Atomism in Philosophy: A History from Antiquity to the Present 9781350107496, 9781350107526, 9781350107502

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ATOMISM IN PHILOSOPHY

Also available from Bloomsbury Skepticism: From Antiquity to the Present, edited by Diego Machuca and Baron Reed The Philosophy of Knowledge: A History, edited by Stephen Hetherington The Bloomsbury Companion to Analytic Philosophy, edited by Barry Dainton and Howard Robinson

ATOMISM IN PHILOSOPHY

A HISTORY FROM ANTIQUITY TO THE PRESENT Edited by Ugo Zilioli

BLOOMSBURY ACADEMIC Bloomsbury Publishing Plc 50 Bedford Square, London, WC1B 3DP, UK 1385 Broadway, New York, NY 10018, USA 29 Earlsfort Terrace, Dublin 2, Ireland BLOOMSBURY, BLOOMSBURY ACADEMIC and the Diana logo are trademarks of Bloomsbury Publishing Plc First published in Great Britain 2021 This edition published in 2022 Copyright © Ugo Zilioli and Contributors, 2021 Ugo Zilioli has asserted his right under the Copyright, Designs and Patents Act, 1988, to be identified as Editor of this work. For legal purposes the Acknowledgements on p. ix constitute an extension of this copyright page. Cover image: Democritus, an influential Ancient Greek pre-Socratic philosopher, painted by Diego Velazquez in 1640 © Aurelian Images / Alamy Stock Photo All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage or retrieval system, without prior permission in writing from the publishers. Bloomsbury Publishing Plc does not have any control over, or responsibility for, any third-party websites referred to or in this book. All internet addresses given in this book were correct at the time of going to press. The author and publisher regret any inconvenience caused if addresses have changed or sites have ceased to exist, but can accept no responsibility for any such changes. Library of Congress Cataloging-in-Publication Data Names: Zilioli, Ugo, 1971- editor. Title: Atomism in philosophy : a history from antiquity to the present / edited by Ugo Zilioli. Description: London ; New York : Bloomsbury Academic, 2020. | Includes bibliographical references and index. Identifiers: LCCN 2020027555 (print) | LCCN 2020027556 (ebook) | ISBN 9781350107496 (hb) | ISBN 9781350107502 (ePDF) | ISBN 9781350107519 (ebook) Subjects: LCSH: Atomism--History. Classification: LCC BD646 .A854 2020 (print) | LCC BD646 (ebook) | DDC 146/.5--dc23 LC record available at https://lccn.loc.gov/2020027555 LC ebook record available at https://lccn.loc.gov/2020027556 ISBN: HB: 978-1-3501-0749-6 PB: 978-1-3503-5505-7 ePDF: 978-1-3501-0750-2 eBook: 978-1-3501-0751-9 Typeset by Deanta Global Publishing Services, Chennai, India To find out more about our authors and books visit www.bloomsbury.com and sign up for our newsletters.

CONTENTS

List of contributors viii Preface and acknowledgements ix Abbreviations and transliterations xi General introduction Ugo Zilioli

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Part I Atomism in ancient philosophy 1 Early ancient atomism: Similarities and differences Andrew Gregory 2 The reception of atomism in ancient medical literature: From Hippocrates to Galen Vincenzo Damiani

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3 Why aren’t atoms coloured? David Sedley

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4 Atoms and minimal ‘Parts’: The originality of Epicurean atomism Francesco Verde

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5 Atoms and universals in Epicurus Attila Németh

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6 Atoms, complexes and simples in the Theaetetus 113 Sophie-Grace Chappell 7 Atomism in Plato’s Timaeus 136 Luca Pitteloud

Part II Atomism in non-Western, medieval and modern philosophy 8 Atoms and orientation: Vasubandhu’s solution to the problem of contact Amber Carpenter, with Sherice Ngaserin

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9 Aggregates versus wholes: An unresolved debate between the NyāyaVaiśeṣika and Buddhist schools in ancient Indian atomism Sahotra Sarkar

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10 Atomism and Islamic thought Francesco Omar Zamboni

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11 Atoms and time I Charles Doyle

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12 Atoms and music in late medieval philosophy Philippa Ovenden

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13 Atomism and the Cambridge Platonists Adrian Mihai

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14 Atomism and society in William Petty Akos Sivado

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15 Atoms, colours and God in Leibniz Alberto Artosi

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Part III Atomism in contemporary thought Section One  Philosophy 16 Logical atomism and Wittgenstein Annalisa Coliva

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17 Atomism and semantics in the philosophy of Jerrold Katz Keith Begley

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18 Atoms and knowledge Nick Treanor

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19 Atoms and time II Mauro Dorato

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20 Atomism and Marxism in Louis Althusser Panagiotis Sotiris

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21 Atomism and liberalism Philip Krinks

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Section Two  Metaphysics 22 Atoms as universals Matthew Tugby

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23 Atoms and extended simples Travis Dumsday

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24 Power gunk, or unlimitedly divided powers Anna Marmodoro and Andrea Roselli

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25 Atoms and tropes Peter Simons

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Section Three  The sciences: Physics and chemistry 26 Atomism and physics-based structuralism Matteo Morganti

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27 Atoms and chemistry I: Not a success story Paul Needham

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28 Atoms and chemistry II: Trusting atoms Robin Findlay Hendry

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Name Index 489 Subject Index 491

CONTRIBUTORS

Alberto Artosi, University of Bologna, Italy. Keith Begley, Trinity College Dublin, Ireland. Amber Carpenter, Yale University at Singapore and York University, Singapore and UK. Sophie-Grace Chappell, Open University, UK. Annalisa Coliva, University of California, Irvine, USA. Vincenzo Damiani, University of Ulm, Germany. Mauro Dorato, University of Rome III, Italy. Charles Doyle, NUI Galway, Ireland. Travis Dumsday, Concordia University of Edmonton, Canada. Andrew Gregory, University College London, UK. Robin Findlay Hendry, University of Durham, UK. Philip Krinks, Centre for Theology and Community, London, UK. Anna Marmodoro, University of Durham, UK. Adrian Mihai, University of Cambridge, UK. Matteo Morganti, University of Rome III, Italy. Paul Needham, University of Stockholm, Sweden. Attila Németh, Institute of Philosophy, Research Centre for the Humanities, Budapest, Hungary. Sherice Ngaserin, University of Michigan, USA. Philippa Ovenden, Yale University, USA. Luca Pitteloud, Universidade Federal do ABC, Sao Paulo, Brazil. Andrea Roselli, University of Durham, UK. Sahotra Sarkar, University of Texas, Austin, USA. David Sedley, University of Cambridge, UK. Peter Simons, Trinity College Dublin, Ireland. Akos Sivado, Hungarian Academy of Sciences, Budapest, Hungary. Panagiotis Sotiris, Hellenic Open University, Athens, Greece. Nick Treanor, University of Edinburgh, UK. Matthew Tugby, University of Durham, UK. Francesco Verde, Sapienza University of Rome, Italy. Francesco Omar Zamboni, Scuola Normale Superiore, Pisa, Italy.

PREFACE AND ACKNOWLEDGEMENTS

This collection grows out from an International Conference on atomism in philosophy that I organized at Hatfield College, Durham, in May 2017, under the auspices of a Marie Curie Fellowship (both the conference and this collection are the main outcomes of the Marie Curie Fellowship n. 656480, Acronym AWWO, 2015–2017). The success of – and the lively discussions at – the conference easily persuaded me that a new exploration and a fresh assessment of the impact and legacy of atomism in philosophy were urgently needed. I was encouraged by the philosophy team at Bloomsbury, Colleen Coalter in primis, to complement the seven original speakers at the conference with other scholars. I thus spent some time recruiting a much larger team of contributors for the collection. I did so either by personal invitation or with a specific call for papers. In both cases, I tried to include contributors coming from different philosophical traditions, as well as scholars at different stages of their career. I am happy to say that the twenty-eight contributors of this collection now represent a varied group of scholars capable of depicting in detail and with perceptiveness the philosophical landscape of atomism in the history of thought, from antiquity to the present time. As the sole editor of such a large collection, the list of people I wish to thank for their help in carrying out such a demanding, yet immensely pleasurable experience is quite long. First of all, I thank all the contributors, who originally accepted my invitation to contribute and who have always been very receptive to all the requests I had for them. I learned immensely from each of them – and not only on atomism. I thank Lucy Manning, Enrico Piergiacomi, Giuseppe Feola, Antonio Di Meo, for having helped the project take shape with their efforts. Robin Hendry and Matthew Tugby have been supportive throughout all the stages of the project, initially by giving papers at the conference in Durham in May 2017, then by suggesting possible names to contact for the collection. I also warmly thank the European Union for having funded the research that gave origin to this project (and to another one still in progress). In the final stages of the revision, which coincided with the Covid pandemic, I was helped by a Grant from the Society of Authors, UK, which I am happy to thank. At Bloomsbury, I wish to thank Colleen Coalter, the best philosophy editor I have ever met, and Becky Holland, the best editorial assistant ever. They have overseen the project with great care and unwavering support. I also warmly thank Zoe Jellicoe and Mohammed Raffi for their hard-copy-editing work on the collection. They both were excellent companions in the final stages of revision of the entire typescript. I also warmly thank the reviewers for their support and suggestions. Ours is an age in which much should be said on how to construct – and foster – a much better, more fruitful dynamic between philosophy and academia. This is not,

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however, the right place to do so: the topic would require at least a book in itself!1 What I wish to highlight here is just one aspect of the whole question. I firmly believe that to carry out her or his philosophical activity with success, pleasure and profit, a philosopher needs to be supported by fellow philosophers and nonphilosophers alike. For different reasons – and it would be too long to explicate them at a fuller extent in each case – over the years and for this project, I have been supported by a large number of friends, philosophers and non-philosophers, who I here wish to thank in a random order: Voula Tsouna, Amber Carpenter, Katherine O’Donnell, Chris Gill, Anna Chahoud, the late Mary Midgley, Alberto Artosi, Mauro Tulli, Aldo Brancacci, Livio Rossetti, Joseph Margolis, Sophie-Grace Chappell, David Sedley, Matt Duncombe, Francesco Verde, Michele Zamboni and his family, Giancarlo ‘Spino’ Chittolini, Nigel and Heather Speight, Heather and Christopher Rowe, Rosie and Nick Garland, all the staff at Tealicious and Crook Hall Cafes (Durham), Fabio and Monica Bonatti, Barbara and Giuliano Copellotti, Roberto Rizzi, Maurizio Pratizzoli, Paolo Giordani. Above all, I wish to thank wholeheartedly my family: my wife Cristiana and our children, Zoe and Delio (and Eros). Without them, I wouldn’t be able to live the kind of life I had always wanted to live: I couldn’t be a philosopher either. Durham (UK)/ Casa di Margherita (Italy)

NOTE 1. The reader interested in the topic could read with much profit Mary Midgley’s last book, What Is Philosphy For?, London: Bloomsbury, 2018.

ABBREVIATIONS AND TRANSLITERATIONS

Authors usually follow the abbreviations, for ancient Greek authors and works, of the Greek-English Lexicon (LSJ), Oxford 1996. For Latin authors and works, they usually refer to P. Glare (ed.), Oxford Latin Dictionary, Oxford 1995. Other specific abbreviations are provided by authors in their papers. Transliterations from ancient Greek and Sanskrit are usually made according to the traditional rules, as these are adopted in the LSJ for ancient Greek and in the English-Sanskrit Dictionary by M. Monier-Williams, New Delhi 2006.

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General introduction UGO ZILIOLI

THIS COLLECTION: SCOPE AND AIMS Atomism is a philosophical term that has some circulation even outside philosophy. If you look it up in the Oxford Dictionary, atomism will be described as ‘a theoretical approach that regards something as interpretable through analysis into distinct, separable, and independent elementary components’ (Oxford Dictionary of English, second edition 2005, entry ‘atomism’). If you look for an entry on atomism in the Stanford Encyclopaedia of Philosophy, you won’t find a single, unique entry on atomism. There are very detailed entries on different kinds of atomism, but no comprehensive definition of atomism in philosophy.1 The Oxford Dictionary gives us a very general definition of atomism that, despite its generality, captures well enough how atomism is understood in the context of the present collection. The different contributors to Atomism in Philosophy: From Antiquity to the Present provide insightful chapters on the different ways in which one ‘thing’ can be analysed and interpreted into separable, more fundamental elements. These elements are ‘atoms’, that is, primary elements that cannot be further reduced to anything more essential.2 The chapters in the collection look at the historical development of atomism since antiquity (when it made its first appearance on the philosophical scene with Democritus and Leucippus) to our times (when it is best understood as a family of interrelated views, widely debated in philosophy and in the sciences). Not only in the collection is atomism to be approached historically, but also – and perhaps more essentially – it is explored from different philosophical angles. The contributors deal with atomism in a variety of philosophical areas: epistemology, metaphysics, ethics, philosophy of language, philosophy of science, philosophy of time, non-Western philosophies, philosophy of music, philosophy of society, philosophy of chemistry, philosophy of physics. The historical breadth of investigation, paired with the variety of philosophical angles through which atomism is here being explored, makes it difficult to provide the reader with a possible definition of atomism that is less general than the one given by the Oxford Dictionary. In addition, some contributors advance some reservations on atomism as a view that (still) has some conceptual legitimacy to commend nowadays. As the editor of the collection, since the very beginning of this demanding but truly satisfactory enterprise, I have adopted a very ecumenical approach. I really wanted the contributors to deal autonomously with the various, often different, at times conflicting ways we come to terms with ‘atomism’ in philosophy. On this aspect, there is one comparison that can help the reader understand the best approach to adopt when reading this collection. In the Preface of the Philosophical Investigations,

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Wittgenstein laments that he was unable to thread his thoughts into a single, coherent whole. He warns the reader to take the sections of the Investigations as ‘a number of sketches of landscapes which were made in the course of [. . .] long and involved journeyings. The same or almost the same points were always being approached afresh from different directions, and new sketches made [. . .]. Thus this book is really only an album’ (Philosophical Investigations, vii). To some extent, the readers of this collection are in the same situation as are the readers of Wittgenstein’s Philosophical Investigations. They will find plenty of sketches showing the philosophical and historical relevance of atomism in one given period or in one specific area of thought. Each sketch has a connection or familiarity with other sketches: between some of them the familiarity is more pronounced, between some others this is less the case. Yet, all the sketches provide readers with a series of landscapes. These latter are the best description the contributors of the collection and its editor could offer for the elusive view that is commonly referred to with the term ‘atomism’. In short, there is no single painting available for atomism, but many sketches aiming to depict it in its different forms and variants. Some words are also to be said about the very nature of these sketches. I haven’t asked the contributors to write merely introductory papers, namely papers aimed at introducing the reader to the main features of atomism as this is discussed in a certain period or philosophical area. Each contribution does also this – the main emphasis in most papers is, however, on drawing an original sketch for atomism, which vividly depicts its importance in a certain historical period or philosophical debate. In other words, the reader is to expect rather colourful and original sketches from the contributors. At the same time, not all the kinds of atomism that have been dealt with throughout the whole history of philosophy are explored in the collection. We have tried to be as fully comprehensive as possible – and, indeed, the terrain we have covered is large and extended. There are some inevitable gaps, the most relevant of which I signal in this introduction.

VARIETIES OF ATOMISM Atomism was born as a metaphysical thesis according to which reality is made up of indivisible entities that are not further reducible to a more elementary level. As such, it was first put forward by Democritus and the elusive Leucippus in ancient Greece. Over the centuries, it became a more ‘scientific’ thesis, when one extended it (or indeed, when one restricts it) to matter. But what nature do atoms really have? What are they like? What does it mean to say that they are the ultimate level of reality and matter? What does current speculation in physics and chemistry say about atoms? If it is a metaphysical view, atomism will also represent a position in the philosophy of language. Since reality is made up of atoms, there is a temptation to conceive of words as referring to those atoms, to be properly and legitimately meaningful. This temptation already acutely affected Plato and his contemporaries (as the Theaetetus witnesses) and, closer to us in time, Wittgenstein too (in the Tractatus Logico-Philosophicus – but not only him: see also the chapter by Begley on

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Katz’s semantics). Is a semantic theory based on linguistic atomism a viable option in philosophy of language? Atomism is also an ethical view. Given that reality is made up of atoms that are always in motion according to some physical rules – as some ancient philosophers thought, is there any room for our actions to be free and not already determined? In other words, is an atomistic universe a place where everything is already determined by the laws of physical causality? Some believe that Democritus thought so, while others highlight the introduction of the notion of swerve into the picture by the Epicureans. The swerve (literally, ‘deviation’) would allow atoms not to follow any pre-fixed trajectory, so that everything cannot be taken as predetermined by physical causality. But atomism is not only a metaphysical, scientific, linguistic or ethical view but also a social view. Not only can reality and matter be conceived of as made up of elementary particles, but also society can be rightly conceived of as composed by ‘individuals’, namely atomic items. How are we to deal with this view? If individuals are ontologically primigenial (when compared to the societies they belong to), how are we to conceive of public, shared rules that are the same for all? Don’t these rules pose a substantial threat to the freedom of the individuals? These are some of challenges that are posed to liberalism, the view in politics and economics that has often made social atomism one of its conceptual kernels (conversely, social atomism has long been resisted by Marxism, but this will not be the end of the story if you read Panagiotis’ chapter). Atomism is also a well-known view in the philosophy of time. If time is not continuous but best conceived of as discrete and ultimately made up of instants, could we think of these instants as the expression of the atomicity of time? Speculations on the nature of time have naturally an impact on how we write and read music. In this collection, for instance, atomism is shown to have played an important role also in the philosophy of music in medieval times. Atomism is an important view also in the philosophy of medicine and epistemology. It is also a view that is restricted neither historically (we have been discussing it since antiquity) nor geographically: it is in fact a philosophical view that had wide circulation in the Arabic and Eastern worlds. Atomism is therefore a view that cuts across the very many different areas of philosophy, in both time and space. This collection aims to reflect the great variety and richness of atomism in philosophy. It is organized historically in three parts, because this has proved to be the best way to help readers navigate more easily through the modifications and alterations that the concept of ‘atomism’ underwent throughout its long history. Part I (‘Atomism in antiquity’) covers the discovery of atomism in Classical Antiquity. Part II (‘Atomism in non-Western, medieval and modern philosophy’) illustrates the diffusion of atomism in the Arabic and Indian world, as well as the reception and transformation of (classical) atomism in medieval and modern times. Part III (‘Atomism in contemporary thought’) deals with the different roles that atomism has played in contemporary thought (with the emphasis being placed mainly but not exclusively on the sciences and on current debates in metaphysics).

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ATOMISM IN ANCIENT PHILOSOPHY Atomism as a doctrine is traditionally understood as being originally elaborated in ancient Greece. Leucippus and Democritus, two philosophers from Abdera, in the north of Greece, are usually taken as responsible for the introduction of the main tenets of atomism in the history of (Western) thought. According to them, what really exists is just a vast array of atoms, which have determinate shapes and sizes and which keep changing in order and position all the time. The movement and change of position for atoms are possible exactly because there is void, namely something that allows the atoms to keep moving according to some (fixed) rules. Democritus’ words according to which ‘the principles of all things are atoms and the void; everything else is believed to be’ (Diogenes Laertius 9.44=DK68A1) is thus a good slogan for (ancient) atomism. As a theory of the material world, ancient atomism is indeed an extraordinary intuition or discovery on the part of two philosophers who wanted to explain the very possibility of change and modification after Parmenides ruled them out as truly impossible (together with Zeno, who made movement unconceivable by means of his sophisticated paradoxes that still haunt our modern mind). Atomism as a theory of reality was so radically innovative that we still find it, opportunely revised, in modern physics (although, as we are going to see in Part III, the influence of ancient atomism on current science is far from being univocal or uncontroversial). The conceptual ramifications of atomism, the ones I briefly mentioned in the preceding section, are already present in ancient philosophy. Another fragment of Democritus goes like this: ‘By convention sweet and by convention bitter, by convention hot, by convention cold, by convention colour; in reality atoms and void’ (DK68B9). Sextus Empiricus, who reports Democritus’ own words, thus comments on them: ‘perceptible things are thought – that is, held by opinion – to be, but it is not these things that truthfully are, but only atoms and void’ (M. 7. 135).3 Ancient atomism thus begins as a metaphysical view about reality, but it soon raises epistemological questions. If reality is made of atoms, how will we be able to come to know it? When we see a red table, is the redness we see substantially inherent to the atoms composing the table (when we see it as red)? Are the atoms that compose the table (we are now seeing as red) red in themselves? If this were the case, how could we explain that those very atoms aren’t red anymore when they happen to compose different material objects on another occasions? The two main forms of atomism that were developed in ancient philosophy – that is, Democritus’ and Epicurus – differ substantially regarding the answer they give to these questions. Democritus appears to have defended the view that secondary qualities such as colour do not belong to atoms as such and are by convention (as his slogan in DK68B9 proclaims), while Epicurus seems to have defended the opposite view.4 Democritus and Epicurus have different views too when atomism is understood as having important consequences for human action and freedom. If atoms move and collide according to some fixed and unalterable rules, as Democritus seems to have thought, it will appear that the course of events is already (pre-)determined. Were this so, our action as free individuals would be seriously jeopardized: our

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freedom would simply be tantamount to accepting what is already there for us to accept, with no possibility to act freely in a substantial way. Again, Epicurus seems to have distanced himself quite decidedly from Democritus’ original atomism by introducing the notion of swerve (literally, ‘deviation’ or clinamen in Latin) into his own atomism. The swerve provides a sort of deviation from Democritus’ alleged determinism by postulating an unpredictable sideways movement for the atoms, that is, a movement that contravenes the fixed rules of the atoms’ fall. That the introduction of the swerve is an injection of unpredictability into Democritus’ apparently deterministic world is out of doubt; how the swerve could account for – and explain – human freedom is another question.5 Both Democritus’ and Epicurus’ versions of atomism, however, identified in imperturbability or tranquillity the final aim of their philosophy (one may wonder how this could be the case – since they seem to endorse alternative conceptions of human action and freedom).6 Ancient atomism is thus a view about the composition of reality and one that provoked a great deal of epistemological and ethical discussions in ancient times. Modern scholarship has been dealing with important aspects of ancient atomism for quite a long time, leaving some other equally important features of atomism partially unexplored. For instance, a comprehensive assessment of the ontological status of material objects in ancient atomism is still lacking. If what exist are only atoms and void, we may well assume that material objects such as tables and chairs are, properly speaking, non-existent. What truly exist are the atoms composing the chair, not the chair itself, which we wrongly assume to be there, in an important sense, in front of us right now. If one could argue that in an atomistic world (as ancient philosophers conceived of it) material items are ultimately non-existent, what could one say about those special material items that are human beings? Are human beings for ancient atomists at the same ontological level of chairs and similar material items? Are human beings given, on the contrary, a special ontological status in ancient atomism? For Democritus, it seems that human beings are on a par with all other material items, while for Epicurus, on some interpretations at least, human beings are provided with emergent existence, that is, with something that, ontologically speaking, goes well beyond their mere atomic composition.7 In any case, these matters have not been investigated with the accuracy they deserve, especially if one aims for a thorough understanding of ancient atomism. Another strand of ancient atomism that has not been possible to explore in this collection concerns the close linkage between ancient atomism and atheism. In light of the materialism so intrinsic to it, atomism in ancient Greece (especially Epicurus) argued for the gods having no role whatsoever in the human world.8 Democritus and Epicurus are therefore the two towering figures of ancient atomism, but this tradition was also richer and more varied than someone may initially expect. An underestimated figure in this rich tradition is surely Diodorus Cronus, a very talented logician and philosopher belonging either to the Socratic school of Megara or to a later group of philosophers usually termed as the ‘Dialecticians’.9 In addition to his celebrated ‘Master Argument’, which aims to prove that ‘nothing is possible that neither is nor will be true’ and that appears to have been debated

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at parties after dinner (!), Diodorus was famous in antiquity for having postulated ‘part-less and minimal bodies’ as the ultimate constituents of the world. How and if Diodorus’ partless bodies differ from Democritus’ atoms and how Epicurus took up the challenge are further evidence of the philosophical vitality of ancient atomism.10 The chapters grouped in Part I aim to reflect such vitality. Andrew Gregory’s ‘Early ancient atomism: Similarities and differences’ provides a wide-ranging exploration of the main atomistic views of Democritus and Leucippus. Gregory insists on the originality of ancient atomism, while he sees it as a reaction to Parmenides’ monism. At the same time, by closely analysing Democritus’ and Leucippus’ ways of presenting their own arguments in favour of atomism (by means of the ou mallon argumentative strategy),11 Gregory highlights the difference between ancient and modern atomism at a fuller extent. Gregory’s chapter can be profitably read in conjunction with two other chapters of Part III, Section Three, dealing with atomism in modern science: Needham and Hendry. In his chapter ‘The reception of atomism in ancient medical literature: From Hippocrates to Galen’, Vincenzo Damiani offers an in-depth investigation into the conceptual and historical linkage between atomism and medicine in antiquity. Apart from his interest in mechanics, Democritus had a strong interest in medicine, especially in embryology and human reproduction. Damiani explores Democritus’ contribution in these fields in close detail, as well as painting a full picture of how this contribution was received, elaborated and contrasted in later times, from Hippocrates down to the Hellenistic and post-Hellenistic period (from Erasistratus of Ceos to Asclepiades and Galen). Once again David Sedley in his chapter ‘Why aren’t atoms coloured?’ writes a highly original contribution to ancient atomism by asking why ancient atoms were thought to be colourless. Sedley assesses the evidence Epicurus gives for atoms being colourless and explains the conceptual legacy that Epicurus’ atomism owes to Democritus’. He does so by providing a very illuminating reading of Democritus’ most famous dictum (DK68B9). Sedley eventually concludes his fascinating exploration of ancient atomism by arguing that the colourlessness of atoms is a feature of ancient atomism that, rather unexpectedly, arrives to Epicurus (via the mediation of Democritus) from the most radically anti-empiricist doctrine to be found in ancient philosophy, that is, Parmenides’ monism. In his insightful chapter ‘Atoms and minimal Parts: the originality of Epicurean atomism’ Francesco Verde assesses the role played by minima in the context of Epicurus’ atomism. The introduction of the concept of ‘minimal parts’ is a very important modification Epicurus makes to Democritus’ atomism (via Diodorus Cronus) and one that goes too often unnoticed. By providing an in-depth analysis of the main available sources, Verde shows that Epicurus didn’t believe the atoms to be partless – indeed, he posited the atoms to have minimal parts. According to Verde, Epicurus held this view to be able to answer some objections (by Aristotle and Plato), as well as to account for the material indivisibility of atoms in their perennial process of destruction and agglomeration. Both Sedley’s and Verde’s chapters can be profitably read in conjunction with all the ‘metaphysical’ chapters of Part III, Section Two. One can also read Sedley’s piece in parallel with Artosi’s chapter in Part II and Verde’s with Sarkar’s chapter, again in Part II.

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In his contribution ‘Atoms and universals in Epicurus’ Attila Németh investigates the role of perception in the context of Epicurus’ atomism. How we come to know things was the main epistemological preoccupation for Epicurus. In his chapter, Németh explains how perceptual knowledge is to be obtained in a world of atoms, such as the one postulated by Epicurus. He does so by explaining Epicurean perception as a three-stage process, where prolepsis plays a unifying role in the epistemological process. In doing so, Németh shows how Epicurus managed to provide a case for knowledge in an atomistic world, thus overcoming the scepticism that appears to be inherent to Democritus’ atomism. The same preoccupation that motivated Epicurus’ attempt to make knowledge possible in a world of atoms is at the centre of the discussion on knowledge that Socrates carries out with the young mathematician Theaetetus in the last part of Plato’s Theaetetus. After the detailed and thorough analysis of the dialogue she provided in 2005,12 in her chapter ‘Atoms, complexes and simples in the Theaetetus’ Sophie-Grace Chappell goes back to the Theaetetus once again, to offer a reading of the dialogue that I dare believe as truly innovative for Platonic scholarship. In his third and final attempt to define knowledge as true judgement with a logos, Socrates refers to a Dream he heard where, among other things, knowledge is described in terms of knowledge of simples (that is, atomic items) and knowledge of complexes (complexes being made up by simples, however difficult the relation of composition is to be properly defined). Socrates rejects this third definition of knowledge too, together with the other two put forward in the preceding parts of the dialogue (at Tht. 151d and at 187b). But Chappell makes the strongest case I know of to take the dialogue as genuinely non-aporetic. She does so by claiming that, for Plato, knowledge is either knowledge of simples or of complexes, or knowledge of how to get from simples to complexes and back again, by way of ‘downwards’ analysis and ‘upwards’ synthesis. And this account of knowledge is itself a list of three examples of knowledge – hence a possible answer to the question ‘what is knowledge?’ that has vexed Socrates and his interlocutor since the very beginning of the dialogue. Both Németh’s and Chappell’s chapters could be read alongside Treanor’s chapter in Part III, Section One. Chappell’s is also well complemented by Coliva’s chapter on Wittgenstein in Part III, Section One (Wittgenstein himself being a careful reader of the last part of the Theaetetus). Lastly, Luca Pitteloud’s piece ‘Atoms in Plato’s Timaeus’ offers a very detailed analysis of Plato’s own handling with atomism in the Timaeus. On different grounds, both Aristotle and Plato opposed atomism as a viable philosophical doctrine. Yet, both philosophers were attracted to it. We have already seen that the problematic relationship between complex and atomic items is at the centre of the final section of the Theaetetus. A more positive approach to atomism is adopted by Plato in the Timaeus. Pitteloud shows how in this dialogue Plato conceives of atomism as a geometrical doctrine and of atoms as two-dimension entities that can be ultimately reduced to two basic shapes of triangles. In this way, Plato thinks he is able to reject the strong materialism he sees as deeply connected to Democritus’ atomism, while fully preserving the idea of ‘geometrical’ atoms as the main constituents of the sort of universe he aims to sketch out in the Timaeus.

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ATOMISM IN NON-WESTERN, MEDIEVAL AND MODERN PHILOSOPHY In the preceding section, I have made a case to grant ancient atomists such as Leucippus and Democritus with the discovery of atomism. Yet, this claim may well generate in the reader the impression that atomism is a doctrine that is somehow exclusive to Western philosophy, stretching somehow tortuously, but straightforwardly from ancient Greece, via the Renaissance, to the main contemporary debates in Western philosophy. But this impression would be utterly misleading. While he refers to a passage of Strabo, which in turn refers to another passage from the Stoic Posidonius, Ralph Cudworth, the most relevant of the Cambridge Platonists (on whom, see the chapter by Adrian Mihai), writes: Wherefore we have made it obvious, that that exceptionally mechanical or atomical rationality, that hath been recen​tly r​eesta​blish​ed vi​a Cartesius and​ Gass​ endus​, with regards to the principle substance of it, was senior to Epicurus, as well as than Plato and Aristotle, nay, than Democritus and Leucippus additionally, its normally presumed dads. Also, thusly we have no motivation to dishonor the report of Posidonius the Stoic, who, as Strabo lets us know, confirmed this atomical rationality to have been ancienter than the seasons of the Trojan war, and first to have been brought into Greece out of Phoenicia [. . .]. What’s more, since it is sure from what we have appeared, that neither Epicurus nor yet Democritus was the primary creators of this physiology, this declaration of Posidonius the Stoic should in motivation to be conceded by us. Presently, what can be more likely than that this Moschus the Phoenician, that Posidonius talks about, is the specific same individual with that Moschus the physiologer, that Jamblichus specifies in the Life of Pythagoras, where he avows, that Pythagoras, living some time at Sidon in Phoenicia, spoke with the prophets that were the successors of Mochus the physiologer, and was told by them [. . .]. He spoke with the prophets that were the successors of Mochus and other Phoenician clerics (The True Intellectual System of the Universe, 12). Mochus of Sidon is credited by Posidonius, Strabo and Cudworth with the discovery of atomism, making thus Leucippus and Democritus as the philosophical developers of a view they didn’t elaborate for the first time. There has been so far no serious attempt to assess whether the story about Mochus’ atomism is reliable or not. As it stands, it is a very interesting story, which locates the birth of atomism much further back in time and not exactly on mainland Greece.13 Geographically speaking, Phoenicia was also much closer to the Eastern world – that is, to the Arabic and Indian worlds. While we know much today on how the Arabic world mediated and reinterpreted ancient Greek and Latin culture in the Middle Age, and are starting to know much more about the possible influence between ancient Greece and India,14 still much is to be learnt about the cultural contaminations between East and West in the Archaic and Classical period (from the tenth century BC to the fifth/fourth century BC). It may well turn out that Cudworth’s account is, after all, true.

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It was long assumed, at least until the nineteenth century, that atomism wasn’t properly debated in the Middle Age.15 This assumption has been proven wrong, since atomism was widely debated in the Middle Age either in the Arabic or in the Jewish world (by the ninth-century Arabic theologians in Baghdad or by the Jewish schools in Egypt).16 In late medieval time, atomism became a topic of great interest for theologians and philosophers working in the Christian tradition, such as those based in Paris, Chartres and Oxford.17 The nature of the atomistic debate in the last centuries of the Middle Age was, however, quite different from the sort of debates developed in ancient atomism. Christian theologians were keen on discussing such topics as the creation and eternity of the world – with the topic of time having prominent relevance in those discussions (on the other hand, Arabic and Jewish discussions of atomistic themes were much closer to the spirit of the ancient debate).18 When the Middle Age ended, the actual discovery in 1417 in the German abbey of Fulda of an important manuscript copy of Lucretius’ De Rerum Natura by the Italian humanist and scholar Poggio Bracciolini fuelled a vital appropriation of atomism in the Renaissance. As Stephen Greenblatt writes, perhaps a bit too emphatically but surely making a point, it is through the swerve that the world became modern.19 The rediscovery of atomism during a time of cultural and societal flourishing had a huge impact on the best minds of the Renaissance. It ignited Boyle’s and Newton’s revolutions in the sciences, which is evidence that atomism was widely debated in the British Isles in that period (as some chapters of Part II claim).20 With their oscillation in either accepting or rebutting atomism, important philosophers such as Leibniz show that atomism was indeed perceived as a central topic to be confronted with in key moments of the Renaissance.21 Part II of the collection shows how atomism is to be understood in the context of non-Western philosophies, as well as in medieval and modern times. The first chapter of Part II is ‘Atoms and orientation: Vasubandhu’s solution to the problem of contact’, by Amber Carpenter with Sherice Ngaserin. Carpenter points out the difference between ancient Greek and Buddhist versions of atomism, suggesting that the latter is best to be interpreted as a (rather austere) theory of tropes. If in Buddhist thought atoms are to be considered as dimensionless, point-particle property-events, two philosophical problems will almost naturally be arising. The first, The Problem of Agglomeration, arises from the fact that dimensionlessness plus any number of additional dimensionlessness bits can only be dimensionless. So dimensionless atoms cannot constitute the basis of the appearance of spatially extended objects. The second, The Problem of Contact, arises from the attempt to agglomerate truly simple, dimensionless atoms into lengths and shapes by placing them adjacent to one another. If they are in contact, either they have parts or remaining partless they wholly touch, and thus coincide completely and end up occupying the same space. By providing us with a very in-depth analysis of the atomism of Vasubandhu (a central figure in Indian Buddhist philosophy), the authors show how these two problems are to be dissolved once one gets closer to a proper understanding of Buddhist atomism. This rich chapter is to be read in conjunction with Sarkar’s chapter, which deals with atomism in Indian (non-Buddhist) thought, and with the chapters by Verde (Part I) and Simons (Part III, Section Two).

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In a sort of a companion chapter, ‘Aggregates versus wholes: An unresolved debate between the Nyāya-Vaiśeṣika and Buddhist schools in ancient Indian atomism’, Sahotra Sarkar shows how atomism flourished among ancient Indian philosophical traditions including the orthodox Nyāya-Vaiśeṣika school and the heterodox Buddhist and Jain schools. For the Nyāya-Vaiśeṣika and the Buddhist schools, atoms were indivisible and imperceptible. But, beyond that, their accounts diverged. According to the Nyāya-Vaiśeṣika school, atoms are eternal, while for most Buddhist atomists, they are not. The Nyāya-Vaiśeṣika school and the Buddhists also diverged on the nature of the composites made up by atoms. Sarkar’s chapter reconstructs a debate in which the Nyāya-Vaiśeṣika school held that composites are wholes, with an individuality that is distinct from their component atoms. His chapter concludes by noticing how there are marked similarities between this old debate in Indian philosophy and twentieth-century debates between reductionism versus emergentism and holism. Again, Sarkar’s chapter can be profitably read in conjunction with the metaphysical chapters of Part III (Section Two). In his chapter ‘Atomism and Islamic Thought’ Omar Francesco Zamboni shows how the atomistic theory of matter represents one of the fundamental themes of medieval Islamic thought. Several Muslim authors tackled the issue from a variety of perspectives. The aim of Zamboni’s contribution is to present a much-needed overview of the atomistic debate in medieval Islam. It is divided into three main parts: the first one presents the elaboration of the atomistic doctrines, considering their doctrinal milieu and their distinctive features; the second part deals with the arguments in favour of atomism and the fundamental assumptions on which they rest; the third and final part tackles the main disputes between Muslim atomists. Zamboni also shows how the mutakallimūn appear to be committed to a ‘minimalist’ and moderately materialistic ontology of atoms. We then move to two very interesting chapters dealing with atomism and time in medieval philosophy. Charles Doyle’s ‘Atoms and Time I’ offers us a fascinating history of time from ancient atomism into medieval philosophy. He focuses on Democritus first, arguing that he may have supposed time to have existence, on a par with atoms and void. He then considers Epicurus’ position on time, suggesting that the Epicureans ended up in denying the existence of time. On the basis of some evidence, it is however possible to claim that Epicurus distinguishes eternity from ordinary time. The notion of a temporal minimum may also follow logically from the Epicurean spatial minimum and the very nature of atomic motion. Doyle then goes on to ask whether there is a conceptual linkage between the atomus in tempore and the Epicurean pars minima of time. He concludes that the medieval atomus in tempore is a reception of corporeal atomism applied to time by Patristic authors in response to a linguistic problem rather than to a problem in natural philosophy. Philippa Ovenden’s ‘Atoms and music in late medieval philosophy’, provides us with a detailed overview of the main issues pertaining to atomism and music in the fourteenth century. After evaluating today’s scholarship about atomism in fourteenth-century philosophy and music, Ovenden introduces the work of two theorists (Johannes Torkesey and Willelmus) who discussed the limits of divisibility of musical time in relation to mensural notation. She then examines different types of indivisibility within the gradus system, which is predicated upon the idea that all

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musical notes can be represented within a range of values and that within this range musical sounds are created that may share the same duration but possess differing functions depending upon their context. According to Ovenden, the idea that continua were composed from the accumulation of discrete, indivisible particles was present in many fourteenth-century music texts. In her wide-ranging analysis of music and the nature of time in the late Middle Age, she argues that music’s integration of a continuum of time and sound with discrete notational values provided an appropriate conduit for philosophical musings about the more-or-less continuous and divisible structure of continua. Both chapters have a close relation with Dorato’s in Part III, Section One (with Doyle to be read alongside Verde’s in part I). Next, we have two chapters showing the proliferation of atomistic approaches in England in the Renaissance. Adrian Mihai’s ‘Atomism and the Cambridge Platonists’ provides us with a detailed reconstruction of the philosophical enterprise of the Cambridge Platonists. They were a most influential seventeenth-century group of philosophers who attempted to unify in a philosophical and theological system freedom, reason, morality and love. They attempted to do so by elaborating on atomism. Mihai gives us a full overview of the main figures of the Cambridge Platonists (Cudworth being the most relevant) and of the different kinds of atomism they discussed. With his careful analysis, Mihai shows the vital impact that the reception of atomism had on modern philosophy (and science) in Renaissance England. Ako Sivado’s chapter ‘Atomism and society in William Petty’ is concerned with the atomistic account developed by an overlooked seventeenth-century English virtuoso, Sir William Petty. Although atomism had been in circulation in early modern England, atomistic accounts of society were scarcely available in the philosophy of that time. Petty wished to remedy that situation in a controversial but highly original way by trying to extend natural philosophical investigations into the realm of social reality. Sivado’s chapter examines Petty’s version of atomism along three main lines. First, it situates his atomism concerning the natural world among the atomistic accounts of his peers; second, it presents the details of how, according to Petty, such an atomism is to serve as the basis of a ’science’ of society; and third, it draws out the potential consequences such a view of society had for the possibility of quantifying social phenomena in the centuries that followed. Sivado’s chapter can be profitably read alongside with Krinks’ in Part III, Section One. Lastly, Alberto Artosi’s ‘Atoms, colours and God in Leibniz’, provides us with a fascinating account of Leibniz’s troubled fascination for atomism. With his usual clarity and verve, Artosi argues that Leibniz’s first writings are considered to offer no conclusive evidence of an atomist phase in his early mechanistic thinking. This view, in turn, made Leibnizian scholars debate whether Leibniz had ever really subscribed to atomism. By relying on Leibniz’s views on colours, God and mechanics, Artosi reaches the conclusion that – even if only for a short time, between 1666 and 1669 – Leibniz effectively endorsed the atoms and the void and that even if this endorsement appears to be temporary, it constitutes a significant chapter not only in Leibniz’s intellectual development but also in the history of seventeenth-century atomism. Artosi’s chapter can be read alongside Sedley’s in Part I and also with the chapters on atomism in the sciences in Part III, Section Three.

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ATOMISM IN CONTEMPORARY THOUGHT Atomism in contemporary thought becomes a conceptual hydra that is hard to come to terms with. The actual specialization of philosophy into many sub-fields and topics on one side and the closer dialogue between philosophy and the sciences on the other are such that atomism was and is discussed in very many areas of contemporary thought and research. The fact that Part III is considerably longer than the other parts is evidence for this fact. If you ask a philosopher about atomism in contemporary thought, you will be likely to get a reply telling you about ‘logical atomism’. This view was first put forward by Bertrand Russell, who argued that the material world is made up of logical atoms – as he puts it, ‘little patches of colour or sounds, momentary things [. . .], predicates or relations and so on’.22 Logical atomism perhaps shines at its best in Wittgenstein’s Tractatus Logico-Philosophicus and survives, modified, in some current positions in analytic metaphysics. But logical atomism is just one of the main kinds of atomism to be discussed in contemporary thought. Important debates on atomism took place – and are still under way – in the philosophy of chemistry and physics, two sciences that underwent a huge theoretical development in the last century and a half. While atomism has become a central topic in less recent as well as current discussions in metaphysics and in the philosophy of the sciences, it is still the case that in contemporary philosophy atomism is also debated in connection with some of the traditional topics dealt with in ancient, medieval and modern times. To map such an unchartered territory, it is useful to divide Part III into three subsections: ‘Atomism in philosophy’ (Section One), ‘Atomism in metaphysics’ (Section Two) and, lastly, ‘Atomism in the sciences’ (Section Three). This partition may induce one to raise one’s eyebrows questioningly. To some extent, it is an unnatural partition. After all, metaphysics is indeed an important part of philosophy – and the distinction between philosophy and some sciences is nowadays not so neat as it was before. Everyone introducing such a division introduces some sort of artificiality. Yet, I couldn’t find a better way than this tri-partition to highlight that the three main areas of contemporary thought where atomism has so far played a vital role are distinct areas of the same family. Section One ‘Atomism in philosophy’ includes six chapters dealing with atomism in contemporary philosophy, from Wittgenstein to Marxism and liberalism; Section Two ‘Atomism in metaphysics’ gathers four chapters dealing with atomism in contemporary analytic metaphysics, from universals to tropes; Section Three ‘Atomism in the sciences’ assembles three chapters tackling atomism in the philosophy of science (especially the philosophy of chemistry and physics).

Atomism in philosophy As for ‘Atomism in philosophy’, we begin with the chapter by Annalisa Coliva, ‘Logical atomism and Wittgenstein’, which deals with Wittgenstein’s handling of atomism. In her chapter, Coliva admirably tackles Wittgenstein’s criticism of his logical atomism in some key sections of the Philosophical Investigations. She shows

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that Wittgenstein needs to discharge his logical atomism because it fails to account for the new conceptions of meaning, language and reality he strongly endorses in the Philosophical Investigations. This is an important chapter in so long as it shows how one of the most important ‘atomists’ of the twentieth century changed his own mind by elaborating a pervasive critique of the very idea of (logical) atoms. The reader can read Coliva’s chapter along with the contribution by Sophie-Grace Chappell on simples and complexes in the Theaetetus and with the next chapter by Begley. In an extremely wide-ranging chapter, ‘Atomism and semantics in the philosophy of Jerrold Katz’ that somehow reflects the richness of Katz’ own philosophical approach, Begley demonstrates how Katz’ innovative views in the philosophy of language remained substantially unaltered throughout his life. These are views that, as Katz himself wished to make it explicit, are to be illuminated by analogies with physical atomism and atomism in chemistry. Begley shows how the analogies with those kinds of atomism allowed Katz to develop a highly sophisticated Democritean semantics for the decompositional representation of simples that, pace Wittgenstein, have a minimal internal complexity and are not ontologically independent and noncomplex. Begley’s chapter can be read in conjunction with those on Democritus by Gregory in Part I and those on atomism in chemistry by Needham and Hendry in Part III, Section Three. Nick Treanor’s chapter ‘Atoms and knowledge’ deals with a question that is often poorly addressed in philosophy, that is, whether knowledge can be conceived of as being composed of atoms (of knowledge). That is, on what grounds are we entitled to think of knowledge as made up of elementary bits of it? This question naturally brings us back to Sophie Grace Chappell’s chapter in the first part of the collection – Treanor examines the problem by dealing with the mereological nature of truth, namely with the question on how it is possible to conceive of truth as the sum of atomic truths. In dialogue with Travis Dumsday’s chapter in Part III, Section Two, Treanor offers important arguments for claiming that truths have extent (but not all the same extent), marginalizing the very idea of cardinality. In doing so, Treanor also accounts for the widespread intuition that truth may be after all atomic, shedding also light on the inevitable confusion and ambiguity that such an intuition carries with it. Mauro Dorato’s chapter ‘Atoms and Time II’ addresses the question whether, in light of the innovation introduced by the discoveries of contemporary (quantum) physics, we may conceive of time as ultimately atomic, that is, as composed of time units, not further reducible to anything more atomic in terms of time. His answer to this question verges towards a ‘No’. Dorato sees two main difficulties in taking time as discrete: (1) the difficulty to understand properly and fully the claim that a unit of time is indivisible; (2) the lack of any empirical evidence for the very existence of an atomic unit of time. Dorato’s chapter can be read alongside Needham’s in Part III, Section Three, which argues for similar views in terms of (the lack of) theoretical success for atomism as a viable view of matter. Dorato’s chapter also reads well with Doyle’s in Part II.

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Panagiotis Sotiris’ chapter ‘Atomism and Marxism in Louis Althusser’, is an important contribution on the role of atomism in Marxist thought. It is well known that Karl Marx was deeply fascinated by the sort of materialism that is implicit in ancient atomism (he devoted to the topic his doctoral dissertation).23 Yet, the possible role that (ancient) atomism played in the adaptation of materialistic views into Marxist thought has often been underestimated (I here think of a very notable exception: Timpanaro 1985). In his chapter, Sotiris analyses the role that ancient atomism (in particular, the Epicurean doctrine of the swerve) plays in the thought of one of the most original Marxist thinkers of the last century, Louis Althusser. Sotiris shows how Althusser originally interpreted the doctrine of the swerve as an anti-theological doctrine, which was pivotal in his development of a sort of materialism that is fully devoid of any intrinsic causality. While Sotiris offers a valuable exploration into the underestimated importance of atomism for Marxist materialism, in his chapter ‘Atomism and liberalism’ Philip Krinks provides us with an assessment of atomism in liberal thought. More than Marxism, liberalism has been thought of as being more receptive to the influence of atomism. Krinks’ chapter shows that this is well the case, but he also demonstrates how varied and multiform that influence is. He argues for social atomism to be further branched into historical, methodological and normative atomism. He then distinguishes a form of atomism that assumes a sort of pre-social individuality from a moderate atomism that does not make that assumption. After having considered possible objections to all these forms of social atomism, in the conclusion Krinks deals with the possibility of further liberal atomist responses to those objections, also highlighting the pressure towards non-atomist responses for liberal thought. Sotiris’ and Krinks’ chapters are nicely read together along with Sivado’s chapter in Part II.

Atomism in metaphysics In the second section of Part III, we have four chapters dealing with the philosophical appeal that atomism has played over contemporary analytic metaphysics. In his chapter ‘Atoms as universals’ Matthew Tugby explores how atomism can be combined with a realist ontology of universals. He proposes a new theory that views an atom as an instance of a simple universal, rather than a bundle of multiple universals. According to this theory, atoms are simple in the strongest sense because they lack inherent qualitative complexity as well as spatial complexity. This theory is motivated, among other things, by the Axiom of Difference. Tugby then addresses what is arguably the most serious objection to the theory he defends, namely that the theory faces difficulties in accommodating talk of ‘fine-grained’ attributes, such being unit negatively charged or having a certain mass. He concludes his chapter by exploring some possible solutions to this problem. In his chapter ‘Atoms and extended simples’ Travis Dumsday articulates atomism into four principal competing theories: (1) atomism version 1 (nature bottoms out at indivisible unextended point-particles); (2) atomism version 2 (nature bottoms out at indivisible extended objects); (3) the theory of extended simples (nature bottoms out at objects that are extended and divisible, but have no actual proper parts); (4) the

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theory of gunk (nature has no bottom level – every object has proper parts that are themselves objects having proper parts that are themselves objects: see the following chapter by Marmodoro and Roselli). Modifying some neglected ideas from within the Aristotelian/Scholastic tradition, Dumsday suggests a new way of thinking about the relationship between atoms and extended simples, according to which unextended point-particles are real and metaphysically primary, but extended simples have a kind of ideological primacy. That is, the idea of an extended simple is still required for the proper conceptualization and modelling of the fundamental level of nature, even though extended simples are not instantiated in nature. The result is a novel hybrid theory in which atomism version 1 is true at the level of ontology but where extended simples have an irreducible role to play in ideology. In their chapter ‘Power gunk, or unlimitedly divided powers’, Anna Marmodoro and Andrea Roselli take atomism as the theory that (1) there is a fundamental ground of being, which (2) comprises elements that cannot be divided into parts that would be more fundamental than the atoms are. They claim that (1) is an expression of metaphysical foundationalism and that (2) is an expression of ontological pluralism. Marmodoro and Roselli deny both and argue that there is no bedrock to the physical world, since there is always something more fundamental than any putative bedrock. In other words, they hold that the world is gunky in so long as its building blocks are such that there are parts of parts of them, ad infinitum. They adopt a powers ontology and develop a model of emergent powerfulness. In place of a plurality of primary powers, they argue for the claim that there is not a fundamental level of powerfulness, since powerfulness ontologically emerges gradually from an infinite series of progressive powers, and epistemologically it emerges as a distinct layer of being interconnected with all the relevant facts around it. They insist that this stance holds promise for tackling some important issues in the philosophy of gunk, such as that gunk has properties that cannot be derived from the parts’ properties. Lastly, in his chapter ‘Atoms and tropes’, Peter Simons offers us a fascinating account of atoms as tropes, which is rich in both historical breadth and philosophical originality (tropes being, as he puts it, ‘thin natures which confer only a single property on the objects that bear them’). After having defined atoms as partless items, Simons shows that tropes lack proper parts and are the ultimate constituents of all entities. Tropes thus fit the traditional role of atoms. Simons then assesses more and less promising examples of tropes; he also deals with all the possible shortcomings that a theory of tropes taken as ultimate constituents of reality could be faced with (these shortcomings have mainly to do with the spatiotemporal extension, the ontological inherence, the occurrence, the ontological simplicity of tropes). Simons offers insightful arguments to hold that if there is a form of metaphysical atomism that is worth having in the sense of being correct, then it should be an atomism of tropes. These four chapters can be read with reference to other chapters that don’t belong to this section but that deal with metaphysical issues in another context or historical period: Sedley, Verde, Németh and Chappell in Part I; Carpenter and Sarkar in Part II; Coliva in Part III, Section One; Morganti and Hendry in Part III, Section Three.

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Atomism in the sciences  The third and final section of Part III comprises three chapters which deal with the important role atomism has played in the sciences since Dalton. The historical continuity that atomism has enjoyed throughout the history of thought until today could make one think that thanks to important developments of modern sciences, we have eventually gained a more refined, ‘scientific’ account of atoms. If this were the case, we would be in a better position to understand the theoretical importance and indispensability of atomism for human thought. According to some of the chapters in this section, this is not the case. In Needham’s chapter, you find arguments for downplaying, if not rebutting, the role played by atomism in modern science. Hendry’s and Morganti’s chapters nicely counterbalance the power of those arguments by offering more positive insights for the theoretical work made by atomism in modern chemistry and physics. In his chapter ‘Atomism and physics-based structuralism’, Matteo Morganti explores the relationship between metaphysical atomism and a form of physicsbased structuralism that has recently become popular among philosophers of science. Prima facie, it looks like there is a clear conflict between the two theses: the material world is either atomic or made up of structures (with partless items playing no role in them). However, this is by no means the case according to Morganti. Once atomism and structuralism are carefully examined, the conflict can in fact be shown to be quite circumscribed and to depend on theoretical preferences, choices and assumptions that go beyond the core claims that characterize the two positions. In particular, Morganti identifies ways in which atomism and physics-based structuralism may coexist (which includes cases in which structures are intended as mereological simples) and ways in which they conflict (most notably, this happens once it is contended that everything is structurally analysable). In his chapter ‘Atoms and chemistry I: Not a success story’, Paul Needham traces the development of atomism in modern chemistry, with Dalton and Duhem. Needham starts by stressing the difference between what ‘atom’ means in ancient philosophy and what it means in modern chemistry with and after Dalton, who is responsible for the greatest innovations in chemical atomism. Yet, Needham argues that Dalton’s atomism faces quite serious problems, which were highlighted by Pierre Duhem. Needham is especially concerned about the problem of substances and compounds – he argues for chemical atomism not being able to provide a satisfactory answer for the problem of mereological composition. In a companion chapter ‘Atoms and chemistry II: Trusting atoms’, Robin Hendry takes issue with Needham’s chapter and makes quite different claims. He highlights that his disagreements with Needham (and with Alan Chalmers) are about the evaluation of atomism in nineteenth-century chemistry. The motivations for Hendry’s disagreements are partly historiographical and partly philosophical. On the historiographical side, he holds that chemical atomism was a new theory and that it is a historical error to saddle it with too many of the problems of its predecessors. (On Hendry’s understanding, it is important to separate those parts of Dalton’s theory that concern the composition and structure of chemical substances from its more ‘mechanical’ parts.) Another disagreement on Hendry’s

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part concerns the distinction between two historical questions: (1) When did atomism become a useful part of structural theory? (2) When did the evidence for the existence of atoms become strong enough for an epistemically responsible person to accept their existence? One can answer the instrumentalist question (1) without addressing (2), the evidential question. Hendry disagrees with Needham on how to answer each question. Hendry conducts his criticism and argues for his views by taking into full account the dialogical relationship between philosophy of science and metaphysics, hence providing us with important insights on how these two branches of philosophy could collaborate more fruitfully (not just for the benefits of the philosophy of chemistry).

NOTES 1. You will find four main entries on atomism in the Standford Encyclopaedia of Philosophy: ‘Ancient Atomism’ by Sylvia Berryman (http​​s:/​/p​​lato.​​stanf​​ord​.e​​du​/en​​tries​​ /atom​​is​m​-a​​ncien​​t/); ‘Russell’s Logical Atomism’ by Kevin Klement (http​​s:/​/ p​​lato.​​stanf​​ord​.e​​du​/en​​tries​​/logi​​ca​l​-a​​tomis​​m/); ‘Wittgenstein’s Logical Atomism’ by Ian Proops (http​​s:/​/p​​lato.​​stanf​​ord​.e​​du​/en​​tries​​/witt​​genst​​e​in​-a​​tomis​​m/); ‘Atomism from the 17th to the 20th century’ by Alan Chalmers (http​​s:/​/p​​lato.​​stanf​​ord​.e​​du​/en​​tries​​/ atom​​i​sm​-m​​odern​/). Honderich (1995) has three entries related to atomism (logical, physical, psychological), while Simon Blackburn’s Oxford Dictionary of Philosophy has a very brief entry on ‘atomism’ (a bit more than half a column). 2. ‘Atom’ comes from the ancient Greek adjective ‘atomos, atomon’, which means ‘uncut’. An atom is ‘uncut’ exactly because it is uncuttable, to the extent that it cannot be further reduced to anything more essential, more elementary. By extension, ‘to atomon’ (as a noun) means the ‘uncuttable element’. See Liddell, Scott, Jones andMcKenzie, Greek-English Lexicon, entry on ‘atomos’. 3. On a new reading of this important fragment, see David Sedley’s chapter in this collection. 4. There has been a prolonged debate among scholars about this. As a start, the reader may want to have a look at these two important contributions: Furley (1993) and O’Keefe (1998). 5. There has been a long debate in recent years on Epicurus’ anti-determinism. A good starting point for the reader interested in the topic is O’Keefe (2005), which argues for the idea that the sort of freedom Epicurus wants to preserve in his philosophy is quite different from our conception of free will. See also Sedley (1983); Hankinson (1999). For two celebrated modern treatments of determinism and free will, see Van Inwagen (1983) and Dennett (2015). 6. See O’Keefe (2005, 110–52). Warren (2002) offers a fascinating genealogical reconstruction of Epicurus’ ethical views by locating them in the context of an ethical tradition that goes from Democritus and Pyrrho to Epicurus himself. 7. On ontological emergence and anti-reductionism in Epicurus, see the seminal paper by Sedley (1988). For a different view, see O’Keefe (2005), chapter 4 and Nemeth (2017), chapter 2.

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8. On ancient atheism, it is now possible to read the very enjoyable account given by Tim Whitmarsh (Whitmarsh 2016). See also Piergiacomi (2017) and the chapter by Mihai in this collection. 9. On Diodorus Cronus, the reader may have a look at the very informative material on him in The Cambridge History of Hellenistic Philosophy, pp. 83–92 (by J. Barnes and S. Bobzien); pp. 356–62 (by David Sedley); pp. 526–9 (by J. Hankinson). 10. On Diodorus’ minima, see Verde (2013) and (2015). 11. On this strategy, see Makin (1993). 12. See Chappell (2005). 13. Phoenicia (Lebanon nowadays) didn’t belong to mainland Greece, although cultural as well as economical exchanges and ties between Phoenicia and the main Greek cities were intense at the time. 14. See McEvilley (2006) and Beckwith (2015). 15. This traditional view was challenged by Lasswitz (1890) and Mabilleau (1895). 16. See Dhanani (1994); Wolfson (1976); Zonta (2002). 17. See Robert and Grellard (2009). 18. The best port of call (quick and very informative) for atomism in medieval time is the entry ’atomism’ by A. Robert in the Medieval Encyclopaedia of Medieval Philosophy (under the editorship of H. Lagerlund, Spinger, 2011). 19. See Greenblatt (2011) and Wilson (2008). Greenblatt’s hardback edition (2011) has the title: The Swerve. How the world became modern, which changes into ’The Swerve. How the Renaissance began’ in the paperback edition (2012). 20. See Chalmers (2009). It has been impossible to include in the collection a piece on atomism and Newton and/or Boyle. Yet, part of the great innovation atomism introduced in modern science is dealt with in the chapters by Needham and Hendry in Part III (Hendry arguing against some of the main claims defended by Chalmers himself). 21. Pierre Gassendi is another Renaissance philosopher who was deeply influenced by atomism. As a Christian theologian, he aimed to show that atomism could be profitably combined with the main dogmata of Christian theology. In her 2008 book, Catherine Wilson calls him ’the foreign parent of British empiricism’, because his atomistic views were deeply appreciated by Newton and Boyle. A good introduction to his thought is Lolordo (2006). 22. Russell (1965, 179). 23. With the title: The difference between the Democritean and Epicurean philosophy of Nature (1841).

REFERENCES Algra, K., Barnes, J., Mansfeld, J. and Schofield, M., eds (1999), The Cambridge History of Hellenistic Philosophy, Cambridge: Cambridge University Press. Beckwith, C. (2015), Greek Buddha: Pyrrho’s Encounter with Early Buddhism in Central Asia, Princeton: Princeton University Press.

GENERAL INTRODUCTION

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Berryman, S., Entry on ‘Ancient atomism’, Stanford Encyclopaedia of Philosophy. https​:/​/ pl​​ato​.s​​tanfo​​rd​.ed​​u​/ent​​ries/​​atomi​​s​m​-an​​cient​/ (accessed April 2020). Chalmers, A., Entry on ‘Atomism from the 17th to the 20th century’, Stanford Encyclopaedia of Philosophy. https​:/​/pl​​ato​.s​​tanfo​​rd​.ed​​u​/ent​​ries/​​atomi​​​sm​-mo​​dern/​ (accessed April 2020). Chalmers, A. (2009), The Scientist’s Atom and the Philosopher’s Stone: How Science Succeeded and Philosophy Failed to Gain Knowledge of Atoms, Dordrecht: Springer. Chappell, S. G. (2005), Reading Plato’s Theaetetus, Indianapolis: Hackett. Dennett, D. (2015), Elbow Room: The Varieties of Free-Will Worth Wanting, 2nd edn, Cambridge, MA: MIT. Dhanani, A. (1994), The Physical Theory of Kalam, Leiden: Brill. Furley, D. (1993), ‘Democritus and Epicurus on sensible qualities’, in J. Brunschwig and M. C. Nussbaum (eds), Passions and Perceptions: Studies in Hellenistic Philosophy of Mind, 53–71, Cambridge: Cambridge University Press. Greenblatt, S. (2011), The Swerve: How the Renaissance Began, London and New York: Vintage. Hankinson, R. J. (1999), ‘Determinism and indeterminism’, in K. Algra, J. Barnes, J. Mansfeld and M. Schofield (eds), The Cambridge History of Hellenistic Philosophy, 513–41, Cambridge: Cambridge University Press. Honderich, T. (1995), The Oxford Companion of Philosophy, Oxford: Oxford University Press. Klement, K., Entry on ‘Russell’s logical atomism’, Stanford Encyclopaedia of Philosophy. https​:/​/pl​​ato​.s​​tanfo​​rd​.ed​​u​/ent​​ries/​​logic​​a​l​-at​​omism​/ (accessed April 2020). Lasswitz, K. (1890), Geschichte der Atomistik vom Mittelalter bis Newton, Hamburg: Voss. Liddell, H. G., Scott, R., Jones, S. and McKenzie, R. (1996), Greek-English Lexicon, with a Revised Supplement, Oxford: Clarendon. Lolordo, A. (2006), Pierre Gassendi and the Birth of Early Modern Philosophy, Cambridge: Cambridge University Press. Mabilleau, L. (1895), Histoire de la philosophie atomistique, Paris: Alcan. Makin, S. (1993), Indifference Arguments, Oxford: Oxford University Press. McEvilley, T. (2006), The Shape of Ancient Thought, New York: Allworth Press. Németh, A. (2017), Epicurus on the Self, London and New York: Routledge. O’Keefe, T. (1998), ‘The ontological status of sensible qualities for Democritus and Epicurus’, Ancient Philosophy 17: 119–34. O’Keefe, T. (2005), Epicurus on Freedom, Cambridge: Cambridge University Press. Piergiacomi, E. (2017), Storia delle antiche teologie atomiste, Rome: Sapienza Università Editrice. Proops, I., Entry on ‘Wittgenstein’s logical atomism’, Stanford Encyclopaedia of Philosophy. https​:/​/pl​​ato​.s​​tanfo​​rd​.ed​​u​/ent​​ries/​​wittg​​enste​​​in​-at​​omism​/ (accessed April 2020). Robert, A. and Grellard, C., eds (2009), Atomism in Late Medieval Philosophy and Theology, Leiden: Brill. Russell, B. (1965), ‘Lectures on the philosophy of logical atomism’, in B. Russell, Logic and Knowledge: Essays 1901-1950, 323–43, London: Allen.

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Sedley, D. (1983), ‘Epicurus’ refutation of determinism’, in G. Pugliese Carratelli (ed.), Suzethesis. Studi sull’Epicureismo Greco e Romano offerti a Marcello Gigante, 11–51, Naples: G. Macchiaroli Editore. Sedley, D. (1988), ‘Epicurean anti-reductionism’, in J. Barnes and M. Mignucci (eds), Matter and Metaphysics, 295–327, Naples: Bibliopolis. Timpanaro, S. (1985), On Materialism, London: Verso. Van Inwagen, P. (1983), Essay on Free Will, Oxford: Oxford University Press. Verde, F. (2013), Elachista. La Dottrina dei Minimi nell’Epicureismo, Leuven: Leuven University Press. Verde, F. (2015), ‘Diodorus on perceptible minima’, in U. Zilioli (ed.), From the Socratics to the Socratic Schools, 134–47, New York and London: Routledge. Warren, J. (2002), Epicurus and Democritean Ethics, Cambridge: Cambridge University Press. Whitmarsh, T. (2016), Battling the Gods: Atheism in the Ancient World, London: Faber and Faber. Wilson, C. (2008), Epicureanism and the Origins of Modernity, Oxford: Oxford University Press. Wolfson, A. H. (1976), The Philosophy of the Kalam, Cambridge: Cambridge University Press. Zonta, M. (2002), La filosofia ebraica medievale. Storia e testi, Rome and Bari: Laterza.

PART I

Atomism in ancient philosophy

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CHAPTER 1

Early ancient atomism Similarities and differences ANDREW GREGORY

There is a great temptation in dealing with the early atomists, especially in a volume like Atomism in Philosophy: From Antiquity to the Present, to overplay the similarities with later and modern atomism and to underplay the differences.1 Even when differences are recognized, there can be a tendency to treat them as philosophically uninteresting or as a product of their time which can easily be ignored or rectified. It is also important not to treat the earlier atomists solely as precursors to later ones. It is common in the literature for the ancient atomists to be seen as predecessors to the mechanical philosophy of the seventeenth century. If the ancients are seen in this manner, there can be a tendency to treat the questions which exercised later atomists as critical to the ancient atomists and there can be a tendency to impose later answers as well. It is often historically more accurate and more enlightening to treat them as reacting to their predecessors and to place them in their own philosophical situation. This chapter will look at the atomism of Leucippus (fifth century BCE) and Democritus (c. 460–c. 370 BCE), who can reasonably be said to have originated the idea of atomism in the Western tradition. It will argue that while there are important similarities with later atomists, there are some interesting differences as well and we need to be cautious about imposing later ideas on Leucippus and Democritus. Key issues here are the motivation towards atomism, how atomism was argued for, what we would term conservation of energy and background assumptions about the nature of atoms and space.

PARMENIDES’ PROBLEMS Leucippus and Democritus are rightly seen as reacting to the philosopher Parmenides (c. 520–c. 450 BCE). Parmenides’ ontology was brutal in its apparent simplicity. He argued that ‘what is’, is and ‘what is not’, is not and ‘what is not’ cannot be thought of or spoken about.2 Parmenides further argued that there can be only one existent thing, which is spherical and incapable of movement or change.3 Exactly how we should understand Parmenides has been discussed ever since,4 but there is no doubting his influence in antiquity. Leucippus and Democritus are seen as in a sense asserting

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the existence of ‘what is not’, in the form of to kenon, the void. They also asserted that ‘what is’ exists as multiple, indivisible entities (Greek atomos, hence atoms). These atoms do not change in themselves, but can become entangled with each other to form more complex entities and can become un-entangled again. These changes at a micro level are what we ultimately perceive and understand as change in our perceptual world. The early atomists also appear to be strongly reductive. Democritus Fragment 9 says: By convention sweet, by convention bitter, by convention hot, by convention cold, by convention cold, but in reality atoms and the void. Here one might say, so far so good. This all looks reasonably familiar and even if the idea of the ‘entanglement’ of atoms looks crude by modern standards, at least there is some account of how atoms come together and come apart again.

FURTHER ISSUES In this context of reacting to Parmenides, there were questions which were important for the ancient Greeks, though they may look a little odd to us. If there is no longer just one entity, then how many entities are there? If there are many entities, what are their shapes? If there are many entities, what is their distribution? On the question of the shapes of the atoms, Simplicius (c. 490–c. 560CE), an important ancient commentator, tells us that Leucippus supposed there to be an infinite number of atoms that are always in motion and have an infinite number of shapes on the grounds that nothing is this rather than such (dia to mêden mallon toiouton ê toiouton einai).5 This is an interesting deployment of an ou mallon, a ‘not rather’ argument. There is no reason to prefer any one shape over another shape (so not rather this shape than that shape) so there are many shapes for the atoms. It is generally held that Leucippus and Democritus thought something similar about the sizes of atoms. There was no reason to prefer any specific size, so there were many sizes.6 On the subject of the distribution of the atoms, Plutarch (c. 46–c. 120 CE), another ancient commentator stated the following: He said that thing exists no more than (mê mallon) nothing, ‘thing’ being the name of body and ‘nothing’ of void, the latter having a nature and substance of its own.7 There is then no reason why this part of space is occupied and this part is not.8 So if asked about the distribution of atoms in the void, the Leucippus and Democritus reply is ou mallon, there is no preferred distribution, no reason why this part of space is occupied rather than that part, no reason why this atom is here rather than there. Aristotle (384–322 BCE) also has this to say about Leucippus and Democritus in his Physics: Why should there be void here rather (mallon) than there? If an entity is in one place, should it not be in all places?9

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Aristotle’s criticism comes, typically, from a strong sufficient reason perspective. Why, if all parts of the void are similar, should there be a distribution of matter, with some here and none there? Sometimes it is thought that Leucippus and Democritus subscribed to the principle of sufficient reason, but that clearly is not Aristotle’s view of them. If their view is that the shapes of atoms and the distribution of atoms are genuinely ou mallon, then it is hard to see how they could support the principle of sufficient reason. This should be seen in contrast to Parmenides who asked, if something were to grow for nothing, when would it do so (why sooner rather than later) and where it would do so,10 given the homogeneity of what is and the nonexistence of what is not.

ATOMS AND VOID AS UNLIMITED? It is standard to say that the early atomists had an infinite number of atoms in an infinite void and that there were an infinite number of shapes and sizes for those atoms. This may be an over-modernization of their thought though. The original Greek here actually derives from the word aperas, literally without limit, the a being an alpha privative and peras meaning limit. This can mean infinite, but it could also mean unlimited. Parmenides was much concerned with limits and used the term peras and its cognates.11 The early atomist reply may be to deny limits rather than to assert infinity. So there are an unlimited number of atoms in an unlimited void and these atoms have an unlimited selection of shapes and sizes. Do we read ‘infinite’ into the early atomists, making them more modern, when ‘unlimited’ might be more appropriate? Interpreting to kenon, the void, as space may also over-modernize the early atomists. So rather than atoms moving in Newtonian space (=the void), it is more likely they thought of atoms (=being) and voids (=not being) moving in space, placing atoms and voids on equal footing in space.12 This would make their reply to Parmenides more coherent. Parmenides did not deny that what exists is somewhere but did deny that there could be somewhere without what exists.13

PLATO ON ATOMISM Plato’s views are often written out of the history of atomism as his strongly teleological and a priori ideas do not sit well with modern conceptions. Nevertheless, his views are of some interest especially in contrast to the views adopted by Leucippus and Democritus. Plato (428–c. 348 BCE) postulated two types of triangle as his basic, unchangeable entities, the 1, 1, √2 triangle and the 1, √3, 2 triangle.14 These could form up into more complex planar entities, four of the 1, 1, √2 triangles into a square and six of the 1, √3, 2 triangle into an equilateral triangle. These in turn could form up into three-dimensional entities, matching the four elements of common early Greek thinking. So there was a cube of air, a tetrahedron of fire, an octahedron of air or an eikosahedron (twenty sides) of water. The two basic triangles are chosen by the demiurge, Plato’s organizer god, who did not create ex nihilo but organized from a primordial chaos. The demiurge chose these two triangles from the unlimited/infinite number of possible

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triangles because he considered them to be the two best types of triangles. While it is less clear why these two triangles are good (are they intrinsically good or good because they can generate the symmetrical Platonic solids which are the elements?), it is clear from the proliferation of the term ‘good’ in the relevant passages that Plato did think these triangles were preferable and so were chosen by the demiurge.15 This was typical of Plato’s approach to such questions. He argued for a single well-designed cosmos against the unlimited multiple worlds by accident view of Leucippus and Democritus.16 He also argued for single well-designed species against unlimited multiple biological accidents generating unviable biological entities but ultimately generating species by accident view of Empedocles. Although he does not name them, it is clear that Plato is critical of Leucippus and Democritus in several passages. The key contrast then is between the unlimited or infinite number of shapes and sizes of atoms in Leucippus and Democritus and the small number of mathematically well-defined ultimate entities in Plato. This is also important in relation to the revival of atomism in the seventeenth century. Often this is portrayed as a revival of the atomism of Democritus, Leucippus and the later ancient atomists, Epicurus and Lucretius. The problem of the shapes of the atoms remained though, and it is interesting to see how it was resolved. Boyle said that The provident Demiourgos wisely suited the fabric of the parts to the uses that were to be made of them.17 Newton said that It seems probable to me that God in the beginning formed matter into solid, massy, hard, impenetrable movable particles, of such sizes and figures and with such other properties and in such proportion in space, as most conduced to the end for which he formed them.18 As with Plato, but contrary to Leucippus and Democritus, a god makes an informed choice of which shapes for the atoms from an unlimited number of possibilities. The atomism of the seventeenth century was framed in terms of Christianity so it can be the Christian god who determines the shapes of the atoms. Note that Boyle is happy to refer to that god as a Demiourgos, the same term that Plato used for his designer god. Gone though is the idea that there is no preference for any shape so there are multiple shapes.19 It is also worth thinking about this in a broader perspective. When we think of the shape of atoms (or whatever we take to be ultimate particles), do we go for a small number of mathematically well-defined shapes or a multitude of different shapes? Think of how to make models of molecules in school chemistry. What shape are the atoms which are joined together by small rods? Then think of the Rutherford–Bohr model of the atom used in school physics and chemistry, with protons and neutrons in the nucleus, orbited by electrons. What shape are the electrons, protons and neutrons? The standard depiction is of spheres and if you agree with that depiction, you side with Parmenides and Plato, with a small number

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of mathematically well-defined shapes against Leucippus and Democritus with an indefinite number of shapes!

LIKE TO LIKE NOT FORCE Leucippus and Democritus invoked the ancient ‘like to like’ principle and this is sometimes thought of as a force which acts between atoms. That overly modernizes the early atomists and their like to like effect is better thought of as a sorting principle, by which like atoms are sorted together when in a vortex, than as a force acting between atoms. The key passage here is by Sextus Empiricus (c.160–c. 210 CE) in Against the Mathematicians VII 116–118, who says: Democritus founds his argument on both animate and inanimate things. For animals, he says, flock together (sunagelezetai) with animals of the same kind – doves with doves, cranes with cranes, and so with the other irrational animals. Similarly in the case of inanimate things, as can be seen from seeds that are being winnowed and from pebbles on the sea-shore. For in the one case the whirl of the sieve separately arranges lentils with lentils, barley with barley, wheat with wheat; and in the other case, by the motion of the waves, oval pebbles are pushed into the same place as oval pebbles, and round pebbles as round as pebbles, as though the similarity in things has some sort of ability for leading things together (hôs an sunagôgon ti echousês tôn pragmatôn tês en toutois homoiotêtos).20 The translation of the last phrase here is critical. Some commentators translate working in the word ‘force’, but actually there is no such word in the Greek here.21 The key point here is that there is no suggestion that if we leave a mixture of lentils, barley and wheat in a sieve, that they will separate out without the sieve being whirled. This was a standard method of separating seeds in ancient agriculture. Nor is there any suggestion that similar stones on the beach will separate out if they are not agitated by the waves.22 We find this in Plato as well, as he advocated a like to like sorting principle which required specific types of motion, with no universal attraction at a distance.23 This effect is critical for Leucippus and Democritus for cosmos formation. When there is a vortex of atoms in the void, then the atoms are sorted like to like and that generates a cosmos. There is no evidence that like to like works outside of vortices or before the formation of vortices. Pseudo-Plutarch tells us that, for the early atomists, The cosmos as it is now was formed in a curved manner in this way. The atomic bodies were in an unprovidential, chance, continuous and extremely rapid motion at the same time, and many of these bodies gathered together, having a variety of shapes and sizes.”24 There is no mention of any like to like attraction here. So too Simplicius tells us that When Democritus says that ‘A vortex of all shapes is separated off from the all’ (how or by what cause he does not say), it appears that this occurs spontaneously or by chance.”25

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According to Aristotle, Leucippus and Democritus viewed the formation of a vortex as a matter of chance: There are some who make chance the reason for the heaven and all of the cosmoi. For from chance arose the vortex and the motion which by separation brought the universe into a state of order.26 Aristotle has a discussion of the motion of particles outside the vortex for Democritus at Aristotle De Caelo 300b8 ff. What Aristotle argues here is that there is no natural motion outside the vortex and this in his view is incoherent. If there was like to like motion outside the vortex, Aristotle would recognize that as natural motion, but he does not even mention like to like outside the vortices.27 There is also the concern that with unlimited past time (there is no beginning to the universe for the early atomists), any like to like principle working outside of vortices would have already brought all like atoms together.28 Like to like then is not a force which acts permanently (or indeed at all) between atoms across empty space. Atoms are sorted like to like in a vortex, rather as they are in a centrifuge, the separation only happening when the centrifuge is in motion and there being no attraction between parts of the same fraction.

TAYLOR AND FORCES Taylor has argued that there is Some evidence that Democritus’ dynamics postulated three fundamental forces, a repulsive force which plays the role of impact in conventional corpuscular theory and two kinds of attractive force, one which draws together atoms of the same shape and another which holds together atoms of a different shape in an atomic aggregate.29 The problem here is the late commentator Philoponus’ concern that if atoms did actually come into contact with one another,30 nothing would separate them and they would coalesce into a single body. So to avoid this possibility, Taylor has suggested that there is a like to like force which brings like atoms together along with a repulsive force to take the place of collision and an attractive force to take the place of the entanglement of atoms which would involve them touching.31 This too overly modernizes the early atomists. There is good evidence though that the early atomists considered their atoms to be solid, as the word they used is nastos, which means ‘close pressed, firm’ and ‘solid’.32 They may have simply taken the view that when atoms collide they rebound, without expressing that explicitly. Aristotle says: Leucippus and Democritus say that the one does not come from many nor the many from one but that all things are generated by entangling (sumplokê) and scattering.33 Here sumplokê means to twine or plait together or to entangle and is used of wrestlers when they become locked together. Simplicius quotes Aristotle’s lost On Democritus to make the same point,34 and in his commentary on Aristotle’s On the

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Heavens has the intertwining of atoms due to their shape, some being hooked, some convex, some concave, etc.35 So while Philoponus raised an interesting philosophical point of what would happen when atoms collide. There is also an interesting issue with the magnitude of any supposed forces. There must be a balance for the interactions between atoms to work. If the repulsive force is too strong, atoms will not come into structures. If the attractive force is too strong, the atoms will already be in structures (there being infinite past time) and will never escape them. So why is there a balance between these forces which allows structures and human life? The explanation cannot be design, as for the early atomists there is no designer. Nor are there many universes with different balance of forces where we might explain our universe as one of an infinite array of universes. For Leucippus and Democritus, there are many cosmoi (plural of cosmos), worlds with an earth, sun, moon planets and stars, but these are all in one universe for them. One might also be concerned that given the way that Leucippus and Democritus deploy ou mallon considerations, what value for each of the forces would be preferred to any other?

MULTIPLE WORLDS AND COSMOGONY The early atomists gave us the first multiple worlds theory, though this needs to be phrased carefully. They believed there was one universe and one space. Within this space there were many differing worlds. So this is a multiple cosmos theory, where a Greek cosmos should be understood to have a central earth, sun, moon, planets and stars. Hippolytus (c. 170–c. 235 CE) tells us that Democritus holds the same view as Leucippus . . . There are innumerable cosmoi, which differ in size. In some of these there is no sun or moon, in some they are larger than ours and in some more numerous. The spaces between cosmoi are not equal, in places there are more and in others less, some are growing, some are in their prime, some declining, some are coming to be and others failing. They are destroyed by falling into each other. There are cosmoi bereft of animals and vegetation and all moisture. In our cosmos the earth was generated prior to the stars, and the moon is the lowest, followed by the sun and then the stars. A cosmos is at its height until it can no longer accrete external material.36 There is no beginning and no end to this process of the generation and ultimate destruction of cosmoi. There was a very clear dichotomy in early Greek cosmogony between those who believed there to be one, well-designed cosmos and those who believed in multiple cosmoi which came about by chance. This is the first clear account we have of multiple cosmoi coexisting at the same time. Empedocles believed in only one cosmos at a time, but each time the cosmos was destroyed, it re-formed again so he believed in multiple successive cosmoi. This is also the first attempt to explain in cosmogony using multiple worlds. We can explain the order of our cosmos as one instance of many differing cosmoi and have no need of a designer to explain its order. In one way, Leucippus and Democritus accept

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Parmenides. These cosmoi are not generated ex nihilo, but by the rearrangement of atoms which themselves have no origin and undergo no change. Against Parmenides though, these cosmoi come into existence without any preferred time or place for them to do so.

