and the external natural world. However, Husserl develops this agitation further, by emphasising the production of geometric relations through the subject’s interaction with others living in the world.9 Thus, Husserl establishes geometric sense-reason through the subject’s embodied relationships, rather than by focusing on aesthetic forms of geometric activity in the single, autonomous subject. Rather, Husserl’s attention to the lost history of geometric temporality also highlights the potential for these relationships to exist between different subjects, in different times and spaces. Intersubjective horizons therefore embody geometric thinking in multiple geometric figures or subjects over a historical or cultural period. In contrast, although Spinoza’s and Leibniz’s multiple geometric figures embody a temporality, they remain abstracted from actual historical space and time.

Horizon II: we-horizons and future-horizons Kant’s and Bergson’s geometric thinking are key event-horizons for Husserl. In each relationship a dynamic retrieval and historical engagement takes place. Thus, Husserl’s project is an intersubjective horizon with these two preceding metaphysics, despite his insistence upon the autonomy of transcendental phenomenology. Intersubjectivity exists between Husserl’s geometric sense-ideas, Kant’s aesthetic judgment and Bergson’s natural intuition. In each case, a temporal futurity and retroactive movement is embodied as an intersubjective form of geometric sense. With respect to Kant, this intersubjectivity exists in the form of the aesthetic purposiveness in the technical geometric intuition of the reflective subject. With respect to Bergson, a future-horizon is constructed in the durational intuition of natural geometry. For Kant, the aesthetic subject is an active and ‘speculative’ power, which is also present in Husserl insofar as he shares Kant’s desire for establishing a critique of scientific speculations. Husserl’s horizons promote the need for new geometric sense-ideas to be constructed that are properly infinite and speculative yet purposive. Husserl critiques the disembodied versions of science and geometric thinking. Rather, history establishes an embodied account of geometry, and his critique of Kant’s theory of spatial and temporal sense-intuitions is legitimate insofar as he focuses on the first Critique where Kant constructs formal limits. However, given the embodied spatiotemporal intuitions and an aesthetic form of geometric thinking that Kant also constructs in the

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Critique of Judgment a clear correspondence between his and Bergson’s method is apparent. In particular, each constructs an embodied geometric sense through the relationship between extended spatiotemporal geometric encounters, such as, the reflective subject’s acts of drawing or geometric explication, and through unextended geometric ideas, such as Historie and the technical imagination.10 However, Husserl’s focus on the Critique of Reason in the Crisis means that he overlooks the potential in Kant’s critical geometry, because he views aesthetic judgment to be an indeterminate kind of thought that is unintelligible or analogous to reason, rather than establishing a pure autonomous form of apodictic reason. His critique of science, not aesthetics, means that he overlooks Kant’s purposive concept of reason, that is, aesthetic judgment.11 However, his geometric sense is nevertheless connected to Kant’s theory of aesthetic judgment; for example, he constructs a concept of geometric sense-ideas without being beholden to the sensibility, yet which are also synthetic a priori. Phenomenology is therefore concerned with the production of knowledge through sense-reason, not the disembodied eidetic transcendental idea, and Husserl’s call for a return to a ‘natural intuitive’ scientific method, a sense-based phenomenology of geometry, is an ethical project. In addition, his horizon is an extended and unextended form of spatiotemporal sense-intuition. Therefore, following Kant’s aesthetic a priori geometric sense, Husserl shows how spatiotemporal relations are reconnected in the horizon, undermining the split between natural forms of the senseintuitions and geometry in the pursuit of ‘exactness’ (ideality), which has led to a crisis in the modern sciences.12 Bergson’s theory of living sense-intuition is another significant ‘horizon’ (i.e., a ‘we-horizon’) for Husserl’s horizons, which bears a striking resemblance to Bergson’s radical aesthetic geometry, in particular, sharing the ambition to retrieve speculative scientific thinking. With the emphasis on the ‘living present’, a retrieval of Bergson’s writing 30 years earlier is possible. Husserl’s attention to the open horizon of phenomenology revisits Bergson’s theory of a properly ‘progressive’ and speculative philosophy. In addition, Husserl’s theory of teleological-historical reflection resonates with Bergson’s theory of pure extensity and duration. Husserl innovates Bergson’s ideas insofar as he locates the spatiotemporal sense-idea of the horizon into a discussion of historical sense-ideas and life. In contrast, however, Bergson’s focus on the corporeality of sense-reason through the uniquely psychic form of memory is distinct. Also, unlike Plato’s Socrates and Bergson’s natural intuition that draws out the geometric figure, Husserl does not refer to the qualitative and

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aesthetic acts of drawing. Nevertheless, each promotes an embodied form of a futural reason-horizon. Thus, if Husserl’s concept of horizon is brought into a tradition of aesthetic geometric existence it too reveals an aesthetic and temporal construction of science. Husserl’s promotion of geometry’s ‘living present’ therefore operates in the aesthetic realm not just in the scientific realm.

Summary Perhaps more explicitly than Spinoza’s, Leibniz’s and Bergson’s philosophies, Husserl examines the value of a legitimate a priori geometric sense that is directed towards re-enlivening scientific forms of geometric reasoning. Undertaking a critique of rational, scientific geometric methods that have mistakenly been taken as ‘pure’ reason, he shows that modern scientific geometric methods must be reconnected to a forgotten tradition of a priori sense-reason. Thus, according to Husserl, scientific forms of geometry cannot be divorced from their original sense-meanings in the living world. Although he rejects a direct association with aesthetics, Husserl’s project has an affinity with aesthetic geometric methods because he is concerned with re-establishing an absolute temporal sense in geometric thinking. In particular, his theory of geometric ‘horizons’ situates him into a tradition of post-Kantian temporal geometric thinking. In addition, his analysis of the temporal ‘intersubjectivity’ of the horizon develops the dynamic and aesthetic sense of geometric ideas. Thus, the project retrieves an ethics of sense-ideas through which modern living geometric science constitutes an aesthetic sense-geometry. Husserl’s project also bears a striking resemblance to Bergson’s theory of duration or living intuition (although he does not actually refer to Bergson in the Crisis), as well as Proclus’ theory of a ‘transitional psychic activity’. Moreover, the Origin connects geometric thinking to a new form of philosophical inquiry, that is, transcendental phenomenology, more explicitly than its role in Bergson’s theory of ‘superior empiricism’ developed in Matter and Memory and Creative Evolution. Yet, like Bergson, Husserl’s post-Kantian research into the philosophical production of pure objective knowledge also leads him to critique Kant’s claims for a total science of the sensibility in the Critique of Reason. For Husserl, an aesthetic-phenomenological geometric life or existence reconfigures symbolic scientific methods. Husserl’s reason-horizons are therefore composed of two particularly temporal forms of horizon: we-horizons and future-horizons. In each case, teleological-historical

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reasoning is embodied in a geometric form of sense; for example, these temporal horizons are materialised in the we-horizons and futurehorizons of Kant’s aesthetic geometry, and Bergson’s aesthetic natural geometric thinking. Through these horizons, geometry is carried forward in a living language, as an absolute and ‘real’ state, rather than as a mere form of representation; it is embodied and sense-based. This also invites a recollection back to Bergson’s natural intuition. However, for Husserl, geometric sense-intuition is always more strongly associated with sense-ideas, not sense-perceptions, and in this respect it has the potential to remain a transcendental idea, rather than a discussion about experience. Nevertheless, geometric historicity radically transforms the potential of geometry to construct a living-world science, rather than a predetermined repetition of disembodied ideas. Husserl’s project is distinct from the other geometric methods explored here, because it is primarily concerned with reinvesting geometric ideas with an apodictic sense-meaning, rather than the focus on perceptual modes of materialisation in the geometric figures that Spinoza and Bergson, in particular, promote. However, this temporal geometric thinking also reveals a close affinity to Kant’s analysis of aesthetic judgment in the Critique of Judgment and to the preceding aesthetic geometries outlined here, especially in his desire to restore the value of a special temporal sense to scientific inquiry. Thus, despite Husserl’s rejection of aesthetics, his retrieval of ‘submerged’ horizons of sense-reason is an important contribution to this study.

Notes Introduction 1. See, for example, P. Rawes, ‘Reflective Subjects in Kant and Architectural Design Education’, Journal of Aesthetic Education, Volume 41, Number 1 (Spring, 2007), pp. 74–89. 2. Colin Rowe’s and Robin Evans’s architectural criticism examines the relationship between geometric ideas and geometric materialisations in architecture. See, for example, C. Rowe, ‘The Mathematics of the Ideal Villa, Palladio and Le Corbusier Compared’, in The Architectural Review, Volume 101, Number 603 (March, 1947), pp. 101–4 and Robin Evans, The Projective Cast: Architecture and Its Three Geometries (Cambridge, MA and London: MIT Press, 2000). 3. Michael Fried and Rosalind Krauss’s art historical criticism about Frank Stella’s paintings are examples of the divergent claims made for geometric figures in modernist art. For Fried, Stella’s geometric forms represent an a priori, yet overtly Formalist, geometric expression. For Krauss, Stella’s work represents evidence of empirical a posteriori Minimalist experiences. In each case, these critics reduce geometry to either an a priori idea or an a posteriori idea. See Michael Fried, ‘Shape as Form’, in Art and Objecthood: Essays and Reviews (Chicago and London: University of Chicago Press, 1998), pp. 77–99, and Rosalind Krauss, Passages in Modern Sculpture (Cambridge, MA and London: MIT Press, 1993). 4. Chapters 5 and 6 will show that Bergson and Husserl are among those who have criticised the first Critique for exemplifying the belief that only scientific or mathematical forms of geometry are adequate a priori geometric intuitions. In histories of geometry, such as, Heath’s Introduction to the Elements, Euclid’s writings are contextualised in relation to his Platonic and Pythagorean sources. Yet, Heath also states that Euclidian geometry is, principally, a revision of scientific geometric method, drawing from De Morgan’s 1848 account of the Elements. Heath cites De Morgan: There never has been, and till we see it we never shall believe that there can be, a system of geometry worthy of the name, which has any material departures (we do not speak of corrections or extensions or developments) from the plan laid down by Euclid. T. Heath, Euclid: The Thirteen Books of the Elements, Volume I. (London: Dover Publications, 1956), p. v Heath expands upon this mathematical emphasis in geometric thinking, writing that much ‘valuable work’ has investigated the axiomatic method subsequently, but that ‘once the first principles are disposed of, the body of doctrine contained in the recent text-books of elementary geometry does not, and from the nature of the case cannot, show any substantial differences from that set forth in the Elements’ (Heath, 1956, p. v).

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1

Drawing Figures

1. An abbreviated form of the titles of the Critiques (CPR and CoJ) is used hereafter in the references. CPR refers to I. Kant, The Critique of Pure Reason, edited and translated by P. Guyer and A. Wood (Cambridge: Cambridge University Press, 1997). CoJ refers to I. Kant, Critique of Judgment, translated by W. S. Pluhar (Indianapolis: Hackett Publishing Company, 1987). 2. Michael Dummett analyses the principle of mathematical form in the synthetic a priori in Kant’s Critique of Pure Reason in, ‘The Philosophy of Mathematics’, in Philosophy 2: Further through the Subject, edited by A. Grayling (Oxford: Oxford University Press, 1998), pp. 126–9. 3. The essay, ‘Concerning the Ultimate Ground of the Differentiation of Directions in Space’ may be described as pre-Critical because it is written before 1770 and 1781. Gary Banham defines Kant’s ‘pre-critical’ writings as those produced ‘prior to 1770 or 1781’, and texts which contain a ‘conceptual’ pre-Criticality, including the Anthropology from a Pragmatic Point of View, from 1798. Banham writes: If a critique is taken to mean an investigation of the limits and status of claims made about an area from within the area in question hence as an immanent questioning, then the writings of Kant’s which do not carry out such a style of investigation must be regarded as ‘pre-critical’. G. Banham, Kant’s Practical Philosophy: From Critique to Doctrine. (Basingstoke: Palgrave Macmillan, 2003), p. 8 4. In Chapter 3 the importance of ‘parts’ or ‘scholia’ will be examined in more detail in relation to Spinoza’s geometric method. 5. Edward Casey provides an alternative reading of this debate. In this particular essay, he suggests that Kant’s theory of spatial relations corresponds with Leibniz’s, writing that the crucial step here taken by Kant is that whereby positions, though declared indispensable for grasping the location of parts of objects and for the relations of objects (‘things’) to each other, are absorbed into regions – which are themselves absorbed into absolute space. Indispensable in one respect, positions are dispensable in other respects, that is, precisely when they cannot be reduced to the sheer relationality of Cartesian ‘external place’ or what Kant simply calls ‘external relation’. E. Casey, The Fate of Place: A Philosophical History. (Berkeley, Los Angeles and London: University of California Press , 1998), p. 189 6. An abbreviated title of the essay is used hereafter in the references. CDS refers to I. Kant, Theoretical Philosophy 1755–1770, Cambridge edition I, edited and translated by D. Walford and R. Meerbote (Cambridge: Cambridge University Press, 1992). 7. Deleuze notes the importance of an internal intuition of space from which external space can be produced in a certain lineage of neo-Kantian thought; he writes: If, in the forms of intuition, Kant recognised extrinsic differences not reducible to the order of concepts, these are no less ‘internal’ even though they cannot be regarded as ‘intrinsic’ by the understanding, and can be represented only in their external relation to space as a whole . . . . In

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9.