COLLISIONS AND MOTION Did the early atomists believe in something like the conservation of energy? As their conception of the universe did not have a beginning or an end and cosmoi in that universe are continually being created and destroyed, it is easy to assume that they did.37 Matters are not quite so simple though. The following passage from Seneca Natural Questions V/ 2 has been much discussed: Democritus says that when in an empty space there are many small corpuscles, which he calls atoms, then there is wind. On the contrary, the air is placid and static when in a large space there are few small corpuscles. For example, when there are not many people in the market or a street, they can walk without tumult, but when a crowd comes together in a small place, they bump into each other and there is a commotion. The same thing happens in the space which surrounds us. When many bodies fill a small space, they of necessity bump and push each other and are knocked back, entwined and compressed. Winds are produced from this, particles jostling with each other and pressing hard for a long time begin to move in one direction. When only a few bodies move in much space, they cannot bump into one another or be impelled. What is clear here is that the extra jostling not only makes them move in the same direction but more motion is also generated as the commotion grows. The implication of this for the atoms is that greater jostling leads to greater speed, or collisions generate motions.38 It was also common in the ancient world, both before and after the early atomists, to believe that motion petered out.39 One can construct an artificial solution here on the idea that perhaps this extra motion from collision balances out any motion lost. However, this would be open to many of the objections to Taylor’s proposed extra forces. Without a designer, how is that balance achieved and maintained, especially with Leucippus and Democritus deploying so many ou mallon considerations? While the conservation of motion is something which concerns us and concerned the seventeenth-century mechanical philosophers, we must not immediately assume that it was a consideration for the early atomists in their reply to Parmenides.

MECHANICAL THINKING? It is often said that Leucippus and Democritus were mechanists. Much here depends on what is meant by ‘mechanist’. The two most useful definitions are that mechanists use mechanical analogies for natural processes, as with the mechanical philosophers of the seventeenth century using clockwork as their key analogy. Secondly, one might

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be a mechanist in the sense of applying mathematics to the motions of the particles. In the first sense here, Leucippus and Democritus were certainly not mechanists. I agree with Berryman that There is little credible evidence before Aristotle’s time that might count as the use of working artefacts in understanding the functioning of nature.40 I also agree with Furley that the available Greek mechanê ‘devices’ of the time did not display the characteristics of regularity or reliability that later machines, in particular clocks, came to exhibit.41 A mechanê for an early Greek was not a machine, but a contrivance. If Leucippus and Democritus had used machine analogies, this would be miraculous in an age when there were no reliable machines to model natural processes on. So what analogues did the early atomists actually use for natural processes? Animals flocking together is a biological analogy, as is humans bumping into each other generating more motion, while seeds being winnowed is an agricultural analogy42 and like-shaped pebbles gathered together on the shore is a maritime analogy. The vortex itself is meteorological or maritime analogy, depending on whether one is thinking of a whirlwind or a whirlpool. Seneca Natural Questions V/ 2 as we have seen has the origin of winds likened to the behaviour of humans in a crowded marketplace, so we have a human analogy. Diogenes Laertius gives this account of early atomist cosmogony: Leucippus holds that the whole is infinite . . . part of it is full, and part void . . . from these innumerable cosmoi come to be and are dissolved into these again. The cosmoi are generated in this manner. By cutting off (apotomên) from the infinite many bodies of all shapes move into a great void, where they are crowded together and produce a single vortex, where colliding with each other and circulating in all manner of ways, they separate out like to like. When, because of their great number they are no longer capable of moving around in equilibrium, those that are fine spread out into the outside void, as if sifted, while the rest hold together and becoming entangled, they unite their motions and create the first spherical structure. This stands apart like a membrane (humena), containing in itself all kinds of bodies. As they whirl around, due to the resistance of the middle, the surrounding membrane (humena) becomes thin, and the close packed atoms flow together due to touching the vortex. In this way the earth came into being, the atoms which had been borne in to the middle remaining there together. Again the surrounding membrane (humena) itself is increased, due to the influx of external bodies. As it moves around in the vortex, it takes in whatever it touches. Some of the bodies which become entangled form a structure which is firstly moist and muddy, but which dries out as it revolves with the vortex of the whole, and then ignites to produce the constitution of the stars.43 So all of the cosmoi have surrounding membranes, humena, which is a ‘thin skin, membrane, caul’ such as those which enclose the brain or heart, so again we have a biological analogy. So there are no machine analogies in Leucippus and Democritus. We get what we might expect from Greeks of their time – biological, agricultural, meteorological, human and maritime analogues.

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One might argue that the winnowing of seeds and the sorting of pebbles on the beach to be mechanical analogies, in that both can be analysed solely in terms of particles bumping into each other. While we in the twenty-first century might analyse in that way, the ancient did not. An important methodological point here. Unguru, in a seminal paper, has argued that we should not treat Greek mathematics, which is usually expressed in a geometrical manner, as modern algebra in a different guise.44 The geometrical form was important to the ancient Greeks and something is lost if we express their mathematics solely in algebraic terms. Doing so is often misleading, imposes anachronistic ideas and can lose important aspects of how a mathematical proof worked for the Greeks. Here, Leucippus and Democritus use a maritime (pebbles and sea) and an agricultural (winnowing basket) analogy. Just as Greek mathematics had a geometrical form which is not reducible to algebra, nor is it algebra dressed in a different language, so too these analogies are not mechanical, and they are not mechanical analogies dressed in a different language. We need to recognize they had a nature of their own appropriate to the thought of the time. That their system was atoms bumping into one another does not make it mechanical to them. Materialism does not imply mechanism, especially in an ancient context. We in the twenty-first century can see how one can turn the system of Leucippus and Democritus into a mechanical one and one might argue that indeed happened in the seventeenth century, but Leucippus and Democritus were not themselves mechanists.45

APPLICATION OF MATHEMATICS? Some have taken the application of mathematics to atoms as critical for a mechanical view. So Lonie says that A mechanistic explanation is one which involves the mathematical application of the science of mechanics to bodies in motion.46 Leucippus and Democritus, as a simple matter of historical fact, did not apply any mathematical mechanics to their atoms. In this sense, their view was not mechanical. I would agree with Lloyd that It is hardly an exaggeration to say that before Aristotle there is nothing that can be called dynamics at all in Greek science.47 Is it possible to ‘retrofit’ a mathematical science of mechanics to the early atomists? I phrase this in such a manner as there are considerable issues of anachronism involved in such an approach. It may look relatively straightforward to do so if we abstract a simple atoms and void view from Leucippus and Democritus. However, if we take their ou mallon considerations about the shape, size and distribution of atoms seriously, then the issue is much more complex. How could one attach numbers to atoms when there are no preferred shapes, sizes or distributions? If motion is gained in collisions or peters out, is it possible to impose a calculus on that? A final comment on the early atomists and the universe as a mechanism. That is a much easier idea for the Christian atomists of the seventeenth century to

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formulate as their god can make the universe act as a machine. A machine, it might be argued, implies some sense of design. That though is antithetical to the approach of Leucippus and Democritus, who denied any role for design in the universe. As Dijkserterhuis has commented, the machine analogy is ‘in every respect the opposite of the Democritean world-picture’.48

DETERMINISM? It has generally been assumed that the early atomists’ system was deterministic, but that too is questionable.49 Was it deterministic in the sense of conforming to Laplace’s demon? If everything about atoms (position, velocity, mass, shape) is known at one moment, can all future states then be determined? The original Laplace is: We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.50 The issue here again is the ou mallon consideration. The shape of atoms is certainly ou mallon and it is likely that their size is too. Their distribution is ou mallon and it is likely that if we were to ask Leucippus and Democritus about their velocities, they would say these are ou mallon too. So how could an intellect know about the relevant data? If we take it that the atoms cannot be perceived, there is no empirical means. If the shape and other attributes of atoms are genuinely ou mallon, it is hard to see how the data can be determined intellectually. There is a further consideration here which is that the formation of vortices and subsequent cosmos formation are also ou mallon with respect to space and time. That is, there are no preferred place and no preferred times for vortex formation. If these are genuinely ou mallon, then there are future states of the system which cannot be predicted. In this sense the system is not deterministic and does not conform to Laplace’s demon. It is of course tempting to read determinism into the early atomists as a precursor of the strongly deterministic mechanical philosophies of the seventeenth century and beyond. Whether Leucippus and Democritus were concerned about determinism is another matter. There are two further interesting differences with Laplace. While Laplace envisaged ‘an intellect which at a certain moment would know all forces that set nature in motion’,51 the early atomists did not have nature being set in motion. There was no first state of the universe for them. Secondly, to Laplace, firmly rooted in a Christian tradition of an omnipotent and omniscient god, the idea of an intellect which would be capable of comprehending and calculating on this sort of scale would come relatively naturally. However, we find nothing of this sort among the early Greeks and certainly nothing of this sort in the early atomists.

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NECESSITY AND DETERMINISM? How does this fit with Leucippus Fr. 2 which allegedly says that everything happens by necessity and nothing happens by chance? To give this fragment its full context: Democritus says everything happens by necessity. This is the same as fate, justice, providence and cosmos making (kosmopoion). Leucippus says that everything happens by necessity, which is the same as fate. He says in his On Mind, ‘Nothing happens at random (matên), but everything for a reason and by necessity.’52 As with the like to like principle, it is important here to distinguish between what happens in vortices and what happens outside of vortices. What happens within vortices, that is, with ‘cosmos making’, is necessary. The generation of vortices, where and when they occur in space and time is not. This agrees with the evidence of Diogenes Laertius, who says that for Democritus: Everything occurs by necessity, the vortex being the cause for the coming into being of all things, and this he calls necessity.53 We should read the first ‘everything’ here in a slightly restricted way as meaning everything in a cosmos rather than everything in the universe. That is justified by the second clause, where the vortex is the cause for everything coming to be. In Physics II 4, Aristotle contrasts the chance formation of vortices with the subsequent formation of animals and plants for them, which he says happens ‘by nature’ and not by chance.54 Simplicius backs Aristotle on this point and says that the ‘ancient theory which denies chance’ refers to Democritus, who made use of chance when explaining the formation of cosmoi, but then says that Democritus did not use chance to explain anything subsequent to cosmos formation, always citing normal causes.55 Later in antiquity, Epicurus (341–270 BCE) and Lucretius (c. 99–c. 55 BCE) introduced the idea of the atomic swerve, an occasional, spontaneous, undetermined motion of atoms. It is thought they did this to avoid the determinism of Leucippus and Democritus and its implications for free will.56 Again, we must distinguish between what happens in vortices and what happens outside them. For Leucippus and Democritus, humans are part of a cosmos and so are in a vortex and so are subject to necessity.57

CONCLUSION The earliest atomists, Leucippus and Democritus had some radical and seminal ideas. They were the originators of the idea of atomism, with multiple, unchangeable, mobile atoms, the interactions of those atoms providing the basis for our perceptual world. Their view was strongly reductive, all perceptual qualities being reduced to the properties of atoms. They originated the idea of multiple worlds and explanation of the order of our world as one in an array of many worlds. There are though important differences with later atomism which need to be recognized, largely stemming from their use of ou mallon considerations. They believed the shapes and in all likelihood the sizes of the atoms to be ou mallon, so too their distribution in

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the void. Vortex and subsequent cosmos formation were also ou mallon with respect to time and space. Like to like was not a force of attraction acting between atoms for them, it was a sorting principle which only operated when certain types of motion were present. It is likely that their view was not ‘atoms and void’ with the void as Newtonian space, but ‘what is’ (atoms) and ‘what is not’ (void) in space. We should not consider Leucippus and Democritus to be mechanists. They used biological, agricultural, human and maritime analogies for natural processes, but no machine analogies. They made no attempt to apply mathematics to motions and collisions of the atoms. Whether they believed in the conservation of energy for atomic motions and collisions is unclear. In an interesting sense they were not determinists, as outside of the vortices the future states of the system, such as time and place of vortex formation, were subject to ou mallon considerations. The early atomists were marvellously original and some of their ideas have been influential throughout the history of Western science. They are though best viewed as reacting to Parmenides, when the differences from later ideas become apparent and comprehensible in that context, rather than as precursors to later atomism when those differences are often glossed over.

NOTES 1. The older standard work on the early atomists, Bailey (1928) and the modern one, Taylor (1999) both do this to varying extents. 2. See Parmenides Fr, 2, 3, 6, & 7. 3. See Parmenides Fr. 8. 4. We have only some fragments of Parmenides’ work which was written in poetry. See Gregory (2014). 5. Simplicius, Physics, 8, 28. 6. Aristotle, On Democritus from Simplicius, De Caelo, 295, 5 brings in differences in size along with differences in shape for the early atomists. 7. Plutarch (1967, 1109a). 8. Barnes (1982, 405). 9. Aristotle (1936, 203b28). 10. Parmenides Fr. 8 6-10, 22-25. 11. See Parmenides Fr. 8 lines 31 & 49. 12. See Sedley (1982). This makes sense of talk of ‘the full’ and ‘the empty’. Plutarch (1967, 1109a). 13. See Parmenides Fr 8 29-30, Sedley (1982). 14. See Plato (1900–1907, 53c ff). 15. See Plato (1900–1907, 53d ff). See Gregory (2000, ch. 8). 16. See Plato (1900–1907, 55c ff). See Gregory (2007, ch. 9). See Pitteloud’s chapter in this collection. 17. Boyle (1772).

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18. Newton (1730, IV, 260). 19. Cf. Garber and Roux (2013) for the complexities of the relation of ancient atomism to that of the seventeenth century. 20. Cf. Pseudo-Plutarch, IV/ 19. 21. Taylor (1999, 5), has ‘as if the similarity in things had a kind of attractive force’. However, sunagôgon does not have this sense of force, rather it is a leading together. At Odyssey XVII/ 218 we find that ‘As always, the god leads like to like’ (hôs aiei ton homoion agei theos hôs ton homoion). Cf. sunagelezetai, to flock together, which is what some animals do. 22. This sieving does work and was agricultural practice. The contents of the sieve are separated out by density. 23. Plato is critical of cosmogony based on this alone, as for him the cosmos is a harmonious blend of opposites, something highly unlikely to be produced from a like to like principle alone, see Laws 889b. At Timaeus 80bc electricity and magnetism are explained as due to contact action and mutual replacement, and there is an outright denial that any attraction (holkê, Timaeus 80c3) is involved. In Plato’s pre-cosmic chaos, the receptacle shakes like a winnowing basket to produce a separation of like to like. 24. Pseudo-Plutarch, I/4. 25. Simplicius (1989–2011, 24, 327). See also Simplicius (1989–2011, 14, 327, 330), Themistius (2003–12, 13, 49), Cicero (2010, I, 24, 66). Simplicius (1989–2011, 14–17, 330) also says that ‘It would seem that “the ancient theory which denies chance” refers to Democritus. For although he appears to use chance in the making of cosmoi, when he is more nuanced he denies that chance is the reason for anything.’ 26. Aristotle (1936, II/4, 196a24 ff). 27. Cf. Aristotle (2002, 641b20-24). 28. This was an ancient concern – Plato was worried that like to like on its own would not produce a cosmos, just a sorting of like things, and Aristotle was concerned that his ideas of natural motion, again working over unlimited past time, would simply have sorted the elements rather than allowing a cosmos. 29. Taylor (1999, 194). 30. Philoponus (1991–1994, 494, 198 ff), Commentary on Aristotle’s On Generation and Corruption, 158, 26 ff. and 160, 7 ff. 31. Taylor (1999, 194). 32. Aristotle Fr. 208, Stobaeus I, 10, 14, and I, 14, 1 and I, 16, 1, Simplicius (2001– 2011, 5, 295), Philo Judaeus (2013), 7, 3 all use nastos. 33. Aristotle, On the Heavens, III/4, 303a7. 34. Simplicius (2001–2011, 11, 295). 35. Simplicius (2001–2011, 24, 242). 36. Hippolytus, Ref. 1, 13, 2. Cf. Aristotle (1936), B4, 196a24 ff. 37. Especially as the atoms are said to be ‘always in motion’, Simplicius (1989–2011, 8, 28). 38. See Balme (1941, 25). Cf. Balme (1939).

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39. See Balme (1941, 24). 40. Berryman (2009, 7). 41. Furley (1987, 13). Cf. Furley (1967). 42. Metrodorus of Chios, in relation to the many cosmoi theory, says that ‘It is strange for one ear of corn to be produced in a great plain, and for one world in the boundless’ which gives us another agricultural analogy. 43. Diogenes Laertius, IX, 31. 44. Unguru (1975). Cf. Netz (1999). 45. Cf. Hirsch (1990). 46. Lonie (1981, 123). 47. Lloyd (1970, 112), cf. (1987, 217). 48. Dijksterhuis (1961, 12). 49. Myself included, see Gregory (2013) which accepted full determinism in the early atomists. 50. Laplace (1902). 51. Laplace (1902). 52. Stobaeus, I, 4. 53. Diogenes Laertius, IX, 45. 54. Aristotle (1936, II/4, 196a25 ff.), cf. (2002, 641b20 ff). 55. Simplicius (1989–2011, 14–20, 330). 56. See Sedley (1983). 57. There is also a current discussion in relation to Aristotle and whether causal determinism and free will are appropriate or anachronistic ideas applied to his views.

REFERENCES Aristotle (1936), Physics, W. D. Ross, Oxford: Oxford University Press. Aristotle (2002), On the Parts of Animals. Translated with an introduction and commentary by James G. Lennox, Oxford: Oxford University Press. Bailey, C. (1928), The Greek Atomists and Epicurus, New York: Russell & Russell. Balme, D. (1939), ‘Greek science and mechanism I. Aristotle on nature and chance’, Classical Quarterly 33: 123–38. Balme, D. (1941), ‘Greek science and mechanism II. The atomists’, Classical Quarterly 35: 23–8. Barnes, J. (1982), The Presocratic Philosophers, 2nd edn, 1982, London: Routledge and Kegan Paul. Berryman, S. (2009), The Mechanical Hypothesis in Ancient Greek Natural Philosophy, Cambridge: Cambridge University Press. Boyle, R. (1772), The Works of the Honourable Robert Boyle: In Six Volumes. To Which Is Prefixed the Life of the Author, London: Rivington.

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Cicero, De Natura Deorum Libri Tres: With Introduction and Commentary (2010). Edited by J. B. Mayor and J. H. Swainson, Cambridge: Cambridge University Press. Dijksterhuis, E. J. (1961), The Mechanisation of the World Picture, trans. C. Dikshoorn, Oxford: Oxford University Press. Furley, D. (1967), Two Studies in the Greek Atomists, Princeton: Princeton University Press. Furley, D. (1987), The Greek Cosmologists, Cambridge: Cambridge University Press. Garber, D. and Roux, S. (2013), The Mechanisation of Natural Philosophy, Dordrecht, Heidelberg, New York and London: Springer. Gregory, A. D. (2000), Plato’s Philosophy of Science, London: Duckworth. Gregory, A. D. (2007), Ancient Greek Cosmogony, London: Duckworth. Gregory, A. D. (2013), ‘Leucippus and Democritus on like to like and ou mallon’, Apeiron 46: 446–68. Gregory, A. D. (2014), ‘Parmenides, cosmology and sufficient reason’, Apeiron 47: 16–47. Hirsch, U. (1990), ‘Was Demokrits Weldbilt Mechanistisch und Antiteleologisch?’ Phronesis 35: 225–34. Laplace, P. S. (1902), A Philosophical Essay on Probabilities. Translated from the 6th French ed. F. W. Truscott and F. L. Emory, New York: Wiley. Lloyd, G. E. R. (1970), Early Greek Science to Aristotle, London: Chatto and Windus. Lloyd, G. E. R. (1987), The Revolutions of Wisdom, Berkeley: California University Press. Lonie, I. M. (1981), ‘Hippocrates the Iatromechanist’, Medical History 25: 113–50. Netz, R. (1999), The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History, Cambridge: Cambridge University Press. Newton, I. (1730), Opticks: or, A Treatise of the Reflections, Refractions, Inflections and Colours of Light, London: Innys. Philo Judaeus (2013), De Plantatione. In Philo, with an English translation by F. H. Colson and G. H. Whitaker, Cambridge, MA: Harvard University Press, 2014. Philoponus (1991–1994), Commentary on Aristotle’s Physics, London: Duckworth. Plato (1900–1907), Timaeus. Platonis Opera. J. Burnet, Oxford: Oxford University Press. Plutarch (1967), Adversus Colotem. Reply to Colotes in Defence of the other Philosophers. Moralia, LCL 428, Cambridge, MA: Harvard University Press. Sedley, D. (1982), ‘Two conceptions of the vacuum’, Phronesis XXVII: 175–93. Sedley, D. (1983), ‘Epicurus’ refutation of determinism’, in Suzetesis, Studi sull’ Epicureismo Greco e Romano Offerti a Marcello Gigante, 11–51, Naples: Gaetano Macchiaroli. Simplicius, Physics Commentary (1989–2011), On Aristotle, Physics, London: Duckworth. Simplicius, De Caelo Commentary (2001–2011), On Aristotle, On the Heavens, London: Duckworth. Taylor, C. C. W. (1999), The Atomists: Leucippus and Democritus, Toronto: Toronto University Press. Themistius (2003–2012), On Aristotle’s Physics. Translated by R. B. Todd, Ithaca: Cornell University Press. Unguru, S. (1975), ‘On the need to rewrite the history of Greek mathematics’, Archive for the History of the Exact Sciences 15: 67–114.

CHAPTER 2

The reception of atomism in ancient medical literature From Hippocrates to Galen VINCENZO DAMIANI

INTRODUCTION Preliminary remarks1 An anecdote reported by Diogenes Laertius (be the details true or not) gives us an idea of the wide diffusion of Democritus’ works already in Plato’s times (Diog. Laert. 9,40): Aristoxenus, in his Historical Commentaries, says that Plato wanted to burn all the copies he could collect of Democritus’ works, and that the Pythagoreans Amyclas and Clinias prevented him, arguing that it would do no good, since by then his books had been widely circulated. (trans. Mensch 2018) The influence of Democritus’ doctrines in various fields of knowledge, and in particular the influence of his atomistic approach to reality, was in fact often controversial but, in any case, far from negligible. According to the atomistic theory, as it can be very schematically reconstructed from the disparate sources available to us (apart from Leucippus’ and Democritus’ fragments,2 Epicurus’3 Letter to Herodotus, his treatise On nature and Lucretius’ On the Nature of Things), reality is composed of incorruptible and indivisible parts (ἄτομα) moving randomly through the void (τὸ κενόν). In the course of their motion, due to the nature of motion itself (a centripetal vortex (δῖνος), according to Democritus)4 or to a deviation from a falling motion in a straight line (according to Epicurus),5 the atoms aggregate with each other. They own few essential qualities: shape (ῥυσμός/σχῆμα), orientation (τροπή/θέσις), arrangement (διαθιγή/τάξις) (Democritus);6 weight (βάρος), size (μέγεθος) and shape (σχῆμα) (Epicurus).7 The qualities of compound bodies are not originally proper to atoms but derived ones,

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that is, they depend on the different possibilities of aggregation of the atoms and are not always perceived by the senses in the same way.8 The processes of generation and corruption are defined as aggregation or disintegration of atoms. As initiated by the relatively obscure Leucippus and later developed by Democritus,9 atomism runs through the history of ancient philosophy with alternate fortunes, but always remains an inescapable benchmark: it is implicitly present in Plato’s reflection on the physical constituents of nature (with a particular view to the dialogue Timaeus), then respectfully criticized by Aristotle, who also delivers a major testimony to Democritus’ physical tenets, eventually taken up, as we have just seen, in a systematic way by Epicurus (through the teachings of the Democritean Nausiphanes) and his followers (in primis Lucretius and Diogenes of Oenoanda). Though, the theoretical impact of atomism in antiquity does not remain restricted to philosophy alone but can also be traced in the so-called Fachliteratur (e.g. mechanics and pneumatics) and especially in medicine, which is the subject of this contribution. The title of the chapter mentions the term ‘reception’. To be sure, we can speak of ‘reception’ in a narrow sense only in a few specific cases, and it will be necessary, at any rate, to distinguish between different modes of reception. As we will see, the relationship that exists – or could have existed – between some of Democritus’ medical–biological doctrines (especially on embryology) and a certain group of writings of the Corpus Hippocraticum should best be excluded from a strict definition of the reception of atomism, as the direction of this very reception cannot univocally be defined nor can a mutual influence be ruled out. In the theories of the Hellenistic physician Erasistratus of Ceos (c. 315–240 BC), in turn, one can actually detect a re-appropriation of atomism which is not confined to single doctrines but looks at it as a whole system. The so-called empiricists, on their part, regard atomism rather as a model of method than as a source of physical explanations.10 It is only with Asclepiades of Bithynia (second–first century BC) that an example of a consequent reception and reuse of the atomistic paradigm shows up, where a corpuscular concept of matter is consistently linked with the aetiology of physiopathological phenomena. Scientists of the imperial age will at times regard Democritus not only as an authority in terms of medical knowledge but also as a technical writer – in any case, hardly as the father of the atomistic theory. We will see that a major exception to this pattern is the physician Galen of Pergamum (AD 129–c. 216): as an exponent of a philosophical eclecticism that makes teleology and providentialism its cornerstones, he turns Democritus’ mechanistic11 view (often coupled with that of Epicurus or/and Asclepiades) into a target of systematic criticism. It is in fact extremely difficult to establish with certainty what can be traced back to Democritus and what cannot: a number of theories reported by the doxographical tradition, while based on a corpuscular view of matter or postulating the existence of void, might not be attributable to the Abderite all the same;12 similarly, not every doctrine claiming a rationalistic (or even seemingly mechanistic) explanation of the physical world must ipso facto be directly echoing a certain Democritean tenet, as rationalism is an issue that actually permeates Ionian thought altogether.13 That being so, a quantum of suspiciousness should temper the temptation of overly optimistic reconstructions.

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Democritus as a medical writer Besides philosophy and natural science, Democritus’ writings show wide interests in other more technical fields of knowledge as well, including medicine itself. Some of the titles listed in the catalogue of his works reported by Diogenes Laertius (following the tetralogical grouping by Thrasyllus)14 patently hint at specifically medical topics and even bear a close terminological similarity to treatises belonging to the Corpus Hippocraticum: that is particularly the case (68 A 33 XII DK) for On Prognosis;15 On Regimen,16 a reference to which Diels–Kranz seem to recognize in CH, Diaet. p. 466 Littré (68 B 26c DK), where the author mentions ‘those who in the past wrote about the human diet’; On the Nature of Man or On Flesh.17 As a matter of fact, we cannot tell with certainty whether such writings actually came out of Democritus’ pen or were rather ad hoc inventions – such as the narrative of Hippocrates having been Democritus’ direct pupil;18 it remains undoubted, at any rate, that Democritus did tackle medical questions in his works.19 According to Clement of Alexandria, Democritus argued that ‘Medicine . . . cures the illnesses of the body, but wisdom removes the soul from its affections’,20 and the very affections of the soul are, in turn, the primary source of illness to the body. His idea of health is chiefly based on conscious observance of a salubrious life regimen:21 men’s soul always has a choice either to guide the body to a healthy conduct or to ruin it by surrendering to pleasure and debauchery, in the same way as a tool can be ruined by those who make a reckless use of it.22 In physical terms, the soul, just like heath, is a material compound of atoms of spherical form (σφαιροειδεῖς . . . ῥυσμοί) that tend to flow out of the body due to the pressure acting from outside (τοῦ περιέ­ χοντος ἐκθλίβοντος). This tendency is countered by the inhalation of air, which itself contains parts of νοῦς and ψυχή. Once respiration cannot compensate the outside pressure anymore, death occurs.23 It seems that Democritus must in some way have treated the problem of the anatomy of vessels and the physiology of pulse. Erotianus, in his glossary of Hippocratic terms, observes:24 He (scil. Hippocrates) called veins not those that are generally referred to by this name, but the arteries; and Democritus too calls φλεβοπαλία (‘beating of the pulse’) the movement of the arteries.25 In addition, Hesychius reports the Democritean use of the term δεξάμεναι indicating the vessels, which more specifically refers to their function of conducting bodily fluids:26 ‘dexamenai: containers for fluids, and the veins [or rather “arteries”? – see previous] in the body. From Democritus’. To an interest in medical terminology as well as in aetiopathogenesis (the causal genesis of disorders), and a possible acquaintance with some form of humoral theory, might testify a passing remark in Soranus’ Gynecology:27 ‘Phlegmone’ (‘inflammed tumour’) received its name from phlegein (‘to inflame’) and not, as Democritus said, from the fact that its cause is phlegm. A passage from the extensive fragment from Theophrastus’ De sensibus could contain a reference to a ‘mixture’ (κρῆσις) of the internal humors:28 ‘Furthermore,

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people themselves change in their mixture according to their affections and age’.29 Democritus’ sceptical views on the possibility of establishing a poor prognosis based on certain signs (e.g. the well-known facies Hippocratica)30 are adumbrated by Celsus when treating the question of whether dead bodies can be still capable of sensation (a question Democritus answered positively):31 Indeed, Democritus, a man of a justly great renown, maintained that there are not even sufficiently certain signs of the ending of life on which doctors could rely; and even less did he accept that there might be any certain signs of a future death. Besides these few scattered remarks, a substantial part of Democritus’ opinions on medical issues regards embryology and the process of human reproduction. Since these are the theories that have been most debated among scholars as to their relationship with the Corpus Hippocraticum, they will be presented and discussed in the next section.

DEMOCRITUS AND THE CORPUS HIPPOCRATICUM Democritus and Hippocrates: A legendary acquaintance? There are a number of allegedly biographical accounts that attempt at drawing a link between Democritus and Hippocrates and at proving that the philosopher and the physician actually got to know each other. Some sources maintain that Hippocrates had been, in a more or less direct way, a pupil of Democritus: the lemma Δημόκριτος of the Lexicon Suda makes Hippocrates a hearer of Metrodorus of Chios, one of the most prominent of Democritus’ pupils;32 in another passage (s.v. Ἱπποκράτης) the young physician is presented as a direct disciple of the old Abderite.33 Diogenes Laertius refers to two anecdotes by the Stoic Athenodorus in which Hippocrates meets Democritus and is struck by his sharpness of observation.34 The most conspicuous instance of such a biographical tradition is the pseudepigraphic Briefroman narrating how Hippocrates was called for by the citizens of Abdera to heal Democritus’ supposed madness. He will find out that the philosopher’s uncontainable laughing at everything in the world was indeed a sign of his unsurpassed wisdom.35 Democritus is portraited tranquilly sitting near a stream, surrounded by books and dissected animals, engrossed in the search of the bodily seat of bile (χολή), the humor that was held responsible of causing folly when exceeding the natural measure (although coming from a spurious testimony, this detail is a further proof of Democritus’ activity in the field of medicine).36 Letter 23 even pretends to offer the full text of Democritus’ writing On the Nature of Man, which actually corresponds to the title of one of the writings attributed to him (see earlier section). To briefly sum up: it is in fact possible that Hippocrates and Democritus were roughly contemporaries (in 68 A 4 DK = P 12 LM their floruit is dated between 436 and 433 BC), but this is also pretty much all we can tell beyond conjecture.37

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Common doctrines The scholarly debate  There has been a lively discussion about how Democritean doctrines might have found their way into some writings of the Corpus Hippocraticum, especially those dedicated to problems of embryology and reproduction, namely On Generation, On the Nature of the Child, On Diseases IV.38 Two main positions have emerged so far in the scholarship, the one claiming a direct reception of Democritus’ biological tenets in those writings,39 the other either arguing for a more cautious interpretation of possible similarities as originated from a common background or even reversing the direction of reception, with Democritus drawing on the observations of an Hippocratic author.40 One point scholars seem to agree on is the absence of substantial traces of elements specifically pertaining to atomic theory in the Corpus;41 what we find there is, at most, the echo of specific doctrines, and in particular,42 (1) the so-called ‘pangenesis-theory’ and the hypothesis that females also produce semen, (2) the explanation of the sexual differentiation of the foetus, (3) the aetiology of twin or multiple births, (4) the phases of development of the embryo and its nourishment in the womb, (5) the relationship between climate and the likelihood of an abortion. Given the controversial character of the relationship between Democritus and the Hippocratic treatises, it will be safer to speak, in the following paragraph, of ‘consonances’ or ‘similarities’ rather than of ‘reuse’ or ‘reception’ in one direction or the other. Embryology and genetics  Chapters 6–8 of On Generation contain a detailed exposition of the aforementioned pangenesis-theory43 and the related (ἐπι)κράτειαtheory (with the former accounting for the transmission of particular somatic characters, the latter for sexual differentiation). Their main points are: (1) both men and women produce both male and female seed; (2) the sex of the child depends on which seed is present in a greater amount (κρατεῖν); (3) the seed comes from the whole body of both men and women (cf. Genit. 3: ἀπὸ παντὸς τοῦ σώματος; Morb. IV 1 = 32 Littré); depending on which body part ‘enters the seed’ (ἔλθῃ ἐς τὴν γονήν) in a greater amount, the child’s single body parts will resemble those of the mother or the father. Some of Democritus’ fragments show a close correspondence with those positions: (1) 68 A 142 DK = D 166 LM – ‘Also the female (καὶ τὸ θῆλυ) produces seed’;44 (2) 68 A 143 DK = 173–174 LM – the differentiation between male and female takes place in the mother’s womb (ἐν τῇ μητρί) and depends on which of the seeds (from the mother and from the father) predominates (κρατήσῃ, κατ’ ἐπικράτειαν);45 (3) 68 A 141 DK = D 165 LM – ‘[The seed comes] from the whole body (ἀφ’ ὅλων τῶν σωμάτων) and its main parts’.46 Democritus consequently refused Alcmaeon’s theory47 according to which seed originates solely from the brain and marrow.48 A somewhat cryptic utterance by the Abderite himself is quoted (probably in a corrupted version) by Ps.-Galen (Definitiones medicae) to support the ‘pangenetic’ hypothesis: ‘One will be humans and a human will be all.’49 There is of course room for conjectures as to how Democritus’ theories can be referred to an atomistic background in the narrower sense – the same can be

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said of the other embryological tenets discussed further. One hypothesis is to think of the seed as a material body50 that is composed, as any other σῶμα, of atoms of different kinds: these re-aggregate during sexual intercourse, forming a new compound;51 one can also conceive of seed as keeping, in a certain measure, the peculiar atomic structures of each parent: the ἐπικράτεια-theory would then explain why and in which way these structures are actually inherited by the child. On the other hand, the fact that these very theories are also extant in some Hippocratic writings rather seems to account for their being independent from any specific physical model of explanation than to confirm their derivation from an atomistic conception of matter.52 The concluding section of the treatise On the Nature of the Child (Ch. 20 = 31 Littré) is concerned with the aetiology of twin and multiple births. The female uterus possesses more than one cavity (κόλποι), and each of them can receive seed. If two cavities do not happen to merge in one, a membrane (ὑμήν) is formed therein, from which the embryo begins to develop. The observation of how animals such as swine and dogs give birth to several offspring at once confirms that a single mating (ἀφ’ ἐνὸς λαγνεύματος) is sufficient for a multiple birth to take place. In Democritus’ fr. 68 A 151 DK (= D 179 LM), reported by Aelianus, exactly the same examples (swine and dogs) are used to explain the physiology of multiple births, although in this case it is clearly stated that a single act of copulation is not enough to fill up all sinuses in the mother’s uterus.53 Besides the general methodological reserves exposed earlier, at least this discrepancy proves the difficulty of establishing a clear relationship between a source account and a derived one.54 Let us now look at the different stages of development of the child as described in On the Nature of the Child (Nat. Puer. 6–9 = 17–20 Littré). As a general principle, the Hippocratic author states that the development of the parts of the embryo (dense, rarefied or moist ones) happens in accordance to the Gleiches zu Gleichemprinciple,55 and that articulation (διάρθρωσις) is made possible by the force of breath that blows into the growing body.56 The single phases can be outlined as follows: (1) swelling of the membrane around the foetus and formation of the placenta; (2) hardening of the bones; (3) development of head, upper and lower limbs; (4) development of mouth, nose, ears, eyes, genital parts; (5) development of the viscera: lungs, intestines, anus and urethra; (6) growth of digits and nails, hairs (hairs first grow in places where the epidermis is rarefied and the necessary moisture is present). Democritus, on his part,57 lets the foetus’ growth begin with abdomen (and the umbilical cord in particular)58 and head, because ‘they have the most void’:59 the appeal to the notion of κενόν provides, for once, a direct explanation in atomistic terms for a biological phenomenon. The consistency of tissue containing a considerable quantity of void could be imagined as comparable to that of the ‘rarefied parts’ (ἀραιόν) in Nat. puer. 6. But the clearest terminological similarities certainly are those between the account on hair growth in Nat. Puer. 9 and Democritus’ tenets on the growth of horns in young animals reported by Aelianus (68 A 153–155 DK = D 190–192 LM). Hairs, as well as horns, need two conditions to be fulfilled in order to correctly develop: (1) they only grow if the surface (respectively, the epidermis or the bone) has an adequately ‘rarefied’ or ‘porous’ consistency (ἀραιότης) and (2)

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they need a sufficient supply of fluid nourishment (ἰκμάς, χυμοί), conveyed through suitably thick (68 A 153 DK: φλέβαι παχύταται) and wide vessels (Nat. puer.: τὰ φλέβια στομοῦται μᾶλλον). The last point to be considered is the alleged correlation between the longterm exposure to north or south winds and the incidence of miscarriages.60 In the Hippocratic Corpus, the doctrine is to be found in On Airs, Waters and Places (ch. 3–4), according to which the constitution of people exposed to warm winds generally is characterized by moisture (ὑγρότης) and lack of vigour (ἀτονία),61 whereas people living where cold winds blow mostly have a rather hard physique (σκληρότης): in the former case, women abort more frequently than in the latter. In Democritus’ own concise wording: ‘in the cold it (scil. the embryo) stays where it is, while in the heat it is most often spat out’ (68 A 152 = D 176 LM).62 That is to say, according to Arrianus’ paraphrase, warm south winds relax the parturient women’s bodies, causing the foetus to be moved in many directions and, eventually, expelled; cold north wind, conversely, contributes to ‘fasten’ the embryo in its position and preserves its natural growth until it is born.63 One case of a linguistic correspondence which could, in fact, hardly be explained without postulating a direct relationship has been pointed out by L. Gemelli Marciano.64 Speaking of the different constitution of men’s and women’s bodies, the former being compact and dense, the latter rarefied and porous, the author of On Glands (ch. 16) uses the term ναστόν to express the thickness of men’s physique (τὸ γὰρ ἄρρεν ναστόν ἐστι καὶ οἷον εἷμα πυκνόν); the adjective ναστός seems to unmistakably pertain to Democritus’ technical vocabulary, indicating the atom’s density in opposition to the void (τὸ κενόν) (67 A 8 = D 32 LM: ‘stating that the substance of the atoms is compact (ναστήν) and full’; 68 A 37 DK = D 29 LM; 47 DK = D 37 LM). According to Gemelli, there is no reason to doubt of a conscious reuse of Democritean terminology: ‘De toute évidence, l’auteur du traiteé Sur les glandes, qui aime le style rechercheé et l’emploi de mots rares, a réutiliseé consciemment l’adjectif démocritéen comme une marque de distinction.’65

ATOMISM IN HELLENISTIC MEDICINE Erasistratus If we exclude the very scanty evidence available on Aegimius of Elis (fourth century BC), perhaps the author of a Περὶ παλμῶν,66 whom some scholars thought to have in some way integrated atomism into his medical theories,67 the dogmatic Erasistratus of Ceos (first half of the third century BC)68 seems to be the first physician who explicitly appealed to general atomistic principles rather than to single biological doctrines more or less clearly traceable back to Democritus.69 Although his thought was certainly receptive of a form of Aristotelian ‘mechanistic’ teleology,70 which Galen did not fail to criticize for its supposed lack of coherence,71 this does not rule out the possibility that Erasistratus and some of his disciples might have nevertheless adopted, on a physical level, a corpuscular concept of matter. Erasistratus’ theory of a ‘trifold connection’ (τριπλοκία) of elemental nerve, vein and artery that cannot be

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observed with the senses but only by means of reason (λόγῳ θεωρητά) and constitute the visible vessels72 comes dangerously close, according to Galen, to admitting the existence of simple, atom-like στοιχεῖα:73 At this point, again, I should like Erasistratus himself to answer regarding this small elementary nerve, whether it is actually one and definitely continuous, or whether it consists of many small bodies, such as those assumed by Epicurus, Leucippus, and Democritus. For I see that the Erasistrateans are at variance on this subject.74 Galen refers to Erasistratus’ doctrine of the τριπλοκία on more than one occasion,75 and we find it summarized in the medical doxographical handbook known as Anonymus Londiniensis76 (col. 21,23–8 Ricciardetto), where the only theoretically conceivable vein, nerve and artery are called ‘first bodies’ (πρῶτα σώματα), a term the sources often use in relation to Leucippean and Democritean atoms.77 Λόγῳ θεωρητά are also, according to Erasistratus and others,78 the invisible particles that emanate in a flux (ἀποφορά) from both inanimate and living bodies (the discussion of the doxography on this issue occupies the last extant section of the Anonymus (col. 31,39–40)). This phenomenon, while not being directly observable through sense-perception, can nevertheless be grasped on an experimental basis: according to Erasistratus, if one leaves an animal for some time without nourishment in a container and weighs it afterwards along with the excrement it expelled, a weight loss can be quantitatively measured that is due to the effusion of invisible particles (Anon. Lond. col. 33, 43–51).79 Emanations of particles from atomic compounds explain, in Democritus’ (as well as in Epicurus’)80 view, the physical dynamics of αἴσθησις: sight is made possible through an ‘impression’ (τυποῦσθαι) onto the air between the pupil and the perceived object (68 A 135,50 DK = D 147 LM), a process that presupposes an ἀπορροή (‘efflux’) from the bodies.81 Lucretius will go a step further to consider perception as a proof of a continuous flux of particles from the objects towards the sense organs, which should confirm the existence of invisible atoms. Empirical evidence is provided in this respect by the phenomenon of consumption/loss of matter that can be noticed in several cases.82 The method of inference from the visible is shared by both Lucretius and Erasistratus, with the latter corroborating empirical observation through the properly experimental parameter of quantitative measurability. Another fragment (68 A 165 DK = D 126 LM) reports Democritus’ explanation of magnetic phenomena: iron moves towards magnet (whose material structure is more rarefied) for three main reasons: (1) atomic effluxes emanate from the objects; (2) similar things move towards similar ones;83 (3) all things move towards the void (εἰς τὸ κενὸν). In 68 A 97 DK (= D 119 LM), according to Aristotle’s account, Democritus explained the cause of seismic activity by stating that the earth, when dried out, ‘pulls’ rainwater from fuller cavities into emptier ones.84 A natural movement towards the void was posited by Erasistratus as well, who spoke of πρὸς τὸ κενούμενον ἀκολουθία (the ‘filling up of vacuum’), a mechanic process that regulates the dynamics of fluids in the body.85 Galen, who also is our main source on this doctrine, harshly criticizes Erasistratus for not having taken into

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account that physiological processes are in fact due to the natural powers (φυσικαὶ δυνάμεις) each organ possesses and cannot therefore be accounted for in mechanistic terms: the movement Erasistratus explains appealing to the πρὸς τὸ κενούμενον ἀκολουθία is in fact caused by the ‘attractive’ power (δύναμις ἑλκτική – cf. supra, 68 A 97 DK), as in the case of the assimilation and distribution of nourishment.86 The ‘filling-up theory’ underpins Erasistratus’ aetiopathogenesis, as disease is said to be caused by a flow of foreign matter into cavities that do not naturally contain it (as it happens, for example if, in consequence of a wound, blood flows out from the arteries, which, according to Erasistratus, normally convey only πνεῦμα).87 A non-qualitative, mechanistic explanation is offered by Erasistratus for the digestive process as well. In opposition to Hippocrates’ theory of πέψις (lat. concoctio, literally a changing that occurs by means of heat, as when cooking food), that presupposes an alteration of substances (Galen will subsequently resume this doctrine in terms of natural powers), Erasistratus considers digestion to be due to a disgregation of food in smaller particles.88 Galen’s On Medical Experience credits him with holding that ‘foods which are easily and quickly ground and pounded are the easiest and quickest of digestion, and the foods that are the reverse of this are difficult of digestion’,89 and Celsus paraphrases his (and his disciples’) view by saying teri in ventre cibum contendunt (‘They claim that food is ground in the stomach’).90

Asclepiades Between the first century BC and the first century AD, Democritus is considered an authority not only on matters of natural philosophy but also on specifically medical issues. Especially in the medical literature of the first century AD, he does not figure as the main representative of atomism, but rather as an author of technical – thus also, if not mainly, medical – treatises (Fachschriftsteller), be they authentic or not from the modern scholar’s perspective.91 In this context,92 the theories of the physician Asclepiades of Prusias ad Mare (floruit end of the second century BC)93 are, after Erasistratus, a major – and rather isolated – instance of a proper reception of atomism and of its consequent integration into a physiopathological and therapeutic system.94 Asclepiades spent at least some part of his career in Rome, where he seems to have achieved a considerable reputation.95 Rejecting the principles of humoral pathology, he recognizes in the atomistic doctrine a paradigm capable of giving account of the aetiology of diseases as well as an indication (ἔνδειξις) for their treatment. Asclepiades conceives of (living) matter as a compound of particles or ‘lumps’ he calls ἄναρμοι ὄγκοι (Galen refers to them as ἄναρμα στοιχεῖα), a designation he apparently shared with the Platonist Heraclides of Pontus (fourth century BC).96 Both qualify the ὄγκοι as παθητοί as well (i.e. ‘masses liable to change’), a characteristic that, according to Sextus Empiricus (our source on this point), distinguishes their version of atomism from that of Democritus and Epicurus, who, on the contrary, posited the ἄτομα to be ἀπαθῆ, that is, ‘unchangeable’.97 Chapter 2 of the treatise Nature of Man (vol. 6, p. 34 Littré), for the most part composed by Hippocrates’ son-in-law Polybus,98 contains a polemic against

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those physicians who claim that the nature of man can be reduced to one and a single element, such as blood or phlegm: this assumption would lead to the absurd conclusion that men cannot feel pain, for there would not be anything else in the body that could affect it producing pain. It is likely that Polybus is here primarily addressing the concept of Being put forward by the Eleatic philosophers,99 but in Galen’s interpretation the polemical spectrum is extended to the atomists as well, since he thinks of each single atom in terms of unchangeable Being in itself (see following section). It is likely that the idea of ὄγκοι παθητοί, that is, constituents of matter that are subject to change, put forward by Heraclides and apparently accepted by Asclepiades also, represented a theoretical reaction to the objections formulated in On the Nature of Man.100 The sources that inform us about Asclepiades’ medical theories are, in fact, not entirely consistent. An important (and also most debated) witness is provided by the fifth-century Latin medical author Caelius Aurelianus in the first book of his translation/paraphrase of Soranus’ lost work on acute and chronic illnesses (Celerum passionum libri 1,105–6 CML VI 1, pp. 80–3). If we stick to Caelius, Asclepiades seems to have distinguished between two different types of corpuscles: (1) the atoms (atomos) and (2) other masses,101 graspable by the mind (intellectu sensa), which come together (or ‘move’, according to a different reading) in an eternal motion, collide and split by mutual impact into fragments (in infinita partium fragmenta soluuntur) but also connect with each other in various ways to form perceptible bodies. This would obviously lead to a patent contradiction with the principle of Democritean atomism and thus fatally undermine the ontological fundament of the atomistic model, for according to it there are no minimal particles that are themselves frangible. In addition to that, the other available sources on Asclepiades seem to attribute to the constituent particles of matter quite disparate characteristics, since they call them sometimes ἄναρμα (with ἀ(ν-) privativum from ἁρμόζω, ‘connect, adapt to’), sometimes ἀμερῆ (‘without parts’), sometimes θραυστά (‘destructible, breakable in fragments’), sometimes, as we have just seen, παθητά, but also ἀμετάβλητα and ἀναλλοίωτα (‘not liable to change’).102 Some of the interpreters have recognized in these divergent denominations a confirmation for the dichotomy adumbrated in Caelius’ report.103 The interpretation of Caelius’ passage is actually more problematic than it might seem at a first glance. The distinction of two types of corpuscles, paraphrased earlier, is based on the following text: primordia namque corporis primo [ed. princ. : prima CML 6,1 p. 80] constituerat atomos, secunda [ed. princ. : secl. ed. Rovilliana]104 corpuscula intellectu sensa sine ulla qualitate solita atque ex initio comitata [ed princ. : concitata Voss] aeternum moventia, quae suo incursu offensa mutuis ictibus in infinita partium fragmenta solvantur. Stückelberger has convincingly argued for the seclusion of the adjective secunda, which he regards as an interpolation made to mistakenly restore a supposed parallelism with primo (which can also mean ‘at first, originally’, without being followed by other temporal adverbs).105 This intervention would remove the dichotomy assumed

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by some scholars and refer the attributes of being intellectu sensa, sine qualitate, ex initio comitata et aeternum moventia to the atomi mentioned at the beginning; the relative clause introduced by quae would thus rather refer to the aggregates than to the atoms themselves (although the verb solvantur is grammatically related to corpuscula through the relative pronoun); the atomic compounds (cf. comitata) are subject to collisions during their motion, which causes them to crumble in fragments (atomi).106 As for the conflicting characteristics attributed to atoms by the other sources (ἄναρμα, ἀμερῆ, θραυστά, παθητά, ἀμετάβλητα/ἀναλλοίωτα), Stückelberger proposes the following explanation: being θραυστά cannot be referred to ἄτομα, since it is a property of atomic compounds, as a passage in Galen’s Περὶ φυσικῶν δυνάμεων seems to confirm (Gal., De fac. nat. 1,13, vol. 2, p. 39 Kühn): According to Asclepiades, however, nothing by nature is sympathetic to anything else, since all substances are divided and broken down into not-permanentlyaggregating elements and absurd ‘lumps’. The accounts in which Asclepiades’ atoms are called θραυστά must therefore have confused the terms of the attribution. But Galen’s statement entails also another question: How shall we intend the adjective ἄναρμα? The translation I have given, ‘not-permanently-aggregating’, relies upon the interpretation of those scholars who understand ἄναρμα as a negation not of the internal cohesion of the atoms (which would essentially make it a synonym of θραυστά), but rather of the stability of the compound bodies, that are per se liable to a continuous alternation of aggregation and disaggregation.107 The masses on which Asclepiades’ concept of matter is based are rightfully qualified as ἀμερῆ, ἀμετάβλητα and ἀναλλοίωτα – such are Democritus’ and Epicurus’ atoms as well. That being so, the meaning of παθητά is yet to be explained. Stückelberger argues that only some of the qualities pertaining to the ὄγκοι do change, without, however, affecting their very substance.108 In this way, as already mentioned, Asclepiades might be responding to the objections moved to the idea of a body made of immutable matter.109 It remains to be seen how Asclepiades’ corpuscular theory relates to his physiological and therapeutic views. The constituents of matter are able to move through void passages (πόροι). That of πόροι is of course a concept not alien to ancient atomism,110 but Asclepiades transfers it to the explanation of the body’s internal processes in order to establish a theory of etiopathogenesis. Depending on the status of perviousness of the πόροι, health or illness follows: if the passages are obstructed, the ὄγκοι are prevented from moving and thus engender a morbid condition.111 Asclepiades does not conceive of void in the same way as atomists do. While for Democritus and Epicurus, void (τὸ κενόν) is a continuous dimension in which atoms are in motion,112 his idea of empty space seems to be closely related with that of Strato of Lampsacus’ κενὸν διαλαμβάνον τὸ πᾶν σῶμα, a ‘void dividing the whole of body’.113 The presence of void also in the bodies is a standard atomistic tenet (cf. 68 A 60 DK = D 40 LM – ‘the void contained in the bodies (ἐμπεριλαμβανόμενον) makes them light’);114 Asclepiades appropriates it, though excluding the possibility of a void extra corpora.

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GALEN The treatment of atomism in the writings of Galen of Pergamum (129– c. AD 216) is as fascinating as contradictory. Although ingenious, it is not at all devoid of arbitrary distortions as well.115 A first and major one is implicit in the very word ‘atomism’. As we have seen so far, there are different accounts and modes of reception of a physical view generally labelled as ‘atomistic’: Democritus’ principles do not remain unaltered in Epicurus’ philosophical system, nor are both authors slavishly received in the medical systems of Erasistratus and Asclepiades. Notwithstanding, Galen does not hesitate to group all of them under one and the same heading when it comes to polemics. It is far from certain that Galen had the possibility to get acquainted at least with Democritus’ works directly. His knowledge of the Abderite must have been somehow mediated by other accounts, and only a few longer literal quotations can be identified in his oeuvre.116 A direct knowledge of Epicurus is, on the contrary, more likely. Galen’s first engagement with Epicureanism dates back to the years of his philosophical training in Pergamum. There he took lessons with a Stoic teacher, for a short time with a Platonist, with a Peripatetic and not least with an Epicurean from Athens (Aff. Dig. 8, vol. 5, pp. 41–2 Kühn).117 In his own catalogue of his works (De libr. propr. 19, pp. 172–3 Boudon-Millot), the titles of (mostly polemical) treatises on specific aspects of Epicurean philosophy are mentioned, which probably implies that he had the occasion to read some of Epicurus’ writings. The theoretical framework of Galen’s critique towards atomism can be outlined as follows: Galen accepts the Peripatetic interpretation according to which atomism can be thought of as a variation on Eleatic principles118 and cleverly leverages this identification in order to undermine (1) its physical postulates and (2) the non-teleological vision these postulates imply.119 The systematic refusal of any corpuscular/mechanistic view of (man’s) nature in favour of a revised revival of Hippocrates’ humoral pathology, combined with Galen’s long-lasting authority, eventually leads to a nearly total ‘eclipse’ of atomism in the dominant medicophysiological paradigm for at least one and a half millennia.120 According to Aristotle (Gen. corr. 325a–b: 67 A 7 DK = D 30 LM), Leucippus (and Democritus) did move from the Parmenidean dichotomy ὄν (Being)/μὴ ὄν (non-Being), but identified the former with a multiplicity of particles, each bearing the main characters of Being (i.e. fullness and incorruptibility), the latter with void (τὸ κενόν). The existence of void cannot be denied – even though void is comparable to the Eleatic concept of μὴ ὄν – for this would make motion impossible. Given this premises, the atomists transfer the explanation of generation and corruption (γενέσθαι, πάσχειν) from the level of the elements (where no change at all is allowed) to that of the compound bodies they form. Aggregation and disaggregation of elemental particles thus account for the vicissitudes of nature that are perceptible through the senses. Galen, though, does not seem inclined to acknowledge this shift in his polemics. Indeed, he strongly stresses the essentially ‘monistic’ character of any atomistic concept. We have already seen (in the previous section) what the objections to

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monism from a ‘Hippocratic’ perspective were (On the Nature of Man). Here is Galen’s interpretation of that argument:121 But a person might say that all things are one in form and power, as Epicurus and Democritus and their followers say of the atoms. And of the same chorus with them are those who postulate elements that are least and unattached and without parts (ἄναρμα καὶ ἀμερῆ). Hippocrates, then, making a common answer to all such persons, proves that the element is not one in form (ἰδέᾳ) and power; and he does not even mention those who say that what is is one in number (ἀριθμῷ), supposing them to be completely mad.122 Whether mentioned by name (Epicurus, Democritus) or hinted at through lexical markers (ἄναρμα καὶ ἀμερῆ), all main exponents of atomism (at least those who Galen recognizes as such) are considered as belonging to the same category.123 The distinction between what is one according to the form (ἰδέᾳ: the atoms) and what is one also according to the number (ἀριθμῷ: the Eleatic Being) is not relevant to Galen’s point.124 He consciously ignores the theoretical distinction between the level of ontology, where atoms count each as an instance of an incorruptible being, and the level of physical causality, where the atoms in movement connect with each other and form transient, perceptible bodies.125 Besides, Galen does not consider the variety of atomic shapes as an argument for the qualitative differentiation of aggregates: in order to discard atoms as the first principles of nature, he has to prove that unchanging atomic ἀρχαί cannot account for the multifaceted qualities of the objects that are perceived through the senses.126 Atomistic physics represent a threat to the Hippocratic–Galenic humoral paradigm not only in terms of physiology and elemental theory but also, and perhaps more dangerously, in terms of teleology. Especially in his treatises On the Utility of the Parts and On the Natural Faculties, Galen is committed to showing how an ‘expert’, demiurgical nature (τεχνικὴ φύσις) determines, down to the smallest detail, the anatomical structures, functions and physiological properties (χρεῖαι and δυνάμεις) within the body.127 This assumption constitutes the basis of Galen’s logical-deductive medical reasoning, as it enables a coherent interpretation of empirical data and thus guarantees reproducibility of therapeutic procedures. The theoretical background traces back to both Plato’s Demiurge in the Timaeus and Aristotle’s causa finalis.128 Inasmuch as it excludes the effect of such a cause, any materialistic explanation of the composition and functioning of the body will turn out to be mistaken.129 Galen considers highly implausible that what he calls an ‘irrational nature’ (ἄλογος φύσις) can lead, by pure chance, to the symmetrical, harmonic and completely function-oriented development of the parts visible in all bodies (cf. De foet. form. 6,3–4, vol. 4, p. 688 Kühn = CMG V 3,3. p. 92).130 At a physiological level, Galen’s finalistic perspective emerges in his theory of the innate powers (δυνάμεις) of individual organs, such as the absorption of similar substances, the secretion of foreign substances and the transformation of nutrients.131 His criticism of the explanation of magnetism given by Epicurus and Asclepiades, which is not testified to anywhere else, can be read against this background.132

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NOTES 1. The abbreviations of writings from the Corpus Hippocraticum (here referred to as CH) as well as those of Galen follow the conventions established in the bibliographical indexes by G. Fichtner, now edited and regularly updated by the Corpus Medicorum Graecorum (URL: http://cmg​.bbaw​.de (20 July 2020)). 2. Respectively collected as nr. 67 and 68 DK; see now the new edition in Laks and Most (2016) as well (in the following references to text passages abbreviated as LM). 3. Epicurus’ attitude towards Democritus is an erratic one: although he proudly remarks his doctrinal independence from his Democritean teacher Nausiphanes (fr. 141 Arrighetti = 114 Usener), allegedly writes a Contra Democritum (fr. 11 Arrighetti = 16 Usener; see also the polemical attack in fr. 101 Arrighetti = 93 Usener) and even asserts the non-existence of a philosopher named Leucippus (67 A 2 DK = R 80 LM = fr. 232 Usener), elsewhere he clearly seems to proclaim himself as a Democritean (fr. 138 Arrighetti). 4. Cf. 67 A 1 DK = D 80b LM. 5. Cf. fr. 280–281 Usener and Arrighetti (1973, 512–13). 6. Cf. 67 A 6 DK = D 31 LM. 7. Ep. Hdt. 44. 8. Cf. 68 B 117 DK = D 24 LM; Epicurus, Ep. Hdt. 54. 9. At present, the tendency is to renounce any attempt to draw a sharp distinction between the two: see e.g. Laks and Most (2016, 3); contra Graham (2008, 333–52). 10. Cf. Deichgräber (1930, 278): ‘Diese speziell für die Medizin bestimmten stark demokriteisch-epikureischen Sätze stehen denen der Empiriker durchaus nicht fern. Die in dieser Lehre verwandten Termini kehren in der empirischen zum großen Teil wieder; das spricht sogar für weitgehende Abhängigkeit der einen Lehre von der anderen.’ Cf. further 68 B 125 DK = D 23a LM; 68 A 171 DK. 11. The adjective ‘mechanistic’ is used here in a general sense, that is, it does not necessarily imply a connection with ‘mechanics’ as a narrow and specific field. See for discussion Berryman (2009, 34–9) and more recently Popa (2018, 14–15), with further literature. 12. Cf. Furley (1986), 541. 13. Cf. Sassi (1978, 183). 14. Diog. Laert. 9,46–9 = 68 A 33 DK; see also 68 B 26b–e. 15. Cf. CH vol. 2, pp. 94–191 Littré (Προγνωστικόν, Prognostic); vol. 5, pp. 574–733 Littré (Κῳακαὶ προγνώσεις, Koan Prognoses). 16. Cf. CH vol. 6, pp. 462–663 Littré (Περὶ διαίτης, On regimen). 17. 68 B 5d DK (cf. 68 C 6). Cf. CH vol. 6, pp. 29–69 Littré (Περὶ φύσιος ἀνθρώπου, On the Nature of Man). 18. 68 A 10 DK; cf. Longrigg (1993, 67). 19. Cf. López Férez (1974, 157–65). 20. 68 B 31 DK = D 235 LM. Transl. Laks and Most.

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21. 68 B 234 DK = D 240 LM. 22. 68 B 159 DK = D 233a LM. 23. 67 A 28 DK = D 132. 136 LM; 67 A 34 DK = D 138 LM; 68 A 106 DK. 24. 68 B 120 DK = D 181 LM. 25. The term φλεβοπαλία also occurs, in an atomistic context, in Epicurus, De natura 34 col. 20,4–5 Leone. 26. 68 B 135 DK = D 182 LM. Transl. Laks and Most. 27. 68 A 159 DK = D 180 LM. 28. 68 A 135,64 DK = D 64 LM. 29. See Sassi (1978, 225). Laks and Most accept the correction of κρήσει in κρίσει (‘judgement’ instead of ‘mixture’), which nevertheless seems to me to be weakened, if not ruled out at all, by a comparison with the text of fr. 68 A 9 DK = D 15 LM. 30. CH Progn. 2. 31. 68 Α 160 = D 141 LM. Transl. Laks and Most. 32. 68 A 2 DK. 33. 68 A 10 DK = P 28 LM, according to the text νέον πρεσβύτῃ. Cf. also Celsus, Med. prooem. 8. 34. 68 A 1,42 DK = P 50 LM. 35. See Philippson (1928, 293–328) for the date of composition; ed. Smith (1990). See also Rütten (1992); Roselli (1998), and Craik (2018, 27–8). 36. CH, Epistulae 17,3, vol. 9, p. 356 Littré; see also Epistulae 19. 37. Jouanna (1999, 20). 38. Lonie (1981) provides a commentary on all three works, that he maintains to be traceable back to a single author; see also Giorgianni (2006). According to Stückelberger (1984, 145), these writings can be traced back to the milieu of the Cnidian medical school: cf. Salem (1996, 231). 39. Wellmann (1929, 297–330); Lonie (1981, 62–70), although with major reservations; Stückelberger (1984, 49–87); Longrigg (1993, 93–7); Salem (1996, 230–52). 40. Heidel (1941, 18); Jouanna (1999, 273–4); Gemelli Marciano (2007b, 313–19); Ead. (2007a, 213); Ead. (2013, 494–5, 514–15). 41. In his De elementis secundum Hippocratem, a treatise attempting at reconstructing an – indeed nowhere explicitly formulated – Hippocratic theory of the first elements (στοιχεῖα), Galen reports a passage of the Hippocratic De natura hominis supposedly directed against the atomistic conception of single, indivisible elements deprived of sensation (vol. 1, p. 483 Kühn = CMG V 1,2, p. 128–30). 42. Cf. Orelli (1996, 39). A circumstance that Stückelberger (1984, 143–5) connects to an oral reception of Democritus’ tenets. 43. On the fortunes of the pangenesis-theory see Longrigg (1993, 68–9). 44. Cf. Gal., De usu part. 14,6, vol. 4, p. 164 Kühn; Salem (1996, 231–4). 45. Cf. Lucr. 4,1209–11. See Taylor (1999, 198). For an interpretation of the ἐπικράτεια-theory as an example of a ‘mechanistic’, ‘non-qualitative’ approach to sexual differentiation see Salem (1996, 237–42).

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46. Cf. scholium ad Epicurus, Ep. Hdt. 67: τό τε σπέρμα ἀφ’ ὅλων τῶν σωμάτων φέρεσθαι; Lucr. 4,1040–4. Cf. Philoponus, in Aristotelis De generatione et corruptione, CAG 14.3 p. 25 Hayduck. 47. See Wilberding (2016, 333). 48. 24 A 13 DK = D 21–22 LM. 49. 68 B 124 DK = D 164 LM. Trans. Laks and Most. See Taylor (1999, 7). 50. 68 A 141 DK = D 165 LM. 51. See Wellmann (1929, 307); Stückelberger (1984, 60–1); Berryman (2007, 355–8). 52. See Taylor (1999, 198); Orelli (1996 56–66). Contra Salem (1996, 234–7). 53. Cf. Erasistratus, fr. 58 Garofalo, who links multiple births with a ‘second (act of) fertilisation’ (ἐπισύλληψις) preceded by the end of the regular menses (ὅταν γὰρ ἡ μήτρα ᾖ κεκαθαρμένη). 54. Cf. Orelli (1996, 51–4). Contra Salem (1996, 242–5). 55. See Müller (1965). Further Democr. 68 A 99a (col. 2) DK = D 123 LM; A 128 DK = D 156 LM; B 164 DK = D 55 LM. 56. An experimental proof of this is offered as well: Nat. puer. 6 = 17 Littré. 57. For an extensive comparison between Nat. puer. and Democritus’ fragments see Salem (1996, 245–7). 58. 68 B 148 DK = D 172 LM. 59. 68 A 145 DK = D 170–171 LM. Transl. Laks/Most. 60. Cf. Diller (1934, 112). 61. Cf. Morb. sacr. 13. 62. Transl. Laks and Most. 63. Gemelli Marciano (2013, 516) supposes an influence of the Hippocratic author on Democritus; contra Salem (1996, 251–2). 64. Gemelli Marciano (2007b, 213–15). 65. Gemelli Marciano (2007b, 215). 66. Gal., Diff. puls. 1,2, vol. 8, p. 498 Kühn. 67. Cf. Wellmann (1929, 687); Horne (1963, 325 and n. 36). 68. See Garofalo (1988, 17); von Staden (1989, 47). 69. Stückelberger (1984, 96). 70. See von Staden (1997); Berryman (2007); Vegetti (2018, 44). 71. See Garofalo (1988, 45–6). 72. See Garofalo (1988, 32–3). The idea of visible structures as composed of invisible constituents responds to the principle τοῖς ἐναργῶς φαινομένων χρηστέον ἐστὶν εἰς τὴν τῶν ἀδήλων πίστιν (fr. 33 Garofalo), already enunciated by Democritus and Anaxagoras (see, respectively: 68 A 111 DK = R 76 LM; 59 B 21a DK = vol. 6, D 6 LM). 73. De nat. fac. 2,6, vol. 2, p. 97 Kühn = p. 153 Brock. Cf. also Erasistratus, fr. 170 Garofalo.

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74. Transl. Brock (1916). 75. See fr. 88–90 Garofalo. 76. P.Lit.Lond. 165, Brit.Libr. inv. 137. 77. Cf. 67 A 14 DK = D 61 LM; 67 A 16 DK = R 42 LM. 78. Like Asclepiades (see section on Asclepiades) and Alexander Philalethes (c. first century BC): cf. Nutton (1990, 247). 79. See Stückelberger (1984, 100). 80. For Epicurus see in part. Ep. Hdt. 46 and De natura 2 col. 8–75 Leone. 81. The same is of course true for other senses as well, cf. 68 A 135,82 DK. 82. Lucr. 1,265–328, cf. Stückelberger (1984, 101). 83. It is the principle known as Gleiches zu Gleichem, see Müller (1965). 84. Cf. Epicurus, Ep. Pyth. 105. 85. This theory shows an evident connection with Straton of Lampsacus’ doctrine of the κενὸν παρεσπαρμένον: see section on Asclepiades. 86. See fr. 74. 93–96. 109–110 Garofalo. 87. Cf. Stückelberger (1984, 119–20); Garofalo (1988, 34–5). This is Erasistratus’ ingenious attempt to comply to the empirical observations that in the section of a corpse the arteries appear empty of blood, while in the living body the incision of an artery produces a considerable flow of blood. 88. Stückelberger (1984, 118). 89. 12,10 p. 107 Walzer (cf. also 11,3, p. 103). 90. Celsus, Med. prooem. 20. 91. Gemelli Marciano (2007a, 223–4). 92. Two passages in Cicero (Hortensius fr. 53 Straume-Zimmermann; Tusc. 1,34,82) inform us about a group of followers of Asclepiades’ doctrine (between the first century BC and the first century AD) who also clearly appealed to Democritus’ authority (Cicero calls them Democritii): see Gemelli Marciano (2007, 7–8). 93. See Vallance (1993, 694). See also Id. (1990) for a comprehensive survey on Asclepiades’ medical thought. 94. Cf. Casadei (1997, 77–106); Serangeli (2017, 33–46). 95. Cf. Caelius Aurelianus, Acutae passiones 2,63. 129. 96. See Ps.-Gal., Hist. philos. 18 (vol. 19, p. 244 Kühn = p. 610 Diels); Sext. Emp., Pyr. 3,32 and Math. 9,364. 10,318. Cf. Heraclides’ fr. 119a–b Wehrli. For Galen’s use see e.g. De elem. ex Hipp. sent. 2,3, vol. 1, p. 500 Kühn. 97. Sext. Emp., Math. 10,318. 98. Cf. Craik (2015, 209). The attribution is based on the fact that a passage on blood vessels from De nat. hom. 11 is cited and ascribed to Polybus in Arist., Hist. an. 3,3 512b12–513a7. Cf. also Gal., In Hipp. De nat. hom. comm. vol. 15, p. 12 Kühn (= CMG V 9,1, p. 8). 99. Cf. Melissus, 30 B 7 DK = D 10 LM. 100. See Stückelberger (1984, 22).

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101. The paraphrase given here relies upon the text of the editio princeps (1533), without considering, for the moment being, the seclusion of secunda in the Editio Rovilliana of 1566. 102. A synoptic table with the extant sources is to be found in Stückelberger (1984, 104–5). 103. See Stückelberger (1984, 107). 104. See footnote 100. 105. Cf. OLD s.v. 106. Stückelberger (1984, 107–10). There are admittedly various other possible interpretations for this passage, which in any case remains very controversial. A detailed critical discussion of recent hypotheses is now offered by Verde (2019, 50–3), who comes to different conclusions from those reported here and assumes a development in Asclepiades’ theory from an atomistic to a corpuscular conception. See also Grimaudo (2019, 9–15). 107. See Stückelberger (1984, 111–12); Morel (1996, 118). 108. Stückelberger (1984, 112–13). 109. The nature of Asclepiades’ ὄγκοι is in fact a much-debated question and the solution schematically outlined here is of course merely hypothetical. For a detailed overview of the terms of the discussion, see Leith (2009, 283–320). 110. Cf. e.g. 68 A 135,80. 165 DK = R 59. D 126 LM; Epicurus, Ep. Hdt. 47. 111. Cf. Gal., Meth. med. 4,4, vol. 10, p. 268 Kühn: Ἀσκληπιάδης ἐν συμμετρίᾳ μέν τινι πόρων τὸ ὑγιαίνειν ἡμᾶς ὑποθέμενος, ἐν ἀμετρίᾳ δὲ τὸ νοσεῖν, ἐπάνοδον εἶναι τὴν θεραπείαν εἰς τὴν ἀρχαίαν συμμετρίαν τῶν πόρων ὑπέλαβεν; Caelius Aurelianus, Acutae passiones 1,106. 112. Cf. 67 A 19 DK = D 39 LM; Epicurus, Ep. Hdt. 39–40. 113. Fr. 30 A Sharples. This is how Strato counters Aristotle’s negation of empty space (cf. Stückelberger (1984, 136). The same was held by Erasistratus as well: see section on Erasistratus on the πρὸς τὸ κενούμενον ἀκολουθία; cf. Stückelberger (1984, 113–16) and fr. 96 Garofalo. On the relationship between Asclepiades’ and Strato see now Verde (2019). 114. Cf. also 68 A 135,62 = D 69 LM. See further Leith (2012, 164–91). 115. See Kupreeva (2014) and Leith (2014, 213–34). 116. 68 B 125–126 DK = D 23a. 196 LM; Gal., De exper. med. 9,5, p. 99 Walzer. See Gemelli Marciano (2007, 9–10) and Ead. (2013, 495). 117. Cf. Alexander (1994, 68–71); Singer (2013, 273). 118. Cf. Morel (1996, 111–12). 119. See Serangeli (2017). 120. Cf. Horne (1963, 326). 121. Cf. Gal., De elem. ex Hipp. sent. 2, vol. 1, pp. 415–16 Kühn (= CMG V 1,2, p. 58). Cf. also De constit. artis med. 6–7, vol. 1, pp. 246–7 Kühn (= CMG V 1,3, p. 72–4). 122. Transl. De Lacy (CMG V 1,2).

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123. Cf. Morel (1996, 117). A direct contact with Eleatism is attested by some testimonies in the case of Leucippus, who is told to have been a pupil of Zeno or Melissus (67 A 5 DK) and possibly born in Elea (67 A 1. 8 = P 1a–b LM). On an epistemological level, Galen nevertheless seems to distinguish Democritus and Asclepiades from each other. In the treatises Subfiguratio empirica and De experientia medica, the former is associated with sceptical empiricism, the latter with dogmatism (see Morel (1996, 120)). 124. The possibility of transferring the properties of the ὄν to a multiplicity of single elements is adumbrated by Melissus, although in the context of a reductio ad absrudum, in 30 B 8 DK (= D 11 LM). 125. Cf. Morel (1996, 126). 126. Cf. Morel (1996, 123). 127. On Galen’s teleology see now Petit (2018) and van der Eijk (2017); Serangeli (2019, 91–117). 128. Cf. Hankinson (2008, 225–9). 129. Cf. De. fac. nat. 1,12, vol. 2, p. 27 Kühn = SM, vol. 3, pp. 119–22. 130. Cf. also De usu part. 1,21, vol. 3, p. 74 Kühn = p. 54 Helmreich. 131. Cf. Vegetti (2018, 307). 132. De fac. nat. 1,14–5, vol. 2, pp. 45–60 Kühn = SM, vol. 3, pp. 133–44.

REFERENCES Alexander, L. (1994), ‘Paul and the Hellenistic schools: The evidence of Galen’, in Troels Engberg-Pedersen (ed.), Paul in His Hellenistic Context, 60–83, Minneapolis: Fortress Press. Arrighetti, G. (1973), Epicuro. Opere, Torino: Einaudi. Berryman, S. (2007), ‘Teleology without tears: Aristotle and the role of mechanistic conceptions of organisms’, Canadian Journal of Philosophy 37, no. 3: 351–69. Berryman, S. (2009), The Mechanical Hypothesis in Ancient Greek Natural Philosophy, Cambridge: Cambridge University Press. Brock, A. J. (1916), Galen: On the Natural Faculties, London and New York: Loeb Classical Library. Casadei, E. (1997), ‘La dottrina corpuscolare di Asclepiade e i suoi rapporti con la tradizione atomista’, Elenchos 18: 77–106. Craik, E. M. (2015), The ‘Hippocratic’ Corpus: Content and Context, London and New York: Routledge. Craik, E. M. (2018), ‘The “Hippocratic Question” and the nature of the Hippocratic Corpus’, in Peter E. Pormann (ed.), The Cambridge Companion to Hippocrates, 25–37, Cambridge: Cambridge University Press. Deichgräber, K. (1930), Die griechische Empirikerschule. Sammlung der Fragmente und Darstellung der Lehre, Berlin: Weidmann. Diels, H. (1879), Doxographi Graeci, Berlin: Reimer. Diller, H. (1934), Wanderarzt und Aitiologe. Studien zur hippokratischen Schrift Περὶ ἀέρων ὑδάτων τόπων, Leipzig: Dieterich.

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Furley, D. (1986), ‘Review of Stückelberger, Vestigia’. Isis 77, no. 3: 540–1. Garofalo, I. (1988), Erasistrati fragmenta, Pisa: Giardini. Gemelli Marciano, M. L. (2007a), ‘Le Démocrite technicien. Remarques sur la réception de Démocrite dans la littérature technique’, in Aldo Brancacci and Pierre-Marie Morel (eds), Democritus: Science, the Arts, and the Care of the Soul. Proceedings of the International Colloquium on Democritus (Paris, 18–20 September 2003), 207–37, Leiden and Boston: Brill. Gemelli Marciano, M. L. (2007b), Democrito e l’Accademia. Studi sulla trasmissione dell’atomismo antico da Aristotele a Simplicio, Berlin and New York: De Gruyter. Gemelli Marciano, M. L. (2013), Die Vorsokratiker, vol. 3, Berlin: Akademie Verlag. Giorgianni, F. (2006), Hippokrates, Über die Natur des Kindes (De genitura und De natura pueri), Wiesbaden: Ludwig Reichert Verlag. Graham, Daniel W. (2008), ‘Leucippus’ atomism’, in Patricia Curd and Daniel W. Graham (eds), The Oxford Handbook of Presocratic Philosophy, 333–52, Oxford: Oxford University Press. Grimaudo, S. (2019), ‘ΟΓΚΟΙ atomici? ancora sulle particelle di Asclepiade di Bitinia’, Galenos 13: 9–15. Hankinson, R. J. (2008), ‘Philosophy of nature’, in R. J. Hankinson (ed.), The Cambridge Companion to Galen, 210–41, Cambridge: Cambridge University Press. Heidel, W. A. (1941), Hippocratic Medicine: Its Spirit and Method, New York: Columbia University Press. Helmreich, G. (1907–1909), Galeni De usu partium libri XVII, 2 vols, Leipzig: Teubner. Horne, R. A. (1963), ‘Atomism in ancient medical history’, Medical History 7, no. 4: 317–29. Jouanna, J. (1999), Hippocrates, Baltimore and London: The Johns Hopkins University Press. (translated by M. B. DeBevoise; Original ed.: Jouanna, Jacques. Hippocrate. Paris: Les belles lettres, 1992). Kupreeva, I. (2014), ‘Galen’s theory of elements’, in Peter Adamson, Rotraud Hansberger and James Wilberding (eds), Philosophical Themes in Galen, 153–96, London: Institute of Classical Studies. Laks, A. and Most, G. W. (2016), Early Greek Philosophy. Vol. 7: Later Ionian and Athenian Thinkers, Part 2, Cambridge, MA and London: Loeb Classical Library. Leith, D. (2009), ‘The qualitative status of the onkoi in Asclepiades’ theory of matter’, Oxford Studies in Ancient Philosophy 36: 283–320. Leith, D. (2012), ‘Pores and void in Asclepiades’ physical theory’, Phronesis 57: 164–91. Leith, D. (2014), ‘Galen’s refutation of atomism’, in Peter Adamson, Rotraud Hansberger and James Wilberding (eds), Philosophical Themes in Galen, 213–34, London: Institute of Classical Studies. Leone, G. (2012), Epicuro: Sulla Natura, libro II, Napoli: Bibliopolis. Longrigg, J. (1993), Greek Rational Medicine: Philosophy and Medicine from Alcmaeon to the Alexandrians, London and New York: Routledge. Lonie, I. M. (1981), The Hippocratic Treatises “On Generation”, “On the Nature of the Child”, “Diseases IV”, Berlin and New York: De Gruyter. López Férez, J. A. (1974), ‘Las ideas de Demócrito sobre salud y enfermedad y su posible influjo en el «Corpus Hippocraticum»’, Asclepio 26/27: 157–65.

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Miller, J., ed., Mensch, P., trans. (2018), Lives of the Eminent Philosophers by Diogenes Laertius, Oxford: Oxford University Press. Morel, P.-M. (1996), Démocrite et la recherche des causes, Paris: Klincksieck. Müller, C. W. (1965), Gleiches zu Gleichem. Ein Prinzip frühgriechischen Denkens, Wiesbaden: Harrassowitz. Nutton, V. ‘The patient’s choice: A new treatise by Galen’, The Classical Quarterly 40, no. 1 (1990): 236–57. Orelli, L. (1996), La pienezza del vuoto. Meccanismi del divenire fra embriologia e cosmogonia nell’ambito dell’atomismo antico, Bari: Levante Editore. Petit, C. (2018), Galien de Pergame ou la rhétorique de la Providence, Leiden and Boston: Brill. Philippson, R. (1928), ‘Verfasser und Abfassungszeit der sogenannten Hippokratesbriefe’, Rheinisches Museum 77, no. 3: 293–328. Popa, T. (2018), ‘Mechanisms: Ancient sources’, in Stuart Glennam and Phyllis Illari (eds), The Routledge Handbook of Mechanisms and Mechanical Philosophy, 13–25, London and New York: Routledge. Ricciardetto, A. (2016), L’Anonyme de Londres, P.Lit.Lond. 165, Brit.Libr. inv. 137. Un papyrus médical grec du Ier siècle après J.-C., Paris: Les Belles Lettres. Roselli, A. (1998), Ippocrate. Lettere sulla follia di Democrito, Napoli: Liguori. Rütten, T. (1992), Demokrit – Lachender Philosoph und sanguinischer Melancholiker. Eine pseudohippokratische Geschichte, Leiden, New York, København and Köln: Brill. Salem, J. (1996), Démocrite: grains de poussières dans un rayon de soleil, Paris: Librairie philosophique J. Vrin. Sassi, M. M. (1978), Le teorie della percezione in Democrito, Firenze: Le Monnier. Serangeli, A. (2017), ‘Significato e retroterra filosofico degli ὄγκοι di Asclepiade di Bitinia’, Technai 8: 33–46. Serangeli, A. (2019), ‘The anti-teleologism of Asclepiades and Epicurus in Galen’s De Usu Partium’, Technai 10: 91–117. Sharples, Robert W. (2011), ‘Strato of Lampsacus: The sources, text and translations’, in Marie-Laurence Desclos and William W. Fortenbaugh (eds), Strato of Lampsacus: Text, Translation, and Discussion, 5–230, Abingdon and New York: Routledge. Singer, P. N. (2013), Galen: Psychological Writings, Cambridge: Cambridge University Press. Smith, Wesley D. (1990), Hippocrates. Pseudepigraphic Writings, Letters – Embassy – Speech from the Altar – Decree. Leiden, New York, København and Köln: Brill. Straume-Zimmermann, L. (1990), Marcus Tullius Cicero. Hortensius, Lucullus, Academici libri, München and Zürich: Artemis. Stückelberger, A. (1984), Vestigia Democritea: die Rezeption der Lehre von den Atomen in der antiken Naturwissenschaft und Medizin, Basel: Friedrich Reinhardt Verlag. Taylor, C. C. W. (1999), Leucippus and Democritus: Fragments, Toronto, Buffalo and London: University of Toronto Press. Vallance, John T. (1990), The Lost Theory of Asclepiades of Bithynia, Oxford: Oxford University Press. Vallance, John T. (1993), ‘The medical system of Asclepiades of Bithynia’, Aufstieg und Niedergang der Römischen Welt 37, no. 1: 693–727.