10.

11.

12.

13.

Notes other words, following certain neo-Kantian interpretations, there is a step-by-step, internal, dynamic construction of space which must precede the ‘representation’ of the whole as a form of exteriority. In the same passage, Deleuze also notes that such an interpretation places Kant much less at odds with Leibniz’s writings on space. G. Deleuze, Difference and Repetition, translated by P. Patton (London: The Athlone Press, 1997), p. 26. Banham examines the development of the different aesthetics in Kant’s three Critiques: for example, assessing the scope of the imagination and synthesis in the first Critique and the ‘productive imagination’ in the third Critique. He summarises the three roles of the imagination in the first Critique as; ‘an empirical rule of reproduction which operates through the presentation of images; a transcendental rule of synthesis whether determined as “figurative” (B-Deduction) or as constitutive of each level of synthesis (A-Deduction); a mediating function between sensibility and understanding via schematism’. Banham also observes that, the imagination’s freedom is amplified by the new transcendental Aesthetic of Reflective Judgment in the third Critique, which brings reflective powers into the operations of the imagination, thereby generating a new set of ‘formal and material purposiveness’. G. Banham, Kant and the End of Aesthetics (Basingstoke: Palgrave Macmillan, 2000), p. 58. However, Kant is also careful to state that art can be generated in the agreement between the understanding and reason in the form of ‘a unique concept’, that is, ‘the concept of nature as art’ (CoJ, pp. 392–3). Geometry can be described as a science of magnitudes, insofar as it is the construction of bodies that are brought about through the division of bodies into parts. Therefore, a link between geometry and feeling may be revealed when both are related to corporeal and living bodies. Daniel W. Smith has examined the relationship between the imagination and figure in Kant’s Critical philosophy in the essay, ‘Deleuze’s Theory of Sensation: Overcoming the Kantian Duality’, in Deleuze: A Critical Reader, edited by P. Patton (Oxford: Blackwell Publishers, 1996), pp. 29–56. Banham’s re-reading of the ‘connections between the imagination, conceptuality and intuition’ the Transcendental Aesthetic suggests, however, that the imagination is a productive form of ‘transcendental psychology’ even in the first Critique, for example, when he writes: The transcendental synthesis of the imagination is here described as productive due to the fact that what is required for it to take place is that the nature of objectivity is itself produced by it. This is not a synthesis of objects, it is rather a synthesis that enables there to be any relation of ‘objects’ to each other such that we can speak of there being a world as it produced the very notion of what an ‘objective’ representation is. The notion that it takes place ‘prior to apperception’ should however, in our view be interpreted as . . . it is directed by the principles of unification that is derived from apperception. The tracing of this synthesis, as a synthesis that brings the unity of apperception to the manifold of intuition, is the subsequent primary work of the Critique. G. Banham, Kant’s Transcendental Imagination. (Basingstoke: Palgrave Macmillan, 2006), pp. 143–4 Kant also examines the nature of relations, in mechanical and dynamic forms, in the 1786 essay, ‘Metaphysical Foundations of Natural Science’. In

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15.

16.

17.

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this text, geometry and its figures tend to be determinate. I. Kant, Philosophy of Material Nature, translated by J. W. Ellington (Indianapolis: Hackett Publishing Company, 1985). Perhaps most strongly expressed in the idea that Euclid’s Elements is an exclusively scientific, mathematical text; for example, T. Heath, Euclid: The Thirteen Books of the Elements, Volume I (London: Dover Publications, 1956), p. v. An abbreviated form of the title of the Anthropology is used hereafter in references. APP refers to I. Kant, Anthropology from a Pragmatic Point of View, translated by V. L. Dowdell (London and Amsterdam: Southern Illinois University Press, 1978). Howard Caygill explains that Kant’s method is particularly metaphysics, developing a ‘proper’ relationship between the intelligible and sensible realities. He writes: The only way to preserve metaphysics is to establish a procedure for determining the proper relation of the sensible and intelligible realms . . . . Kant offers another analogy, but one which this time he fully develops. He offers the example of spatial orientation, and the nature of directionality. In order to orient ourselves spatially we must make a distinction between left and right; but how can this distinction be made? In Kant’s words, is it transcendental or empirical, in Heidegger’s is it ontic or ontological? We shall see in the next section that Heidegger’s decision in Being and Time to assign this distinction to ontic determination does Kant an injustice, making the difference empirical: it isn’t, but then neither is it transcendental . . . . Caygill continues: ‘Spatial orientation rests on a difference which is in a sense outside of and yet underlying spatial orientation. Dropping the spatial metaphors, it assumes a procedure or activity of distinction . . . ’. Caygill refers back to the CPR to ask if spatial orientation is an activity that is not yet defined by the faculties, and he suggests that it is a production of space that is, in some ways, prior to conceptual knowledge. In addition, he observes that it is a different kind of judgment; ‘because this differentiating activity cannot be represented in intuition that Kant calls it a feeling, or an “affection” of the subject. This indicates that it does not form part of either the sensible or intelligible realms, but is yet essential for this proper calibration . . . ’. H. Caygill, Art of Judgment (Oxford: Blackwell Publishers, 1989), p. 198. Caygill has noted that Kant’s commitment to the relationship between understanding and the intuitions is, ultimately, underwritten by an absolute division between intuition and God, the world and the soul, which means that critique is always determined by an external drive or difference. H. Caygill, Walter Benjamin: The Colour of Experience (London: Routledge, 1998), p. 2. Kant’s Critical philosophy therefore enables geometry and aesthetics to be understood as aspects of the understanding and intuition. Yet it also allows an immanent sufficiency to be located in the subject. Brian Massumi examines discontinuity and infinity in drawing and geometry in his essay, ‘The Diagram as Technique of Existence’. He writes: Let the clean blackboard be a sort of Diagram of the original vague potentiality, or at any rate of some early stage of its determination. This blackboard is a continuum of two dimensions, while that which it stands

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for is a continuum of some indefinite multitude of dimensions. I draw a chalk line on the board [ . . . ]. For this white chalk mark is not a line, it is a plane figure in Euclid’s sense, a surface, and the only line that is there is the line which forms the limit between the black and the white surface. This discontinuity can only be produced upon that blackboard by the reaction between two continuous surfaces into which it is separated, the white surface and the black surface. (Massumi, 2001, p. 163) 19. Phaedo, 73b (cited in Proclus, 1992), p. 37. The following chapter will examine Proclus’ attention to the relationship between the discursive nature of the mathematical diagram and memory; for example, in the first part of the Prologue, Proclus refers to the Phaedo and the Meno as examples of Plato’s theory that recollection is the understanding or ‘a part of the soul’ that unfolds the ideas it already contains, Proclus, A Commentary on the First Book of Euclid’s Elements, translated by G. R. Morrow (Princeton: Princeton University Press, 1992), p. 37.

2

Folding-Unfolding

1. In Chapter 4 we will see that Leibniz also emphasises the importance of ‘intermediacy’ in a series of figures, but in an analytic form. 2. All quotations from the Commentary are taken from Proclus, A Commentary on the First Book of Euclid’s Elements (AD 410–485 ), translated by G. R. Morrow (Princeton: Princeton University Press, 1992).This translation is developed from Gottfried Friedlein’s 1873 text, Procli Diadochi in Primum Euclidis Elementorum Librum Commentarii ex Recognitione Godofredi Friedlein, Leipzig, 1873, and is based upon the Greek text by Simon Grynaeus, Basel, 1533. The title is abbreviated to CEE in references hereafter. 3. The Commentary’s Prologues and Definitions offers most insight to the metaphysical nature of the geometric method and its figures but it is also a step by step explication of Book 1 of the Elements. 4. Deleuze notes Proclus’ definition of ‘series’ in relation to the Pythagorean divine notions of the one, many, limit and unlimit. G. Deleuze, The Fold: Leibniz and the Baroque, translated by T. Conley (London: The Athlone Press, 2001), pp. 23 and 146. Later in this chapter I elaborate on these principles, and in Chapter 4 I discuss the infinite and multiple in relation to Leibniz’s geometric method and figures. 5. Éric Alliez analyses the fold and neo-Platonism in relation to the construction of the soul and time in Plotinus’ philosophy. He distinguishes between three different kinds of folding: a. Greek folding (pli) of the forces engaged in the relation to others that is constitutive of the relation to self . . . . b. the Greco-Roman unfolding (dépli) of the relation to self in power relations . . . . c. the neoplatonic refolding (repli) or the self within the whole that puts it outside itself . . . . É. Alliez, Captial Times: Tales from the Conquest of Time, Theory Out of Bounds, Volume 6, translated by G. van den Abbeele. (Minneapolis and London: University of Minnesota Press, 1996), p. 73

Notes 189 6. Plato’s theory of the Divided Line in the Republic, Book VI, 510–510e, is demarcated by the upper, transcendental realm of Being and the lower sensible realm of Becoming, emphasising the division between the faculties of reason and understanding from the faculty of imagination and sense opinion. See Plato, Collected Dialogues of Plato, Including the Letters, edited by E. Hamilton and H. Cairns (Princeton: Princeton University Press, 1989), pp. 745–6. 7. Dimitri Nikulin’s study of geometry, matter, and the imagination in Plotinus, Proclus, and Descartes observes that ‘The notion of the intermediary plays an important role in Platonic ontology as present in Proclus’ commentary to the first book of Euclid’s “Elements”. In this approach, geometrical objects are considered intermediate between ideal objects (notions) and their physical images, being irreducible to any of them.’ D. Nikulin, Matter, Imagination and Geometry: Ontology, Natural Philosophy and Mathematics in Plotinus, Proclus and Descartes (Burlington, VT: Ashgate Publishing, 2002), p. xiv. 8. Ian Mueller outlines three diagrams that show the different realms of metaphysical schema which inform the Commentary. They are: figure 1, the order derived from the Divided Line, Book IV, Republic; figure 2, the Neo-Platonic order, and figure 3, an alternative order of the soul mediating between the non-sensible and sensible realms (CEE, pp. xvii–iii). 9. Proclus, however, attributes creative power to the soul and reproductive power to the understanding. This discussion is also an interesting precursor of philosophical discussions concerned with notions of ‘life-force’, for example, Freud’s examination of the ‘to-and-fro’ movement between pleasure and displeasure, or the forces of Eros and Thanatos, in the essay ‘Beyond the Pleasure Principle’ (1920). See S. Freud, The Standard Edition of the Complete Psychological Works, Volume 18, edited and translated by J. Strachey (London: Hogarth Press and the Institute of Psychoanalysis, 1955). Also see Baudrillard’s discussion of the economics of force between Eros and Thanatos. J. Baudrillard, Symbolic Exchange and Death, translated by Iain Hamilton Grant (London: Sage Publications, 1993). In each case, concepts of life and death produce notions of production that are characterised by a twofold movement. 10. In the Timaeus, Plato describes the world’s soul as a mixture of mathematical matter and cosmic powers, a ‘strip’ divided into parts to form the cosmos and the metaphysical principles of the Existent, the Same and the Different, in constant, autonomous movement. It is conceived as being immaterial and partless, and corporeal and divided (Plato, 1989, pp. 1165–7). 11. Also see Nikulin’s examination of imagination in Proclus’ Commentary (Nikulin, 2002, pp. 234–7). 12. The diagram is re-examined in Deleuze’s study of Francis Bacon’s paintings, when he writes: The diagram is thus the operative set of asignifying and nonrepresentative lines and zones, line-stokes and color-patches. And the operation of the diagram, its function, says Bacon, is to be ‘suggestive’. Or, more rigorously, to use language similar to Wittgenstein’s, is to introduce ‘possibilities of fact’ . . . . Because they are destined to give us the Figure, it is all the more important for the traits and color-patches to break with

190 Notes figuration. This is why they are not sufficient in themselves, but must be ‘utilized’. They mark out possibilities of fact, but do not constitute a fact (the pictorial fact). In order to be converted into a fact, in order to evolve into a Figure, they must be reinjected into the visual whole; but it is precisely through the action of these marks that the visual whole will cease to be an optical organization; it will give the eye another power, as well as an object that will no longer be figurative. G. Deleuze, Francis Bacon: The Logic of Sensation, translated by D. W. Smith. (London and New York: Continuum, 2003), pp. 101–2 13. Nikulin writes: Limit for Proclus is related to the unlimted as substance is related to pure potency, dynamis. He further distinguishes two potencies, the one of the productive principle, which has to be associated with the One [ . . . ], the other of pure receptivity, which has to be linked to matter. Since Proclus does not accept any potentiality in the intelligible, all matter should be excluded from here. But since the intelligible is constituted for him by both principles of limit and the unlimited, being has to exemplify a certain potency, which is the infinite potency of unceasing thinking and production; although this potency of being is not potentiality and in this way is different from the potentiality of coming to be. (Nikulin, 2002, p. 143) 14. Robin Durie’s essay, ‘The Strange Nature of the Instant’, examines this discussion. See R. Durie (ed.), Time and the Instant: Essays in the Physics and Philosophy of Time (Manchester: Clinamen Press, 2000), pp. 1–24.

3 Passages 1. The term ‘expression’ is used repeatedly by Spinoza in the Ethics to describe the immanence of God in modes of substance; for example, in Part I, Definition 6, he writes: ‘By God I mean an absolutely infinite being; that is, substance consisting of infinite attributes, each of which expresses eternal and infinite essence.’ B. Spinoza, Ethics, Treatise on the Emendation of the Intellect and Selected Letters, edited by S. Feldman (Indianapolis and Cambridge, MA: Hackett Publishing Company, 1992), p. 31. This chapter is also informed by Deleuze’s examination of the term which emphasises the importance of the internal movements of thought in Spinoza’s and Leibniz’s post-Cartesian philosophies see G. Deleuze, Expressionism in Philosophy: Spinoza, translated by M. Joughin (New York: Zone Books, 1992). Martin Joughin’s Preface succinctly summarises this argument when he writes: Spinoza and Leibniz: two different expressions of ‘expressionism in philosophy’ characterized in this book as a system of implicatio and explicatio, enfolding and unfolding, implication and explication, implying and explaining, involving and evolving, enveloping and developing. Two systems of universal folding: Spinoza’s unfolded from the bare ‘simplicity’ of an Infinity into which all things are ultimately folded up, as into a universal map that folds back into a single point; while Leibniz starts from the infinite points in that map, each of which enfolds within its

Notes 191

2.

3.

4.

5.

6.

7. 8.

9.

10.

infinitely ‘complex’ identity all its relations with all other such points, the unfolding of all these infinite relations being the evolution of a Leibnizian Universe. (Deleuze, 1992, p. 5) This chapter does not extend into an evaluation of whether Descartes’ geometric writings also constitute a ‘forgotten’ geometry. However, see Chapter 6 for a discussion of Husserl’s engagement with Descartes. Later in this chapter I refer to Spinoza’s criticism of Descartes’ ‘occultist’ union between the mind and body. In addition, Chapter 5 notes Bergson’s frustration with Descartes’ scientific method that not only leads him to acknowledge Descartes’ skill as a physicist but also to criticise his dependency upon the ‘symbolic’ limits of modern rational science. Deleuze notes that Spinoza constructs the geometric method in relation to a ‘way of being’, which is reflected in his practical work of polishing optical lenses. Deleuze writes: In Spinoza’s thought, life is not an idea, a matter of theory. It is a way of being, one and the same eternal mode in all its attributes. And it is only from this perspective that the geometric method is fully comprehensible . . . . The geometric method ceases to be a method of intellectual exposition; it is not longer a means of professorial presentation but rather a method of invention . . . . G. Deleuze, Spinoza: Practical Philosophy, translated by R. Hurley. (San Francisco: City Lights Books, 1988), pp. 13–14 Also see Bergson’s description of Spinoza’s approach as having the impact of a ‘dreadnought’, in K. Ansell Pearson and J. Mullarkey (eds), Henri Bergson: Key Writings (London: Continuum, 2002), and cited in Chapter 5, note 9, below. All citations from the Ethics are taken from B. Spinoza, The Ethics, Treatise on the Emendation of the Intellect, Selected Letters (1677), edited by S. Feldman and translated by S. Shirley (Indianapolis and Cambridge, MA: Hackett Publishing Company, 1992). The title is abbreviated to E in references hereafter. References give the following: Title, Part, Proposition or Definition, Corollory or Scholium, and page number; for example (E: I, Prop. 7, p. 34). See, for example, Martin Joughin’s discussion about enfolding/unfolding and implicatio/explicatio (Deleuze, 1992, pp. 5–7). Seymour Feldman writes: Spinoza’s Ethics is perhaps the first purely philosophical treatise that presents its conclusions consistently and completely in an axiomatic manner. In this respect it is the paradigm of the hypothetical-deductive method suggested by Aristotle in his Posterior Analytics as the model for a scientific theory, which until Spinoza was only exemplified by Euclid’s geometry. (E, p. 7) Richard Arthur provides an excellent survey of the development of atomistic theories in which Spinoza and Leibniz participated. See R. Arthur (ed. and trans.), The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672–1686, G W Leibniz, The Yale Leibniz Series (New Haven and London: Yale University Press, 2001). Deleuze notes the confluence between the geometric plan and immanent plane in Spinoza’s emphasis on the modal nature of ‘life’: What is involved is no longer the affirmation of a single substance, but rather the laying out of a common plane of immanence on which all bodies, all minds, and all individuals are situated. This plane of immanence or

192 Notes

11.

12.

13. 14.

15.

16.

17.

18.

19.

consistency is a plan, but not in the sense of a mental design, a project a program; it is a plan in the geometric sense: a section, an intersection, a diagram. Thus to be in the middle of Spinoza is to be on the modal plane, or rather to install oneself on this plane – which implies a mode of living, a way of life. (Deleuze, 1988, p. 199) Spinoza can be said to be ‘materialist’ insofar as he anticipates the modern concern with biophysical definitions of matter. See, for example, Deleuze (1988) pp. 56–7, and Seymour Feldman’s introduction to the Ethics (E, p. 12). Spinoza continues this discussion into an extended examination of the motion of extended bodies and their constitution as divisible parts, motion, internal and external qualities and capacity to affect other bodies (E, pp. 72–6). See Part II, Proposition 40, Scholium 1 (E, p. 89). Spinoza’s common notions may have some correspondence to Deleuze and Guattari’s notion of the ‘percept’ when they write that the definition of art lies in its attempts to ‘create the finite that restores the infinite: it lays out a plane of composition that, in turn, through the actions of aesthetic figures, bears monuments or composite sensations’. G. Deleuze and F. Guattari, What is Philosophy? translated by H. Tomlinson and G. Burchell (New York and Chichester: Columbia University Press, 1994), p. 197. Deleuze writes: But joyful passions lead us closer to this power [of action], that is, increase or help it; sad passions distance us from it, that is, diminish or hinder it. The primary question of the Ethics is thus: What must we do in order to be affected by a maximum of joyful passions? (Deleuze, 1992, p. 273) Deleuze notes the historical ‘pantheist’ tradition in the relationship between the implication and explication (implicaito/explicatio) that produce a synthetic unity – ‘complicatio’ – that is underscored by the neo-Platonic principles of ‘multiplicity in the One, and of the One in the Many’ and noting that the principles of implication and explication do not therefore constitute opposition but synthesis (Deleuze, 1992, p. 16). Deleuze uses the concept of ‘speeds’ to register the multiple kinds of activity that are generated in the body by the emotions; for example, of the modes, he writes: ‘For, concretely, a mode is a complex relation of speed and slowness, in the body but also in thought, and it is a capacity for affecting or being affected, pertaining to the body or to thought’ (Deleuze, 1988, p. 124). Deleuze writes that the Ethics is a twice-written book; the first book is the formal geometric method, the second ‘subterranean’ book is the ‘broken chain of the scholia, a discontinuous volcanic line, a second version underneath the first, expressing all the angers of the heart and setting forth the practical theses of denunciation and liberation’ (Deleuze, 1992, pp. 28–9). Isabelle Stengers, for example, discusses whether it is possible to think of an ethics of science that might reflect feminist practice or radical politics and suggests rethinking the scope of the scientific method towards an ethical and critical practice that reflects Bergson’s inquiry. See I. Stengers, Power and Invention: Situating Science, Theory Out of Bounds, Volume 10 (Minneapolis and London: University of Minnesota Press, 1997).

Notes 193

4

Plenums

1. See, for example, Leibniz’s writings on the problem of bodies, motion and rest in, ‘On Matter, Motion, Minima, and the Continuum’, 1675 (Arthur, 2001, pp. 30–41). In his introduction, Arthur notes that Leibniz gave up ‘the ontology of perfect solids and perfect fluids’ during this period, and suggests that he develops a different ontology in which ‘matter has varying degrees of resistance to division, [and] a given body can respond to the actions of the plenum by differing internal divisions, manifested as elasticity’ (Arthur, 2001, p. lxv). Arthur’s commentary on the problem of the Continuum informs discussions about geometric methods, in particular, in his analysis of Leibniz’s conceptual development of substance, infinite divisibility, and the ‘unassignables’ or ‘indiscernibles’. Arthur also shows how Leibniz’s ontology of the Monad is developed in conjunction with the intensive debates about physical and mathematical sciences in the seventeenth century, whilst recognizing the inherent labyrinthine nature of his writings in which internal disagreements, correspondence with other writers and progressive changes of opinion and contradiction, come together to form a discontinuous, yet continuous philosophy. Arthur notes, for example, that Leibniz’s philosophy of infinity is intimately related to the operations of Geometry, such as, his essay, ‘De usu geometriae’ (1676), in which Leibniz considers geometry to be the basis for discussions about the ‘Continuum’, when he writes: ‘Only Geometry . . . can provide a thread of the Labyrinth of the Composition of the Continuum, of maximum and minimum, and the unassignable and the infinite, and no one will arrive at a truly solid metaphysics who has not passed through that labyrinth’ (cited in Arthur, 2001, p. xxiii). 2. In his ‘Treatise on Calculus’ (1675–1676), Leibniz defines calculus as ‘every curvilinear figure is nothing but a polygon with an infinite number of sides, of an infinitely small magnitude’. Arthur explains, ‘according to this conception any curve can now be represented as an infinite “sum” of such differentials . . . . Similarly, the area can be represented as an infinite sum of the products of each ordinate and a differential . . .’ (Arthur, 2001, p. liv). 3. This is reflected in the different metaphysical ‘levels’ in which the two texts begin; the Ethics begins with a definition of the infinite, yet indivisible, substance or God, whereas the Monadology begins with an explication of the infinite divisibility (i.e., magnitude) of the Monad. 4. As a result, the relationship between quality and quantity becomes the central condition of production, not as an opposition of forces but as a variation in degrees of intensity in the Monad. Deleuze examines these relations in Leibniz’s, Kant’s and Maïmon’s theories of qualitative difference (Deleuze, 1997, pp. 170–6). 5. Leibniz’s correspondence with Samuel Clarke between 1705–16 contains an expanded discussion about Kant’s Newtonian understandings of space and time versus Leibniz’s theories of geometry, space and time. See, H. G. Alexander, The Leibniz-Clarke Correspondence (Manchester: Manchester University Press, 1956). This debate is too large to be addressed here, but it is worth noting that Deleuze suggests that the differences between Kant’s and Leibniz’s positions are mediated through Salomon Maïmon’s ‘reformulation’

194 Notes

6.

7.

8.

9. 10.