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van der Eijk, P. (2017), ‘The place of disease in a teleological world-view: Plato, Aristotle, Galen’, in Julius Rocca (ed.), Teleology in the Ancient World. Philosophical and Medical Approaches, 217–41, Cambridge: Cambridge University Press. Vegetti, M. (2018), Scritti sulla medicina galenica, Pistoia: Petite Plaisance. Verde, F. (2019), ‘Asclepiade tra Epicuro e Stratone di Lampsaco’, Technai 10: 45–67. von Staden, H. (1989), Herophilus: The Art of Medicine in Early Alexandria, Cambridge: Cambridge University Press. von Staden, H. (1997), ‘Teleology and mechanism: Aristotelian biology and early Hellenistic medicine’, in Wolfgang Kullmann and Sabine Föllinger (eds), Aristotelische Biologie: Intentionen, Methoden, Ergebnisse, 183–208, Stuttgart: Steiner. Walzer, R. (1944), Galen: On Medical Experience, London, New York and Toronto: Oxford University Press. Wehrli, F. (19692), Herakleides Pontikos (Die Schule des Aristoteles, vol. 7), Basel and Stuttgart: Schwabe. Wellmann, M. (1929), ‘Spuren Demokrits von Abdera im Corpus Hippocraticum’, Archeion 11: 297–330. Wilberding, J. (2016), ‘Embryology’, in Georgia L. Irby (ed.), A Companion to Science, Technology and Medicine in Ancient Greece, and Rome, 329–42, Chichester: Wiley.

CHAPTER 3

Why aren’t atoms coloured?1 DAVID SEDLEY

The term ‘atom’ has enjoyed a virtually uninterrupted lineage in physics: from the original conception of an ‘unsplittable’ body by Leucippus and Democritus in the fifth century BCE; via Epicurus’ revised and updated atomist system c. 300 BCE; on through the later rehabilitation of this latter in Renaissance and early modern physics; and all the way down to the by no means unsplittable ‘atoms’ of today’s physics. So radically has physics altered that today’s atoms share with their ancient counterparts little more than a name. There is, however, one further linking thread. Individual atoms, ancient and modern alike, are devoid of such properties as colour. There are no blue or red atoms. In what follows I shall concentrate on the paradigmatic case of colour, although I expect my conclusions to apply, broadly speaking, to other sensible properties too. That atoms are colourless (as also odourless, temperature-free, etc.)2 is among the most inspired insights of the ancient atomists. But how did they arrive at such a conviction? As with most aspects of atomism, a fuller articulation of the underlying arguments will be found in the surviving Epicurean texts, especially of Epicurus himself and of Lucretius, than in the indirect testimonies for their late fifth-century forerunners. Generally speaking, Epicurean physics is following its Democritean model when it assumes some degree of continuity between the properties of macroscopic and microscopic bodies. Thus, for example, we can work out that the atoms composing a liquid must be smooth and round, because at the macroscopic level smooth round bodies are the most mobile, and there is no reason not to assume that the same explanatory principle operates, below the threshold of perception, all the way down to single atoms. Why then should the same not apply to colour? That very question was to be put to a later generation of Epicureans, around 100 BCE, by the Stoic Dionysius.3 Unfortunately, Epicurus’ own justification of the thesis in the condensed summary at Letter to Herodotus 55 is not easy to decipher:4 Moreover, the atoms themselves must be considered to exhibit no quality of things evident, beyond5 shape, weight, size, and the things necessarily included in the nature of shape. For all quality changes. But the atoms do not change

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at all, since something solid and indissoluble must survive the dissolution of the compounds to ensure that the changes are not into, or out of, the nonexistent, but result from rearrangements within many things, and in some cases from additions and subtractions of certain things. Hence those things that do not admit of [internal] rearrangement must be indestructible, and must lack the nature of that which changes. And their own peculiar masses and shapes must survive, since this is actually necessary. After all, also in familiar objects which have their shape altered by shaving, it can be ascertained that in the matter which undergoes change, as it is left, shape remains whereas the qualities do not remain but vanish from the entire body. So these properties which are left are sufficient to bring about the differences of the compounds, given the necessity for some things to be left and not be destroyed into the non-existent. (Emphasis in original) Atoms, then, have shape, weight and size, but no ‘qualities’ (ποιότητες), because all qualities change, whereas atoms cannot. It seems that the term ‘qualities’, rare elsewhere in Epicurus, does not cover all attributes, but those that are intrinsically volatile – just such sensible properties as colour, in fact.6 The basic contention is that if an atom could undergo a change of, for example, colour or temperature, it itself would be undergoing change, and not merely relative change but an intrinsic change which would undermine its indestructibility.7 But why would a change of such properties entail the atom’s destructibility? Because, at least according to this text, changes of qualities like colour are the result of rearrangement within atomic complexes, whereas an atom itself is for obvious reasons immune to internal rearrangement: (1) All colours change. (2) If a body changes colour, that is due to rearrangement of its constituent parts. (3) The parts of an atom cannot be rearranged. (4) Therefore individual atoms are colourless. Premise (1) is no doubt presented as an uncontroversial datum of experience and could be further reinforced by Epicurus’ recorded contention that colours do not exist in the dark,8 which implies that colours can in most cases9 also undergo existential change, passing out of and back into existence. Premise (2) will look equally secure in the eyes of an atomist: since colour changes are causally dependent on underlying changes at the atomic level,10 in the absence of any bodily change at the microscopic level, macroscopic changes of quality would surely be ruled out too. As for premise (3), it is fundamental to the very concept of an atom. Hence the argument may seem successful. If individual atoms had colours, those colours would undergo change, which would in turn require a physical impossibility, the rearrangement of the atoms’ parts. But what if colour turned out, like weight and magnitude, to be a property that can be traced all the way down from macroscopic bodies to individual atoms, perhaps with (in the simplest cases) red objects made of red atoms, yellow objects of

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yellow atoms and so on? If that were so, premise (2) need not be endangered. When, for example, the setting sun gradually changes from yellow to red, could that not be because its atoms are being so rearranged that red ones come to the surface, hiding the yellow ones? It would therefore be entirely possible that premise (1) is false: no doubt macroscopic colours are always subject to change, but might it not be that atoms themselves have entirely immutable colours. Indeed, a rival model of this kind seems to be targeted by Lucretius when he writes (2.757-75):11 Besides, if the primary particles are colourless, and possess a variety of shapes from which they generate every kind of thing and thus make colours vary – since it makes a great difference with what things and in what sort of position the individual seeds are combined and what motions they impart to each other and receive from each other – it at once becomes very easy to explain why things which a little earlier were black in colour can suddenly take on the whiteness of marble, as the sea, when its surface has been churned up by great winds, is turned into waves whose whiteness is like that of gleaming marble. All you need say is that what we regularly see as black comes to appear gleaming white as soon as its matter is mixed up, as soon as the ordering of its primary particles is changed, as soon as some particles are added and some subtracted. But if the sea’s surface consisted of blue seeds, there is no way in which they could turn white. For things that are blue could never change to the colour of marble, no matter how you were to jumble them up. (Emphasis in original) At issue here are the relative explanatory merits of the two rival postulates, that of coloured atoms and that of colour-free atoms. On the Epicurean thesis, ‘it at once becomes very easy to explain’ why visible things suddenly change colour, namely, thanks to atomic rearrangement. The rapid volatility of observed colour, as when the sudden onset of wind turns the sea’s surface from blue to white, is more satisfactorily and economically explained by postulating the instant rearrangement of colourless atoms, than a sea composed of blue atoms. For reasons already outlined earlier, this invites a simple rejoinder. Why should the wind not be, with similar rapidity, bringing white atoms to the surface? My point here is not to decry the Epicurean explanation, which mutatis mutandis has fared outstandingly well in the subsequent history of atomic theory, and rests on a powerful insight. It is, rather, to express a doubt as to whether so meagre an explanatory advantage over its rival can be enough to help us understand what motivated the insight. When we survey the various supplementary Epicurean arguments for atoms’ lack of secondary qualities,12 including colour, it becomes clear that, despite their cumulative weight, none of them is close to being cogent in its own right. Many of them belong broadly to the class of ‘non-contestation’ (ouk antimarturēsis) arguments – the typically Epicurean procedure whereby a theoretical claim about things inaccessible to sensory inspection (ta adēla), already arrived at on theoretical grounds, is corroborated by appeal to its consistency with observed phenomena.13

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Consider an argument of Epicurus’ already quoted earlier (Ep. Hdt. 55): After all, also in familiar objects which have their shape altered by shaving, it can be ascertained that in the matter which undergoes change, as it is left, shape remains whereas the qualities do not remain but vanish from the entire body. We would be able to make very little of this crabbed summary but for the good fortune of having Lucretius’ fuller exposition of the same argument or at any rate of one specific application of it (2.426-33): Moreover, the tinier the pieces into which any object is shredded, the more you can see the gradual disappearance and blotting out of its colour. This happens, for example, when purple cloth is pulled apart into little pieces: when it has been dismantled thread by thread, the purple and the scarlet colour, by far the brightest there is, is completely wiped out. So you can recognise from this that fragments breathe away all their colour before they are reduced to the seeds of things.14 Colour, that is, should not be expected to be an ineradicable property of body, because when a strongly coloured body is progressively fragmented, you can see the colour gradually fade. Hence, it is implied, by the time this process of fragmentation has been extended all the way down to single atoms, it is only to be expected that the colour will have disappeared altogether. But while this reasoning helps make it unsurprising, and consistent with phenomena, that atoms should be altogether colourless, it falls far short of proving that they in fact are. For the opposed assumption, that atoms do have colour, could accommodate the same reasoning with apparently equal facility. For example, it might be argued, when a strongly coloured body is progressively fragmented, however much the colour fades there is always, so long as we can administer a visual check, some colour left; hence, it would be unsurprising if even at the very end of that progression, when single atoms have been reached, a modicum of colour should remain. Much the most powerful of the Epicurean arguments assembled by Lucretius in favour of colourless atoms is the following (2.737-56):15 For particles of matter have absolutely no colour, whether like or unlike that of the objects. If you think that the mind cannot be projected onto such particles, you are quite wrong. For given that those who are blind from birth and have never seen the sun’s light nevertheless from their first day know bodies by touch without colour as an inseparable property, you can be sure that our mind too can form a conception of bodies without any coating of colour. In fact, we ourselves sense as colourless everything that we touch in blind darkness. This finds a partial parallel in Lucretius’ Epicurean contemporary Philodemus: Likewise, familiar bodies have colours, not in so far as they are bodies – for in so far as tangible things resist touch they are bodies, but they do not, in so far as they are tangible, manifest any colour. At any rate, objects in the dark do not have colour, yet are bodies.16

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That bodies in the dark have no colour is a thesis reported to have been advanced by Epicurus in book 2 of his lost work Against Theophrastus. We know nothing of the original context,17 but the aforementioned twin passages from Lucretius and Philodemus suggest that our current topic, the colourlessness of atoms, was at issue there too. As we saw Epicurus say in the Letter to Herodotus, atoms have ‘shape, weight, size, and the things necessarily included in the nature of shape’, but no additional qualities. Shape, weight and size correspond to the bundle of permanent properties (ἀίδιον συμβεβηκότα) that jointly constitute body as such,18 and without which atomic bodies cannot even be conceived. The excellent point is being made that colour could not legitimately have been added to the list. First, people blind from birth may never conceive of colour, but thanks to their other senses, especially touch, they certainly acquire the concept of body. Second, even the sighted can replicate that same austere conceptualization of body by reflecting on their periodic exposure to bodies in the dark when no feature of body qua body is absent. The fact thus established, that body can lose all colour but still exist, guarantees that colour stands to body as an accidental property (σύμπτωμα, = eventum in Lucretius). Those who think it is a permanent property have made an understandable error: colour is a permanent property, not however of body as such, but of visible body.19 Since atoms are not visible bodies,20 there is absolutely no reason why they should be coloured. A doubt still remains. Epicurus has by now shown good reasons why atoms, as invisible bodies, need not be coloured, but he has not shown us why they must be colourless. To put the problem slightly differently, we have seen his reasoning in favour of the colourlessness thesis, but not what motivated or inspired his commitment to it in the first place. To understand this motivation, we must inevitably turn to Democritus, whom Epicurus deeply respected, if not as an infallible authority, at any rate as a primary precursor. Epicurus’ system is an updated version of Democritus’, and what he inherited from his forerunner included the colourlessness of atoms. In Democritus’ famous dictum (SE M 7.135 = B9 DK = D14 LM): By convention (nomōi) sweet and by convention bitter, by convention hot, by convention cold, by convention colour. In reality (eteēi) atoms and void.21 Colours, along with other such properties, are a matter of ‘convention’ or ‘custom’, thereby standing in stark contrast to ‘reality’, which is attributable to atoms and void alone. This Democritean dictum, a version of which we know to have been attacked by Epicurus’ pupil Colotes as making life itself unliveable, is one on which Epicurus clearly had to take a carefully nuanced stance, since his own metaphysics agreed on the fundamental reality of atoms and void yet avoided the tempting Democritean inference that sensible properties like colour are unreal epiphenomena: they are real, according to Epicurus, but exist at the macroscopic level only.22 It therefore seems overwhelmingly likely that Epicurus’ denial of coloured atoms was developed in the course of his reflections on the celebrated Democritean dictum. He endorsed the colourlessness of atoms but resisted its negative implications for the reality of colours.

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It is therefore of the utmost importance to work out what Democritus had meant by nomōi. Although the term regularly means ‘by convention’, in contrast to physei, ‘by nature’, Democritus can hardly be appealing to any kind of consensus – to a claim, that is, that perceivers in general agree among themselves as to which things should be called sweet and which bitter, which white and which yellow and so on. For according to Democritus, it is at least partly because perceivers again and again disagree on these questions that true beings (i.e. atoms and void) can be inferred to be neither sweet nor bitter, neither white nor yellow and so on. ‘From the fact that honey tastes bitter to some but sweet to others, Democritus said that it is neither sweet nor bitter, Heraclitus that it is both’ (SE PH 2.63). This inference relied on his trademark ‘no more’ (ou mallon) slogan: when there is no more reason to assent to some attribution than to a contrary attribution, it would be arbitrary to choose one over the other. In Aristotle’s words (Metaphysics Γ 5, 1109b7-12):23 Also, both many other animals and we ourselves have opposites appear about the same things, and even each individual in relation to himself does not always have the same sensory appearances. Hence which of these are true or false is unclear. For no more are these ones true than those ones: they are of equal status. That is why Democritus, for his part, says that either none is true, or to us at least it is unclear. We should not, if we can avoid it, deprive Democritus of this characteristic piece of reasoning.24 To appeal to consensus would be, besides, to associate nomōi with comparatively stable agreement, whereas it more often connotes subjective instability: conventions, customs and laws vary from place to place, from time to time and from context to context. If instead consensus did play some part in his reasoning, we might expect the majority view, for example, that honey is sweet rather than bitter, to earn preference. But preference is precisely what Democritus’ iconic ou mallon dictum seems to deny. Could Democritus’ use of nomōi then instead be his way of stressing that the many and various perceivers each arbitrarily privilege their own viewpoint and thus not unlike legislators – ‘by law’ being another possible translation of nomōi – impose their own subjective viewpoint on appearances, some insisting that the wind itself is warm, or that the wine is bitter, just because that is how it appears to them? If so, the eteēi colourlessness of atoms and void could be arrived at as a means of conflict resolution at the nomōi level. Such an interpretation, based on conflicting appearances, fits well with the two pairs of opposites – sweet and bitter, hot and cold – cited in the dictum. But when Democritus adds that ‘colour’ too – and not, for example, ‘black and white’ – is nomōi, he seems to be referring less to conflicting appearances as to which colour something has than to colour as such, saying that the very fact of things appearing coloured (flavoured, etc.) at all is where they fail to correspond to reality. To read the dictum as a unified whole, then, we should take it to be saying that the attribution to bodies of flavour, temperature and colour as such – and not just of this or that flavour, temperature or colour – fails to correspond to reality.

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This still leaves at least two competing options. On the first option, Democritus thinks that each of our words for flavours, colours and so on does refer to something external, namely to such and such a kind of atomic structure in the perceived body, or such and such a kind of interaction between the perceived body and the sense organ; and that this way of speaking departs from reality just to the extent that both species-words like ‘blue’ and genus-words like ‘colour’ alike misrepresent what kind of thing that structure or interaction really is: not a colour at all, that is, but an arrangement – whether static or dynamic – of atoms and void. Such a use of nomos would be close to that of Empedocles (B9 DK = D54 LM), who warns us that in accordance with a deceptive convention (nomos) he will speak of ‘generation’ and ‘destruction’, even though what is really happening is not coming-to-be at all but the combination and separation of enduring components. Thus interpreted, both Empedocles and Democritus assume that each ‘conventional’ name (for Empedocles ‘birth’ and ‘death’; for Democritus ‘sweet’, ‘blue’, ‘colour’, etc.) succeeds in referring again and again to a corresponding objective reality (mixture and separation; this or that arrangement of atoms), but misrepresents its real nature.25 Although the relevant body of evidence is too large to explore here, it is unclear whether what I have said so far adequately captures Democritus’ considered position, which is why a second option must be considered. Although the extensive report in Theophrastus’ De sensibus26 shows that when explaining sensory mechanisms Democritus did tend to write as if each colour or flavour corresponded, as on option 1, to a specific atomic property of the perceived object, it also emerges there that he sometimes favoured a more overtly sceptical27 approach, founded largely on the relativity of any percept to the perceiver’s condition. From the evidence for the existence of conflicting appearances he is said to have inferred that what property is perceived depends primarily on the current state of the perceiver. Thus, the second option differs from the first in not necessarily assuming any systematic correlation between perceptions and their external objects. It is hard to be sure which of these two options, if either, represents the mode Democritus was in when he introduced the celebrated dictum according to which colour is nomōi. We may nevertheless make progress by examining the dictum in its own right. Consider first Sextus’ version:28 Democritus at times eliminates sensory appearances, and says that none of these appears truly but only in opinion (doxa), and that the truth (alētheia) in the things that exist is that atoms and void exist. For, he says, ‘By convention (nomōi) sweet and by convention bitter, by convention hot, by convention cold, by convention colour. In reality (eteēi) atoms and void.’ That is, although perceptibles are believed (nomizetai) and opined (doxazetai) to exist, these things do not exist in truth, but only atoms and void do.29 The nomōi . . . eteēi antithesis is here being interpreted as mere ‘believing’ (nomizein) versus ‘truth’. Is this plausible? The derivation of nomizein from nomos, although etymologically correct, can be semantically misleading, since the verb’s familiar epistemic connotation, ‘believe’, is not directly inherited from the noun.

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The transition from nomos to nomizein seems to be more gradual than that. The verb’s basic meaning is ‘make customary’, as most commonly in the passive voice, nomizesthai, ‘be customarily practised’. As a result, a phrase like nomizein tous theous comes to cross the semantic boundary between the non-epistemic ‘follow custom (nomos) with regard to the gods’ and the epistemic ‘believe in the gods’ or ‘believe that the gods exist’. Nevertheless, it seems doubtful that the simple nomōi ever followed suit and came to mean ‘according to belief ’. In this passage Sextus, or his source,30 manifests detailed knowledge of Democritus’ writings, and it is not hard to guess what led him to link Democritus’ nomōi to nomizein. It looks as if Democritus himself had elucidated the term nomōi, as used in his dictum, by introducing a new bit of jargon, nomisti. This piece of self-commentary by Democritus has as far as I know attracted no attention in the modern scholarship on him. Consider first Marcus Aurelius, Meditations 7.31: He [i.e. Democritus] says that all things are nomisti, and that in reality there are just the elements. It is enough to remember that the whole world is nomisti.31 There is no possible doubt that Marcus is paraphrasing the first half of the Democritean dictum. One might be tempted to conjecture on this evidence that the very rare term nomisti is what Democritus actually wrote there, and that its replacement with the more familiar nomōi is a later banalization. But this faces at least two objections. First, the contrasting term of the pair, the Ionic eteēi, has been faithfully preserved in all the full citations of the dictum that have come down to us, and it at least, far from being a banalization, is known to have been a regular part of Democritus’ terminology, while virtually unattested outside it.32 It would be odd, therefore, if the term nomisti, paired with it in the very same sentence, had suffered a quite different fate. Second, there is independent evidence that nomisti, rather than being the word used in the dictum itself, was Democritus’ own added gloss, elucidating the term nomōi that he had used in it. This evidence is found in Galen, On the elements according to Hippocrates 1.417-18 K: For all of these [particle theorists] posit that the primary element is qualityless, possessing neither intrinsic whiteness, blackness or any colour whatsoever, nor sweetness, bitterness, heat or coldness, nor any other quality whatsoever. For ‘by convention (nomōi) colour, by convention sweet, by convention bitter, but in reality (eteēi) atoms and void’ says Democritus, who thinks that all the perceptible qualities are brought into being, relative to us who perceive them, by the combination of atoms, and that by nature nothing is white or black or yellow or red or sweet or bitter. For the expression ‘by convention’ (nomōi) means the same as what one might call nomisti and relative to us, not according to the nature of things themselves. But what is by contrast ‘in reality’ (eteēi) he so calls by deriving the term from ‘real’ (eteon), which means ‘true’. The general idea of his theory would be along the following lines. A thing is believed (nomizetai) among humans to be white, black, sweet, bitter and everything else like that, but in truth ‘thing’ (den) and ‘nothing’ is all there is. For this too is his own term: he

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named atoms ‘thing’ (den), and void ‘nothing’ (mēden). So all atoms, which are small bodies, are without qualities.33 Even though it contains what may well be accretions from the later interpretative tradition, including the assimilation of nomōi to nomizein and the association of this in turn with relativity,34 it remains a remarkably rich report. And Galen is virtually explicit in presenting nomisti not as part of the dictum, but as a gloss intended to clarify its meaning. My reason for adding that the gloss was in all probability supplied by Democritus himself is that the tiny handful of attestations of this very unusual term occur exclusively in the context of the Democritean dictum.35 Clearly, it was deemed to have the power to elucidate the meaning of nomōi there, but that can only have been for readers already versed in Democritean terminology. It is hard to see who, if not Democritus himself, would have used it for that purpose. What then does nomisti mean? The sources reflected in the passages we have seen from Sextus and Galen give the impression of deriving this very rare word from nomizein, ‘believe’. But that is unlikely to be correct. In prose, and frequently in verse too, the -isti termination standardly means ‘in such and such a language (or dialect)’. Thus, Hellēnisti means ‘in Greek’, Attikisti ‘in Attic’, barbaristi ‘in a foreign language’, Rōmaïsti ‘in Latin’ and so on. Not only is this the termination’s overwhelmingly dominant usage in Greek literature as a whole, it is also the sole use of it in the most directly comparable prose texts, for example, in its eight occurrences in Herodotus, Democritus’ Ionian contemporary. It seems, then, that with his gloss nomisti Democritus was coining a term that we might render as ‘in the language of convention’.36 More appropriately to his characteristic terminological inventiveness, let us create the translation ‘in conventionese’. In mentioning Democritus’ terminological creativity, I have in mind his readiness to invent a term where Greek did not already supply one. The best-known example in fact occurs at the end of the passage already cited from Galen, which alludes to Democritus’ creation of the word den,37 ‘thing’ (= body), as the positive counterpart of mēden, ‘no-thing’ (= void).38 Democritus is, moreover, reported to have cited the phenomenon of missing words as one of his four proofs that existing language is not natural but the result of human imposition or contrivance (thesei, a term close in sense to nomōi).39 So when he introduced new technical terms like den, did he deem himself to be improving conventionese – adding missing words in order to bring it one step closer to mapping accurately onto reality, despite the fact that the gap could never be altogether closed? Or was he, rather, starting to build a new language, one that we might on his behalf think of as ‘real-ese’? Either way, we are seeing here reasons to interpret his nomōi dictum as one that he himself explained as making a primarily linguistic point: when, following convention, we40 speak of colour as such, or of this or that colour, we are not speaking the language of reality.41 Such a language, if we could ever learn to speak it fluently, would not include colour terms at all, replacing them with talk of atoms and void: the various arrangements of these in external atomic structures, and their interactions with atomic structures in our sense organs.

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The final question must be where this conviction of Democritus’ came from. If we were to assume its origin to have been internal to his atomic theory, it might prove hard to find reasons for the colourlessness thesis that were any more compelling than we earlier found those of Epicurus to be. After all, if atoms had turned out to be in reality coloured, the language of colours could, like the language of shape, have been legitimately included in Democritus’ vocabulary for his preferred language, realese. The likelier answer is that this division between conventional language and reality was an integral part of Democritus’ Parmenidean heritage. To cut a long story short, Parmenides had argued that reality is indivisible and unchanging, since there is no conceivable not-being to punctuate or vary its pure being; and that all talk of division and change (explicitly including changes of colour), is just that, mere ‘naming’ (B8.38-41), based on sense-perception, not reasoning (B7). Democritus is credibly understood as trying to preserve as much as possible of this Eleatic ontology.42 To achieve that, he sacrifices just one Parmenidean tenet, the inconceivability of not-being, an entity which he instead vindicates, identifying it with void.43 That each body is internally indivisible (= atomic) is established by an expanded version of Parmenides B8.22, ‘nor is it divisible, since it is all alike’: since all being is alike, if a being were internally divisible anywhere it would be divisible everywhere, which, since the resultant parts must each either have or lack magnitude, would have the unwelcome consequence of making the whole either infinitely large or altogether sizeless.44 Given this overall Eleatizing strategy, there should be no surprise if Democritus hoped to follow Parmenides in another way too, by vindicating the latter’s exclusion from reality of all colour differentiation, along with other sensible properties. And his atomism proved to lend itself remarkably well to the task, at a stroke reducing colours, temperatures and the like to combinations of being and the now legitimized not-being. If my main contentions have been right, the colourlessness of Epicurean atoms, for all its rich legacy to the later history of physics, had a curiously hybrid ancestry. Thanks to the intermediacy of Democritus, it seems, the doctrine of colourless body, having drawn its original inspiration from the radical anti-empiricist Parmenides, in time found its way into Epicureanism, antiquity’s leading empiricist philosophy. Epicurus had no really compelling argument for the thesis, but, having inherited it from Democritus along with the other main principles of the latter’s atomism, he judged it both conceptually defensible and explanatorily powerful, as indeed it is. In adopting it he faced the danger that, as Democritus had already seen, colours and other observer-dependent qualities might have to be sidelined as unreal. He resisted that danger by developing an anti-reductionist metaphysics,45 one that unlike its Democritean predecessor recognized equally authentic, and irreducibly different, realities at the microscopic and the macroscopic level. If Democritus did not himself adopt such an escape route from the temptations of scepticism about the senses, that was ultimately because his atomism was a pluralized form of Eleatic monism, just as committed as Parmenides had been to excluding volatile sensible qualities from his core ontology.

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NOTES 1. My thanks for helpful comments from Gábor Betegh, from Ugo Zilioli, from an anonymous referee and from audience members at the May 2017 Durham meeting on atomism. 2. For the uncertain case of temperature, see Taylor (1999, 75, n. 63). 3. Philodemus, On signs 17.28-18.16. 4. καὶ μὴν καὶ τὰς ἀτόμους νομιστέον μηδεμίαν ποιότητα τῶν φαινομένων προσφέρεσθαι πλὴν σχήματος καὶ βάρους καὶ μεγέθους καὶ ὅσα ἐξ ἀνάγκης σχήματος συμφυῆ ἐστι. ποιότης γὰρ πᾶσα μεταβάλλει· αἱ δὲ ἄτομοι οὐδὲν μεταβάλλουσιν, ἐπειδήπερ δεῖ τι ὑπομένειν ἐν ταῖς διαλύσεσι τῶν συγκρίσεων στερεὸν καὶ ἀδιάλυτον, ὃ τὰς μεταβολὰς οὐκ εἰς τὸ μὴ ὂν ποιήσεται οὐδ’ ἐκ τοῦ μὴ ὄντος, ἀλλὰ κατὰ μεταθέσεις ἐν πολλοῖς, τινῶν δὲ καὶ προσόδους καὶ ἀφόδους. ὅθεν ἀναγκαῖον τὰ μὴ μετατιθέμενα ἄφθαρτα εἶναι καὶ τὴν τοῦ μεταβάλλοντος φύσιν οὐκ ἔχοντα, ὄγκους δὲ καὶ σχηματισμοὺς ἰδίους· ταῦτα γὰρ καὶ ἀναγκαῖον ὑπομένειν. καὶ γὰρ ἐν τοῖς παρ’ ἡμῖν μετασχηματιζομένοις κατὰ τὴν περιαίρεσιν τὸ σχῆμα ἐνυπάρχον λαμβάνεται, αἱ δὲ ποιότητες οὐκ ἐνυπάρχουσαι ἐν τῷ μεταβάλλοντι, ὥσπερ ἐκεῖνο καταλείπεται, ἀλλ’ ἐξ ὅλου τοῦ σώματος ἀπολλύμεναι. ἱκανὰ οὖν τὰ ὑπολειπόμενα ταῦτα τὰς τῶν συγκρίσεων διαφορὰς ποιεῖν, ἐπειδήπερ ὑπολείπεσθαί γέ τινα ἀναγκαῖον καὶ εἰς τὸ μὴ ὂν φθείρεσθαι. 5. πλήν here must, somewhat irregularly, mean something like ‘over and above’, rather than ‘except’. Epicurus could hardly list the shapes, sizes and weights of atoms among the ‘qualities’, yet go on to add that all qualities change while atoms do not. 6. For its background we should not be looking to, for example, chapter 8 of Aristotle’s Categories, where ‘qualities’ include long-term dispositions, but rather to Plato, Theaetetus 182a4-c5, where the word ποιότης was in fact first coined in the course of elucidating the radical instability of sensory properties. 7. For changelessness as the defining property of Epicurean atoms, see Betegh (2006). 8. See below, pp. 64–5. 9. I add this qualification to allow for things that may well be constantly illuminated so long as they exist, for example the sun. 10. This causal dependence of colour on the atomic (re)arrangement of bodies need not be threatened by an additional dependence acknowledged by Epicurus, colour’s relativity to the eye (Plut. Col. 1110C). All perceptions are true, he holds. Hence, the eye sees a colour either correctly or not at all. 11. praeterea si nulla coloris principiis est / reddita natura et variis sunt praedita formis, / e quibus omne genus gignunt variantque colores,/ propterea magni quod refert, semina quaeque / cum quibus et quali positura contineantur / et quos inter se dent motus accipiantque,/ perfacile extemplo rationem reddere possis,/ cur ea quae nigro fuerint paulo ante colore,/ marmoreo fieri possint candore repente,/ ut mare, cum magni commorunt aequora venti,/ vertitur in canos candenti marmore fluctus;/ dicere enim possis, nigrum quod saepe videmus,/ materies ubi permixta est illius et ordo / principiis mutatus et addita demptaque quaedam,/ continuo id fieri ut candens videatur et album./ quod si caeruleis constarent aequora ponti / seminibus, nullo possent albescere pacto;/ nam quo cumque modo perturbes caerula quae sint,/ numquam in marmoreum possunt migrare colorem . . . 12. Esp. Lucr. 2.730-990.

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13. I defend this interpretation in Sedley (1992, 44–55). 14. quin etiam quanto in partes res quaeque minutas / distrahitur magis, hoc magis est ut cernere possis / evanescere paulatim stinguique colorem; / ut fit ubi in parvas partis discerpitur austrum: / purpura poeniceusque color clarissimus multo, / filatim cum distractum est, disperditur omnis; / noscere ut hinc possis prius omnem efflare colorem / particulas, quam discedant ad semina rerum. 15. nullus enim color est omnino materiai / corporibus, neque par rebus neque denique dispar./ in quae corpora si nullus tibi forte videtur / posse animi iniectus fieri, procul avius erras. / nam cum caecigeni, solis qui lumina numquam / dispexere, tamen cognoscant corpora tactu / ex ineunte aevo nullo coniuncta colore, / scire licet nostrae quoque menti corpora posse / vorti in notitiam nullo circumlita fuco. / denique nos ipsi caecis quaecumque tenebris / tangimus, haud ullo sentimus tincta colore. 16. Philodemus, On signs 18.3-10: ὁμοίως δὲ χρώματ᾽ ἔχει τὰ παρ᾽ ἡμῖν σώματα οὐχ ἧι σώματ᾽ ἐστίν· τὰ γὰρ ἁπτὰ καθὀ μὲν ἀντιτυπεῖ τ̣ὴν ἁφὴν σώματ᾽ ἐστίν, καθὸ δ᾽ ἁπτά ἐστιν οὐδεμίαν ἐμφαίνει χ̣ρόαν. τὰ γοῦν ἐν τῶι σκ̣ότει χ̣ρ̣ό[α]ν μὲν [οὐ]κ ἔχει, σώματα δ᾽ ἐστί[ν]. 17. It is presumably no accident that it should have been in a work against Theophrastus that Epicurus argued for the colourlessness of bodies in the dark. An Aristotelian would be likely to say that such circumstantially hidden colour remains present, as a first actuality. 18. See Betegh (2006, 280–1). 19. This is clearly implicit in the closing lines of Ep. Hdt. 68. 20. The invisibility of atoms was argued by Epicurus on independent grounds (cf. Ep. Hdt. 56) and not just inferred from their colourlessness. 21. νόμῳ γάρ φησι γλυκὺ καὶ νόμῳ πικρόν, νόμῳ θερμόν, νόμῳ ψυχρόν, νόμῳ χροιή· ἐτεῇ δὲ ἄτομα καὶ κενόν. I here adopt the most reliable version, reported by Sextus Empiricus M 7.135 in a passage full of verbatim Democritean quotations. The Epicurean Colotes (Plutarch, Col. 1110E) included ‘compound’ (σύγκρισις) in the list of νόμῳ items (see Kechagia 2011, 180–5). I agree with those who hold this to be Colotes’ own addition, without which he could not have gone on, as he did, to accuse Democritus of making life impossible by requiring us to disbelieve in our own existence. (For considerations in favour of Colotes’ version, see Wardy 1988, 139–40.) 22. Sedley (1988), Furley (1993). 23. ἔτι δ​ὲ καὶ​  πολλ​οῖς τ​ῶν ἄλ​λων ζ​ῴων τ​ἀναντ​ία πε​ρὶ τῶ​ν αὐτ​ῶν φα​ίνεσθ​αι κα​ὶ ἡμῖ​ν,  κα​ὶ αὐτ​ῷ δὲ ἑκάστ​ῳ πρὸ​ς αὑτ​ὸν οὐ​  ταὐτ​ὰ κατ​ὰ τὴν​  αἴσθ​ησιν ​ἀεὶ δ​οκεῖν​.  ποῖ​α οὖν​  τούτ​ ων ἀλ​ηθῆ ἢ​  ψευδ​ῆ, ἄδ​ηλον·​  οὐθὲ​ν γὰρ​  μᾶλλ​ον τά​δε ἢ ​τάδε ​ἀληθῆ​,  ἀλλ​'  ὁμο​ίως. ​διὸ Δ​ ημόκρ​ιτός ​γέ φη​σιν ἤ​τοι ο​ὐθὲν ​εἶναι​  ἀληθ​ὲς ἢ ​ἡμῖν ​γ' ἄδ​ηλον.​ 24. See Makin (1993, esp. 65–84). 25. For this interpretation, including the Empedocles comparison, see Furley (1993, 76–81). 26. E.g. Sens. 60–1, 63–4, 67–70. For a very judicious exegesis see Lee (2005, 200–16). For the specifics of Democritus’ account of colour, see Rudolph (2019).

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27. E.g. Theophrastus Sens. 64, 68; SE M 7.135-7. On the question of Democritus’ scepticism cf. esp. Taylor (1999, 216–22). 28. SE M 7.135. The Sextan report (ib. 135–40) is almost certainly derived in its entirety from Posidonius’ treatise On the criterion, see Sedley (1992, 27–43). The passage quoted from it in the next note is from the sceptically inclined remarks collected there, which are followed by an allegedly more optimistic quotation from a different Democritean work, The Canons: ‘Of knowing there are two forms, the one genuine, the other bastard. And of the bastard kind this is the complete list: sight, hearing, smell, taste, touch. The one which is genuine, but separate from this one, is when the bastard one is no longer able either to see in the direction of greater smallness, nor to hear or smell or taste or sense by touch other things in the direction of greater fineness.’ For the Greek text, which in my view contains no lacuna, see Sedley (1992, 39–43). 29. Δημόκριτος δὲ ὅτε μὲν ἀναιρεῖ τὰ φαινόμενα ταῖς αἰσθήσεσι, καὶ τούτων λέγει μηδὲν φαίνεσθαι κατ᾿ ἀλήθειαν ἀλλὰ μόνον κατὰ δόξαν, ἀληθὲς δὲ ἐν τοῖς οὖσιν ὑπάρχειν τὸ ἀτόμους εἶναι καὶ κενόν. “νόμῳ” γάρ φησι “γλυκὺ καὶ νόμῳ πικρόν, νόμῳ θερμόν, νόμῳ ψυχρόν, νόμῳ χροιή· ἐτεῇ δὲ ἄτομα καὶ κενόν.” ὅπερ ἔστι, νομίζεται μὲν εἶναι καὶ δοξάζεται τὰ αἰσθητά, οὐκ ἔστι δὲ κατ᾿ ἀλήθειαν ταῦτα, ἀλλὰ τὰ ἄτομα μόνον καὶ τὸ κενόν. 30. See note 27 above. 31. ἐκεῖνος μέν φησιν ὅτι “πάντα νομιστί, ἐτεῇ δὲ μόνα τὰ στοιχεῖα”, ἀρκεῖ δὲ μεμνῆσθαι ὅτι τὰ πάντα νομιστὶ ἔχει. 32. The TLG records 33 Democritean occurrences of έτεῇ, and only one nonDemocritean. 33. ὑπόκειται γὰρ ἅπασι τούτοις ἄποιον εἶναι τὸ πρῶτον στοιχεῖον οὔτε λευκότητα σύμφυτον ἔχον ἢ μελανότητα ἢ ὅλως ἡντινοῦν χροιὰν οὔτε γλυκύτητα ἢ πικρότητα ἢ θερμότητα ἢ ψυχρότητα οὔθ’ ὅλως ἡντινοῦν ἑτέραν ποιότητα. ‘νόμῳ γὰρ χροιὴ νόμῳ γλυκὺ νόμῳ πικρὸν, ἐτεῇ δ’ ἄτομα καὶ κενόν’ ὁ Δημόκριτός φησιν ἐκ τῆς συνόδου τῶν ἀτόμων γίγνεσθαι νομίζων ἁπάσας τὰς αἰσθητὰς ποιότητας ὡς πρὸς ἡμᾶς τοὺς αἰσθανομένους αὐτῶν, φύσει δ’ οὐδὲν εἶναι λευκὸν ἢ μέλαν ἢ ξανθὸν ἢ ἐρυθρὸν ἢ γλυκὺ ἢ πικρόν. τὸ γὰρ δὴ νόμῳ ταὐτὸ βούλεται τῷ οἷον νομιστὶ, καὶ πρὸς ἡμᾶς, οὐ κατ’ αὐτὴν τῶν πραγμάτων τὴν φύσιν. ὅπερ δ’ αὖ πάλιν ἐτεῇ καλεῖ παρὰ τὸ ἐτεόν, ὅπερ ἀληθὲς δηλοῖ, ποιήσας τοὔνομα. καὶ εἴη ἂν ὁ σύμπας νοῦς αὐτοῦ τοῦ λόγου τοιόσδε· νομίζεται μέν τι παρὰ τοῖς ἀνθρώποις λευκὸν εἶναι καὶ μέλαν καὶ γλυκὺ καὶ πικρὸν καὶ τἆλλα πάντα τὰ τοιαῦτα, κατὰ δὲ τὴν ἀλήθειαν ὲν καὶ μηδέν ἐστι τὰ πάντα. καὶ γὰρ αὖ καὶ τοῦτ’ εἴρηκεν αὐτὸς ὲν μὲν τὰς ἀτόμους ὀνομάζων, μηδὲν δὲ τὸ κενόν. αἱ μὲν οὖν ἄτομοι σύμπασαι σώματ’ οὖσαι σμικρὰ χωρὶς ποιοτήτων εἰσὶ . . . 34. At any rate, the passage’s terminology for relativity, πρὸς ἡμᾶς, οὐ κατ’ αὐτὴν τῶν πραγμάτων τὴν φύσιν, does not correspond either to anything reported from Democritus, even among his verbatim dicta at SE M. 7.135-40, or to anything of comparably early date. It does on the other hand strongly recall the denial of Protagorean relativism at Plato, Crat. 387d1-2, οὐ πρὸς ἡμᾶς οὖσαι, ἀλλ᾽ αὑτῶν τινα ἰδίαν φύσιν ἔχουσαι, referring back to 386d8-e4. 35. The one not already quoted is Diog. Oen. 7 II 2-8 Smith: ἐσφάλη δ’ ἀνα̣ξίως ἑαυτοῦ καὶ Δημόκριτος, τὰς ἀτόμους μόνα̣ς κατ’ ἀλήθειαν εἰπὼν ὑπάρχειν ἐν τοῖς οὖσι, τὰ δὲ λοιπὰ νομιστεὶ ἅπαντα: ‘Democritus too let himself down, when he said that only

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atoms truly exist in reality, and that everything else is nomisti.’ Here νομιστεί represents νομιστί, the itacism being regular orthography in the inscription (see Smith 1992, 117). 36. Cf. Lucian, De morte Peregrini 3.2, . . . γὺψ ἀναπτάμενος ἐκ μέσης τῆς φλογὸς οἴχοιτο ἐς τὸν οὐρανόν ἀνθρωπιστὶ λέγων . . ., where ἀνθρωπιστί means ‘in human language’. 37. The Galen MSS twice have not δέν but ἕν here, but the emendation – whether it be to Galen’s text or to that of his source – is a near-certainty. For the coinage δέν see Plutarch, Col. 1109A = D33 LM. 38. See Democritus D33-8 LM for a few further examples of his terminological independence. 39. Procl. In Plat. Crat. 6.20-7.6 = Democritus B26 DK = D205 LM. The examples of linguistic anomalies in this passage do not go back to Democritus, but the names for the four proofs evidently do, including the archaic term for the argument from missing words, νώνυμον, which (appropriately?) has no other prose attestations. 40. It is possible that, rather than ‘we’, it was the senses themselves that were represented as speaking conventionese. For the senses speaking (though in this case presumably not in conventionese), cf. Galen, Med. exp. 15.7 = B125 DK = D23a LM, where Democritus is reported to have represented the senses as speaking back against the mind’s criticism: ‘Poor mind, you get your evidence from us and then you condemn us. The condemnation is your own downfall.’ 41. I am not suggesting that in the dictum eteēi means ‘in realese’: in its recorded Democritean contexts it unmistakably refers to reality itself, and there is no evidence that Democritus ever coined a term like eteïsti. 42. Wardy (1988), Sedley (2008). 43. Sedley (1982). 44. Sedley (2008). 45. For this admittedly controversial thesis, see Sedley (1988), and the counterarguments of O’Keefe (2005, ch. 4).

REFERENCES Betegh, G. (2006), ‘Epicurus’ argument for atomism’, Oxford Studies in Ancient Philosophy 30: 261–84. Furley, D. (1993), ‘Democritus and Epicurus on sensible qualities’, in J. Brunschwig and M. Nussbaum (eds), Passions and Perceptions: Studies in Hellenistic Philosophy of Mind, 72–94, Paris and Cambridge: Cambridge University Press. Kechagia, E. (2011), Plutarch against Colotes, Oxford: Oxford University Press. Lee, M.-K. (2005), Epistemology after Protagoras, Oxford: Oxford University Press. Makin, S. (1993), Indifference Arguments, Oxford: Blackwell. O’Keefe, T. (2005), Epicurus on Freedom, Cambridge: Cambridge University Press. Rudolph, K. (2019), ‘Democritus’ theory of colour’, Rhizomata 7, no. 2: 269–305. Sedley, D. (1982), ‘Two conceptions of vacuum’, Phronesis 27: 175–93. Sedley, D. (1988), ‘Epicurean anti-reductionism’, in J. Barnes and M. Mignucci (eds), Matter and Metaphysics, 295–327, Naples: Bibliopolis.

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Sedley, D. (1992), ‘Sextus Empiricus and the atomist criteria of truth’, Elenchos 13: 19–56. Sedley, D. (2008), ‘Atomism’s Eleatic roots’, in P. Curd and D. W. Graham (eds), The Oxford Handbook of Presocratic Philosophy, 305–32, Oxford: Oxford University Press. Taylor, C. C. W. (1999), The Atomists: Leucippus and Democritus, Toronto: University of Toronto Press. Wardy, R. (1988), ‘Eleatic pluralism’, AGPh 70: 125–46.

CHAPTER 4

Atoms and minimal ‘Parts’1 The originality of Epicurean atomism FRANCESCO VERDE

FOREWORD Despite Marx’s dissertation about the Differenz der demokritischen und epikureischen Naturphilosophie nebst einem Anhange (1841), one of the most common misunderstandings made by those who have only a cursory knowledge of ancient materialism and, more generally, of the history of ancient thought is to believe that Epicurus’ philosophy amounts to nothing more than a repetition of Leucippus and Democritus’ atomism in the Hellenistic age. Those who believe this are generally unaware of the theoretical efforts made by Epicurus to clearly criticize certain Democritean doctrines (e.g. physical determinism).2 Not only that, but they also fail to recognize that the attempt to downplay the originality of Epicurean philosophy is rooted in antiquity. If we take the first book of Cicero’s De finibus,3 we find that in these pages Epicurus is considered a mere ‘copyist’: his physical doctrine would depend entirely on Democritus, his ethics on Aristippus. In ancient thought it is very common for those who intend to dismiss or fiercely criticize a particular doctrine to say that it is not original but depends on previous philosophers and philosophies. In short, most ancient authors cling to the principle that what comes before is always better than what comes after. Of course, this principle means that in ancient philosophy, we almost never find a neutral and objective understanding of the historical past, but rather a deliberate and partial use of the past through which new philosophical identities are built.4 Many examples of this attitude could be adduced; I will just mention two. From Diogenes Laertius5 we learn that the Peripatetic philosopher Aristoxenus of Tarentum6 accused Plato’s Republic of plagiarism: all its teaching had already been expounded in Protagoras’ Antilogies. What is also of particular interest is Athenaeus’ testimony7 on the historical figure of Theopompus of Chios.8 After having trained at Isocrates’s school, in his (lost) work Against the Teaching of Plato (Κατὰ τῆς Πλάτωνος διατριβῆς), Theopompus stated that most of Plato’s dialogues depended on the diatribes of Aristippus and the writings of Anthistenes9 and Bryson of Heraclea. Of course, the examples could be multiplied. Something similar occurred in Epicurus’ case, as we know from Cicero’s De finibus, as well as from a passage from

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Diogenes Laertius’ Life of Epicurus.10 Epicurus was accused of having plagiarized the physics of Democritus and the ethics of Aristippus.11 But while it is an undisputed fact that the historical and doctrinal origin of Epicurean atomism lies in the philosophy of Leucippus and Democritus (something Epicurus himself was aware of),12 this does not mean that Epicurus’ philosophy is nothing more than a ‘copy’ of the doctrines of earlier atomists. Of the many arguments that could be put forward to prove this statement, one of the most important concerns the doctrine of minimal ‘parts’ (ta elachista), which will be the subject of this chapter. It is a genuinely Epicurean theory which, at least on the basis of the ancient sources we possess, cannot be attributed to either Leucippus or Democritus. As we will see in this chapter (where I will be dealing with the theory of elachista only in relation to Epicurus and Lucretius, leaving aside later developments), the doctrine of minima has often been neglected, but actually constitutes a highly original innovation compared to the physics of earlier atomists.13 This teaching had absolutely crucial consequences for Epicurus’ philosophy, as well as for the history of ancient materialism more generally.14

THE DEMOCRITEAN BACKGROUND AND EVIDENCE FROM EPICURUS’ LETTER TO HERODOTUS In H. Diels and W. Kranz’s edition of the fragments of the so-called Presocratics, the editors report a problematic testimony from Dionysius, the Christian bishop of Alexandria,15 according to which Democritus (unlike Epicurus) supposed that very large atoms exist. Diels and Kranz here refer to paragraph 55 of the Letter to Herodotus, where Epicurus – taking up what he has written in paragraphs 42–43 of the letter – affirms that atoms cannot be of any size at all: for this is absolutely denied by the crucial ‘criterion’ of enargeia or perceptive self-evidence.16 Phenomena refute the idea that there are atoms so big as to be visible. On the basis of the reconstruction proposed by Diels and Kranz, most scholars have concluded that when Epicurus asserts that atoms, despite having an inconceivable (yet not unlimited) variety of shapes, cannot be of all sizes, he is being critical of the Democritean position. This conclusion may be acceptable, but remains rather controversial (for it is likely that Dionysius’ report expresses not Democritus’ own position but rather Dionysius’ own criticism of Democritean atomism).17 Here I would like to underline at least two points: (1) the ‘very large atoms’ (megistas [. . .] atomous) supposed by Democritus according to bishop Dionysius’ testimony are not necessarily perceivable; (2) in a passage from Aristotle’s (unfortunately lost) work On Democritus, reported by Simplicius,18 we read that Democritus admitted the most varied shapes and arrangements for atoms (pantoias morphas kai schemata pantoia). Since immediately before Aristotle refers that the Democritean atoms are so small as to elude our senses, here I would assign the adjective pantoios the meaning of ‘more varied’ or ‘more different’. However, the issue becomes even more complicated if we take into account another Aristotelian passage from On Generation and Corruption,19 in which the author establishes a clear terminological distinction in relation to Leucippus and Plato (the reference is to the Timaeus). Leucippus is said to call his indivisible bodies

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‘solid’ (sterea), whereas Plato is said to define them as ‘surfaces’ (epipeda).20 It is well known that behind this distinction there lies the deep (and perhaps sincere) regard in which Aristotle held the atomists, by contrast to Plato and his ‘geometric reductionism’: although for Aristotle the atomistic hypothesis is completely senseless (even if Aristotle himself is frequently very respectful of Democritus – see e.g. Aristot. GC I 8, 324b 35-325a 2 – although he thinks that his arguments fail), the philosopher recognizes that the atomists have investigated nature from a correct methodological point of view (i.e. in conformity with nature), by theorizing physical principles (i.e. bodies and not two-dimensional geometric surfaces)21 to explain the totality of things.22 Unlike Plato (who in the Timaeus admits only two indivisible figures, i.e. the ‘original’ rectangular triangles),23 according to Aristotle, Leucippus argues that each of the indivisible solids (atoms) is defined (horisthai) by infinite shapes (apeirois [. . .] schemasi).24 If the unlimitedness of atomic shapes is connected to the unlimitedness of sizes, then visible atoms must exist; from this point of view, therefore, the Epicurean position can (also) be interpreted as standing in polemical contrast to ancient atomism (i.e. more specifically to Democritus). At any rate, it is clear that for Epicurus atoms, although of infinite number, do not possess all shapes and, consequently, all sizes: phenomenal self-evidence confirms this. There is an inconceivable yet finite spectrum of atomic shapes and sizes, although for each form the number of atoms is infinite. Had Epicurus admitted the existence of only a few shapes for atoms, he would have encountered considerable difficulties in explaining the great variety of aggregate bodies. Depending on their shapes, atoms will aggregate in different ways: it is precisely these different modes of aggregation (which, of course, depend on the differences in shape of the atoms) that justify the equally inconceivable variety of aggregate bodies. From Aristotle25 we learn that there are three differences in atoms according to Leucippus and Democritus: (1) the configuration or shape (rhysmos/schema); (2) the mode of contact or order (diathige/taxis); (3) the mode of turning/revolving or position (trope/thesis). The first Greek term in parenthesis in all likelihood belongs to the technical lexicon of the atomists, while the second one is typical of Aristotelian terminology: one could conjecture that the second word is the Aristotelian ‘translation’ of a term that must have been difficult to understand – already in Aristotle’s day. It is clear that while the name probably used by the atomists describes a dynamic state of affairs, the Aristotelian ‘translation’ makes the original terms more static (and hence more distant from their original physical meaning). According to Epicurus, atoms – which in themselves do not possess qualities such as colour, which only belongs to the aggregate body26 – possess three characteristics of their own: (1) shape (schema); (2) weight (baros); (3) size/magnitude (megethos).27

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Ancient doxography attributes to Democritus the idea that atoms only had two characteristics: size (megethos) and shape (schema).28 Epicurus, unlike his atomistic ‘predecessor’, would have added weight (baros), so as to explain the necessary condition for the movement of atoms. The Letter to Herodotus confirms that according to Epicurus, atoms had three intrinsic features: shape, weight and size. In my opinion, it is rather difficult to believe that the Democritean atoms – compact (nasta)29 and endowed with an indissoluble solidity (alytos sterrhotes)30 – had no weight; it is likely, instead, that the doxographic tradition attributed such a distinction to the philosophers in order to justify two different modes of movement. By depriving the atoms of weight, Democritus would have explained movement by referring to the mutual impact between atoms (kat’ allelotypian),31 whereas Epicurus would have seen weight, or more correctly the impact or thrust due to weight (tei tou barous plegei),32 as the precondition for the movement of atomic bodies. Of course, it is also conceivable that Epicurus added weight as an atomic characteristic to explain the mutual impact between atoms that, in his opinion, Democritus had been unable to justify with the elimination of weight. In any case, although Democritus considered shape to be the primary characteristic of atoms, on the basis of the ancient sources, it seems that the philosopher had not clarified the reason for differences in shape between atoms. In other words, why does atom a have shape x and atom b shape y? In this regard, Epicurus introduces a complex doctrine, that of atomic minima (ta elachista). It is a complicated theory that in my view must be studied in terms of ‘doctrinal evolution’: for it underwent considerable development within the Garden. On the basis of our sources, it is possible to observe a sort of ‘transposition’ of the elachista theory from the physical sphere (as witnessed by the Letter to Herodotus) to the more specifically geometric one.33 We should not forget the fact that Epicurus and/or the Epicureans theorized time, space and movement in ‘granular’ terms, that is, as discontinuous quantities consisting of indivisible ‘particles’, as we know from late sources and perhaps also from the Epicurean Demetrius of Laconia.34 In relation to the doctrine of the clinamen (the swerve), this view could explain why the angle of deviation of atoms travelling through void is of a minimum size.35 Here, however, I shall deal exclusively with how the doctrine of minima is presented in the Letter to Herodotus. This work transmitted by Diogenes Laertius in Book 10 of his Lives of the Eminent Philosophers offers the most detailed treatment of the theory of atomic minima, often misunderstood or superficially interpreted. It is well known that the Letter to Herodotus is a compendium of Epicurus’ science of nature (physiologia) (even if in this text some doctrines belonging to the canonic – the first part of the philosophical system, dealing with epistemology – are summarized). If Epicurus decided to dedicate part of a short epitome to the atomic minima, this clearly means that the doctrine in question plays a decisive role in his physics. Despite the fact that it is a rather long text, I think it is worth quoting paragraphs 55–59 of the Letter to Herodotus, in which Epicurus explains what minima are, how they are known and what their function is: Furthermore, we should not believe that atoms are of all sizes, lest the evidence prove us wrong; instead we should admit that there are some variations in

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size. For this will make it easier to explain what, according to our feelings and sensations, actually happens. 56. But the notion that every magnitude exists does not help to explain the differences of quality, since in that case visible atoms should have reached us, which has not been seen to occur; nor is it possible to conceive how an atom could become visible. Besides this, we must not believe there can be an unlimited number of masses, no matter how small, in any finite body. Accordingly, not only must we reject unlimited division into smaller pieces, lest we make everything weak, and in our conceptions of compound things be forced to squeeze existing things into nonexistence; we must also not believe that the passages in finite bodies can be divided infinitely or into smaller and smaller increments. 57. For it is not possible, once one says that the masses in an entity, however small they may be, are infinite in number, to conceive how the entity could be limited in size. For it is clear that the unlimited number of masses are of some size; accordingly, no matter how small they are, their aggregate would be infinitely large. And given that when something is finite its limit is distinct, even if one cannot observe it, it is not possible not to think of another such entity placed beside it; and it is therefore possible, when mentally adding one such entity to the next, to arrive in thought at infinity. 58. We must conceive of the smallest perceptible mass as neither similar to one that can be traversed, nor as wholly dissimilar to it, but as having something in common with those that can be traversed, though parts cannot be distinguished in it. But whenever, by reason of the resemblance created by this common property, we think we will distinguish something in it – one part here, the other there – we must be encountering something else of equal size. We discern these one after another, beginning with the first, and not as occupying the same space or as touching each other’s parts; instead we see them measuring out magnitudes in their characteristic way, more of them measuring out a larger magnitude, fewer of them a smaller. We must consider that this analogy also applies to the smallest part in the atom. 59. For clearly, only in its minuteness does it differ from what is observed by the senses, though the same analogy applies. For precisely because of this analogy we have asserted that the atom has magnitude; we have merely projected something small onto a larger scale. Furthermore, we must regard the smallest and indivisible parts as the limits of lengths, furnishing from themselves as units the means of measuring the lengths of larger or smaller atoms, when we mentally contemplate invisible realities. For what the atoms have in common with things that do not admit of any passage is sufficient to take us this far. But it is not possible for these smallest parts to form compounds, supposing that they possessed motion. (Transl. by P. Mensch) Epicurus rules out the existence of visible atoms; atoms have different sizes, but it is not possible for them to be large enough to be visible. To introduce the theory of minima, Epicurus states that a limited physical body cannot include an infinite number of corporeal masses: as we shall see, this is an important clarification to

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understand the relationship between atoms and minima. If within a limited body there is a finite number of corporeal masses (obviously smaller than the body they form), division to infinity is impossible: for Epicurus, the limitedness of bodies excludes division to infinity. Division would be infinite if the body included an infinite number of masses, but this is impossible because what is limited cannot contain what is unlimited. Now, according to Epicurus, each limited body must have an extremity (akron) that can be distinguished (dialepton) from the rest of the body; in short, the extremity of which Epicurus speaks is the limit of a limited visible body that, as such, will have an equally limited number of limits/extremities. As Epicurus writes at the end of paragraph 57, this extremity/limit is distinct (dialepton), even if one cannot observe (theoreton) it; the difference is subtle but clear. It is possible to distinguish the limit of a body – that is, to recognize the end of a body – but it is impossible to isolate this end from the body: it is not possible to observe and to isolate the limit of a body because the limit is a limit if and only if it remains united to the body. From this point of view, we can distinguish the extremity of a body only if this extremity remains united to the body: the extremity or limit of a body isolated from the body itself is a senseless concept, while nothing excludes that the next extremity of the body is similar to the first one. In this way the body itself will be made up of extremities/limits which are distinguishable but not observable in themselves.36 This is another important clarification for understanding the theory of minima. By remaining faithful to the fundamental principle that entities eluding perception (ta adela/ta aorata) can be admitted to exist and investigated only by analogical means (i.e. by starting from phenomena and then proceeding through the use of analogy),37 Epicurus begins his analysis with a minimum size which is perceptible and recalls the extremity of the body which the philosopher has just discussed. The perceptible minimum (to elachiston to en tei aisthesei: Hrdt. 58) is a minimum magnitude that, despite its size, is still perceptible: the perceptible minimum is the last size that sense-perception is able to grasp (or the first, depending on the point of view). If this perceptible minimum is actually the last (or the first) magnitude we are still able to perceive, by observing perceptible minimum a and consecutively perceptible minimum b, it will be possible to ‘cover’ the entire surface of the body that is the object of our perception. Each perceptible minimum, therefore, will be a unit of measure of magnitudes because – considered in its own individuality, as the limit beyond which our perceptive capacity cannot go – it is neither coincident with the next perceptible minimum nor in contact with it by means of parts: a minimum magnitude cannot have distinguishable parts; if it did have parts, they would be smaller than the minimum, which is a contradiction. After describing the characteristics of the perceptible minimum, Epicurus introduces the minimum in the atom (to en tei atomoi elachiston); the atomic minimum differs from the perceptible minimum by its smallness (mikrotes: Hrdt. 59), but still stands in a relationship of analogy to it. This means that the atomic minimum analogically has the same characteristics and performs the same function as the perceptible minimum. The mikrotes mentioned by Epicurus marks the fundamental difference between the

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perceptible minimum and the atomic minimum: the latter really is a minimum (material) magnitude that lacks parts, whereas the former only appears to our senses to be lacking parts. In other words, a magnitude of minimum size appears to be devoid of parts to our sense-perception, but if the atomic minimum also exists and differs from the perceptible one by its smallness, this means that only the former will really be a minimum magnitude devoid of parts. The atomic minimum is a material magnitude (a ‘piece of matter’ in all respects) of minimum dimensions: according to Epicurus, in nature there is no physical magnitude smaller than the minimum in the atom. The minimum indissolubly belongs to the atomic body. In paragraph 59 of the Letter, Epicurus explains a fundamental point by reasoning by analogy: as it is thanks to the perceptible minimum that visible bodies have size, the same analogy is valid for the atomic minimum, thanks to which the atom has size (megethos). In other words, the (physical and material) size of the atom would be nothing if there were no atomic minima, just as any body deprived of its perceptible minima would not have any size. If the minimum accounts for the size/dimension of the atom, it is the unit of measure (katametrema) of the (atomic) size. Thus, depending on the (obviously limited) number and the disposition of these minima, the atom will have a certain size, shape and weight. The question of the nature of atomic minima has long been debated; some scholars have hypothesized that minima are entities of a mathematical–theoretical sort.38 This hypothesis seems very problematic to me because in the Epicurean universe everything is body, with the exception of void; therefore, it is hardly conceivable that Epicurus inserted some theoretical elements inside the atom, which is the body par excellence. Textual evidence of the materiality of the atomic minima is found in the fact that Epicurus, in paragraph 59 of the Letter to Herodotus, explicitly declares that atomic minima differ from perceptible minima in terms of smallness: since, if we are able to perceive perceptible minima, this means that they are physical material, the same will apply to the atomic minima, with the difference that they are much smaller than the perceptible minima (which are themselves made up of further atoms). Using the rational theory about what is invisible (dia logou theoria epi ton aoraton), that is, the rational activity based on perceptible phenomena which investigates by analogy the field of what eludes sense-perception, Epicurus states that minimum magnitudes devoid of parts (ta elachista kai amere) are limits (perata) and units of measure (katametrema) of the dimensions of atoms. Before moving on to the treatment of infinity (to apeiron) in paragraph 60, Epicurus underlines a fundamental point: even if minima were endowed with movement, they could not form compound bodies. I will return to the characterization of minima as limits, as well as to the impossibility for minima to form a compound. Let us now examine Lucretius’ testimony on this theory.

LUCRETIUS’ EVIDENCE Lucretius devotes two parts of the De Rerum Natura to atomic minima: the first is in Book I of the poem (599–634), the second in Book II (481–499).

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Here is the first piece of evidence in the continuous translation given by M. F. Smith: Then again, since each of those ultimate particles that are beneath the ken of our senses has an extreme point, that point is evidently without parts and is the smallest existence; it never has had and never will be able to have an independent, separate existence, since it is itself primary and unitary part of something else. Then rank upon rank of similar parts in close formation provide the ultimate particle with its full complement of substance and, since they cannot have an independent existence, they must cling so fast to the whole atom that they cannot by any means be wrenched apart from it. The primary elements are therefore solid and simple, being formed of smallest parts packed solid in a closely cohering mass; 610. they are not compounded as a result of the assembly of those parts, but rather derive their power from their everlasting simplicity; nature does not allow anything to be torn away or subtracted from them and so preserves the seeds of things. Moreover, if there is no smallest point, every minutest body will be composed of an infinite number of parts, since a half of a half will always have a half and there will be no limit to the possibility of division. If this is the case, what will distinguish the whole universe from the smallest thing in it? 620. Nothing; for, no matter how fully infinite is the whole universe, the minutest objects will equally be composed of an infinite number of parts. But since sound judgment loudly protests against this conclusion and denies that the mind can believe it, you must admit defeat and acknowledge the existence of points that have no parts and are the smallest things; and this being so, you must also acknowledge the existence of solid and everlasting primary elements. Lastly, if it had been creative nature’s way to compel all things to be resolved into their smallest parts, 630. she would no longer be able to renew anything out of them, because objects that are insufficiently bulky to have any parts cannot possess the essential characteristics of generative matter, namely the variety of interlacements, weights, collisions, concurrences, and movements that cause all things to happen. (Transl. by M. F. Smith) Like Epicurus, Lucretius deals with the extremity (I 599: extremum cacumen), only this time the body is one we cannot perceive through sensation; this is a notable difference compared to the Letter to Herodotus. At any rate, this extremity is of minimal size, without parts (I 601: sine partibus) and cannot exist separately from that of which it is the extremity. These ‘parts’, one after the other, complete the nature of the body and cannot in any case be isolated from the body to which they necessarily belong. Having described the role played by the extremity, Lucretius – unlike Epicurus – introduces a crucial point for Epicurean physics: atoms should not be considered aggregate bodies, but rather simple and unitary bodies (609– 610), even though they are made up of the smallest parts. The existence of the minimum and the fact that the primordia (i.e. atoms considered as limited bodies) are absolutely simple (i.e. not formed by minima in the same way as atoms form aggregate bodies) exclude division to infinity: nothing exists beyond minimal parts. Lucretius clearly points out that the natura creatrix (629) has not made it possible

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for all things to be reduced to minima – for it would be a contradiction to assert that atoms are divisible into the minima – because they are not able to generate any aggregate body, as atoms instead are. The second passage in which Lucretius deals with the doctrine of minima is found in the second book of the poem (478–499): Now that I have demonstrated this truth, I will at once append a proposition that is a corollary of it and an inference from it: 480. the number of atomic shapes is limited. If this were not so, the bulk of some seeds would inevitably be of infinite magnitude. For within the narrow compass of any single atom, there cannot be scope for much variety of shape. Suppose that the ultimate particles consist of three smallest parts, or even increase this number by a few more. Try all those parts of a single atom in every combination, interchanging top and bottom, right and left, 490. so that you see what form is given to the whole atom by each arrangement. Thereafter, if you should wish to vary its shape further, you must add more parts. Subsequently, if you should wish to vary its shape still further, the arrangement will call for more parts, as before. So acquisition of new shapes involves increase of bulk. Therefore you cannot believe that the seeds differ in form to an infinite degree, or you will compel some of them to be of immeasurable magnitude, which, as I have demonstrated above, is inadmissible. While in the first quoted passage, Lucretius mentions the doctrine of minima to confirm the idea of the solid simplicity and indivisible unity of the atomic body, in Book II the main argument concerns the fact that the number of forms of atoms is finite. The counter-evidence is the same as that known from the Letter to Herodotus and other ‘parallel’ sources: if atoms had infinite shapes, they would be visible, which is denied by perceptible self-evidence (enargeia). These Lucretian verses are decisive because they clarify a point which is only implicit in the Letter: the shapes of atoms are limited, because they depend on the number and arrangement of minima in the atom, and minima cannot be numerically infinite since the atom is a limited body. The example given by Lucretius concerns an atom made up of three minimal parts (485–486: minimis e partibus esse/corpora prima tribus): the combinations of these three parts cannot be infinite; for this reason, the atoms of three minimal parts will certainly not all have the same shape, but rather different yet limited shapes (depending on the combinations of minima). Therefore, in Lucretius, the doctrine of minima is brought into play because it explains, on the one hand, why the atom is an indissoluble and indivisible body (i.e. it does not only generically refer to its solidity, as with Leucippus and Democritus) and, on the other hand, why atoms have a limited number of shapes.

CONCLUSION: THE MEANING AND ROLE OF ATOMIC MINIMA One of the possible objections to this sophisticated theory concerns the indivisibility of the atom. If the atom is materially/physically constituted by these

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minima (for without minima the atom would have no size), it must be divisible into the minima forming it; but this stands in contradiction to the fact that the atom is by definition a totally indivisible body. It is not by chance, therefore, that in the closing lines of paragraph 59 of the Letter to Herodotus we read that minima could not constitute any aggregate (symphoresis) through their movement (i.e. even if they moved). As we have just seen in Lucretius, atoms are endowed with solida simplicitas, even though they are material agglomerates (DRN I 610: stipata) of minima; the significant point is that atoms, precisely because they are endowed with indissoluble simplicity, cannot be regarded as aggregated bodies of minima. However, the difficulty lies in understanding why atoms, despite being constituted by minima (but not in the same way an aggregate body is constituted by atoms), cannot be reduced to them and hence are divisible. The solution of this problem still depends on paragraph 59 of the Letter; here we read that it is necessary to consider these minimal and partless (amere) magnitudes (elachista) as limits (perata). Atomic minima are limits of the atom; obviously the atom, being a limited body, must possess limits/boundaries coinciding with the minimal magnitudes. The term peras/limit is the crucial notion used by Epicurus; it very probably derives from Aristotle’s Physics.39 Aristotle – for example, in his treatment of time in Physics Book IV – draws a clear distinction between the concept of part/meros and that of limit/peras.40 Whereas a part is able to measure (and, therefore, to be a unit of measurement), a limit cannot measure any magnitude. Indeed, a part can detach itself from that of which it is a part, whereas if a limit were separated from that of which it is a limit, it would no longer be a limit at all. Moreover, in the context of the treatment of time, Aristotle argues that the ‘now’ (to nyn) is not a part but a limit of time; this means that the now is not a unit of measurement of time, precisely because it is not its part: time is not constituted by ta nyn. Although on this point Lucretius does not follow Epicurus (the poet speaks of minimal parts and not of limits),41 it is very likely that Epicurus was aware of the Aristotelian distinction between meros and peras and adopted it, while really bending its meaning compared to the original Aristotelian definition.42 This is an example of the fact that it is very likely that Epicurus knew the so-called ‘unpublished’ works of Aristotle;43 and it makes it possible to explain an aspect of this doctrine which I can only hint at here. In my opinion, the elachista theory can be fully and correctly understood if one takes into account the influence on Epicurus not only of Zeno of Elea’s paradoxes on the existence of movement and of Aristotle’s Physics but also of Xenocrates’ indivisible lines (considered as the smallest magnitudes), of Diodorus Cronus’ perceptible minima and paradoxes on movement,44 and – in all likelihood – also of the section on the so-called ‘dream theory’ in Plato’s Theaetetus (201d–202c). Envisaging atomic minima as limits, Epicurus states that, just as the limit can never detach itself from that of which it is the limit, elachista can never detach themselves from that of which they are limits (i.e. from the atom itself). Unlike Aristotle, Epicurus believes that although minimal magnitudes are limits, they act as units of measurement and are able to constitute a physical quantity (a function

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which Aristotle instead attributes to the part). In doing so, Epicurus justifies the notion of the indivisibility of the atom: while admitting that the latter is constituted by elachista, he argues that minima, as limits, will never be able to detach themselves from the atom. For this reason, the atom cannot be reduced to the limits that also constitute it and its indivisibility/indissolubility is fully justified. Unlike atoms, which form an aggregate, minima cannot constitute parts of an atom, which as such can be separated from the atomic body. To sum up: in the Letter to Herodotus Epicurus deals with the question of the size of atoms starting from immediate empirical data – for example, the fact that in any delimited body it is impossible for there to be unlimited masses or masses of any size.45 The philosopher proceeds according to his usual analogical/semiotic method: in order to test a certain theory that cannot be directly evaluated at the level of experience, he starts from an empirically observable phenomenon from which he derives by analogy the existence of an ‘entity’ that is not immediately evident. In this case analogy is presented in terms of a proportion: just as a delimited body (which we can observe) cannot contain an unlimited number of masses, so atoms cannot be constituted by an unlimited number of elements. The same procedure is used to show the existence of minima in the atom; Epicurus starts from the (empirically evident) existence of perceptible minima, that is, the smallest physical–spatial quantity which the eye is able to see. Perceptible minima are consecutive and, in this way, exhaust the surface of the body in question; and even though they are not in contact (being minima, they cannot have parts, since these would necessarily be smaller than the minimum itself, which is impossible), in their singularity (idiotes) they are able to measure different magnitudes.46 Immediately afterwards, Epicurus introduces the minimum in the atom, which differs from the perceptible minimum by its smallness. This means that the minima in the atom are not theoretical–mathematical entities – as much of the critical literature has long argued47 in order to show the deep influence of some academic (primarily Xenocratean) views on Epicurus48 – but rather material magnitudes (thus not 2D boundaries: see on this crucial point e.g. Vlastos 1965, 138, n. 86, and Sedley 2007, 161) far smaller than either the perceptible minima or the atom itself. Epicurus clarifies this point, arguing that it is precisely because of minima that the atom has magnitude. Probably borrowing (and modifying) the terminology of Aristotelian physics, the philosopher maintains that it is necessary to consider atomic elachista to be partless limits that act as the units of measurement of magnitudes. In order to ‘prove’ all this, Epicurus resorts to the rational theory about what is invisible, that is, to an analogical procedure capable of linking phenomenal data to what is not empirically evident. So, what is the most intrinsic meaning of the Epicurean doctrine of elachista? In brief, why does Epicurus feel the need to introduce an enigmatic theory like this, which – to the best of our knowledge – is completely absent in Leucippean– Democritean atomism? At the end of paragraph 59 of the Letter to Herodotus, Epicurus states that, even if these minimal bodies could move, they would not be able to constitute an aggregation: this is stated in order to exclude that atomic minima have the same capacity as atoms to move about in order to form a compound

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body. Therefore, elachista cannot form aggregate bodies as atoms do, but must be considered 1.) entities without parts, since, being minimal bodies, they cannot have smaller parts (note that Epicurus does not consider the atom a body without ‘parts’, that is, an ameres, precisely because of the existence of minima); 2.) limits of the atom, that is, magnitudes which cannot detach themselves from the atomic body (the limit of a body is such when it is inseparable from it); 3.) units of measurement of the different (and always limited) magnitudes of atoms. With the elachista theory, Epicurus, unlike Leucippus and Democritus, is able to physically justify the primary differences between atoms and their unalterability too. To put it more clearly: the limited sizes, weights and shapes of atoms depend on the (limited and fixed) number and arrangement of minima within the same atom: this involves a fundamental consequence, that the frequency/probability that atoms always combine with each other in the same or similar way is quite high and this depends on the limitation of the atomic forms. Of course, the number of possible combinations between atoms remains inconceivable and stunning but still finite. At the cosmological level, for example, this means that the possible ‘types’ of world are limited, and it is very likely that worlds similar to ours in which there is life will continue to be formed49 and that the order of the cosmos will be preserved.50 Without the doctrine of atomic minima, this would be impossible. With the introduction of additional material (yet partless) bodies into the atom, Epicurus responds to Aristotle’s objection (in Physics VI 10,51 although the argument can be traced back to Plato’s Parmenides 138d-e) against earlier atomists, namely that a body without parts cannot move.52 But there is also another decisive reason why Epicurus posits elachista. Minima do not account for the indivisibility of the atom merely because they prove its material solidity (as with the earlier atomists); rather, to assert that elachista would not be able to form a symphoresis even if they moved is 1.) to deny that an atom is an aggregate body that, like those formed by atoms in motion, can disintegrate: an atom, on the contrary, is a whole body, which, despite being made up of minima, is not divisible because minima are its constituent limits but not its parts. 2.) to affirm that minima, as limits without parts, cannot for any reason detach themselves from the atom, which will be absolutely indivisible, even if it is materially constituted by a certain (always limited) number of partless physical entities,53 on which the size, weight and shape of each atom depend. The analysis I have conducted in this chapter has shown how, in the field of the science of nature, the theory of atomic minima constitutes a significant innovation

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with respect to the physics of the earliest atomists: this is certainly also due to the influence of the many post-Democritean philosophical traditions (most notably, Aristotle, Xenocrates and Diodorus Cronus) with which Epicurus remained in constant and critical dialogue in the Hellenistic age.

NOTES 1. I am very grateful to Ugo Zilioli for the invitation to participate at this prestigious scientific project. Thanks also to the anonymous reviewers for their ideas and suggestions. 2. See Epicur. Men. 134, and Diog. Oen. Frr. 6 II 9-III 1; 54 II 3-III 9 Smith with Morel (2000). 3. Cic. Fin. I 5, 13-7, 26. 4. On this topic, see Cambiano (2013, 165–209) and, more generally, Boardman (2002). 5. Diog. Laert. VP III 37 = 80 B 5 DK = VIII 31 R1a LM. 6. 67 Wehrli. 7. Athen. Deipn. XI 118, p. 508 C-D. 8. FGrHist 115 F 259. 9. SSR V A 42. 10. Diog. Laert. VP X 4. 11. Of course, Epicurus was accused of plagiarism too: according to Diogenes Laertius (VP X 14 = F 25 Stork et alii), in his Life of Epicurus Aristo (perhaps the Stoic philosopher from Chios, although other scholars identify him with the Peripatetic philosopher of Ceos) claimed that Epicurus had written his Canon by plagiarizing Nausiphanes’ Tripod. 12. Plutarch’s testimony in Adversus Colotem (1108E = part. 234 Usener), in which it is said that Epicurus called himself as a ‘Democritean’ because Democritus had been the first to identify the principles of nature, is very significant. According to ancient doxography too (Aët. Plac. I 3, 18 (Dox. 285) = 68 A 47 DK = VII 27 R94 LM = 267 Usener), Epicurus philosophized in agreement with Democritus (kata Demokriton philosophesas). The Democritean philosopher Nausiphanes of Theos may have played an important role in the ‘atomistic education’ of Epicurus. On the relationship between Epicurus and Democritus, see Silvestre (1985). 13. See Betegh (2006, 266). 14. On the theory of minima, its background and its development in Epicurus’ school, see Verde (2013a); a recent examination of this doctrine is offered by Blanco Rodríguez (2018) and Noller (2019, 151–61). See too Purinton (1994) for a very different analysis. I will only briefly point out that the theory of elachista is probably to be seen (at least since Pines 1936) to lie at the origin of the doctrine of minima in the Kalām (see Dhanani 1994). It is not possible to sum up the critical reception of this doctrine in a footnote; I will limit myself to recalling that the notion of elachista had a certain influence on Giordano Bruno (De triplici minimo: see Albanese 2001) and probably on Galileo Galilei (see Galluzzi 2011).