11.

of the Critique of Pure Reason by means of a Leibnizian form of qualitative ‘difference’. Deleuze outlines how Maïmon overcomes the external difference that constitutes ‘the Kantian duality between concept and intuition’ by ‘showing how inadequate the point of view of conditioning is for a transcendental philosophy: [so that] determinability must be itself conceived as point towards a principle of reciprocal determination’ (Deleuze, 1997, p. 173). The Monad’s internal infinity, difference and magnitude also suggests a strong precedent to the Jena Romantics concept of ‘fragment’, whose magnitude is an excessive unity that challenges the notions of finite extension and agency. See, for example, P. Lacoue-Labarthe and J-L. Nancy, The Literary Absolute: The Theory of Literature in German Romanticism, translated by P. Barnard and C. Lester (Albany, NY.: SUNY, 1988). All citations from the Monadology are taken from the edition, G. W. Leibniz, Discourse on Metaphysics, Correspondence with Arnauld, Monadology, translated by G. R. Montgomery (La Salle, Ill.: Open Court Publishing Company, 1973). The title is abbreviated to M in references hereafter, followed by the section number and page number. Arthur writes that the problem of the atom and the void is ‘a tangled thread’ throughout Leibniz’s writings, which develops from his theories of atoms as ‘unextended “indivisibles” to; “the insensibly small, very hard particles” such as the “bullae” and the “terrellas” [which are] akin to the “chemical” atoms of Sennert [and represent] “units of formation” or “action” in his writings before 1676’ (Arthur, 2001, pp. xliii–xliv). After 1676, however, Arthur writes that Leibniz embraces ‘atomism’, positing the ‘necessity’ of an ‘indestructible core’. But with respect to the Monadology we find that it goes a step further, reflecting Leibniz’s subsequent rejection of atomism for substance, which he calls the ‘substantial atom’, ‘the combination of soul and body’, or the ‘corporeal substance’ in which there is the indivisible soul (Arthur, 2001, p. xlviii). Also see below for Leibniz’s definition of the immensum from ‘On the Origin of Things from Forms’ (1676), and cited in Arthur (2001), p. 121. Keith Ansell Pearson writes that Bergson’s critique of knowledge is a ‘philosophy of life’, which recalls Spinoza’s philosophy of a ‘practical way of living’. K. Ansell Pearson, Philosophy and the Adventures of the Virtual: Bergson and the Time of Life (New York and London: Routledge, 2002), p. 17. Ansell Pearson also notes that both Spinoza and Bergson are linked by a commitment to a univocal and transcendental immanence (Ansell Pearson, 2002, p. 103). However, there are also significant differences between Leibniz’s and Bergson’s concepts of perception and memory: first, Leibniz provides a particularly strong concept of intensity, whereas Bergson considers perception and its intensity in relation to a much more explicitly defined spatiotemporal order. In this respect, Bergson brings Leibniz’s project into a more fully formed topological geometric project. Second, Bergson analyses memory and the actions of the individual in relation to an embodied and transcendental ‘intuition’. Bergson’s thinking therefore displays a more psychological mode of interpretation than Leibniz’s logic, despite the sufficiency of the Monad that is derived from perception and appetite.

Notes 195 12. Robin Durie, for example, highlights the extent to which a ‘potential’ theory of infinity as magnitude is problematic, since it does not admit the corporeality of being (entelechia). He cites Aristotle’s discussion about the reality of a ‘potential’ magnitude and the Entelechy in the Physics: ‘To be’ means to be potentially [dynamei] or to be actually [entelechia]; and the infinite is either in addition or in division. It has been stated that magnitude [megethos] is not in actual operation infinite [i.e., there is a limit to the actual size things can be]; but it is infinite in division – it is not hard to refute indivisible lines – so that it remains for the infinite to be potentially [dynamei]. We must not take ‘potentially’ here in the same way as that in which, if it is possible for this to be a statue, it actually will be a statue, and suppose that there is an infinite which will be in actual operation. (Aristotle’s Physics III, 206a, 14ff, cited in Durie, 2000, p. 13) 13. The seventeenth-century notion of substance provides a distinct shift in the concepts of infinity and magnitude from those that existed in the ancients’ philosophy. Durie states that, according to Zeno, a magnitude of an infinite aggregate is impossible, or insufficient, since it is only potentially given (Durie, 2000, p. 13). For Leibniz, however, the notion of aggregate or incompossibility is not just demonstrated as a logical idea, but is established as the definition of an intensive and thinking substance that is sufficient. 14. See, for example, The Fold in which the ‘amplitude’ of the soul (its intension and inflection) is similar to Bergson’s memory in which the ‘living present’ or act is ‘essentially variable in both extension and intensity’ (Deleuze, 2001, p. 70). 15. Deleuze writes: ‘This procedure of the infinitely small, which maintains the distinction between essences (to the extent that one plays the role of inessential to the other), is quite different to contradiction. We should therefore give it a special name, that of “vice-diction”’ (Deleuze, 1997, p. 46). 16. Arthur suggests that ‘metaphor of the net presages that of the folds of matter’ (Arthur, 2001, p. 402). 17. The relationship between space and matter is also evident in the use of the term ‘plenum’ in architecture, which refers to an interstitial space or void between the floor and the ceiling; and plenums are also chambers for the circulation of air in combustion engines. Also note Deleuze’s opening sentence about the operation of folding in relation to the baroque architectural form of interconnected chambers, floors and layers in The Fold; ‘the Baroque differentiates its folds in two ways, by moving along two infinities, as if infinity were composed of two stages or floors: the pleats of matter, and the folds of the soul’ (Deleuze, 2001, p. 3). 18. Arthur writes that Leibniz was committed to a plenistic physics from the beginning, largely under the influence of Hobbes. But this was the dominant view of his contemporaries, shared by the Cartesians and even atomists like Huygens. It was not displaced in continental Europe until the spread of Newtonianism in the latter part of the eighteenth century. (Arthur, 2001, p. 460) 19. Arkady Plotnitsky explores a rather polarised geometrico-topological route from Leibniz, Riemann to Deleuze, and does not include Bergson in this

196 Notes lineage. A. Plotnitsky, ‘Algebras, Geometries and Topologies of the Fold: Deleuze, Derrida and Quasi-Mathematical Thinking (with Leibniz and Mallarmé)’, in Between Deleuze and Derrida, edited by Paul Patton and John Protevi (London and New York: Continuum, 2003), pp. 98–119.

5

Envelopes

1. By contrast see, for example, art historical studies of Duchamp’s concept of the ‘ready-made’, his interest in geometry and ‘l’infinite’, in The Writings of Marcel Duchamp, Marchand du Sel edited by M. Sanouillet and E. Peterson (Cambridge, MA and New York: Da Capo Press, 1973). See also L. Dalrymple Henderson, Duchamp in Context: Science and Technology in the Large Glass and Related Works (Princeton: Princeton University Press, 1998). 2. Developments in mathematics and geometry in the nineteenth century focus on the break away from Euclidian geometry, and become increasingly concerned with the mathematical possibility of a spatial ‘fourth dimension’ and non-Euclidian principles of space-time. In each case, these geometries sought to disrupt the apparently teleological determination of Euclidian geometry towards ideal truths. Thus, non-Euclidian geometry challenges the axiomatic a priori conception of mathematics, in particular, focusing on the possibility of alternative solutions to the ‘truth’ of Euclid’s fifth postulate: ‘That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles’ (Heath, 1956, p. 155). Gauss’s, Lobachevsky’s and Bolyai’s theories of lines that will necessarily meet – the lines on a sphere – in the 1820s and 1830s lead on to Beltrami’s ‘pseudosphere’ in the 1860s which provided an accessible diagram of curved space. Other non-Euclidian geometries took the issue of congruence as evidence that Euclid’s geometry did not fully describe spatial relations (e.g., the non-congruence of the left and right hand). See, for example, L. Dalrymple Henderson, The Fourth dimension and Non-Euclidian Geometry in Modern Art (Princeton: Princeton University Press, 1983). Most complex of these was Riemann’s theory of topological manifolds (explained in his lecture ‘On the Hypothesis that Lie at the Foundations of Geometry’ of 1854 and published in 1867), which distinguished between bounded and infinite space, and congruence, diverging significantly from Euclidian principles of transformation to produce geometric figures that are ‘locally’ and ‘globally’ differentiated. Lawrence Sklar, for example, explains that a ‘topological structure’ is evident in the intrinsic difference between two surfaces, such as a plane and a cylinder, determined at a local and a global level. At a global level ‘the two surfaces differ, even neglecting their embedding in three-space, in the global properties of connectivity determinable by a geometer confirmed to the surface and ignorant of the embedding’. At a local level they differ if we are given ‘free mobility throughout the surface’; for example, if we start from a point on a plane and travel along any geodesic (straight line) through that point without ever reversing direction, then we will never return to our initial point. On a cylinder, however, through each

Notes 197

3. 4.

5.

6. 7.

8.

9.

point there is a geodesic (the circle around the cylinder through that point) such that if we travel along that geodesic, never reversing direction, we will sooner or later return to our starting point. This shows that the intrinsic identity of a [figure, such as a] cylinder and a plane is a local matter. L. Sklar, Space, Time and Spacetime. (Berkeley, Los Angeles and London: University of California Press, 1977), pp. 41–2 (my emphasis) See, for example, Ansell Pearson’s examination of the multiplicity in Bergson’s notion of the ‘virtual’ (Ansell Pearson, 2002). In his essay ‘Laughter: An Essay on the Meaning of the Comic’ (1900), Bergson suggests that the imagination abstracts from the particular perception: ‘[the] artistic imagination [ . . . ] simply reveals what we have hidden from ourselves in our perceptual power of condensation which is at the same time an abstraction from the individual to the general’. Cited in J. Mullarkey, Bergson and Philosophy (Edinburgh: Edinburgh University Press, 1999), p. 59. All quotations are taken from H. Bergson, Matter and Memory, translated by N. M. Paul and W. Scott Palmer (New York: Zone Books, 1991). The title is abbreviated to MM in references hereafter. Mullarkey defines the image as a universal designation of ‘the objects of every type of perception’ (Mullarkey, 1999, p. 32). For an interesting ‘psychical’ interpretation of topology and the body see Bernard Burgoyne’s discussion in his essay ‘Autism and Topology’, in which he examines the ‘weak’ topological structures of the autistic child. In note 32 to the essay he writes: Equally topological structure can be obtained by considerations of boundary or frontier. There exists a range of topological notions, all of which can be demonstrated to be equivalent in having this power to generate the structure of a space: where there are limitations of the equivalence they raise questions about the foundations of topology and the foundations of mathematics. The equivalent notions include the concepts of neighbourhood, interior, closure, closed set, net, limit, filter and ideal. B. Burgoyne (ed.), Drawing the Soul: Schemas and Models in Psychoanalysis. (London: Rebus Press, 2000), p. 215 (my emphasis) Mullarkey, for example, discusses Bergson’s development of heterogeneous notions of space in Matter and Memory, in distinction to the ‘homogenous’ notion of space in the earlier essay ‘Time and Free Will’ (1888) (Mullarkey, 1999, p. 13). See, for example, Bergson’s impressive analysis of Spinoza’s geometric method in the Ethics, in the essay ‘Philosophical Intuition’, in The Creative Mind (1933), which Bergson suggests has ‘behind’ it the ‘subtle’ ‘lightness’ of intuition that elides the conceptual weight of his method. He writes: Nevertheless I know of nothing more instructive than the contrast between the form and the a matter of a book like the Ethics: on the one hand those tremendous things called Substance, Attribute and Mode, and the formidable array of theorems with the close network of definitions, corollaries and scholia, and that complication of machinery, that power to crush which causes the beginner, in the presence of the Ethics, to be struck with admiration and terror as though he were before a battleship of the Dreadnought class; on the other hand, something subtle,

198 Notes very light and almost airy, which flees at one’s approach, but which one cannot look at even from afar, without becoming incapable of attaching oneself to any part whatever of the remainder, even to what is considered essential, even to the distinction between Substance and Attribute, even to the duality of thought and Extension. What we have behind the heavy mass of concepts of Cartesian and Aristotelian parentage is that intuition which was Spinoza’s, an intuition which no formula, no matter how simple, can be simple enough to express. (Ansell Pearson and Mullarkey, 2002, pp. 236–7) 10. All quotations are taken from ‘Introduction to Metaphysics’ (1903) (reprinted in Ansell Pearson and Mullarkey, 2002). The title is abbreviated to IM in references hereafter. 11. On intuition and geometry, Mullarkey writes: ‘Bergson believes there is no “simple and geometrical definition of intuition”’. He cites the Creative Mind in which Bergson writes that a changing reality requires ‘views of it that are multiple, complementary and not at all equivalent’. Mullarkey continues: ‘Intuition entails whatever is required by a subject in a particular context to adjust to the full alterity of that situation as it extends beyond the confines of [the subject’s] perspective’ (Mullarkey, 1999, p. 159). 12. All quotations are taken from H. Bergson, Creative Evolution, translated by A. Mitchell (London: Macmillan and Company, 1964). In references, the title is abbreviated to CE hereafter.