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15. This testimony is transmitted by Eusebius, PE XIV 23 2-3 = 68 A 43 DK = VII 27 R96 LM; see too Aët. Plac. I 12, 6 (Dox. 311) = 68 A 47 DK = VII 27 D62. On Dionysius of Alexandria, his Peri physeos and the Epicureanism, see Fleischer 2016. 16. On this concept, see Ierodiakonou (2011); it’s very interesting the occurrence of enargeia in Aristot. De an. II 7, 418b 23-24. 17. See Laks-Most (2016, 117, n. 1) (D62). For an alternative ‘Posidonian’ explanation, see Gemelli Marciano (2007, 231–4). 18. Simpl. In Aristot. De cael. 249, 33-295, 22 = Aristot. Democr. Fr. 208 Rose = F 642 Gigon = 68 A 37 DK = VII 27 D29 LM. 19. Aristot. GC I 8, 325b 25-33 = 67 A 7 DK = VII 27 R18 LM. 20. See too Aristot. GC I 2, 315b 25-316a 10. 21. See e.g. Aristot. De cael. III 7, 306a 23-26. 22. See Aristot. GC I 8, 324b 35-325a 3 = 67 A 7 DK = VII 27 D 30 LM. 23. See Plat. Tim. 53c-d. 24. See too Aristot. De an. I 2, 403b 31-404a 10 = 67 A 28 DK = VII 27 D 132 LM. 25. Aristot. Metaph. A 4 985b 4 = 67 A 6 DK = VII 27 D31 LM. 26. See Simpl. In Aristot. Phys. 1318 33 = 68 A 58 DK = VII 27 D36 LM. 27. See Epicur. Hrdt. 54. 28. See Aët. Plac. I 3, 18 (Dox. 285) = 68 A 47 DK = VII 27 D51 LM. 29. See Aët. Plac. I 12, 6 (Dox. 311) = 68 A 47 DK = VII 27 D37 LM. 30. See Dionys. ap. Eus. PE XIV 23, 2-3 = 68 A 43 DK = VII 27 R96 LM. 31. See Aët. Plac. I 12, 6 (Dox. 311) = 68 A 47 DK = VII 27 D53 LM. 32. See Aët. Plac. I 3, 18 (Dox. 285) = 68 A 47 DK = VII 27 R94 LM. 33. It is not possible here to investigate in detail the big question of whether there is such a thing as Epicurean geometry (see the classical and pioneering study by Vlastos 1966); in my opinion, the Epicureans not only demolished the geometry of their time but also tried to create an alternative geometry based on the notion of minimum. This also implied a criticism of Aristotle: in a famous passage of the De caelo (I 5, 271b 9-11), for example, Aristotle states that the introduction of a minimum magnitude would be incompatible with the fundamental principles of mathematics. For more recent studies on this topic, see Verde (2013b), Netz (2015), and the response to Netz’s conclusions in Verde (2016b). 34. See Verde (2011), Gœury (2013), and Verde (2015a). According to Konstan (2018), the elachista theory and the ‘granularization’ of space and time have led contemporary physics to re-evaluate Epicurus’ science of nature (see too Konstan 1982). 35. See Lucret. DRN II 244; see also Verde (2013a, 309–16). 36. On Epicurus’ notion of the infinite in thought (Hrdt. 57), see Verde (2015b, 143–4). 37. See for example, Philod. Sign. (PHerc. 1065) col. XXXVII 1-29 De Lacy-De Lacy. For a first overview, see Allen (2001, 194–241), Manetti (2012), and Sedley (2018, 115–20). 38. Recently, see, for example, Leith (2012, 190), Polito (2013, 125), Asmis (2016, 95) (I do not fully understand the reasons that lead Asmis to affirm that the minimum is

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a boundary contributing extension to the whole of which it is a part but that it has no extension in itself), and Noller (2019, 151, 154, 158 and 160). 39. On Aristotle’s notion of limit, see now Pfeiffer (2018, 89 ff). 40. See Aristot. Phys. IV 10, 218a 6-24; see too Phys. IV 11, 220a 14-24 e De an. I 3, 407a 10-12. 41. See Lucret. DRN I 610, 616, 622, II 485. 42. See O’Brien (2007). 43. See Verde (2016). 44. See Verde (2013a, 128–237). 45. Epicur. Hrdt. 56. 46. See Verde (2015b, 145, n. 17). 47. See n. 38 above. 48. See at least Furley (1967, 104–10), Krämer (1971, 231–57), and Isnardi Parente (1991). 49. See for example, Lucret. DRN III 854-858 with Sedley (2007, 155–66). 50. On this specific point, see again the recent monograph by Noller (2019). 51. See too Aristot. De an. I 4, 409a 1-5. 52. See Konstan (1987), and Verde (2013a, 196 ff). 53. Of course, it is not easy to understand how elachista can coexist within the atomic body; for more information on this point and for further bibliographical references, see Sedley (2007, 155–66), and Verde (2013a, 238–47).

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Furley, D. (1967), Two Studies in the Greek Atomists, Study I: Indivisible Magnitudes, Study II: Aristotle and Epicurus on Voluntary Action, Princeton: Princeton University Press. Galluzzi, P. (2011), Tra atomi e indivisibili: La materia ambigua di Galileo, Florence: Olschki. Gemelli-Marciano, M. L. (2007), Democrito e l’Accademia: Studi sulla trasmissione dell’atomismo antico da Aristotele a Simplicio, Berlin and New York: De Gruyter. Gœury, M. (2013), ‘L’atomisme épicurien du temps à la lumière de la Physique d’Aristote’, Les Études Philosophiques 107: 535–52. Ierodiakonou, K. (2011), ‘The Notion of enargeia in Hellenistic Philosophy’, in B. Morison and K. Ierodiakonou (eds), Episteme, etc.: Essays in Honour of Jonathan Barnes, 60–73, Oxford: Oxford University Press. Isnardi Parente, M. (1991), ‘La prima Accademia e la fisica di Epicuro’, in Ead., Filosofia e scienza nel pensiero ellenistico, 171–95, Naples: Morano (already  ‘L’atomismo di Epicuro fra Democrito e Senocrate’, in F. Romano (ed.), Democrito e l’atomismo antico, Atti del Convegno Internazionale, Catania 18–21 aprile 1979, Siculorum Gymnasium 33/1980, 367–91). Konstan, D. (1982), ‘Ancient Atomism and Its Heritage: Minimal Parts’, Ancient Philosophy 2: 60–75. Konstan, D. (1987), ‘Points, Lines, and Infinity: Aristotle’s Physics Zeta and Hellenistic Philosophy’, Proceedings of the Boston Area Colloquium in Ancient Philosophy 3: 1–32. Konstan, D. (2018), ‘Lucrezio e la scienza moderna: Alcuni punti di contatto’, Griseldaonline 5/17: 1–12. https​:/​/gr​​iseld​​aonli​​ne​.un​​ibo​.i​​t​/art​​icle/​​view​/​​9396/​​9169 (06 July 2019). Krämer, H. J. (1971), Platonismus und hellenistische Philosophie, Berlin and New York: De Gruyter. Laks, A. and Most, G. W., eds (2016), Early Greek Philosophy: Later Ionian and Athenian Thinkers, Part 2, In Collaboration with G. Journée and Assisted by L. Iribarren, Cambridge, MA and London: Harvard University Press. Leith, D. (2012), ‘Pores and Void in Asclepiades’ Physical Theory’, Phronesis 57: 164–91. Manetti, G. (2012), ‘La semiotica salvata(si) dal Vesuvio: Il dibattito tra Epicurei e Stoici (?) sull’inferenza da segni nel De signis di Filodemo’, Blityri 1: 135–76. Morel, P.-M. (2000), Atome et nécessité: Démocrite, Épicure, Lucrèce, Paris: Presses universitaires de France. Netz, R. (2015), ‘Were there Epicurean Mathematicians?’, Oxford Studies in Ancient Philosophy 49: 283–319. Noller, E. M. (2019), Die Ordnung der Welt: Darstellungsformen von Dynamik, Statik und Emergenz in Lukrez’ De Rerum Natura, Heidelberg: Universitätsverlag Winter. O’Brien, D. (2007), ‘Démocrite à l’Académie?’, in A. Brancacci and P.-M. Morel (eds), Democritus: Science, the Arts, and the Care of the Soul, 239–63, Leiden and Boston: Brill. Pfeiffer, C. (2018), Aristotle’s Theory of Bodies, Oxford: Oxford University Press. Pines, S. (1936), Beiträge zur Islamischen Atomlehre, Berlin (Studies in Islamic Atomism, Engl. transl. by M. Schwarz, Ed. Tzvi Langermann, Jerusalem: The Magnes Press 1997).

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Polito, R. (2013), Asclepiades of Bithynia and Heraclides Ponticus: Medical Platonism?, in M. Schofield (ed.), Aristotle, Plato and Pythagoreanism in the First Century BC: New Directions for Philosophy, 118–38, Cambridge: Cambridge University Press. Purinton, J. S. (1994), ‘Magnifying Epicurean Minima’, Ancient Philosophy 14: 115–46. Sedley, D. (2007), Creationism and Its Critics in Antiquity, Berkeley, Los Angeles and London: University of California Press. Sedley, D. (2018), ‘Epicurean Theories of Knowledge: From Hermarchus to Lucretius and Philodemus’, (in F. Verde and M. Catapano (eds), Hellenistic Theories of Knowledge) Lexicon Philosophicum, Special Issue 2018, 105–21. http:​/​/lex​​icon.​​cnr​.i​​t​/ind​​ex​.ph​​p​/ LP/​​artic​​le​/​vi​​ew​/56​​3​/416​ (06 July 2019). Silvestre, M. L. (1985), Democrito e Epicuro: Il senso di una polemica, Naples: Loffredo. Verde, F. (2011), ‘Minimi in movimento? Note sulle coll. XLVIII-L Puglia del PHerc. 1012 (Demetrii Laconis Opus incertum)’, Cronache Ercolanesi 41: 49–61. Verde, F. (2013a), Elachista: La dottrina dei minimi nell’Epicureismo, Leuven: Leuven University Press. Verde, F. (2013b), ‘Epicurean Attitude Toward Geometry: The sceptical account’, in S. Marchand and F. Verde (eds), Épicurisme et Scepticisme, 131–50, Rome: Sapienza Università Editrice. Verde, F. (2015a), ‘Testimonianze tardoantiche sulla fisica di Epicuro’, in D. De Sanctis, E. Spinelli, M. Tulli and F. Verde (eds), Questioni epicuree, 179–95, Sankt Augustin: Academia Verlag. Verde, F. (2015b), ‘Diodorus Cronus on Perceptible Minima’, in U. Zilioli (ed.), From the Socratics to the Socratic Schools: Classical Ethics, Metaphysics, Epistemology, 134–48, London and New York: Routledge. Verde, F. (2016a), ‘Aristotle and the Garden’, in A. Falcon (ed.), Brill’s Companion to the Reception of Aristotle in Antiquity, 35–55, Leiden and Boston: Brill. Verde, F. (2016b), ‘Ancora sulla matematica epicurea’, Cronache Ercolanesi 46: 21–37. Vlastos, G. (1965), ‘Minimal Parts in Epicurus’, Isis 56: 121–47 (repr. in Id., Studies in Greek Philosophy, Vol. II: Socrates, Plato, and their Tradition, ed. by D. W. Graham, Princeton: Princeton University Press, 1995, 285–314). Vlastos, G. (1966), ‘Zeno of Sidon as a Critic of Euclid’, in L. Wallach (ed.), The Classical Tradition: Literary and Historical Studies in Honor of Harry Caplan, 148–59, Ithaca and New York: Cornell University Press (repr. in Id., Studies in Greek Philosophy, Vol. II: Socrates, Plato, and Their Tradition, ed. by D. W. Graham, Princeton: Princeton University Press, 1995, 315–24).

CHAPTER 5

Atoms and universals in Epicurus1 ATTILA NÉMETH

Epicurus famously held that every perception is true or real.2 He took the phenomenal world as the starting point of his epistemology, claiming that our perceptual awareness of it is devoid of any reasoning, and to this extent irrational, because sense-perception is not moved by itself and it cannot add or subtract anything when it is itself moved by something else. Since all knowledge is dependent on and derived from sense-perception, the truth or reality of sense-perception is only confirmed by the fact of our own awareness of it. But what is the content of this perceptual awareness, and how is it articulated? Naturally or rationally? If the content of our perceptual awareness were rationally articulated, this would contradict Epicurus’ thesis of the irrationality of senseperception, because it would mean that we could not perceive anything without the participation of the mind. The former alternative, that it happens naturally, seems however to lay the foundation of a purely empirical theory, since it holds that the contents of our perceptual awareness are articulated by experience alone. But what are exactly these contents of perceptual awareness on which the mind places its judgement according to Epicurus? It seems fairly clear that they are the raw perceptions of the different, separate sense-organs, combined with the affections we undergo simultaneously in perception. But if this is the case, how are these separate perceptions combined into a unified perceptual awareness? If it happens in the mind, this would contradict Epicurus’ initial assumption that senseperception is irrational. Are, then, these simultaneous perceptive experiences also connected into a unified perceptual awareness by some natural means devoid of rational reasoning? I wish to argue in this chapter that, according to Epicurus, the content of our perceptual awareness is indeed naturally unified by the process he calls prolépsis.3 To a certain extent, perceptual content undergoes a certain form of elementary, natural classification, thanks to the product of a proleptic process, the typos.4 In my interpretation of Epicurus’ conception of perceptual awareness is a three-stage process. The external world directly affects our separate sense-organs, which are all vitalized by the irrational part of the soul. The irrational part of the soul is responsible for a complex, non-rational natural process of the sensual data for the

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rational part of the soul,5 and the rational soul part, consequently, makes judgements based on the following three elements of our perceptual awareness: (1) the different sense-perceptions of the disparate sense-organs (the phantasia of the immediate, visual representation of an object or that of an immediate olfactory perception or sound or touch or taste sensation); (2) the typos, which is the product of a proleptic process; and (3) the pathé or the affections we undergo during our perception. For example, when we make such a judgement that ‘There is a horse’, we may have a visual, olfactory or other perception of a horse as it appears to us; an act of precognition (prolépsis) that presents the rational mind with a typos of a holistic recognition composed of these direct sensory perceptions; and some affective state (an unpleasurable feeling because I’m scared of horses). Based on these three elements, we are able to make the rational judgement that ‘This is a horse’. Prolépsis has the function in this account of unifying the perceptual experience of our different senseorgans, each of which has its own, limited area of discrimination and effectiveness. This process of natural unification – which might, as in the current example involve visual, tangible, olfactory and aural sense-perception of something recognized as such and such, for example, as a horse – produces a general recognition of that object in the form of a typos. This typos in turn, I wish to argue, fulfils the function of a sort of natural universal through whose form prolépsis presents a naturally and empirically classified perceptual recognition of the world, based on which – together with the other elements involved in sense-perception – we can make judgements about the world. My interpretation is built on the following chain of argument: in the first section, I will first clarify the problem of universals, which is, simply put, the question of how a plurality of things can share one and the same property, and of the ontological status of that shared property. As we will see, the problem of universals was tangled up, philosophers in antiquity, in a variety of different problems.6 In the second section, I will show that Epicurus’ discussion of the ontological status of properties in his Letter to Herodotus is strongly connected to his understanding of our perceptual awareness of the world. It would seem that the way in which we perceive the world through the complex conception of properties already provides us with a naturally classified awareness of things. At this point in the argument, it will be unclear whether Epicurus can justify the position that this natural classification is not the result of a rational operation, but rather as a result of perceiving the world through, as it were, a net of naturally formed universal conceptions of bodies and place and properties and accidents thereof. Therefore, in the third section, I will turn to the secondary textual evidence of Diogenes Laertius (D. L. X 33), who seems to fill the gaps in the process by which sense-perception translates into human awareness by providing an account of the Epicurean conception of prolépsis. Based on this and some other primary evidence that I argue corroborates my interpretation, I shall attempt to piece together the terminological puzzles of Epicurus’ conception of perceptual awareness, which will not only explain how the contents of the separate sense-organs are unified in a single perceptual awareness structured along a natural classification of the world but also have certain definite and profound consequences for our interpretation of Epicurus’ atomism.

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THE QUESTION OF UNIVERSALS What is the problem of universals?7 Charles Sanders Pierce in the nineteenth century originally discussed the question in the field of semantics. If you take the following semantic example, The  the

and ask how many words there are in the box, it is obvious that the question has two good answers: there are two words; or there is only one. According to Pierce, there are two tokens of one type. If we consider it, the distinction between tokens and types can be applied not only in semantics but also to almost everything. There are many tokens of certain types of atoms, chairs or horses, for example. The philosophical problem appears when we start talking about sameness of type, as in this example: The horse in front of me has the same shape and size as all the other horses in the herd.

Many different particular things – in our example many horses – can be of the same type: as such, they are tokens of the same shape and size. Does that make them identical? If we take these tokens to be identical, then the two definite articles in our first example are in some sense also identical. Yet, the tokens are completely separate in both examples, after all they occupy, for example, different locations in space. In what sense can they said to be identical? Some philosophers (e.g. Plato, St. Augustine, St. Anselm, St. Thomas Aquinas) thought that we can say truly that tokens are of the same type, if in respect of their sameness the tokens are strictly identical. The horses, for instance, have the same shape and size. The same shape and size are the constituents of their common properties. In the opinion of these philosophers, this identity can only hold if the universal properties in virtue of which these tokens are identical are real, and thus, philosophers of this view are normally called the Realists. There are others (e.g. John Buridan, William of Ockham, John Stuart Mill) who do not take sameness in the sense of strict identity but conceive it rather in a loose sense. Philosophers of this view normally think that a number of tokens of the same type are non-overlapping parts of some larger whole or unity, and these larger unities do not have an independent existence. As John Locke put it, ‘all things that exist are only particulars’. Therefore, universals do not exist in the same way as particulars, and philosophers of this opinion are generally called the Nominalists. If we rewind history back almost to the end of antiquity, we find Galen, a medical doctor of the second to early third century AD, using just the same type-token distinction when discussing the letters of the alphabet (Meth. med. 2.7-8 = X 131-2 Kühn). This example actually already had a precedent in Aristotle: [T]here is nothing to stop there being many alphas and betas, as with the elements (stoicheión) of speech (tés phónés), without there also being over and above the many (para ta polla) a certain ‘alpha itself ’ and ‘beta itself ’ (auto alpha kai auto béta). (Met. M.10, 1087a7–10)

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Aristotle, who thought that universals are inherent in things (universalia in rebus), uses this example to distance himself from Plato’s belief that the distinction between universals and particulars requires a separately existing, transcendent Form: ‘the F itself ’. In the first book of the Metaphysics (A 6, 987a32-b7), Aristotle tells the story of Plato’s background assumptions which may have led him to the theory of Forms. It goes as follows: Plato, in his youth, became familiar with Cratylus and with Heraclitus’ view that all sensible beings (ta aisthéta) are always in flux and, thus, cannot be the object of knowledge – the ontological claim concerning the nature of sensible beings – they are always in flux – implying the epistemological thesis that the sensible beings cannot be the objects of scientific knowledge, since they are always in flux. According to Aristotle, Plato retained these views in his philosophy, maintaining that if there is anything that is the object of knowledge and definition, this thing cannot belong to the domain of sensible beings. Therefore, the objects of knowledge and definition must be different from sensible beings. What, then, are these objects? Plato – the story continues – in order to resolve the inherent epistemological consequences, took over Socrates’ method in ethical matters, which focused on what is universal (to katholou) and aimed to provide definitions of general terms. By extending the application of this Socratic practice, Plato arrived at the Forms or Ideas as the objects of knowledge and definition. Although some commentators question the authenticity of this story,8 I agree with those who think that since Aristotle was Plato’s most eminent pupil, he was in a very good position to know, if he wished to, about the sources of Plato’s theory.9 Nonetheless, it is not clear to what extent Forms can be identified here with universals. Even if we accept that Forms are shared predicates and have their own ontological class – a rather controversial issue on its own – it is fairly clear that Plato did not postulate Forms as universals, since the development of the theory of Forms was motivated by other considerations, and thus appears to have been unrelated to the problem of universals.10 Nonetheless, according to Aristotle’s story, Forms were conceived by Plato as the proper objects of definition; and consequently, it seems plausible that everything that can be defined has a corresponding Form, which can amount to a general term or universal. In defining the problem of universals, I have proceeded backwards from modern formulations towards ancient conceptions, to demonstrate that as we approach closer to the ancient roots of the realist/nominalist debate, we find that the issue is embedded in many diverse philosophical considerations. As Alexander Mourelatos has effectively argued, it may have even had earlier formulations than Plato’s – perhaps in Parmenides, Empedocles, the Pythagoreans and Democritus.11 I think this short, reversed overview will serve not only to justify the complexity of the issues at stake in Epicurus’ grappling with the problem but also as a useful point of reference for articulating how it appears in his philosophy.

PERCEPTION OF BODIES AND PLACE According to Epicurus, the totality of things is bodies and place – the latter of which we also call void, room and intangible nature.12 Sensation itself testifies to the fact

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that bodies exist (Ep. Hdt. 39-40). Beyond bodies and place, nothing can even be thought of as having a complete nature (ὡς καθ’ ὅλας φύσεις), but only as accidents (συμπτώματα) and properties (συμβεβηκότα) of these two things (Ep. Hdt. 40).13 The claim that accidents and properties are not independent natures is repeated later in the same work (Ep. Hdt. 68-71), and the language used to express this opinion (οὔθ' ὡς καθ’ ἑαυτάς εἰσι φύσεις δοξαστέον) is sharply pointed. Plato describes his eternally unchanging Ideas as αὐτὰ καθ’ αὑτά, as ‘themselves in themselves’ (e.g. Symp. 211b1, Phd. 78d5-6, Parm. 129d7-8), that is, as per se beings, indicating by this formula that they exist independently of any other kind of item, in the intelligible realm, as opposed to the continuously changing, sensible objects which participate in them. In sharp contrast, Epicurus attributes an independent status to body and place as existing in their own right: ὡς καθ’ ὅλας φύσεις.14 This distinction implies a further, significant difference: while on the Platonic view the objects of experience are constantly becoming and changing and, are thus multiform (πολυειδές, Phd. 80b4) and have no consistent unity in opposition to the simple and uniform existence of Forms (cf. e.g. the Form of Beauty as μονοειδές, Phd. 78d5), for Epicurus bodies in our experience are considered distinct unites or wholes.15 It is important to note that at paragraphs 39-40 in the Letter to Herodotus, where he identifies bodies and place as autonomous wholes, Epicurus is talking about bodies in our experience: only after his argument for atomism does he start describing atoms in these terms as bodies. Thus, Epicurus talks about bodies at least in two senses: (1) ‘bodies’ as wholes made up of their properties; and (2) ‘bodies’ as aggregates of atoms or as aggregates of aggregates (Ep. Hdt. 69). In the first case, the properties which make a body the kind of body it is – a stone, a horse or a man – create the unity of body: a position which, in contemporary jargon, could be best described as a ‘bundle theory’.16 Yet, this is only with the important restriction that a body is not an aggregate of its properties, since bodies exist as distinct unites or wholes (ὡς καθ’ ὅλας φύσεις). The properties of a body constitute the nature of the body as a whole, or as Epicurus elaborates it, T1 (1) ἀλλὰ μὴν καὶ τὰ σχήματα καὶ τὰ χρώματα καὶ τὰ μεγέθη καὶ τὰ βάρη καὶ ὅσα ἄλλα κατηγορεῖται σώματος ὡς ἀεὶ συμβεβηκότα – ἢ πᾶσιν ἢ τοῖς ὁρατοῖς καὶ κατὰ τὴν αἴσθησιν αὐτοῖς γνωστοῖς –, οὔθ’ ὡς καθ’ ἑαυτάς εἰσι φύσεις δοξαστέον (οὐ γὰρ δυνατὸν ἐπινοῆσαι τοῦτο)· (2) οὔτε ὅλως ὡς οὐκ εἰσίν· οὔθ’ ὡς ἕτερ’ ἄττα προσυπάρχοντα τούτῳ ἀσώματα· οὔθ’ ὡς μόρια τούτου, (3) ἀλλ’ ὡς τὸ ὅλον σῶμα καθόλου μὲν τούτων πάντων τὴν ἑαυτοῦ φύσιν ἔχον ἀΐδιον – οὐχ οἷόν τε εἶναι, συμπεφορημένων ὥσπερ ὅταν ἐξ αὐτῶν τῶν ὄγκων μεῖζον ἄθροισμα συστῇ ἤτοι τῶν πρώτων ἢ τῶν τοῦ ὅλου μεγεθῶν τοῦδέ τινος ἐλαττόνων –, ἀλλὰ μόνον ὡς λέγω ἐκ τούτων ἁπάντων τὴν ἑαυτοῦ φύσιν ἔχον ἀΐδιον. (4) καὶ ἐπιβολὰς μὲν ἔχοντα ἰδίας πάντα ταῦτά ἐστι καὶ διαλήψεις, συμπαρακολουθοῦντος δὲ τοῦ ἀθρόου καὶ οὐθαμῇ ἀποσχιζομένου, ἀλλὰ κατὰ τὴν ἀθρόαν ἔννοιαν τοῦ σώματος κατηγορίαν εἰληφότος. (1) Further, the shapes and colours and sizes and weights and all the other things which are predicated of body as properties, – either of all [bodies] or of visible ones, and are known by sense-perception itself – these things must not be thought

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of as independent natures (for that is inconceivable). (2) Nor [must it be thought] that they [i.e. properties] are altogether non-existent, nor that they are distinct incorporeal entities inhering in [the body], nor that they are parts of it. (3) But [one should think] that the whole body throughout (καθόλου) derives its own permanent nature from all of these [properties] – though not in such a way as to be a compound [of them], just as when a larger aggregate is produced from the masses themselves, whether the primary ones or magnitudes smaller than the whole object – but only, as I say, deriving its own permanent nature from all of these. (4) But all of these [are known by] their own peculiar forms of application and comprehension, always in close accompaniment with the aggregate and in no way separated from it, but with the body receiving its predication according to the complex conception. (Ep. Hdt. 68-9)17 In this passage Epicurus is interested in the properties of three kinds of bodies: (1) body as such, (2) perceptible body and (3) visible body.18 Colour, for example, belongs only to the third group, whereas the other (permanent) properties listed at the beginning – shape, size and weight – belong inseparably to all three groups of bodies. The way in which properties create a body is not the same as how an aggregate of atoms constitutes a body. I assume the difference can simply be explained by the distinctions between the kinds of bodies listed at the beginning of this passage: although atoms are the constituents of all the bodies there are, since they are invisible, the way in which properties constitute the nature of a body is by making bodies perceptible and visible. How is Epicurus’ theory different from that of Democritus, who made properties unreal, since he held that they are only the results of interactions between a sense organ and its sensible object, and consequently held that the way we describe the world is just the result of rational conventions?19 I think Epicurus had the following idea: a shape and size of a horse, for example, is not visible as an atomic aggregate on the phenomenal level, but as such and such bodily shape, which exists as a distinct and whole nature. In a similar way, the horse’s colour is not visible as an atomic aggregate, but as such and such a colour of a perceptible body, which is perceptible because it is colourful. The reality or existence of perceptible properties means that it is because of them that certain things become perceptible as they are, since if properties make independent bodily natures perceptible in the way in which they are perceptible, these properties must exist and be real as well.20 This explanation does not answer, however, the question of how a bundle of different properties creates the unity of a body, as it is stated in T1(3). In T1(4) Epicurus points out that properties are conceived according to ‘their own peculiar forms of application and comprehension’ and that whatever is predicated of a body is based on the ‘complex conception’ of a body. These remarks point in the direction that it is possibly through one’s perception that a bundle of properties give the impression of a unified body. It is this claim that Epicurus seems to be repeating after his discussion of accidents (συμπτώματα) as well.21 How permanent and accidental properties create a unified impression of a perceptible body thus seems to depend on their complex conception. This could mean

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that although properties themselves are real, the unified nature of a body depends on the observer’s mental capacity to recognize them as properties of a single body. Such an understanding seems to threaten Epicurus, however, with the consequence that his theory will eventually become very similar to that of Democritus, since even if a bundle of properties is real, the unity of the things they represent would depend on how the mind interprets these bundles. This sounds just like a rational convention. But this is clearly not what Epicurus is saying, since such an explanation would make a body ultimately an aggregate of its properties, which possibility Epicurus openly denies by the introduction of his καθ’ ἑάυτας φύσεις language applied to perceptible and visible bodies and place in T1(1). As we have seen, he introduces this terminological language already before establishing his argument for atomism. In my understanding, this insistence marks out the epistemological primacy of the phenomenal level (Ep. Hdt. 39-40). Epicurus takes it as a given that the totality of things is composed across the board of bodies and place. The universally witnessed phenomenal world is primary because perceptions are the starting points of our awareness and the foundation of knowledge. If in our perceptions, we primarily become aware of bodies and place, what seems to be at stake here for Epicurus is whether his thoroughly empiricist theory can explain why we naturally perceive bodies and place as whole natures (ὡς καθ’ ὅλας φύσεις) without the involvement of a rational convention of the mind. If his theory can justify that we naturally perceive properties as properties of whole natures, then he can secure his claim that the way in which we describe the world is not just a result of some linguistic convention, but that it is the way the world is. For that, Epicurus has to show that human being’s mental ability to recognize properties as properties of whole and distinct natures is natural; or, in other words, that our perceptual experience is articulated through a naturally, and pre-rational, classified awareness of the world. Thus, in order to find out how and why we perceive bodies and place ὡς καθ’ ὅλας φύσεις, I will proceed with examining how Epicurus and later Epicureans describe the workings of the soul in sense-perception.

THE FUNCTION OF PROLÉPSIS As Lucretius makes it clear, every sense has its proper perception that cannot be corrected by the other senses because each sense has its own sphere of discrimination (see DRN IV 479–99 and D. L. X 32). The shape perceived by sight, for example, is not the shape one differentiates by touch. The former is merely the outline of colour, an effect brought about in the perceiver; thus, sight does not represent the actual shape of the object but only that of colour. That is to say, sight does not merely discriminate colour, but everything related to colours: for example, the shape and size of colour, its distance and so forth. The apparent tension of a possible contradiction between sight and touch representing the same object differently – for example, a stick seemingly bent in water but appearing straight to the touch – is only the result of a mistake based on a false analogy, as an anonymous Epicurean pointed out,22 given that shape is the proper object of touch, while sight only discriminates the shape of colour.

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Thus, the senses do not have a common sphere of discrimination. This leads to the following difficulty: if individual senses perceive only the proper objects of their own discrimination, what is there that unifies the separate perceptual experience of the five senses (cf. Plato’s Tht. 184-6)? This question corresponds to our earlier question about how a bundle of properties can create the unity of a body. If a stinky, neighing horse kicks me, what is it that unifies my visual experience of the domesticated animal with a flowing mane and tail with its related aural and olfactory sensations, as well as the visual and eventually tactile experience of the approaching hooves? What makes all these separate sensations part of one and the same experience? The straightforward answer seems to be that it is the mind which places its judgement on the sensations of perception based on the listed elements which compose my sensations and feelings, along with prolépsis, which is the third criterion of truth. But does that mean that the disparate sensations appropriate to the different sense-organs appear as part of one and the same perceptual awareness as a result of a rational combination? If this Platonic solution (Tht. 186b) were the case, this would mean that we are unable to attain perceptual awareness of anything without the mind and that would contradict Epicurus’ requirement that sense-perception must be irrational. Or is it rather the case that the mind places its judgement on an already unified awareness of perception? In my understanding, Epicurus’ notion of prolépsis denoted a process of recognition, which, as an activity, precedes to the rational judgement of the mind. Etymologically, prolépsis is a ‘taking/grasping before’ (i.e. before the intervention of the mind): as a process, it either unifies the various proper contents of the individual sense-organs or supplements certain missing sense-perceptions if an object is perceived only by one of the sense organs (e.g. I smell the horse with my eyes shut). This unification of the information provided by the senses by a proleptic process produces a general outline or typos of the thing perceived, so that the mind can relate the apparent sensations and feelings to a certain empirically formed taxonomy of bodies, in order then to place the seal of its rational judgement on the perceived object. I have argued elsewhere at length for this interpretation,23 and since the jigsaw puzzles of Epicurus’ ideas can be assembled only from a large number of disparate sources, I have this time decided to restrict myself to a brief presentation of the idea, in a nutshell (admittedly still a rather complicated nutshell), in order to present what I think is Epicurus’ solution to the problematic status of properties related to καθ’ ὅλας φύσεις. Epicurus taught that the soul is divided into rational and irrational parts.24 The whole human body is permeated by soul: the irrational parts of the soul vitalize the body, while the rational aggregate directs it from its dwelling place in the chest. The reason why this distinction is important is that sense-perception is in itself devoid of reason and of interpretation, since reasoning itself depends on our perceptual awareness. The information the sense organs provide is, in this respect, thus irrational; consequently, only the irrational part of the soul can vitalize them. This on its own, however, does not explain how sense-perception translates into a person’s ideas or thoughts. It seems to me that Diogenes Laertius meant to fill

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in this gap when he augmented this account with the Epicurean description of prolépsis. T2 (1) Τὴν δὲ πρόληψιν λέγουσιν οἱονεὶ κατάληψιν ἢ δόξαν ὀρθὴν ἢ ἔννοιαν ἢ καθολικὴν νόησιν ἐναποκειμένην, τουτέστι μνήμην τοῦ πολλάκις ἔξωθεν φανέντος, οἷον τὸ Τοιοῦτόν ἐστιν ἄνθρωπος· (2) ἅμα γὰρ τῷ ῥηθῆναι ἄνθρωπος εὐθὺς κατὰ πρόληψιν καὶ ὁ τύπος αὐτοῦ νοεῖται προηγουμένων τῶν αἰσθήσεων. (3) παντὶ οὖν ὀνόματι τὸ πρώτως ὑποτεταγμένον ἐναργές ἐστι. (4) καὶ οὐκ ἂν ἐζητήσαμεν τὸ ζητούμενον εἰ μὴ πρότερον ἐγνώκειμεν αὐτό· οἷον Τὸ πόρρω ἑστὼς ἵππος ἐστὶν ἢ βοῦς· (5) δεῖ γὰρ κατὰ πρόληψιν ἐγνωκέναι ποτὲ ἵππου καὶ βοὸς μορφήν. (6) οὐδ’ ἂν ὠνομάσαμέν τι μὴ πρότερον αὐτοῦ κατὰ πρόληψιν τὸν τύπον μαθόντες. (7) ἐναργεῖς οὖν εἰσιν αἱ προλήψεις· (8) καὶ τὸ δοξαστὸν ἀπὸ προτέρου τινὸς ἐναργοῦς ἤρτηται, ἐφ’ ὃ ἀναφέροντες λέγομεν, οἷον Πόθεν ἴσμεν εἰ τοῦτό ἐστιν ἄνθρωπος; (1) Prolépsis, they [the Epicureans] say, is something like apprehension (κατάληψις), or right opinion (δόξα ὀρθή), or conception (ἔννοια), or a stored general thought (καθολικὴ νόησις ἐναποκειμένη), that is, a memory of that which has often appeared from outside, for example, that man is this sort of thing. (2) For at the same time that ‘man’ is spoken, immediately in accordance with prolépsis the outline (ὁ τύπος) of man also is thought of, as a result of preceding perceptions. (3) In the case of every name, then, that which is first subordinate is evident. (4) And we would not have sought what we seek if we had not previously been aware of it. For example, is the standing thing in the distance a horse or a cow? (5) We must have learned at some time by prolépsis the form of a horse and a cow. (6) Nor would we have named anything if we had not previously been aware of its outline (ὁ τύπος) by prolépsis. (7) Prolépseis, then, are evident. (8) Further, an object of belief depends on something prior that is evident, by reference to which we state [the belief]; for example, how do we know whether this is a man? (D. L. X 33)25 Let me restrict my discussion of this passage to its structure and to a few of the most immediate questions and comments that concern the possible function of prolépsis. (1) Prolépsis is something like (οἱονεί) katalépsis, doxa orthé, ennoia and noésis, – but importantly it is not identical with any of these.26 (2) For once the word ‘man’ is uttered, immediately by means of prolépsis its typos comes to mind, as a result of preceding perceptions. According to T2(2), the reason why prolépsis is something like katalépsis, doxa orthé, ennoia and noésis is that as soon as we hear the word ‘man’, there is an automatic process that brings the basic features of ‘man’ to one’s mind in the form of a typos. Prolépsis seems to be the motor of this process; or, rather, the process itself between hearing an utterance and realizing that utterance in the form of a typos. What is a typos? The noun ὁ τύπος comes from the verb τύπτω (‘to beat, strike’), and it literally means (to quote LSJ sv)27 (I.) ‘a blow’ or ‘beat’, for example, of horses’ hooves. But it also means (II.) ‘the effect of a blow or of pressure’ and hence an ‘impression’ of a seal, a ‘print’ (see Ep. Hdt. 46 and PHerc. 1191), an

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(VII.) ‘archetype, pattern, model’, capable of exact repetition in numerous instances (Ep. Hdt. 49), from which it is just a step further to gain the meaning of ‘general character’, or (VIII.) ‘general impression’ and thus ‘general type or schema’, ‘outline’ (PHerc. 1056) or even ‘outline account’ (see Ep. Hdt. 35–6 and 68). It is difficult to pin down what exactly a typos is, but Epicurus uses the term in two dominant meanings, as a general description, or as the basic features of a kind understood through something like a mental image.28 As it will turn out, Henry Dyson’s latter characterization of the term perfectly describes the way in which it is used in T2. (3) Thus, the thing primarily denoted by every name is evident (enarges).29 Why is this the case? The reason can be grasped fairly explicitly from this passage – because of the context and T2(7). What are subordinate to names are, it seems, prolépseis. According to David Glidden, however, Diogenes plays on an ambiguity when he does not clarify whether prolépseis are subordinate to names as their meaning (i.e. when uttering ‘man’ we signify some conception of ‘man’), or whether they are subordinate to names in the way that our words fundamentally refer to real things we can name when talking about them: for example, the word ‘man’, when spoken, signifies some perceived form in the world.30 Stephen Everson has shown convincingly based on Plutarch’s and Sextus’ relevant evidence that for the Epicureans, it is names themselves that have semantic properties, and not something else that underlies them. It would have been unnecessary for Epicurus to introduce a set of meanings in order to show how words are meaningful, because for words or sentences to be meaningful for him there do not need to be separate sets of corresponding meanings other than the words themselves. Thus, according to Everson, the semantic value of a word, for example, ‘man’, is only correlated with a prolépsis, to the extent that the function of a prolépsis is to help one recognize the object of one’s sense-perception.31 But how, exactly, are we to understand the characteristics of the underlying things? If we accept that there is a prolépsis which is subordinate to every name, Diogenes’ argument reads as follows: (1) Prolépsis is like things that are ‘such and such’. (2) For by prolépsis we get the basic features of a kind through a mental image (typos) as soon as a name is uttered. (3) Thus, the primarily denoted prolépsis is evident: (enarges): it has a certain vividness or clarity. Point (2) does not say that prolépsis refers to the product of a cognitive process, nor that we think of prolépsis when we hear the word ‘man’ uttered, but that by the means of prolépsis we think of a certain typos, some representation of the general characteristics of a thing (ἅμα γὰρ τῷ ῥηθῆναι ἄνθρωπος εὐθὺς κατὰ πρόληψιν καὶ ὁ τύπος αὐτοῦ νοεῖται προηγουμένων τῶν αἰσθήσεων).That is to say, what is enarges is the process that produces this representation in the form of a typos. If this is correct, it also follows that a typos certainly cannot be something like a representation formed from only one of the sense organs, because a single representation of one of the

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sense organs does not require a process (cf. Ep. Hdt. 50). A typos, therefore, must be a unified, holistic (and generic) representation of the sense-experience for the rational part of the soul. Hence, Dyson’s characterization of what a typos is in this context, as the basic features of a kind through a mental image, perfectly fits my interpretation.32 I think Epicurus thus thought of perception as a tripartite transitive process which begins with the activation of the sense organ. In Diogenes Laertius’ example, the uttered word ‘man’ stimulates one’s ears vitalized by one’s irrational soul, which given the sensory recognition of the utterance of the word ‘man’ initiates a proleptic process of recognition. This, finally, presents a typos of the perception for the mind. One important question whose answer is not evident from T2 is whether such a proleptic process is only initiated by a sensory recognition (epaisthésis) through of one of the specific sense-organs or, alternatively, is initiated through the stimulation of many simultaneous sensory recognitions of the relevant sense-organs.33 The consequent proleptic process thus P(a) either complements the information present through one of the sense organs – in the case of hearing the word ‘man’, the proleptic process supplies all the relevant extra information for the representation of the basic features of what man is in the form of a mental image (typos) – or P(b) the proleptic process connects and presents the separate sensory recognitions of sense organs in a holistic outline or typos. A combination of P(a) and P(b) is also possible. And there is also the alternative non-P(c) that there is no sensory recognition (epaisthésis), for example, if you are a barbarian and do not understand the word ἄνθρωπος; in this case there is no proleptic process initiated. Instead, the perceiver must make a judgement of hearing something not on the strength of a proleptic process, but with (or without) the help of one’s separate stock of conceptual apparatus. How is the immediate association between the uttered word and the consequent proleptic process in T2(2) guaranteed? If prolépsis is a process based on many previous sense-perceptions that leave physical traces in us,34 and if it is a requirement of Epicurus’ theory to give names to things (T2(6)), then there is an essential connection between experience and language:35 prolépseis are something like physical memories of former experiences and are thus directly associated with the language of one’s social environment. Thus, the theory seems to take it for granted that the condition for acquiring the language we speak is that our natural and social environments affect each of us in very similar ways. As the individuals of a group, how we use and understand language is based on such proleptic processes associated with words that have a similar causal history resulting in similar outlines or typoi for all the members of a community.36 This connection between language and prolépsis becomes more evident in the second part of Diogenes’ evidence, so let me continue with the reconstruction of T2(4-8): (4) Inquiry into something presupposes prior awareness (egnókeimen) of the inquired thing. (5) We had become aware (egnokénai) of shapes (morphé) of things formerly by means of prolépsis. (6) Naming presupposes learning the mental image (typos) of things by means of prolépsis.

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(7) Thus, prolépseis are enargeis. (8) The object of the doxaston depends on something prior and evident, by reference to which we state the belief. In T2(4-8), prolépsis appears as an indispensable starting point of inquiry, and consequently, it has been generally conceived as Epicurus’ empirical answer to Meno’s paradox, namely that you could not inquire about something unless you already knew what it was (Meno 80d-e). Plato’s solution in his Phaedo was to argue for the immortality of the soul and claim that the soul during its prenatal acquaintance with the transcendent Forms of things already acquires some knowledge, which fades at birth, and thus, an inquiry is always a recollection of the relevant former piece of knowledge. Epicurus’ apparent reaction to Plato has often drawn interpreters to take the evidence of T2 as concerning philosophical inquiry, because of the language used here (καὶ οὐκ ἂν ἐζητήσαμεν τὸ ζητούμενον) – also cf. Ep. Hdt. 37-8 –, which also conforms to the slave boy resolving a mathematical theorem in the Meno. Diogenes’ immediate example – ‘Is the standing thing in the distance a horse or a cow?’ – however makes it clear that Epicurus was rather interested in discussing the proleptic function of our perceptual awareness in connection with τὰ δοξαζόμενα, that is to say, the fundamental elements of how we recognize things. In this part of the argument, point (6) makes it clear that prolépseis are prior to names37 since if naming presupposes – as (6) indicates – learning the basic features of a kind through a mental image (typos) by means of prolépsis, a proleptic process must precede the act of verbally designating something. Consequently, the wide consensus in the scholarship that Epicurus’ prolépseis are true beliefs – given Diogenes’ characterization of prolépseis as doxa orthé, though seemingly corroborated by various passages (Ep. Men. 123–4; On Nature XXVIII) – is seriously questionable. However, it appears that the processes of sensation create physical traces in us, based on which prolépseis come about by repeated sense-perceptions, and it is through these that we acquire the basic features of a kind in a mental image (typos). If point (5) in my reconstruction indicates a primary awareness of things, meaning that we first only become aware of visually appearing shapes (morphai) by prolépsis, and only later through a typos as in (6), then a typos, indeed, needs to be something more complex, at least concerning naming, if giving a name is to be based on a holistic and generic outline of a thing. Once we have learned a set of constantly and, to some extent, coherently uttered sounds in relation to objects repeatedly perceived in various ways, these utterances become fixed as names based on our repeated experience with things and, therefore, these names are naturally connected with the end products of proleptic processes, the typoi. Consequently, T2 seems to present prolépsis as a criterion in two ways: according to T2(1-3), if I hear someone uttering the word ‘man’, the function of prolépsis is to provide the typos of man, so that even if I do not see a man I can understand (recognize) what is referred to; in T2(4-8) prolépsis, however, appears as the precondition of naming and consequently as the precondition to uttering any names, and in this sense it discriminates my perceptual experience in direct connection

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to my linguistic abilities. In both cases, I discriminate or recognize my current experience by the proleptic outcomes, some typoi, and hence, prolépsis functions as a criterion without which the world would not appear as naturally classified as it does. It seems obvious that if I am to form a judgement about something, that judgement cannot concern something that I have not recognized as ‘such and such a thing’. If I see a horse, in order to utter or think a statement such as ‘This is a horse’, I have to recognize to a certain extent what I am looking at. Falsehood does not arise concerning what I see because of the perception, prolépsis or pathé. The truth or the falsehood of the propositional content of the mind’s judgement simply depends on the strength of some further evidence, also presented by the three criteria in a later instant of time (e.g. when the horse ‘seen’ through thick fog eventually turns out on closer examination to be a cow). It is not the case then that former or latter perceptions and related proleptic and affective states corroborate or falsify one another, but rather that many sequential perceptions, affective states and their related prolépseis corroborate or falsify the propositional content of the mind’s judgement concerning one or the other perceptual evidence formed at an earlier point in time.38 Prolépsis characterized as a kind of memory, in my interpretation, is a physical process of recognition stimulated by the senses. There is strong textual evidence in book XXV of On Nature for memory being a physical process, which, although formulated in a different set of Epicurus’ technical language, I find very similar to how I have characterized the proleptic process: Text M:39 [μ]νήμ̣η̣ ἢ τὸ τει [μνή]μηι πάθοϛ ἀνάλογον ὧν ἔδει μᾶλλον ἐνεγείνετο πρὸϛ τὸ ὡρισμ̣ένον καὶ τὰ πάντα ἐξελέγχον τῆϛ ἀναφορᾶϛ γινομένηϛ καὶ οὐ πρὸϛ ἀόριστα καὶ κρίσεωϛ προσδεόμενα.* αὕτη δ’ αὖ πάλιν ἡ τούτου μνήμ̣η ἢ ἀνάλογοϛ μνήμηι κίνησιϛ τὰ μὲν συνεγεγέν[νη]το εὐθύϛ, τὰ δ’ ηὔξητο τὴν ἀρ̣χ̣η̣ν ἔχουσα καὶ τὴν αἰτίαν ἧι μ̣ὲν τει πρώτει συστάσει τ̣ῶν τε ἀτόμων ἅμα καὶ τοῦ ἀ̣πογεννηθέντ[ο]ϛ, ἧι δὲ τει ἐ[παυ]ξομένει̣, ε[ἶ π]άντα δρῶ[με]ν̣, τ̣[ῶ]ν ἀ̣τ̣όμω̣ν̣ ἅμ̣α καὶ αὐτοῦ τοῦ ἀ̣πογε[γεν]νημένου ε[̣̓ ξ] α[̣̓ νά]γκ[ηϛ ἀ]ν̣τίξουν ἐπ’ ἐνίων [τοῖϛ] ἀπ̣[ογ]ε̣ν̣νήσ̣ασιν ... . . . the memory or the affection (pathos) proportionate to the memory of the more necessary things came to be/exist within (ἐνεγείνετο) in consequence of the welldefined and that is used to test all things when the reference-back [to the standard] happens and not in consequence of things that are undefined but need judgment. This memory of that, or the movement proportionate to memory, was again in one aspect co-generated immediately, and under another it had grown, being the beginning and the cause for, in the first case, the first constitution of both the atoms and what is produced (τοῦ ἀπ̣ ογεννηθέντ[ο]ϛ [i.e. the occurrent mental state]), in the other case, for the on-growing [constitution], by means of which we perform all our actions, of the atoms and the product itself [i.e. the occurrent mental state itself] that in some cases is necessarily opposed to what produced . . . This passage is about how a memory comes to be, first, in consequence of what is well-defined and all refuting: that is, because of one of the criteria of truth.

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The causal connection between this criterion – most likely sense-perception – and memory-formation is just like the connection I have suggested between senseperceptions and proleptic processes, as a case of P(a) or P(b) or a combination of the two. Actually, a version of non-P(c) also seems to be apparent in Text M, in line 2, which again strengthens the connection between Text M and my interpretation of prolépsis. And in so far as a prolépsis (i.e. physical proleptic process of recognition) is conceived of as a kind of memory (T2(1)), the affection or movement proportionate to the emerging memory also corroborates the connection between a proleptic process and another criterion of truth, the pathos, since the rearranged atomic constitution of a memory also brings about the affection of the perception or thought remembered. The interpretation of Epicurus’ technical term, the ἀπογεγεννήμενα is out of scope here,40 but it is easy to notice that the two different grammatical tenses of the verb ἀπογεννᾶν can be easily harmonized with the general framework of my interpretation: the passive aorist participle (τοῦ απ̣̓ ογεννηθέντ[ο]ϛ) can be taken as the first constitution of a mental state in which the product of prolépsis, the typos appears for the mind’s judgement, and the passive perfect participle (τοῦ ἀ̣πογε[γεν]νημένου) can be understood as the further processed mental state already including the mind’s verdict. One reason why this terminology is missing here is simply the context of discussion where this fragment appears, which concerns psychology and not epistemology. Let this much suffice here for the connection between my explanation and a primary evidence. Let me now, instead of pursuing this possible explanation, wrap up and harvest the consequences of our findings.

CONCLUSIONS Epicurus’ empiricist theory, in my understanding, placed the classification of the apparent world – that is the problem of universals – to a natural and automatic level. Perception itself testifies that bodies and place exist (Ep. Hdt. 39). Visible bodies knowable through sense-perception have properties and accidents, which themselves are known by their own peculiar way of focusing and comprehension. That is to say a perceptible body as a whole derives its whole nature because of how its properties and accidents are predicated (κατηγορεῖται) of it according to the complex conception (κατὰ τὴν ἀθρόαν ἔννοιαν) of these properties and accidents (Ep. Hdt. 68-69). As we have seen, predication is the result of a proleptic process, the criterion of how we recognize the perceptual world. As I have argued Epicurus thought that we recognize natural kinds and their properties and accidents in the way we do because of proleptic processes that produce typoi. Even though our sense organs separately receive their proper objects, proleptic processes of recognitions connect these properties into a unified perceptual awareness, without any participation by the mind. The results of these processes, in the forms of universally conceived typoi, naturally classify our repeated sensory experiences. This natural classification, based on the proleptic processes of recognition, automatically shapes our experience (independently of the mind) so

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that we perceive the world with the distinction between καθ’ ὅλας φύσεις and their properties and accidents. Without this natural proleptic connection, the criterion of recognition, the totality of things would not appear primarily as of bodies and place and their properties and accidents. For example, if I have seen multiple horses in my life, upon hearing something neighing behind a bush, it is due to my proleptic process of recognition initiated by the sound I hear that I can form an opinion of what shape or olfactory sensation I can expect upon closer inspection, and what sort of tactile impression I should avoid. But the way in which neighing initiates the proleptic process in me, placing the typos of a horse before my mind, picks out the outline of a natural kind: namely the horse as a whole and distinct nature, with such and such properties or accidents because as a human being my perceptual awareness is formed in such a way that my disparate sensations are also represented as a unified whole. If they were not, aisthésis alone would only provide the phantasia of colours or odours and so on, but I would be short of a unified perceptual awareness, which implies the recognition of the totality of things in a certain way. Consequently, the concomitant of a prolépsis, the typos, provides the foundation for the nominal definition of a type, and it classifies our experience naturally and automatically along Epicurus’ distinction between καθ’ ὅλας φύσεις and their properties and accidents. Thus, a typos of a horse, for example, is not a substantial definition or categorical proposition necessary for demonstration and understanding proper, but it is a naturally occurring mental image. To the extent that it picks out the basic features of a type, it fulfils the function of a universal because it explains why people perceive the world naturally in similar patterns. The consequent mere nominal definitions of types do not create a problem for Epicurus’ semantic theory, because he held that understanding only requires the ability to apply words correctly to types and their properties: nominal definitions are merely reminders by which we bring these types and properties to mind. Moreover, Epicurus did not find substantial definitions useful because his atomist explanations are reached not by deductive reasoning from definitions but by analogical reasoning from sense-data,41 and the nominal denotation based on our typoi is sufficient to ground the meanings of our terms in these inquiries. In closing, let me just examine one last strand in the texture. I have intentionally avoided speaking the language of reductionism versus anti-reductionism versus nonreductive physicalism in my explanation of Epicurus. Still, my reconstruction of his ideas lends strong support for interpreting Epicurus’ atomism as non-reductive physicalism.42 Undoubtedly for Epicurus, everything on the atomic level is an interaction of atoms moving in void in certain patterns and so on. However, the way in which he thought we naturally perceive the world does not lead through atomic structures, but rather through full bodies with καθ’ ὅλας φύσεις and their properties and accidents. Therefore, our perception and awareness of the world is, first and foremost, causally influenced by whole and distinct natures, their properties and their accidents. Their interaction can be described in atomic terms, but this, in the framework of Epicurus’ epistemology, does not shift the causal priority of the phenomenal level to that of the atomic.

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NOTES 1. I would like to thank Ugo Zilioli for the invitation and for his precious comments. I also wish to thank David Sedley and Francesco Verde for their invaluable remarks and Péter Agócs for his constructive reflections. This chapter was written with the support of the Hungarian NKFI-128651 and NKFI-120357 research grants. 2. Both understanding of aléthés – true or real – runs into problems, cf. Striker (1977), Taylor (1980) and Furley (1993). 3. For the most recent discussions of the concept cf. Tsouna (2016), Verde (2016) and Németh (2017). 4. Cf. Lembo (1981–1982). 5. This vitalization and the consequent process obviously cannot imply any rational activity how Verde misreads my position in Verde (2018a, b, c). 6. Cf. Sedley (2013). 7. Cf. Armstorng (1989). 8. Cf. Kirk (1996), and Kahn (1997); see also Yonezawa (2017). 9. Cf. Sedley (2003, 17). 10. Cf. Sedley (2013). 11. Mourelatos (2006); also cf. his (2003). 12. For the vexed questions of textual certainty cf. Verde (2018d). I accept Usener’s emendation as opposed to Gassendi’s in Ep. Hdt. 39, because philosophically it makes the best sense to suppose that Epicurus, building his argument for atomism through a number of steps, begins with the perceptual reality of bodies and place, and that he would not pair, instead, the natural conception of bodies with the theoretically loaded term of ‘void’ right at the beginning of his argument for the two basic elements of his theory, that is, the atoms and void. For Philodemus’ evidence on the basic distinction between bodies and place cf. Obbink (1996, 136–7 and also 338–9). 13. Sedley argues convincingly based on the evidence of Demetirus of Laconia that συμπτώματα is a sub-class of συμβεβηκότα, and thus their difference can be conceived as that between permanent (συμβεβηκότα) and accidental (συμπτώματα) properties in Sedley (1988, esp. 304–12). 14. Epicurus later describes void as καθ’ ἑαυτό in Ep. Hdt. 67. 15. Cf. Bronowski (2013). 16. Cf. Betegh (2006, 280). 17. Text: Dorandi (2013). Translation – with minor modification: Inwood and Gerson (1994). 18. Sedley (1988, 312). 19. Cf. Furley (1993) and O’Keefe (1997). 20. These properties, however, do not need to be intrinsic, that is to say, observer independent. If the blueness of the sky is a dispositional property, the reality of that property is that it causes people to perceive it as blue in the most case, that is under certain circumstances.

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21. ‘They [i.e. accidents] are observed just as sense-perception itself presents their peculiar traits’ (Ep. Hdt. 71). 22. Cf. Monet (1996). 23. For a comparatively exhaustive discussion, see Németh (2017, ch. 1). 24. Cf. Epicurus’ distinction for the mental aggregate in PHerc.1420 corn.2 z.2 = 6 II = 3 N =Arr. [35.10], quoted for example, in Németh (2017, 38) as Fr.(b); the scholion to Ep. Hdt. 66 and Lucretius’ distinction between the animus (and mens) and anima, as well as the evidence of Demetrius of Lacionia (P.Herc. 1012). 25. Text and translation from Asmis (1984), with modifications in the translation. I do not follow the punctuation of the text in (1) – putting a comma after τουτέστι μνήμην – as in Long and Sedley (1987, vol. II, 92), and I agree with their comment that omitting the comma weakens the further definitions of prolépsis, which suits my interpretation of section (1) well. 26. For my extensive discussion of the Stoic terminology applied by Diogenes to characterize the Epicurean concept of prolépsis as katalépsis, doxa orthé, ennoia see Németh (2017, 34–7). Briefly, the Stoic idea of apprehension (katalépsis) as a grasp of a necessarily true presentation of a thing is very close to my interpretation of prolépsis as recognition; doxa orthé is in inherent tension with T2 (8), and the most that seems to follow from T2 is that there are some self-evidently true opinions, in harmony with the idea of prolépsis being self-evident (on this also see Verde (2018b), who draws attention to the possible connection with Plato’s Meno); as I have shown, following the lead of Glidden (Glidden 1985, 175–217; contra Glidden cf. Hammerstaedt 1996), there was an important difference between the Stoic and Epicurean understanding of ennoia, which is, in short, a difference between its de dicto and de re understanding, and the latter, in Epicurean terms, represents the process of awareness in recognition. Of course, Diogenes may have just meant to characterize prolépsis in the sense of the Stoic’s terminology, but if there is a difference between the Stoic and Epicurean understanding of ennoia, that is a good reason to distance ourselves from trying to understand the Epicurean notion of prolépsis along its Stoic meaning. 27. Cf. LSJ, 1865. 28. Cf. (Dyson 2009, 86). 29. The nominal phrase in (3) – ‘παντὶ οὖν ὀνόματι τὸ πρώτως ὑποτεταγμένον ἐναργές ἐστι’ – immediately brings to mind the first principle of methodology in the Letter to Herodotus and the related questions concerning Epicurus on meaning, however, perhaps simply because it is a well-pointed emendation by Gassendi. In four codices we have ‘ἐπιτεταγμένον’, from the verb ‘ἐπιτάσσω’, which would simply mean that the connection between the thing – as it will turn out between a prolépsis – and a name is clear, which would perfectly – and actually more easily – suit my interpretation as well. 30. In Glidden (1985, 181). In Glidden (1983, 187) he says: ‘Epicurean linguistics show little concern for how words acquire meanings and thereby describe what is happening in the world.’ 31. Cf. Everson (1994, 100–1 and 106). Or as Everson puts his conclusion: ‘we should cautiously accept that the semantic values of the words a speaker utters are a function of his prolépseis.’

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32. A typos is distinct from phantasia in the sense that a phantasia is a representation in consequence of one of the sense organs affected by the external world (cf. Ep. Hdt. 50), while a typos is a mental image combining these representations as one. A typos as a mental image has its physical foundation in the proleptic process. 33. Cf. Ep. Hdt. 52-3 and DRN IV 553-62, and for my interpretation of epaisthésis, see Németh (2017, 17–24). 34. ‘Preceding perceptions (προηγουμένων τῶν αἰσθήσεων)’ in T2(2) seems to stand for the natural, empirical encounters, which determine proleptic processes. Also cf. Diogenes of Oinoanda, Fr. 9 II 9-VI 3. 35. Cf. Ep. Hdt. 75. 36. For contesting the plausibility of such uniformity cf. Long (1972, 121). For my rejection of Long see Németh (2017, 14–24). 37. Cf. Scott (1995, 167), describing prolépsis as pre-verbal. 38. Cf. K. D. 24. 39. Text M = 1191 corn. 4 pz. 2 z. 1 = –24 inf./1191 corn.7 pz.1 z.2-3 = –23 sup., and 697 corn.2 pz.2 z.4 (=col.5), and 1056 corn. 5 z. 1, in Laursen (1997, 16–17), modified slightly in Németh (2017, 48–50). In this most current representation of the text, I have slightly modified my previous translation. 40. Cf. Németh (2017, 86–92). 41. Cf. Dyson (2009, 88). 42. I formulate the arguments for this interpretation in Németh (2017, ch. 2).

REFERENCES Armstrong, D. M. (1989), Universals: An Opinionated Introduction, San Francisco and London: Westview Press. Betegh, G. (2006), ‘Epicurus’ argument for atomism’, Oxford Studies in Ancient Philosophy 30: 261–83. Bronowski, A. (2013), ‘Epicureans and Stoics on universals’, in R. Chiaradonna and G.  Galluzzo (eds), Universals in Ancient Philosophy, 255–98, Pisa: Springer. Dorandi, T. (2013), Diogenes Laertius, Lives of Eminent Philosophers, Cambridge: Cambridge University Press. Dyson, H. (2009), Prolepsis and Ennoia in the Early Stoa, 86, Berlin and New York: Walter de Gruyter. Everson, S. (1994), ‘Epicurus on mind and language’, in S. Everson (ed.), Language, Companions to Ancient Thought 3, 74–108, Cambridge: Cambridge University Press. Furley, D. J. (1993), ‘Democritus and Epicurus on sensible qualities’, in J. Brunschwig and M. Nussbaum (eds), Passions and Perceptions, 72–94, Cambridge: Cambridge University Press. Glidden, D. K. (1983), ‘Epicurean semantics’, in ΣΥΖΗΘΗΣΙΣ. Studi sull’ Epicureismo greco e latino offerti a Marcello Gigante, 2 vols, vol. i, 185–226, Naples: Macchiaroli. Glidden, D. K. (1985), ‘Epicurean Prolépsis’, Oxford Studies of Ancient Philosophy 3: 175–217.

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Hammerstaedt, J. (1996), ‘H, Il ruolo della ΠΡΟΛΗΨΙΣ epicurea nell’interpretazione di Epicuro, Epistula ad Herodotum 37 sg’, in G. Giannantoni and M. Gigante (a cura di), Epicureismo greco e romano, Atti del Congresso Internazionale, Napoli, 19–26 maggio 1993, 3 vols, vol. I, 221–37, Naples: Bibliopolis. Inwood, B. and Gerson, L. P. (1994), The Epicurus Reader, Indianapolis and Cambridge: Hackett Publishing Company. Kahn, C. H. (1997), Plato and the Socratic Dialogue, Cambridge: Cambridge University Press. Kirk, G. S. ([1951] 1996), ‘The problem of Cratylus’, American Journal of Philology 72: 225–53. Laursen, S. (1997), ‘The later parts of Epicurus, On Nature, 25th Book’, Cronache Ercolanesi 27: 5–82. Lembo, D. (1981–1982), ‘ΤΥΠΟΣ e ΣΥΜΠΑΘΕΙΑ in Epicuro’, Annali della Facoltà di Lettere e Filosofia dell’Università di Napoli 24: 17–67. Monet, A. (1996), ‘[Philodème, Sur les sensations]: P.Herc.19/698’, Cronache Ercolanesi 26: 27–126. Mourelatos, A. (2003), ‘Democritus on the distinction between universals and particulars’, Philosophische Analyse 9 (Monism; Festschrift in honour of Andreas Graeser): 43–56. Mourelatos, A. (2006), ‘The concept of the universal in some later pre-platonic cosmologists’, in M. L. Gill and P. Pellegrin (eds), A Companion to Ancient Philosophy, 56–76, Oxford: Blackwell. Németh, A. (2017), Epicurus on the Self, London and New York: Routledge. O’Keefe, T. (1997), ‘The ontological status of sensible qualities for democritus and epicurus’, Ancient Philosophy 17: 119–34. Obbink, D. (1996), Philodemus On Piety. Part 1, Oxford: Clarendon Press. Scott, D. (1995), Recollection and Experience, 167, Cambridge: Cambridge University Press. Sedley, D. N. (1988), ‘Epicurean anti-reductionism’, in J. Barnes and M. Mignucci (eds), Matter and Metaphysics, Fourth Symposium Hellenisticum, Elenchos 14, 295–327, Naples: Bibliopolis. Sedley, D. N. (2003), Plato’s Cratylus, 17, Cambridge: Cambridge University Press. Sedley, D. N. (2013), ‘Plato and the one-over-many principle’, in R. Chiaradonna and G. Galluzzo (eds), Universals in Ancient Philosophy, 113–38, Pisa: Springer. Striker, G. (1977), ‘Epicurus on the truth of sense impressions’, Archiv für Geschicte der Philosophies 59: 125–42. Taylor, C. C. W. (1980), ‘All perceptions are true’, in M. Schofield, M. Burnyeat and J. Barnes (eds), Doubt and Dogmatism: Studies in Hellenistic Epistemology, 105–24, Oxford: Oxford University Press. (Repr. in C. C. W. Taylor, Pleasure, Mind, and Soul: Selected Papers in Ancient Philosophy, Oxford: Oxford University Press, 2008). Tsouna, V. (2016), ‘Epicurean preconceptions’, Phronesis 61: 160–221. Verde, F. (2016), ‘Epicuro nella testimonianza di Cicerone: La dottrina del criterio’, in M. Tulli (ed.), Testo e forme del testo: Ricerche di filologia filosofica, 335–68, Pisa-Roma: Fabrizio Serra editore.

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Verde, F. (2018a), ‘I pathe di Epicuro tra epistemologia ed etica’, Elenchos 39, no. 2: 205–30. Verde, F. (2018b), ‘Review of Attila Németh, Epicurus on the Self, London-New York: Routledge, 2017’, Rhizomata 6, no. 2: 236–44. Verde, F. (2018c), ‘Ancora Sullo Statuto Veritativo Della Sensazione in Epicuro’, LPh, Special Issue: 79–104. Verde, F. (2018d), ‘Once again on Epicurus’ Letter to Herodotus §§ 39–40’, Classical Quarterly 68: 736–9. Yonezawa, S. (2017), ‘Aristotle’s testimony regarding Plato’s philosophical development’, Rheinisches Museum für Philologie 160: 276–98.

CHAPTER 6

Atoms, complexes and simples in the Theaetetus1 SOPHIE-GRACE CHAPPELL

ὅθεν ἐλήλυθε τὰ μάλιστα καθόλου στοιχεῖα εἶναι, ὅτι ἕκαστον αὐτῶν ἓν ὂν καὶ ἁπλοῦν ἐν πολλοῖς ὑπάρχει ἢ πᾶσιν ἢ ὅτι πλείστοις. Aristotle, Metaphysics 1014b6

I You may perhaps have heard of the philosophy examinee taking the Aristotle’s Ethics paper, who saw the essay question ‘What would be a case where courage degenerates into mere rashness?’, and answered ‘This.’ Or again you may have heard of the philosophy job applicant who was asked at interview ‘What is your worst personality trait?’: ‘I tend’, she answered, ‘to grasp the semantics of utterances but miss their pragmatics’. ‘Can you give an example?’ they asked her. ‘Yes.’ Untutored common sense, relying perhaps on phenomena like the fixing of paint- or fabric-hues by colour-swatches – maybe with a side-order of Dr Johnson’s ‘refutation’ of Berkeley – apparently inclines to think that definition by example is the simplest kind of definition that there could possibly be. ‘What is X?’ we ask. ‘That is’, says common sense, ostending it; or ‘Well, X is anything like this’ – again with an ostension of some clear and obvious example of X. Socrates rejects this view. On the contrary, according to him, the problem with definition by example – the reason why really there is no such thing as definition by example – is that ‘definitions’ by example lack all real unity and simplicity; they lead us into all sorts of disorganized variety and complexity (ta pantodapa, ta poikila; in Republic V, ta polla kala). But this is a view that Plato comes to reconsider and revise. The way he does so and his reasons for his change of mind take us deep into the heart of what Plato has to say about the complex and the simple, the one and the many, the element and the compound – especially in that prize exhibit for the mature Plato’s reconsideration of his Socratic inheritance, the Theaetetus. My ‘what Plato has to say’ is, of course, a phrase that should evoke a reaction. [In] a dialectical context . . . someone puts up a thesis, but the opponent argues that this thesis generates an absurd conclusion. So long as the inferences are

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valid, this strategy forces the exponent of the original thesis to rethink his premisses, and thereafter to deny one or more of them – the opponent hopes that he will deny the thesis. Arguments like this are indirect. (MM McCabe, Plato’s Individuals, 19) I have myself argued at book length (Chappell 2005) that Plato’s strategy in the Theaetetus is an indirect one. In much of the dialogue he confronts opponents whom we would classify as ‘empiricists’ and/ or ‘naturalists’ – the atomists, for example – and argues that on their assumptions there can be neither knowledge nor even belief. I stand by that general thesis about the Theaetetus as a whole. But the last section of the dialogue is particularly indirect and presents us with particularly interesting problems, problems that this chapter, in its last section, will take a fresh look at. So a lot of this chapter will be detective-work about ‘what Plato is getting at’, and hence necessarily both speculative and presumptuous. If this tries the patience of more cautious and fastidious scholars, I apologize in advance.

II At the beginning of the dialogue named after him, the brilliant young mathematician Theaetetus does two things about defining knowledge. First, he gives a definition by example of knowledge (Tht 146c-d). This is a list of skills, and it draws on him a familiar, and familiarly withering, Socratic response. Then second – and this is less frequently remarked – he offers an example of a definition of knowledge. And the case he cites veritably is an example of a definition of knowledge; it is this, precisely because the definition of knowledge that it involves is not a definition by example of knowledge. Theaetetus defines knowledge by the example of his own and his colleagues’ use of plane and solid geometry to develop a classification of the integers (Tht 147d-148b). This case’s status as an example of a definition of knowledge that is not a definition by example of knowledge is clearly marked by Plato, both before and after its presentation: Probably, though, you are asking for something like what occurred to us ourselves, just now in our discussions (οἷον καὶ αὐτοῖς ἡμῖν ἔναγχος εἰσῆλθε διαλεγομένοις) . . . (147c-d) . . . And yet, Socrates, what you ask about knowledge, I wouldn’t be able to answer as I did about length and power. But that seems to be the kind of thing (τοιοῦτόν τι) that you are looking for. (148b) What Theaetetus says in advancing his second example of definition is right, as far as it goes. And what he and his colleagues grasp about geometry and its relation to the integers really is knowledge. (I say more about what exactly they know in the third section.) So by describing their knowledge of geometry and numbers, Theaetetus really is succeeding in giving an example of knowledge. And a definition by example? As we all know, definitions by example are not acceptable to Socrates2 as formal definitions; but a formal definition such as

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Theaetetus and his friends excogitate for the numbers is acceptable. So the rest of the dialogue is a search for a definition of knowledge that, like Theaetetus and friends’ definition of those numbers, is not merely a definition by example. But the search has – apparently – not succeeded by the ending of the Theaetetus: if it is an aporetic dialogue, this ending is what makes it so. So what is wrong with definition by example? It depends, of course, what you mean by ‘definition by example’. But three3 obvious examples of definition by example, in the sense that Plato is against it, come in Euthyphro, Meno and Theaetetus.4 Euthyphro 6d-e: Socrates. When I asked you about piety, as to what exactly it is, you did not give me a sufficiently informative answer. No, what you said was that, as it happens, this is piety, the thing that you are doing now: prosecuting your father for murder. Euthyphro. And what I said was true, Socrates. Socrates. Maybe, Euthyphro. But you agree that there are many other pious acts? Euthyphro. Yes, there are. Socrates. So do you remember that this isn’t what I asked you to do, to teach me some one or two of the multitude of things that are pious? Rather, I asked for that particular respect (ekeino auto to eidos) in which all the things that are pious, are pious. After all, you did say that that it was by some single aspect (idea) that impious things are impious, and pious pious. Euthyphro. Yes, I remember. Socrates. So teach me about this aspect, what exactly it might be, in order that I may look towards it and use it as a paradigm (paradeigma), so that if anything is like it among the things that you or anyone else does, then I can call that deed pious, and if it is not like it, not call it pious. [. . .] Euthyphro. Piety, then, is what is dear to the gods, and impiety is what is not dear to them. Socrates. Entirely excellent, Euthyphro; now you have given me the sort of answer that I was looking for. Meno 71d-72a: Socrates. Let’s leave Gorgias out of it, since he’s not even here . . . What do you say virtue is, Meno? [. . .] Meno. But Socrates, it’s not hard to say what virtue is. If you want the virtue of a man first, that’s easy. A man’s virtue is to be man enough to run his city’s affairs, and to run them so as to benefit his friends and harm his enemies – and make sure that no such harm ever comes to himself. Then if you want a woman’s virtue, that’s not hard to state either. What she has to do is keep house well, looking after the property and obeying her husband. A child’s virtue is another thing, and it is different again depending on whether we mean a boy’s virtue or a girl’s. And there is a specific virtue for an old man, with further differences depending on whether he is a free old

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man or a slave. There are any number of other varieties of virtue too. So it’s hardly a problem, Socrates, to say what the definition of virtue is. There is a specific virtue for each sort of activity and age, for each of us, in whatever we do. And likewise, I suppose, for vice. Socrates. I do seem to be having the most remarkable luck today, Meno. I was only after one virtue, but what I have found lurking within you is a whole swarm of virtues. Theaetetus 146c3, c7-d2: Socrates. What do you think knowledge is? Theaetetus. I think the things that you might learn from Theodorus are kinds of knowledge (epistêmai): geometry and the other examples you listed just now [cp. 145d1, where Socrates listed ‘astronomy, harmony, arithmetic’]. Also cobbling and the other manual labourers’ crafts (dêmiourgwn technai). All of these are nothing other than knowledge; each of these is a kind of knowledge. Socrates. How open-hearted and generous of you, my friend! You were asked for one thing and you gave me lots of things – and multifarious things too, instead of something simple (polla kai poikila anth’ haplou). The evidence of these passages is that the kind of definition by example of X that Plato rejects is the kind that merely lists examples of X. (There are other kinds, as we shall see.) So what is wrong with defining X in this way, by merely listing examples of X? Here are the three dialogues’ answers in turn: Euthyphro 6c-d (already quoted earlier): Socrates. Do you remember that this isn’t what I asked you for, to teach me some one or two of the multitude of things that are pious? Rather, I asked for that very respect (eidos) in which all the things that are pious, are pious. After all, you did say that that it was by some single aspect (idea) that impious things are impious, and pious pious . . . Teach me about this aspect, what exactly it might be, in order that I may look towards it and use it as a paradigm (paradeigma), so that if anything is like it among the things that you or anyone else does, then I can call that deed pious, and if it is not like it, not call it pious. Meno 74d-e, 77a-b: We keep coming back to pluralities, but don’t offer me anything like that. You call them all by the one name, and say that each of them qualifies as a shape, even though they contrast with one another. So – what is it that the curved shape has, just as much as the straight-sided shape? What is it that you call shape no less in the curved ones than in the straight-sided ones? . . . [In the same way] you’ve got to tell me about virtue as a whole, what it is. And you must stop making a many out of a unity, as the jokers say whenever someone breaks a vase or something. No: let virtue be a safe, unfragmented unity, and say what it is.

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Theaetetus 147b-c: If someone asks you ‘What is knowledge?’ it is ridiculous to reply merely by giving the name of some craft. Your reply mentions knowledge about something. But that isn’t what you were asked . . . And then, this sort of answer is interminably roundabout when it could have been brief, and even banal. For example, there is a banal (phaulos) and straightforward answer to the question ‘What is clay?’. We could just have said that clay is earth mixed with moisture, and passed over the question whose clay it is. Mere list definitions by example fail as definitions in four ways: they lack plainness, brevity, explanatoriness and projectibility. Their lack of projectibility is their failure to tell us ‘how to go on’. There is nothing in any finite list of examples of X, just as such, which tells what else we should count as an example of X. Given the list of numbers 1, 2, 4, 7, 11, 16 . . . , there is not – despite all those ‘intelligence tests’ we all did at school – any such thing as the next number in this list. Those six numbers are an infinitesimal portion of infinitely many infinitely long lists generated by infinitely many algorithms; so there are infinitely many next numbers. Indeed, even just for any one number – and also, for any number – there are infinitely many explanations of why it should be the next number. A mere list of examples of X is not projectible, because nothing tells us how to project it – what we should or shouldn’t count as a new example of X. And that point moves us from projectibility to explanatoriness: if we know what the algorithm is that generates the list of examples of X, then we will know both what counts as a further example, and also, in a non-circular way, why anything counts or fails to count as a further example. Hence, we can also get from projectibility to plainness and brevity. The algorithm that tells us how to project ‘X’ can be quite short and plain (perhaps even banal). But the algorithm’s power – one which amazes Plato, excites suspicion about ‘infinite rails’ in the later Wittgenstein,5 and ought to both amaze and puzzle us – is that, without in any way losing its own unity and simplicity, it can be used to generate an indefinitely large multiplicity of further examples of X. Further testimony to the plainness and brevity of the kind of definitions that Plato is looking for is given by his own examples of Platonically acceptable definitions. Given how hard Socrates’ interlocutors find it to locate such definitions, and given the perhaps off-putting mystical tendencies of some of the middle-period dialogues, especially the Republic, we easily get the impression that there is something elusive or abstruse about them. But in fact, as Plato himself says, they can often be very easily located indeed: as Socrates assures Meno, ‘there can be a general account of anything’ (Meno 74b). We have already seen a definition that Plato himself describes as ‘brief and banal’ – but by no means, pace Parmenides 130c-d, as inadequate. This is Socrates’ definition of ‘clay’ as ‘earth mixed with moisture’ at Tht 147c. Another is the second definition of ‘shape’ that Socrates offers at Meno 76a: ‘I define shape to be that in which the solid is bounded; or, more concisely, the limit of solid.’ A third example is Socrates’ exceedingly succinct formula for defining virtue at Meno 87b-89c: ‘Virtue is knowledge.’ As the Meno’s discussion shows, whatever other problems this proposal may face, it does not face any objection based on its form.

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The definition of virtue as knowledge is at least formally correct – and I think Plato thinks it is also just outright correct. A fourth example is given by the four geometrically characterized classes of integers that Theaetetus defines at 147e-148e: ‘a square number is a number that can be formed by multiplying a number by itself ’, ‘an oblong number is a number that can only be formed by multiplying a number by some other number’, ‘the side of a square that represents a square number is a length’ and ‘the side of an oblong that represents an oblong number is a power’. (It is no accident that the Tht just echoes the Meno by including a geometrical discussion; indeed the geometrical topic discussed is itself a faint echo of the Meno. The Meno’s question is how specifically to get from a square to its double, via the diagonal, and the Meno’s inquirer stumbles over this relatively undemanding task despite Socrates’ persistent help; the Tht’s question is how in general to use geometrical form to classify the numbers so as to bring out their interrelations perspicuously, and its brilliant inquirer presents Socrates with a tour de force without any help at all from Socrates.) Perhaps a fifth Platonically acceptable definition is ‘courage = wise endurance’ (Laches 192b-194b). And perhaps a sixth comes up at Euthyphro 12e: Euthyphro. My view, at any rate, Socrates, is that the Reverent and the Pious is this part of the Just: it’s the part concerned with care for the gods, while the rest of the Just is the part which is concerned with care for humans. Socrates. And my view, at any rate, Euthyphro, is that you give a good answer. Socrates clearly approves of this definition. Admittedly, of course, he said something approving of Euthyphro’s first attempt too: ‘Piety, then, is what is dear to the gods, and impiety is what is not dear to them’ (6e). His immediate response to that was ‘Entirely excellent, Euthyphro’, even though he goes on straight after that to tear this attempted definition to pieces. The point is that both definitions have the correct form – and Euthyphro’s second attempt, I suggest, has not only the right form but the right content too. What it takes to define the Pious can be expressed, with some anachronistic terminology, as this: a grasp of the essence of the Pious, and of its part–whole relations, not to accidental properties like being loved by the gods (or come to that, commanded by them), but to other essential properties, or essences, of the same sort as the Pious itself. When Euthyphro says that the Pious is simply what we call the Just in its application to humans, only applied to the gods, and that like any other application of the Just, what the Pious involves is care (therapeia) for its objects, this is precisely what he is supplying. If this is right, then Socrates has reason indeed to praise Euthyphro for his second answer. For maybe Euthyphro does know after all how to define the Pious in a Platonically acceptable way. He just doesn’t know that he knows. The suggestion is, then, that Euthyphro is someone who sometimes hits, perhaps by ‘divine lot’ (theiai moirai, Meno 99d), on truths about definition that are always important to Plato, and especially in his mature metaphysics. This is a suggestion that is perhaps confirmed by something that Plato did with the Euthyphro, some number of years after he wrote it: namely, he set up the stage-directions in the Theaetetus so as to intercalate the Euthyphro into the fictional timeline of his own

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dialogues in between the Theaetetus and the Sophist. As all modern scholars have agreed at least since Auguste Diès (Parménide 1923), the Theaetetus ends with a clear forward-reference to the probably-not-yet-written Sophist. It also ends – and this has not been noticed so often6 – with a clear back-reference to the surely-alreadylong-written Euthyphro: And now I must answer Meletus’ indictment at the King’s Porch (basileôs stoan). But tomorrow morning, Theodorus, let’s meet here again. (Theaetetus 210d) What novelty is this, Socrates, that you have left your hang-outs (diatribas) in the Lyceum, and are now hanging out (diatribeis) here, at the King’s Porch (basileôs stoan)? (Euthyphro 2a) In line with yesterday’s agreement, Socrates, here we ourselves come in good order, and we bring with us this man, some guest of ours. (Sophist 216a) Incidentally, this not only situates the Euthyphro in time, in between the Theaetetus and the Sophist. (The Cratylus happens in the same gap: Cratylus 396d.) It also situates the Theaetetus and Sophist in place: Euthyphro has just told us that the setting for both is the Lyceum, Socrates’ ‘usual haunt’. This location is not inconsistent with the one that commentators usually give to the Theaetetus, and therefore to the Sophist also, namely some (unidentified) gymnasium.7 What the text of Theaetetus 144c suggests is that there was a(n outdoor) gymnasium immediately outside the temple of the Lyceum. And this suggestion has recently been confirmed by Greek archaeologists working on the site.8 The Lyceum was the setting for at least two of the Socratic dialogues and the (as it were) sanctified intellectual wrestling that they involve: Lysis 208a, Euthydemus 271a. So that the fact that we return there for the Theaetetus and its sequels evidently fits with the familiarly ‘retro’ presentation of the dialogue, as a studied return on Plato’s part to the old Socratic method,9 and to the old Socratic concerns with – for example – the inadequacy of definitions by example.

III The lesson of the dialogues just cited is that Platonically acceptable definitions need not be abstruse or recondite. What they do have to be is plain, brief, projectible and explanatory. And what they do have to do is reduce the complex to the simple. They do this by classifying the complex in terms of the simple. So clay is that part of earth which is mixed with moisture; piety is that part of justice which is concerning the gods; shape is the limit aspect of solid; the square numbers are that subset of the numbers that can be represented by square figures and so on. Now this conception suggests a quite general philosophical programme: reduce the complex to the simple everywhere. Look at all points for the most general classes in which things come to us, the broadest and most basic structural categories of reality; and work out how those most general classes are divided up into more specific and particular categories or classes in the world around us – that is to say, how the simple compounds constitute the complex.

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One obvious difficulty that faces this programme is to say when we have reached the simple, and how we will know when we’ve got there. Plato’s example of clay may well be chosen deliberately because Plato thinks it gets us to the simple very quickly indeed. According to Timaeus 31b, he thinks that water and earth are two of the four physical elements, alongside fire and air, and that clay is just a differentiation of one of them by the other. Maybe he takes justice, solids, and numbers, in their contrasting way, to be simples as well: these too are basic ingredients of reality, relative to which other parts of reality are always compounds, and so, in a sense, ontologically secondary. (Or maybe not, because, after all, at least one Platonic dialogue is devoted to an attempt to define justice, as ta heautou prattein. The suggestion that one wants to reach for here, to help out the proposal, is that we take complexity and simplicity to be fixed only relative to particular inquiries/explanations: cp. Wittgenstein, Philosophical Investigations I, 59.10 On the whole, it seems easier to find this idea in Aristotle than in Plato: see, for example, Mph 1014a26-b15, where that distinction is surely implied (note especially ἐκεῖνα δὲ μηκέτ᾽ εἰς ἄλλα εἴδει διαφέροντα at 1014a34), and the Physics’ and Metaphysics’ recurrent distinction between ‘the immediately underlying’ and ‘prime matter’. However, note (1) that the Timaeus (53d) shows us how we might analyse the four elements ‘into particles identifiable with four regular solids, and . . . those solids into a set of absolutely primary triangles’ (Sedley MWP, 72), which seems to be a (double) example of just this sort of inquiryrelative simplicity; and (2) that at least some encouragement for the idea of inquiryrelative complexity/simplicity comes in Part Three of the Tht, where, for instance, the ‘simple parts’ of a wagon are clearly not themselves simple in any absolute sense. More about that part of the Tht further.) Here, in any case, we come to a second and perhaps less obvious difficulty for the philosophical programme that I am sketching. This arises from the fact that, following Socrates, the middle-period Plato frequently affirms that knowledge of any X is impossible without a definition of X. But we’ve just seen reasons for thinking that definition always means reducing a complex to its simple elements. It seems to follow that there can only be definitions of complexes and not of simples, and so that there can only be knowledge of complexes and not of simples. But that looks like a serious setback for the programme, the whole point of which, you might have thought, was to ground knowledge. For we might say – as I shall argue Plato certainly does – that nothing can base knowledge that is not knowledge itself. (On this, though not on much else in the Theaetetus, I am in full agreement with Fine PR 1979; she calls this principle KBK.) To this problem, one solution that suggests itself immediately is to divide knowledge into two sorts – the knowledge that is of simples, and the knowledge that is of complexes. Knowledge of complexes will itself be complex because it will involve a statement of how the complexes to which it relates reduce to their simple parts. Knowledge of simples will itself be simple – it will be an immediate awareness of the simples to which it relates. (In NE VI, Aristotle seems to be on to something like this idea when he distinguishes directly apprehending nous from the complex and structured forms of knowledge that are worth calling episteme and sophia.)

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The difficulty for this solution seems to be that, as before, Plato is seeking unities, and a theory of knowledge that explains how we are to arrive at unities. So it might seem awkward for him to be forced, by the requirements of his unifying theory of the objects of knowledge, to postulate a disunity in the nature of knowledge itself. It is (someone might complain) as if we are merely moving the bump in the carpet: the more we strip out complexity from the world that knowledge accesses, the more complexity we need to build in to our conception of knowledge. If the knowledge of complexes is so different in nature from the knowledge of simples – the complaint would continue – then that is at the least a dilution of the sense in which both of them are knowledge. I think this complaint can be answered, in at least two ways. (My initial examination of these two answers here is provisional: in subsequent sections, there will be more to say about both answers after this initial statement.) First, Plato might respond that the duality in his conception of knowledge is, after all, only a duality – it is not as if he is making knowledge indefinitely complex – and that that duality is worth having, because it mirrors a duality in the world: the objects of knowledge are either simples or complexes, so it is actually rather neat that knowledge too, correspondingly, is either simple or complex. Or the second answer (and this is not just a response to this last complaint but to the whole line of thought that has led us up to it): Plato can say that at the deepest level he is not actually arguing for a duality in knowledge at all. What he is really arguing for is more like a duality in the notion of a logos, a definition. When a thing is complex, definition for it means, as before, an analysis of it that shows how it is combined out of its constituent simples. Now (unless or until we admit inquiry-relative simplicity/complexity) it is true that those simples are not combined out of anything; that’s the point of calling them simples. But that does not mean that there can be no definition of them. When we cannot further analyse a simple, we can still display its place within a system. And it is a striking fact about the Theaetetus that, almost nonchalantly, it shows us Theaetetus himself doing exactly this – not just once, but twice. The first time he does so is in a passage I have already mentioned, the example of real knowledge given at Tht 147d-148b, where the simples that Theaetetus and his colleagues classify – under plain, brief, explanatory and infinitely projectible classifications – are the integers. The classification is achieved by the brilliant device of treating the integers as geometrical figures in either two or three dimensions – a device that gives us, among other things, an interesting and vividly visual way to explore the relations between integers with more than one pair of factors/ oblongs with various-length sides and also between squares and cubes (e.g. 72 = 2 × 36 and = 3 × 24 and = 4 × 18 and = 6 × 12 and = 9 × 8 and = (3 × 3) × (2 × 2 × 2)). It also gives us a handle on the notion of a prime number, by defining the prime numbers as those that can only be represented by an oblong with a short side of 1 unit. (Note, by the way, that the opposite of ‘prime number’ is ‘composite number’: there is a tradition of thought about prime numbers that goes all the way back to Pythagoras, that takes the prime numbers to be the elements, the simple

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parts, of the number-system. Come to that, another kind of elementality is indicated by the word ‘integer’.) Theaetetus’ second foray into the classificatory definition of simples comes when his discussion of the Dream Theory with Socrates reaches this point, at Tht 203a-c: Soc. Your definition (logos) of the syllable SO is ‘sigma plus omega’? Tht. Yes. Soc. Well then, say in the same way what your definition is of sigma. Tht. And how can anyone state a definition of the elements of an element? After all, Socrates, sigma is just one of the letters that don’t have a voice of their own. All sigma is is a noise – so to say, a hissing of the tongue. Other consonants, like beta and indeed most of the letters, aren’t even noises. So it is entirely appropriate to say that these letters have no definition. For even the seven most explicable of them, the vowels, have only voices, but no sort of account at all. For the purposes of this discussion, the letters of the alphabet are the elements. (Indeed, historically the letters are the first and original elements: the Latin word elementum comes from L-M-N-tum.) But anyone who is as dogmatic as Theaetetus is made to appear in this passage that the elements cannot be defined needs only to look at what he actually says to see how blatantly Plato is teasing us here. Despite professing his inability to give any definition of sigma, Theaetetus has plenty to tell us about its nature and classification, and something too about the other letters, and the principles by which he or anyone else can classify and contrast them (cp. the Philebus’ distinction between ‘limited’ and ‘limitless’ letters, and indeed the whole discussion of Phb 17b-19b). Sigma is (a) among letters, a consonant not a vowel – it ‘has no voice of its own’; but (b) among consonants, it is a continuant not a stop – sigma is a ‘noise’, not a non-noise like beta or tau; and (c) among consonants that are continuants not stops, sigma is, distinctively, made with the tongue, not with both lips like mu, nor with the nose, tongue and palate like nu. Theaetetus could have added, as a modern phonologist would, that sigma is also differentiated from the other continuant in classical Greek that is made with the tongue, namely rho, by (d) being a whistling noise rather than a rattling noise. If he had also added (e) that sigma is not ‘voiced’ in the modern sense – that would give us our z sound – and (f) that, like rho, sigma is aspirated – then by way of these six ‘divisions’, he would have given what even modern phonological science would count at least as a passable shot at a complete classification of sigma. This classification, a modern linguist might further add, is not explanatorily inert. It is not just a harmless exercise in philatelic tidiness to classify sigma, or as we would say /s/, this way. Consider how a classification of sounds can help us to understand how some of them can combine with one another, and others can’t, either in some particular language, as in English very few of us can (or at least, usually do) really pronounce ‘sixths’, or in any language, as for example, the consonant-cluster ‘gmpr’ is pretty well physically insurmountable for any human mouth; (cp. Sophist 254d, ‘some of the gene can combine with each other and others cannot’). Consider again how phonological change itself is, obviously enough, a principal driver of larger-

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scale linguistic change. Two important phonological changes that can happen to /s/ are that it softens into /h/ when the aspiration remains but the tongue stops touching the palate (contrast Latin Sabrina ‘the Severn’ and Welsh Hafren, or Latin sus ‘pig’ and Greek hys) and that it hardens into /r/ when the tongue-shape making it changes (compare Latin mus ‘mouse’ in the nominative and the genitive of the same Latin word, muris, or English was and German war, or r as an ending in Norwegian and Icelandic masculine nouns, reflecting the older s endings found in, for example, Gothic). Part of what we need to understand these and other phonological changes is, precisely, a full classification of the relevant languages’ phonemes and of their propinquities and distances from each other. And this too Theaetetus’ first shot at the phonology of /s/ sets us on the way to achieving. Thus, Theaetetus’ explanatory and contrastive classification of sigma has a lot in common with the classification of /s/ that a modern phonologist would provide. Maybe Plato could have had him take it even further than he does; notice, for instance, how Theaetetus casually throws in the complete list of the seven Greek vowels as well. Perhaps the only reason why Theaetetus doesn’t take his listing and classification of the letters any further is because Plato feels that he, Plato, is already laying it on with a trowel. (As Aristotle for one surely noticed: plausibly, Aristotle’s lexicon entry for stoicheion at Mph 1014a16-b15 is written with the Theaetetus very much in mind.) At the outset I signalled my agreement with McCabe that much of what goes on in many of Plato’s dialogues, particularly one like the Theaetetus, is one species or another of indirect argument; and this clearly applies here. The point Theaetetus, or Plato, is indirectly making is – surely – the following. While Theaetetus is right that no one can ‘state a definition of the elements of an element’, that doesn’t mean that no one can state a definition of an element. When we touch rock-bottom in our analytic system, by reaching the elements, we can carry on doing something worth calling definition, simply by starting to move upwards again: by showing how those elements can be classified into their natural groups and kinds. (This is, if you like, a cyclical movement, and it implies something like a coherentist picture of our knowledge, but it is not, not at least in Fine 1979’s sense, a circular movement: exploring and/or displaying the structure of a ‘hierarchical’ epistemic system is not the same as travelling back and forth around the nodes in an ‘egalitarian’ one.) At the beginning of the dialogue, Theaetetus and his colleagues gave a classification of the integers; so now at 203a Theaetetus says enough to make it very clear how a classificatory account can be provided for the elements that are the letters of the alphabet – and the classification he offers is strikingly similar to the one that modern phonology gives. He also says enough to make it clear that Plato would count that masterpiece of modern chemistry, the periodic table, as a paradigm of definition in this classificatory sense. After all, if anything is the canonical classification of the elements of reality in the physical world, the periodic table is.11 One consequence of this line of thought is that, at least to an extent, it ends up rehabilitating the procedures of definition by examples. It remains true that the mere listing of a plurality of Xes can never be sufficient for a full definition of the unitary notion of an X. But strikingly, when we get down to the level of the simples, we do

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see that a kind of listing is a necessary part of Plato’s definitional procedure. The sort of classification of the letters of the alphabet on which Theaetetus, apparently by accident, makes such a useful start in the speech last quoted – a classification like this starts out as a mere list; then, by the application of distinctions and divisions to it, it becomes an ordered list. To cite such a classification of the elements is, among other things, to recite a list of examples of the elements. So when you are trying to define X, giving examples of X may not be forbidden by Plato after all. And perhaps Wittgenstein was overstating the case when he famously said (The Blue and Brown Books, 20) that ‘When Socrates asks the question, “What is knowledge?”, he does not regard it even as a preliminary answer to enumerate cases of knowledge.’ If I am right, Socrates does, pace Wittgenstein, think that examples are at least a starting point for definitions. What matters, according to Socrates, is what you do with those examples – how you analyse them, in the original sense of the word: that is, how you take them to be composed, or if incomposite, what kind of simples you take them to be. (Of course there are other things we can do when defining; for example, a wide-ranging stock of examples of X is a useful source of counterexamples, that is, a test for intuitions about how the definition of X should go given the full range, or something like it, of things that we call X. No one could accuse the early Socratic dialogues of being unalive to this point. Yet Socrates does not seem ever to enounce it as a general principle showing how philosophically useful examples are.) One surprising place where we see Socrates, or Plato, beginning to adopt this more accommodating attitude to the use of examples in definition is Protagoras 329c-e, where Protagoras defends something distinctly like a definition of virtue by examples. Protagoras says that virtue is one thing with the particular virtues – justice, temperance, piety and so on as parts (ἑνὸς ὄντος τῆς ἀρετῆς μόριά ἐστιν ἃ ἐρωτᾷς, 329d). Pressed by Socrates on how these parts relate to each other and to the whole thing virtue, Protagoras replies ‘in the same way as the parts of the face relate to the whole face’ (ὥσπερ τὰ τοῦ προσώπου μόρια ἔχει πρὸς τὸ ὅλον πρόσωπον, 329e). Socrates quickly points out that this means that each of the individual virtues must have its own distinctive δύναμις (330a) and goes on to fashion an objection to Protagoras’ conception of the virtues that depends on the claims (a) that they interpredicate and (b) that if they inter-predicate, then all of them must have exactly the same one δύναμις in common (330a-333a). This is a subtle, interesting and ingenious argument on Socrates’ part against Protagoras’ conception of the virtues. What it quite obviously isn’t is a straightforward insistence, parallel to the three quotations that I gave in the second section, that Protagoras cannot define virtue simply by enumerating examples of virtue. Such an insistence is what we might expect, if we come to the Protagoras fresh from the Euthyphro and the Laches – and the opening of the Theaetetus. It is, after all, precisely how Socrates rebuffs Meno’s attempt to offer him a list of the virtues (Meno 73e-74a): Socrates. Virtue, Meno, or a virtue? Meno. What do you mean?

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Socrates. I mean exactly what I would mean in any other case. A circle, for example, is a shape and not simply shape: that’s what I should say, because there are other shapes. Meno. Quite right. And that is exactly what I am saying about virtue – that there are other virtues as well as justice. Socrates. What are they? Tell me the names of them, as I would tell you the names of the other shapes if you asked me to. Meno. Well, I think courage, temperance, wisdom, and magnificence are virtues; and there are many others. Socrates. Yes, Meno; and now we are back in the same place as before, though this time by a different route. We’ve been searching for one virtue, and we’ve found many; but the general notion of virtue that runs through them all – that we cannot find. To make this move, of rejecting the listing of examples, against what Protagoras says would merely beg the question against him. For – though I freely admit that this is not fully spelled out – the whole point of Protagoras’ conception of the virtues seems to be that he thinks he can give a definition, a logos, of virtue, precisely by enumerating the most important examples of virtue and saying how they interrelate. What he offers is an account of the particular virtues as complementary parts that together constitute an anhomoiomerous whole. It follows that we can say what virtue is just as we can say what a face is: by enumerating its different, anhomoiomerous parts, and showing how those different parts fit together to constitute a whole which is also different from any of its parts. So, on Protagoras’ account, the enumeration of the parts of overall virtue – the listing of examples of individual virtues – is at least part of what it takes to define virtue, just as the listing of the parts of the face is part of what it takes to give a full account of the face. Perhaps it is even the main part of defining virtue. After all, while specifying the spatial interrelations of the parts of a face is something clearly distinct from specifying the parts of the face, it definitely looks like a smaller and simpler task. Can’t Socrates protest, along the lines of the aporetic dialogues, that nothing in Protagoras’ account tells us what it is to be virtue? The analogy of the face and its parts brings out that this question is, on Protagoras’ account, ambiguous. For Protagoras – unlike Socrates – there is one question about what it is to be virtue overall (cp.: a face), and another about what it is to be a particular virtue (cp.: a face-part). Taken the first way – as ‘Can’t Socrates protest that nothing in Protagoras’ account tells us what it is to be virtue overall?’ – the answer is simple. No he can’t, because Protagoras has already addressed that question at considerable length, in the mythos of 320d-323d. His answer to it is zoological naturalism: Protagoras holds that ‘overall virtue’ is, simply, the conjunction of whatever qualities or dispositions members of the human species need to have in order to flourish in their natural environment. Taken the second way, Socrates’ protest is that nothing in Protagoras’ account tells us what it is for any particular trait to be a virtue. But to this, Protagoras’ reply

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need be no more than that the particular virtues are, by definition, just those traits that together make up overall virtue, just as the particular parts of the face are, by definition, just those parts that together make up the overall face. For Protagoras the parts of virtue can be as different from each other – as anhomoiomerous – as you like; the only thing that must unify them is that they need to be genuine members of the list of examples of particular virtues that go together to make up the face. (So, by implication, they must be compatible. But to say that the virtues are (ideally) compatible is to come a long way short of any classic version of the thesis of the ‘unity of the virtues’. In the face of Socrates’ critique, Protagoras can and should stick to his guns about the difference of courage from the other virtues; indeed he should be prepared to go further still and at least consider the possibility that other virtues too are or could be different from the other virtues in something like the way that courage is.) I suggest then that the Protagoras deserves attention as an important staging-post in Plato’s rethinking of Socrates’ opposition to definition by examples. It is the first place where we see Plato take seriously the possibility that is so important by the time of the Philebus, Theaetetus and Sophist: the possibility that a good definition of a complex will be an orderly statement of how its parts constitute it as a whole, and conversely, that a definition of a simple will be an account that says what kind of simple it is and how to classify it. The Phaedrus (265d-266a), the Cratylus (389d) and the Theaetetus (202b6) all make it clear that there are right and wrong ways to do both the collection and the division parts of this task of definition. So, and most emphatically of all, does the Sophist (253b8-d3): EV. If someone is to give a correct demonstration of which of the kinds accord with which other kinds, and which do not, mustn’t that person have a sort of knowledge of how to go through the logoi? And [likewise, if he is to show] whether there are some [kinds] that persist through all things, in order that they can mix together, and again [some other kinds, found in] the separations of things, these other [kinds] being the universal causes of separation? Tht. How could he not need science – maybe almost the greatest science? EV. . . . Well, by Zeus! Have we stumbled without realising it on the science of free men [cp. Tht 172d2]? Have we perchance found the philosopher first, when we were looking for the sophist? . . . The ability to divide by kinds, and avoid the mistake of confusing one Form with another, or the other with the first – shall we not say that this ability belongs to the science of dialectic? Making the correct collections and divisions, analyses and syntheses, is itself a matter of skill – or as we may also say, a matter of knowing how. As we shall see before the end of this chapter, it is not an insignificant fact about Theaetetus’ first attempt to define knowledge by examples, that all the examples of knowledge that he gives there are, in fact, examples of knowledge how.

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IV The interpretive proposal that I have been pursuing is that for Plato definition, and so also knowledge, consists both in a ‘downward’ movement of analysis, dividing our way down towards the simples or elements, and also in an ‘upward’ movement of synthesis, combining our way from those simples up to the complexes that they compose. A thesis closely associated with this proposal is that the reality that we know is also like this. And Plato accepts this thesis too. For him, the structure of reality is isomorphic with the structure of our understanding of it: ‘the elements, so to speak’, are what ‘we and the other things are composed of ’ (201e); our logos, our explanatory discourse, about reality is a symploke eidwn because reality itself is a symploke eidwn. We might almost say that the mature Plato is the last and the greatest of the pre-Socratics. From his middle period on, Plato reverses Socrates’ ‘ethical turn’. He repudiates his master Socrates’ resolution to look only into moral and moral– psychological questions and recuperates the study of the entire range of questions, not only ethical but physical too, that all Greek philosophers before Socrates had always taken to be their concern. As much as the Milesians, the mature Plato’s basic questions are: ‘What exists?’, ‘What things are there, and what are those things made of?’. He is aware of other answers to this question, such as Democritus’ atomism and the ‘twin theory’ of ‘the Heracleiteans’, whoever they were, that he describes and criticizes in Part One of the Theaetetus, and like the monistic idealism of Parmenides and Zeno that he confronts in the Parmenides and the Sophist. Like the pre-Socratics, Plato is looking for the elements of the world, the ultimately simple constituents of everything else. But for Plato the elements of the world are not physical atoms like Democritus’, nor sense-data like the Heracleiteans’; nor is ‘the world’ for him what it is for Parmenides, a shadowy illusion behind which there hides the only true reality there is, the undifferentiated One. For Plato, the elements of the world are the Forms. So the story that we need to tell if we are to explain the world’s complexity on the basis of those simples is the story of how the Forms combine to constitute reality (and somehow do this without each of them losing its own elemental simplicity; how that can be, is one of the major questions of the Parmenides). In the Timaeus, this story generates a cosmogony and the fundamentals of a physical science; in the Sophist, it generates a general metaphysics that is capable, among other things, of showing the errors both in the Parmenidean metaphysics of absolute monism and in the atomism or quasi-atomism of Socrates’ Dream. The heart of the Sophist’s metaphysics is the ‘five greatest kinds’; it is by their interweaving that everything else is constituted, including all the other Forms, and then, by successively more complex interweaving of these and the five greatest kinds, everything else as well. Now an obvious problem that the pre-Socratics’ various fundamental theories of reality had all had to confront was the problem of spatiality. If, for example, Democritus’ atoms are only finitely small, then there is a puzzle about why we should call them a-toma, indivisibles. If on the other hand the atoms are infinitely

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small, then there is a problem about how they can ever compose anything that is not infinitely small. How can any accumulation of infinitely small millet-seeds ever itself become larger than an infinitely small accumulation? There is also a problem about how they can ‘interlock’, as Democritus crucially claims they do in order to form complexes. To interlock, the atoms need to have shapes. But to have shapes, they need to be larger than infinitely small. It is the problem of spatiality, as it arises for the theory of Forms, that Parmenides is pressing when he urges on Socrates the ‘sailcloth’ objection of Parmenides 131b. A Form is a unity, a simple. Yet it is supposed to be present – somehow – in infinitely many particular things. How can this unity be present at once in infinitely many different places and still be a unity? This is a problem about universals that modern metaphysicians too often profess not to be able to answer; when, indeed, they make it part of the definition of a universal, as they often do, that a universal is something that can present in lots of places at once yet still be a unity, they are not so much facing up to this problem as stipulating the persistence of the problem. The theory of Forms will indeed face difficulties – as the dialectic of the Parmenides shows – if the Platonist simply accepts the sailcloth metaphor as a way to think about Forms and their instantiation.12 But Plato has a better metaphor to help us visualize the kind of relation that he has in mind between Forms and their instances, and this metaphor has perhaps not been taken as seriously by scholars as it deserves to be.13 Confronted by Parmenideans with the question how it is possible for the whole of a Form to be present in more than one place, Plato’s best answer is not the sailcloth metaphor of Parmenides 131b but the interweaving metaphor of Sophist 259e, Tht 202b6 and Cratylus 388-389.14 The idea is to think of particular things as symplokai, interweavings, the Forms as the threads that are thus interwoven.15 Then the question whether a thing X has a property Fness is the question whether the Fness thread is present in the interweaving that composes X. And the key difference between the interweaving and sailcloth metaphors lies in this: that the sailcloth is a finite area that is divided into different parts, whereas the thread of the interweaving is – or at any rate Plato wants us to think of it as – an indefinite resource, any instance of which is qualitatively identical with any other. When we say ‘Weave the scarlet thread into this tapestry’, it is as if we are asking for a quality to be added to the tapestry, not for a subpart of the length of a continuous coloured string to be inserted into it. The interwoven threads have instances; but they do not have parts. Of course, this is precisely not the case with a literal weaving. At that point – an opponent might say ‘that crucial point’ – Plato’s metaphor runs out. But that it is this thread metaphor he is thinking of, and that he wants us to extend it at this point to enable us to visualize the case where we are talking about an indefinite resource of a pure quality, a bobbin that never runs out, seems to me indisputable. (If you want a bad joke at this point, feel free to describe Plato as the original inventor of string theory.) The Forms as Plato conceives them are simple qualities of no physical quantity and no physical location. The same one can show up anywhere, and in arbitrarily large or small quantities, without any one instance, however ‘big’ an instance it may be, in any way draining ‘the reservoir of being’ (see Symp 210d) from which it takes its source; and in any such instance it remains one and the same simple

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Form. The Forms are the fabric of the universe; yet they do not come in bales. No amount of use of that fabric can ever deplete the stock of it, as a dressmaker making a skirt might need more ocean-blue taffeta, or a seamstress frilling a petticoat-hem might need more lace trim. The Forms, like Yeats’s Mother Eire, are always young. To repeat the key point: the interwoven threads have instances, but they do not have parts. And why on earth should we believe any such theory? Plato’s answer to that, I think – though this too is speculation and detective-work on my part – is ultimately just experience. As a matter of experience, we do find ourselves predicating properties of things in a way that (he thinks) presupposes just his picture of an inexhaustible supply of thread always available to be so predicated. His picture, too, is related in what he takes to be the right way to our experience and our understanding of expertise, scientific knowledge and the analysis and synthesis that this involves: He who is able to [perform divisions accurately] has an adequate sense of how a single idea is stretched right through many things, each of them lying apart from the others; [he sees too how] many ideas different from each other can be encompassed from the outside in a single idea, and again [how] a single idea can be composed by the twining together as a whole made of many ideas, and how many ideas are separated from all the other ideas at every point. (Sophist 253d-e) One of various things that the interweaving metaphor is intended to show, as the sailcloth metaphor cannot, is how it can be the same Form that is present in indefinitely many different things, even when those things are widely disjoint from each other in space. Intuitively there is a familiar sense in which both in a scarlet ball-gown in Dundee or Durham and in a scarlet ball-gown in Dushanbe or Durban, one and the same scarlet thread can be present. It is in this sense of ‘one and the same’ that one and the same Form can be interwoven into indefinitely many spatially separated things, without this plurality of instantiation in the least undermining that Form’s unity. It is not even a weakness of the interweaving metaphor that the Forms stand in relations to each other: even a thread is a weaving-together of thinner threads. For Plato there are absolute simples, because there are, he thinks, minimally thin threads. There are exactly five of them, and these five occur in everything: as I’ve already suggested, he calls them ta megista gene. It is in this sense, then, that the Forms are supposed by Plato to be simples; and yet in a way that he thinks renders them invulnerable to the problems of spatiality and unity. Of course, a determined opponent of Platonism – and in particular, a physicalist such as an atomist – might press Plato harder at this point. Plato’s metaphor says that Forms are threads, and particular things are interweavings of these threads. But particular things occupy space, and Forms (apparently) don’t. How then can any combination of the non-spatial result in the spatial? The difficulty looks analogous in one way to the atomists’ sorites problem earlier, in a second to a problem faced by some forms of modern ‘bundle’ or ‘trope’ metaphysics and in a third way to one of the main difficulties facing Empedocles’ conception of the rhizwmata pantwn (a conception to which Plato may well be indebted). Plato’s answer to this difficulty depends, I think, on two moves in particular: one is his

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famous equation of reality with causal potency at Sophist 247e, and the other is the notion of matter as ‘receptacle’ that is so central to the argument of the Timaeus. Enough of this for now. My main purpose in addressing these issues is just to bring out the extent to which that metaphysics is, as I said at the beginning of this section, just as much an essay peri tou ontos as the work of pre-Socratic ‘natural philosophers’ such as Empedocles or Parmenides or Anaxagoras – or Democritus; and how, for Plato as much as for them, it is a central part of his inquiry to identify simples and complexes, to ask what the universe is fundamentally made of and how its fundamental constituents combine to give us the reality that we know.

V I want now, in conclusion, to build on these findings by asking how the inquiries of Part Three of the Theaetetus fit into the picture that I have been building up of Plato’s overall philosophical project. I focus on Part Three, first because there is just too much to say about Parts One and Two for me to be able to say it at all well in this short chapter, and second because I have already written at length about those parts of the dialogue elsewhere. But very briefly, both Parts One and Two do seem to be concerned with the relations of the simple and the complex, in something like the way I believe those relations are central to Part Three. Part One explores a metaphysics antithetical to Plato’s own, based on a quite different kind of simple: the simple percept/perception pairs of the Twin Theory, a theory which if I understand it aright shows some striking similarities to Russell’s ‘neutral monism’. In his exploration of that theory, Plato is concerned to show not only its unfeasibility but also the centrality to it of questions about the simplicity or complexity of the psyche. Plato had long argued (cp. e.g. Phaedo 78c) that the soul was incomposite and simple; and it is a central concern of the argument of Tht 184c-186d to insist on this simplicity, as the condition, now, for the existence of any kind of unity of consciousness. Yet how can the soul be simple when so many different thoughts – and experiences – are possible for it, and when it seems – as the Phaedrus and Republic both acknowledge – to have parts? Those questions about the soul become central towards the end of Part One of the Tht, especially, as I say, in the final argument of 184-7. They remain central in Part Two, where Plato’s focus is on the question of the complexity or simplicity of the objects of belief, especially when that belief is false. The simplicity of the objects of false belief considered in Part Two is of a kind that I described in Chappell 2001 as a ‘pebble theory’ of judgement: the problem about false belief arises because of a problematic account of belief, as a simple grasping of a simple object, and the puzzle that Plato is trying to resolve is how to devise a better account of belief. (The ‘pebble theory’ of judgement is no doubt one of the main sources of the difficulties that drive Meno’s paradox, too (Meno 80d); it is interesting and important to read Part Two of the Theaetetus in the light of that Menonian background.) One main upshot of Parts One and Two’s explorations must indeed be, as I argued in Chappell 2005, the incoherence and inadequacy of any non-Platonist metaphysics and epistemology – the unstated lesson being that we should prefer a Platonist approach, of the kind

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that Plato immediately goes on to flesh out in the Sophist. (I hope that this present essay does something to show how the Sophist’s approach is well understood as a ‘Platonist approach’.) But there are other upshots too, and they come out most clearly of all in Part Three of the Theaetetus, to which I now turn. In Part Three, we are offered three pictures of what it might mean for knowledge to be ‘true belief with an account’. The first is Socrates’ Dream (201e-206e), the key feature of which is that the elements are objects of perception, their complex combinations objects of knowledge. The second (which like the third, is presented as an interpretation specifically of the logos part of ‘true belief with an account’) is ‘going through the elements to the whole with true belief ’ (206e-208b): the key feature of this is that it is true belief with which one goes through the elements, not any more exalted epistemic state. And the third is being able to add, to one’s true belief about something, a logos consisting in a specification of what it is that distinguishes that thing from everything else (208c-210b); that is, a feature that locates the thing at a terminal point in the division that leads us from complexes to simples. One thing to note about this discussion is the way in which it reflexively relates to the Theaetetus as a whole. In the Theaetetus overall, we discuss first the proposal that knowledge is perception, then that knowledge is true belief and then that knowledge is true belief with an account. Likewise here in Part Three of the Theaetetus we discuss first a version of the proposal that ‘knowledge is true belief with an account’ – the Dream Theory – in which knowledge depends upon and presupposes perception; then a version of that proposal – featuring the logos as the way to the whole through its parts – in which knowledge depends upon and presupposes true belief; and finally a version of it – featuring the logos as ‘the statement of the difference’ – in which (as is shown by the circularity that this version leads to) knowledge depends upon and presupposes true belief with an account. Consider too the ways in which these three versions of the proposal are refuted. The last version is condemned for its circularity (209d-e), or rather its regressiveness: to have knowledge is to have true belief plus an account, but then we need knowledge of the account. This is a condemnation that signals a kind of ring-composition in the dialogue, since it should immediately remind us of Socrates’ condemnation of Theaetetus’ very first attempt at defining knowledge, right at the other end of the dialogue (147c-d); where we were to get a definition of knowledge by listing examples of knowledge, for each of which we then need a further definition of knowledge. The second version is condemned in a way strongly reminiscent of the Juryman argument of 201a-c (and behind that of the Road to Larisa passage in the Meno, 97a ff.), for failing to exclude cases where someone can state ‘the way to the whole through the elements’ with true belief, and yet clearly not possess knowledge – as when someone spells a name right, but accidentally (207d-208b). As for the first version of ‘knowledge is true belief with an account’, Socrates’ Dream, this is condemned, in 202d-206e, for three main reasons. First, no amount of perception can ever add up to knowledge: the gap between them is a categorical one (203d). Second, when a whole is understood to be complex, it is known at one and the same time as a unity and as a collection of parts. But then, by the

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Dream Theory’s own assumptions, the same thing will be at one and the same time the object both of perception (qua unity) and of knowledge (qua collection of parts). This contradicts Socrates’ Dream, which posits that simples are objects only of perception, complexes objects only of knowledge (204a-205e).16 Third, it is a matter of familiar experience that when we learn, say, writing or music, we learn to know – not merely perceive – simples like letters and notes first. Equally familiarly, it is only because of our prior and better knowledge – not mere perception – of these simples that we ever attain, beyond them, to any knowledge of complexes at all (206a-c). So the moral of the discussion of Part Three is that what we need in order to have an account of knowledge is not true belief about the object plus true belief about the account of its differentia, but knowledge of it plus knowledge about the account of its differentia; and not ‘a way to the whole through the elements’ with true belief, but ‘a way to the whole through the elements’ with knowledge; and not a knowledge of complexes that is based on perception of simples, but a knowledge of complexes that is based on knowledge of simples. Across the board, knowledge has to rest, not on perception, or true belief, or true belief with an account or anything else, but on knowledge. There is just no recipe whereby we can cook up knowledge out of ingredients like those. No doubt such simple ingredients do or can form complexes of various sorts – but none of those complexes is knowledge. Knowledge is, in fact, itself a kind of unanalysable simple. As Timothy Williamson (Knowledge and its Limits Ch.1) would put a well-known thesis of his that is in important ways somewhat similar: knowledge is prime. Or rather – as Williamson would certainly not say – knowledge is for Plato more than one unanalysable simple, in fact at least three. There is the knowledge which is knowledge of complexes, and there is the knowledge which is knowledge of simples and there is also the knowledge of how to get by analysis from complexes to simples and by synthesis back again from simples to complexes. (Self-referential sidebar: How does this trichotomy of kinds of knowledge map onto the tetrachotomy of knowledge that I have argued for elsewhere (see Chappell 2013, 2014)? The answer is that knowledge how is the same in both analysis and synthesis, while objectual and propositional knowledge are both exemplified both by knowledge of simples and of complexes. (And, pace Fine 1979 and McDowell 1973, 114–16 and Bostock REF, Plato does take objectual knowledge, not propositional, to be primary, and is not blundering either in that, nor in moving ‘Knowing X’ > ‘Knowing X, what it is’ > ‘Knowing that X is F’, nor in taking the three steps of this movement to be systematically and logically connected: cp. Chappell (2013).) Experiential knowledge, however, seems on the evidence of most of the Theaetetus, as I am now reading it, not to be counted by Plato as knowledge at all. Apparently, he would now call it perception, and while he grants that it is a source for knowledge, he would staunchly resist the idea that it is knowledge. The crucial evidence here is 184c-186d, arguing that knowledge cannot be in the senses; so perception is never sufficient for knowledge. Contrast the juryman argument of 201b-c, which apparently concludes that perception is often necessary for knowledge. End of sidebar.)

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Now notice what has happened here. Plato’s question in the Theaetetus is ‘What is knowledge?’; and by the end of the dialogue, if I am right, his answer is ‘Knowledge is knowledge of complexes, and knowledge is knowledge of simples, and knowledge is knowledge of how to analyse complexes into simples, and how to synthesise simples into complexes’. This definition of knowledge is (a) circular and (b) a definition by examples. It does not give a formula analysing the essence of knowledge in the terms of some deeper range of essences; instead, it lists three varieties of knowledge. By Theaetetus 210b, where Socrates says that their last attempt to define knowledge has turned out ‘completely naïve’, pantapasi euethes, the discussion – or so we might say – has got no further than it already had by Theaetetus 147b, where Socrates says that a list of examples of knowledge is a ‘laughable response’, a geloia apokrisis, to someone who wants to know what knowledge is. Or so we might say. But actually – I want to close by suggesting – this circularity, and this listing of examples of knowledge, is not the defeat of Plato’s project in the Theaetetus, but exactly where it has been aiming all along. Plato’s list of these three examples of knowledge is not meant as a mere list of examples. Rather, it is meant to be an ordered catalogue, a dia stoicheiou diexodos (207c), of the three simple kinds of knowledge that together constitute the complex reality that is knowledge overall. To know each of these three simple kinds of knowledge is to know the elements of knowledge; to know the complex thing that is their conjunction and to know how to combine the three simple kinds to form this conjunction, is to know knowledge overall. If our aim is to answer the question ‘What is knowledge?’, there is no conception or definition of knowledge that can tell us any more than this account can.17 Now despite everything else that’s going on in the Theaetetus, it is Plato’s aim that the dialogue should answer ‘What is knowledge?’. And when he arrives at the threefold account of knowledge that I have just described, his work in answering that question is complete.

NOTES 1. For encouragement and for comments on this chapter, which is nonetheless not their fault, I am grateful to my Open University colleagues and to audience members at a conference on atomism in ancient and contemporary philosophy in Durham in May 2017; in particular to Alex Barber, Sarah Broadie, Amber Carpenter, Sean Cordell, Jakub Jirsa, MM McCabe, Lucy Fay Manning, Carolyn Price, Christopher Rowe, Ugo Zilioli and Nicolas Zucchetti. 2. Or, as for tidiness’ sake I shall say from here on, Plato. I take it that the views about definition that I am describing were Socrates’ first, but they were also Plato’s later. And my concern in this chapter is really with Plato. 3. For a fourth, see Laches 191e: courage as a virtue sometimes counsels retreat as at other times it counsels standing firm. So it is no good to say that courage just is standing firm. 4. All the Plato translations are my own. The Theaetetus ones come from Chappell 2005 and the Meno ones from Chappell 2017, with occasional revisions.

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5. Philosophical Investigations I, 218. I take it Wittgenstein’s puzzle is, in the first instance, simply about the idea of infinite projectibility; it is not primarily about the more arcane possibility of a ‘bent’ projection that exercises Kripkenstein, though certainly that concern arises too. 6. Though it is picked up by, for example, Sedley, MWP, 66. 7. So Ruby Blondell, The Play of Character in Plato’s Dialogues, 317: Tht, Soph, Stm take place in a ‘gymnasium’ or ‘palaestra’ of uncertain location, but apparently in Athens. And C. J. Rowe, Theaetetus and Sophist, 3 fn: ‘We are evidently to picture the conversation taking place in a gymnasium.’ And Loeb, ad 144c4: ‘The scene is evidently laid in a gymnasium.’ Levett in Burnyeat, 261: ‘The scene of the dialogue itself is a gymnasium or wrestling school in Athens.’ And Lesley Brown in Waterfield, 112: ‘[“rubbing themselves with oil”] indicates that the setting is a gymnasium.’ 8. See http:​/​/www​​.cafe​​babel​​.co​.u​​k​/ath​​ens​/a​​rticl​​e​/a​-n​​ew​-ar​​chaeo​​logic​​al​-si​​te​-fo​​r​-ath​​ens​-t​​ he​-l​y​​ceum-​​of​-ar​​istot​​le​.ht​​ml (accessed 20 April 2017). 9. Sedley 2004 is the definitive study of the Theaetetus as an exercise in this sort of retro-ness. 10. ‘ “A name signifies only what is an element of reality. What cannot be destroyed; what remains the same in all changes.” – But what is that? – Why, it swam before our minds as we said the sentence! This was the very expression of a quite particular image: of a particular picture which we want to use. For certainly experience does not shew us these elements. We see component parts of something composite (of a chair, for instance). We say that the back is part of the chair, but is in turn itself composed of several bits of wood; while a leg is a simple component part. We also see a whole which changes (is destroyed) while its component parts remain unchanged. These are the materials from which we construct that picture of reality.’ 11. With, of course, the qualification that in the case of the periodic table we can ‘state a definition of the elements of any element’: that’s what atomic physics does. Here as elsewhere, as I point out in the main text, it looks right to say that at least some simplicity/complexity contrasts are inquiry-relative. 12. In the next couple of paragraphs, I recapitulate some of the argument of my ‘Making Sense of the Sophist’, Symposium Platonicum Pragense 2011. 13. Though occasionally they also take it too seriously: Ryle claims in ‘Plato’s Parmenides’, Mind 1939, that in the Sophist talk of methexis is completely replaced by talk of koinônia. And this is just untrue: see, for example, Sophist 255e6. 14. In responding to this challenge, Plato also frequently resorts to the metaphor of letters – Sophist 253a, Tht 202-3, Stm 285c-d. This metaphor might seem to face the objection that it takes for granted what it is supposed to explain, the type-token or Form-particular distinction. That objection seems misplaced to me: what the example of letters shows is rather that there are familiar cases where we are very happy to work with something like the Form-particular distinction, hence that this distinction is not as mysterious or puzzling as Parmenideans will want to suggest. 15. And in parallel to this, think of names too as interweavings of Forms (Cratylus 388b13-c1): ‘A name, then, is a kind of instructive instrument, which separates being as a shuttle separates a web.’

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16. A sufficiently alert defender of the Dream Theory could meet this objection by invoking the inquiry-relativity of simplicity and complexity, and saying that the very same thing can be simple-relative-to-Inquiry-A, and therefore a (possible) object of perception for Inquiry A, while being complex-relative-to-inquiry-B, and so a (possible) object of knowledge for inquiry B. Perhaps this move is in Aristotle, but it isn’t in the Theaetetus. 17. As both Ugo Zilioli and Amber Carpenter have independently suggested to me, perhaps something similar happens to the account of pleasure in the Philebus: perhaps Socrates there ends up accepting a list of pleasures as a possible definition of pleasure. See Phb 12c4-8 (tr. Zilioli): ‘as to pleasure, I know that it is complex and, just as I said, we must make it our starting point and consider carefully what sort of nature it has. If one just goes by the name it is one single thing, but in fact it comes in many forms that are in some way even quite unlike each other.’

REFERENCES Blondell, R. (2002), The Play of Character in Plato’s Dialogues, Cambridge: Cambridge University Press. Bostock, D. (1988), Plato’s Theaetetus, Oxford: Oxford University Press. Burnyeat, M. (1990), The Theaetetus of Plato, Indianapolis: Hackett. Chappell, T. (2001), ‘The puzzle about the puzzle of Theaetetus 187a’, Bulletin of the Institute of Classical Studies. Chappell, T. (2005), Reading Plato’s Theaetetus, Indianapolis: Hackett. Chappell, T. (2011), ‘Making sense of the Sophist’, Symposium Platonicum Pragense, ed. M. Szlesak. Chappell, T. (2013), ‘Varieties of knowledge in Plato and Aristotle’, Topoi. Chappell, T. (2014), Knowing What To Do, chapter 11, Oxford: Oxford University Press. Chappell, Sophie Grace (2017), Plato’s Meno, Chicago: Open University Press. Fine, G. (1979), ‘False belief in the Theaetetus’, Phronesis 24, no. 1: 70–80. McDowell, J. (1973), Plato’s Theaetetus, Clarendon: Oxford. MM McCabe (1999), Plato’s Individuals, Cambridge, MA: Princeton University Press. Rowe, C. J. (2015), Theaetetus and Sophist. Cambridge Texts in the History of Philosophy, Cambridge: Cambridge University Press. Ryle, G. (1939), ‘Plato’s Parmenides’, Mind, 302–25. Sedley, D. (2004), The Midwife of Platonism: Text and Subtext in Plato’s Theaetetus, Oxford: Oxford University Press. Waterfield, R. (1987), Plato’s Theaetetus, London: Penguin Classics. Williamson, T. (2003), Knowledge and Its Limits, Oxford: Oxford University Press. Wittgenstein, L. (1951), Philosophical Investigations, Oxford: Blackwell’s. Wittgenstein, L. (1958), The Blue and Brown Books, Oxford: Blackwell’s.

CHAPTER 7

Atomism in Plato’s Timaeus LUCA PITTELOUD

Aristoxenus of Tarentum, in a rather satirical anecdote, suggests that Plato wanted to collect the writings of Democritus and have them burned before he was stopped by two Pythagoreans who argued that this attempt would be futile since copies of the books were already widely available (DL, IX, 40). This story seems to point out a certain jealousy of Plato towards Democritus, a philosopher he never quotes in his dialogues. However, in the Timaeus, Plato describes how the corporeal entities of our universe are constituted by the combination of invisible basic particles. The resemblance between Plato’s Timaeus and Leucippean–Democritean atomism was pointed out by Aristotle and seems to be based on the fact that both theories investigate the ultimate stoicheia from which the universe is composed.1 More generally, it seems plausible that Plato’s Timaeus should be understood as a critical dialogue with the physiologoi, especially Anaxagoras, Empedocles and the Atomists.2 Two main characteristics must be kept in mind when comparing Plato’s Timaeus with Presocratic cosmologies: first, in Timaeus’ eikôs mythos, a divine craftsman, the Demiurge, looking an intelligible model, builds the universe by imposing order to a pre-existing chaotic milieu. Second, two complementary dimensions must be considered in order to provide a complete explanation of what the cosmos is, namely intellect (nous) and necessity (ananke). Timaeus’ description of the constitution of the four elements in 53b-61c (fire, air, water, earth – FAWE) can only make sense if it is understood within the context of the whole dialogue in which the Demiurge persuades an unordered necessity by means of his intellect.3 In this context, Plato will introduce a geometric atomism that aims to describe how the elements FAWE are themselves constituted by more basic elements which he identifies with two basic triangles. In this way, both Plato and Democritus are committed to the idea that, as Vlastos notices, ‘the unobservables we postulate to account for properties of observables need not themselves possess those same properties’.4 In other words, neither philosopher commits the ‘fallacy of division’, that is, attributing to the parts the properties of the whole, at least without providing any justification. However, unlike Democritus’ atoms which are indivisible solids possessing an infinite multiformity, Plato’s geometrical atoms consist of planes limited in number.5 But before getting there, some contextualization has to be

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provided in order to be able to understand what kind of atomism Plato introduces in the Timaeus. I will divide my investigation into two parts: first, in order to understand how a geometric atomistic view of reality emerges in the Timaeus, a close look will be given to the structure of Plato’s story about the constitution of the universe. More precisely, the narration of the myth will be distinguished from its doctrinal content. Second, a description of Plato’s geometric atomism will be offered by considering its specific context. To conclude, following the results of the two first parts, I will briefly address three puzzles: the origin of motion, the nature of the Receptacle and the relationship between Plato’s geometric atomism and the Theory of Forms as it appears in the Timaeus.

PRINCIPLES AND EVENTS IN THE TIMAEUS A myth and its principles Plato’s geometrical atomism appears in the second part of the Timaeus (53b-61c): the first part describes the works of intellect (29d-47e), the second the action of necessity (47e-69a) and the third the cooperation between intellect and necessity (69a-92c). These three parts are preceded by the introduction of premises on which all the cosmological discourse is founded (27d-29d).6 In this context, a divine Demiurge (29d-30c), looking at an intelligible Model (30c-d), will constitute the cosmos. Plato qualifies the discourse he is offering as a probable account (eikôs mythos7). Although this interpretation has been criticized,8 Timaeus seems to suggest that a discourse which will concern the universe and its origin can only reach the level of likelihood: this is a consequence of considering the cosmos an image of an intelligible Model. A discourse about the intelligible Model can be true, whereas an account about its sensible image will be, at most, plausible (29c-d). Consequently, Timaeus’ story should not be read as a scientific account of the coming to be of the universe, but as the best possible discourse considering the human nature of the narrator (29d1). Furthermore, Timaeus, from 29d7, qualifies his discourse as a myth about the fashioning of the universe. This implies (i) a specific temporality (from a noncosmos at t1 to a fully constituted universe at t2), (ii) characters (endorsing different roles: the Demiurge (Father 1), the Forms (Father 29), the Receptacle (Mother) and the sensible Universe (Child) (50d)) and (iii) a dramaturgy (the proper constitution of the universe). Should the reader take this story seriously? Was the universe literally fabricated by a divine craftsman? Or should the account be demythologized and translated into a more rational explanation of the ontological constituents of the universe?10 Without either getting into the details of this difficult question, or deciding which alternative must be assumed, the following comment might suffice for my investigation. Plato himself leans towards a certain demythologization: the order of the events described by Timaeus does not actually follow the order of what should be the myth of the creation of the Universe if expressed in a strictly chronological progression. As a matter of fact, Timaeus insists that the story he is telling (i) did not happen in the same order as the one he has chosen (34c) and (ii) necessitates different points of departures (48a-b). That is, due to Timaeus’ and

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the reader’s human limits (34c3), the myth offered is, in fact, already somehow demythologized. The main division of the discourse between (1) the works of intellect (30a2-6) and (2) necessity (47e5-48a7) is actually a good exemplification of Plato’s didactical intention: the story must describe the origin of the universe from two points of view which are not chronologically but ontologically differentiated. It is directly after the second point of departure (2) that the description of the constitution of the four elements (FAWE) starts. However, right after the first beginning (1), Timaeus has already described the fabrication of the World Body by the Demiurge (31b-34a). In order to understand better the complexity in which Plato’s geometric atomism appears, it might be relevant to distinguish between the cosmological principles (P) on which the whole discourse is built and those which are introduced during the course of the myth and the events (E) of the constitution of the universe in their chronological order. Let’s start with a list of the principles (P) supposed by Timaeus:11 P1: The Universe is the combination of intellect (1) and necessity (2). P2: The constituents (also called the three kinds) of the Universe are: (a) the intelligible Model, (b) the Receptacle and (c) the sensible being, itself an image of the Model (48e-49a). P3: The Receptacle is a neutral milieu in which the images (mimêmata 50c7) of the Forms of the Four Elements appear. P4: The Receptacle possesses both a spatial (in which) and a constitutive (of which) dimension.12 P5: Since (i) the cosmos must possess intellect, (ii) the place of intellect is the soul and (ii) for sensible entities a soul is united with a body, then the cosmos will be a living animal constituted by a body and a soul (30a-b).13 P6: The Model of the cosmos is the Form of the Living Creature, a Form which is all-inclusive (which possesses all the intelligible species) (30c-d).14 P7: The cosmos is both an image (reflection) of the intelligible Model and a demiurgic fabrication.15 P8: The cosmos is the best possible realization.16 These are the principles which do not depend on any chronological ordering, and they justify why the universe is the way it is. However, Timaeus’ discourse exposes the coming to be of the cosmos in a certain order. This order is not completely chronological, for it is based on the two points of view: intellect (1) and necessity (2). It might be useful to try to reconstruct the chronology of the universe,17 according to Plato’s myth.

The chronological events The first event  The demiurge does not create the universe from nothing. He organizes all that is visible and finds a pre-existing disordered milieu (30a2-6), which implies that something already existed before the demiurgic work. According to P2, P3 and P4, the disordered pre-cosmic state of the universe implies the existence of

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the Model (intelligible Forms), the Receptacle (milieu) and the images (mimêmata) of the Forms (becoming), even before the Heaven came into being (52d4). E1: Before the beginning of the constitution of the ordered cosmos, there was a chaotic pre-cosmic state of affair. This state of the affair is described at 52d-53c and represents a moment of the story when a deity is absent from it (53b3-4).18 In that condition, all that exists, says Timaeus, can be found in a state of disorderedly motion. The text (52d4-53b5), which is rather obscure, depicts how the Receptacle was before the Demiurge’s intervention and follows a long description of the ‘wandering cause’ (which starts at 47e). This description aims to show what the state of the four elements FAWE was before the generation of the Heaven (48b3-4). This description implies the introduction of a third kind, the Receptacle, in addition to the Model (paradeigma) and the sensible copy (mimêma) of the Model (48e-49a). The new distinction between these three kinds involves a certain difficulty when it comes to refer to the sensible four elements, since they are the appearances of the intelligible Forms of the Four Elements (whose existence is justified in 51b-52a) in the Receptacle: this difficulty (49c7-50b5) concerns the fact that only the Model and the Receptacle possess the permanence which allows them to be designated with the demonstrative ‘this’ (touto).19 The phenomenal appearances of the Forms of the Four Elements are deprived of stability as they are constantly entering and leaving the Receptacle. Consequently, the sensible appearance of fire, for example, in the pre-cosmic chaos, should not be called this fire but ‘suchlike’ (toiouto) fire.20 In this context, the Receptacle is described as a milieu in which the images of the Forms of the Four Elements appear and disappear.21 Timaeus affirms that the Receptacle must be always called the same (50b6-7) and concludes that it (the Receptacle) appears to have different qualities at different times; while the things that pass in and out are to be called copies of the eternal things, impressions taken from them in a strange manner that is hard to express: we will follow it up on another occasion.22 After having introduced two other analogies,23 Timaeus asserts that even though the Receptacle partakes of the intelligible in some obscure way and is very hard to apprehend (51a6-b2),24 it is an essential entity which needs to be supposed in order to guarantee the existence of the sensible as an image of the intelligible (52cd). This fairly long reasoning leads to the description of the pre-cosmic chaos, a description (52d4-53b5) which implies that before the Demiurge initiates his work, the images of four elements already existed and appear in the Receptacle. These images are called traces (ichnê) of the elements. They are not yet configured by the Demiurge and they appear as affections (pathê) with powers (dunameis) that are neither alike nor evenly balanced, without proportion and measure (alogôs kai ametrôs). Nevertheless, says Timaeus, they are visible, although no living creature has been yet constituted and could actually look at them! One might wonder how these forms (morphia) would appear in the Receptacle if someone could look at them? As traces, those pre-elements could either, or not

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yet, or no longer, be images of the elements.25 If it is true that strictly speaking those are traces ‘of the elements, not of the models of the elements’,26 it nevertheless seems that the setting of the pre-cosmic chaos just after the description of the four elements, which are called images (mimêmata) of the Forms of FAWE (especially in 52c), strongly suggests that the traces should be identified with the images of the Forms. Those traces are said to be without proportion and measure (69b5), and when Timaeus returns to talk about them in 69b, he claims that they should not even be called elements (69b7). Furthermore, these traces are to be found in a disorderly motion.27 How can this motion be understood? First, Plato seems to describe a mechanical motion caused by powers that are not evenly balanced, because there is no principle of measure and proportion to be found yet. This causes a shaking of the Receptacle and all its contents (the traces). The Receptacle is said in turn to shake the pre-elements. Later in his discourse, when Timaeus gives a description of motion and rest in the universe, at 57d-58c, he insists on the fact that a necessary condition for motion is inequality: inequality, according to Timaeus, implies heterogeneity, and heterogeneity is itself necessary in order for motion to occurs: no motion could ever to occur in a state of complete homogeneity.28 This description, contrary to what is said at 52d453b5, presupposes that the four elements have already been geometrized by the Demiurge. However, in both cases, the motions described are chaotic and unordered since the World Soul, which is the principle of all ordered motions, has not yet been united with the World Body. In the case of the pre-cosmic chaos, the pre-elements are neither alike nor evenly balanced (52e2): this diversity seems to be somehow what causes the motion of the Receptacle which, in turn, will shake and move the irregular traces. The pre-elements, Timaeus says, due to their similar affections, will be attracted towards each other and occupy different regions. This seems to imply that each of the four kinds of pre-elements (i.e. pre-fire, pre-air, pre-water and pre-earth) will agglutinate and occupy a place in the Receptacle. The reader must thus imagine a chaos, where pre-elements are without proportion or measure (53a8), and yet a kind of pre-order appears since each kind of the pre-element will occupy a different region. The passage at 57d-58c completes this picture: now the elements have been properly constructed by the Demiurge and a spherical shape has been given the cosmos. Still, a chaotic motion will also appear in the case of the geometrized elements: inequality between the shapes given to the elements by the Demiurge will cause a chaotic motion which will be followed by a natural tendency for the like to be grouped with the like (fire with fire, air with air, etc.). This should lead to a state of complete rest by the formation of four homogeneous masses, as it appeared in 52d4-53b5.29 Consequently, inequality is a necessary but not sufficient condition for motion for a cosmos which is not yet ensouled. This is probably why Timaeus adds that a limit must be given to the World Body (58a-c) in order to prevent motion to stop. Let us now examine the next chronological events as they should be reconstructed. The following events  After E1, the demiurgic work begins, based on P5, P6, P7 and P8. His reasoning implies that the cosmos will be one living creature with a body

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and a soul in order to be the best possible realization.30 Let’s qualify briefly the event in a chronological order: E2: The Demiurge constitutes the World Soul Contrary to the order of Timaeus’s discourse, it is clear that the World Soul was fashioned before the World Body since the soul is ontologically prior to the body.31 The World Soul is an intermediary between the intelligible and the sensible made out of sensible and intelligible Being, Sameness and Difference (35a-b).32 The World Soul possesses a mathematical structure (35b-36b) and is responsible for the motions of the planets (36b-d and 37c40d) in the Universe.33 It also possesses a cognitive function (36e-37c) and can consequently access the sensible and the intelligible. E3: The Demiurge geometrizes the traces of the four elements. He constitutes the elements in order to justify their reciprocal transmutation, the different grades of corpuscles, the variety of each elements and the mechanism of sensations (53b-69a). This is what is called ‘Plato’s atomism’ and will be described in the next section. E4: The Demiurge constitutes the World Body (31b-34a). From the four geometrized elements, the Demiurge fashions the World Body. Since (i) what comes to be must be bodily, and so visible and tangible, and nothing can be visible without fire or tangible without earth, (ii) two things alone cannot be satisfactorily united without a third, then, for there must be some bond between them drawing them together. And of all bonds the best is that which makes itself and the terms it connects a unity in the fullest sense; and it is of the nature of a continued geometrical proportion to effect this most perfectly.34 Geometrical proportion implies only three terms (A/B=B/C); however, as the cosmos is a three-dimensional solid, two middle terms are necessary in order to reach more unity (35a6).35 This is Timaeus’ justification of postulating air and water between fire and earth.36 As the World Body contains the whole of all the four primary elements (32c-33b),37 and is self-sufficient and everlasting, it is then the best possible realization.38 Furthermore, it is a sphere, without organs or limbs, rotating on its axis (33b-34a). E5: The Demiurge unites the World Body and the World Soul The union of the Body and Soul of the Universe is described in the following way: And in the centre he set a soul and caused it to extend throughout the whole and further wrapped its body round with soul on the outside; and so he established one world alone, round and revolving in a circle, solitary but able by reason of its excellence to bear itself company, needing no other acquaintance or friend but sufficient to itself.39 E6: The Demiurge instructs the young gods to continue his work and retires.

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When it comes to creating the inhabitants of the cosmos, as the universe must be as complete as possible, the Demiurge decides that it must contains the four following species: the heavenly gods; winged things whose path is in the air; all that dwells in the water; all that goes on foot on the dry land.40 (39e-40). The Demiurge constitutes the heavenly gods and, from the mixture made in order to fashion the World Soul, he constructs the immortal part of human souls, asking the young gods to take care of the rest of the creation (mortal parts of human soul, human bodies and the rest of the species), giving them precise instruction how to do so (41a-d). After that discourse, the Demiurge retires (42e5) and lets his helpers continue the constitution of the cosmos.

TRIANGLES AS ATOMS E3 aims to expound the composition and transmutation of FAWE.41 This will be undertaken through a description of the Demiurge’s geometrization of the traces of the four elements. The account, which assumes that the four elements are not the ultimate and simplest constituents of the World Body, will admit the following premises: (i) FAWE are bodies, (ii) bodies have depth, (iii) depth must be bounded by surface, (iv) every surface, which is rectilinear, is composed of triangles (53c). This implies that the most basic components of FAWE are triangles. Two triangles are chosen: Now all triangles are derived from two, each having one right angle and the other angles acute. Of these triangles, one has on either side the half of a right angle, the division of which is determined by equal sides; the other has unequal parts of a right angle allotted to unequal sides.42 For a reason which is not disclosed, Plato chooses, as the two ultimate types of basic components which constitute the four elements, two kinds of triangles, namely the right-angled isosceles and the right-angled scalene. However, these two kinds of triangle are not the ultimate principles of reality, as Timaeus immediately adds: This we assume as the first beginning of fire and of the other bodies, following the account which combines likelihood with necessity; the principles yet more remote than these are known to Heaven and to such men as Heaven favours.43 The text suggests that the introduction of the two triangles is a sufficient condition (53e8) to explain what is the nature of the most perfect (53e8) four types of body and how their construction from these triangles can explain their transmutation. Timaeus points out that there is only one sort of isosceles triangles, whereas the scalene triangles are of an endless number. The criterion adopted in order to determine which scalene triangle will be chosen is the following: it must be the most beautiful (54a3). Although Timaeus leaves open the possibility that a friend (45a5) could find a better kind of triangle and contradict his theory (54b1-2), he indicates that, for him, the two best triangles are the following: the half-square isosceles and the half-equilateral scalene, which has the greater side triple in square of the lesser (Figure 7.1).44

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FIGURE 7.1:  The two most beautiful triangles.

FIGURE 7.2:  The two plane surfaces (the square and the equilateral triangle) which will constitute the so-called Platonic solids.

From these two basic triangles, the Demiurge will constitute more complexes plane surfaces in order to construct five different solids. It might have been expected that two exemplars of each basic triangle would be sufficient for the construction of a square and an equilateral triangle (as in Figure 7.1); however, the Demiurge will opt for another combination (Figure 7.2): Respectively, four isosceles and six scalene triangles will be put together in order to constitute a square and an equilateral triangle. Those two plane surfaces will then be combined in order to obtain four regular solids: from the square, the demiurge will build the cube, and from the equilateral triangle, he will build the pyramid (tetrahedron), the octahedron, the icosahedron (Figure 7.3). Each of these four solids will be then assigned to a primary body (55d-56c).

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FIGURE 7.3:  The Platonic solids which will be associated with the four elements.

This is, to say the least, a very elegant way to describe the nature of the four elements. As one enters into the details of this account, some puzzles will arise and, as Plato did not develop all the aspects and consequences of his theory, the reader must try to answer the difficulties involved. For the present purpose, I wish to make a few comments. (1) Plato’s attempt to build up the four regular solids by means of putting together two basic triangles might seem like ‘a child puzzle’45 or a ‘Legoland’,46 forcing mathematics into physics (which appeared to have caused Aristotle a great deal of stress (De Caelo 299a6-9 and Physics VI, 1)). The ultimate motive of this account is soon revealed. In 56c-57d and 58c-61c, Plato shows how the transformation of the elements will occur between them according to three premises: (i) because they are made from two different basic triangles, isosceles for earth, scalene for fire, air and water, only transformations between those three later elements will occur; earth cannot be transformed into any other element and reciprocally;47 (ii) the four regular solids (and their respective parts) are not, as such, visible because of their smallness (56c1); however, when a certain number of the solids is aggregated, the masses constituted by them can be seen; (iii) the transformation between the elements can be easily calculated since it depends on the number of basic triangles. Accordingly, we have Fire

4 equilateral triangles

24 scalene triangles

Air

8 equilateral triangles

48 scalene triangles

Water

20 equilateral triangles

120 scalene triangles

Earth

6 squares

24 isosceles triangles

Consequently, the transformation between fire, air and water is established according to the following proportions: 1 particle of fire = 4 equilateral triangles (e.t.) 2 particles of fire = 2 × 4 e.t. = 1 particle of air = 8 e.t. 1 particle of fire + 2 particles of air = 4 e.t. + 2 x 8 e.t. = 1 particle of water = 20 e.t. 2 ½ particles of fire = 2 ½ × 8 e.t. = 1 particle of water = 20 e.t. (2) If for Democritus, atoms are solid particles of minimum size and definite shape (characterized by an infinite variety of figures), Plato’s atomism implies that the

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basic constituents of the World Body can be reduced to four figures which are constituted by two types of triangle. In this way, Plato’s atoms are not three- but two-dimensional entities which can be reduced to two basic shapes. This geometrical atomism was designed to improved or correct Democritus materialism.48 Giving precise structure and configuration to his atoms will allow Plato to support the idea that their respective nature is not due to chance but, on the contrary, is a direct consequence of the work of nous. Intelligibility, given through mathematization, implies here economy and simplicity. (3) Unfortunately, Plato’s account of the two fundamental triangles is incomplete and various problems arise which are not addressed in the text. Indeed, when commenting on the choice of scalene and isosceles triangles, Timaeus affirms that a proper justification of his choice would take too much time (54b1). Still, why does he specifically choose those two triangles as the most beautiful? Within Timaeus’ geometrical account, no specific size is given to the sides of the two basic triangles (only the two proportions for their sides of, respectively 1, 2, 3 and 1, 1, 2). One interesting property of the two basic triangles is that they can be indefinitely subdivided into parts of the same type as themselves (Figure 7.4). With these premises, the reader is left to attempt to answer the following questions: first, why does the Demiurge choose triangles and not the faces of the solids as basic constituents? Second, why those specific kinds of triangles? Third, why building up the faces of the solids with four (for the square) and six (for the equilateral) triangles when two of each of them would have sufficed (see Figure 7.2)?49 Various lines of enquires have been pursued to answer those questions. It must be noted that the choice of the two specific triangles allowing a division ad infinitum into the exact same kind of triangle implies that not only regularity but also symmetry is preserved.50 That is exactly what the presence of the god should assure in the traces (69b5: analoga kai summetra). This of course does not explain why the surfaces of the solids are divided into four and six triangles.51 It has to be observed

FIGURE 7.4:  Division and symmetry of the most beautiful triangles.

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that P7 and P8 seem to imply that in E2, E3 and E4 the same strategy is adopted: looking for the most beautiful bond, allowing for the stable construction made by the Demiurge (either of the World Soul or the World Body or the four elements) to be the best possible realization. Shapes are consequently chosen (circle for the World Soul and triangles – the simplest rectilinear figures – for FAWE), and with the use of numbers (arithmôn), the Demiurge will establish the relevant proportion (arithmetic, harmonic and geometric) in order to maintain unity in his work. Note that, as the relationship between the sensible particulars and the intelligible Forms is that of deficient pluralities (particulars) towards a perfect unity (a Form,), the demiurgic work recreates the same relationship between unequal and plural parts (the triangles) towards unity (from the one found in one particle of an element to the one of the unique universe). In this way, both the Forms and demiurgic mathematics act like two sorts of glue, one metaphysical and one physical. This allows beauty (up to a certain point) to emerge from disordered multiplicity (the one of E1.)

THREE PUZZLES One important result from the geometrization of the four elements is that it allows a theory which explains the transformation of the elements on the basis of the diversity of shapes, combinations and transformations of one body into another (61c). As the universe is spherically limited and the quantity of the four elements is determined (E3),52 the reader is left with a complete picture of how materiality is conceived of in the Timaeus: from basic scalene and isosceles triangles (A) complex surfaces (equilateral triangles and squares) are derived (B), which in turn constitute the elementary polyhedra (C). Those polyhedra are invisible corpuscles which, when combined in sufficient number, form the four elements (D). Finally, those elements are mixed themselves by juxtaposition and form Plato’s particulars (E).53 From A to E, it appears likely that visible qualities supervene on mathematical quantities.54 Three difficulties should be addressed in conclusion: (a) What is the source of the motion of the triangles? (b) What is the relationship between the triangles and the Receptacle? and (c) What are the implications of geometric atomism for the Theory of Forms? As for the first difficulty, 56b7-c6 suggests that the qualitative changes between the elements are inherently related to their local changes.55 The complete picture might be the following: (i) in E1, the traces, which are affections (pathê) with powers (dunameis), are found in a state of motion in the Receptacle.56 Consequently, Timaeus states that the traces move the Receptacle and are in return moved by it; (ii) this would come to a stop according to the natural attraction of the like for the like, in the form of four regions characterized by the four pre-elements sorts; (iii) in E3, the geometrizing of the traces into the forms of two basics triangles opens the possibility of transformation of the elements, that is, it injects even more heterogeneity (based on the inequality of shapes between the solids constructed from the triangles) which will counterbalance the attraction of the like for the like and allow motions of elements between the four regions (the process of transformation will constantly modify the tendency of the elements to form four regions as described in E1); (iv)

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E4 specifies that the elements are in a limited number within a limited space (the sphere of the World Body) which will prevent the mechanical motions described in i–iii to ever cease; and finally (v) these mechanical motions will be ordered when the World Soul is united to the World Body in E5. Consequently, even if motion is described in the Timaeus, as going from an unordered state (E1) to an ordered one (E5), it must be noted that the unordered motion (from i to iv) already implies the work of intellect (the geometrization in E3 and the fashioning of the World Body in E4) in order to subsist. Without the action of the Demiurge, chaotic motion, although it is described as having its source in itself, would cease to be. Must, then, the Receptacle be reduced to a spatial milieu in which free-floating triangles could be found during the cycle of elementary transformation? In that case, what would the triangles be made of? Would they, qua geometrical two-dimensional shapes, be filled with some matter? As suggested earlier,57 Plato is extremely careful when discussing the Receptacle within his eikôs muthos. Let’s recall the teachings of the gold analogy (50a5-b5): the Demiurge modified the Receptacle and gave it the shape of the two basic triangles. Moreover, in 58a7, the existence of void is excluded. However, since the transformations between the different grades of elements occur in the Receptacle, it appears necessary to suppose small interstices (58b5) between the polyhedra formed by the triangles.58 If indeed, inequality and heterogeneity must be supposed to guarantee motion and change, then it is impossible to suppose that the whole World Body should form a compact solid mass. It might be possible to conclude from that that the Receptacle must be considered as a kind of matterspace: let’s imagine that the Receptacle is a sort of undetermined gold substrata: it must not be so solid that it would prevent any kind of motion between the forms imposed on it and, as such, it might be compared to a sort of thick, stable fluid.59 From this fluid are formed the different shapes of the regular solids, and between these solids, remain interstices which could be filled by solids of smaller grades. Obviously, as the division into smaller grades of particles is limited, the minimal size of the interstices will also be limited. In any case, the Receptacle must fill a double role: one of a substrata guaranteeing the existence (a) of the traces as images of the Forms reflecting in it (in E1) and (b) of the solids made out of basic triangles constituted of it (in E3);60 and one of a space in the very specific sense of a milieu in which (a’) the traces and (b’) the regular solids can move and change their positions given to the existence of small interstices. Timaeus’ account of geometric atomism finally seems to bear interesting consequences for the Theory of Forms. More especially, it should be asked what is the relationship between E1 and E3 when it comes to understand the participation of the sensible to the intelligible? This, of course, is related to the question of the more remote principles (57d7) beyond the two basic triangles: are they intelligible Forms? If so, what kinds of Forms? Could they be the Forms of the Four Elements (FAWE)? Or does Plato suppose the existence of Forms of Triangles? As noticed, the text remains silent on this matter. Perhaps, a few thoughts, to conclude, could be proposed. Some of the difficulties here are related to the question of the Demiurge: if he merely is a literally tool, a symbol of the intellective work in the universe, it seems possible to somehow identify E1 and E3, without having to distinguish two

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degrees of participation. This would also imply conceiving the traces of the elements as indeterminate anticipations of sensible participation.61 However, E1 and E3 seem to be distinguished by Timaeus, not only chronologically but also conceptually. The description of the traces is explicitly associated with the nature of the sensible as an image of the intelligible model (52c2-5), and the Demiurge geometrizes this sensible image in E3. Another possible reading62 of Timaeus’ account seems to imply a distinction between two moments of participation. Even if the proto-participation of E1 might be problematic,63 it appears not completely possible to do away with two types of imitation: (a) the mirroring of the images of the Forms of the Four Elements in the Receptacle and (b) the mathematization of these traces by the Demiurge in order for them to acquire a greater degree of perfection.64 In case of (a), the model of the traces are the Forms of the Four Elements which must be found in the intelligible Model.65 In the case of (b), it is less clear what the model that the Demiurge looks at in order to mathematize the pre-elements is. Since the result of this work generates basic triangles, it might be likely that this model is of a mathematical kind. This does not necessary imply the existence of Forms of Triangles. What seems plausible is that, in the Timaeus, Plato differentiates two kinds of likeness: (i) the one which comes for the appearance and disappearance of the images of the Forms in the Receptacle and which allows explaining the ontological difference between a model and its images, as well as the imperfection of the latter in relation to the former and (ii) the one which justifies a greater resemblance between the sensible and the intelligible in terms of mathematical properties and which explains how the images possess a likeness of the model.66 This, as a matter of fact, does imply, in my view, a revised version of the Theory of Forms which should specify how participation takes place as the geometrization of imperfect traces in the Receptacle. If this is correct, then Plato’s atomism, although founded on the same distinction between micro and macro properties, should clearly be distinguished from any kind of materialistic atomism. In that sense, Plato’s atomism could well be an essential characteristic of his Theory of Forms.67

NOTES 1. De Caelo III 2, 300b16; III 8, 307a16; IV 5, 312b21, De Gen. et Corr. I 8, 325b30. 2. See Taylor (1928) and Vlastos (2005, 66–8). 3. 48a2. 4. Vlastos (2005, 68). 5. Vlastos (2005, 70) and Cornford (1937, 210), who speaks of a ‘deliberate correction of Democritus’ atomism’. 6. Following Cornford (1937, 21), the axioms are ‘(1) The eternal is the intelligible; what comes to be is the sensible. Since the world is sensible, it must be a thing that comes to be. (2) Whatever comes to be must have a cause. Therefore, the world has a cause-a maker and father; but he is hard to find. (3) The work of any maker will be good only if he fashions it after an eternal model. The world is good; so its model must have been eternal. Finally, the conclusion is drawn: any account that

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can be given of the physical world can be no better than a “likely story”, because the world itself is only a “likeness” of unchanging reality.’ 7. 29b3-d3, 48c2-e1, 54a2-5, 56b3-c7, 57d4-6, 68b6-8, 68c7-d2. 8. See Burnyeat (2008) and Brisson’s answer (2012). 9. See note 17. 10. For an account of the distinction between a literal and a didactic reading of the Timaeus, see Pitteloud (2017, 197–8). For the didactic reading, see Cornford (1937, 26). Each of the two readings must deal with specific difficulties (e.g. on the literal account: does a time exist before the creation of time (37c-38c)? What is the cause of motion (52d-53c) before the constitution of the World Soul? On the didactic account: what could be demythologized in Timaeus’ discourse? Could the Demiurge be identified with the nous of the World Soul, or with the Model, or with the Form of Good?). See Cornford (1937, 209–10) and Vlastos (2005). 11. Beyond these principles, the premises pointed out by Cornford (1937), see n. 6 above, must also be accepted. Furthermore, some metaphysical principles (like the hypothesis of Forms, see 51b-e) are admitted too. Obviously, each of these principles would require a great deal of discussion, which goes beyond the scope of this chapter. 12. The Receptacle (hupodokhê: 49a6, 51a5) is described as a mother (mêtêr: 50d3), a nurse (trophos: 88d6, tithênê: 49a6, 52d5, 88d6), a place (khôra: 52a8, 52b4, 52d3, 56a6, topos: 52a6, 52b4, 57c3, edra: 52b1, 53a6). It is described in ways that might make think of it as space and matter (Timaeus uses the metaphor of gold (50a5-b5), an impress or mould (50c2-3) and an odourless base of perfumed ointments (50e8-51a1)). Aristotle believes Plato mistakenly identified the two concepts (Phys. 4.2.209b11-12). For a discussion of Aristotle’s criticisms of Plato’s khôra and the relationship with his own hulê, see Brisson (2011). See also Harte (2006, 247–64) and (2010). 13. At least, for visible objects, nous must be found ‘within’ a soul, which does not prevent the possibility for the Demiurge to be a pure transcendent nous (see Menn 1995). 14. On the different sorts of model, see O’Meara (2017, 41–64). 15. Both ‘images’ are to be found in the Timaeus: the appearance of the mimêmata of the Forms of FAWE in the Receptacle is distinguished from the artisanal constitution of the cosmos by the Demiurge. In this way, the cosmos possesses two different kinds of fathers: the Demiurge (29e-30b) and the Model (50d3). 16. This implies that the Demiurge must bestow some properties of the Model upon the cosmos (self-sufficiency, independence, indissolubility). See on that O’Meara (2017, 60–4). 17. For the present purpose, I will only reconstruct the story of the cosmos until the end of the Demiurge’s work. 18. This moment could be either a proper stage of the development of the universe (on the literal reading) or a thought experiment of what would be the world without demiurgic order (on the didactic reading). As the Forms are already reflected in the Receptacle, it might be safe to conclude that the intelligible realities are only necessary, but not sufficient, conditions for the ordering of the cosmos. On this

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question, see Vlastos (2005), Cherniss (1954), Brisson (1994, 298) and Cornford (1937, 198–210). 19. A ‘much misread passage’ according to Cherniss (1954b). See on that Brisson (2011, 4–7) and Pitteloud (2017, 286–90). 20. It has been suggested that the appearances of the Forms in the Receptacle could be understood as tropes. See Buckels (2018). 21. The analogy with gold (50a5-b5) is introduced to help understand this point: we should imagine someone who never ceases forming different shapes (triangle, square, . . .) from some gold substrate. It would be absurd, in order to refer to one of these shapes, to use the demonstrative ‘this’, since at the very moment we would do so, the form would have already been transformed into another one. Only the gold substrate could be designated with the demonstrative ‘this’ as it is what remains through the transformations. For the comparison between the Receptacle and a kind of matter or medium, see Harte (2006, 255–6). 22. 50c3-6. On the ‘another occasion’ (eis authis), scholars don’t agree if this promise has been fulfilled or not by Timaeus. If it has, does it refer to the pre-demiurgic appearance of the Forms into the Receptacle as described in 52c-d (which does not seem to be a real explanation) or to the imposition by the Demiurge of geometrical shapes to the elements (53c4). On that see O’Meara (2017, 60). 23. See n. 14. 24. The Receptacle seems to be apprehended without the senses by a sort of abstraction, called a bastard reasoning (52b2). 25. For a suggestion that the term ichnon does not refer to the relation between a particular and a Form, see O’Meara (2017, 60–1). For an occurrence of this term in Plato, and the difference between the heuristic and causal aspect of a trace, see Harte (2010, 133, n. 6). 26. O’Meara (2017, 60). 27. This is a vexed issue: as the World Soul has not been fashioned yet, it seems that some motions are not caused by the soul. Timaeus’ discourse seems to differentiate between two kinds of motions: an irrational-unordered motion which takes place in the Receptacle and a rational-ordered one which is caused by the World Soul. Both are also distinguished by the presence or absence of the god (53b3-4). This of course can be understood in a literal or didactic sense. In both cases, the existence of the chaotic motions in the Receptacle must be justified. If one believes that all motions are caused by the soul in Plato’s philosophy, then one of two following options will emerge: i) the hypothesis of a pre-cosmic irrational soul or ii) an irrational part of the World Soul (as Cornford (1937) defended on pp. 209–10). 28. See 58a1. Unfortunately, Timaeus does not explain much this idea. It seems to imply that motion involves alterity which can only occur when it is associated with heterogeneity (inequality, dissimilarity and difference appear to be somehow related in this complex question). 29. I will come back to that motion in the next part. 30. In 31a-b, the Demiurge seems to think that since the model is unique, and in order to bestow more perfection upon the image, the world must also be unique. 31. 34c4–35a1.