6

Horizons

1. David Carr distinguishes the Crisis from Husserl’s earlier major ‘introductions to phenomenology’, the Ideas (1913), the Cartesian Meditations (1931) and Formal and Transcendental Logic (1929), in particular, drawing attention to the ‘unmistakable and passionate expression, though in the most general terms, to his position on the turbulent events of the time.’ E. Husserl, Crisis of European Sciences and Transcendental Phenomenology: An Introduction to Phenomenological Thought (1954), translated by David Carr from the posthumous German edition by Walter Biemel (Evanston: Northwestern University Press, 1970), p. xvi. 2. Husserl’s Crisis of European Sciences and Transcendental Phenomenology is abbreviated to CES in references hereafter. 3. Of course, there are other significant phenomenological intersubjective horizons with Husserl’s work; including, Husserl’s difficult relations with his former student, Heidegger’s phenomenology, Merleau-Ponty’s theory of ‘life-world’ and Derrida’s analysis of ‘historicity’ in Husserl’s Origin of Geometry: An Introduction (1962) translated by J. P. Leavey (Lincoln and London: University of Nebraska Press, 1989). Also see Carr’s introduction to the Crisis (CES, pp. xxv–xxvii, and note 20, p. xxx). 4. David Carr rejects the association between Husserl’s teleological Geist and Hegel’s phenomenology, stating that although each examines the production of reason in a European context, Husserl’s earlier phenomenological writings show that he does not follow the causal progression of historical events that Hegel promotes. However, Carr does recognise that the Origin has most

Notes 199 affinity with these arguments because of its ‘inquiry into the essence of history as such rather than one concerned with facts directly’ (CES, pp. xxxiii– xxviii). Also see Paul Ricoeur’s discussion of Husserl’s concept of sense and historical analysis in the Crisis in, ‘Husserl and the Sense of History’, in P. Ricoeur, Husserl: An Analysis of His Phenomenology, translated by E. G. Ballard and L. E. Embree (Evanston: Northwestern University Press, 1967), pp. 161–7. 5. Citing Gurwitsch, Carr observes that many commentators have described Husserl’s project as a ‘history of philosophy’ and ‘a philosophy of history’. A. Gurwitsch, ‘The Last Work of Edmund Husserl’, in Studies in Phenomenology and Psychology (Evanston: Northwestern University Press, 1966), p. 401. Cited by Carr (CES, p. xxxiii). 6. Deleuze, however, suggests that Husserl’s concept of the idea follows Riemann’s topological idea of the ‘multiplicity’, thereby offering a more positive connection between Husserl’s theory of the idea and modern geometric science, in contrast to Husserl’s anxiety that modern geometry ignores the potential for univocal transcendental ideas. Deleuze writes: Everything is a multiplicity in so far as it incarnates an Idea. Even the many is a multiplicity; even the one is a multiplicity. That the one is a multiplicity (as Bergson and Husserl showed) is enough to reject back-toback adjectival propositions of the one-many and many-one type. Everywhere the differences between multiplicities and the differences within the multiplicities replace schematic and crude oppositions. Instead of the enormous opposition between the one and the many, there is only the variety of multiplicity – in other words, difference. (Deleuze, 1994, p. 182) Here Deleuze appears to refer to Husserl’s Ideas, and Cartesian Meditations (but does not state this explicitly) to show that Riemann, Husserl and Bergson are linked in their concern with infinite univocal ideas. So, if we agree that Bergson and Husserl also construct multiplicities or ideas through their discussions of sense and perception, this potential may also be extended to include Husserl’s investment in the geometric idea in the Crisis and the Origin. 7. This potential is also reflected in Deleuze’s and Guattari’s observation that Kant’s Critical philosophy and Husserl’s Cartesian Meditations and Ideas are connected, writing: A transcendental logic (it can also be called dialectical) embraces the earth and all that it bears, and this serves as the primordial ground for formal logic and the derivative regional sciences. It is necessary therefore to discover at the very heart of the immanence of the lived to a subject, that subject’s acts of transcendence capable of constituting new functions of variables or conceptual references: in this sense the subject is no longer solipsist and empirical but transcendental. We have seen that Kant began to accomplish this task by showing how philosophical concepts are necessarily related to lived experience through a priori propositions or judgment as functions of a whole possible experience. But it is Husserl who sees it through to the end by discovering, in nonnumerical multiplicities or immanent perceptivo-affective fusional sets, the triple root of acts of transcendence (thought) through which the subject constitutes first of all a sensory world filled with objects, then an

200

8.

9.

10.

11.

12.

Notes intersubjective world occupied by the other, and finally a common ideal world that will be occupied by scientific, mathematical or philosophical concepts. (Deleuze and Guattari, 1994, p. 142) See Ricoeur for a discussion of Kant’s Critique of Reason as an implicit phenomenology. Ricoeur re-reads Kant through Husserl’s phenomenological method and ends his chapter on Kant and Husserl, writing: ‘Husserl did phenomenology, but Kant limited and founded it’ (Ricoeur, 1967, p. 201). Interestingly, Ricoeur does not consider intersubjectivity in relation to the reflective subject, but writes that Kant comes ‘closest’ to a proper ‘theory of intersubjectivity’ in the Anthropology (Ricoeur, 1967, p. 196). Danielle Lories argues that Husserl’s Ideas Volume I (§11) presents an ‘aesthetic attitude’ which corresponds with Kant’s theory of aesthetic disinterestedness and reflective judgment in the Critique of Judgment. She proposes re-reading the Critique of Judgment in light of Husserl, ‘as a phenomenology of the aesthetic attitude’, with particular reference to Kant’s theory of reflection: It is, I think, in this description of the reflection specific to aesthetic judgment that the main interest of a phenomenology of the aesthetic attitude that can be found in Husserl resides. By describing the double movement that unifies the aesthetic feeling and gives the object its aesthetic colour or tone, Husserl discards the interpretations of the Kantian form that consider it to be empty (which it never was!). D. Lories, ‘Remarks on Aesthetic Intentionality: Husserl or Kant’, International Journal of Philosophical Studies, Volume 14, Number 1. (March, 2006), p. 45 Derrida writes that Husserl cannot promote a ‘poetic language’ because its significations are not transcendental ‘objects’ and are therefore outside the realm of shared linguistic translatability. Derrida’s observation suggests that Husserl considers the poetic act of aesthetic thinking to be a weak transcendental project because it does not deliver clear univocal knowledge (Derrida, 1989, p. 82). Derrida notes that Husserl radicalises Kant’s transcendental intuition in his theory of the horizon: Would not, then, his original merit be to have described, in a properly transcendental step (in a sense of that work which Kantianism cannot exhaust), the conditions of possibility for history which were at the same time concrete? Concrete, because, they are experienced [vécues] under the form of horizon. The notion of horizon is decisive here: ‘horizon-consciousness,’ ‘horizon-certainty,’ ‘horizon-knowledge,’ such are the key concepts of the Origin. Horizon is given to a lived evidence, to a concrete knowledge which, Husserl says, is never ‘learned’ . . ., which no empirical moment can then hand over, since it always presupposes the horizon. Therefore, we are clearly dealing with a primordial knowledge concerning the totality of possible historical experiences. Horizon is the always-already-there of a future which keeps the indetermination of its infinite openness intact (even though this future was announced to consciousness). As the structural determination of every material indeterminacy, a horizon is always virtually present in every experience; for it is at once the unity and the incompletion for that

Notes 201 experience – the anticipated unity in every incompletion. The notion of horizon converts critical philosophy’s state of abstract possibility into the concrete infinite potentiality secretly presupposed therein. The notion of horizon makes the a priori and the teleological coincide’. (Derrida, 1989, p. 117)

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Index Note: Bold entries refer to key concepts/names in the book and bold numbers refer to pages that contain key definitions of concepts. abstract transcendental thought, xv acts aesthetic, 1, 5–10, 19–21, 57, 70, 107, 134, 160, 199 intuitive, 42–5, 62, 122–9, 141–51, 176, 181–2 see also aesthetics; enactments; technical activities actuality, 134, 137–9, 151, 162, 167–8, 173–7 aesthetics, xiii–xvi, 1, 3, 6, 9, 12, 19, 28, 34, 65, 68, 155–6, 165, 172, 181–3, 186, 187 aesthetic geometry, 1, 6, 29, 32–5, 41, 62, 65, 89, 103, 118–19, 124–5, 137–40, 155, 181, 183 aesthetic judgment, 3, 9, 11, 12, 19–27, 34, 119, 130, 156, 165, 172, 180–3, 200 see also figures; judgment affects, 7, 67–9, 71–87, 96–7, 103–8, 116 aggregate, 23, 99, 102, 118–19, 124, 129–31, 135–7, 195 agitation, 23–7, 159, 169, 172, 180 agreement, 14, 22, 42, 50, 69, 73–86, 97–8, 110, 186 Alliez, Éric, xvii, 188 amplification, 14, 42, 48, 55, 84, 87–8, 92, 99, 141, 186 analytic geometry, xiv, 6–7, 13–15, 34, 41, 49, 59–65, 68, 76, 86, 90–9, 103, 108, 110–11, 115, 117–18 123–4, 145, 169, 188 Ansell Pearson, Keith, 191, 194, 197, 198 apodictic knowledge, 154–6, 159–61, 163, 165, 168–70, 172, 177, 181, 183

apperception, 104, 186 see also consciousness appetite, 87, 91, 93–4, 98, 103–6, 119, 194 approximate, 47, 102, 109, 111, 118, 138, 162 architecture, xiii–xiv, xvii, 2–3, 11, 20, 184, 195 Aristotle, 44, 191, 195 arithmetic, 13–15, 38–9, 49–50, 59 art, xiii–iv, 184, 186, 192, 196 artistic, 10, 21–2, 24–5, 32 Arthur, Richard, 95, 99, 111–15, 191, 193–5 atoms, 68, 95, 99–100, 102, 191, 194–5 attention to life, 43, 129 attributes, 18, 54, 67–8, 71–4, 76, 81–2, 85, 88, 97, 108, 190–1 autonomy, 21, 32, 38, 58–9, 74–5, 93, 98, 108, 120, 164, 180 axiomatic method, 5, 7, 31, 34, 36, 52, 54, 62–3, 65, 67–9, 72, 74, 77, 81–2, 84–7, 99, 150, 161, 166–7, 184, 191, 196 axioms, 49, 55–6, 63, 82, 85–6, 111, 167 Bacon, Francis, 189 Banham, Gary, ix–xii, 185–6 Battersby, Christine, xiv, xviii Baudrillard, Jean, 189 Baumgarten, Alexander Gottlieb, 12 beauty, 47 becoming, 38, 76, 165, 189 being, 38, 40, 42, 44, 48, 53, 60, 64, 66, 69, 70, 72–3, 75–6, 85–6, 89, 99–100, 108, 112, 117, 123, 132, 158, 164, 168, 178, 187, 189–91, 195

207

208 Index Beltrami, Eugenio, 196 Bergson, Henri, xiii–xviii, 2, 8, 11–12, 16, 37, 39, 53, 61, 68, 70, 74, 80, 86, 90, 92, 102–5, 107, 109, 114–15, 118–53, 155, 159, 165, 175, 180–4, 191–2, 194–5, 197–8 Berkeley, George, 155, 170 biological, 43, 70 bodies, xiii, 4, 6, 11, 18, 58, 68–72, 74–5, 92, 96–7, 100, 104, 112, 115, 133, 135–6, 140, 142, 162, 186, 191–3 bodily, 9, 26, 80, 106, 141, 164 body image, 128–9 body-in-process, 67 body-limit, 132–5 body surface, 133, 135 mathematical bodies, 39, 87, 164–5, 69, 172, 175, 186 Bolyai, János, 196 boundaries, 16, 24, 29, 51, 57, 59, 121 Braidotti, Rosi, xvii Burgoyne, Bernard, 197 Butler, Judith, xiv, xvii Calculus, 91, 96, 109, 193 calibration, 93, 109 Carr, David, 198 Cartesian philosophy, 62, 68, 94, 96, 98, 100, 102, 104, 106, 119, 122, 124, 138–9, 150, 152, 159, 169, 185, 195, 198–9 post-Cartesian, xiv, 1–2, 11, 61, 94, 169, 171, 190 pre-Cartesian, 116 Casey, Edward, xiv, xvii, 185 Cavalieri, Bonaventura, 95 Caygill, Howard, ix, 187 centers of real action, 127, 135 circles, 25, 41, 45, 50, 57, 59, 74, 88–9, 93, 95, 99, 118, 162, 173, 197 semi-circles, 99 Classical geometry, see geometry cogito, 159, 169–70 see also cognition cognition, 7, 9, 12–13, 15–16, 19–21, 24, 26–7, 30–2, 40, 44–5, 54, 68, 78, 104, 130, 141, 173

Colebrook, Claire, xiv, xvii common notions, 67, 72, 74, 77–9, 81, 87, 89, 97, 124, 192 compass, 4, 25 complexity, 28, 35, 37, 55–7, 59, 72, 79, 89, 97, 102, 125, 155, 191–2 comportment, 63, 87, 89, 141 conatus, 70, 76, 102, 105, 109, 123 concepts, xv, xvii–xviii, 2, 3, 9, 11, 14–15, 17, 21–2, 24–5, 27, 34, 41, 94, 109, 113, 118, 121–3, 136, 145–6, 152, 167, 185 congruency, 19, 196 see also incongruency consciousness, 27, 104–5, 107, 110, 118, 129–30, 132–4, 147, 149, 159, 172, 175, 200 coconscious, 99 see also apperception construction acts of construction, 1, 19, 28, 42, 56, 122 aesthetic, 31, 33 geometric, 14, 25–6, 32, 36, 45, 55, 57, 76, 92–3, 98–9, 114 spatial, 18–19, 22 see also acts; geometry; space continental philosophy, xv–xvi continuity, 6, 8–9, 12, 16, 18–19, 36, 43, 54–5, 59–60, 74, 93, 97, 103, 107, 114–17, 121, 127, 134–5, 137, 140, 144–5, 149, 167–8 see also discontinuity continuum, 9, 28, 48, 57, 59–61, 66, 69, 71, 91, 93–4, 96–7, 101–2, 108–11, 113–18, 131, 144, 149–50, 187, 193 see also continuity contraction, 105, 115 contradictory relations, 56, 111, 127 see also vice-diction corporeal magnitude, 91, 98–9, 103, 124 see also incorporeal magnitude corporeality, 7, 18–19, 27, 66–8, 79, 81, 88, 91, 98–101, 103, 108, 117, 119, 142, 151, 164, 172, 175, 181, 186, 189, 194 see also incorporeality