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32. All these claims should be discussed and contextualized. See Pitteloud (2019). 33. The World Soul is divided according harmonic intervals which by means of inequality maintains its unity since the nature of Difference is hard to mingle (35a78). See Cornford (1937, 66–72). 34. 31c1-4. That is: the best bond between three things is through geometrical proportion: for example, for the numbers 2, 4 and 8, it gives: 2/4=4/8, 8/4=4/2, 4/8=2/4, 4/2=8/4. 35. See Brisson (1994, 232, n. 136) and Cornford (1937, 46–52). 36. The proportion resulted is: as fire is to air, so is air to water, and as air is to water, so is water to earth. 37. As all particles geometrized will be used, there is nothing left outside the World Body. As the four elements appear in the Receptacle, which is identified with space, there is no place beyond the limit of the World Body, which constitutes the whole universe. 38. Since the Model is all-inclusive (it contains the four species). See Cornford (1937, 52). 39. 34b3-8. For a comparison between this description and Empedocles’ cosmogony, see O’Brien (2003). 40. Each of them corresponds to one of the four elements. 41. For a detailed account of the process of mathematization of the traces, see Cornford (1937, 210–39), Opsomer (2012), Artmann and Shäfer (1993) and Pohle (1971). 42. 53c8-d4. 43. 53d4-7. What are those ultimate principles? Lines? Points? Numbers? The One? For an overview of the different possibilities, see Karfik (2007, 149, n. 138). 44. Figures are taken from (1) Cornford (1937); (2) Artmann and Shäfer (1993); (3) Friedlander (1964–1975), Vol. 1; (4) Cornford (1937). 45. Cornford (1937, 213). 46. O’Meara (2017, 75). 47. Plato does not justify why this is the case. Either it is a limit of the mathematical account which leads to a displeasing consequence, or it could be the case that Plato believes that empirical experience shows that earth cannot be changed into any other elements. See Vlastos (2005, 78–86). Again, Aristotle got displeased by this (De Caelo 360a1-7). 48. Vlastos (2005, 70): ‘On the Democritean theory an atom of fire, for example, could never change its shape or size in any way whatever – hence never change into an atom of air or of water. And this is precisely the sort of change Plato wants to insure. He wants corpuscles which will be susceptible of two types of radical transformation.’ That is (i) a transformation between the elements and (ii) between varieties of each of the primary elements. See also Cornford (1937, 210). 49. For some answers to those specific questions, see Harte (2006, 239–54). 50. Symmetry can be defined in different ways: (i) due proportion, (ii) bilateral correspondence with regard to a reference, and (iii) our present-day mathematical

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definition. On those distinctions and for a rejection of our present-day conception of symmetry in the Timaeus, see Paparazzo (2015a). 51. Cornford (1937, 230–9), argues that this issue is related with the question of the grades of the primary bodies. 57c-d seems to imply that there are different varieties of the four elements, and this is due to the different size of the (microscopic) particles composed by triangles. That is, the sizes of the scalene and isosceles triangles are unique as 56b2 suggests. What varies in size (as affirmed in 57d1-2) are the two surfaces (square and equilateral triangle) constituted by these two triangles, and these variations depend on the number of basic triangles which constitute them. This allows to explain the transformation of the elements of different grades between them (although the exact proportions for these transformations are not to be found in the text). The choice to introduce surfaces with four and six triangles is meant to indicate, according to Cornford, the possibilities of different grades constructed by basic triangles of the same size. Thus, the two surfaces of figure 4 will constitute corpuscles of intermediary types (neither the smallest nor the biggest). On this issue, see (1971), Brisson (2001, 302–6), Artmann and Shäfer (1993), O’Meara (2017, 74–5) and Paparazzo (2015b). 52. Plato indicates neither the size of the sides of the two basic triangles nor the exact quantity of the four elements. What is given are the respective proportions involved. Nothing seems to suggest that it is possible to work out the details of one proportion from the other. 53. It appears that the relationship between A and B, and B and C is of parts/whole. The relation between C and D seems to be of identity (a certain number of identical particles become visible) and the one between D and E of juxtaposition. See for a complete analysis, Opsomer (2012, 148–55). 54. See Opsomer (2012, 155–7), for a list of Aristotle’s criticisms. 55. ‘Moreover, in the course of suffering this treatment, they are all interchanging their regions. For while the main masses of the several kinds are stationed apart, each in its own place, owing to the motion of the Recipient, the portions which at any time are becoming unlike themselves and like other kinds are borne by the shaking towards the place of those others to which they become like.’ 56. See above on pp. 130–40 for an explanation of the cause of these motions within the Receptacle. Cornford (1937, 228–30) defends that the motions of the traces are due to ‘blind irrational impulse in the soul that animates the body of the world’.  57. See above pp. 140. Timaeus of Locri’s interpretation considers the basic triangles to be compound of (Aristotelian) form and matter. See Ulacco and Opsomer (2014). 58. Such interstices represent space for smaller particles of elements. Note that interstices should not appear between particles of earth, since the combinations of cube cannot allow it. As Cornford (1937) rightly points out, n. 1, p. 245: ‘It is not explained where earth comes into the scheme. There is nothing to show what sizes the earth cubes have, as compared with the other bodies. Cubes can be packed so as to leave no interstices; yet at 60e we hear that interstices between earth cubes are so large that fire or air particles can slip into them without disturbance.’ 59. See Brisson (2001, 257, n. 456) for a similar analogy. 60. 52c2-5. ‘whereas for an image, since not even the very principle on which it has come into being belongs to the image itself but it is the ever-moving semblance of

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something else, it is proper that it should come to be in something else, clinging in some sort to existence on pain of being nothing at all’. 61. See O’Meara (2017, 62): ‘Of course there remains the mystery of what the predemiurgic “traces” of the elements might be. Here Timaeus gives us little help. We might suppose that the traces are indeterminate, confused je ne sais quoi which the demiurge will work up and give determination and shape. Later, at 69b6, it seems that the pre-elements might have shared to some extent, by chance (tuchê), in the order which the Demiurge will give them, by anticipation, as it were. [. . .] In this case, the trace would be an adumbration, a chance anticipation of, not something produced by, that of which it is a trace. [. . .]. Another possibility, which I think is far less likely, would be that the traces had been produced by elements, then these elements would have existed earlier than the traces, sometime in the past. The word “trace” can also have this sense, the sense of something left from a past era, a memory.’ 62. A reading that does not need to be literal in any case. 63. All the order of the sensible must come from the work of nous; however, this work cannot start from nothing, since in that case nothing would exist to be worked upon. The traces appear to point out an imperfect pre-demiurgic order. This pre-order must somehow be a first degree of order, for if it were not, it would be a nothing. See Vlastos (1939, 77). The text is indeed explicit on this fact: the Demiurge does not initiate his work in E3 on a completely indeterminate milieu but bestow mathematical order upon the images of the four elements. 64. As O’Meara notices (2017, 59, n. 61): ‘The reverse of such a sequence is described in the Republic, where an artisan (demiurge) makes a couch, according to “idea” of a couch (596b), and then a painter makes an image, an appearance, of the couch, as if in a mirror (596ce).’ 65. The model is described as containing the four-living species. At no point, the text suggests that it must contain the Form of the Four Elements. However, as each specie is linked to a region of the world which in turn is associated with an element, it might be possible to suppose that the Forms of the elements can somehow be found within the Model. 66. It has to be noted that it is not a physical resemblance: as 61d-62a indicates, this is the piercingness of the sharp angles of the pyramids which constitute fire particles that explains the pain feeling it causes. On that see Morrow (1968, 26–7). 67. The research for this paper has been supported by FAPESP grant 2019/07210-7 and CAPES grant 88881.169828/2018-01.

REFERENCES Artmann, B. and Schaäfer, L. (1993), ‘On Plato’s ‘fairest triangles’ (Timaeus 54a)’, Historia Mathematica 20: 255–64. Baltzly, D. (2009), Proclus: Commentary on Plato’s Timaeus. Vol. 4. Book 3, Part 2, Cambridge: Cambridge University Press. Brisson, L. (1994), Le Même et l’Autre dans la Structure Ontologique du Timée de Platon, Sankt Augustin: Akademia Verlag.

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Brisson, L. (2001), Timée, Critias, Paris: GF Flammarion. Brisson, L. (2011), ‘La “matière” chez Platon et dans la tradition platonicienne’, in Giovannozzi Delfina e Veneziani Marco (a cura di), Materia, XIII Colloquio Internazionale, Roma, 7-8-9 gennaio 2010, 1–40, Firenze: Leo S. Olschki editore, Lessico Intellettuale Europeo. Brisson, L. (2012), ‘Why is the Timaeus called an Eikôs Muthos and an Eikôs Logos’, in C. Collobert, P. Destrée and F. J. Gonzalez (éds), Plato and Myth: Studies on the Use and Status of Platonic Myths, 369–91, Leiden: Brill. Buckels, C. (2018), ‘Triangles, tropes, and τὰ τοιαʋ̃τα: A Platonic trope theory’, PLATO JOURNAL: The Journal of the International Plato Society 18: 9–24. Burnyeat, M. F. (2008), ‘Eikôs muthos’, in Catalin Partenie (ed.), Plato’s Myths, 167–86, Cambridge: Cambridge University Press. Cherniss, H. (1954a), ‘The sources of evil according to Plato’, Proceedings of the American Philosophical Society 98, no. 1: 23–30. Cherniss, H. (1954b), ‘A much misread passage of the Timaeus: (Timaeus 49 C 7–50 B 5)’, American Journal of Philology, 75 298: 113–30. Cornford, F. M. (1937), Plato’s Cosmology, London: Routledge & Kegan Paul; reprinted, Indianapolis: Hackett Publishing Co. Friedlander, P. (1964–1975), Platon, Berlin: Gruyter. Harte, V. (2006), Plato on Parts and Wholes: The Metaphysics of Structure, Oxford: Clarendon Press. Harte, V. (2010), ‘The receptacle and the primary bodies: Something from nothing?’, in Richard D. Mohr and Barbara Sattler (eds), One Book, the Whole Universe: Plato’s Timaeus Today, 131–40, Las Vegas: Parmenides Publishing. Karfík, F. (2007), ‘Que fait et qui est le deémiurge dans le « Timee » ?’ Etudes Platoniciennes 4: 129–50. Menn, S. (1995), Plato on God as Nous, Carbondale: Southern Illinois University. Morrow, G. (1968), ‘Plato’s theory of the primary bodies in the Timaeus and the later doctrine of forms’, Archiv für Geschichte der Philosophie 50, nos. 1–2: 12–28. O’Brien, D. (2003), ‘Space and movement: Two anomalies in the text of the Timaeus’, in Plato Physicus: Cosmologia e antropologia nel Timeo, 121–48, Amsterdam: Adolf M Hakkert. O’Meara, D. J. (2017), Cosmology and Politics in Plato’s Later Works, Cambridge: Cambridge University Press. Opsomer, J. (2012), ‘In defence of geometric atomism: Explaining elemental properties’, in J. Wilberding and C. Horn (eds), Neoplatonism and the Philosophy of Nature, 147–73, Oxford: Oxford University Press. Paparazzo, E. (2015a), ‘Does present-day symmetry underlie the cosmology of Plato’s Timaeus’, Apeiron 48: 2. Paparazzo, E. (2015b), ‘It’s a world made of triangles: Plato’s Timaeus 53B-55C’, Archiv Fur Geschichte Der Philosophie 97, no. 2: 135–59. Pitteloud, L. (2017), La séparation dans la métaphysique de Platon: Enquête systématique sur le rapport de distinction entre les formes et les particuliers dans les dialogues, Sankt Augustin: Academia Verlag.

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Pitteloud, L. (2019), ‘Why is the world soul composed of being, sameness and difference?’, in L. Pitteloud and E. Keeling (eds), Psychology and Ontology in Plato, 85–108, Philosophical Studies Series, 139, Cham: Springer. Pohle, W. (1971), ‘The mathematical foundations of Plato’s atomic physics’, Isis 62: 36–46. Taylor, A. E. (1928), A Commentary on Plato’s Timaeus, Oxford: Clarendon Press. Ulacco, A. and Opsomer, J. (2014), ‘Elements and elemental properties in Timaeus Locrus’, Rheinisches Museum Für Philologie 157, no. 2: 154–206. Vlastos, G. (1939), ‘The disorderly motion in the Timaios’, The Classical Quarterly 33, no. 2: 71–83. Vlastos, G. (2005 (First Published 1975)), Plato’s Universe, Las Vegas: Parmenides Publishing.

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PART II

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CHAPTER 8

Atoms and orientation Vasubandhu’s solution to the problem of contact AMBER CARPENTER, WITH SHERICE NGASERIN

INTRODUCING VASUBANDHU, BUDDHIST ATOMIST Vasubandhu (fourth century CE) was a Buddhist philosopher working, originally at least, in the Abhidharma tradition.1 The Abhidharma philosophers made it their ambition to make more precise and systematic the various epistemological and metaphysical claims the Buddha made in the course of his informal discourses. Ᾱbhidharmikas held a wide range of different views, debating among each other as well as against non-Buddhists and, from about the first century CE, against Mahāyāna Buddhists. They were broadly united by a metaphysics of dharmas, classified in various ways according to different purposes. Naturally, what exactly a dharma is, and which dharmas there are, was a matter of dispute. Vasubandhu, a celebrated master of debate, contributed the most precise and comprehensive statement of one of the most well worked out Abhidharma Buddhist positions of his time (that of the Vaibhāṣikas), as well as some of the most trenchant critiques of that position. He did so in what is effectively the same work, namely the Abhidharmakośabhāṣya – the Treasury of Abhidharma (Abhidharmakośa) with commentary (bhāṣya).2 Vasubandhu’s metaphysics has been likened to trope theory,3 and it is not hard to see why. His position in the Abhidharmakośabhāṣya, like most Abhidharma positions (and all those taking their cue from the Vibhāṣā4), eschews any form of universals; and moreover, it does so by eliminating from the root the substanceproperty metaphysics which gives rise to the need for universals in the first place.5 What really exists are occurrences of particular properties – and that is all. They are ‘occurrences’ because, for reasons Vasubandhu gives at AKBh. VI.2b-3b, these property-particulars do not endure or perdure or persist in any way.6 Their very existence is the cause of their ceasing to exist, and so to arise is to pass away, making all existences of ultimately real things strictly momentary. Take this as our first indication that this is not trope theory as we know it. The argument for momentariness relies on the presumption that a property-particular cannot unfold its nature over time.7 As far as Vasubandhu is concerned, being a trope,

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or a property-particular or dharma, in this sense means neither bearing properties nor inhering as a property in a substance. From this follows the impossibility of a thing changing, with the appearance of change secured by the difference in which dharmas exist at different moments.8 Such dharmas may be either mental or physical in kind – nothing about the view, or the arguments for it, rely on the corporeality of dharmas, and Vasubandhu (again like all his Abhidharma peers) took it that there were both mental and non-mental dharmas. Particular moments of mental properties will be just as momentary as the physical dharmas and for the same reason. Vasubandhu is, then, a strict atomist, and it is the atomist principle that leads to his understanding of dharmas as property-particular occurrences, rather than property-bearing substances.9 Vasubandhu’s commitment to the atomist principle is articulated at AKBh. VI.4, which is regularly taken to be the definitive articulation of his understanding of the distinction between what we might call fundamental (and he would call ultimate) reality and everything else – what he calls ‘conventional’ reality, or the way everything that is not a dharma (but is composed of dharmas) exists. There is no idea of a pot when it is broken, and none of water when it is analysed by the mind. That which is like the pot and water is conventionally real. The rest is ultimately real.10 It is criterial of existing ultimately, or fundamentally, that an existent ‘survives analysis’ – that is, no division can make it cease to exist. Vasubandhu specifies two sorts of divisibility: physical and conceptual. Physical divisibility is straightforward enough, and Vasubandhu does not specify whether it means divisible by part or within a homogeneous mass,11 nor whether it means divisible in practice (as a thing a centimetre long) or divisible in principle (as a thing a nanometre long). He does not need to, because he has a second sort of divisibility ready to hand: divisible ‘in the mind’, or conceptual divisibility. According to Vasubandhu, even what is conceptually divisible is thereby shown to be merely conventionally, and not ultimately, real. This conceptual divisibility is not just the sort that applies to lengths – any mathematician can halve the nanometre, whether or not the two halves can be physically prised apart. Vasubandhu’s conceptual divisibility applies, for instance, to the distinction one can make between the coolness of the water and its translucence. This is what ‘analysing’ water with the mind involves. Such analysis applies equally to mental things (if any there be) as well as to concepts. My burning resentment over an unwarranted slight may indeed cause me to act; but it consists wholly in an interacting set of judgements, intentions, recollections, perceptions, desires and conceptualizations – and its reality as a single thing is merely conventional, not ultimate. This is an absolute atomism, far beyond what Democritus envisaged, with his atoms of multiple shapes, and his presumption of souls as unified loci of multiple experiences.12 Such an atomism is not primarily animated by concerns about extension – although the principle of divisibility will apply there, too. Wherever a conceptual distinction may be drawn, this suffices as demonstration that there are as many distinct entities as are distinguished, and the analysand, if it is not identical

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to some one of the final analysata is not an ultimately existing thing – for what is analysable by mind is thereby shown to have been put together by the mind.13 If this is trope theory, it is a very austere trope theory indeed.14 Any collection of dharmas is not ultimately real – whether that collection is notionally or physically or temporally aggregated, whether of the same kind or not (AKBh. I.20ab), whether mutually conditioning (as the person-constituting aggregations of dharmas, AKBh. IX15) or not. Vasubandhu’s immediate Ᾱbhidharmika associates (often called Vaibhāṣikas for their core text, the Vibhāṣā, or sometimes, more specifically, Sarvāstivādins after their core commitment) had already recognized the impossibility of anything like real locomotion, once the momentariness of dharmas is agreed. But with his principle of ‘surviving analysis’, Vasubandhu goes them one better and explicitly argues against shape’s being ultimately real (AKBh. IV.3c). For any shape, whether physically instantiated or not, is divisible notionally – it has distinguishable parts and is therefore only conventionally real. ‘Shape is not in an atom. There is no atom of length’ (AKBh. IV.3c).16 Vasubandhu then tacitly appeals to the principle cited earlier (but stated only later in the Abhidharmakośabhāṣya) concerning the conventionality of what dissolves under analysis: as it diminishes, length is no longer cognized; this demonstrates, Vasubandhu argues, that length is not something ultimately real.17 But of course anything of any dimensions at all must necessarily have some shape – it must have, at least, length, which Vasubandhu treats as a kind of shape and equally unreal for the same reasons. Unlike Epicurus, then, Vasubandhu cannot have an atomism of a sort that asserts minima as constituting the fundament of physical reality, bulking it up through their accumulation.18 Vasubandhu’s physical dharmas – the paramāṇus – must be dimensionless point-particle property-events.19 Thus, regarding the physical dharmas, Vasubandhu’s atomism suffers acutely the twin problems of agglomeration and contact. The Problem of Agglomeration arises from the fact that dimensionlessness plus any number of additional dimensionlessness bits can only ever be dimensionless. So dimensionless atoms cannot, it seems, constitute the basis of even the appearance or illusion of spatially extended objects. The Problem of Contact arises from the attempt to agglomerate truly simple, dimensionless atoms into lengths and shapes by placing them adjacent to one another: if they are in contact, then either they have parts (the parts touching while the rest does not) or, remaining partless, they wholly touch and thus coincide completely and end up occupying the same space. Vasubandhu addresses these related objections in Book I of the Commentary on the Treasury of Abhidharma.

THE PROBLEMS OF CONTACT AND AGGREGATION The Problem of Contact is raised in a discussion about the way that the non-distal senses – taste, touch, smell – operate. Unlike the senses of sight and hearing, which operate at a distance, the non-distal senses are said to attain their objects. That is, unlike sight and hearing, senses such as touch cannot operate at a distance. They must, as we say, make contact with their objects in order for perception along that modality to arise.

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This poses an immediate problem for atomism, at least for the sensible atoms constituting the organs and objects of perception, and designated by the term paramāṇu.20 If the organ must be in contact with its object, then presumably contact between the constituent atoms of organ and object must be in principle possible. But such contact seems to be inconsistent with atomism, as had already been recognized by Vasubandhu’s predecessors, referred to here as ‘the Kaśmīris’, who conclude that atoms ‘do not touch’ because ‘[i]f they were to touch completely, things21 would coalesce. Then suppose instead that they were to touch at one spot. There would be the unwanted result that they have parts – and atoms do not have parts’ (AKBh. I.43d(4)). The dilemma is this: if atoms were to touch, they must either touch in their entirety or touch at a point (or in part). But atoms cannot touch in their entirety. If they did, they would be entirely in the same place (coalesced). And in this way, there could never be aggregation, nor contact of a sense organ with its object to produce a perception. The sense organ would occupy the space of the object without ever making contact.22 And in general any agglomeration of atoms-in-contact would end up with atoms simply occupying the same spot. But neither can we avoid coalescence by supposing atoms touch at a point instead; for if one atom touched another at one point, but not at another, then this atom would have two points, genuinely distinct from one another. The points must be genuinely distinct, for if the point-of-contact of atoms a and b were not different from the rest of atom a and the rest of atom b, it would be identical to the whole of both atoms and we have the problem of coalescence. But comprising multiple distinct parts already violates the simplicity required of atoms. The option of avoiding the dilemma by appeal to indivisibles comprising multiple distinct points (extended minima) is not entertained by Vasubandhu and is indeed precluded by his insistence at AKBh. IV.3 that shape, including length, is not ultimately real. And even supposing we could make sense of dimensionless points touching partially, this would violate the principle of atomism as Vasubandhu has laid it out. For there would be even so a conceptual distinction between the in-contact part and the not-in-contact part of an atom, and once these were distinguished, the atom itself would no longer be in either place (or, conversely, the atom requires taking the two parts together as a single thing). Such analysability – or such taking one thing to be two things – is, on the Abhidharma view, an indication that genuinely distinct things have been put together by the mind. Even dimensionless atoms partially in contact and partially not (if such there could be) would have parts and not therefore be atoms. So atoms can touch neither partially nor wholly. If they cannot touch, they cannot aggregate to form familiar macro-objects, so atomism cannot explain the apparent objects of everyday perception that it was supposed to ground. Further, if atoms cannot touch, there is no discernible difference between the distal and the non-distal senses: both do not attain their objects (alternatively, the distinction remains, but atomism cannot account for non-distal perception). Yet it is agreed that if the tongue does not attain its object, there is no taste; and in general, where non-distal senseorgans do not attain their objects, no sensation of that sort arises.

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The canonical Vaibhāṣika response to this last difficulty seems to involve rejecting the presumption that in order for macro-objects – the organ and its object – to touch, their constitutive atoms must touch. There is no error, Vasubandhu has them say, in the claim that ‘agglomerations touch because they have parts’ (AKBh. I.43d(8)). Macro-objects are partite, they consist of multiple atoms; so there is no problem about them making contact in part. Vasubandhu has little patience for this evasion, saying simply, ‘it is not the case that agglomerations are anything other than atoms’ (AKBh. I.43d(12)). If atoms cannot touch, neither can agglomerations of atoms, on pain of the agglomeration being something over and above its atomic constituents; conversely, if agglomerations make contact, they can do so only by their constituent atoms touching.23 Moreover, the supposition that agglomerations touch while atoms do not would not address the prior difficulty of agglomeration, closely associated with the Problem of Contact; without agglomeration, there are no macro-objects for which partial contact could be unproblematic. Vasubandhu does, however, have more time for the basic bullet-biting stance of the Vaibhāṣika position: atoms do not touch. If senses are to attain their objects, it will have to be by juxtaposition without interval – yet without touching. ‘What is this term “attain”? Occurring without interval.’24 This is the proposal as Vasubandhu inherits the discussion, and the challenge is to adjudicate between different understandings of ‘without interval’ and discover one that permits nondistal perception and aggregation of physical atoms. This is no small challenge, for ‘contactless immediate juxtaposition’ is not exactly perspicuous. Indeed, it might seem more a description of the problem than an actual solution. Vasubandhu canvasses two different options of how this trick might be turned, depending on what ‘without interval’ means. The suggestion Vasubandhu inherits and rejects looks promising enough. According to the Vaibhāṣikas, ‘There not being anything in the middle is indeed a state of non-interval for these [organs]’ (AKBh. I.43d(7)) – that is, where there is no third paramāṇu between two paramāṇus, these two are without interval with respect to each other. This interpretation of ‘without interval’ is the more promising as space is not itself considered a thing (AKBh. I.5c; AKBh. II.55d) – thus, empty space between two atoms is the same as there being nothing intervening between them.25 On this view, non-distal sense-organs reach their objects when no thing intervenes. They do not in fact touch because there is a gap between them – an unoccupied interval ensuring against the fracturing of atoms contact would imply.26 The sense organ may make contact with its object without coalescence, for both macro-objects are admittedly partite, so they may touch in part without being wholly collocated. Their respective constituent atoms, however, do not touch, but rather remain all at an empty interval from each other, so that they may aggregate without compromising their atomicity. Vasubandhu rejects this straightforward solution which takes ‘without interval’ as ‘without any thing intervening’, an empty gap sparing atoms any actual contact, while at the same time allowing that aggregations (such as sense organ and sense object) may touch. He does this for two reasons. First, he rejects asymmetry. Aggregations of atoms cannot touch without their constituent atoms touching – unless, of course,

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one takes the aggregate to be something other than its aggregated atoms. Naturally that is a possible position; but it is not an Abhidharma position, and anyone willing to endorse that will not have the problem of contact in the first place, since they evidently do not suppose that being complex, or ‘analysable’, is tantamount to being only conventionally real, and not ultimately, mind-independently so. If partite entities could be ultimately real, then there would be no need to posit an asymmetry between touching aggregates and their contactless constituents, for the constituents themselves would not be vulnerable to the Problem of Contact in the first place. But the gappy interpretation of ‘without interval’ was meant to save anti-holist atomism – a view that rejects the independent reality of any partite entities; and it can only do that by positing an inconsistent asymmetry between aggregates and their constituents – or else by conceding that neither atoms nor aggregations of them make contact with one another. Conceding that neither aggregations nor atoms make contact may not seem particularly significant. After all, any version of the view under discussion is redefining what ‘contact’ means in such a way that it does not involve actual contact. At least, that is in effect what has happened at the atomic level – an organ ‘attains’ its object when there is no intervening thing between organ-atoms and object-atoms, all of which stand with decent intervals of empty space between them; accepting that it follows from this that the macro-objects do not really touch one another does not seem a lot more to swallow.27 But Vasubandhu does not rest his critique of this interpretation of ‘without interval’ on this. His second ground for rejecting this Vaibhāṣika solution goes for the gap. Although the position can be stated without reference to a void or gap between atoms, the ‘no intervening thing’ view of ‘without interval’ is distinguished from its alternative by holding that instead of an intervening thing there is an unoccupied interval surrounding each atom. Without such a void, we are returned to the puzzle of how to understand immediate adjacency (atoms being without interval) in such a way that it is not in fact just plain old contact – and thus we are returned to the problem of contact itself: either adjacent atoms touch at a point and so have parts, or they make contact in their entireties and thus occupy the very same space (coalesce). But such a void, Vasubandhu claims, is impossible. And therefore the whole interpretation collapses. Vasubandhu’s argument against the void is very compressed and relies on claims and commitments argued for in other passages. The first of these is the claim that space is not absolute. Space, he argues at AKBh. II.55cd, is not itself a thing (dravya, vastu); if it were, it would be an extended thing and thus not fundamentally real anyway. Rather space is simply the absence of obstruction to a paramāṇu arising (AKBh. I.5d). The only obstructions to atoms arising are other atoms (AKBh. I.13); so space is, as such, just an absence of such atoms preventing other atoms from arising. It is no positive entity in its own right, and it has no causal powers (AKBh. II.55cd). But this means, as Vasubandhu observes here in our current passage concerning contact, that space certainly cannot prevent an atom from arising – ‘were atoms to have space in between them, by what would their going into the empty intervals be restrained?’, Vasubandhu asks rhetorically (AKBh. I.43d(11)), since resistance is

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needed to block the arising of atoms (AKBh. I.43d(12)), and space by definition does not resist or obstruct the arising of atoms. Indeed, the only thing that offers resistance is another atom; but if an atom obstructed the arising of another atom, then the space was not empty after all. Therefore, there can be no void, and so the interpretation of ‘without interval’ as ‘without intervening thing’ collapses. This dispatches the first interpretation of ‘without interval’. There remains a second. This is the much more obvious construal of ‘without interval’ as without interval. That is, there is nothing at all intervening between atoms – not another atom, not an interval of atom-free space. This is what Vasubandhu takes to be the view of his Ᾱbhidharmika predecessor, Vasumitra, who seems to have put it forward in response to a different worry, and not in relation to the problem of contact.28 According to Vasumitra, atoms ‘do not touch; but there is the cognition of touch when they are without interval’ (AKBh. I.43d(10)).29 This, Vasubandhu affirms, is the correct position.30 The only difficulty is in making out what the position is – or rather, more importantly, how it could possibly be thought to solve the problem of contact, rather than introduce it. As the Ᾱbhidharmika Saṃghabhadra puts it in his Nyāyānusāra, ‘If one says that atoms absolutely are without any intermediate space between them, and yet are not mixed one with another, they must have parts: a false opinion. Otherwise, if nirantara signifies “without interval” (anantara), how is it that the atoms do not touch one another?’31

RESISTANCE IS USELESS! ... OR IS IT? Vasubandhu is shockingly reticent here. He has endorsed the claim that atoms do not really touch, that the correct position to take is that we say atoms touch, or cognize them as touching, under certain circumstances – namely when they are immediately adjacent. This is a conception or cognition, saṃjñā, that we supply when experiencing what is, taken in itself, merely immediate adjacency, or atoms being without interval. This adjacency must be properly immediate – not even empty space intervening; for Vasubandhu insists such empty spaces are not possible. Space, being no positive thing, is merely the absence of obstruction to dharmas arising. And where there is no obstruction to arising, there is the arising of dharmas. So Vasubandhu’s interpretation of ‘without interval’ must be one that is indistinguishable from actual touching. How then has he not just returned us to the original Problem of Contact? Directly addressing the problem in his own voice, Vasubandhu remarks briefly and without further comment: regarding atoms: if division according to directions is being supposed, then there is the unwanted result that they have parts – whether they touch or not.32 If not supposed, then even when they do touch, there is no unwanted result. (AKBh. I.43(13)) This unelaborated conclusion to the discussion is almost shocking, for it seems to be dismissing the Problem of Contact out of hand, rather than offering a solution

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to it. ‘If you think contact is a problem for atomism’, Vasubandhu seems to say, ‘then it is; but if you don’t, then it isn’t’. This looks like an appeal to tablebanging and strength of mind – ‘simply do not concede partition in the first place, then no amount of contact and touching could force you to do so’ – and these are poor ways of meeting a metaphysical argument. One has sympathy for Saṃghabhadra’s complaint that Vasubandhu has simply missed the point of the Problem of Contact. Now there is an important insight in the first part of Vasubandhu’s observation. If one supposes – for whatever reason – that atoms have a left side, say, and a right side, and these are not identical, then one is already taking them to be partite entities. Once you have done this, no amount of theorizing about intervals and voids will restore their original unity. Let there be ever so much unfilled space between atoms; this will not prevent the left side being different from the right side, if they are in fact different. Moreover, if the left side and the right side of an atom have to be genuinely distinct (even if collocated at the same point) in order to avoid coalescence, then these directional parts must be genuinely distinct in order to avoid coalescence even if there is no contact. Even if a void were possible, addressing the Problem of Contact by positing a gap only causes the problem to re-emerge as the more refined Problem of Directionality, a version of the problem which makes it clear that the threat of partition in no way depends upon the notion of extension. Orientation itself suffices to split the atom, on Vasubandhu’s principle of analysability, if upwards of an extensionless atom must be genuinely distinct from downwards. The Vaibhāṣikas thought that if atom a touches another atom b at a point, a must have a b-touching part and a non-b-touching part, and similarly for atom b, thus impugning the simplicity of both atoms. Any atom-in-contact would have as many parts as there were atoms with which it was in contact. Their solution was to insert a space between a and b. If there is no contact between distinct atoms, there is no need for any to have a point-of-contact distinct from its other part, then there will be no problematic partitioning of atoms. But Vasubandhu observes that they have misdiagnosed the problem: if the partitioning of atoms is a concern, it is not a problem caused by contact, but by directionality alone.33 For atoms which are not in contact must still have directional orientation, at least so long as there is more than one atom (and only a theory of multitudinous atoms could account for the macro-objects of everyday perception). Even with an empty interval between atoms, any given atom must have a ‘rightwards’ that is genuinely distinct from its ‘leftwards’, and equally its ‘upwards’ and ‘downwards’ must each be genuinely distinct from the others. If they are not properly distinct, then there is no difference between the interval on the left and the interval on the right, and coalescence threatens again. If, however, they are genuinely distinct, then partition threatens just as much as if the atoms touched. Thus, if there is a problem at all, it is much bigger than the Problem of Contact (and its proposed solution) supposed, for there is nothing special about contact that forces partition. Simple juxtaposition with respect to one another suffices to make atoms partite. Once the magnitude of the difficulty is clear, there is motivation

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enough to reject the antecedent and deny altogether that there is a genuine problem. The question remains, on what grounds Vasubandhu can legitimately do this. The reformulated problem focuses attention back on aggregation. If atoms exist contemporaneously in some spatial relations to one another, then their tops must be genuinely distinct from their bottoms – not merely conceived by us as such – in order to avoid the atom above occupying the same place as the atom below, namely the same place as the atom in the middle. A commitment to absolute space might resolve this, for then position could be fixed absolutely, without reference to other atoms. The top of a point may be identical to its bottom, but the point-particle atoms above and below it occupy their respective positions independently of this, and there is, as Vasubandhu observes, no need to suppose atoms must be partite in order to avoid the collapse of the universe, the coalescence of all into a single point (or perhaps all into every point?). But Vasubandhu and his atomist brethren reject absolute space, Vasubandhu arguing particularly emphatically that space, being an unconditioned dharma, is acausal and not a thing in any sense, but rather is simply the absence of obstruction.34 So we must search elsewhere for some basis for Vasubandhu’s confidence that the Problem of Directionality is a non-problem. The place to look is in Vasubandhu’s immediately preceding argument against the void. His explanation there for why there cannot be an empty interval of space includes the reminder that ‘resistance is needed’35 to obstruct the arising of paramāṇus, or rūpa dharmas. This ‘resistance’ cannot be provided by space, but can only be provided by other rūpa dharmas, because this is just what it is for something to be rūpa (a sensible property). Earlier in the text, at Abhidharmakośabhāṣya I.13, Vasubandhu canvassed two different understandings of rūpa, the latter and more promising of which is ‘resistance’ (AKBh. I.13).36 This understanding of rūpa as resistance is endorsed several verses later, in a discussion of the basic elements of reality (AKBh. I.28-29). ‘The ten [elements] which have the characteristic of rūpa have resistance’,37 which is to say that they ‘obstruct the arising of another in its place’,38 and so are in this sense capable of colliding (AKBh. I.29bc). Whatever is rūpa is such that it would, by virtue of its existing, prevent anything else rūpa from being where it is.39 Vasubandhu then appeals implicitly to this point about rūpa in his discussion of the Problem of Contact. In the course of defending the claim that there is no actual contact among atoms, Vasubandhu must meet the objection that without contact sound could not arise. His table-turning reply, which he is simply relaying from the Vibhāṣa itself, asserts that if there were real contact, there would be no sound, for ‘if they were to touch, a hand which strikes at a hand, a rock which strikes at a rock, would become attached’ (AKBh. I.43d(5)). Precisely because there is no contact, he says, sound is possible. This is only a reply to the objection if we import the point from I.28-29, that material dharmas resist other material dharmas. ‘Just as a hand strikes at a hand or a rock at a rock, these also [strike back].’40 Note how references to two hands clapping tie the two passages together. When one hand strikes at another, they do not become attached or coalesce. For the hands (or the two rocks), being rūpa, are said to resist or ‘strike back’ – a hand repels another hand simply by preventing it from occupying its own place and thus, at AKBh. I.43d, giving rise

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to sound. This, Vasubandhu claims, is what it means to say material dharmas resist each other.41 So the claim that atoms prevent each other from arising where they are is already in the background when Vasubandhu goes on to claim, in his argument against the void, that an atom is the only thing that precludes another atom from arising. Space is acausal (AKBh. II.55d) and, as a mere absence, can offer no resistance to the arising of material dharmas. By contrast, preventing another atom from being where it is is what any atom does just by its very existence. But if that is what an atom does just by occurring, then the Problem of Contact dissolves. Paramāṇus are dimensionless point-particle occurrences of properties. They are the converse of space: for an atom to be is to preclude or resist some other atom being just there where it is. If that is just what it is for an atom to be, then there need be no appeal to their ‘sides’ or directions, nor even to absolute space, in order for aggregation to be possible. Atoms aggregate, rather than coalesce, because any atom being here now precludes any other atom being here now. Coalescence is avoided not by a specific kind of contact or lack thereof, but by resistance. For an atom to be is for it to prevent the arising of any other atom at this point of occurrence. Nothing in this explanation requires that there be a point of contact distinct from the whole atom, or space between atoms to prevent collapse into one point. It is simply atoms happening that prevents that outcome. If we hold fast to this, then indeed we do not need to suppose that either points-of-contact or directional ‘parts’ are inevitably implied as ultimately real, whether by gappy locatedness or by immediate adjacency. Do these atoms therefore actually touch? Being able to answer this question negatively is essential to Vasubandhu’s dissolution of the problem – for actual touching, as that is commonly understood, does require a point of contact and would therefore imply partition. Yet Vasubandhu’s negative answer to this does not take the form of asserting the opposite, that the atoms are not in contact, as if floating in space at a safe distance from each other could solve the problem. No such distance could be safe, for if ‘sides’ are really real, then so too by the same reasoning must directions be real parts of atoms. This is the point at which Vasubandhu’s interpretation of Vasumitra is relevant. Vasumitra was the one, Vasubandhu approvingly tells us, who observed that ‘[atoms] do not touch; but there is a cognition of touch when they are without interval’ (AKBh I.43(10)). Atoms are without interval inasmuch as they resist each other and nothing else can offer an obstacle to an atom’s existing. But touching, or ‘being in contact’, is a relation between atoms, not something that belongs to any particular atom. As a relation, it is partite, does not survive analysis and is therefore at best conventionally and not ultimately real. So, similarly, being above or below are relations that we draw in the course of ordering our perceptions, not really existing perceptible things. To note that relations are for this reason conventional is to acknowledge that relations arise from our relating things – that is, from our cognizing essentially independent individuals as standing in some relation to one another. This is the significance of Vasumitra’s observation that the cognition ‘in contact’ or ‘these atoms are touching’ arises under certain circumstances – namely when

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atoms are (as indeed they always are) without interval. When we perceive macroobjects as without interval, the cognition that they are in contact arises. So similarly, when we consider mentally that each dimensionless point has an atom located there, a cognition that the adjacent atoms are touching arises.42 Because the relating of two (or more) atoms as ‘in contact’ or ‘touching’ is something we are doing in order to construct an intelligible reality to navigate, there is no danger that this should in any way affect the atoms themselves, which remain dimensionless points of resistance, regardless of what is going on around them or how we conceive of that. When relating these atoms to each other, we could think of the resisting atom as resisting upwards, resisting downwards and so on. But this is not a perception of what is happening independently of our conceiving it. An atom is not engaging some special force of ‘resistance’ which it then deploys outwards in various directions. The atom simply is – for it to be is for it to resist or obstruct other atoms being where it is. As with touching, thinking of an atom as having multiple directional parts which each resist in a different direction, then, is a way we conceive it, not a way it has of being. The atoms no more have directional parts than they have points of contact. Even when the Problem of Contact is revealed to be a Problem of Directionality, Vasubandhu argues that it is a non-issue: these directional parts are not real parts, but ex post facto ways we have of ordering phenomena. This is not so much tablebanging, then, as an invitation to be very precise about distinguishing cognitions, even inevitable cognitions of the world from how the world must in fact be. Thus, by the end of the discussion of atoms in the Abhidharmakośabhāṣya, Vasubandhu can legitimately dismiss the Problem of Contact as a non-problem. Atoms – dimensionless, directionless simple occurrences of properties – exist in immediate juxtaposition to each other, and exist unproblematically in this way. Aggregation is thus equally unproblematic, as are collision and cohesion between macro-objects. Concerning the original question of non-distal perception which gave rise to the discussion, Vasubandhu’s determination that ‘touch’ and ‘contact’ are conventional or conceptual realities enables him to distinguish distal from nondistal perception according to whether or not the macro-objects of sense organ and object are cognized as immediately juxtaposed – when they are, we say the organ is in contact with its object; when not, not.

THE UNEXPECTED ENCORE This elegant atomist account of contact and aggregation did not please all of Vasubandhu’s Abhidharma peers; but they seem not to have got the point that directionality partitions atoms every bit as much as contact, and so they persist in supposing the atomist cause would be helped by the introduction of empty spaces between atoms. More interesting is that Vasubandhu himself became dissatisfied with this atomist defence against the Problem of Contact. In his Twenty Verses, Vasubandhu gives the anti-atomist argument from contact a second outing – but this time he comes down on the other side and takes it to show that atomism is incoherent.43 Between the time of writing the two texts, Vasubandhu is said to have converted from the realist Abhidharma to the idealist Yogācāra – these

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labels are approximate and disputed, but adequate for our purposes.44 Vasubandhu had myriad reasons, nothing to do with partition and contact, for rejecting atomism simply as a form of realism. But to reject atomism as one species of realism, there is no need to revisit the particular problems special to atomism – one might do better to elaborate concerns over representationalism, say. Nevertheless, in verses 11-15 of the Twenty Verses, and the commentary on these, Vasubandhu reintroduces this argument from contact to argue specifically for the incoherence of atomism. What is it that has changed his mind about this argument in particular? Vasubandhu offers no critique of his previous solution, or dissolution, of the Problem of Contact/Directionality. The aggregation of atoms as such still may not tell decisively against atomism. But Vasubandhu has discovered a related phenomenon that does. While the immediate juxtaposition of atoms necessary for aggregation may be made possible by the resistance each atom offers to being collocated with another, such resistance cannot explain the opacity of such aggregations (Twenty Verses 14b), and so it cannot explain their visibility. Opacity requires that light falling on one side of a thing not be falling on the opposite side. When we think of macro-objects, such as trees, this is clear and apparently unproblematic. But Vasubandhu has already observed in the Abhidharmakośabhāṣya that no proper atomist can admit really existing qualities of the whole that are not qualities of its constituents. How is it possible for an aggregation of atoms to be opaque, to cast a shadow and block light, if none of its constituent atoms does so? But if an atom blocks light, then it would have to be lit on one side and shaded on another – and thus it would have sides. These sides, moreover, would have to be really distinct parts and not ex post facto described thusly by us for our convenience. There must in fact be light on part but not all of the atom, and that lit part must be distinct from the shaded part.45 Even if an atom is a point-particle and these two sides are in the same place, the lit side and shaded side are analysable without the original unit remaining – and therefore the atom is no atom in the Ᾱbhdiharma sense required in order to be ultimately real. The resistance that any atom has to anything else being where it is can account for aggregations, but not for the visibility of these aggregations, for the resistance implied by the occurrence of an atomic property-particular does nothing to account for the asymmetry of opacity. Such asymmetry is fatal to the simplicity of atoms. If Vasubandhu had worried about opacity when writing the Abhidharma­ kośabhāṣya, there is no evidence of it in the text; and the worry does not seem to have been raised in Abhidharma circles elsewhere. Its role in the Twenty Verses is to follow up arguments for the non-necessity of mind-independent reality with arguments for the incoherence of mind-independent reality as the Buddhist realists thought of it.46 However, while the Problem of Opacity was not likely Vasubandhu’s reason for discarding realism, it is very hard to see what Vasubandhu of the Abhidharmakośabhāṣya could have offered in defence of atomism, had he been presented with the argument from opacity. His ingenious dissolution of the Problem of Contact, and of the subtler Problem of Directionality which persists when contact is avoided by positing a gap, draws on the resistance any rūpa dharma offers, just by its existence, to there being another rūpa dharma just here. This resistance is not a property in addition to the specific characteristic event which is a

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rūpa dharma. It is just what ‘being blue here now’ or ‘solidity happening here now’ is. X occurring here now just is for nothing but X to be occurring here now. This wins Vasubandhu aggregation, collision and the avoidance of coalescence (or the collapse of the universe to a single point). But this is the most that Vasubandhu could hope to get from resistance-existence. No such argument can be made to account for the asymmetry inherent in opacity which gives three-dimensional aggregations their apparent depth.

IN CONCLUSION Abhidharma atomism is so severe it takes even conceptual distinguishability to indicate non-simplicity. It thus faces the problem of contact in a particularly acute form. Vasubandhu recognizes that no attempt to resolve the difficulty by appeal to an unfilled void between atoms is adequate, because the problem of contact is only a crude form of the subtler problem of directionality – and directionality persists through gaps between atoms. Extension and numerical divisibility are irrelevant to the refined problem of directionality, for even extensionless points must be oriented with respect to each other, and these respects must be distinct from each other within each atom of an aggregate. Vasubandhu’s deft solution is to insist that at the metaphysical level, atoms do not coalesce or become partite because the occurring of a rūpa dharma just is its exclusion of other there and then. If we, considering several such dharmas together, determine to draw a relationship of ‘above’ or ‘below’ between them, then this is a matter of how we are conceiving things and does not impugn the simplicity of the atom itself. Thus ‘touching’ is what we call macro-objects that we perceive, or atoms that we conceive, as having no interval between them. This elegant dissolution of the problem of directionality, however, cannot stretch to explaining opacity and shadow. Even dimensionless point-particulars must have genuinely distinct (if collocated) sides, one lit and the other not, if rūpa dharmas or their aggregations are to be perceptible. The real threat to atomism does not come from partition due to aggregation; the real threat arises rather from the perceptibility of the macro-objects that atoms were meant to explain.47

APPENDIX: TEXT AND TRANSLATION OF ABHIDHARMAKOŚABHĀṢYA I.43D Sanskrit text of AKBh. I.43d (1) śeṣaṃ tu ghrāṇa-jihvā-kāyâkhyam. 1.43d. trayam anyathā. (2) prāpta-viṣayam ity arthaḥ. ghrāṇaṃ kathaṃ prāpta-viṣayam. nirucchvāsasya gandhâgrahaṇāt.48 kêyaṃ prāptir nāma. nirantarôtpattiḥ. (3) kiṃ punaḥ paramāṇavaḥ spṛśanty anyônyam āhosvin na. (4) na spṛśanti iti kāśmīrakāḥ. kiṃ kāraṇam. yadi tāvat sarvâtmanā spṛśeyur miśrībhaveyur dravyāṇi. atha eka-deśena sâvayavāḥ prasajyeran. niravayavāś ca paramāṇavaḥ. (5) kathaṃ tarhi śabdâbhiniṣpattir bhavati.

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ata eva yadi hi spṛśeyur hasto haste 'bhyāhataḥ sajyeta upalaś ca upale. (6) kathaṃ citaṃ pratyāhataṃ na viśīryate. vāyu-dhātu-saṃdhāritatvāt. kaścid vāyu-dhātur vikiraṇāya pravṛtto yathā saṃvarttanyāṃ kaścit saṃdhāraṇāya yathā vivarttanyām iti. (7) katham idānīṃ nirantara-prāptyā prāpta-viṣayaṃ trayam ucyate. tad eva eṣāṃ nirantaratvaṃ49 yan madhye nāsti kiṃcit. (8) api khalu saṃghātāḥ sâvayavatvāt spṛśanti ity adoṣaḥ. (9) evaṃ ca kṛtvā ayam api grantha upapanno bhavati vibhāṣāyām. kiṃ nu spṛṣṭa-hetukaṃ spṛṣṭam utpadyate āhosvid aspṛṣṭa-hetukam iti praśnayitvā āha ‘kāraṇaṃ prati. kadācit spṛṣṭa-hetukam aspṛṣṭam utpadyate yadā viśīryate. kadācid aspṛṣṭa-hetukaṃ spṛṣṭaṃ yadā cayaṃ gacchati. kadācit spṛṣṭa-hetukaṃ spṛṣṭaṃ yadā cayavatāṃ cayaḥ. kadācid aspṛṣṭa-hetukam aspṛṣṭaṃ yadā vātâyatanaraja’ iti. (10) yadi paramāṇavaḥ spṛśeyur uttara-kṣaṇâvasthānaṃ syād iti bhadantavasumitraḥ. na spṛśanti. nirantare tu spṛṣṭa-saṃjñā iti bhadantaḥ. bhadanta-mataṃ ca eṣṭavyam. (11) anyathā hi sântarāṇāṃ paramāṇūnāṃ śūnyeṣv antareṣu gatiḥ kena pratibādhyeta. yataḥ sapratighā iṣyante. (12) na ca paramāṇubhyo 'nye saṃghātā iti. ta eva te saṃghātāḥ spṛśyante yathā rūpyante. (13) yadi ca paramāṇor digbhāgabhedaḥ kalpyate spṛṣṭasya aspṛṣṭasya vā sāvayavatva-prasaṅgaḥ. na50 cet spṛṣṭasya apy aprasaṅgaḥ.

English translation of AKBh. I.43d (1) But as for the rest, called ‘smell’, ‘taste’, and ‘touch’: I.43d. The three are otherwise.51 (2) That means that they attain their objects. In the case of smell, how is it that the object is attained? Because there is no grasping of an odour when there is no breathing in.52 What is this term ‘attain’? Occurring without interval. (3) But do atoms touch one another or not? (4) The Kaśmīris say, ‘they do not touch’. What is the reason? If they were to touch completely, things would coalesce. Then suppose instead that they were to touch at one spot. There would be the unwanted result53 that they have parts – and atoms do not have parts. (5) Then how is there the production of sound? For that very reason. For if they were to touch, a hand that strikes at a hand, a rock that strikes at a rock, would become attached. (6) How does a heap that is struck not break apart? Because it is in a state of being held together by the air element (vāyu-dhātu).54 A certain air element acts for the purpose of scattering, just as in the destruction of the world; a certain one for the purpose of holding together, just as in the creation of the world. (7) How in this case can it be said that the three [organs] attain their object by reaching without interval? There not being anything in the middle is indeed a state of non-interval for these [organs]. (8) Moreover, the statement that ‘agglomerations touch because they have parts’ is without error. (9) And supposing thus, this section in the Vibhāṣā is correct: Having asked the question ‘Now does a thing-in-contact arise caused by a thing-incontact or caused by a thing-not-in-contact?’, it then answers: ‘It depends on the cause. Sometimes, a thing-not-in-contact arises caused by a thing-in-contact when it breaks. Sometimes a thing-in-contact is caused by a thing-not-in-contact when it goes to a heap. Sometimes a thing-in-contact is caused by a thing-in-contact55 when

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heaps are combined. Sometimes a thing-not-in-contact is caused by a thing-not-incontact, as when dust stays in the air.’ (10) Bhadanta Vasumitra says, ‘if atoms were to touch, they would stay a moment later’. Bhadanta says, ‘they do not touch; but there is a cognition of touch when they are without interval’. And the opinion of Bhadanta ought to be accepted. (11) For otherwise, were atoms to have space in between them, by what would their going into the empty intervals be restrained? For atoms need to have resistance.56 (12) And it is not the case that agglomerations are anything other than atoms. Those very things [i.e. atoms] can be made an object of touch insofar as they are agglomerations, just as they can be made an object of perception. (13) And regarding atoms: if division according to directions is being supposed,57 then there is the unwanted result that they have parts – whether they touch or not. If not supposed, then even when they do touch, there is no unwanted result.

NOTES 1. For an excellent overview of Vasubandhu’s contested biography and the full range of his philosophical contributions, see Gold (2018). 2. We are not well served for English translations of this central Buddhist philosophical text. The standard complete translation into English is by Leo Pruden (Berkeley, CA: Asian Humanities Press, 1988); but this is a translation from the French of the great Buddhologist Louis de la Vallée Poussin’s L’Abhidharmakośabhāṣya de Vasubandhu (1923–1931) translation from the Sanskrit. De la Vallée Poussin is not careful to translate the same Sanskrit term with the same French term, nor does he indicate when he chooses not to do so; and he often interpolates text, whether from commentary or supplying his own expansions, without marking them as such. These features are directly carried over into Pruden’s English translation. Accordingly, all quotations from this text are our own translation, unless otherwise noted. Our translation of the target passage of this discussion from Pradhan’s Sanskrit edition may be found in the appendix to this chapter. 3. Charles Goodman (2004) offers detailed philosophical examination along these lines. See also Siderits (1997, 455–78), and Ganeri (2001, 101–2). 4. The Māhavibhāṣā (Great Compendium), or Vibhāṣā for short, is a compilation of Abhidharma Sarvāstivāda debates. Each debate begins with a statement introducing the issue, followed by a compilation of the views and arguments from various Abhidharma scholars and anonymous sources. Among these scholars is Vasumitra, who is often endorsed by the Vibhāṣā. Vasubandhu often agrees with Vasumitra as well, though he disputes the Vibhāṣā’s understanding of Vasumitra (see Potter 1998, 111–19). 5. It is a mistake, however, to suppose (as Warder (1971) does and, following him, Goodman (2004)) that the rejection of substance-property metaphysics is a rejection of substance in every sense of it. Substantia translates the Greek ousia, and all subsequent philosophical discussion of ‘substance’ and its cognates inherits not only connotations of ‘standing under’, but also more centrally and ineliminably connotations of being, including individuation and identity. Being properly

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individuated is crucial enough to substance for Aristotle to dismiss matter’s claim to be substance – although it ‘stands under’ everything – on the ground that it is not an individual (a ‘this-something’), in Metaphysics Ζ.3. And conversely, substance does not, as Goodman seems to think (2004, 397), require connotations of ‘stuffyness’ – indeed even for Aristotle (the father of substance-property metaphysics) what is most a substance, mostly completely a being, has no matter of any sort whatsoever. 6. Ronkin (2005, ch. 2) argues that this event metaphysics evolved out of an earlier process metaphysics. 7. It also relies on a clever argument about whether absences, or non-existents, can be caused by a presently existing thing. 8. In fact, the context for Vasubandhu’s momentariness argument is an argument against the reality of motion. 9. Goodman (2004) rejects the atomist label for Vasubandhu’s position on two misguided grounds. The first is the assumption that to reject substance as ‘that in which properties inhere’ is eo ipso to reject substances as discrete individuals with their distinct identities. Dharmas are well-individuated items and are moreover that of which everything large or complex consists. In these ways, dharmas do play the substance role in the same way Democritean atoms do. See note 5. The second ground Goodman offers for rejecting the attribution of atomism is textual. We address this in notes 17 and 19. 10. yatra bhinne na* tad-buddhir anyâpohe dhiyā ca tat. ghaṭâmbu-vat saṃvṛtisat paramārthasad anyathā (AKBh. 6.4) – with Pradhan’s instrumental ‘bhinnena’ emended to a locative absolute and negation, ‘bhinne na’ (Pradhan 1975). 11. Scade (2013, 80–105) offers valuable discussion of the distinction made by the Stoics between divisibility according to real parts and mere in principle divisibility within a mass. 12. In  Metaphysics Α4, 985b6, Aristotle reports that the ‘differences [between the atoms] are three – shape, arrangement and position’ (KRS 555); Simplicius (de caelo 295) reports that the atoms ‘have all sorts of forms and shapes and differences in size’ (KRS 556, DK 68A37). Democritus seems to have thought there was a soul, but it is not clear he had a good atomist account of this. Was it an agglomeration or a single atom? 13. A quick defence of this principle would run along the lines: it is absurd to think that two (or multiple) things are one thing, although of course we can always consider many things as one (as for instance, with an army or a forest), if it is useful. All that is actually there are the several individuals, each of which is not any other but itself. As with the soldiers of an army, so with the properties of a conventional object: blueness and heaviness are two quite distinct things, and nothing can make them a single thing – except, of course, so considering them because it is useful; but this is no guide to reality (see Carpenter 2014, 35–47, for more detailed discussion). Democritus appeals to this thought – in Barnes’ (1982, 269) translation of Simplicius, de caelo 295.11 (KRS 583, DK 68A37), ‘It is absolutely silly to think that two or more things could ever become one’ – though he does not pursue the point as thoroughly as Vasubandhu’s metaphysics does. The Buddhists appealed explicitly (sometimes extensively) to the one-cannot-be-many principle in later texts (Śāntarakṣita’s Madhyamaka-Alaṃkāra is a well-known case).

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14. It has also been called a kind of reductionism (Siderits 1997, 2003) and a kind of fictionalism (Matilal 1970, 83–110, 93; Garfield 2006, 1–7; Siderits 2009; D’Amato 2013); see recent valuable discussion of both in Sauchelli (2016). These labels are not necessarily incompatible with characterizing the Abhidharma position as an atomistic trope theory. Any such categorization of the view must, however, be sure to accommodate the fact that Vasubandhu goes on here to insist that conventional reality is not false (‘So if one says, conforming to convention, ‘There is water’, ‘There is a pot’, one speaks truly and not falsely’, AKBh. VI.4). Ronkin (2005) offers insightful discussion of a range of evolving Abhidharma views in terms of atomism and momentariness. 15. In an appendix, Kapstein (2001b) offers a more recent and reliable translation of the whole of Abhidharmakośabhāṣya IX; discussion relevant to this point can be found in what he marks as section 4. 16. So at least runs Pruden’s bald translation. More accurately, the verse says, ‘It is not in the atom. [Comment:] Nor is shape or length and the rest perceived in the atom’ (na ca anṇau tat. na ca saṃsthāṇaṃ paramāṇau vidhyate dīrghâdi). It is context – namely that the whole passage is arguing against the ultimate reality of length – that warrants Pruden’s (or rather de la Vallée Poussin’s) freedom here. 17. Goodman (2004) thinks that the absence of a similarly reductive argument for ‘derived form’ (molecules composed of atoms) implies that Vasubandhu takes such derived form to be ultimately real, and so he cannot be an atomist. According to Goodman, Vasubandhu thinks ‘tropes that are derived form really do exist; they are merely less fundamental than the basic physical tropes’ (the mahā-dhātu; 2004, 399). But the textual grounds for this interpretation are weak (see note 19). Vasubandhu’s own definition (AKBh. I.12ab) of the difference between primary (mahat) and ‘derived’ or ‘secondary’ (upādāya) specifies that it hinges not only on the former ‘supporting’ (dhāraṇa) the latter but also on what is primary having svalakṣaṇa (distinctive characteristic), later glossed as svabhāva (proper nature). But having svabhāva is tantamount to being ultimately real, so that Vasubandhu is effectively saying that primary elements differ from derived by being ultimately real. There is a further philosophical difficulty with Goodman’s account, for Vasubandhu has no vocabulary, nor conceptual space, for ultimately real things of varying fundamentality, which Goodman’s non-reductive trope theory would require. 18. Contra Kapstein (2001a, 191–4), who supposes that Vasubandhu had available to him (and presumably himself held in the AKBh.) only a ‘pre-modern minimal part atomism’ (191), we see that Vasubandhu himself must have supposed that non-mental dharmas were extensionless. Nor does Vasubandhu equate resistance with extension (2001a, 193–4), either in the Abhidharmakośabhāṣya or in his later Twenty Verses (the relevance of this will become evident in what follows), and there is therefore reason to doubt Kapstein’s further claim that ‘a cogent point-particle theory would mitigate severely the force of ’ Vasubandhu’s later criticism brought to bear on his own atomist picture, if only he could have considered it. 19. Goodman (2014) takes Vasubandhu’s paramāṇus to be complex, not simple – molecules and not atoms at all. Coseru (2012, 80) agrees, taking this as a Sautrāntika view adopted by Vasubandhu: ‘the Sautrāntikas conceive the atom (paramāṇu) not as a substantial impartite entity, but rather as the subtlest collection of material elements (rupasaṃghāta) [sic]’. They both base this claim

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on AKBh. 2.22. But this passage first of all attributes this use of the word ‘paramāṇu’ to another text, the Kāmadhātu; and then distinguishes the meaning of the word in that text from Vasubandhu’s own. ‘In Kāmadhātu, a paramāṇu which is without sound and without an organ is made of eight things (dravyas). Because it is completely minute, an aggregate of forms (rūpa-saṃghāta) is called “paramāṇu” – from which it should not be understood as the other one’ (kāme ‘ṣṭa-dravyako ‘ś́ abdaḥ paramāṇur anindriyaḥ. sarva-sūkṣmo hi rūpa-saṃghātaḥ paramāṇur ity ucyāte. yato na anyataro vijñāyeta. AKBh. II.22). That is to say, the Kāmadhātu calls basic molecules ‘paramāṇu’, because they share with actual paramāṇus that they are tiny (and physically inseparable, see below); but they should not for this reason be confused with actual paramāṇus, discussed previously, which are not composed of several things. It is only after urging his readers to keep this caveat in mind that he takes up the Kāmadhātu’s use of ‘paramāṇu’ when talking about molecules. Saṃghabhadra confirms this construal. In his comment on this passage (Nyāyānusāra, quoted by Pruden 332, n. 95), Saṃghabhadra confirms that paramāṇu are simple, not divisible conceptually or physically; he then distinguishes between ‘paramāṇu’ (primary/ultimate unit) and ‘saṃghātāṇu’ (aggregated unit) and clarifies that the Kāmadhātu is actually talking about saṃghātāṇu when it says ‘paramāṇu’. The reason it is an ‘aṇu’ (unit) is because it is ‘not susceptible of disaggregation’ (even though it is not partless like paramāṇu). 20. Hereafter, ‘atom’ will be used specifically to designate paramāṇus, or rūpa dharmas – sensible property-particular events such as ‘this warmth here’ or ‘that blue there’. 21. dravyāṇi, namely the atoms themselves. Using the generic word ‘thing’ here picks up on the close association in Abhidharma thought between dharma and dravya (thing) to denote ultimately existing objects and the general association of fundamentally or ultimately real (paramārthasat) with substantially real (dravyasat). 22. Thus, on behalf of the Kaśmīris, Vasubandhu replies to the objection that without contact there would be no sound, by observing that with contact there would be no sound. In a world of atoms, contact would be tantamount to co-occupation of the same space – or coalescence; and coalescence cannot cause sound. ‘For if they were touching, a hand that strikes at a hand, a rock that strikes at a rock, would become attached’ (AKBh I.43d(5)). 23. There is, however, a way in which Vasubandhu could happily endorse the claim, by stressing that there is no error in cognizing that agglomerations touch. In fact, it is possible to read the Sanskrit as saying just that, since the particle of quotation ‘iti’ conveys that the claim in quotes is being thought or said by someone. And Vasubandhu would happily acknowledge the very same at the atomic level; see discussion below. 24. keyaṃ prāptir nāma. nirantarôtpattihḥ, AKBh. I.43d(2). 25. This might look as if the Vaibhāṣikas are perversely interpreting ‘without interval’ as ‘with only an interval’, for there is nothing but empty space between two atoms. Though antara consistently means ‘interval’, the prefix nis- can be read as either a negation or a strengthening particle. So, as Saṃghabhadra – Vasubandhu’s Abhidharma contemporary and rival – points out, nir-antara can mean ‘without an interval’, but it could also mean the opposite, ‘certainly having an interval’. Saṃghabhadra endorses this: ‘the word nirantara signifies “close”. The prefix

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nis signifies “certitude”. As there is certainly an interval, the atoms are nirantara, “possessing intervals” ’. (Nyāyānusāra, quoted by Pruden 1994, 149, n. 191 (following de la Vallée Poussin), whose translation this is). We could, he goes on to say, take the nir- prefix to indicate absence, provided this is understood as absence of an intervening perceptible dharma. 26. This additional claim, that there is unfilled space between the atoms that are in non-separation, is not explicitly included here by Vasubandhu; but it is explicit in Saṃghabhandra’s defence of the view, and it is the aspect that will concern Vasubandhu when he critiques the position. 27. Though it may be too much to swallow, after all. If the agglomerations are after all only ‘in contact’ in some attenuated sense – not contact, but at an unfilled distance from each other – then there is no difference between distal and nondistal perception; both ‘attain’ their objects when there is nothing obstructing the intervening space between organ and object. 28. His worry, for the record, is that ‘if atoms were to touch, they would stay a moment later’ (AKBh. I.43d(10)). It is not clear to us exactly why this would be, nor how his solution addresses his particular worry. Unravelling this does not (so far) seem to bear on the Problem of Contact, nor Vasubandhu’s solution to it. 29. ‘the cognition’ renders saṃjña, a word used in five-skandha analyses for mental events which can be true and false, as distinct from perceptions-sensations which are pleasant, painful and neutral. 30. In word (only), Vasubandhu’s Vaibhāṣika opponents agree; they claim the ‘no intervening body’ construal represents Vasumitra’s position. Technically, Vasubandhu is here disputing their reading of Vasumitra. 31. As quoted by Pruden 1994, vol. 1, 149, n. 191. 32. This will be more obvious in the Sanskrit where the neutral word for ‘direction’ is digbhāga, a compound word dig-bhāga, literally ‘direction-parts’. Pruden glosses ‘digbhāga’ as ‘spatial division’, which is not entirely accurate, and obscures the Problem of Directionality that Vasubandhu is introducing here (see below). It also risks seeming even more like a non-answer to the Vaibhāṣika worry than it already appears to be. 33. Stefan Anacker sees this correctly in his note on Vasubandhu’s Twenty Verses: ‘any atom’s being in a positional relation to another, implies that the atom has parts, and thus is not really an atom’ (Anacker 1984, 177, n. 15). 34. AKBh. II.55d, where Vasubandhu reiterates (in Pruden’s translation) that ‘What is called ‘space’ (ākāśa) is solely the absence of any tangible thing, that is, the absence of a resistant body. Persons say, in their obscurity, that there is space when they do not encounter any obstacle’. 35. sapratighā iṣyante, AKBh. I.43d(11). 36. The other is ‘breakability’ or liability to destruction and deterioration. The passage is difficult and inconclusive – an objection is brought which seems in its content to apply to the ‘breakability’ definition, but in its grammar to refer to the ‘resistance’ definition. Vasubandhu answers the objection, but it is unclear whether he does so in his own voice or whether he is here just offering the received answer to known objections.

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37. sapratighā daśa rūpiṇaḥAKBh. I.29bc. 38. sva-deśe parasya utpatti pratibandha, AKBh. I.29bc. 39. We make an effort to be careful with the formulation of this, for we would not want to commit Vasubandhu to the view that material dharmas have resistance as a property, for this would partition atoms just as surely as contact would. Vasubandhu is clear that ‘having resistance’ is nothing other than a property-occurrence precluding the co-occurrence of another. For further unpacking of this, see note 41. 40. yathā hasto haste pratihanyate upale vā. upalo 'pi tayoḥ, AKBh. I.28. 41. This basic conception of rūpa as resisting may also be implicit in Vasubandhu’s later canonical statement of the distinction between conventional and ultimate reality at AKBh. VI.4, where he offers rūpa as the prime example of non-conventional, ultimately existing reality: ‘For example, rūpa: if one breaks it into atoms, one can remember smell and other dharmas in the mind, but the comprehension of the nature of rūpa persists.’ One can dissolve any ‘material’ object (form, rūpa) into its constituent atoms; each atom would be rūpa, so that rūpa remains even after analysis bottoms out. Contrast this with water, which is not to be found in any of its constituents (liquidity, coolness, wetness). Each constituent atom of a rūpa macroobject remains itself rūpa – not ‘breakable’, as the first definition of rūpa at AKBh. I.13 has it, but resisting the existence of another perceptible quality, by being the occurrent perceptible quality it is. 42. And this might even be a more plausible interpretation of Vasumitra than the Vaibhāṣika interpretation. For it is unclear why we would cognize that these atoms touch when they are ‘without interval’ in the Vaibhāṣika sense. After all, ‘without interval’ in this case means that atoms are separated by space. It is strange to think that we would come to cognize that these atoms are touching when they are without interval in this way. We cannot defend this by claiming that atoms are tiny and the gap is small, causing us to perceive them as touching. For atoms are imperceptible – it is extended objects comprised of atoms that we see. Thus, this can only be meant in a conceptual sense, and here the Vaibhāṣika interpretation fails to explain why we would conceive that they are in contact when there is a space between the atoms. In Vasubandhu’s view, however, the presence of atoms at every dimensionless point would understandably cause us to cognize them as touching. 43. In fact, it is this later discussion of atoms and aggregation that is by far the better known among scholars. Stefan Anacker (1984, rev. 2005) offers a reliable and complete translation of verses and auto-commentary as well as the Sanskrit in an appendix. More recently, Das (2018) translates the Twenty Verses, though incompletely. 44. In the mid-twentieth century, some scholars also disputed the received view that the same person wrote texts in defence of these different positions (Frauwallner 1951). Broad, though not complete, consensus has settled on the view that they were indeed written by the same person (see Gold 2015), with some even going so far as to suppose that this person never endorsed the Abhidharma view he articulated in the Abhidharmakośabhāṣya (Kritzer 2003, 331–84 and 2005). 45. It is not clear, though probably relevant, what Vasubandhu thought light actually was. If it is just another atom, this will not help the atomist picture; and if it is not, what should it be instead?

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46. See Carpenter (2014, 142–5) for analysis of the structure of the Twenty Verses, and Carpenter (2014, 147–50) and Kapstein (2001a) for discussion of its anti-atomist arguments. 47. We would like to thank participants of the Durham conference on the History of Atomism for discussion of the initial ideas behind this paper, and Ugo Zilioli in particular for the invitation which has led to such fruitful and interesting lines of inquiry, well beyond the scope of the current paper. Our thanks also go to David Brick and Malcolm Keating for their invaluable comments and exchanges on this translation, as well as to Ng Sai Ying for reading the translation for clarity. 48. It is grammatically ambiguous whether this compound should be taken as ‘gandhāgrahaṇāt’ (‘because there is the grasping of an odour’) or ‘gandhā-agrahaṇāt’ (‘because there is no grasping of an odour’). But the latter reading of the compound makes more philosophical sense, and Paramārtha and Xuánzàng’s Chinese translations, as well as Jinamitra and dPal brtsegs rakṣita’s Tibetan translation of the text all translate this line with the negation. 49. Pradhan: niruttaratva. Pradhan provides ‘nirantaratva’ as an attested alternative at 32n22, and we have corrected the text from ‘niruttaratva’ to ‘nirantaratva’ – ‘niruttaratva’ would mean ‘a state of having no superior’ rather than ‘a state of non-interval’. 50. Pradhan: no. We have emended ‘no’ to the negation ‘na’, which Pradhan provides as an attested alternative at 33n3. 51. The preceding line of the verse reads: ‘sight, hearing, and the mind do not attain their objects’. Skt: cakṣuḥ-śrotra-mano 'prāpta-viṣayaṃ. 52. Put simply, smell is said to attain its object because grasping an odour requires there to be breathing in. It is therefore rightly categorized under the non-distal senses. 53. The verb is ‘prasajyeran’, which implies a negative or undesirable result. 54. Vāyu-dhātu roughly translates as ‘air element’ and is one of the four primary (mahat) elements which ‘support’ all other rūpas (AKBh I.12). 55. Pruden’s translation has this as ‘Sometimes a thing-in-contact arises from a thingoutside-of-contact, as when agglomerations come together’ (121, italics mine). Since the word here is ‘spṛṣṭa-hetukaṃ’ and not ‘aspṛṣṭa-hetukaṃ’ with the privative a-, the italicized phrase should read ‘thing-in-contact’ rather than its negation. 56. The Sanskrit is ambiguous here between meaning (1) that atoms need to have (intrinsically) the characteristic of resistance or (2) that atoms need to have resistance from something else, that is, that they need to be resisted by something else. 57. Here, the verb translated as ‘being supposed’ is the passive causative kalpyate. Note that the root kḷp immediately suggests for the Buddhists the act of conceptual construction, and so the conventional reality of the thing being conceptualized – in this case, division according to directions.

REFERENCES Anacker, S. ([1984] 2005), Seven Texts of Vasubandhu, the Buddhist Psychological Doctor, Delhi: Motilal Banarsidass.

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Barnes, J. (1982), The Presocratic Philosphers, London: Routledge. Carpenter, A. D. (2014), Indian Buddhist Philosophy, Durham: Acumen (now Routledge). Coseru, C. (2012), ‘Cognitive awareness and its object’, in Perceiving Reality, 80, New York: Oxford University Press. D’Amato, M. (2013), ‘Buddhist fictionalism’, Sophia 52: 409–24. Das, N., trans. (2018), ‘Vasubandhu’s twenty verses with auto-commentary’, in G. Rosen, A. Byrne, E. Harman, J. Cohen and S. Shiffrin (eds), The Norton Introduction to Philosophy (Second Edition), New York: W. W. Norton. Frauwallner, E. (1951), On the Date of the Buddhist Master of the Law Vasubandhu, Rome: Istituto Italiano per il Medio ed Estremo Oriente. Ganeri, J. (2001), Philosophy in Classical India, 101–2, New York: Routledge. Garfield, J. L. (2006), ‘Reductionism and fictionalism: Comments on Siderits’s Personal Identity and Buddhist Philosophy’, APA Newsletter on Asian and Asian-American Philosophy 6, no. 1: 1–7. Gold, J. (2015), Paving the Great Way, New York: Columbia University Press. Gold, J. (2018), ‘Vasubandhu’, in Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Summer 2018 Edition). https​:/​/pl​​ato​.s​​tanfo​​rd​.ed​​u​/arc​​hives​​/sum2​​018​/e​​ntrie​​​ s​/vas​​uband​​hu Goodman, C. (2004), ‘The Treasury of Metaphysics and the physical world’, Philosophical Quarterly 54, no. 216: 389–401. Kapstein, M. (2001a), ‘Mereological considerations in Vasubandhu’s “Proof of Idealism”’, in Reason’s Traces, 191–4, Somerville: Wisdom Publications. Kapstein, M. (2001b), Reason’s Traces, Boston: Wisdom Publications. Kritzer, R. (2003), ‘Sautrāntika in the Abhidharmakośabhāṣya’, Journal of the International Association of Buddhist Studies 26: 331–84. Kritzer, R. (2005), Vasubandhu and the Yogācārabhūmi: Yogācāra Elements in the Abhidharmakośabhāṣya, Studia Philological Buddhiaca XVIII, Tokyo: International Institute for Buddhist Studies. La Vallée Poussin, Louis de, trans. (1923–31), L'Abhidharmakosha de Vasubandhu, 6 vols, Paris: Paul Geuthner. Matilal, B. K. (1970), ‘Reference and existence in Nyāya and Buddhist philosophy’, Journal of Indian Philosophy 1: 83–110. Potter, K., ed. (1998), Encyclopedia of Indian Philosophies Vol. VII: Abhidharma Buddhism up to 150 A.D., 111–19, Delhi: Motilal Banarsidass Publishers. Pradhan, P., ed. (1975), Abhidharmakośabhāṣyam of Vasubandhu, rev. 2nd edn, Patna: K.P. Jayaswal Research Center. Pruden, Leo M., trans. (1994), Abhidharmakośabhāṣyam of Vasubandhu, vol. 1. Berkeley: Asian Humanities Press. Ronkin, N. (2005), Early Buddhist Metaphysics, New York: Routledge. Sauchelli, A. (2016), ‘Buddhist reductionism, fictionalism about the self, and Buddhist fictionalism’, Philosophy East and West 66: 1273–91. Scade, P. (2013), ‘Plato and the Stoics on limits, parts and wholes’, in A. G. Long (ed.), Plato and the Stoics, 80–105, Cambridge: Cambridge University Press. Siderits, M. (1997), ‘Buddhist reductionism’, Philosophy East and West 47: 455–78.

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Siderits, M. (2003), Personal Identity and Buddhist Philosophy: Empty Persons. Burlington: Ashgate. Siderits, M. (2009), Is reductionism expressible? in M. D’Amato, J. L. Garfield and T. J. F. Tillemans (eds), Pointing at the Moon: Buddhism, Logic, Analytic Philosophy, 57–69. Oxford: Oxford University Press. Warder, A. K. (1971), ‘Dharmas and data’, Journal of Indian Philosophy 1, no. 3: 272–95.

CHAPTER 9

Aggregates versus wholes An unresolved debate between the Nyāya-Vaiśeṣika and Buddhist schools in ancient Indian atomism SAHOTRA SARKAR

INTRODUCTION Scientific disputes within classical Indian philosophy have received relatively little critical attention in contemporary philosophy compared to their idealistic metaphysical counterparts. This chapter is intended as a modest antidote. But, before we begin, we should note that what constitutes Indian philosophy is a matter of dispute, particularly among South Asian scholars. Here, following Chatterjee (2016, 494), Indian philosophy will refer ‘to those systems of thought that were originated in India within the first 1500 years of CE and their extensions’. But, as Chatterjee also notes: ‘Indian philosophy is a myth created by Indologists and orientalists. Just as there was no Indian nation before 1947 [which marks independence of the subcontinent from British rule], similarly there was no monolithic body of thought called “Indian philosophy” before the advent of the orientalists’ (494). Nevertheless, this flawed but useful history-dependent characterization will suffice for the purposes of this chapter since the texts it will rely on date no later than the first few centuries CE.1 In both research and pedagogy, the orientalists are also responsible for the focus of attention on idealistic metaphysical systems of Indian thought, especially Vedanta. Nevertheless, thanks to the efforts of scholars, starting over a century ago with Seal (1915), it is also now widely acknowledged that naturalistic and, in that limited sense, scientific accounts of the world formed a major part of the philosophical corpus of ancient India. What is far less clear is how these Indian discussions compare and relate to contemporaneous developments elsewhere or, for that matter, to modern disputes that occupy philosophers today.2 This is as true of classical Indian discussions of atomism as of other scientific matters broached in that corpus. The purpose of this chapter is to analyse one scientific debate within classical Indian atomism, that between the orthodox Nyāya-Vaiśeṣika school and the heterodox Sautrānika and Vaibhāṣika Buddhist schools, though perhaps only from a

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Nyāya-Vaiśeṣika perspective because of the absence of the original Buddhist sources. While both sides to this debate endorsed atomism in the sense that they accepted that objects of the everyday material world consist of minute ultimate units, they differed about how these units co-occurred in the composite entities that constitute the macroscopic world.3 Nyāya-Vaiśeṣika held that composites are wholes that have an individuality distinct from their component atoms; the Buddhists held that they are aggregates that had no distinct individuality beyond the components. Much of this debate centred around the interpretations of the emergence of composite properties that were not shared by the parts and on how the composite whole is related to its component parts. Though there is a large corpus of later commentary, the core of this debate occurred before or shortly after the beginning of the Common Era and the central text is the Nyāya-sūtra of Gautama which will occupy most of this chapter. There was, of course, no possibility of empirical resolution of this dispute at the time (even if there had been any special veridicality attributed to empirical data by both sides – and there was not). Arguments were supposed to be decided on the basis of logic and metaphysical plausibility. Nevertheless, the debates that occurred can inform to some extent our contemporary discussions about reduction, holism and emergence. The section that follows provides some background to classical Indian atomism. Since exegesis-for-itself is not a goal of this chapter, it is short. The third section notes the major tenets of Nyāya-Vaiśeṣika atomism that are relevant to the debate between Nyāya-Vaiśeṣika and the Buddhists, including the role of Kanāda’s Vaiśeṣika-sutrā. The fourth section, the core of this chapter, turns to Gautama’s Nyāya-sūtra and the commentary (Bhāṣya) by Vātsyāyana which provided its first (surviving) systematic exposition. These works are the only extant sources of early Buddhist atomism.4 The final section tries to draw out the modern contemporary implications of the debate.

BACKGROUND: ATOMISM IN CLASSICAL INDIAN PHILOSOPHY While there is no canonical chronology of periods of Indian philosophy, consistent with how ‘Indian philosophy’ was temporally delimited in the previous section, classical Indian philosophy will be construed here at beginning around 600 BCE and lasting until 1500 CE (see, however, Ganeri 2001) though nothing much will hinge on the exact dates. According to orientalist tradition, there were six orthodox or āstika systems (those that accepted the authority of the Vedas as eternal and infallible).5 Of these, the Vedanta, the Sāmkhya and Yoga clearly did not accept atomism in the sense of the existence of minute ultimate units of the material world; in contrast, Mīmāṣā, Vaiśeṣika and Nyāya clearly accept such atomism, though all such claims are subject to the usual interpretive controversies (Gangogpadhyaya 1981, 3–4).6 Atomism was central to the doctrines of both the Vaiśeṣika and Nyāya systems (unlike Mīmāṣā). The Vaiśeṣika and Nyāya systems began as independent systems well before the Common Era. Of these, the former has traditionally been regarded as the older

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with some scholars (perhaps somewhat implausibly) dating it back to 1300 BCE (Chatterjee 2011). In the Common Era, the two systems came to be treated together because of shared tenets (samānatantra).7 For instance, Chatterjee (2011, 112) notes six of these: a common-sense realism; a pluralist ontology; a belief that the world is created from material atoms; that these atoms were conjoined by ‘God’; an account of causation in which an effect is produced by a cause rather than being a manifestation of that cause; and that liberation is the cessation of suffering. Nevertheless, there were important differences. Vaiśeṣika doctrine held the world to be composed of seven categories: substance (dravya), quality (guṇa), action (karma), universal (sāmānya), ultimate differentiator (viśeṣa), inherence (samavāya) and absence (abbhāva). In contrast, Nyāya doctrine held that there were sixteen categories: valid means of knowing (pramāṇa), objects of knowing (prameya), doubt (saṃśaya), aim (prayojana), example (dṛṣṭānta), established conclusion (siddhānta), constituents of arguments (avayava), argument (tarka), decision (nirṇaya), honest debate (vāda), tricky debate (jalpa), destructive debate (vitaṇḍa), fallacy (hetvābhāsa), underhanded trick (chala), false rejoinder (jāti) and defeat (nigraha-sthāna). The former accepted two modes of cognition or proof: perception (pratyakṣa) and inference (anumāna); the latter admitted four by also embracing comparison (upamāna) and testimony/ authority (śabda). The syncretic Nyāya-Vaiśeṣika system emerged when the Nyāya system was enlarged to embrace the seven Vaiśeṣika categories. Cohabitation was achieved at the cost of ontological profligacy – but a critique of that and of the heterogeneity of the categories will be left for another occasion. The source of Nyāya-Vaiśeṣika atomism was the Upaniṣadic doctrine of pañcamanābhūtas (or five elements): earth, water, fire, air and vyom/ākāśa. These are five of the nine members of the category of substance (dravya), the other four being space, time, self and mind. In Nyāya-Vaiśeṣika texts, two terms were used for atoms: aṇu and paramāṇu. The former was used in the Upaniṣads as both a noun (for any very small entity) and an adjective (meaning very small). It began to be used in the sense of atom only later, in the tradition of sutrās (aphorisms) commenting on the Vedas and thereby establishing the various systems of philosophy. Paramāṇu refers to the ultimate small entities of matter, that is, atoms in general (Gangopadhyaya 1981, 2). According to Nyāya-Vaiśeṣika, atoms are eternal, whereas their composites (‘wholes’) are not because they are ultimately destroyed, for instance, by decomposition into their components (and their subsequent recombination into new composites). There will be more on Nyāya-Vaiśeṣika atomism in the next and subsequent sections. Besides the orthodox systems, classical Indian philosophy included three heterodox (nāstika) families of systems. Of these, no reliable texts survive for the Cārvāka systems and it is impossible to classify them credibly as atomist or not. The Jaina systems accepted atomism uniformly.8 The Buddhist systems included both those that endorsed atomism and those that were opposed to it. These systems can broadly be divided into two groups: the Hīnayāna, comprised of the Sautrānika and Vaibhāṣika systems; and the Mahāyāna, comprised of the Mādhyamika and Yogācāra systems. The two Hīnayāna systems, which are historically earlier, are typically regarded as realist (because they endorse the reality of the external world); both endorse atomism. The Mahāyāna systems reject the reality of the external world

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and, also, atomism with the Yogācāra system being comparatively strident in its rejection of atoms. Though both the Sautrānika and Vaibhāṣika systems endorse the reality of external entities (and of atoms), they differed insofar as the former held that these entities can only be inferred, whereas the latter claimed that they could be directly apprehended. It is impossible to state the Sautrānika doctine with any confidence because no seminal text has survived. In the case of the Vaibhāṣika system, the situation is somewhat but not much better in our context. An important text, the Abhidharma-kośa of Vasubandhu has survived along with a well-known commentary by Yaśomitra.9 However, the Abhidharma-kośa is from the fifth century CE (and Yaśomitra’s commentary is from the sixth century CE), whereas the debate between the Buddhist and Nyāya-Vaiśeṣika atomic doctrines discussed further are from no later than the second century CE. We will assume that some basic doctrines of the Abhidharma-kośa date back to this earlier period. Given these problems with sources, the extent to which Sautrānika and Vaibhāṣika atomism differed remains unclear. According to the latter, if we draw on the Abhidharma-kośa, atoms are the smallest units of the rūpa-skandha (collections of forms that constitute the material world). These include the five sense-organs, that is, the visual, the auditory, the olfactory, the gustatory and the cutaneous. Each of these consists of atoms with a distinctive shape: those of the visual organ have the shape of an ajājī (cumin) flower; those of the auditory organ, the shape of a bhūrja (Himalayan birch, Betula bhojapatra) leaf; those of the olfactory, the shape of slender iron sticks (śalākā); those of the gustatory organ, the shape of the halfmoon; and those of the cutaneous organ, the shape of the body itself. Atoms are thus somewhat indirectly related to sensory experience. There is a sharp contrast here with Nyāya-Vaiśeṣika thinking which treats the sensory modalities under the distinct category of quality (guṇa) which is not recognized by the Buddhists. For the Buddhists, the four material elements, earth, water, fire and air, also consist of atoms or each of their types. These have both natural (svabhāva) and derived (upādāya) properties. The natural properties of earth, water, fire and air are solidity, viscidity, heat and motion, respectively, and thus give them distinct capacities: earth can support things, water can cause cohesion, fire causes chemical changes and air causes displacement and growth (Gangopadhyaya 1981). The objects of the first four senses are derived properties of these elements; there are others. More importantly, Buddhist atomists agreed that atoms were indivisible, imperceptible and momentary, continually undergoing qualitative changes. (Some Sautrānika writers seem to have believed that they were a dynamic force or energy rather than particles of matter [Chatterjee 2017].) The Abhidharma-kośa states that atoms are always found in aggregates and never alone. Several lines of argument are supposed to support this conclusion including the claim that single atoms cannot produce an awareness which is what material objects must produce. The general thrust of the argument is that individual atoms have at best a fleeting existence. A considerable amount of Buddhist text is spent discussing the minimum number of atoms there must be in an aggregate that can be distinguished by the senses. While no agreement was reached, in the earlier texts there is some convergence to the view

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that it takes seven or eight atoms. Unfortunately, there was also very little explicit concern in the extant texts about the ontological status of these aggregates. This would be the source of perhaps the best-known dispute between the Buddhist and Nyāya-Vaiśeṣika schools (as opposed to their arguably more important differences about the sensory modalities).