Index critical, xiii–vi, 1–2, 6, 24, 27, 32, 62, 90, 96, 121, 125, 130, 151–2, 156, 158–9, 170, 181, 186–7, 192, 199–201 pre-critical, 17, 19, 28, 185 critique, xiv–xv, xvii in Bergson, 92, 121–2, 125, 128–30, 136–7, 139–42, 145–6, 151, 194 in Husserl, 154–6, 163, 165, 169–71, 180–2 in Kant, 1–32, 35, 37, 46–7, 69, 75, 79, 97–8, 119, 125, 129–30, 141–2, 150–1 in Leibniz, 68, 100–11 in Spinoza, 68 cross-disciplinary, xiii–xiv Curley, Edwin, 63–5 curves, 93, 95–6, 109, 193, 196 De Morgan, Augustus, 184 de Vogel, Cornelia, 49 Deleuze, Gilles, xiii–xviii, 111, 185–6, 188–93, 195, 199 demonstration aesthetic, 56, 59, 76, 86 geometric, 7, 25, 29, 38, 55, 86–7, 93 practical, 67, 89 see also acts; methods Derrida, Jacques, xviii, 195–6, 198, 200 Descartes, René, x–xi, xiv, 7, 41, 62–8, 84–5, 95, 99, 139–40, 159, 169–71, 189, 191 desire, xvi–xvii, 3, 74, 78, 84, 104, 106–7, 130, 145, 147, 164, 180, 183 see also love diagrams, xvii, 2, 5, 25, 29, 32, 46, 105, 122, 137–8, 142, 149–50, 187–9, 192, 196 dialectic, 50, 53–4, 67–9, 71, 81, 123, 125, 199 diaonetic thinking, 37, 42, 50 see also discursivity difference external, 14–19, 60, 98, 111, 185 geometric, 18, 20, 32, 61, 69, 92, 94, 96, 98, 102, 118, 123, 126, 129 internal, 19, 61, 96, 98, 108, 111, 115

209

irreducible, 19, 103, 109, 132–5, 194, 199 qualitative, 18, 57, 86, 98, 116, 119–24, 193 quantitative, 78 differential, 59, 91, 94–5, 97, 102, 113, 115, 117–18, 193 dimensionality, 17, 51, 170, 187–8, 196 directions, 10, 14, 17–19, 40, 45, 68, 145, 150, 160, 185, 187, 196 disagreement, 130, 193 discrete, 6, 49, 60, 64, 71, 73, 86, 92–3, 95–6, 98, 101, 108, 113, 117–20, 130, 160 discontinuity, 55, 187–8 discursivity, 5–7, 16–19, 29–31, 33–43, 48–57, 59–60, 66–7, 69, 74, 81, 86–8, 117, 119, 124, 128, 142, 148, 150, 160, 166–9, 174, 175, 176, 188 displeasure, 12, 23–4, 26, 28, 47, 189 see also pleasure disruption, 1, 48, 74, 87, 89, 130, 145, 196 divisibility, 7–8, 37–9, 44–5, 50, 52–4, 68–9, 72, 79, 91–2, 96–103, 105, 111, 113–14, 117, 119–20, 123, 138, 192–3 see also indivisibility dizziness, 109 double movement, 35, 40, 93, 138, 149 drawing, xvii, 2–6, 8–10, 19–20, 24, 26, 28–9, 31, 33, 56–7, 89, 141, 143, 146–7, 149, 168, 181–2, 187 see also acts; aesthetics; construction dualism, 94, 96, 125–7, 129–30, 136 Duchamp, Marcel, 196 Dummett, Michael, 185 duration, 2, 11, 70, 76–9, 81, 91–2, 102, 105, 107, 109–10, 120–8, 132–4, 136–50, 152, 159, 175, 180–2 Durie, Robin, 190, 195 dynamic, xiv, xvi, 22–3, 40, 47, 50, 68, 79, 122, 140, 157, 159–61, 174, 180, 182, 186

210

Index

eidetic, xv, 158, 160, 166, 179, 181 elements in Bergson, 127, 133 in Euclid, 5–6, 34–5, 37, 53, 55, 99, 161, 184, 187–8 in Husserl, 167 in Leibniz, 93–5, 97–102, 104, 111, 115–16 in Proclus, 5, 10, 13, 14, 34–7, 42, 45, 50, 54–7, 60, 67–8 in Spinoza, 67–8, 72, 80–1, 84, 87 ellipse, 95 embodied reason, 20, 72, 84, 89, 112, 119 see also embodiment; reason; sense embodiment in Bergson, 125, 128, 140–1, 133, 136, 138, 141–50 in Husserl, 152–3, 161–3, 165, 167, 169, 172–6, 178–83, 194 in Kant, 9–12, 17–18, 20–4, 26–34 in Leibniz, 91, 100–1, 103–8, 110, 112–15, 117–21 in Proclus, 34, 36, 43–5, 47–8, 51–2, 54, 60 in Spinoza, 65–9, 71–2, 74–81, 83–90 emotions, 7, 22, 43, 47, 65–9, 72, 74–90, 97, 103, 124, 192 empiricism, xvi, 12–15, 17, 20, 27, 38, 40, 44, 142, 155, 161–3, 165, 170, 178, 184, 186–7, 199–200 enactments, xi, 2, 5–7, 24–6, 28–33, 89 see also acts encounters, xv–xvi, 2, 7–8, 11, 31–2, 42, 155, 181 endeavour, 70, 109, 147 see also conatus energy, 83, 122, 159, 160, 171 entelechy, 97, 107, 110, 117, 159, 195 see also monad envelopes, 2, 4–5, 8, 93, 115, 120–2, 133, 135, 141, 143, 145–6, 150, 152 essence, 42, 59, 70, 72, 74, 77, 83, 87, 100, 190, 195, 199 ethics, 7, 43, 62–3, 65–8, 82–3, 89, 157, 181–2, 192

Euclid, xiv, 2, 5–7, 34–5, 48, 51, 57–8, 60, 92, 99, 161, 184, 187–9, 191, 196 Evans, Robin, 184 events, 4–5, 135, 141, 143, 152, 161, 164, 177, 198 exactness, 29, 95, 162–5, 181 see also ideality excessiveness, 12, 27–8, 44, 48, 67, 194 existence, 4, 43–4, 50, 64–5, 67, 69–73, 76–8, 80, 82, 96, 100–1, 104, 107, 110–11, 114, 128, 131, 146, 154–61, 164–9, 173–9, 182, 187 see also extensity expansion, 105, 111 see also explication experience aesthetic, xvii, 2, 12, 20–3, 26, 89 embodied, xiii, 5, 10, 15–17, 66, 79, 81–2, 152, 163–5, 199–200 sense, 1, 3, 27, 162, 170 spatiotemporal, xiv, 4, 20, 134, 143, 162 explication (explicato), 39, 166, 169, 174 see also expansion expression, xii–xv, 1–3, 5–8, 184, 187 in Bergson, 122–3, 128, 130–2, 134–7, 139–43, 147, 149, 151–2 in Husserl, 156, 167, 169, 170, 173, 176, 178, 184, 187, 190 in Kant, 14, 20–1, 24, 26, 31 in Leibniz, 93–7, 99–101, 105–6, 109–11, 113, 115–20 in Proclus, 38, 42–4, 49, 52–4, 59 in Spinoza, 62, 65–9, 71–8, 80, 82–9, 192, 198 extensity, 93, 100–2, 119, 121–8, 133, 137–9, 141, 143–4, 146–7, 152, 181 see also existence; inextensivity exterior, 69, 80, 93, 108, 127, 135, 186 faculties, 23, 26–7, 32, 34, 37, 39, 44–6, 51–2, 67–8, 78–9, 107, 137, 141–2, 145–8, 165, 172, 187, 189 feelings, 12, 18, 22–4, 26, 28, 32, 65, 82, 106–7, 133, 170, 172, 186–7, 200 feminist philosophy, xiii, xvi–xvii, 192

Index fictional, 47, 98, 109, 111–12, 114–15, 118, 138 figure-subjects, xiv, 5–6, 8 see also figures figures, xii–xiv, 1–10, 184, 188, 196, 197 in Bergson, 121–3, 125, 128, 132, 135–8, 141–3, 146–8 in Husserl, 156, 160, 167–8, 172, 174–6, 178–83 in Kant, 10, 12–13, 15, 17, 19, 22, 24–6, 28–33, 186–7 in Leibniz, 91–6, 98–102, 104, 107–20, 193 in Proclus, 34–39, 41–8, 50–61, 188–90 in Spinoza, 62–4, 66–8, 77–82, 84–9, 192 see also figure-subjects; geometry finitude, 2, 7–8, 14, 16, 29, 35–6, 61, 69–71, 73–4, 79, 85–6, 92–3, 95–6, 99–103, 112–14, 116, 118–20, 123, 161, 192 see also infinity folding, xiv, 34–7, 42, 47, 59–60, 188, 190, 195 enfolding, 36, 40, 46–8, 57, 79–81, 87, 94, 105, 112, 117, 136, 142, 159, 174, 190–1 see also unfolding forces, 70, 87, 91, 97, 102–6, 109, 115–20, 124, 188–9, 193 forgetting, 6, 10, 51, 62, 122–3, 125, 132, 137–8, 143, 146, 152, 156, 171, 182, 191 forms aesthetic, 1, 34, 156, 180 embodied, 3, 26, 44, 110, 176 geometric, 5, 183, 184 ideal, 3, 16, 20, 35, 38, 46, 51, 155, 175 spatial, 2, 22 Freud, Sigmund, 189 Fried, Michael, 184 fulcrum of action, 128, 137, 140 futural, 105, 182 see also futures futures, 139, 159, 177–8, 180, 182–3, 200

211

Galilei, Galileo, 157–8, 161–4, 170 Gatens, Moira, xiv, xvii Gauss, Johann Carl Friedrich, 196 generatrix, 42 genetic, 5, 7, 12, 16, 18, 28, 34, 36, 39, 43, 48, 50, 74, 86, 88, 94, 114, 117, 119, 149, 152, 155, 159–60, 176, 179 geometry (geometric) definitions, 57, 59, 69, 111, 167, 188 ideas, xv, 1, 4, 50, 109, 155, 160–2, 164–6, 174–9, 181–2, 184, 199 life, 6, 182 objects, 5, 41, 50, 54, 160–1, 167, 173–4 texts, 2, 4, 7, 34, 82, 85, 115–16, 143, 184, 187 see also aesthetics; figures; method; thinking Grosz, Elizabeth, xiv, xvii Gurwitsch, Aron, 199 Heath, Thomas, 184, 187, 196 Hegel, Georg Wilhelm Friedrich, 198 Heidegger, Martin, 187, 198 Henderson, Linda Dalrymple, 196 heterogeneity, 4, 9–10, 13–14, 17, 25, 58, 68, 98, 115, 118, 125, 133, 137, 140, 197 history, xi, xiv, xvii, 121, 145, 151, 154–6, 158–60, 169, 171, 174, 176–7, 180, 185, 199–200 historie, 154–6, 198, 161, 174, 176, 178–9, 181, 184 Hobbes, Thomas, 195 homogeneity, 98, 115, 130–3, 137–8, 140–4, 146, 150, 197 horizons, 2, 4, 5, 8, 11, 115, 136, 153–6, 160, 164, 169, 174–83, 198, 201 Hume, David, 155, 169, 171–2 Husserl, Edmund, xi, xiv, xv, 2, 8, 11, 31, 37, 61, 68, 86, 90–2, 106, 113, 115, 119, 136, 153–83, 184, 191, 198–200 Huygens, Constantyn, 195 hypothesis, 37, 40, 57, 111, 191, 196

212

Index

idealism, 7, 9–10, 26, 35–9, 45, 50–1, 66, 88–9, 95, 113–14, 121, 127–30, 136, 139, 144–5, 149, 155, 157–8, 160–9, 173–5, 178–9, 181, 184, 189, 196–7, 200 ideality, 160–8, 173–5, 179, 181 see also exactness identity, 14, 36, 50, 56, 58, 61, 81, 89, 93–5, 97, 102, 114, 118, 173, 191, 197 imagination, 3, 5–12, 31–8, 41–8, 50–3, 56, 58, 60, 66–8, 71, 74–5, 78–80, 98, 105, 114, 126, 137–41, 180–1, 186, 189, 197 productive imagination, 19, 22–6, 42, 47, 186 immanence, xviii, 12, 28, 34–5, 37, 48–50, 54, 59, 67–71, 74–5, 85–9, 97, 100, 112, 115, 117–18, 122–3, 131–2, 141, 143, 148, 150, 163–4, 170, 185, 187, 190–1, 194, 199 immateriality, 9, 11, 30, 38, 40–1, 45–6, 49–51, 54, 58, 65–9, 71, 78–9, 81, 93, 96, 102–3, 110, 114–17, 119–20, 161, 173, 189 immensum, 100, 102, 113, 94 implication, 36, 40, 44–5, 51, 54, 59, 80, 104, 112, 115, 150, 159, 169, 174, 176, 190–2, 200 implicato, 169, 174 incompossibility, 38, 110–11, 115, 119 incongruency, 18–19, 196 incorporeal magnitude, 7, 91, 98, 103, 106, 113, 124 incorporeality, 106, 110, 119 indetermination, 9, 20–1, 23, 26, 127, 132, 167, 177, 181, 200 indivisibility, 37–41, 44–5, 48, 51–4, 66–73, 78–9, 90–2, 94, 96–8, 100–2, 119, 123, 193–5 inextensivity, 126–7, 134, 137 see also extensity infinitessimals, 102, 113 infinity, 7–8, 14, 187, 190, 196, 199 in Bergson, 121, 123–4, 126, 128, 138, 148–9, 163 in Husserl, 167, 171, 180, 200–1 in Kant, 29

in Leibniz, 91–103, 105, 107–9, 111–20, 188 in Proclus, 35–6, 49–50, 53–4, 59–61 in Spinoza, 64–73, 78–9, 85–8, 190–2 see also finitude inscription, 42, 46 instruments, 23, 25–6, 136 intelligibility, 3, 7, 27, 34–5, 37–9, 43, 45, 47, 50, 52, 54, 58, 60, 187, 190 unintelligible, 39, 151–2, 181 intensity, xv, 7, 14, 23, 26, 28–9, 60, 63, 66, 69, 76, 78–84, 87, 90–1, 93–4, 96–7, 101–9, 111, 113 interior, 69, 80, 93, 107, 125, 135, 142, 151–2, 196–7 intermediary, 11, 15, 34, 38, 50, 56, 61, 66, 75, 81, 92, 94, 122, 124, 126, 130, 136, 144, 149, 188–9 interruption, 48, 87 intersubjectivity, 8, 90–1, 156, 173–6, 178–80, 182, 198, 200 intervals, 52–3, 57, 59, 145, 149 intrapsychic, 174, 176, 178 intuition embodied, 6, 136, 142–3, 147 geometric, xiv, 2–5, 15, 17–18, 23, 30, 92, 144, 150, 162–3, 165, 172, 174, 176, 180 natural, 141, 171, 180–1, 183 pure, 1, 4, 6, 11, 16, 19, 24, 144, 169, 175 sense, xv, 1–2, 4–5, 8–9, 11, 16, 17–18, 24, 142, 161, 164, 169, 171–2, 175, 177, 180–1, 183 spatial, 3, 15, 148 see also sense invention, xii, xiv–xvi, 27, 85, 92, 96, 109, 114, 128, 143, 161, 164, 171, 191 Jena Romantics, 194 Joughin, Martin, 190–1 judgment, 9–11, 13–16, 20–2, 27, 30, 32, 167, 187, 199 reflective judgment, 6, 20–5, 31–3, 153, 159, 172, 186, 200 see also aesthetics

Index Kant, Immanuel, xi, xiv–xv, xvii, 1–35, 37, 44, 46–7, 56, 62, 65, 68–9, 75, 78–9, 90–2, 97–8, 107, 119, 121, 125, 129–30, 136–8, 141–2, 144, 146, 150–3, 155–6, 159, 165, 169–72, 175, 180–7, 193, 199, 200 Kepler, Johannes, 95 kinaesthetic, 172 Krauss, Rosalind, 184 Lacoue-Labarthe, Philippe, 194 language, 54, 160, 166, 169, 173–6, 178–9, 183, 199, 200 Latta, Robert, 95, 115–16 Leibniz, Gottfried, Wilhelm, xi, xiv–xv, 2, 6–7, 11–12, 14, 17, 32, 36–7, 41–3, 47, 49, 52–5, 58–6, 67–9, 74–6, 78, 90–126, 131, 135, 138, 142, 150–1, 155, 163, 171, 175, 180, 182, 185, 188, 190–1, 193–5 lemma, 56–7 see also intervals Lennon, Kathleen, xiv, xviii life, xiii, xvii, 100, 103, 105, 108, 129, 134, 137, 150, 152, 155–7, 170–2, 181–2, 191–2, 198 life-giving, 39–40, 43–5, 48, 116, 119, 121–4, 140–7, 159–63, 174–7, 189 see also living subject limit, xvi, 6–8, 10–12, 16–17, 23–4, 27–9, 32–7, 49–55, 58–61, 66–73, 79–84, 91–6, 98–103, 106–9, 113–14, 117–23, 125–6, 128–9, 132–3, 137–8, 140–3, 151–2, 161–3, 190–1, 197 living body, see bodies living present, 178, 181–2, 195 living science, 8, 157, 160, 167–9, 178 living subject, 1, 4, 8, 36, 66, 72, 83, 117, 121, 122–3, 125–7, 131–3, 139, 141–2, 144, 147, 152, 154–5, 170–3, 175–6, 181, 186, 192, 194 see also life Lloyd, Genevieve, xiv, xvi–xviii

213

Lobachevsky, Nikolai, 196 Locke, John, 155, 170, 172 Lories, Danielle, 200 love, 43, 74, 78, 84 see also desire magnitude, 7–8, 14–15, 17, 22–3, 28–9, 45, 49–50, 56–9, 69, 78, 91–103, 105–11, 113–15, 117–20, 123–4, 126, 155–6, 186, 193–5 see also corporeal magnitude; incorporeal magnitude Maïmon, Salomon, 193 manifold, 40, 109, 130, 141, 171, 186, 196 Massey, Doreen, xvii Massumi, Brian, xvii, 187 materialism, 9, 11, 30, 38, 40–1, 45–6, 49–51, 54, 58, 65–9, 71, 78–9, 81, 93, 96, 102–3, 110, 114–17, 119–20, 161, 173, 189 matter extended, 10–13, 15–19, 29, 32, 38, 40, 44–7, 50–3, 56, 58–9, 66–7, 69–76, 79–81, 87–9, 93, 96–7, 100–2, 106, 115–16, 121, 123, 125–6, 128–9, 131, 133, 137, 139, 150, 181, 192, 199 unextended, 10–11, 13, 15–17, 19, 50, 56, 66–7, 69, 71–3, 79, 89, 91, 93, 97–8, 101–2, 106, 119, 121, 123, 125–8, 133, 137, 139, 181, 194 mathematics, xiv, 11, 13–16, 32, 35, 37, 38–44, 47–50, 94–6, 111, 141, 158, 161, 165–6, 170, 189, 196–7 measurement, xi, 2–3, 14, 39, 57, 59, 109, 127, 137–9, 158, 162–3 mechanics, 2, 18, 25, 40, 44, 69, 94, 96, 102, 104–6, 108, 116, 127, 139, 161, 186 memory, 5–6, 8, 10, 28–33, 42, 68, 80, 92, 101, 103–7, 110, 112–15, 117, 120–8, 130, 132–43, 147–50, 152, 159, 166, 181–2, 188, 194–5 see also duration Merleau-Ponty, Maurice, 198

214

Index

metaphysics, 2, 4, 8, 11, 13, 16, 31, 33–7, 39, 48–9, 51–2, 60, 65–9, 71, 82, 89, 94–6, 102, 112, 120–3, 125–6, 128–9, 134, 136–43, 145–6, 150–3, 157, 166, 169–70, 187–9, 193 method, 1–2, 4–8, 10–12, 14–15, 25–6, 28–35, 37, 39, 41, 43–4, 48–9, 50, 52, 54–7, 59–69, 71–2, 74–8, 80–2, 84–99, 101, 103 107–8, 111, 113, 116–26, 138–40, 142–3, 147, 150–1, 155–61, 163–70, 172, 174–5, 177–9, 181–5, 187–8, 191–3, 197, 200 see also geometry mind, 62, 64–7, 70, 72, 77, 79–81, 83–4, 87, 90, 92–4, 96–7, 100, 104, 107, 110, 112, 116, 126, 129–30, 132, 139, 142, 145, 147, 149–50, 191, 197–8 mixture, 48, 54, 59–61, 78, 96, 98, 111, 133, 136, 189 modern thinking, x–xi, xiii, 8, 136, 140, 142, 145–6, 150–1, 155–8, 161, 163–4, 166, 169, 171–2, 175, 181–2, 191–2, 199 pre-modern, 122, 152 modes, xiii, xvii, 1–2, 5, 7, 10, 14, 17, 28, 55, 103, 105, 118, 120, 122–3, 141, 183 in Spinoza, 62, 64–5, 67–74, 76, 78, 80–1, 83–7, 89–90, 92, 96–7, 108, 190, 192 monad, 7–8, 52–3, 69, 76, 91, 93–4, 97, 99–110, 112–20, 124, 135, 193–4 see also entelechy Mueller, Ian, 43, 189 Mullarkey, John, 197–8 multiplicity, xiii, xv, 8, 37, 44, 49, 56, 59–60, 67, 69, 74, 80, 82–3, 87, 96–7, 98–9, 102–3, 109, 115, 119, 144, 192, 197, 199 Nancy, Jean-Luc, 194 nature, 18, 21–2, 24–5, 28–31, 42–6, 57–8, 66–74, 78, 82, 85, 87–9, 99, 103, 112, 118, 123, 129, 151, 161–3, 173, 186

neo-Platonic philosophy, 1, 6, 11, 32, 35, 37–9, 44, 48, 58, 69, 80, 87, 94, 97–8, 121, 188–9, 192 net, 113–14, 120, 197 Newton, Issac, 193, 195 Nikulin, Dimitri, 189–90 nous, 36–46, 50–2, 54, 60, 142, 159 ontology, xiii–xvi, 1–4, 6, 8, 48, 71, 102, 122–3 operation, see acts; geometry optics, 40, 44, 190–1 orientation, 17–18, 152, 187 origination, 13–14, 23, 27, 31–2, 35–6, 39, 41, 43–5, 47, 49–51, 60, 80–1, 94, 101, 113, 116, 143, 151, 156–68, 171–4, 176–82, 187, 200 parallels, 67, 74, 80, 96, 125 parts, xviii, 14, 17–19, 46, 51–4, 58–9, 71, 79, 82, 89, 92–3, 96, 98–102, 105, 107–9, 114–16, 123, 135, 128–9, 131–5, 139, 148, 154, 185–8, 192, 198 see also unities passages, 2, 4, 5, 7, 25, 43, 62, 66–8, 74–5, 77–8, 80, 82, 84–7, 89–90, 101, 105, 116, 132, 138–9, 160 passions, 7, 76–7, 87, 192 see also emotions perception, 8, 12, 15, 18–20, 23, 26–7, 32, 38–41, 44, 53, 75, 80–1, 91–4, 97–8, 103–7, 109–14, 118–24, 126–37, 139, 141–4, 147, 149–50, 155, 163, 165, 175, 186, 194, 197, 199 see also apperception; sense percepts, xv phenomenology, xvi, 3, 8, 154–6, 169, 172, 180–2, 198–200 planes, 18, 51, 54, 87, 92, 131, 137, 144, 149, 188, 191–2, 196–7 Plato, x–xi, xiv, 6–7, 10, 28–33, 35, 38, 40–3, 48–50, 56, 117, 181, 189 Platonic philosophy, 152, 161, 184, 189 pleasure, 12, 23–4, 26, 28, 47, 66, 75, 77, 79, 189 see also displeasure plena, 162–3, 171