NYĀYA-VAIŚEṢIKA ATOMISM According to Nyāya-Vaiśeṣika doctrine, earth, water, fire and air come in two forms: eternal and non-eternal. Atoms of these four substances are eternal; their composites are non-eternal since every composite (of any kind) is eventually destroyed (by having its atoms separated). In mature Vaiśeṣika doctrine (post-500 CE), atoms, besides being eternal, are indivisible, have the smallest possible magnitude (aṇuparimāna) and are spherical (parimṇḍala). (We have none of the fancy shapes of Buddhist atomism.) There is a complex hierarchical account of composite formation. First, two atoms of the same kind must form a dyad (dyaṇuka). This is the stage at which creation starts since atoms are eternal and cannot be created. Three dyads of the same kind combine to form a triad (tryaṇuka) which is the smallest perceptible entity. Finally, triads can form larger composites with each other, and in various combinations, to give rise to macroscopic objects. Nyāya-Vaiśeṣika doctrine had an array of arguments for the existence of atoms and their properties and these rules of composition, but these later arguments need not detain us here.10 What matters here are arguments about atoms and the relation of composition as they were conceived from the earliest stages of Vaiśeṣika doctrine. Compared to the Buddhist schools, textual resources related to early NyāyaVaiśeṣika are plentiful. The earliest basic text of Vaiśeṣika to have survived is the Vaiśeṣika-sutrā of Kanāda which is taken by convention to be the original Vaiśeṣika text. It cannot be dated precisely, but most scholars agree that it cannot have been composed before 400 BCE and is probably no later than 200 BCE. (Nothing biographical is known about Kanāda.) The Vaiśeṣika-sutrā is a difficult text to explicate; there is uncertainty about the number of sutrās and how they should be interpreted. Almost all the early commentaries that are referred to in the subsequent literature have not survived and most interpretations have to rely on a very late (1425 CE) commentary, the Upaskāra of Śaṇkara Miśra. That said, the text definitely contains a developed atomic theory even if some of the arguments supporting its claims remained murky until much later when they were filled in through a vast corpus of commentary. The presence of this atomic theory in the foundational text underscores the centrality of atomism in Vaiśeṣika and, subsequently, NyāyaVaiśeṣika thinking. In the following exposition, attention will be restricted to those sutrās that are germane to the features of atoms that determine how composites are formed. In Kanāda’s work, the word for atom is aṇu meaning very small (rather than paramāṇu, the more unambiguous term for atom). As noted earlier, in Vaiśeṣika doctrine, the four substances of earth, water, fire and air exist in both eternal atomic and non-eternal composite (gross) form. For Kanāda, air in its atomic form is a

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substance because it has no substance to which it inheres (ii.1.11) and also because it has movement (kriyā) and qualities (guṇa) (ii.1.12). What is true of an atom of air is implicitly presumed to be also true of atoms of earth, water and fire. Inherence is one of the Vaiśeṣika categories and one that played an increasingly central role in Nyāya-Vaiśeṣika thinking over the centuries. Since the concept will also play an important role later in this chapter, it will be worth our while to explore inherence further. As Kronen and Tuttle (2011) have noted, its subtleties are such that it never received a satisfactory formal definition in the Nyāya-Vaiśeṣika corpus. Kanāda’s discussion makes it clear that there is an essential asymmetry between what inheres and what it inheres in. Further, the relationship of inherence is more intimate than (mere) conjunction in which the parts retain their exact identities even after separation from each other: later Nyāya-Vaiśeṣika thinkers explicate this idea by insisting that the relationship is permanent in the sense that if it is destroyed, one of the relata must have ceased to exist. (There is a kind of parity here in the relata that stands in tension with the basic asymmetry of inherence.) Returning to Kanāda’s account, an atom is eternal because it does not inhere in a substance (ii.1.13). What Kanāda presumes here is that destruction consists of the disappearance of material parts or disintegration of their formal arrangement. Therefore, atoms, having no parts (and thus no arrangement of such parts) cannot be destroyed. Further, movement in atoms is caused by a specific invisible force (adṛṣṭa). Unfortunately, the Vaiśeṣika-sutrā is silent about the nature of this force. Atoms have qualities (iv.1.3) because, as causes, they produce qualities seen in their effects in gross matter. Atoms are not visible; consequently, visibility and especially the perception of colour require the presence of many substances as inherent causes (iv.1.6). This can be interpreted to mean that compositeness, grossness and colour are required for visibility. Consequently, there is a difference between atomic magnitudes and gross magnitudes (vii.1.10). This claim is naturally interpreted as indicating that composite units have properties that need not inhere as properties in their component parts. This interpretation, in turn, suggests that there is a sense in which composites are more than mere aggregates of their composite parts, an issue that would distinguish Nyāya-Vaiśeṣika from Buddhist atomism (see the next section). Whether qualities of substances are eternal or not depends on whether the substances themselves are eternal or not (vii.1.3). Because atoms are eternal, qualities present in atoms of earth, water, fire and air are also eternal (vii.1.4). The qualities residing in earth are accorded a sūtra by themselves (vii.1.6) because they undergo transformation by fire: they are colour, taste, smell and touch. Kanāda simply asserts that atoms are spherical (parimaṇḍalya) (vii.1.20), an assumption that remains unquestioned in subsequent Nyāya-Vaiśeṣika doctrine. Later works that are a commentary on Kanāda include the Padārtha-dharmasaṃgraha of Praśastapāda which includes a discussion of the creation and the destruction of the physical world. The world is created from atoms and ultimately dissolves back into them. Chemical change is a result of heat changing features of individual atoms. This text is impossible to date with any certainty beyond it being earlier than 700 CE. Most commentators believe it to have been composed around 500 CE and nothing is known about the biography of Praśastapāda.11 The

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Padārtha-dharma-saṃgraha is aptly regarded as the first expression of the mature form of Vaiśeṣika docrine. (It includes the theory of composition through dyads and triads summarized at the beginning of this section.) Later Vaiśeṣika commentaries on Praśastapāda include the Kiraṇāvali of Udayana (which is precisely dated to 984 CE), the Nyāyakandali of Śridhara (from 980–981 CE) and the Setuṭikā of Padmanābha Miśra (thirteenth–fourteenth century CE).12 However, from the perspective of this chapter, the most important original development of Nyāya-Vaiśeāika atomism after Kanāda is to be found in a text that is also hard to date but is believed to have been composed shortly after the Vaiśeṣikasutrā, namely the Nyāya-sūtra of Gautama. It is certain that it is earlier than 500 CE, the time of Vātsyāyana who left an extensive Bhāṣya (commentary) on the Nyāyasūtra. The text of Nyāya-sūtra may have evolved over centuries; Ganeri (2001, 10) has recently dated its redaction to the first or second century CE. The Nyāya-sūtra consists of five adhyāyas (parts or, roughly, books). Each adhyāya consists of two āhnikas (roughly, chapters). The basic tenets of Gautama’s atomism emerge late in the text, in the second āhnika of the fourth adhyāya. The exposition of Gautama’s views that follows will make systematic use of Vātsyāyana’s Bhāṣya but not of later sources that expand Nyāya-Vaiśeṣika beyond the resources available to its Buddhist opponents in the debate discussed further. Using Vātsyāyana’s elaboration of the Nyāya-sūtra in his Bhāṣya is critical from this perspective because, presumably, it addresses how contemporary Buddhists (in lost works) responded to Gautama. In the Nyāya-sūtra, because atoms are eternal, and all things are composed of atoms, an absolute non-existence of all things is not possible (iv.2.17). These atoms are encountered when things are further subdivided beyond the level of the triads (iv.2.17). Recall that the triads are the smallest perceptible units. Thus, for Gautama, atoms are encountered just beyond the edge of perception. The text expends much effort to address the objection that atoms cannot be partless because they must be penetrated by akāsa.13 Gautama answers this objection by characterizing internal parts of an entity as those that have a causal role in producing properties of the composite unit (iv.2.20). But, since an atom is not a produced substance (kāryadravya), the term ‘inside’ is not applicable to it. Because there are no atoms causing properties of an atom, it has no parts even if it is pervaded by akāsa. Gautama’s exposition of atomism is detailed and expands significantly, though not very systematically, the fragmented remarks in Kanāda’s Vaiśeṣika-sutrā. It sets the stage for the subsequent systematic theory of Praśastapāda. But what is more pertinent here is how Gautama’s discussion reconstructs and responds to an ongoing dispute between Nyāya-Vaiśeṣika and the Buddhists. (Indeed, it is a major source for the views of the Buddhists given the unavailability of original material that has survived.)

COMPOSITES AS WHOLES: THE CENTRAL ARGUMENTS OF THE NYĀYA-SŪTRA Parts of the second and fourth adhyāyas of the Nyāya-sūtra provide a record of the Nyāya-Vaiśeṣika dispute with Buddhist atomism, beginning with disagreements

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about the part–whole relation. However, Gautama’s own exposition of atomism, that is, his positive theory of atoms, is only found in the fourth adhyāya well after the beginning of the text’s engagement with Buddhist counterarguments. This indicates that elucidating the proper part–whole relation is more fundamental to Gautama than establishing an atomic ontology (on which there was at least superficial agreement with the Buddhists, that is, agreement about the existence of atoms though not about their properties). The structure of each of Gautama’s arguments follows what had become the standard Nyāya methodology: critical exposition begins with the statement of a doubt. In these parts of the Nyāya-sūtra, the doubt expressed a position held by the Sautrānika or Vaibhāṣika school of Buddhism. Typically, a single sūtra expresses this doubt. The sūtras that immediately follow then address that doubt so as to refute the position it endorses.14 Recall that the Nyāya-Vaiśeṣika school views a composite as a whole, whereas the Buddhists (both the Sautrānika and Vaibhāṣika schools) hold it as ‘merely’ an aggregate of the components, that is, the atoms. The second adhyāya of the Nyāyasūtra begins a critical examination of the concept of the whole. Gautama launches the process by examining a ‘doubt about the existence of the whole because it is “not yet proved”’ (ii.1.33). His response: ‘If the existence of the whole is denied, then there can be no knowledge of anything’ (ii.1.34). Note that the claim is an epistemological one, and Vātsyāyana subsequently fills out the reasoning in his Bhāṣya: a mere assemblage of atoms cannot be seen because each atom is too small to be visible. What holds for sight presumably holds for all other modes of perception. Two separate epistemological claims are at stake here. The weaker (and uncontroversial) one is that we do have perceptions of the external world and the Buddhist position is supposed to entail that such perception would be in principle impossible: if atoms are all there is (because wholes are just assemblages of atoms), and atoms are imperceptible, the wholes would also be imperceptible. But we do have perceptions and, thus, the Buddhist argument leads to a contradiction. The stronger thesis concerns what must at least implicitly be assumed when we move from perception to knowledge: that perception plays a necessary role in the generation of knowledge (even if it is left open whether all knowledge ultimately arises only in perception). What is also at stake is a rudimentary form of emergence: interactions between the parts, that is, the atoms generate properties that are perceptible and, hence, the composite is more than a mere aggregate of the parts. Recall that Kanāda in the Vaiśeṣika-sutrā had argued that atoms are not visible and that the visibility and colour perceptibility of a composite (gross) unit require the participation of multiple substances as causes. Gautama implicitly draws on this argument to refute the Buddhists who, one presumes, were responding to Kanāda. (What emergence means will be left vague here; it will be more systematically discussed in the next section.) The next sūtra develops the same line of reasoning in a different way: some composite objects can be gripped (dhārana) and pulled (ākarṣaṇa), whereas atoms cannot. In the Bhāṣya, Vātsyāyana notes that mere aggregation by itself cannot explain these properties: a handful of sand cannot be gripped or pulled because it is a mere aggregate of the constituent particles of sand. Moreover, when a composite

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is gripped or pulled, yet another emergent feature is manifested: when an entity is pulled or gripped, that which does the pulling or gripping (say, a hand) is only in contact with some of the atoms that comprise it. Yet, the entire (whole) composite is gripped or pulled; thus, the whole must exist beyond the existence of these atoms at least in the sense of having an emergent capacity to be pulled. Nevertheless, the argument is not entirely definitive and Vātsyāyana implicitly acknowledges this: consider a lump of sand held together with lac. It can be gripped and pulled but, even by Nyāya doctrine, it is not a whole though it has a collectivity that is different from a mere aggregation as in the case of the grains of sand (without lac). After noting this objection, Vātsyāyana lets it stand.15 This is a typical NyāyaVaiśeṣika move: while they provide what they took to be strong reasons to reject the Buddhist arguments, there is no point at which it is claimed that those positions have been definitively refuted. It is in this sense, as the title of this chapter notes, the dispute between the Nyāya-Vaiśeṣika and the Buddhist schools remained unresolved (at least when it comes to aggregates vs. wholes). The Nyāya-sūtra then turns to yet another possible objection to the claim that the emergence of perceptibility is evidence of the existence of a whole: that it is possible to have perceptions of aggregates even if the parts are imperceptible (ii.1.36). Vātsyāyana elaborates: a forest is perceptible even from a distance at which individual trees are imperceptible. In the subsequent Nyāya-Vaiśeṣika corpus, this argument received extensive attention over centuries. Vātsyāyana’s first response was that in such situations the parts are not perceived only because of a special confounding cause, distance in the case of the forest. In contrast, atoms are intrinsically imperceptible and, thus, their aggregates should also be imperceptible. Evidence that distance acts as a confounding cause in this fashion comes from the fact that details of the individuals, for instance, the leaves of the trees are lost to the perception of the forest. Vātsyāyana extends this argument further: if it is presumed that an aggregate of atoms is somehow more than a mere collection of the atoms, that is, it exhibits some new feature, then this argument would concede the case for the existence of wholes that are more than the sum of their parts. In his words: ‘Since the atoms are intrinsically imperceptible, their collectivity also must be so. If the objector argues that this collectivity of the atoms is something more than their individuality, then he will commit himself to the doctrine of the whole.’ There is no record of a Buddhist answer to this objection. Moreover, Vātsyāyana goes on to argue, atoms within their composites are not properly analogous to trees in a forest to which many other different properties inhere and any argument from such a poor analogy cannot be decisive. Further, the argument simply assumes that the forest is being perceived as a whole rather than as consisting of individual trees and thus presuming what was to be proved. (Distance is the special confounding cause that leads to this false perception of forest.) An extensive discussion of the epistemology of the perception of a whole follows but the details are not germane here.16 A new set of Buddhist objections to the reality of the whole are addressed in the fourth adhyāya. The first of these is that there is no logical basis to any claim that a

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part occupies a whole or any segment (ekadeṡa) of the whole (iv.2.7). That it cannot occupy the whole in its entirety follows from the difference in the magnitudes of the two: a part must be smaller than the whole. That leaves open the possibility that the part occupies a segment of the whole. The trouble is that the whole qua whole has no segments to be occupied. (Indeed, according to Vātsyāyana, the only segments of a whole are the parts themselves. Thus, there can be no question of occupation.) This still leaves open a third possibility, that the whole occupies the part. (After all, anyone positing ontological primacy to the whole may well choose to demote the parts to such a subsidiary status.) The next sūtras of the Nyāya-sūtra duly note the objections, first, that ‘the “whole” does not exist, because the “whole” cannot be present within the parts’ (iv.2.8); second, in the sūtra that follows, that the whole ‘is present neither within something other than the parts’ (iv.2.9). Gautama finally proceeds to answer these objections with a dose of hairsplitting metaphysics relying on linguistic usage: ‘The question [of parts occupying a whole] does not arise, because there being no difference within a single entity, the use of “words signifying difference” (bheda-ṡabda) is not [logically] justified’ (iv.2.11). In the Bhāṣya, Vātsyāyana points out that the word for ‘entire’, kṛtsna, and the word for ‘segment’, ekadeṡa, both assume differences which are logically inadmissible in this context because they presume a part–aggregate distinction. They cannot be used with respect to a whole. By itself, this argument would not generate much confidence in the Nyāya-Vaiśeṣika position. Luckily, Gautama does not stop here and attempts to elaborate an alternative positive account of the part–whole relation. Gautama’s intended conclusion to be drawn from the dispute with the Buddhists seems to be that the whole is connected to its parts through the relation of inherence mentioned earlier in this chapter and that this relation must not be explicated using relations such as that of being an entirety or a segment. The whole and the parts cannot exist independent of each other. There is no other part to a whole other than its component parts (iv.2.12). Though this is far from clear from the text, commentators (including Vātsyāyana) now interpret Gautama, following Kaṇāda, to hold that the whole must be contained in the parts, that is, the whole is the superstratum (āśrita) with the parts as the substratum (āśraya). From a textual perspective, a word of caution is in order here. It is natural to interpret the superstratum–substratum relationship invoked by Vātsyāyana to be that of inherence and most modern commentators make this move. That move is supported by the fact that the concept of inherence goes back to Kanāda. Nyāya commentators began invoking inherence explicitly to interpret Vātsyāyana’s intent at least as early as the sixth century CE (probably starting with Uddyotakara). The argument is straightforward: there is obviously some relation between whole and part. This relation cannot be that of conjunction because it is required that the whole would no longer be a whole, and a part would no longer be a part, if the part–whole relation ceases to exist. Thus, the whole must inhere in the parts. Nevertheless, Gautama does not use the word for inherence (samavāya) in this context even though he uses it in the third adhyāya of the Nyāya-sūtra. Similarly, Vātsyāyana also does not use the word in this context in his Bhāṣya.

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Nevertheless, there should also be some caution about reading too much into the non-use of a word that only later became part of the received technical vocabulary of the Nyāya-Vaiśeṣika school. Given Kanāda’s use of the concept of inherence in the Vaiśeṣika-sutrā, and that it seems to be a straightforward interpretation of what Vātsyāyana was invoking in the superstratum–substratum relation when commenting on Gautama, the next section will use inherence as part of an assessment of the contemporary relevance of the dispute between Nyāya-Vaiśeṣika and the Buddhists.

REFLECTIONS Kronen and Tuttle (2011) have recently defended a modified version of the Nyāya Vaiśeṣika theory of parts and wholes based on the relation of inherence though the version of the theory they use is of much later vintage than the one being considered here. (According to them, the Nyāya-Vaiśeṣika account of substance is superior to the Aristotelian account. Establishing this claim is the main purpose of their paper, but this is an issue that is beyond the scope of this chapter.) In the process of formulating their defence, Kronen and Tuttle correctly note the relevance of the Nyāya-Vaiśeṣika account to contemporary discussions of reductionism.17 However, that discussion is short on detail with respect to reductionism which they identify with the Buddhist view of wholes being assemblages of parts and then reject on the metaphysical ground of a difference in identity of a whole and a set of the parts. From their perspective, Nyāya-Vaiśeṣika theory provides an account of ‘structural’ explanation of wholes that is superior to Buddhist reductionism. What they mean by ‘structural’ remains unclear, though their remarks imply that a mere set does not have structure in the intended sense. This can be interpreted to mean that a whole must be a class with some additional internal relations between their members beyond being elements of the class qua set. If the Buddhist conception of aggregate refers only to a set, then the Buddhists are denying any additional internal structure. Whether such an explication constitutes a coherent interpretation of the texts remains questionable since the texts make no mention of internal relations. In general, a much more nuanced treatment is necessary to assess if the Nyāya-Vaiśeṣika account has any positive contribution to make to discussions of reductionism and its alternatives. Let us turn to such an assessment. By reductionism, here, we will mean the view that all features of a whole (which will be construed spatially18) can be accounted for in terms only of features of its parts.19 Reductionism, so construed, has a long history in the Western philosophical tradition, going back to the mechanical philosophy of the seventeenth century. In explicating ‘accounting for’ we must distinguish between epistemological and ontological interpretations. By and large, twentieth-century philosophy of science interpreted ‘accounting for’ epistemologically: reduction was construed as a type of explanation in which features of the whole were explained entirely by features of the parts.20 In contrast, the standard construal of ‘accounting for’ in metaphysics is ontological: reduction is a claim of exclusive upward causation (or determination) from the constituent parts to the whole.

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Of course, the two interpretations are not disjoint. For instance, it is often assumed that explanation may consist of an identification of causes. If this view of explanation is adopted, the epistemological and ontological interpretations of reductionism are intimately intertwined. The intent of the Nyāya-Vaiśeṣika account of the part–whole relation is clearly causal and what follows will follow that interpretation. However, we will assume that there is a sufficiently close connection between causes and explanation so that contemporary epistemological discussions of reductionism from the philosophy of science can credibly be given an ontological gloss based on the Nyāya-Vaiśeṣika part–whole theory without misinterpretation. Because of the continuity between the mechanical philosophy and modern science, the success of science has often been regarded as a triumph of reductionism. But there always have been sceptics.21 During the twentieth century, three sceptical movements have challenged reductionism, in particular, whether it was ever useful in science and, even if it had once been useful, whether it has outlived its utility and is no longer the best methodology for further scientific progress. The best known of these sceptical movements is organicism22 which has largely been confined to discussions of biological phenomena. Organicism posits that the role of parts in a whole cannot be properly understood without understanding the functioning of the whole. Even though the Nyāya-Vaiśeṣika account of the material world does have a normative content, and thus permits discussions of functions and teleology, these considerations played no role in its account of the part–whole relation and in the debates with the Buddhists. In what follows, we will ignore organicism on the ground that it is not relevant to our concerns. The other two sceptical movements, emergentism and holism, remain relevant to our context. Emergentism embraces the doctrine that wholes have features that their parts do not.23 This ontological claim is supposed to entail an epistemological thesis that even a complete knowledge of all features of the parts will not allow explanation, let alone, prediction of all features of the whole. The relevant features of the whole are ‘emergent’ in this way (and emergentism is sometimes called the doctrine of emergence). In Nyāya-Vaiśeṣika doctrine, the ontological thesis dates back to the discussion of perceptibility in Kanāda’s Vaiśeṣika-sutrā. It is the reason why wholes are supposed to exist beyond the aggregate parts. Holism is a more murky thesis, officially dating back only to the 1920s when the term was introduced.24 According to holism, the parts of a system (as characterized within the system) do not exist independent of the whole and, thus, in this sense, wholes are more than an aggregate or the ‘sum’ of their parts. The quantum world provides an uncontroversial example: holism is characteristic of what are called nonseparable systems.25 These are systems with multiple interacting parts where some features of each part can only be characterized by also referring to other parts. The parts thus lose their individual identity. In that sense properties of parts depend on those of the whole. To go into more detail: in quantum mechanics, each entity is represented by a state vector belonging to a mathematical structure called the Hilbert space for that entity.26 If two entities interact, they are represented as a state vector in a more complicated mathematical structure consisting of a tensor product of the Hilbert

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spaces of each of the entities. Since the two interacting entities then become inseparable, this state vector may be in an entangled state, that is, it is not a tensor product of component state vectors with each such component belonging to exactly one of the Hilbert spaces of the constituent entities.27 Thus, in an entangled state, we cannot refer to either of the entities without invoking the other. In this sense, neither entity can exist as it does without the presence of the whole. Both emergence and holism are relevant to the Nyāya-Vaiśeṣika account of the part–whole relation; what makes it interesting to recast that account using these contemporary resources is the role played by the relational category of inherence. From the Nyāya-Vaiśeṣika perspective, emergence allows the whole to exist beyond being a mere aggregate of its parts as the Buddhists maintained. Now, as the case of the forest and the trees shows, besides genuine emergence there are misleading cases of apparent inherence. These are detected by showing confounding external causes misleading the senses. In the case of genuine emergence, the parts have a much more intimate relationship with each other (beyond conjunction or aggregation). Thus, in holistic systems, the parts exist as such because of some feature of the whole. Inherence captures that feature: the whole inheres in its parts. But inherence does even more: it underscores holism by the requirement that, if the relation of inherence ceases to hold, either the whole or the part (or, of course, both) ceases to exist. Now, recall that inherence is an asymmetric relation. Thus, there is a sense in which parts are privileged over the whole.28 Thus, this account captures the intuition of the reductionists that there is a sense in which parts are more important than the whole. But the most important role inherence plays in this story is that, by insisting that we look beyond mere aggregation of parts, it forces a focus on structure or arrangement: second-order features of the parts that, so to say, construct the whole qua whole.29 This is what the Buddhist doctrine of mere aggregation misses. It is also what any plausible twentieth-century reductionist account must also have accepted.30 It remains part of our discussions of reductionism today. In this way, the Nyāya-Vaiśeṣika framework facilitates an interesting reconstruction of contemporary discussions of reductionism, holism and emergence.

NOTES 1. A discussion of the ideological implications of this characterization will also be left for some other occasion. It excludes, for instance, the political philosophies that emerged among Indian scholars as a result of colonial encounters. This is a large body of work that has not received critical philosophical attention (see, however, Brodov 1984) only part of which can be regarded in any plausible sense as extensions of work earlier than 1500 CE. In practice, if not in theory, it also excludes all philosophical work in India with non-indigenous (e.g. Islamic) roots even if it predates 1500 CE; see, however, Ganeri (2011) which is a welcome exception, at least insofar as Islamic-Indian thought is concerned. 2. For a welcome counterexample, see Kronen and Tuttle (2011) which will be used and further discussed below.

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3. A note on the texts used in this chapter: All of the most relevant sources on classical Indian atomism have been collected with commentary in a seminal work by Gangopadhyaya (1981). Except when explicitly noted otherwise, all primary sources cited in this chapter are from that work. This chapter mostly but not exclusively uses the translations offered there (which are mainly from standard works) but some translations have been modified for the sake of accuracy using the original Sanskrit texts which are also included in Gangopadhyaya’s sourcebook. 4. For a discussion of later Buddhist atomism, but still within the classical period as characterized here, see Carpenter and Ngaserin’s contribution to this volume. 5. But even the category of āstika system is not without controversy: Śaṃkarācārya regarded the Vaiśeṣika system as ardhavaināśika or ‘antagonistic’ to Vedic culture, as emphasized by Chatterjee (2011, 112). Potter (1961) attempted a radical reclassification of Indian philosophical systems but his scheme has never achieved traction. 6. For instance, Seal (1915) interprets the Sāmkhya to be atomists. 7. There is a large corpus of recent philosophical work on Nyāya-Vaiśeṣika; for good accounts, see Matilal (1977), Potter (1977), and, especially, Phillips (1985, 41–74). 8. Mishra (2006) attempts a comparison of Nyāya-Vaiśeṣika and Jaina atomism from the perspective of what he takes to be modern science. 9. Some independent texts, such as the Bāhyārthasiddhi-kārikā of Śubhagupta (eighth century CE), have also survived. 10. See Gangopadhyaya (1981) and Chatterjee (2017). 11. See Gangopadhyaya (1981, 46). 12. Details of these works and relevant excerpts from them, as well as some biographical information on the authors can be found in Gangogpadhyaya (1981). 13. The ubiquity of akāsa is central to Nyāya doctrine; hence, the import of this objection. 14. This structure is characteristic of the entire Nyāya-sūtra. 15. Moreover, it is left standing even though Gautama has a response available: the lac is a confounding special cause, a line of reasoning he later used to argue that forests are mistakenly perceived as wholes rather than aggregates – see below, p. 191. 16. What follows in Vātsyāyana’s Bhāṣya includes a long (and interesting) digression on the properties of sound as a whole. It is beyond the scope of this chapter. 17. They call it ‘reductivism’ but that is merely a terminological difference. 18. This means that the whole exists as an entity in space, however that is explicated. It thus already has features beyond being merely a set of its parts. For a discussion of categories of reductionism that do not embrace spatiality, see Sarkar (1998). 19. For more details on this view of reductionism and its history, see Sarkar (2008) and the references therein. Sarkar et al. (2017) provide an annotated bibliography. 20. The locus classicus of this view is Nagel (1961); for an analysis, see Sarkar (2015). 21. There have always been good grounds for this scepticism. As was recognized in the West by the end of the seventeenth century, the mechanical philosophy denied action-at-a-distance, whereas the most predicatively precise and accurate theory

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that had ever been formulated was Newton’s theory of gravitation which embraced action-at-a-distance. There were many attempts to provide a mechanical basis for Newtonian gravitation in subsequent centuries and they failed comprehensively until the project seems to have been abandoned in the nineteenth century. 22. See Beckner (1974) and Gilbert and Sarkar (2000). 23. A good discussion, though limited to physics, is in Butterfield (2011). Nagel’s (1961) earlier analysis continues to be relevant. 24. The term goes back to Smuts (1926). 25. There is a vast literature on this topic. Shimony (1987) and Jaeger and Sarkar (2003) provide useful entries into the literature. 26. Technically, the entity is represented by a ray of vectors rather than a single vector but that complication does not affect our argument. 27. Once entities become inseparable in this way, standard quantum dynamics does not allow them to evolve into separable ones represented by a tensor product of individual state vectors. One example of this problem is quantum measurement where one entity is the measuring apparatus and the other the system being measured. The well-known quantum measurement problem is just a special case of the more general disentanglement problem. 28. In twentieth-century work, a similar role can be attributed to supervenience. But that is a type of discussion that does not form part of the Nyāya-Vaiśeṣika account of the part–whole relationship that is being discussed here. 29. This seems to be what Kronen and Tuttle (2011) recognized. 30. And, indeed, they typically do – see, for instance, Sarkar (1998).

REFERENCES Beckner, M. (1974), ‘Reduction, hierarchies and organicism’, in F. J. Ayala and T. Dobzhansky (eds), Studies in the Philosophy of Biology: Reduction and Related Problems, 163–77, Berkeley: University of California Press. Brodov, V. V. (1984), Indian Philosophy in Modern Times, Moscow: Progress Publishers. Butterfield, J. (2011), ‘Emergence, reduction and supervenience: A varied landscape’, Foundations of Physics 41: 920–59. Chatterjee, A. (2011), ‘Nyāya-Vaiśeṣika philosophy’, in W. Edelglass and J. L. Garfield (eds), Oxford Handbook of World Philosophy, 112–26, Oxford: Oxford University Press. Chatterjee, A. (2016), ‘Naturalism in Indian philosophy’, in K. J. Clark (ed.), Blackwell Companion to Naturalism, Malden: Blackwell. Chatterjee, A. (2017), ‘Naturalism in classical Indian philosophy’, Stanford Encyclopedia of Philosophy (Winter 2017 Edition). https​:/​/pl​​ato​.s​​tanfo​​rd​.ed​​u​/arc​​hives​​/win2​​017​/e​​ ntrie​​s​/nat​​​urali​​sm​-in​​dia/ Ganeri, J. (2001), Philosophy in Classical India, New York: Routledge. Ganeri, J. (2011), Lost Age of Reason, Oxford: Oxford University Press. Gangopadhyaya, M. (1981), Indian Atomism: History and Sources, Atlantic Highlands: Humanities Press.

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Gilbert, S. and Sarkar, S. (2000), ‘Embracing complexity: Organicism for the 21st century’, Developmental Dynamics 219: 1–9. Jaeger, G. and Sarkar, S. (2003), ‘Coherence, entanglement, and reductionist explanation in quantum physics’, in A. Ashtekar, R. S. Cohen, D. Howard, J. Renn, S. Sarkar and A. Shimony (eds), Revisiting the Foundations of Relativistic Physics: Festschrift in Honor of John Stachel, 523–42, Dordrecht: Kluwer. Kronen, J. and Tuttle, J. (2011), ‘Composite substances as true wholes: Toward a modified Nyāya-Vaiśeṣika theory of composite substances’, Canadian Journal of Philosophy 41: 289–316. Matilal, B. K. (1977), Nyāya-Vaiśeṣika, Wiesbaden: Otto Harrassowitz. Mishra, A. K. (2006), ‘Atomism of Nyāya-Vaiśeṣika vs Jainism—A scientific reappraisal’, Indian Journal of History of Science 41: 247–61. Nagel, E. (1961), Structure of Science: Problems in the Logic of Scientific Explanation, New York: Harcourt, Brace, & World. Phillips, S. (1985), Classical Indian Metaphysics, Chicago: Open Court. Potter, K. H. (1961), ‘A fresh classification of India’s philosophical systems’, Journal of Asian Studies 21: 25–32. Potter, K. H. (1977), Tradition of Nyāya-Vaiśeṣika up to Gaṇgeśa, Delhi: Motilal Banarasidas. Sarkar, S. (1998), Genetics and Reductionism, New York: Cambridge University Press. Sarkar, S. (2008), ‘Reduction’, in S. Psillos and M. Curd (eds), Routledge Companion to the Philosophy of Science, 425–34, London: Routledge. Sarkar, S. (2015), ‘Nagel on reduction’, Studies in History and Philosophy of Science Part A 53: 43–56. Sarkar, S., Love, A. and Wimsatt, W. C. (2017), ‘Reductionism in biology’, in D. Pritchard (ed.), Oxford Bibliographies in Philosophy, New York: Oxford University Press. http:​/​/ www​​.oxfo​​rdbib​​liogr​​aphie​​s​.com​​/view​​/docu​​ment/​​obo​-9​​78019​​53965​​77​/ob​​o​-978​​01​953​​ 96577​​-0359​​.xml Seal, B. (1915), Positive Sciences of the Ancient Hindus, London: Longmans, Green & Co. Shimony, A. (1987), ‘The methodology of synthesis: Part and wholes in low-energy pysics’, in R. Kargon and P. Achinstein (eds), Kelvin’s Baltimore Lectures and Modern Theoretical Physics, 399–423, Cambridge, MA: MIT Press. Smuts, J. C. (1926), Holism and Evolution, London: Macmillan.

CHAPTER 10

Atomism and Islamic thought FRANCESCO OMAR ZAMBONI

INTRODUCTION The atomistic theory of matter represents one of the fundamental themes of medieval Islamic thought. Several Muslim authors tackled the issue from a variety of perspectives, debating the very existence of atoms, their nature and their features. The aim of this chapter is to present an account of such debates, considering the doctrinal milieu of Islamic atomism, its supportive arguments and its peculiar characteristics. The present chapter consists of three parts: ●●

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the first section presents the elaboration of the atomistic doctrines, considering their doctrinal milieu and their distinctive characteristics; the second section deals with the arguments in favour of atomism and their fundamental assumptions; the third section considers the main questions of dispute between Muslim atomists.

ISLAMIC ATOMISM AND ITS MILIEU The emergence of Islamic atomism Islamic thought can be subdivided in two main traditions: kalām (lit. ‘discourse’, ‘speech’) and falsafa (from the Greek philosophia). Kalām designates a specific form of Islamic theological speculation which involves both scriptural and purely rational discussion. The mutakallimūn (the practitioners of the kalām) should be qualified as ‘theologians’ because they aimed to defend the fundamental tenets of Islam, but their discipline encompassed all sorts of subjects of inquiry (physics, metaphysics, ethics, epistemology, etc.). Falsafa, on the other hand, designates the Neoplatonizing strand of Aristotelianism we attest in authors such as Kindī (d.873), Fārābī (d.950) and Ibn Sīnā (Avicenna, d.1037). The two traditions were polarized on the issue of the atoms: the majority of the mutakallimūn asserted their existence, whereas the

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majority of the falāsifa (the practitioners of falsafa) rejected it, following Aristotle’s account of matter as divisible ad infinitum. The atomism of the kalām shares some basic features with its Greek counterpart, especially with Epicurus’ version of atomism. However, it is substantially impossible to trace Islamic atomism back to its putative Greek sources on the basis of textual evidence, because such evidence is not extant, and perhaps never existed in the first place. In face of this striking absence, Alnoor Dhanani suggested that atomistic theories reached the mutakallimūn orally, via debates with various sorts of adversaries: Christians, Dualists and Naturalists (the so-called ‘Eternalists’, dahriyya).1 The very nature of the oral medium could justify the high degree of originality we see in Islamic atomism. All of this notwithstanding, it is still possible to provide a meaningful contextualization of the emergence of atomism in the kalām, since it is possible to locate it in the broader context of the debates among Muslim thinkers between the mid-eighth and the mid-ninth centuries, the period when the kalām itself emerged as a definite discipline, and the positions of individual mutakallimūn began to crystallize in actual schools of thought. On a preliminary ground, it is useful to consider the issue of the nature of bodies from an abstract perspective, in order to understand how the different accounts of corporeality stand with respect to one another. Sense-perception attests that bodies exist. Bodies differ from one another in some properties (heat, reflectivity, colour, etc.) and share other properties. It is evident that the most basic property common to all bodies qua bodies is the fact of being a non-empty volume (i.e. solidity together with three-dimensionality): I call this property ‘corporeality’. In view of what I mentioned, any corporeal object can be said to support two kinds of division. The first kind (α) is the qualitative division between corporeality as such, which belongs to all bodies as an invariant property, and the other variant qualities. At this point, the following question needs to be considered: What is the structure of the relation between bodies, their corporeality and their variable qualities? Islamic thinkers present four alternatives. ●●

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(α1) Every variant quality possesses corporeality on its own (i.e. every variant quality is a corporeal substance). (α2) All variant qualities are essential parts of the corporeal substances they qualify. (α3) Some variant qualities are essential parts of the corporeal substances they qualify, whereas others are external to them and inhere in them as accidents. Corporeality is an essential part of corporeal substances. (α4) All variant qualities are external to the corporeal substances they qualify and inhere in them as accidents. Pure corporeality is the only essential constituent of corporeal substances.

The second kind of division (β) concerns corporeality itself. In fact, corporeality consists in solidity together with three-dimensionality. Three-dimensionality is the conjunction of three extensions, and extensions can be measured and divided

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with respect to quantity. This means that any corporeal thing can be said to possess quantitative parts. We attest three positions concerning the quantitative parts of bodies. ●●

(β1) They exist in actuality and are finite in number.

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(β2) They exist in actuality and are infinite in number.

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(β3) They exist in potentiality and are infinite in number.

We find four main doctrines concerning corporeality. Each one of them can be considered the conjunction of one α-thesis and one β-thesis. The first doctrine consists in the coupling of (α2) and (β1). It is ascribed to the mutakallimūn Ḍirār ibn ʿAmr (d.815), Ḥafṣ al-Fard (early ninth century) and Abū al-Ḥusayn al-Naǧǧār (mid-ninth century). Abū al-Ḥasan al-Ašʿarī (d.936) – the most important source on the doctrines of the early kalām, since none of the writings of the early mutakallimūn is extant – informs us that they believed bodies to be nothing but bundles of primary qualities, such as colour, heat, softness and so on.2 Qualities are parts of the corporeal substance and do not inhere in it as external accidents. This doctrine blurs the line between the qualitative and the quantitative parts of bodies, for Ḍirār and his companions are credited with the following ancillary theses: ●●

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bodies do not possess other kinds of parts in addition to their qualitative parts; the qualitative parts of bodies are adjacent to one another, and their ‘compenetration’ (mudāḫala) is impossible; the minimal body must possess ten qualitative parts (this means that those parts can be quantified).

The second doctrine is the conjunction of (α1) and (β2). Hišām ibn al-Ḥakam (d.795) and Ibrahīm al-Naẓẓām (d.845) are credited with this position, which reverses the perspective of the previous one: qualities themselves are bodies, in the sense that they possess corporeality.3 The separation between corporeal substances and accidental qualities is rejected once again, for qualities are said to be bodies in their own right: the complex objects we experience via sense-perception are the results of the penetration of simple corporeal qualities into one another.4 However, Hišām and Naẓẓām maintain a clear distinction between qualitative and quantitative parts, since the qualitative parts of bodies are bodies on their own, and thus possess corporeality, which is subject to quantitative division. These two thinkers argued that the division of bodies is endless, and thus corporeal substances must be said to possess an actual infinity of minimal parts.5 The third doctrine is the conjunction of (α3) and (β3) and can be traced back to Aristotle’s physics: there are four genera of elementary bodies each of which emerges from the combination of two fundamental couples of opposite qualities (heat/coldness, humidity/dryness), and bodies are infinitely divisible in potentiality.6 The fourth position is atomism itself, which consists in the coupling of (α4) (all variant qualities are accidental) and (β1) (bodies have finite quantitative parts which exist in actuality). The Muʿtazilite Abū al-Huḏayl al-ʿAllāf (d.840) – one of the most

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prominent figures of the early kalām – is often credited with the introduction of the atomistic doctrine in Islamic thought. Such a claim is hard to validate, but it is sure that he was one of the main proponents of atomism and that his positions became particularly widespread.

The basic features of Islamic atomism Abū al-Huḏayl is credited with the following theses: ●●

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any body consists in an aggregation of a finite number of ‘parts that are not partitioned’ (aǧzāʾ lā tataǧazzaʾu), namely atoms; a single atom has no length, no width and no depth, for only the aggregation of two or more atoms can be said to have dimensions; atoms, either individually or in aggregation, are the substrates for all classes of accidents, like perceptible qualities, life, mental qualities and accidents related to location (aggregation and separation, movement and rest).7

The assertion that atoms cannot be partitioned must be understood in the most radical way: they are indivisible both physically and conceptually.8 Furthermore, atoms qua atoms are homogeneous, in the sense that they do not differ from one another in their essence, even though they may still differ in other respects. Another implicit feature of Islamic atomism is that atoms possess solidity, in the sense that they cannot penetrate into one another. Finally, unlike Greek atomists, Muslim atomists believed atoms not to be eternal: they were brought into existence by God, who has also the power of annihilating them. Almost all of the mutakallimūn accepted the basic tenets of Abū al-Huḏayl’s doctrine, refining them and debating on various derivative questions (the third part of this study will consider the topics of those debates, together with the different stances assumed by the main authors on each one of them).9 As for the refinement of the basic tenets of atomism, an important contribution came from the Baṣran Muʿtazilite master Abū ʿAlī al-Ǧubbāʾī (d.915): according to Ašʿarī, he explicitly asserted that atoms are homogeneous (a thesis that is only implicitly present in Abū al-Huḏayl), in that they share a single genus and a single essential nature, whereas they are different from one another in number.10 Another noteworthy elucidation concerns the very definition of the atoms. Abū al-Maʿālī al-Ǧuwaynī (d.1085) presents three definitions given by his predecessors, without mentioning the exact sources: ●●

the atom is what receives accidents;

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the atom is ‘what occupies a secluded space’ (mā šaġala l-ḥayyiza);

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the atom is the minimal part of a body.11

The last definition is trivial. The first highlights that, for the mutakallimūn, the property of being a substrate is restricted to corporeal objects: accidents are not capable to be substrates for second-level accidents.12 The second definition highlights another intrinsic property of atoms, namely solidity: each atom occupies an ‘enclosed’ or ‘secluded’ portion of space no other atom can occupy.

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The doctrine which emerges from these considerations not only presents an account of matter as fundamentally discrete but also asserts the predominance of pure corporeality (i.e. solidity and three-dimensionality) over the variant qualities of bodies.13

THE CASE FOR ATOMISM The atomism of the kalām can be described as the conjunction of two fundamental theses: ●●

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(α4) pure corporeality constitutes the substrate for the variant qualities of bodies; (β1) bodies are composed out of a finite number of indivisible parts.

In this section, I aim to present an overview of the mutakallimūn’s arguments for both tenets, which are essentially connected to the refutation of the competing accounts of corporeality I outlined before, in particular: ●●

(α1) the doctrine that qualities are bodies on their own;

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(α2) the doctrine that corporeality derives from the aggregation of qualities;

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(β2) the doctrine that bodies are composed out of an actual infinity of parts;

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(β3) the doctrine that bodies are potentially divisible ad infinitum.

The homogeneity of bodies Naẓẓām and the falāsifa presented two different accounts of the relation between qualities and corporeality. However, they agreed on the basic thesis that bodies, while sharing corporeality, belong to different genera: Naẓẓām claimed that (α1) all perceptible qualities are bodies on their own, whereas the falāsifa asserted that (α3) there are four kinds of elementary bodies (fire, water, air, earth).14 The mutakallimūn rejected Naẓẓām’s position by claiming that all corporeal substances must be homogeneous, namely identical in their essence.15 They did not explicitly attack the falāsifa on this point, but it is evident that a similar thesis contradicts their position. There are three main arguments for the claim that corporeal substances are homogeneous. The first is the argument from perception. It is structured as follows: ●●

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any two bodies are perceptually equivalent, in the sense that the perceptual experience related to the corporeality of X is the same as the perceptual experience related to the corporeality of Y; when perception grasps a thing, it grasps it according to its essential characteristic; it follows that any two bodies are essentially equivalent (i.e. homogeneous).

The claim that perception grasps things in their essential characteristics appears doubtful. However, some kind of corroboration might come from an analogy with

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perceptible qualities, like colours and tastes: perception grasps the very essence of whiteness, or that of sourness, and not just some of their accidental properties.16 The second proof argues that if bodies were essentially different, then there would be some fundamental property which shows such an essential difference. This is not the case, since all bodies share the same fundamental properties: all of them occupy a secluded space, and all of them can receive the same accidents. This argument is found wanting, for it needs to establish that all bodies share all their fundamental properties: the falāsifa would reject this, claiming that celestial bodies lack most of the fundamental properties of terrestrial bodies, like being subject to corruption, or being capable of receiving a number of perceptible qualities.17 The third proof argues that bodies are essentially different from other things (accidents, God). Occupation of space is both what sets bodies apart from the other things and what is shared by bodies qua bodies: it follows that bodies must possess the same basic essence, which implies the property of being space-occupying.18 This argument is also wanting, for it might be possible to object that several different essences are connected to a single accidental feature, that is, space-occupation.

The substantiality of corporeality Ḍirār and Ḥafṣ are credited with the thesis that (α2) bodies are nothing but aggregates of perceptible qualities (taste, smell, colour, etc.) which are not bodies on their own: they become a body only in the state of aggregation, and in that state they can be said to be a substrate for accidents such as movement, rest and so on.19 The atomists called those perceptible qualities ‘accidents’ as well, but such a denomination is fundamentally misleading in this case, for the very point at stake concerns whether perceptible qualities should be considered as accidents (i.e. things which inhere in a substrate) at all. The atomists directed two main arguments against (α2). The first proof is built on two premises. ●●

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Two things can be said to be aggregated in one of three ways: because they are spatially contiguous to one another, because all of them inhere in a single substrate, or because one of them inheres in the other. Every substrate must be said to be space-occupying.

The assumption of both these premises invalidates the thesis that corporeality emerges from an aggregation of qualities which do not possess corporeality on their own. In fact, such aggregation would consist in their spatial contiguity or in their inherence in a common substrate, or in their inherence in one of those qualifies: in all cases, the occupation of space would be the very condition for the aggregation of those qualities, since this is a necessary condition for spatial contiguity and for substrates qua substrates. It follows that corporeality would be prior to the aggregation of qualities and not posterior to it (as the view of the adversaries implies).20 The second main argument is based on the assumption that occupation of space is a property which is intrinsically connected to the very essence of the thing it is predicated of.21 In that case, thesis (α2) would require perceptible qualities to change

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their essence when acquiring corporeality: in isolation they would be essentially non-space-occupying; in aggregation they would become space-occupying, thus acquiring another essence. This is what the mutakallimūn called ‘the subversion of the genera’ (inqilāb al-aǧnās), which is absurd, for a thing would lose its very essence and acquire the essence of an intrinsically different thing (e.g. whiteness becoming blackness, substance becoming accident).22 In addition to these two main arguments, there are three ancillary arguments. ●●

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The argument from movement: if an aggregation of qualities were to move, then the accident of movement would inhere in one quality or in all qualities; if it inhered in all qualities, then a singular accident would inhere in many substrates, which is absurd; if it inhered in only one quality, the aggregate could not be said to move at all, which contradicts the hypothesis.23 The argument from contraries: if an aggregate of qualities were said to be capable to receive contrary accidents, those contraries would either inhere in a single thing or in many things; the adversaries cannot assert that the substrate of contraries is a single thing (for it is nothing but an aggregate of qualities), and contrary accidents which inhere in many things cannot be said to be actually contraries (for the existence of one does not prevent the other from existing).24 The argument from accretion: if corporeality as such consisted in a bundle of a certain number of qualities, then the addition of more qualities would require the body to increase in size, and this is absurd.25

The finiteness of the parts of bodies For the early mutakallimūn (ninth–tenth centuries), the main adversary on the issue of the quantitative division of bodies was Naẓẓām, who upheld that bodies are subject to an infinity of divisions. Naẓẓām’s doctrine can be understood in two ways: ●●

(Na1) no minimal parts exist;

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(Na2) each body includes an actual infinity of minimal parts.

(Na1) draws near to the falāsifa’s own doctrine, the only element which sets the two apart being the concept of potential existence. It is not completely clear whether Naẓẓām actually upheld one thesis or the other, even though (Na2) seems the most probable26. What is clear is that the atomists attacked both theses, arguing for (1) the existence of minimal parts and for (2) their finiteness in number. Here I aim to present the main arguments for (2), leaving the arguments for (1) to the next section. The first proof for the finiteness of minimal parts is based on the impossibility to traverse an infinity. The argument reframes Zeno’s arrow paradox, imagining an ant crawling over a surface: if bodies were divisible in an infinity of parts, the ant could not traverse any given portion of that surface. In fact, before reaching a certain point, the ant would need to reach a point which is placed before it and so

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on ad infinitum: since each one of these infinite steps must correspond to an instant of time, the ant would traverse the surface in an infinite amount of time. It should be noted that this argument implicitly assumes a discrete account of time: if time were said to be a continuous quantity, then it would be infinitely divisible just like matter, and the ever-decreasing portions of the ant’s movement would correspond to the ever-decreasing non-discrete portions of time (no infinite amount of time would be required). Naẓẓām, however, did not present any similar counterargument and reverted instead to the theory of the leap (ṭafra). The ant can traverse infinite steps in a finite amount of time because it does not ‘traverse’ them at all: it ‘leaps’ over that infinity, reaching the end of its movement without passing through all the infinite intermediate steps. Every spatial translation would thus require the body to traverse a finite number of intermediary steps and jump over an infinite number of other steps, thus encompassing proper instances of movement and improper instances of movement (leaps). The atomists answered that this doctrine defies sense-experience: it would be possible for a thing to ‘leap’ from the far east to the far west in no time, and it would be impossible for a quill to trace a line over a sheet of paper (for it would ‘jump’ over most of the parts of the paper).27 The second argument is based on the impossibility for an infinity to be greater or lesser than another infinity: if bodies were constituted out of an infinity of parts, it would be impossible for any object to be greater (or lesser) than another in quantity, for each one of them would consist of infinite parts, and one infinity cannot be greater (or lesser) than another.28 The third argument is dialectical, for it is built on the assumption – shared by all the mutakallimūn, Naẓẓām included, but not by their adversaries – that it is not possible for the world to be eternal ex parte ante: in other words, the chain of past events cannot go on to infinity.29 The atomists argued that since Naẓẓām’s doctrine requires the possibility of a regressus ad infinitum in the process of division of bodies, it should also assert the possibility of the regressus in the chain of past events. Moreover, infinite parts would receive an infinity of variable accidents (in case they moved, for example, all their infinite accidents of location would change), and this would contradict the basic kalām tenet that an infinity of things cannot be brought into existence, be that in one stroke or by succession.

The actuality of the parts of bodies The main adversary of the early atomists on the issue of the division of bodies was Naẓẓām. However, the situation changed around the mid-tenth century, when the hylomorphic doctrine of the falāsifa (Fārābī, Avicenna, and their followers) began to acquire increasing pre-eminence as an account of the nature of corporeal substances. The falāsifa’s doctrine differs from both interpretations of Naẓẓām’s doctrine in that it introduces the notion of potential existence: the quantitative parts of bodies exist only in potentiality, in the sense that a homogeneous body is something unitary whose parts come to exist in actuality only after the operation of division (be it physical or mental) has occurred.30

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The argumentative strategy of the late mutakallimūn (eleventh century onwards) consisted in asserting that the aforementioned account is not substantially different from Naẓẓām’s, either because the thesis of the infinite potential divisibility entails that it is possible for some active cause to exhaust the infinite process of division or because the minimal parts of bodies must exist in actuality before the process of division takes place. Most of the arguments I am going to consider were originally devised against interpretation (Na1) of Naẓẓām’s doctrine (minimal parts do not exist at all), and then reframed in order to be used against the falāsifa as well. The most important of those arguments is the proof from junction.31 Its fundamental premise is that a body is capable to receive quantitative division on account of the fact that an accident of junction (or aggregation) inheres in its parts: when that accident is annihilated, being replaced by its contrary (i.e. separation), the body undergoes division. The crucial point behind this assertion is that what accounts for the possibility to divide bodies is not space-occupation itself – which is an essential property of bodies – but rather an external accident: it follows that not everything which is space-occupying must be divisible, and that there must be some indivisible space-occupying object which is the substrate for the accidents which account for aggregation and division. Thus, an infinite or finite number of minimal parts must exist. The falāsifa argued that this account of division is erroneous, and that the meaning of the assertion that bodies are potentially divisible ad infinitum is simply that any corporeal substance, regardless of magnitude, is such that an active cause can divide it and make it two things. This does not entail that the infinite process of division might terminate and that an infinity of minimal parts might come to exist in actuality: the infinite process of division is a set encompassing infinite elements, and the assertion that each one of those elements is potentially existing does not entail that the set as a whole is potentially existing. The atomists answered that an infinite number of divisions is imaginable and thus that it must be intrinsically possible, for otherwise we could be able to imagine logical impossibilities (e.g. the conjunction of the contraries). Thus, if such an infinite division is intrinsically possible, then an omnipotent active cause will be capable to operate it, either physically or mentally: God can ‘discretize’ the continuum. Moreover, they argued that there is a necessary implication between the potential existence of each of the infinite divisions and the potential existence of the whole in view of the fact that none of the divisions prevents any number of the others from existing (unlike in the case of the parts of time). The other arguments do not aim to demonstrate that infinite potential divisibility implies the potential exhaustion of the process of division (i.e. that minimal parts can exist in actuality), but rather that minimal parts do exist in actuality. ●●

The argument from the limits of bodies: corporeal objects are limited by surfaces, surfaces can be limited by lines and lines can be limited by points. Points must be indivisible in all respects, for otherwise there would be no way for us to set them apart from lines, surfaces and bodies.32 The falāsifa rejected both this and the next argument by arguing that points (as well as lines and surfaces) are accidents of bodies and not their parts. The atomists

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retorted that a real accident inheres in its substrate as a whole, whereas points do not. ●●

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The arguments from geometrical figures. There are several proofs of this kind. The most famous was perhaps Ǧuwaynī’s argument from the sphere and the surface: a sphere can be in contact with a perfectly even surface in one point only; were that point to be divisible, then either the sphere would not be a sphere or the surface would not be perfectly even.33 The argument from substantial generation and corruption: if the quantitative parts of bodies existed only in potentiality, then the division of a simple homogeneous body (e.g. a mass of water) into two parts would require the whole to undergo complete substantial corruption, while two new substances would be generated in its place; this is intuitively absurd.34

THE DEBATES AMONG THE ATOMISTS Even though Muslim atomists agreed on some basic tenets, they debated on a large number of derivative questions. I gather the most noteworthy of them in seven groups, according to thematic criteria.

Atoms and corporeality The first group of questions concerns whether some properties of bodies can be said to pertain to the atoms as well and how so.

1) The magnitude of the atom.

Abū al-Huḏayl upheld that atoms cannot be said to have length, width and depth and that three-dimensional extension derives from the aggregation of several atoms. Abū al-Ḥusayn al-Ṣāliḥī (ninth century) is the only atomist who explicitly rejected this.35 However, the mutakallimūn diverged on a closely related issue, namely on whether an atom can be said to possess ‘a portion of dimension’ (ḥaẓẓ min al-masāḥa). Some authors – the Baṣran Muʿtazilite Abū ʿAlī al-Ǧubbāʾī and the Baġdādian Abū alQāsim al-Kaʿbī (d.931), as well as Ašʿarī – rejected this, probably on the basis that it would compromise the absolute indivisibility of the atom. Later authors – both Baṣran Muʿtazilites like Abū Hāšim al-Ǧubbāʾī (d.933), and Ašʿarites, such as Abū Bakr al-Bāqillānī (d.1015) and Abū al-Maʿālī al-Ǧuwaynī (d.1085) – upheld this thesis, arguing that it would be absurd for a body to consist in a sum of parts devoid of magnitude (the sum of a finite number of elements devoid of magnitude is also devoid of magnitude).36

2) The shape of the atom.

Some early Muʿtazilites and the Ašʿarites (Bāqillānī, Ǧuwaynī) rejected the possibility for the atom to have any shape at all or even to be describable as analogous to what has shape. Most of the Muʿtazilites, on the other hand, accepted such an analogy, describing atoms as analogous to squares (Baṣrans) or to regular triangles (Baġdadians), probably referring to cubes and tetrahedrons: the disagreement

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between the two positions boils down to how many atoms can be in contact with the same atom at the same time (six, as the sides of cube, or four, as the sides of a tetrahedron).37

3) The sides of the atoms.

The Muʿtazilites understood the side of an atom to be that thing by means of which the atom encounters another atom. They diverged on whether the side of an atom is that very atom (Abū Hāšim), or something different from it, presumably an accident (Abū ʿAlī and Kaʿbī). Those who affirmed the latter thesis argued that there must be some difference between two sides of a single atom, for one of them is in contact with a certain atom and the other is in contact with another. Their adversaries stated that an atom must be in contact with multiple other atoms by means of itself and not by means of something other than itself (probably because this would imply divisibility). Ašʿarī upheld a different definition of ‘side’, thus concluding that a single isolated atom does not have sides at all and that its sides are the atoms that might encircle it.38

4) The weight of the atom.

Abū ʿAlī, Kaʿbī and Ašʿarī upheld that weight is an essential property of the atoms, just like the occupation of space. Abū Hāšim, as well as some Ašʿarites, rejected this, arguing that weight is just an accidental quality.39

The constitution of bodies There are two issues related to how atoms make up bodies.

1) The nature of junction.

The Baṣran Muʿtazilites affirmed that in order for a bunch of contiguous atoms to constitute a body, an accident of junction (taʾlīf) needs to be present in them: junction is additional to pure proximity in location and contiguity, for sense experience attests that some bodies are hard to separate, whereas others are not, and so there must be some accident which accounts for that difference. It follows that bodies do not arise from the mere spatial juxtaposition of atoms but require some additional accident which keeps them together. The Baġdadians rejected this thesis, stating that junction is merely proximity in location and contiguity. The Ašʿarite position is not completely clear, but it seems that they did not criticize the Baṣrans on this point.40

2) The substrate of junction.

Ṣāliḥī upheld that junction can inhere in a single atom in complete isolation. The Baṣran Muʿtazilites, on the other hand, affirmed that the same accident of junction inheres in more than one atom at the same time. Kaʿbī, Ašʿarī and all the other mutakallimūn rejected both theses, stating that an atom can be said to possess junction only in aggregation with other atoms and that each atom receives its peculiar accident of junction, since every occurrence of an accident must inhere in only one substrate.41

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The void Does an intra-cosmic void exist? The mutakallimūn phrased this question as whether two atoms can be separate with no third atom between them. The Baṣran Muʿtazilites deemed the intra-cosmic void to be not only possible but also necessary, for otherwise movement would be impossible (since atoms cannot penetrate into one another). The Baġdādian Muʿtazilites, on the other hand, rejected the void, claiming that air can be the medium for movement because, being fluid, it is able to occupy the position that was occupied by the moving object at the beginning of its movement.42

Atoms and space The mutakallimūn considered the issue of whether space itself is discrete, just like matter, in the form of a thought-experiment: Could a single atom be placed right above the point where two other atoms are connected? Those who answered that that is possible, like Abū Hāšim and perhaps ʿAbd al-Ǧabbār al-Hamaḏānī (d.1025), accepted that space is continuous, since the possibility for one atom to end up right above the point of connection between two other atoms requires space itself to accept measures of subatomic magnitude.43 The majority of the mutakallimūn, however, maintained that an atom cannot move right above the point of connection of two other atoms, thus implying that space must be discrete just like matter.44

Atoms and accidents The mutakallimūn debated on how atoms receive different accidents, and in particular on whether a single isolated atom can receive all classes of accidents, or some accidents require the atoms they inhere in to be arranged in some particular structure (what the mutakallimūn called the ‘physical constitution’, binya). First of all, it is necessary to know that they classified the accidents in four basic categories: (1) junction (for those who deemed it to be something additional to mere spatial juxtaposition); (2) the so-called ‘accidents of location’ (akwān), which include movement and rest, proximity and remoteness, aggregation and separation; (3) secondary perceptible qualities, like heat, colour, smell and so on; (4) the accidents of animation (life and death, capacity and incapacity, sensorial perception) and mental qualities (knowledge, will, desire, thought). The issue of the substrate of (1) junction has already been considered. As for the other three classes, the atomists presented four main positions. ●●

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Some early Muʿtazilites are credited with the thesis that no kind of accident can inhere in a single atom inasmuch as the latter is isolated, not even (2) the accidents of location: a single atom cannot be said to move, or to be at rest, or to be in isolation. Abū al-Huḏayl believed that an isolated atom can support (2) the accidents of location (with the exception of aggregation), but it does not support (3) secondary perceptible qualities and, presumably, (4) qualities of animation and mental qualities.

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Later Baṣran Muʿtazilites accepted both (2) and (3) for isolated atoms, while claiming that the inherence of (4) requires some specific physical structure, or ‘constitution’ (binya), and thus cannot inhere in an isolated atom. Ṣāliḥī and the Ašʿarites claimed than an isolated atom can support all kinds of accidents and thus could be said to be alive, hearing, knowing, willing and so on: Ṣāliḥī brought this theory to the extreme, claiming that even the accident of junction can inhere in an isolated atom and that life is not a condition for the inherence of mental qualities (will and knowledge can inhere in what is not alive); the Ašʿarites rejected both these claims.45

Atoms and geometry The relation between atomism and Euclid’s geometry was matter of dispute. In particular, the atomists debated on whether some basic Euclidean figures may exist in an atomistic world. Ǧuwaynī admitted that atoms can constitute a sphere and even built a proof for the existence of atoms assuming the existence of spheres. Ašʿarī, and perhaps Abū ʿAlī, rejected this possibility. Abū Hāsim probably admitted the existence of triangles and squares made out of atoms, for he claimed that space is continuous. The late Ašʿarite Faḫr al-Dīn al-Rāzī (d.1210) rejected the existence of circles, triangles and squares since all of those figures imply the existence of subatomic magnitudes. He explicitly recognized that the very foundations of Euclidean geometry are at odds with the existence of atoms and claimed that in order to save atomism, one must reject Euclid’s assumption of ‘coincidence’ (taṭbīq), namely the idea that two lines (or surfaces) can share the same point (or segment), as well as his third postulate, namely the assertion it is possible to construct a circle out of a segment rotating around one of its endpoints.46 This points to the fact that the atomists were in the process of elaborating a kind of discrete geometry in opposition to Euclid’s continuous geometry. However, at least until Rāzī, such an elaboration appears not to be fully developed, being prone to some major setbacks (e.g. Ǧuwaynī’s claim that atoms can constitute a sphere, Abū Hāšim’s claim that space is continuous).47

Atoms and time The mutakallimūn considered movement and rest to be accidents which inhere in corporeal objects and primarily in atoms. However, their account of temporal succession is not restricted to this, for they considered persistence in time (baqāʾ) to be something different than both movement and rest. In particular, they asked themselves whether atoms and accidents can be said to persist in time and, if so, what accounts for that persistence. The Baṣran Muʿtazilites upheld that atoms persist in time by themselves and not by means of some additional accident: they are intrinsically inclined to remain in existence until God actively annihilates them. Abū ʿAlī claimed that this is true for all accidents except movement. Abū Hāšim asserted that all accidents qua accidents must be said to be persistent by themselves.

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The Baġdādian Muʿtazilites and the Ašʿarites stated that no accident persists in time: they are all are continuously recreated instant by instant. Atoms, on the other hand, do persist. However, what accounts for that persistence is not their very essence, but rather an inessential accident of persistence (which, being an accident, does not persist): the only thing which maintains the atoms in existence is the continuous re-creation of non-persistent accidents which inhere in them. This doctrine suggests an atomistic conception of time, for the ‘extension’ of the existence of non-persistent objects must be point-like. The seeds of a similar conception were also present in the Baṣran doctrine.48 Faḫr al-Dīn al-Rāzī developed these intuitions and elaborated a full-fledged atomistic theory of movement, claiming that the former consists in a succession of different point-like events which occur within a succession of point-like instants. This discretization of movement and time, in turn, represented one of the foundations of Rāzī’s case for atomism itself.49

CONCLUDING REMARKS This chapter presented an account of the atomistic doctrine of the mutakallimūn. In conclusion, it is appropriate to add some notes on how atomism integrates into the broader picture of their metaphysics. The mutakallimūn appear committed to a ‘minimalist’ and moderately materialistic ontology. They admit only three kinds of entities: atoms, which are corporeal; accidents, which are not corporeal but inhere in what is corporeal; and God, who is the only truly incorporeal being. The atomists do not postulate incorporeal self-subsistent substances (i.e. souls) in order to explain how human beings possess knowledge, perception and life: biological and mental faculties are accounted for by appealing to the inherence of specific accidents in the atoms. Two divergent attitudes can be outlined concerning this issue. The Baṣran Muʿtazilites defend a kind of organicism, asserting that liferelated accidents require their substrates to be arranged in certain structures. On the other hand, the Ašʿarites completely reject organicism, claiming that any accident (with the exception of junction) can inhere in any substrate, regardless of whether the latter is inserted in some structure or not. Despite the aforementioned disagreement, we see a clear reductionist tendency in the mereology of the kalām: the parts precede the whole and ‘exhaust’ it. According to Ġazālī (d.1111), when the parts of a whole exist, the whole must exist as well.50 For the late Muʿtazilite Ibn al-Malāḥimī (d.1141), such an immediate existential implication becomes complete semantic identification: the whole consists in the sum of its parts, the difference between the former and the latter being merely verbal.51

NOTES 1. Dhanani (1993, 5–6, 1997, 139–40). 2. Ašʿarī (1954, 15.21-16.2). See also Ibn Fūrak (2005, 215.7); Ǧuwaynī (1969, 148.13-14).

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3. Ašʿarī (1954, vol. 2, 9.18-21); Ǧuwaynī (1969, 148.12-13). One of Ašʿarī’s sources attributes an almost identical position to Hišām, even though Ašʿarī himself is dubious about the attribution – (1954, vol. 1, 125.3.5). 4. Ibn Fūrak (2005, 215.15-17). 5. Ašʿarī (1954, vol. 2, 6.12-14, 16.6-7). On Hišām see Ašʿarī (1954, vol. 1, 124.5-8. 6. Ašʿarī (1954, vol. 2, 9.15-16, 16.8-9). 7. Ašʿarī (1954, 13.17–14.4). 8. Ibn Fūrak (2005, 211.5-7); al-Rāzī (1978, vol. 6, 19.11–20.9). 9. See Infra, Section ‘The debates among the atomists’. 10. Ašʿarī (1954, vol. 2, 9.7-9). Abū al-Qāsim al-Kaʿbī (d.931) is the only atomist which credited with the thesis that homogeneity is not necessary for the atoms, namely that atoms may belong to the same genus and may not – al-Nīsābūrī (1979, 29.3-7). 11. Ǧuwaynī (1969, 142.5-19). For a thorough discussion of this issue, see Dhanani (1993, 55-ff). 12. Ibn Fūrak (2005, 219.22-23). 13. The idea that corporeality is the only essential feature of bodies has great importance for the mutakallimūn, since it represents the fundamental premise of an argument for the existence of God as the incorporeal agent who provides bodies (or rather an essentially undifferentiated corporeality) with their differentiating accidents. On this see Davidson (1987, 174–93). 14. Mattawayh (mid-eleventh century) attributes Naẓẓām’s theory also to Naǧǧār (1975, 49.10). 15. Mattawayh (1975, 49.3-13); Ǧuwaynī (1969, 153.19–154.8). 16. Nīsābūrī (1979, 29.8–33.4). 17. Nīsābūrī (1979 , 33.5–34.16); Ǧuwaynī (1969, 154.9-22). 18. Nīsābūrī (1979, 34.17–6.21). 19. Mattawayh (1975, 49.3-6, 15. 20. Mattawayh (1975, 50.8-16; Ǧuwaynī (1969, 149.8-18). Mattawayh presents a variation of this argument: the aggregation of qualities cannot be necessary, for aggregated things must have the possibility to separate; since it must be contingent, there must be some accident which accounts for that aggregation, and accidents must inhere in space-occupying substrates (1975, 50.17–51.7). It should be noted that there is a counterexample against the assumption that all substrates must be space-occupying: the human mind, which could be said to be a non-corporeal substrate for various accidents. Such an account contradicts the doctrine of the atomists (mental qualities inhere in the atoms and not in an incorporeal substance), but the point at stake is precisely whether the mutakallimūn can validate their tenets in a non-circular way. 21. See Supra, Section ‘The homogeneity of bodies’. 22. Mattawayh (1978, 49.15–50.3); Ǧuwaynī (1969, 149.19–151.1). A variation of this argument states that if space-occupation emerged as a consequence of the aggregation of qualities, then space-occupation should be the effect of an external agent, and this is absurd because essential properties of things cannot be the effect of

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an agent (1978, 50.4-7). Mattawayh notices that this reformulation of the argument is sound only if one shares the doctrine of his school (the Baṣran branch of the Muʿtazila) on the active cause and its efficiency. 23. Ǧuwaynī (1969, 151.4-13). 24. Ǧuwaynī (1969, 151.14–152.6). As Ǧuwaynī rightly notices, the Baṣran Muʿtazilites cannot use both the ancillary arguments because for them a single accident can inhere in more than one substrate (see Infra, Section ‘The constitution of bodies’). 25. Ǧuwaynī (1969, 152.7-17). 26. Ašʿarī credits Naẓẓām with an assertion which draws near to (Na1) (Ašʿarī 1954, vol. 2, 6.12-14). However, many arguments directed against Naẓẓām aim to prove the finiteness of minimal parts and not their existence. Faḫr al-Dīn al-Rāzī explicitly credits him with thesis (Na2) (Rāzī 1978, vol. 6, 20.10-12). 27. Mattwayh (1978, 169.2–170.12); Ǧuwaynī (1969, 143.14–146.4); Rāzī (1978, vol. 6, 69.7–71.14). 28. Mattwayh (1978, 170.13–171.8); Ǧuwaynī (1969, 146.4-21; Rāzī (1978, vol. 6, 75.5-20). 29. Mattawayh (1978, 171.9–172.12); Ǧuwaynī (1969, 148.1-9). 30. On this point see Avicenna (1983, 219–23). 31. Mattawayh (1978, 163.16–169.1); Ǧuwaynī (1969, 147.1-9); Rāzī (1978, vol. 6, 61.8 – 63.6). 32. Mattawayh (1978, 163.1-7). 33. Šahrastānī (?), Masʾila fī iṯbāt al-ǧawhar al-fard, apud Id. (1934, 507.7-16); Rāzī (1978, vol. 6, 47.5–52.15). Rāzī criticized this argument in view of the fact that it assumes the existence of circles, which is at odds with atomism itself. 34. Rāzī (1978, vol. 6, 22.18–23.17). 35. Ṣāliḥī stated that atoms are bodies, in direct opposition to Abū al-Huḏayl, who explicitly discriminated between bodies and atoms (Ašʿarī (1954, vol. 2, 8.15-20); Rāzī (1978, vol. 6, 21.14-16). 36. Ibn Fūrak (2005, 214.10-11); Mattawayh, Taḏkira, 181.9-10; Nīsābūrī (1979, 58.56); Ǧuwaynī (1969, 142.17-19, 156.4-11). 37. Ǧuwaynī (1969, 158.20– 59.14); Rāzī (191978, vol. 6, 21.9-13). 38. Ibn Fūrak (2005, 212.3-7); Nīsābūrī (1979, 59.24–60.14). 39. Ibn Fūrak (2005, 214.20 – 215.6); Mattawayh (1978, 183.13-16). 40. Nīsābūrī (1979, 219.4-11). 41. Ašʿarī (1954, 5.15-16); Ibn Fūrak (2005, 212.13-14); Mattawayh (1978, 503.3-4); Nīsābūrī (1978, 219.4-5). It should be noticed that Abū ʿAlī is credited with the thesis that accidents of other kinds (like colours) may inhere in more than one substrate at once. 42. Mattawayh (1978, 116.16–124.3); Nīsābūrī (1979, 47.9–51.13). This debate appears related to that concerning whether God has the power of annihilating some atoms while leaving others in existence, for another possible solution to the puzzle of movement is precisely that of admitting the annihilation of the atoms which

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occupy the terminus ad quem of the movement. On that point, see Nīsābūrī (1979, 87.22–94.24). 43. In order to end up above the point of connection between two atoms, an atom should move for a distance which is not an integer multiple of the magnitude of an atom. 44. Rāzī (1978, vol. 6, 21.5-7). Rāzī tells us that ʿAbd al-Ǧabbār deemed space to be continuous, but his testimony is contradicted by other sources (Dhanani (1993, 190). 45. Ašʿarī (1954, 10.2–13.4); Ibn Fūrak (2005, 213.18-21); Ǧuwaynī (1969, 165.15– 166.3); Rāzī (1978, vol. 6, 21.16 – 22.2). 46. Šahrastānī (?), Masʾila, 507.7-16; Rāzī (1978, vol. 6, 139–57) (in particular 155.5-9). Rāzī claimed that Euclidean geometry as a whole contradicts atomism – (1978, vol. 6, 166.14-16). 47. Dhanani claimed that various assertions of the mutakallimūn imply the assumption of some kind of discrete geometry (1993, 133-ff.). On a speculative level, this claim is completely justified, in the sense that the atomism of the kalām is at odds with the very foundations of continuous geometry. On a historical level, however, I would say that the mutakallimūn were in the process of untangling their atomistic doctrines from the basic intuitions which constitute the foundations of Euclidean geometry (like the third postulate). Only in Rāzī, such process reached completion and gained self-awareness. I believe that this perspective helps us explain the inconsistencies and setbacks that we encounter in multiple authors. 48. Mattawayh stated that an object cannot be said to be in movement or at rest with respect to the very instant of its creation, for both rest and movement acquire meaning with relation to the previous instant of its existence: if the object remains in the same position, then it is at rest; if it changes position, it is in movement – ʿAbd al-Ǧabbār (sic) (1965, 33.5-11). 49. Rāzī (1978), vol. 6, 29–46. 50. Ġazālī (2000, 124). Cf. Rāzī (2004, 374). 51. Ibn al-Malāḥimī (2008), 28–31.

REFERENCES ʿAbd al-Ǧabbār, Abū al-Ḥasan (1965), Al-Maǧmūʿ fī al-muḥīṭ bi-al-taklīf, vol. 1, Beirut: al-Maṭbaʿa al-Kāṯūlīkiyya. al-Ašʿarī, Abū al-Ḥasan (1954), Maqālāt al-islāmiyyīn wa-iḫtilāf al-muṣallīn, vol. 2, Cairo: Maktabat al-Naḥḍa al-Miṣriyya. al-Ǧuwaynī, Abū al-Maʿālī (1969), al-Šāmil fī uṣūl al-dīn, Iskandariya: Manšaʾat alMaʿārif. al-Ġazālī, Abū Ḥāmid (2000), The Incoherence of the Philosophers, Provo: Brigham Yound University Press. al-Nīsābūrī, Abū Rašīd (1979), al-Masāʾil fī al-ḫilāf bayna al-baṣriyyīn wa-al-baġdadiyyīn, Beirut: Maʿhad al-Inmāʾ al-ʿArabī.

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al-Rāzī, Faḫr al-Dīn (1978), al-Maṭālib al-ʿAliya, vol. 6, Beirut: Dār al-Kutub al-ʿArabī. al-Rāzī, Faḫr al-Dīn (2004), Šarḥ al-Išārāt, vol. 2, Tehran: Anǧoman-e Āṯār va Mafāḫer Farhangī. al-Šahrastānī, Masʾila fī iṯbāt al-ǧawhar al-fard in al-Šahrastānī (1934), Nihāyat al-iqdām (sic.), pp. 505–14. al-Šahrastānī (1934), Nihāyat al-iqdām (sic.), London: Oxford University Press. Avicenna (1983), Al-Šifāʾ – al-Ṭabīʿiyyāt, vol. 1 (al-Samāʿ al-Ṭabīʿī), Cairo: al-Hayʾa alMiṣriyya al-ʿĀmma li-l-Kitāb. Davidson, Herbert A. (1987), Proofs for Eternity, Creation and the Existence of God in Medieval Islamic and Jewish Philosophy, 174–93, Oxford, New York and Toronto: Oxford University Press. Dhanani, A. (1992), The Physical Theory of Kalam: Atoms, Space, and Void in Basran Muʿtazilī Cosmology, 5–6, Leiden and New York-Koln: Brill. Dhanani, A. (2016), ‘Atomism in islamic thought’, in Seline Helaine (ed.), Enciclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, 139–40, New York: Springer Science&Business Media. Ibn al-Malāḥimī, Rukn al-Dīn (2008), Tuḥfat al-mutakallimīn fī al-radd ʿalà al-falāsifa, Tehran: Iranian Institute of Philosophy & Institute of Islamic Studies - Free University of Berlin. Ibn Fūrak, Abū Bakr (2005), Muǧarrad maqālāt al-šayḫ Abī al-Ḥasan al-Asʿarī, Cairo: Maktabat al-Ṯaqāfa al-Dīniyya. Mattawayh, Abū Moḥammad (1975), Taḏkira fī aḥkām al- ǧawāhir wa-al-aʿrāḍ, Cairo: Dār al-Ṯaqāfa li-al-Ṭibāʿa wa-al-Našr.