Index plenitude, 43, 50, 52, 54, 76, 81, 93, 100–1, 112–17, 119–20, 163 plenums, 2, 4–5, 8, 91, 93–4, 102, 113–20, 193, 195 Plotnitsky, Arkady, 195 plurality, 40, 59, 109 point, 14, 37, 51–6, 99–100, 102, 135, 190, 196–7 politics, xvii–xviii, 192 polygon, 95, 111, 114, 193 porism, 56–7 post-Cartesian, see Cartesian philosophy post-Kantian, see Kant postulates, 37, 55–6, 63, 68, 111, 196 potentiality, 4, 7, 18, 22, 24, 26, 35–7, 42–3, 52, 55, 58, 65–6, 69–70, 75–6, 78, 81–2, 100, 118, 123–4, 130, 145, 152, 156–7, 163–4, 167–9, 172–3, 179–81, 190, 195, 199, 201 powers, xvii, 1, 3, 5–7, 9–12, 19–22, 24–8, 31–8, 40–9, 51–2, 58, 66, 69, 71–2, 74–6, 78–81, 85–6, 89, 96–8, 100, 104, 113, 118–19, 130–1, 142–3, 147–8, 150–1, 168, 179, 186, 189 praxis, 6, 8–11, 70, 77, 114, 118, 142, 154, 156, 165–8, 173, 175–7, 179–80, 182–3 problems, x–xi, xv, 15, 37, 55–7, 11, 121–2, 138, 155, 158, 160, 168, 175, 177–2, 193–4 procedure, see geometry Proclus, xi, xiv–xv, 2, 5–7, 11, 26, 29, 33–62, 65–8, 71, 74, 78–9, 86–7, 90, 94, 99, 105, 108–9, 117, 119, 123, 137, 142, 159, 161, 169, 188–90 progressive philosophy, 70, 89, 118, 121–2, 124, 143, 145, 149, 152–3, 161, 181 projections, 26, 41–2, 45–7, 171 proofs, 18, 56–7, 70, 83, 86, 88, 111 properness, 77, 145–6, 151–2, 159, 180–1, 187, 200 proposition, 16–17, 25, 29, 35, 37, 55–6, 64, 71, 75, 77, 79, 82–3, 86–8, 111, 125, 167, 199

215

psychic, 5, 8, 28, 39, 43, 48, 51, 60, 68, 72, 76, 92, 101, 103–4, 121, 124–7, 129–30, 12, 147, 149–50, 152, 158–60, 170, 173–7, 178, 181–2, 197 psychology, 47, 60, 65, 67, 82, 96, 106, 124, 147, 186, 189, 194, 199 purposiveness, 21, 23, 158, 160, 180–1, 186 Pythagorean philosophy, 6, 33, 35–7, 43, 45, 48–50, 52–3, 58–60, 65, 87, 95, 98 qualitative, xvii–xviii, 7, 59, 83–4, 91, 93–4, 96, 98, 100–1, 103, 105–6, 110, 116, 118–19, 124, 136, 138, 140, 181, 193–4 quantitative, xviii, 88, 98, 100–1, 110, 116, 118, 123, 125, 132–3, 142, 193 Rajchman, John, xiv, xviii ratio, 41, 49–50, 60, 75, 77, 90, 93–4, 98, 101–3, 106, 109–10, 113, 119, 124, 126, 177 rationalism, xvi, 7, 34, 41, 107, 110, 112, 118, 126, 129, 142, 157, 161, 163–4, 169–71, 182, 191 reactivation, 154–6, 158, 160, 166–8, 172, 174, 179 reading, 7, 62, 66, 82, 185–6, 200 reality, xiii, 2, 17, 27–8, 37–40, 47, 49–50, 64–5, 72–3, 77, 83–4, 86–102, 116, 121–2, 127, 129–31, 134, 138, 140, 143–6, 149, 172, 187, 195, 198 reason embodied, 20, 22, 72, 84, 89, 112, 119, 180 geometric, 5, 17, 40, 32, 155–7, 160, 182 pure, 2–3, 9–17, 1–20, 30, 113, 151, 163, 169, 171 sense, 4–6, 8, 20, 28, 124, 153, 155–6, 158–9, 164–5, 169–70, 172, 175, 177–83 sufficient, 47, 75–6, 88, 93–4, 101, 104, 106–7, 109–13, 115–19, 124

216

Index

recollection, 6, 8, 10–22, 26, 28–33, 36, 42–3, 47, 125, 136–7, 147, 155, 160, 174, 177, 183, 188 reflective subject, 1–5, 9–12, 15, 21–4, 26, 28, 31–2, 34, 44, 47, 69, 85, 90–1, 98, 119, 129–30, 136, 180–1, 184, 200 see also judgment relations, xiii–xvi, xviii, 1–4, 8, 184 in Bergson, 121–7, 129–53, 196 in Husserl, 154, 159, 161, 163–4, 170, 172–8, 180–1, 198 in Kant, 9–25, 28, 30–1, 185–8 in Leibniz, 93, 95, 97–101, 104–6, 108–18, 193–5 in Proclus, 34–7, 39, 41, 44–6, 51–2, 58, 188 in Spinoza, 62, 66, 68–71, 74–6, 78, 80–9, 191–2 repetition, xi, 1, 125, 134, 136, 151, 158, 160–1, 166, 173–4, 179, 183, 190 reproduction, 27, 40, 43, 167, 179, 186, 189 retrieval, 1, 6, 8–10, 19, 29, 32–3, 115, 123, 132, 136, 141–2, 145–6, 152, 154, 157–9, 161, 165, 168–9, 172, 177, 180–3 Ricoeur, Paul, 199–200 Riemann, Bernhard, 140, 195–6, 199 Rowe, Colin, 184 ruler, 3, 25 sand, 10, 29, 31, 42, 148 scholia, 24, 48, 62–3, 67–8, 71, 73, 82, 85, 87–8, 104, 185, 192, 197 self-evidence, 166–8, 171, 173–4, 178–9 self-same identity, 14, 98, 160, 164, 166 self-sufficiency, 52, 58, 60, 66, 163, 166 sense geometric, xiv, 2, 4–5, 8, 136, 154, 156, 160–1, 165–9, 172–83, 191 ideas, viii, 5, 8, 68, 154, 156, 158, 161–2, 167, 172–5, 177–83 intuition, xv, 1–2, 4–5, 8–9, 11, 16–18, 24, 142, 161–2, 164, 169, 171–2, 174–5, 177, 180–1, 183

perception, 5, 8, 32, 38, 41, 44, 67, 81, 134, 144, 172, 174, 183 see also reason sensibility, xvi, 2–3, 11–13, 16–19, 21–2, 24, 26–8, 32, 65, 67, 78, 82, 84, 96, 114, 141, 148, 155–6, 170–2, 181–2, 186 sexed subject, xii–iv, 3 shape, 29, 44–5, 49, 51–3, 59, 114, 156, 162–4, 184 simultaneity, 15, 36, 52–3, 160, 163, 166, 170, 173 Sklar, Lawrence, 196 Smith, Daniel, 186 smooth space, xv Socrates, 6, 8, 10, 28–31, 33, 42 soul, 7, 10, 15, 30–1, 34, 36–47, 49–53, 58, 60, 66, 82, 93, 95, 97, 100–1, 103–7, 109–10, 112–13, 116–19, 121, 124, 126, 133, 136–7, 139, 142, 151, 170, 187–9, 194–5 space, xii–xviii, 4, 44, 48, 52–3, 93, 108, 113–18, 158, 160, 162, 164, 180, 193, 195 in Bergson, 122–8, 130–53, 196 in Kant, 1–3, 6, 9–13, 15–20, 22–8, 31–2, 185–7 space-time, 132, 136, 140, 156, 164, 196 spatial thinking, xiii–xvii, 9–11, 15, 19, 118, 142–3, 148 spatiotemporality, xv, 2, 4–5, 9–13, 17, 26, 114, 116–17, 121–2, 128, 131, 144, 146–7, 149, 152, 162–5, 167, 173, 175–7, 180–1, 194 Spinoza, Baruch, xi, xiv–xv, 2, 7, 11, 26, 32, 36–7, 41, 43, 47–8, 54–5, 58–9, 61–92, 94, 96–7, 101–3, 105–9, 116, 120–6, 141–2, 150–2, 155, 175, 180, 182–3, 185, 190, 191–2, 194, 197 spirit, 127, 133–4, 139, 155, 157–61, 164, 173–5 Stella, Frank, 184 Stengers, Isabelle, 192 stoic philosophy, 6, 10, 12, 29, 34–5, 45, 68, 96, 119, 161, 163

Index subjectivity, xiii, xvi–xviii, 1, 5, 7–8, 10, 66, 133, 154–5, 171–2, 174, 178, 180, 182, 200 see also intersubjectivity sublime, 12, 22–3, 25, 27–8, 98–9 substance, 36, 43, 51, 54, 64–74, 77–82, 84–6, 88–9, 92–3, 96–106, 108–10, 112–13, 115–16, 119, 123, 130, 190–1, 193–5, 197–8 sufficient reason, 47, 75–6, 88, 93–4, 101, 104, 106–13, 115–19, 124 see also reason superior empiricism, 39, 182 surfaces, 18, 92, 133, 135, 144, 188, 196 synthetic, 6–7, 12–16, 18–19, 21, 27, 32, 34, 36, 47, 55–6, 59–63, 65–6, 68, 73–4, 76, 78, 86, 90–1, 94–100, 111, 123–4, 156, 165, 172, 181, 185, 192 techne, 2, 21, 25 technical activities, 1–3, 5–7, 10, 21–6, 28–9, 31–3, 35, 42, 46, 56, 66, 136, 156, 162–5, 169, 172–3, 180–1 teleology, xi, 8, 23, 97, 154–61, 164–5, 172, 175, 177, 181–2, 196, 198, 201 temporality, 6, 8–11, 70, 77, 114, 118, 142, 154, 156, 165–8, 173, 175–7, 179–80, 182–3 see also spatiotemporality theorem, 37, 54–7, 63, 197 thinking, xiv–xv, 1–3, 5, 10, 11–12, 15, 17, 24–5, 43, 52, 56, 62–3, 65, 68, 83–4, 91, 94, 99, 103, 108, 121, 141, 146, 151, 153–6, 159–62, 165–9, 172, 174–80, 182–4 see also method; modern thinking thresholds, 82, 100, 169 time, xiii, xvii–xviii, 187 in Bergson, 122–5, 127–8, 132, 134, 136–9, 146–7, 149–53, 197 in Husserl, 156, 158, 164–5, 180 in Leibniz, 114, 118–19, 193 in Kant, 2–4, 9–12, 15–17, 20, 27, 30, 47 in Proclus, 44, 48, 52–3, 188

217

in Spinoza, 86 see also space-time; spatiotemporality topology, xiv, xvii, 8, 93, 120, 123–4, 135–6, 140, 143–4, 152, 194–7, 199 totality, 38–49, 131, 161, 166, 169–70, 173, 200 tradition, x, xv, 1–2, 11, 68, 121, 140, 146, 150, 153, 155–8, 160–1, 167–9, 172, 177, 182, 192 transcendental, xiii–xv, 2–4, 8–9, 11–13, 16–17, 19–21, 23, 27–8, 36–7, 47, 50, 59, 66, 68–71, 76, 81, 86, 92, 124, 142, 154–7, 161, 166, 171–5, 178, 180–3, 186–7, 189, 194, 199–200 triad, 53–4, 72–3 triangle, 13, 56, 59, 86, 88, 148, 162, 173 understanding, xii, 7, 10, 12, 15, 17, 19–23, 25, 27, 30, 32, 34, 36–47, 50, 53–5, 64, 66, 78–9, 82, 130, 146, 152, 157, 185–9 see also faculties unfolding, 5–6, 10, 30, 34–51, 54, 58–60, 62, 66–7, 72, 79–81, 94, 105, 119, 136–7, 159–60, 174, 188, 190–1 see also folding unities, 18, 26, 32, 36, 38, 51, 53, 58–9, 61, 65–7, 72–81, 83–4, 92, 96–7, 99–103, 105–10, 114–17, 119, 122–4, 127, 130–2, 139, 141, 145, 148, 168, 186, 192, 194, 200–1 universal, 18, 21, 41, 50, 55–6, 79, 87, 113, 115, 138–9, 155, 158, 177, 190, 197 univocal, xv, 68, 71–4, 80, 97, 101, 194, 199–200 unlimit, 6–7, 10, 12, 29, 34–7, 42, 44–5, 48–52, 54–5, 58–60, 66–72, 74, 76–7, 81, 86, 91–2, 94–5, 98–103, 123, 125–6, 142, 161, 188, 190

218 Index vice-diction, 52, 101, 110–11, 115, 195 violence, 145, 147 virtuality, xvi–xviii, 104–5, 112, 123, 127, 131, 135, 137–9, 148–9, 194, 197, 200 visual arts, xiv, 3, 10 see also art vitality, 26, 85, 144, 148–9, 152, 158, 160, 163, 168, 176–7 see also life voice, xvi, 87 void, 114, 116, 194–5 Wittgenstein, Ludwig, 189 Wood, David, xviii

world, x, xvi, 3, 5, 8 in Bergson, 123, 127, 129, 131–2, 135, 141, 149–50, 152 in Husserl, 154–5, 162–5, 169–73, 175–7, 180, 182–3, 198–200 in Leibniz, 104, 112, 114, 116–17 in Kant, 9, 12, 15, 17–18, 20–2, 30, 186–7 in Proclus, 37–41, 43, 45–6, 48–9, 52, 54, 56, 60 in Spinoza, 67, 80, 82 Zeno, 195