CHAPTER 11

Atoms and time I CHARLES DOYLE

Time is not the easiest subject to discuss in terms of ancient atomist physics. This is primarily due to the dearth of information left by the atomists concerning the nature of time in their world systems. From the information which we do have, we can discuss the general outlines of time among the Presocratic and Hellenistic atomists and explore how time functioned within their wider world views. Discussing the specifics is more challenging. However, there arises the possibility from the Epicurean concept of time that it was composed of discrete temporal quanta which constituted the briefest possible time at the level of atoms. Although nothing in the ancient sources is said about such a minimum, during late antiquity and the Middle Ages the reception of atomism gave rise to the atomus in tempore, a temporal atom which did not admit further division. While the medieval atom in time was controversial in its day, it has not received much attention from scholarship. This chapter will set out what we know about time among the ancient atomists and trace the history of the atomus in tempore, with the aim of establishing what connections, if any, there are between the ancient and medieval atomic theories on the nature of time.1

THE ANCIENT ATOMISTS ON TIME There is precious little to examine when it comes to atomist thought on time in the ancient world. Thanks to Diogenes Laërtius, we do have a first-hand summary of Epicurus’ opinion on the matter (Lives 10.72-3). However, since we lack a comprehensive account of the opinions of Democritus and Leucippus on time, we are left to examine the reports of others for evidence of what they wrote on the topic. If little can be said of Democritus’ physical doctrines, then there is even less to say about Leucippus’ teachings generally, let alone on the particular topic of time. The few remarks that survive concerning the Presocratic atomists on this subject were made specifically about Democritus. There is more to be said about Epicurus and his school than the Presocratic atomists with respect to time; however, primary sources on the topic are also scarce. That we have no statement of Democritus on the nature of time will come as no surprise. The majority of fragments attributed to him are ethical maxims, rather than his physical teachings.2 However, his teachings on nature were impactful enough in antiquity that other authors, notably Aristotle, Sextus Empiricus and Simplicius,

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commented on them. It is from these critiques and commentaries that we may glean some insight into Democritus’ thought on time. Aristotle comments that among his predecessors, everyone, with the exception of Plato, held that time did not come to be but was eternal (Physics 251b14-17). Democritus drew on this premise to make the case for his atomist cosmos. The argument runs as follows. Time has always existed and will always exist. Assuming there is nothing exceptional about time, is there any reason why this cannot apply to other beings? If one thing is so, why can the same not be said for everything else? Why not atoms and the void? Simplicius reads this comment in a similar light: ‘Democritus was so convinced that time was eternal that he used the premise that time did not come into being as something obvious in order to demonstrate that not everything came into being’ (In Physicam 1153.22-4).3 This says very little about the nature of time itself in the view of Democritus, but it does suggest that he at least held that time was eternal and existed per se. Elsewhere Simplicius portrays time and motion has having a close relationship in Democritus’ world system. He applies this to the issue of the multiplicity of worlds, problematizing the Democritean cosmos, stating the problem thus: ‘If there are many heavens, i.e. worlds, as Democritus and his followers posit, then the rotation of each of them would be time, and so there would be many simultaneous times, which is impossible. For it is possible that there should be simultaneous motions, but not simultaneous times’ (In Physicam 701.30-4).4 This critique of Democritus’ teaching that there are many worlds is grounded in Aristotelian views of the nature of time and in Simplicius’ own critique of his master Damascius. Time for Simplicius was a continuum and infinitely divisible, rather than composed of infinite discontinuities across the universe (Urmson 2014, 88–9). From his perspective then, Democritus’ multiplicity of worlds and thus their respective independent motions splinter the temporal continuum into an infinity of distinct times. Sextus Empiricus comments on the atomists’ opinions on time, stressing the dependence of time on sense-perception. He summarizes their thought of time as ‘an appearance in the form of night and day’ (Against the Mathematicians 10.181).5 Through our senses we notice the movements of the heavenly bodies most noticeably by the presence or absence of the sun, but undoubtedly by the motion of the moon, planets and constellations. What can we say then about Democritus’ thought on time? First, that he held time to exist in and of itself, rather than a secondary characteristic of matter. Secondly, that despite its existence, the passage of time was likely considered highly dependent both on motion and on sense-perception. Lastly, Simplicius’ objection to the simultaneous times of multiple worlds might suggest that time was composed of discontinuous moments rather than a universal continuum, but within the context of the commentary this seems to be a consequence of Democritus’ world system which Simplicius himself is positing, rather than a doctrine of the philosopher himself. The extent to which Epicureanism diverges from Democritus was the subject of debate even in antiquity (Morel 2009, 65–85; Sedley 2020, 61–75). With regard to time, in his famous doctoral dissertation, Karl Marx argued that they diverged from each other in that Democritus’ teaching implies that while time was removed

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from essential being (i.e. atoms) it was nevertheless eternal (2013, 56–60). On the other hand, Epicurus took this removal from atoms to relegate time from any meaningful existence to being an appearance. Epicurus outlined his thoughts on time in his Letter to Herodotus, suggesting that time is heavily dependent on our experiences and sense-perception (Diogenes Laërtius, Lives 10.72-3). While the letter as a whole is an insightful epitome of Epicurus’ teachings, the section on time is notable for its ambiguous language and lack of clarity (Morel 2002, 195). Exhorting the addressee to be cautious in his approach to investigations into time, Epicurus advises Herodotus not to scrutinize time in the same manner as other phenomena. As Long and Sedley describe it: ‘Time is a special case, being discernible not in bodies themselves but in certain of bodies’ accidents, typically motion and rest. Paradoxically, it is something self-evident, yet can only be understood by “analogical reasoning” – first drawing directly on experience to collect an appropriate set of accidents, then abstracting time as the common measure of them all’ (Long and Sedley 1987, 37). Our experience of the passage of time leads us to distinguish the length of periods of time; Epicurus considers these distinctions sufficient for the purposes of inquiry into nature, meriting no substantial change to the language with which we describe it. He hints that there may be some substantial reason why these terms derived from our everyday experience are not accurate or even real, but he does not delve into the matter in more detail. For that, we must look to other writers. Reporting an account from the Epicurean philosopher Demetrius of Laconia, Sextus Empiricus provides an overview of Epicurean time. He concludes that Epicurus held that time was incorporeal, though in a different sense to Stoic incorporeality, calling it a property of properties. Time does not exist in and of itself, but it is dependent on sense-perception and the motion of bodies. He writes that Epicurus said ‘that time is a property of properties which accompanies days and nights and hours and feelings and absences of feeling and motions and states of rest. For all of these are accidental properties of certain things, and since it accompanies all of these, time would not unreasonably be called a property of properties’ (Against the Mathematicians 10.219).6 In other words, time does not exist per se, but it is a secondary quality of the matter, understood through perception. It has no tangible existence on its own like atoms and the bodies made thereof, but rather it is discerned through our senses and marked by the changes in the positions of bodies, especially celestial bodies, relative to where they once were. We see this view of time expanded upon in more detail by Lucretius in book one of the De Rerum Natura (1.458-61). ‘Time also exists not of itself, but from things themselves is derived a sense of what has been done in the past, then what thing is present with us further what is to follow after.’7 Once again we see that time is inextricable from sense-perception. The evidence suggests that Epicurus and Democritus differed somewhat on the nature of time. Democritus seems to have held time to exist and to be ungenerated, while Epicurus and his school denied the existence of time per se. Motion and rest, presence and absence and the discerning of change through sense-perception appear to be common to the understanding of time by both the early and later atomists.

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However, other features of Epicurean physics and their impact on time merit some examination. Namely, the Epicurean kanon and the so-called minimum in the atom (Verde 2013, 138–40). Epistemologically, the Epicureans affirmed the truth of all perceptions, and we should reflect on the impact which this has on their view of time (Long and Sedley 1987, 78–86). In addition, one of the Epicurean innovations to atomic theory was the introduction of the minimum in the atom, a lower limit of size which nothing, not even an atom, could be smaller than. This innovation can be understood as having consequences for the nature of time. Epicurean time may well have developed a minimum in tempore, analogous to the other Epicurean partes minima in the atom and in perception. Verde argues that while it is not possible to trace such an idea to Epicurus himself, there may be more circumstantial evidence from which temporal minima follow as a consequence (Verde 2013, 122–8). Epicurus states that we should not investigate time according to our prolepses about it. Verde takes this as implying two consequences. The first being that time itself lacks prolepsis and that it has a particular (stricto sensu) nature. The second consequence is dependent on later evidence rather than Epicurus’ own testimony, especially that of Lucretius. Time, according to Epicurus, lacks a prolepsis, yet through sense-perception we can nevertheless discern the passage of time. Somehow time, despite lacking ‘autonomous reality’, is inferred by observation. It is no wonder then, Verde argues, that later Epicureans would try to study and define time. One such definition survives in the testimony of Demetrius, that of time as ‘an accident of accidents’ as distinct from a property, as Lucretius treats it (Sextus Empiricus Against the Mathematicians 10.219-27). Lucretius distinguishes things which happen per se from things which do not, and which depend on the things which exist per se, being properties (coniuncta) and accidents (eventa), respectively (De Rerum Natura 1.455-82). Demetrius’ account presents a different taxonomy. Things either exist per se like atoms and the void or are properties.8 Through comparison of the movement of bodies at an absolute speed, we can discern that motion over a small space takes less time than motion over a longer space. This movement is not time, but time accompanies the motion which happens in quanta and thus can be understood itself as a quantum or minimum. Morel examines some of the issues around Epicurean time and explores potential solutions and explanations for these ambiguities (Morel 2002, 204–6). Most usefully, he draws a distinction between ‘aiôn/aidion’ and ‘chronos’, which we might term ‘real’ or ‘objective’ time in contrast with the sense of time perceived by the senses. ‘Chronos’ has no real existence, but it arises from our perceptions of the motions of bodies. This time, Morel argues, is the time which Lucretius said not to exist per se.9 The dismissal of time per se by Lucretius would seem to settle the matter, were it not for a certain problem arising from Epicurus’ description of atoms in the Letter to Herodotus. If time does not exist at all, then how can the statement ‘atoms move continuously and eternally’ have any meaning? Morel argues from this point that this motion of the atom necessitates a reality for time; not the time which we perceive at a macrocosmic level but time at the level of atoms. As he puts it ‘sa mobilité relève de l’aiôn plus authentiquement que du chronos’ (2002, 205).

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Morel goes on to look at the ethical consequences for the nature of time within an Epicurean system and, like Verde, makes the case that atomic time follows as a consequence of these doctrines. If the minimum within the atom exists as distinct from the minimum in perception, he infers that a minimum in time may exist as distinct from the continuum of time perceived by sense-perception. Furthermore, this minimum time must be parcelled out in quanta because the things on which its status as an accident depends, namely the speed with which an atom has moved across the spatial, is also parcelled out in indivisible quanta. However, he notes that it is not proper to speak of Epicurean time atoms because time is not corporeal, but nevertheless this ‘punctum temporis’ (the imperceptible length of time which it takes for an atom to have moved at the minimum speed over the distance of the minimum) effectively constitutes the basic unit of time. The accounts of the ancient atomists on the nature of time are less than clear. While the sole statement which we have concerning Democritus’ views suggests a distinction between his and those of the later atomists, it is difficult to speak authoritatively on the Epicurean stance, given the ambiguities within the evidence. While the arguments set out by Verde and Morel are certainly persuasive, they do have to contend with the lack of an explicit statement of temporal atomism by the authors in question. Nevertheless, such a reading of the evidence is valid and, as I will explore further, gained acceptance in late antiquity and the Middle Ages.

ATOMS AND TIME IN LATE ANTIQUITY AND BEYOND Scholarship on atomism in late antiquity and beyond tends to focus on its reception and development in Islamic philosophy as well as its subsequent reintroduction to Western Europe during the High Middle Ages (Marenbon 2007, 88–91). Less has been said on the afterlife of atomic theory in Latin literature in the period between the decline of the Epicurean school and the rise of Scholasticism. With reference to the kind of non-providential cosmos it relies on, atomism was discussed often in Patristic literature and often attacked for its association with the Epicureans, although the relationship between Christianity and Epicureanism is more complex than pure antagonism. In the Middle Ages, there emerged a tradition of presenting certain individual things as ‘atoms’. This discussion of atoms as discontinuities in body, time, sound and number has been held up as example of the loss of Greek learning during the so-called dark ages. Pyle dismisses these as ‘garbled third or fourth hand accounts’ without any engagement with the texts themselves (1995, 210). G. B. Stones concludes that ‘the idea [of atomism] is still there, though it is frequently misunderstood’ (1928, 445–6). This chapter aims to plot the development of the atom as a unit of measurement of time and show that its origins are not as far removed from ancient atomism as they may seem at first glance. To clarify, I am not suggesting that the medieval atomus in tempore was sophisticated by any means. However, I am arguing that the atomus’ lack of philosophical rigour does not diminish its significance as part of the Nachleben of atomist philosophy.

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Before outlining the history of the atomus in tempore, we must first ask what exactly it is. Put simply, it is a theoretical minimum time which features in late antique and early medieval literature, encyclopaedias and computistics. Years can be divided into months, months into days and so on, until one reaches a moment so brief that it admits no further division into smaller parts. This unit varies in its length and in its relationship to other temporal divisions, but it was frequently posited as the shortest possible quantum of time. It is not that time itself is an indivisible continuum but that it is composed of indivisible moments. While the development of the atomus in tempore was influenced by the reception of ancient philosophy in late antiquity, its origins appear to lie in Christian biblical exegesis and a particular problem of translation of the Bible. In the early days of the Christian church, Greek was a lingua franca for much of the Mediterranean world. However, in particular for the non-hellenophone populations of North Africa and Gaul, the need to translate the Bible into Latin grew with the spread of the religion. This gave rise to the Vetus Latina or Old Latin versions of the Bible, though many writers who knew Greek like Tertullian and Augustine engaged in their own ad hoc translations as needed. We must remember that Jerome’s Vulgate was not written until late in the fourth century and not promulgated by the Roman Catholic Church until many centuries later. In the epistles of Paul of Tarsus, we find a reference to atomism. In 1 Corinthians, Paul discusses the resurrection of the dead saying ‘ἰδοὺ μυστήριον ὑμῖν λέγω: πάντες οὐ κοιμηθησόμεθα, πάντες δὲ ἀλλαγησόμεθα, ἐν ἀτόμῳ, ἐν ῥιπῇ ὀφθαλμοῦ, ἐν τῇ ἐσχάτῃ σάλπιγγι’ (1 Cor. 15.51-2).10 Although rare, Paul’s use of the word ἄτομος here in a temporal sense had precedents in Classical Greek. For example, Aristotle used ἄτομος as an adjective describing time three times (Physics 236b6, 263b27; De Sensu 447b18). However, the word had been part of Latin philosophical vocabulary since at least the time of Cicero, who introduced it as a loanword for the discussion of Epicurean physics. Its use outside of a philosophical context seems to have caused some issues for readers, which needed an explanation. Subsequently, Latin Patristic authors engaged with the letter, variously interpreting the ἄτομος of Paul either literally or metaphorically. The word is used in a temporal sense by Tertullian in his Adversus Marcionem and also in the pseudoCyprian computus of 243 AD.11 Jerome of Sidon translated it as ‘in momento’ in the Vulgate and offers an explanation for this translation in one of his letters. It receives its most detailed explanation in a sermon of Augustine of Hippo. Of these, the latter is the most relevant to the history of the atomus in tempore, but I shall comment briefly on the others. Tertullian and Jerome understand atomus in a temporal sense, and the author of the computus does not elaborate on its nature but clearly understands it as a unit of time, if not the smallest unit of time. Tertullian, in discussing 1 Cor. 15.51-2, paraphrases the passage once (Adv. Marc. 3.24) and quotes it with his own translation (Adv. Marc. 5.10). In the first passage he states ‘demutati in atomo in angelicam substantiam’ (we shall all be changed in an atom into angelic substance).12 In the second he quotes Paul saying ‘et nos mutabimur in atomo, in oculi momentaneo motu’ (and we shall all change in an atom, in the momentary moving of an eye).13 Tertullian opted to retain the word ‘atom’ in his

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translation and paraphrase, but he clearly understands it to have a temporal meaning rather than a material one. Jerome abandoned the word ‘atom’ altogether in his translation, rendering the meaning plainly as ‘in momento’. Writing to two monks in Toulouse, he offers more detail on his understanding of Paul’s use of the word atom: ‘Atomus autem punctum temporis est, quod secari et diuidi non potest; unde et Epicurus ex suis atomis mundum struit et uniuersa conformat’ (However, an atom is a point of time, the cutting and division of which is not possible. It is from this that Epicurus established a world out of his atoms and fashioned a universe).14 Two things are striking about this interpretation. First, his definition of the atomus as temporal, rather than corporeal, and secondly his attribution of such a definition to Epicurus. While this statement from Jerome can be taken to support Morel and Verde’s arguments for reading temporal atomism into Epicurean physics, Jerome is not writing for a philosophical audience. His phrasing here hints that there is a distinction to be drawn between atoms in reality, which are temporal and attested by scriptural authority, and Epicurus’ own atoms which are corporeal and conjectured by a philosopher. It is in the writings of Augustine that we find the most detailed Patristic account of the nature of the atomus in tempore (Pabst 1994, 40). Augustine’s Sermon 362 explains the nature and manner of the resurrection of the dead, establishing that it is literal rather than metaphoric and addresses the possibilities for how it will unfold. In answering these questions, he cites scriptural authority and 1 Corinthians features prominently. Like his fellow countryman Tertullian, Augustine makes use of the loanword atomus, necessitating an explanation to his Latin-speaking congregation: Many do not know what an atom is. An atom is said to be from τομή, which means a cutting, and in Greek ἄτομος means something which cannot be cut. It is said that there are bodily atoms and temporal atoms. It is said that a bodily atom (if such a thing could be discovered) is something that it is not possible to split, a tiny body so minute that it does not have anywhere where it can be divided. However, a temporal atom is a short moment, which has nowhere to be split. I will give this example for those of you with minds too slow to grasp what I am saying; there is a rock, divide it in parts and those parts into pebbles, then the pebbles into granules, then they are sand, and again divide the grains of sand into the most fine dust, until you can arrive at something so small, that it is of a quality which cannot be divided further. This is the atom in the body. The atom in time is to be understood in this manner. A year, for example, is split into months, months are divided into days, and days can be split into hours and now hours can be lead out into certain parts of hours which admit division, up until you arrive at such a point in time and a certain droplet of a moment so that no further parts can be drawn out of it and so it cannot be divided: this is the atom in time.15 Augustine was no stranger to the atomism of Epicurus and was doubtlessly familiar with the ordinary usage of atomus in Latin.16 By his own admission, he would have considered Epicureanism to have the superior ethics of all schools of philosophy, but he rejected it because of its affirmation of the mortality of the soul (Confessions

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6.26). It is clear from this sermon that the Epicurean atom is in his mind when he read the word in Paul’s letter. Augustine explains the word, foreign to Latin ears, by way of etymology as an indivisible before describing the nature of the atom. After outlining how there are atoms in bodies (albeit with some reservations), he explains that a corresponding particle exists in time. Through this approach, Augustine resolves any potential conflict as an orator trying to convey a complex idea to his audience. By means of a simple analogy, Augustine explains the nature of atomic matter and, by extension, atomic time to his audience. With the statement on the matter from a Patristic authority, the atomus in tempore was poised to spread into the Latin literary traditions of late antiquity. An eccentric treasure trove of late antique Latinity, the fifth-century De Nuptiis Philologiae et Mercurii of Martianus Capella presents time as a discontinuity. The book, which describes a union of learning and eloquence through the figures of pagan gods, features this description of time: ‘First let us take up the tempus [the basic unit of time], which, like the atom, admits of no cutting into parts or particles. It is comparable to the point of the geometricians or the monad of arithmeticians, i.e. a certain singularity and comprised of itself ’ (De Nuptiis 9.971).17 However, it is in the works of Isidore of Seville that we find the most influential account of the atomus in tempore. The influence of Isidore of Seville on the Latin West in the Middle Ages cannot be understated. Drawing on all available sources to him, he compiled in his De Natura Rerum and the Etymologies, summaries of ancient knowledge which went on to be the definitive authority on many topics. Both works address the divisions of time, but only the Etymologies discusses the atom as such a division. The thirteenth book of the Etymologies, De Mundo et Partibus, presents a synthesis of scripture and science across its first three chapters. The book opens with a short description of Christian cosmogony and is followed by chapters with a more philosophical slant, incorporating atoms and the four elements into this world view. His account features multiple categories of atoms, including the atom in time:18 Atoms (atomus) are what the philosophers call certain corporeal particles in the world that are so tiny that they are not visible to sight, and do not undergo τομή, that is, ‘splitting’, whence they are called ἄτομοι. They are said to fly through the void of the entire world in unceasing motion and to be carried here and there like the finest dust motes that may be seen pouring in through the window in the sun’s rays. Some pagan philosophers have thought that all trees and plants and fruits have their origins from these particles, and that from them fire and water and the universe were born and exist. There are atoms in bodies, in time, and in number. In a body, such as a stone. You may divide it into parts, and the parts into grains, like sand; then divide the grains of sand themselves into the finest dust, until, if you can, you will reach a certain minute particle, which no longer can be divided or split. This particle is the atom in bodies. With reference to time, the atom is understood in this way: you may divide a year, for example, into months, months into days, days into hours. The parts of hours still admit division until you come to a point of time and a speck of an instant such that it cannot be

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extended through any small interval, and thus can no longer be divided. This is an atom of time.19 The description of the atomus in corpore and in tempore is taken almost verbatim from Augustine’ Sermon. The only noticeable difference between the presentations is the absence of Augustine’s tentative reservations about corporeal atomism. Additionally, Isidore alludes to Lucretius in his account of atoms.20 He provides us with an image of the motion of atoms in space: ‘Hi per inane totius mundi inrequietis motibus uolitare et huc atque illuc ferri dicuntur, sicut tenuissimi pulueres qui infusi per fenestras radiis solis uidentur.’21 The image of dust moving through the air seen through sunlight in a dark room is certainly adapted from Lucretius.22 Isidore’s account synthesizes disparate sources into a new account of atoms, one which would have wide-ranging influence in Medieval Europe. In the Middle Ages, the atomus was often understood through Isidore’s etymologizing lens, which analysed it to its literal meaning of indivisible. The atomus in tempore is the indivisible in time, raising the question of whether time in this context was understood as an indivisible continuum or as a series of discontinuous instants. The medieval discourse surrounding it appears to understand time to be composed of many divisions, with the atom as the primary part of time rather than time itself to be indivisible. While Isidore’s summary of atomism was influential in the areas of encyclopaedias and grammatical texts, it was in the field of computistics that we see the greatest influence. Computus, the medieval science of chronology, developed out of a need to calculate the date of Easter. In the early centuries of Christianity, the celebration of Easter varied regionally, with some communities following the Jewish celebration of Passover, while others selected various dates in the spring, leading to a number of paschal controversies within the church. Blackburn and Holford-Strevens provide a comprehensive overview of these disputes which led to the need for standardization (1999, 791–800). Isidore’s readership compiled together many divisions of time from his Etymologies and De Natura Rerum, spanning from the atomus as the smallest unit to the mundus, the totality of time between the world’s creation and destruction. While the Computus of 243 is chronologically the first such text to mention the atomus as a unit of time, it was systematized by the generations of computists who followed after Isidore (Mosshammer 2017). The seventh- or eighth-century Computus Einsidlensis lists the atomus as the basic unit of time, albeit with some reservations about whether it or the momentum is the fundamental unit of time (85). The Carolingian Munich Computus goes into some detail on the atomus (Warntjes 2010, 4–8). The anonymous text quotes Isidore, but also responds to him, listing four, rather than three categories of atomi, including of course the atomus in tempore. The temporal atom is here defined as being one-fifteenth of a momentum. The mid-seventh-century De Ratio Computandi also casts some doubt as to whether or not the atomus is truly the fundamental unit of time (Walsh and Ó Cróinín 1988, 113–213). The author of the Munich Computus never questions the status of the atom as the shortest moment. This dispute between atomus and momentum is

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perhaps best traced to the different accounts presented by Isidore, who notes the atomus as the shortest time in book thirteen but lists the divisions of time in the fifth book as ‘Tempus autem momentis, horis, diebus, mensibus, annis, lustris, saeculis, aetatibus diuiduntur’23 (Etymologies 5.29.1).24 Bede wrote two chronological texts concerned with the calculation of Easter, De Temporibus around AD 703 and a much longer work De Temporum Ratione over twenty years later. The latter was a more comprehensive treatment of the topic of chronology and had a lasting legacy during the Carolingian period. Bede lists the divisions of time at the outset, among them the atomus. However, he notes that the term is synonymous with both punctus and momentum and defines it partially with reference to 1 Corinthians: The Apostle uses the term for this kind of time in a better sense, to suggest the swiftness of the Resurrection, stating ‘We shall all rise, but we shall not all be changed, in a moment, in the twinkling of an eye, at the last trumpet.’ This deserves our attention, because although computists make a strict distinction [between these terms], many writers indiscriminately call that tiniest interval of time in which the lids of our eyes move when a blow is launched [against them], and which cannot be divided or distributed, either a momentum, a punctus or an atom.25 Through Bede’s testimony we see that in the computistical tradition, the atom remained a unit of time, but its status was disputed, as was its relationship to larger temporal units. Hrabanus Maurus also uses this definition of the atomus in connection with 1 Corinthians, likening its brevity to the speed of the blink of an eye (De Computo 1.11.26). Computuses influenced by these Irish and British sources began being produced on the continent in the eighth and ninth centuries AD. Textbooks on the subject were produced in France, Germany and Italy in the ninth and tenth centuries. Following increased contact with the Arabic world in the eleventh century onwards, the nature of these chronological works began to shift (Wartnjes 2010, lv–lvi). As mechanized means of time-keeping were developed in the High Middle Ages, the hour was divided in a geometrical system of minutae primae (1/60) and secundae (1/3600), ultimately the source of our modern minutes and seconds which eventually superseded the earlier temporal divisions (Blackburn and Holford-Strevens 1999, 663–4). Outside of the computus, the atom in time permeated other genres during the Carolingian renaissance, often with less precise definitions. For example, in certain grammatical works attributed to Peter of Pisa, Charlemagne’s personal tutor from AD 774–790, we find reference to the atomus in tempore as a day rather than a smaller subdivision (Krotz and Gorman 2014, 337–42). Although lacking the sophistication of earlier atomism, the atomus in tempore played a role in the intellectual life of the Middle Ages. Is it possible to argue that, despite its simplicity, the atomus in tempore fundamentally refers to a continuity from ancient atomist thought rather than being a later innovation? Does it relate to the proposed temporal minima which Verde and Morel argue are implied by

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Epicurean physics or is fundamentally a misunderstanding of ancient physics? I am hesitant to speculate that Jerome and Augustine had access to Epicurean sources which explicitly attest the existence of temporal minima, but also I cannot rule out that they did not. While Verde and Morel’s arguments follow from evidence from ancient sources, I do not think that this medieval atomic time was taken directly from atomist teaching. Having traced the development of the atomus in tempore to Paul via Augustine and Jerome, I think we can conclude that the origin is based on their understanding of Epicureanism as non-Epicureans with a certain degree of technical knowledge. Their arguments for the atomus are those of exegetes and translators, not of natural philosophers. Thus, even if they were drawing on Epicurean doctrine, we cannot necessarily trust them to do so impartially and honestly. Augustine himself, as I read it, expresses doubt about the existence of the corporeal atom but sees the temporal atom as being sanctioned by the Apostle Paul by his usage of the word in the letter. Jerome, as noted earlier, may have drawn a distinction between Epicurus’ atoms and the atom in time attested in scripture. While nothing in this paper rules out the possibility of genuine continuity from the Epicureans, we do have the simpler explanation that Jerome and Augustine took the existence of the atom in time from religious authority rather than from secular philosophical doctrine. In addition to providing practical instruction on the mathematics and arithmetic required to compose and interpret Easter tables, computus also provided theoretical instruction to users about the divisions of time and it is here that the atomus in tempore finds a natural home as the basic unit of time within a wider theoretical framework of chronology. In some cases, it is even precisely defined. In the Munich Computus and Bede’s De Temporum Ratione we are told that there are fifteen atomi in a momentum, while a ninth-century treatise De Divisionibus Temporum reckons that a momentum contains 564 atomi. The atom as a unit of time is present in the Computus of AD 243 but it was not defined specifically.26 At 2.14, the computist simply describes the units of time in ascending order from smallest to largest ‘ad athomum per dies et annos singulos’ (to the atom through the day and single year).27

CONCLUSION We may never know precisely what Democritus and Leucippus thought on time, or how influential these teachings of the early atomists were on Epicurus and his school. We can say that Democritus is likely to have treated time as having an existence on a par with matter and void, whereas Epicurus and his school in some sense denied time’s existence per se. However, owing to the ambiguous language in Epicurus’ own account, it is possible to distinguish eternity from ordinary time, although later evidence may contradict this interpretation. As discussed, a temporal minimum may follow logically from the Epicurean spatial minimum and the nature of atomic motion. Looking at the later reception of atomism, we must ask what connection, if any, is there between the temporal quantum of the atomus in tempore and the potential Epicurean pars minima of time. Its development, clearly shaped by philosophical

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vocabulary, testifies to its connections with the Epicurean tradition, but the evidence does not suggest direct continuity here. Rather, the medieval atomus in tempore is a reception of corporeal atomism applied to time by Patristic authors in response to a linguistic problem rather than a response to a problem in natural philosophy. Nevertheless, we can say that the medieval understanding of this temporal atomism certainly lends some weight to Verde and Morel’s argument that atomist time may be understood as being composed of discontinuities. At least one of these Patristic authors who elaborated on the doctrine was, after all, familiar with the Epicurean school and well versed in the natural philosophy and science of his day.

NOTES 1. This chapter is partially adapted from my doctoral thesis (Doyle 2018, 135–90). I wish to thank the Irish Research Council and the College of Arts, Social Sciences and Celtic Studies at NUI Galway for their support of this research. 2. Most of the evidence for his physical teachings comes to us from Aristotle, who is reported to have written a lost monograph on Democritus. The majority of the ethical fragments were recorded by Stobaeus. 3. Taylor (2010, 90). 4. Taylor (2010, 90). 5. Taylor (2010, 90). 6. Inwood and Gerson (1997, 88). 7. Lucretius (2002). 8. As Verde puts it, ‘Se il tempo è un symptoma, evidentamente non ha il carattere “eterno” e “inseparabile” della proprietà; il tempo, inoltre, è un accidente che si accompagna ad altri accidenti, ossia a eventi e ad avvenimenti che, non sussistendo di per se stessi e non avendo in sé alcuna autonomia, già di per sé sono accidentali’ (Verde 2013, 128). 9. ‘Le temps n’est donc pas le mouvement de l’univers, non seulement parce qu’il n’existe pas par soi, mais aussi parce qu’il doit être défini par rapport à la representation que nous en avons’ (2002, 201). 10. NSRV: ‘Listen, I will tell you a mystery! We will not all die, but we will all be changed, in a moment, in the twinkling of an eye, at the last trumpet’. 11. This text is the earliest extant Christian computus, once attributed to Cyprian. The text argues for the use of a sixteen-year cycle within a wider 112-year period of seven sixteen-year cycles for the calculation of Easter. For more see Mosshammer (2017). 12. Tertullian (1954, 609). Translation my own unless otherwise stated. 13. Tertullian (1954, 609). 14. Jerome (1912, 450). 15. Augustine (1846). 16. Erler (92009, 46–64). 17. Capella (1977, 373).

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18. The etymologizing of atomus leads to other individuals being classified as atomi. Isidore counts the number of types of atomus as three, the atomus in corpore, in tempore and in numero. He alludes to other atom-like indivisibles including letters (i.e. phoneme/graphemes) and geometrical points. His account leads to subsequent types of atoms in the Carolingian period including the atomus in oratione/litteris and the atomus in sole. 19. Isidore (2006, 271–2). 20. Cf. De Rerum Natura 2.114-17: contemplator enim, cum solis lumina cumque / inserti fundunt radii per opaca domorum: / multa minuta modis multis per inane videbis; and 2.125-8: Hoc etiam magis haec animum te advertere par est / corpora quae in solis radiis turbare videntur, / quod tales turbae motus quoque materiai / significant clandestinos caecosque subesse. The phrase per inane is also characteristically Lucretian, occurring no fewer than thirteen times in the second book alone. 21. ‘They are said to fly through the void of the entire world in unceasing motion and to be carried here and there like the finest motes of dust that may be seen pouring through the window in the sun’s rays’ (Barney et al. 2006, 271). 22. Although the image itself as an analogy for the motion of atoms is considerably older than Lucretius, plausibly dating back to Democritus. See Aristotle De Anima 404b31-404a4 and Strohmaier (1968, 1–19). 23. ‘Intervals of time are divided into moments, hours, days, months, years, lustrums, centuries and ages’ (Trans. Barney et al. 2006, 125). 24. It is tempting to speculate whether this issue ultimately reflects beyond Isidore to the translation of 1 Corinthians. However, the extant Vetus Latina texts of the letter do not attest in atomo as a translation, making it more likely that this dispute was grounded in Isidore’s differing accounts. See Houghton et al. (2018, 293). 25. Bede (1999, 16). 26. Anonymous (1847, 557). 27. The usage of the term is similar to Tertullian’s in the Adversus Marcionem. As a Christian text, we can presume that that it is based on Paul’s usage in 1 Corinthians. Apart from this metaphorical usage, there is no further elaboration on the atom in this computus.

REFERENCES Anonymous (1847), ‘De Computo Paschali Libri Duo’, in Jacques Paul Migne (ed.), S. Gelasius I Papa, S. Avitus, S. Faustinus, Joannes Diaconus, Juliabus Pomerius, Duo Anonymi, Aurelius Prudentius, 543–60, PL 59, Paris: Migne. Augustine (1846), Opera Omnia, ed. Jacques Paul Migne, PL 39, Paris: Migne. Bede (1999), Bede, The Reckoning of Time, ed. F. Wallis, Liverpool: Liverpool University Press. Blackburn, B. J. and Holford-Strevens, L. (1999), The Oxford Companion to the Year, Oxford: Oxford University Press. ‘Computus Einsidlensis’ (n.d.), Einsiedeln. Stiftsbibliothek 321 (645).

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Diogenes Laërtius (1925), Lives of the Eminent Philosophers: Books 6–10, trans. R. D. Hicks, vol. 2, 2 vols, LCL 185, Cambridge, MA: Harvard University Press. Doyle, C. (2018), Studies in the Latin Christian Reception of Early Greek Materialism, Galway: National University of Ireland, Galway. Erler, M. (2009), ‘Epicureanism in the Roman empire’, in J. Warren (ed.), The Cambridge Companion to Epicureanism, 46–64, Cambridge: Cambridge University Press. Houghton, H. A. G., Kreinecker, C. M., MacLachlan, R. F. and Smith, C. J. (2018), The Principal Pauline Epistles: A Collation of Old Latin Witnesses, Leiden, The Netherlands: Brill. Inwood, B. and Gerson, L. P. (1997), Hellenistic Philosophy: Introductory Readings, 2nd edn, Cambridge: Hackett Publishing Company. Isidore (2006), The Etymologies of Isidore of Seville, ed. S. A. Barney, W. J. Lewis, J. A. Beach and O. Berghof, Cambridge: Cambridge University Press. Jerome (1912), Opera: Epistulae LXXI-CXX, ed. Isidor Hilberg, CSEL 55, Vienna: Tempsky. Krotz, E. and Gorman, M. M., eds (2014), Grammatical Works Attributed to Peter of Pisa, Charlemagne’s Tutor, Biblotheca Weidmanniana, Hildesheim: Weidmann. Long, A. A. and Sedley, D. (1987), The Hellenistic Philosophers: Volume 1, Translations of the Principal Sources with Philosophical Commentary, Cambridge: Cambridge University Press. Lucretius (2002), De Rerum Natura, ed. Martin Ferguson Smith, trans. William Henry Denham Rouse, LCL 181, Cambridge, MA: Harvard University Press. Marenbon, J. (2007), Medieval Philosophy: An Historical and Philosophical Introduction, London and New York: Routledge. Martianus Capella (1977), Martianus Capella and the Seven Liberal Arts, trans. William Harris Stahl, Richard Johnson and Evan Laurie Bruge, vol. 2, 2 vols, New York: Columbia University Press. Marx, K. (2013), ‘Differenz der demokritischen und epikureischen Naturphilosophie’, in Hans-Joachin Lieber (ed.), Werke, Schriften, 8–106. Darmstadt: Lambert Schneider. Migne, J. P., ed. (1850), ‘De Divisionibus Temporum’, in Venerabilis Bedae Opera Omnia, Patrologiae Cursus Completus Series Latina 90, Paris. Morel, P.-M. (2002), ‘Les ambiguïtés de la conception épicurienne du temps’, Revue philosophique de la France et de l’étranger 127, no. 2: 195–211. Morel, P.-M. (2009), ‘Epicurean atomism’, in James Warren (ed.), The Cambridge Companion to Epicureanism, 65–83, Cambridge: Cambridge University Press. Mosshammer, Alden A. (2017), ‘Towards a New Edition of the Computus of AD 243’, in I. Warntjes and D. Ó Cróinín (eds), Late Antique Calendrical Thought and Its Reception in the Ealry Middle Ages: Proceedings of the 3rd International Conference on the Science of Computus in Ireland and Europe, Galway 16-18 July 2010, Studia Traditionis Theologiae: Explorations in Early and Medieval Theology 26, Tournhout: Brepols. Pabst, B. (1994), Atomtheorien Des Lateinischen Mittelalters, Darmstadt: Wissenschaftliche Buchgesellschaft. Pyle, A. (1995), Atomism and Its Critics: Problem Areas Associated with the Development of Atomic Theory of Matter from Democritus to Newton, Bristol: Thoemmes Press.

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Sedley, D. (2020), ‘Why aren’t atoms coloured?’ in U. Zilioli (ed.), Atomism in Philosophy, 61–75, London: Bloomsbury Academic. Stones, G. B. (1928), ‘The atomic view of matter in the XVth, XVIth and XVIIth centuries’, Isis 10, no. 2: 445–65. Strohmaier, G. (1968), ‘Demokrit Über Die Sonnenstäubschen’, Philologus 112: 1–19. Taylor, C. C. W. (2010), The Atomists, Leucippus and Democritus: Fragments: A Text and Translation with a Commentary, Phoenix Pre-Socratics 5, London: University of Toronto Press. Tertullian (1954), Opera I: Opera Catholica. Adversus Marcionem, ed. E. Dekkers, J. G. P. Borleffs, R. Willems, R. F. Refoulé, G. F. Diercks and A. Kroymann, vol. 1, 2 vols, Corpus Christianorum Series Latina 1, Tournhout: Brepols. Urmson, J. O. (2014), Simplicius: Corollaries on Place and Time, London: Bloomsbury Publishing. Verde, F. (2013), Epicuro, Rome: Carocci editore. Walsh, M. and Ó Cróinín, D., eds (1988), Cummian’s Letter De Controversia Paschali, Toronto: Pontifical Institute of Mediaeval Studies. Warntjes, I. (2010), The Munich Computus: Text and Translation: Irish Computistics between Isidore of Seville and the Venerable Bede and Its Reception in Carolingian Times, Sudhoffs Archiv/Beiheft 59, Stuttgart: Steiner.

CHAPTER 12

Atoms and music in late medieval philosophy1 PHILIPPA OVENDEN

INTRODUCTION The above bars in Figure 12.1 present the infamous opening motif of Beethoven’s Fifth Symphony. In unison, the orchestra plays three quavers, followed by a minim. A quaver rest signals a momentary break before the motif returns. This ominous rhythm is well known – ta ta ta taaaa | ta ta ta taaaaaaaa.2 The practice of notating musical rhythms – providing symbols that represent spans of time – has a complex and intriguing history. It is bound to the ways in which we think about musical sound and its being in time. In the aforementioned example, the first 2/4 bar contains four quaver pulses, and the second one minim. This is because we view musical time as being innately duple – notes are divided into two by default. The rhythm reads horizontally, since we conceive of time as a linear flow. We adopt a standard pattern for the representation of pitch and rhythm – a set of clefs, a fiveline stave and an array of rhythmic signs. These practices emerged over hundreds of years. But the passage also provokes questions that are more abstract – are the spans of time represented by the minims divided into shorter timespans, such as the quavers? Conversely, do shorter timespans, such as the quavers, group together to form the minims? Is the time of the passage continuous or discrete? How does the physical reality of these sounds influence their notation? Are musical notes limited in brevity? If so, how is this determined? In the fourteenth century, the notation of musical rhythm was in a state of flux. Notes could be divided interchangeably into two or three parts, not just two; the number of lines of a stave could vary; notes were black and diamond-shaped, or square, not white and round. Yet like today, the writing of musical rhythm was a mirror to human experience of the world. Theorists interrogated the nature of the

FIGURE 12.1:  Ludwig van Beethoven, Symphony No. 5, mvt. 1, mm. 1-4.

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noteshapes they wrote and the sounds they sang. They pondered whether longer notes were made of groups of shorter notes, or whether shorter notes were created through the division of longer notes. Some asked what the unit of measurement for musical time might be, and whether this unit was a musical note or an atom of time.3 In his Ars cantus mensurabilis – a substantial treatise outlining an innovative system for the notation of rhythmic, or mensural music, probably written in the late thirteenth century – Franco of Cologne had described the breve ■ as a ‘minimum in plenitudine vocis’, (least in the fullness of sound).4 It was divisible into shorter notes – semibreves ♦ – of which there were various kinds. However, since the breve was believed to represent the briefest complete sound, semibreves were first rationalized in an aggregate, appearing among groups of other notes to complete units of musical time. The appearance of a semibreve by itself was unthinkable to some theorists, for whom semibreves were by nature parts of the breve, not autonomous notes.5 Over time, semibreves became more widely accepted and were ultimately acknowledged as independent figures. By the early fourteenth century, the note depicting the shortest duration of musical time was the semibrevis minima, least semibreve or minim . This note came to embody extra-musical significance. Presumed by many theorists to signify the sound of briefest duration that could be sung physically, it was also deemed to be the minimal conceptually quantifiable duration of musical sound. Once the minim had become established, notes shorter than it soon appeared. They were assigned a number of names, but are today referred to most commonly as semiminimae (semiminims) .6 For many authors semiminims were controversial, since their existence challenged the idea that the minim figure was synonymous with a minimum extent of musical sound. The invention of noteshapes portraying spans of time briefer than the minim compelled theorists to reinvent a belief system about musical sound, time and its parts that had been contrived with the minim as the minimal limit. Theorists discussed the legitimacy of these new notational practices in relation to the physical limitations of human capacity to sing short notes, as well as the conceptual limits of mathematical systems that were used to quantify note lengths. Concepts articulated in canonical texts, such as Boethius’ De institiutione arithmetica, as well as the many translations of Aristotle’s writings that had become ubiquitous in the thirteenth century, provided philosophical justification for these ideas. The unitas (the unit or unity), as described by Boethius and Aristotle, was invoked by music theorists in conflicting (and at times contradictory) ways in order to promote the idea that mensural hierarchies were formed of grouped indivisible particles.7 This chapter provides an introduction to the prominent issues pertaining to atomism and music in the fourteenth century from the perspective of a number of music theorists. After an overview of today’s scholarship about atomism in fourteenth-century philosophy and music, I introduce the work of two theorists – Johannes Torkesey and Willelmus – who discussed the limits of divisibility of musical time in relation to mensural notation. Through the writings of Jean des Murs, I examine different types of indivisibility within the gradus system. The gradus system

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is predicated upon the idea that all musical notes can be represented within a range of values, limited at its minimal and maximal extent, and that within this range musical sounds are created that may share the same duration but possess differing functions depending upon their context. The concluding section considers the work of Johannes Vetulus de Anagnia, who wrote the most comprehensive account of fourteenth-century musical atomism. Although specific references to atoms occur infrequently in fourteenth-century music theory, the idea that continua were composed from the accumulation of discrete, indivisible particles was inherent in many fourteenth-century music texts.8 Music’s integration of a continuum of time and sound with discrete notational values provided an appropriate conduit for philosophical musings about the more-or-less continuous and divisible structure of continua, as well as the physical limitations of performance.

FOURTEENTH-CENTURY ATOMISM It has traditionally been acknowledged that fourteenth-century atomism was a peripheral field of enquiry that arose as a by-product of anti-Aristotelian critiques of late medieval indivisibilist mathematics. Atomists were critical of the revival of the idea proposed in Physics Book VI that a continuum could not be composed of points, instead rationalizing the existence of minimal parts of the continuum – atoms.9 Unlike Aristotelian thinkers, who believed that continua were infinitely divisible, and that small parts of the continuum were arrived at through division, atomists argued that the continuum was derived from the accumulation of indivisible points or atoms. As such, atoms were integral and prior to the structure of the continuum itself.10 While some atomists theorized durational atoms containing parts, most believed that atoms had no extension – they were conceived as dimensionless points.11 Despite this, the majority of fourteenth-century atomists also rejected the existence of empty space between atoms, or void, which had been an essential component of ancient Greek atomism.12 This provided fodder for the many critics of atomism, who quoted the Aristotelian dictum that a continuum cannot be composed of dimensionless points, since they would all occupy the same space.13 Some late medieval atomists, namely Michel of Montecalerio and Henry of Harclay, appear to have focused on the mathematics of atomism alone. Others, such as Walter Chatton, William Crathorn, Gerard of Odo, Nicholas of Autrecourt and John Wyclif, considered both the metaphysical and physical implications of their atomistic approaches. They asked whether atoms existed not only in mathematics but also in physical reality itself; whether time, space and substances were composed of atoms; and the ways in which atomism could describe the nature of the soul and humanity’s connection with God.14 The melding of Neoplatonist metaphysics with neo-Pythagoreanism, in which atoms were described as elemental particles, resulted in the transmission of atomism to the later Middle Ages.15 Boethius’ De institutione arithmetica, a loose translation of Nichomachus of Gerase’s Art of Arithmetic, played a prominent role in establishing this tradition.16

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Most music historians write about musical atoms in reference to Italian theorist Johannes Vetulus de Anagnia, author of Liber de musica (A Book about Music), an eccentric treatise about mensural theory believed to have been written in the midlate fourteenth century.17 The majority of the work is devoted to a comprehensive method for the calculation of the duration of musical notes and their proportional relationships with one another, through atoms of time worth 5/36 of a second. Frederick Hammond’s edition is the most complete study of the treatise to date. It includes an overview of the durational atoms and notes within Vetulus’ mensural hierarchy.18 Peter M. Lefferts has also provided a useful overview of the treatise. His work draws attention to parallels between Liber de musica and the gradus system, as well as the language of English theoretical sources.19 More recently, Karen Desmond has written a brief account of Liber de musica, assigning the treatise a place within her typology of fourteenth-century music theory treatises, as well as discussing its numerous tree diagrams.20 To date, Dorit Tanay has provided the most detailed discussion of atomism in fourteenth-century music theory. Tanay’s overview of the intellectual history of fourteenth-century music theory details scholastic influences within Jacobus’ Speculum musicae, and Jean des Murs’ Notitia artis musicae, among many other theorists’ work.21 Although she rejects the idea that music theorists conceived of general time in terms of atoms, she observes that musical time was sometimes viewed as an aggregate of minimal, indivisible particles of sound, which she terms atoms.22 She identifies parallels between the neo-Pythagorean tradition of music theory and the idea that short notes group to form larger mensural structures, describing such practices as ‘atomistic’ at times.23 As Tanay herself notes, her documentation of the scholastic context of late medieval music theory makes general claims about atomism in fourteenth-century music.24 Much work remains to be done to examine the relationship between atomism and fourteenth-century music theory. This chapter develops research that will appear in my dissertation to illustrate the ways in which fourteenth-century music theorists theorized musical time in terms of accumulated indivisible particles.

THE UNITAS AS INDIVISIBLE MINIMAL UNIT In his well-known definition of number from De institutione arithmetica, Boethius writes of the punctum (the point, which he earlier termed the unitas).25 According to Boethius, the word unitas is a Latin equivalent to the Greek µονáҫ – the monad, unity, singularity or atom.26 The unitas is an imperceptible and indivisible particle that has no extension. Since it is dimensionless, it is said to be a perpetual potential becoming of a line or space: Constat punctum ipsum sine ulla corporis magnitudine vel intervalli demensione, cum et longitudinis et latitudinis et profunditatis expers sit, omnium intervallorum esse principium et natura insecabile, quod Graeci atomon vocant, id est ita deminutum atque parvissimum, ut eius pars inveniri non possit. Est igitur punctum primi intervalli principium, non tamen intervallum, et lineae caput, sed nondum linea.27

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A point exists without the magnitude of a body or the dimension of an interval, since it is bereft of length, width, and depth. It is the beginning of all intervals and indivisible by nature, and the Greeks call it atom; it is so diminished and very small that parts of it cannot be found. Therefore the point is the beginning of the first interval, but it is not an interval; it is the head of the line, but not yet a line.28 From this definition, Boethius proposes two alternative ways of understanding mathematical reality – multitude and magnitude.29 Multitude, which pertains to arithmetic, is formed from the accumulation of an infinite number of discrete, indivisible particles, commencing from the singularity (atom). Magnitude, which pertains to geometry, is an infinitely divisible continuum that is limited nevertheless at its maximal extent.30 The unitas serves as the unit of the multitude – the number ‘1’. It is also the durationless coming-to-be of a line and the beginning of magnitude – the number ‘0’. Table 12.1 provides a reproduction of Boethius’ table of unitates from De institutione arithmetica. Here, the unitas as the numeral ‘1’ serves as a unit for numbers that are multiples of two and three.31 The idea that the unitas, and thereby number, constituted the beginning of things, was ubiquitous in treatises of the medieval neo-Pythagorean tradition. That this idea was incorporated into music theory directly is evinced by a number of sources, including English theorist Johannes Torkesey’s Trianguli et scuti declaratio de proportionibus musicae mensurabilis (An Exposition of the Triangle and the Shield about the Proportions of Mensural Music).32 Six types of note appear in the treatise – the larga , longa , breve , semibreve , minim  and simpla . The hierarchical relationships of these notes are codified in tabular form, in imitation of Boethius’ table, as is shown in Figure 12.2. In fourteenth-century mensural hierarchies, some notes contained three notes of the next-shortest kind. These were termed ‘perfect’. Others contained two notes of the next-shortest kind and were termed ‘imperfect’. Thus, the duration of all notes was determined through groupings of multiples of two and/or three. Proceeding from the simpla at the top of the triangle in Figure 12.2, notes to the left are multiplied by two, and notes to the right by three. Torkesey’s diagram introduces dots to annotate the triple groupings of notes. The number of triple groupings within a given note determines the extent to which it is ‘perfect’. A dot above a note shows that it is perfect at one degree (i.e. one of the groupings of notes it contains is triple), below TABLE 12.1  Boethius’ Arithmetic Multiples of Two and Three 1

2

4

8

16

32

3

6

12

24

48

9

18

36

72

27

54

108

81

162

Source: Boethius (1983, 124).

243

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FIGURE 12.2:  Torkesey’s triangle. Willelmus and Torkesey (1966, 61).

the note that it is perfect at two degrees, both above and below by three degrees and with two dots below by four degrees. Thus, all notes to the right of the triangle are uniformly perfect. They bear a single dot to their right, demonstrating that they contain notes that are all multiples of three. All notes to the left of the diagram are imperfect. As Figure 12.2 demonstrates, Torkesey’s triangle presents a hierarchy of note shapes which are systematized as multiples of two and three.33 Students who wish to use the triangle can trace a path through the diagram to visualize the mensural hierarchy they wish to employ, considering its relationships with other paths that lead through the triangle. Such lines must be traced either straight down the graph

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FIGURE 12.3:  Translation of the dotted path of Figure 12.2 into mensural notation.

or follow the lines diagonally to the right. Figure 12.3 provides a translation of the dotted path traced through Figure 12.2. Torkesey explains that the unitas of his diagram is the simpla, which is indivisible. He writes: Cum sex sunt species notarum simplicium prout ostensum est in scuto, sciendum est quod praeter simplam quam impartibilem dico quia solam unitatem significat, quinque aliae species variantur secundum diversam appositionem punctorum et puncti carentiam.34 Since there are six species of individual notes, as is shown in the shield, it should be known that apart from the simpla, which I say is indivisible because it signifies a single unitas, the five other species vary according to the different positions of the dots and lack of dots. The Breviarium regulare musicae, a contemporary English source, also adopts Torkesey’s model for the representation of musical time. Its author, Willelmus, reuses Torkesey’s triangle (with some modifications) and reflects on the conceptual underpinnings of the mensural hierarchy. Willelmus adds one additional note to the system above the level of the larga – the largissima. He renames Torkesey’s minim the minuta and provides two alternative names for the simpla – crocheta and minima. Willelmus demonstrates his loyalty to the idea that the simpla is indivisible by permitting changes that result in the increase in its size by a multiple of itself, but prohibiting any changes to the note that might result in its division or increase in size by a fraction of itself.35 The practice of doubling the length of a note, referred to as alteration, is allowed, presumably because it does not presuppose division of the simpla (altered  = +).36 On the other hand, dotting of the simpla, which would increase the length of this note by half of its value, is proscribed. Since Willelmus’ musical time is formed through the grouping of discrete particles, dotting of the simpla would presuppose the existence of a note lasting half of its duration, which is impossible if the simpla is indivisible (. = ++).37 Similarly imperfection, which occurs when one-third of the value of a perfect note is removed from it, is also prohibited for the simpla (perfect  = ++; imperfect  = +). Perfection, like dotting, results in half of the value of a note being added to it and is also forbidden for the simpla.38 In all of these examples, Willelmus allows the simpla to be increased by a multiple of itself, but divided neither through increase in

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its size by part of itself nor through decrease in its in size. The simpla retains its indivisibility throughout. Despite this, Willelmus tells us, following Aristotle, that musical sound is in theory infinitely divisible. This is because Willelmus’ belief that musical sound is limited minimally does not apply to the infinitely divisible continuum of sound in general.39 Although musical sound is constructed through the accumulation of prior indivisible particles, the time within which the notes exist is not atomistic.40 Willelmus renames the simpla the minima to reflect the customs of his contemporaries, who believe that this note represents the physical limitations of vocal technique. Any note that is shortest within the mensural hierarchy should bear this name, since the word ‘minima’ itself implies that it is the briefest sound that can be sung. Because time is infinitely divisible, the only plausible objection to the existence of a note shorter than the minim is that such a note would be too brief to be sung. Thus, although Willelmus’ simpla is the indivisible ‘principium’ (beginning or foundation) of all notes, he explains that even its brevity may be surpassed through practice and artifice.41 The theorization of an indivisible note facilitates the creation of a mensural hierarchy in which all notes share a common unit. It implies mathematical, but not physical indivisibility of musical sound.

‘IT IS IMPOSSIBLE TO GIVE LESS TO THE LEAST’ According to Willelmus, his contemporaries’ objections to notes shorter than the minim were rooted in the fact that the minim represented the sound of briefest duration that could be sung. However, many other music theorists of the fourteenth century offered an alternative explanation, appealing to variations upon the pseudoAristotelian maxim ‘impossibile est dare minimum minimo’ (it is impossible to give less to the least).42 This phrase does not actually appear in any of the Latin translations of Aristotle’s works, although it is in keeping with his philosophical position.43 In Book I of the Physics, Aristotle details his disdain for Anaxagoras’ belief that everything in the world is formed of infinitesimally small parts of the same substance, mixed together. Aristotle rejects this idea, insisting instead that all naturalia (natural substances) are limited in their size both maximally and minimally and are composed of parts, limited similarly in their maximal and minimal extents.44 A substance’s size is limited to the extent that if it should become larger than its maximal extent or smaller than its minimal extent it will continue to exist, but in a different form. It will no longer be the same substance. The idea that all natural substances were limited maximally and minimally made its way into the medieval Latin discussions of indivisibles through the socalled ‘latitude of forms’ thesis. According to this thesis, all substances exist within a latitude, that is, ‘a range within which a given form, complexio, quality, or quantity can vary’.45 Since all substances operate within a varying range, they possess more than one substantial form. A supervening form determines the being of an object as a whole. An object may also embody a variety of other, accidental forms to facilitate variation within the limits of its own latitude.46 The formal minimal and maximal limits of substances came to be known as minima naturalia (natural minimums) and

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maxima naturalia (natural maximums), respectively. Although minima naturalia were described in a number of different ways in the Middle Ages, most fourteenthcentury philosophers believed that the minimum naturale constituted the lower limit of a latitude of a substance.47 The theorization of the latitude of forms thesis is believed to have influenced music theory through the work of Jean des Murs, who argued that each note possessed two forms – a natural form that was the sound itself, and a mathematical form that constituted its measurement.48 Des Murs described musical sound as a latitude, limited at its natural minimal extent. Quoniam ergo vox tempore mensurata unionem duarum formarum, naturalis scilicet et mathematicae, comprehendit, licet quod ratione alterius fractio non cessaret, tamen propter aliam vocis divisionem necessarium est alicubi terminari. Nam sicut omnium natura constantium positus est terminus et ratio magnitudinis et augmenti sic parvitatis et diminuti. Demonstrant enim naturales, quod natura ad maximum et minimum terminatur. Vox autem est per se forma naturalis iuncta per accidens quantitati. Igitur oportet eam habere terminos fractionis, quorum latitudinem nulla vox quantacumque frangibilis valeat praeterire. Hos autem terminos volumus comprehendere ratione.49 Seeing, on the other hand, that sound measured by time consists in the union of two forms, namely the natural and the mathematical, it follows that because of the one its division never ceases, while because of the other its division must necessarily stop somewhere; for just as nature limits the magnitude and increase of all material things, so it also limits their minuteness and decrease. For natural things demonstrate that nature is limited by a maximum and a minimum. Sound, moreover, is in itself a natural form to which quantity is attributed accidentally. Therefore, it is necessary for there to be limits of division beyond which no sound, however fractionable, may go. These limits we wish to apprehend by reason.50 Musical sound consists of a unification of the natural and mathematical. Sound, as a natural substance, is a latitude that is limited at its minimal and maximal extent. Time, as a mathematical continuum, is limitlessly divisible. Muris’ explanation for the minim’s status as the shortest of notes is ‘non est minimo dare minus’ (less cannot be given to the least).51 The minim serves as a natural minimal extent of musical sound writ large, but not of time.52 Des Murs’ devotion to the latitude of forms thesis is highlighted by his use of the gradus system. As outlined in his Notitia artis musicae, this is based upon the principle that notes can be categorized into different groups or gradus. The shortest note of each successive set of notes (or gradus) is equivalent to the longest of the next set of notes. The longest note in each gradus is worth three of the shortest notes of its own gradus. The note between the shortest and longest notes in a gradus is worth two of the shortest notes in its gradus. Table 12.2 presents des Murs’ gradus system.

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Table 12.2  Des Murs’ Gradus System Duration in minims

First gradus

81

Longissima 

54

Longior 

27

Longa 

Second gradus

Third gradus

Fourth gradus

Perfecta 

18

Imperfecta 

9

Brevis 

Brevis 

6

Brevior 

3

Brevissima 

Parva 

2

Minor 

1

Minima 

Source: Murs (1972, 79) (modified).

As Table 12.2 demonstrates, the minim is the formal limit of the mensural hierarchy in its entirety. In addition to this, each gradus possesses its own formal limitation. The longa is the minimal limit of the first gradus – no notes in this gradus can be shorter than it. Once the longa changes its form and becomes the perfecta, it serves as a maximal limit of the second gradus and is no longer a unit of measurement. Similarly, the breve serves as the minimal threshold of the second gradus, but once it enters the third gradus and is divided, it is no longer a minimal threshold of the second gradus. Its form has altered. Des Murs establishes what appears to be a set of minima naturalia for each gradus.53 But des Murs also states that the shortest note of each gradus is an unitas. The longa, serving as the minimal threshold of the first gradus, can also serve as a unit of measurement for its own gradus. The breve serves as the unitas of the second gradus, while the brevissima is the unitas of the third gradus. The minim is the unitas of the mensural hierarchy in its entirety; it is the foundational unit of measurement of all notes, as well as of the fourth gradus. The other unitates are units only for their respective gradus.54 In his application of the term ‘unitas’, des Murs describes two different kinds of indivisibility. The unitas as primary number or unit remains implicit throughout and is equivalent to the minim. This is similar to the usage that would be found in Torkesey’s and Willelmus’ later diagrams. At the same time, the shortest note in each gradus operates both as a minimal threshold and as a unit of measurement for its given gradus. Des Murs is able to make this claim because he associates sound with number only ‘accidentally’. While its natural, sounded form may not change, a note’s accidental, mathematical form may alter depending upon its function and context.

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Despite his fleeting reference to the minim as the minimal threshold and unit of the mensural hierarchy, des Murs was not an atomist. In part, this is because des Murs’ conceptualization of the gradus system is predicated upon the idea that longer notes are divided into shorter notes. This is highlighted both by his placement of longer notes at the top of the mensural diagram and by his hierarchical predilection towards longer notes. The continuum of time as a whole is prior to its parts. Des Murs’ unitates sometimes act as mathematically indivisible units, but they also serve as minimal limits, or minima naturalia, of the musical continuum. Time itself is still infinitely divisible.

THE ATOMISM OF JOHANNES VETULUS DE ANAGNIA Among fourteenth-century theories of mensural music, the most explicitly atomistic is that of Johannes Vetulus de Anagnia. His book Liber de musica provides only tangential clues about his personality and education; nothing else is known about Vetulus’ biography. Although he may have been the notary of the same name identified by Alberto Gallo in a document dated 1372, there is no direct evidence connecting him with this man.55 In Liber de musica, Vetulus cites Boethius as his principal authority, paraphrasing passages from both the Consolation of Philosophy and De institutione arithmetica. He follows Boethius in associating the unitas with the minimal and indivisible particle of his system – the atom. He justifies his atomism by paraphrasing a ubiquitous definition of number from De instutione arithmetica and applying it to music: ‘Numerus est secundum philosophum collectio de unitatibus congregata. Et ita secundum musicum est congregatio notarum vel atomorum in uno corpore.’56 (According to the Philosopher, a number is an assembled collection of unitates. And thus according to the musician, it is a collection of notes or atoms in one body). Vetulus’ atoms are durational. Calculated through the division of the year into months, weeks and days, they are worth 5/36 of a modern second. According to Vetulus, days contain four quarters, each worth six hours. Hours are divided into four points, which contain ten moments made up of twelve ounces, each worth fiftyfour ‘atomi’ (atoms).57 Although the origin of Vetulus’ calculation of the specific value of the atom remains obscure, the atomist who I have identified whose model is closest to Vetulus’ is Bartholomaeus Anglicus, who divided the hour into four points, ten moments, twelve ounces and forty-seven atoms.58 Andrew Pyle suggests that Bartholomaeus’ system, which adopted the figure for atoms per hour most commonly accepted in the Middle Ages, may have originated in occult numerology.59 If Vetulus was aware of the tradition to which Bartholomaeus is referring, he might have altered the model to account for fifty-four atoms per ounce, rather than the customary forty-seven. This would have enabled him to apply the division of time to the mensural hierarchy more easily. Divisible by two, three and nine, the ounce of fifty-four atoms can be used to represent one of the most important notes in his system – the medium tempus or lesser perfect breve of the greater subdivision (see Table 12.4).

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The shortest note in Liber de musica is not of equal duration to the atom. Instead, Vetulus describes two types of minimal note shape, one worth two atoms, and another worth three atoms, named the ‘improper’ minim of the least subdivision, and the ‘proper’ minim of the least subdivision, respectively.60 These minimally short notes, both stemmed minims , group together to form Vetulus’ mensural hierarchy.61 He also acknowledges the existence of semiminims, describing them as minims ‘mutentur . . . in figura’ (altered . . . in form) but expresses trepidation about their use.62 Vetulus’ system is founded upon the Italian trecento divisions of Marchetto of Padua’s Pomerium. In this system, breves contain between two and twelve semibreves. Each type of breve is assigned a name that corresponds to the number of semibreves that it contains, namely binaria (two), teraria (three), quaternaria (four), octonaria (eight), novenaria (nine) and duodenaria (twelve). Two methods for dividing the breve into six semibreves are described. Senaria perfecta (or ytalica) divides a breve into three imperfect semibreves, each worth two semibreves minimae (least semibreves, or minims). Senaria imperfecta (or gallica) divides a breve into two perfect semibreves. Unlike Marchetto, Vetulus describes the division of largae as well as breves into between two and twelve parts.63 Each of the three divisions of larga, breve and semibreve – greater, lesser and least – is divided into subdivisions – also greater, lesser and least. Least semibreves are synonymous with minims. In its entirety, Vetulus’ system comprises over sixty notes, each of which is assigned a precise value in atoms. Table 12.3 illustrates Marchetto’s divisions. Tables 12.4 and 12.5 provide Vetulus’ divisions and subdivisions of the breve for comparison with des Murs’ gradus system. Divisions are shown in the top row (shaded in grey), while the subdivisions are shown in the columns. This system, which is far more elaborate than des Murs’, also organizes notes within a latitude of degrees.64 Vetulus’ greater, lesser and least divisions of notes

TABLE 12.3  Marchetto’s Divisions of the Breve Ternaria

Binaria 







Senaria perfecta (ytalica) Novenaria

Quaternaria







       

             

    

Duodenaria

Octonaria

Senaria imperfecta (gallica) 

        Duodenaria







                                 

                   

                                  

Table adapted from Padova (1961, 102–85), Apel (1953, 368–84) and Stoessel (2002, 249).

Least

36

Greater

Lesser

Greater



3 Least See Hammond’s edition for tables of the largae and longae. A more detailed explanation of the differences between my interpretation of Vetulus’ divisions of notes and Hammond’s will appear in my dissertation. Anagnia (1977, 21–2).

4

Least

6

Greater



Lesser

Least

Lesser

Greater



8

9

Least

Least

Lesser

12

Least

Greater



Lesser

Least

Lesser

Greater



Lesser imperfect Least imperfect breve Greater Lesser Minim (least breve senaria semibreve semibreve semibreve) quaternaria imperfecta

16

Lesser

Greater

18

Least

Lesser

Greater

24

27

32

Lesser

Greater









Greater imperfect breve octonaria

Greater Lesser Least perfect perfect breve perfect breve breve senaria duodenaria novenaria perfecta

48

54

72

Duration in Atoms

TABLE 12.4  Vetulus’ ‘Proper’ Divisions and Subdivisions of Notes

243

Least

Lesser

Greater

Least

Lesser

Greater

Least

Lesser

Greater

Least

Lesser

Greater



Greater



Minim (least semibreve)

Least

Least

Lesser



Lesser semibreve

2

Least

Lesser

Greater

Greater



Least imperfect Greater breve semibreve Quaternaria

Lesser

Least

Lesser

Greater



Lesser imperfect breve senaria imperfecta

3

4

6

8

9

12

16

18

24

27

Least

Lesser

36

32

Greater









48

Atoms

Greater imperfect breve octonaria

Least Lesser perfect Greater perfect perfect breve breve breve duodenaria novenaria senaria perfecta

TABLE 12.5  Vetulus’ ‘Improper’ Divisions and Subdivisions of Notes

244

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can be compared with des Murs’ brevis, brevior and brevissima. And again like des Murs, Vetulus applies a minimal limit to his continuum of musical sound. Simplex minima quoad vocem est sicut atomus quoad tempus. Et sicut per atomum recolitur tempus, sic per minimam simplicem mensurae vocum de gradu ad gradum reducuntur ad maiores.65 A single minim is to sound as an atom is to time. And just as time is cultivated by the atom, so are the measures of sounds grouped by the simple minim from degree to degree to larger ones. Although Vetulus compares his shortest note to the atom, it is nevertheless divisible. This divisibility is inherent not only within his atomistic model but also within the mensural system itself. Vetulus’ proper minim of the least subdivision (see Table 12.4) is worth three atoms, or 5/12 of a second, while the note of shortest duration of the whole system – the ‘simplex minim’, or improper minim of the least subdivision – is a ‘particle of sound’ worth two atoms (5/18 of a second, see Table 12.5).66 Since the duration of these two minimal notes would sound at a ratio of 2:3 to one another, neither can be viewed as a fundamental unit of measurement for the mensural system as a whole, only for their own group of divisions (proper and improper). If the two notes are superimposed, the metrical conflict between them – known as a hemiola – implies that they are potentially divisible, since they move in and out of alignment with one another, creating an implicit, underlying pulse – the atom.67 Figure 12.4 demonstrates this relationship. Unlike the simpla of Torkesey’s and Willelmus’ systems, which was a physical, sounded unit for the whole system, Vetulus’ atom is a mathematical quantity. The briefest sound that can be notated is not the atom, but the improper minim of the least subdivision, worth two atoms. When these minims are divided, a new type of entity is arrived at – the atom itself. As I noted earlier, minima naturalia were believed to be the minimal limits of given natural substances, but not of the world, which was believed to be infinitely divisible. As such, the division of the

FIGURE 12.4:  Proportional relationship between atoms, proper minims of the least subdivision and improper minims of the least subdivision. Although Vetulus does not align his notes in a graph of this kind, his tree diagrams imply the superposition of proper minims of the least subdivision and improper minims of the least subdivision. He depicts this through the merging and splitting of branches which represent the process of transitioning from the division of notes into two or three parts. For example, a semibreve worth six atoms can be divided in two ways – into two proper minims of the least subdivision (worth three atoms each), or three improper minims of the least subdivision (worth two atoms each). This particular relationship is portrayed visually a number of times in the tree diagrams. A more detailed explanation of this process will appear in my dissertation.

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minimum naturale results in the creation of a new form and with it a new substance. The division of Vetulus’ shortest notes also results in a change in their form. When divided, minims become atoms. His system therefore arguably rationalizes an atomic layer of particles of musical time worth 5/36 second that lies below a latitude of musical sounds, arranged in overlapping gradus. The minimal limit of the latitude of improper notes is the minim, worth two atoms. Vetulus says that this note is an ‘atom’, presumably because it provides a unit of measurement for the improper notes (Table 12.5). The minimal limit of the latitude of proper notes, on the other hand, is the proper minim of the least subdivision worth three atoms (Table 12.4). Both of these notes are functionally distinct from the temporal atom, which provides a unit of measurement for both the proper and improper notes and thereby all musical time in Liber de musica.68 Music theorists of the fourteenth century devised mensural hierarchies that befitted the music they wished to notate, sing and speculate about. The idea that musical sound was composed of indivisible particles – at times referred to explicitly as atoms – played a prominent role in the theorization of minimal note shapes. Indivisible minimal units (often referred to as unitates) at times came to imply a sense of hierarchy among all notes within a system, as evinced in Torkesey’s and Willelmus’ work. In des Murs’ gradus system the unitas was associated with the minimal thresholds of overlapping latitudes to describe the indivisibility of a note within a given context. Johannes Vetulus alone appears to have adopted an atomistic model which accounted for a layer of temporal atoms below the level of the shortest note. His work draws on both atomistic doctrine and, like des Murs, the latitude of forms thesis. Various interpretations derived from both Aristotelian and Boethian thought were interchanged freely to lend philosophical prestige to models such as these. Music-theoretical adaptations of philosophical ideas, at times contradictory to Aristotle’s and Boethius’ original texts,69 demonstrate the flexibility with which fourteenth-century music theorists engaged with philosophical ideas. Underlying all studies of the nature of small particles of musical time was the question of the relationship between minimal particles of sound and the mathematics of the continuum. Theorists sought to reconcile the physicality of singing with their beliefs about the structure of musical time. Music notation was a visual representation of musical sound that could also embody philosophical significance. Through their visualization of the mensural hierarchy, theorists revealed their thoughts about the relationship between time, sound and mathematics. Considering the underlying philosophical beliefs of music theorists provides a glimpse into how scholars of the past made sense of reality as they knew it.

NOTES 1. I would like to express my gratitude to Michael Scott Cuthbert for kindly allowing me to use his Ciconia font in this chapter. 2. For the purposes of this illustration, I am ignoring the fermatas. 3. Fourteenth-century music theorists were not the first to describe musical rhythm in relation to indivisible atoms. In the fourth century BCE, Aristoxenus theorized

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a chronos protos, or ‘primary time unit’, that formed the basis of the three types of rhythmic activity he identified: speech, melos and bodily movement. Rowell (1979, 72). 4. Colonia (1974, 34). 5. Jacobus de Ispania, author of the most substantial medieval music treatise – the Speculum musicae, probably written sometime between c. 1320s–1350 – objected to lone semibreves. Karen Desmond has argued that was because he preferred the old Franconian way of writing rhythms using groups of semibreves undifferentiated from one another by stems, rather than the imperfection of the breve, that is, the removal of one-third of the value of a breve by an adjacent semibreve. A more detailed description of imperfection, the practice to which Jacobus objected, can be found on p. 237 of this paper. See: Desmond (2018a, 403–16). Marchetto of Padua also wrote about the incompleteness of short notes in his Pomerium, the most substantial source for Italian trecento music theory (dated to 1321-6 by Joseph Vecchi). Marchetto believed that the sound of the perfect breve was produced through the complete inhalation and exhalation of the lungs. For him, the existence of semibreves relied upon their appearance in groups that added up to the time of a breve. See Table 12.3 for Marchetto’s divisions. Padova (1961, 27, 78). On the dating of the Speculum musicae, see: Zayaruznaya (2020, 95–148). 6. These notes were also called fusae (also commonly fusiel by English theorists), crochetae, simplae and so on. Karen Cook’s dissertation provides a history of the semiminim. Cook (2012). 7. Gilles Rico has shown that the first two books of Boethius’ De institutione musica (which was influenced heavily by his De institutione arithmetica) were part of the curriculum in the Arts faculty of the University of Paris in the thirteenth and fourteenth centuries. It is also known that fourteenth-century theorist Jean des Murs was familiar with Boethius’ De institutione musica, since his own Musica speculativa is a recension of it. Despite this, it is difficult to establish whether specific music theorists had actually read the Boethian texts to which they referred. As such, we can assume that some theorists evoked ideas prevalent in contemporary philosophy (including those that were transmitted into the later Middle Ages via Boethius’ and Aristotle’s writings) in order to justify their notational endeavours, even though they did not possess a comprehensive understanding of the philosophical doctrine they claimed to cite. Rico (2005, 35, 64–75). 8. Some fourteenth-century theorists who write of indivisible particles of time or pitch using the term ‘atomus’ or a variant thereof include Jacobus de Ispania, Johannes Vetulus de Anagnia and Arnulf of St. Ghislain. See: Ispania (1973, vol. 1, 69, 106), Ispania (1973, vol. 4, 55–6, 74), Ispania (1973, vol. 7, 36, 85) and Page (1992, 16). 9. Murdoch (1974, 11). Murdoch (2009, 17). 10. Sander W. de Boer describes an interpretation of atomism in which continua are believed to be composed of atoms, rather than atoms representing mere parts of continua, as the ‘strong’ reading of fourteenth-century atomism. He identifies this reading in Gerard of Odo’s writings on the nature of the continuum. de Boer (2009, 89). 11. Following Democritus, fourteenth-century French atomists Nicholas of Autrecourt and Nicholas Bonet claimed that atoms had an extension. Grellard (2004, 193). 12. Robert (2017, 206). See also Robert’s discussion of void and atomism in the fourteenth century in Robert (2012, 67–98).

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13. A number of mathematical proofs were also brought to bear in the fourteenth century to disprove atomism. Best known among these can be found in the writings of John Duns Scotus (1265/66–1308). Jung and Podkoński (2009, 66–71). Later in the fourteenth century, Thomas Bradwardine adopted Scotus’ proofs in his famous attack on contemporary atomism. Murdoch (1974, 18–19); Murdoch (1987, 103–37). 14. See, for example, Emily Michael’s discussion of John Wyclif ’s tripartite structure of the soul. Michael (2003, 358). 15. Murdoch (1974, 31). Grellard and Robert (2009, 7). 16. Aurélien Robert has proposed that the influence of the neo-Pythagorean and Neoplatonic traditions in Boethius’ writings played an important role in the establishment of atomism in fourteenth-century philosophy. Robert (2017, 181– 206). For Robert’s discussion of the influence of Boethius on the atomism of John Wyclif, see: Robert (2018, 107–31). 17. In a recent article, Francesca Manzari and Jason Stoessel offer a new dating of Biblioteca Apostolica Vaticana, Barb. lat. 307, the manuscript containing the only complete copy of Liber de musica from the Middle Ages. They suggest that this codex was compiled in the 1350s or 1360s. Manzari and Stoessel (2019, 283–331). A discussion of the dating of this treatise can also be found in Anagnia (1977, 16). 18. Anagnia (1977, 21–2). 19. Lefferts (1990, 238–9). See Table 12.2 for Jean des Murs’ gradus system and Tables 12.4–12.5 for Vetulus’ application. 20. Six tree diagrams appear in Liber de musica, each of which details the author’s numerous divisions of a single note. Desmond (2018b, 195–7). 21. Although Jean des Murs’ Notitia artis musicae is less substantial than the Speculum musicae, it has been credited with exercising considerable influence over notational practices of the early fourteenth century. Ulrich Michels has dated the work to 1321. Michels (1970, 2). 22. Tanay (1999, 114–24). 23. For example, she discusses atomism in relation to John of Tewkesbury’s Quatuor principalia musicae, a large mid-century source of English mensural theory, and Marchetto of Padua’s Pomerium. She argues that these works are atomistic, comparing them briefly to Vetulus’ Liber de musica. Tanay (1999, 4). 24. ‘While I make no claim for a profound and exhaustive scrutiny of any of these fields, I do plead for the merit and fruitfulness of a broad view of possible interconnections between different cognitive spaces of the late Middle Ages. It is here, I admit, that my approach is most vulnerable: it sacrifices meticulous analysis and comprehensive exposition of a single issue or discipline in order to advance beyond a rephrasing of theoretical discourse and reach the level where one can suggest explanations of the cultural sensibilities underlying theorists’ intentions and modes of shaping musical concepts, within the conceptual universe that constitutes late medieval scientific reflections.’ Tanay (1999, 4). 25. Est igitur unitas vicem obtinens puncti, intervalli longitudinisque principium; ipsa vero nec intervalli nec longitudinis capax, quemadmodum punctum principium quidem lineae est atque intervalli, ipsum vero nec intervallum nec linea. Boethius

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(1867, 87). Therefore, unity has the potential of a point, the beginning of interval and longitude; it is not itself capable of interval or longitude, just as the point is the beginning of the line and the interval, although it is itself neither interval nor line. Boethius (1983, 129). 26. The concept of the monad first appeared in Plato, although it was commonly associated with Pythagorean mathematics. Boethius and Pythagorians (among others), sometimes referred to the unitas in place of ‘0.’ Boethius (1983, 129). 27. Boethius (1867, 89). 28. Boethius (1983, 130) (modified). 29. Aristotle also rationalizes a unity (unum or unitas) from which multitude is derived in Metaphysics Book X I-IV. He states that this originated in the Platonic and Pythagorean traditions. However, he does not include Nichomachus’ diagram discussed here. Aristotle (1995, 195–206). 30. Boethius’ discussion of multitude and magnitude originated in Nichomachus’ Introduction to Arithmetic. Medieval music-theoretical commentators were familiar with Nichomachus’ representation of proportions largely through the Boethian tradition. For Boethius’ descriptions of multitude and magnitude, see: Boethius (1983, 128–30). Boethius (1989, 14–15, 52). As Andrew Hicks has shown, this was largely thanks to translations by Calcidius, Macrobius and Martianus Capella. Hicks (2014, 422). 31. As Table 12.1 illustrates, the indivisible unitas can be grouped together into multiples of two and three. Numbers in the first row of the table represent multiples of two, and numbers at the bottom of the table multiples of three. Adjacent numbers in rows represent a proportion of 1:2, and in the columns 2:3. This diagram also originated in Nichomachus’ Introduction to Arithmetic. For a discussion of applications of Nichomachus’ arithmetic to proportional systems, including this table, see: Kappraff (2000, 41–55). 32. Laurie Koehler has provided a summary of the similarities between Torkesey’s triangle and the Neoplatonic and neo-Pythagorean traditions. See: Koehler (1990, 49–50). Tanay’s discussion of Torkesey’s and Willelmus’ use of the unitas can be found in Tanay (1999, 126–8). An edition of the treatise can be found in: Willelmus and Torkesey (1966, 58–61). Anne Stone has argued that the idea that musical time was built upon a common unit (unitas) manifested itself in the minim equivalence of fourteenth-century French theory. This can be compared with Italian trecento theory in which notes were derived from the division of the breve into smaller parts. Stone suggests that the melding of these two ideas offers one explanation for the appearance of complex notations in the late fourteenth century. Stone (1996, 22). 33. Richard Cohn has argued that this model provides a graphic method for the representation and generation of musical metres based on the combination of duple and triple pulses that anticipates the metric structures found in later music of the mid-to-late nineteenth century. Cohn (2016, 241–3). 34. Willelmus and Torkesey (1966, 58). 35. ‘Simpla neque perfecta dicitur neque imperfecta sed principium indivisibile omnium subsequentium.’ Willelmus and Torkesey (1966, 28). (The simpla is said to be neither imperfect nor perfect, but is the indivisible beginning of all subsequent [notes]).

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36. Alteration occurs typically when two notes of the same type are placed between two longer notes. The second of the two notes is lengthened in order that together the two notes will be worth three beats. For example, the following sequence of notes     (breve, semibreve, semibreve, breve), will be worth 3 1 2 and 3 semibreves, respectively. In music notation today at a reduction of 2:1, this could also be notated as | w. H w w. |. 37. This is one among a number of uses of the dot in the Middle Ages.

38. Willelmus also prohibits plication. Like imperfection, perfection and dotting, this results in the division of a note into parts. Opinions over the exact signification of the plica differ, but in general terms it is a small stroke that appears typically on a longa or breve. It signals the insertion of an ornament similar to a passing note. 39. Willelmus also cites Aristotle as an authority for the simpla’s status as a measure for all other notes, writing ‘minimum in quolibet genere est mensura omnium eiusdem generis’ Willelmus and Torkesey (1966, 23). (The least in any genus is the measure of all other things within that genus.) 40. This reading supports Cohn’s view that the triangle may be viewed as a collection of pulses, since it is not the absolute duration of a musical note that is crucial to the structure of the mensural hierarchy, but rather the overall relationship among the notes in the triangle. 41. ‘Unde ut conformem me modernis, pono crochetum seu simplam vel minimam. Non quia ea minor non possit esse, sed quia data mensura debita longarum, brevium, non bene humana voce minor pronuntiatur perceptibilis. Et ex hoc patet solummodo obiectio modernorum. Quia arguunt contra crochetum per hoc quod minima nulla est minor. Respondeo quod Odington non vocavit illam notam minimam sed minutam, quia posuit quod minor possit esse. Vel aliter respondetur quia tunc dicebatur minima illo tempore divisa, sed nunc voco crochetum minimam, licet iam artificio et usu cantores moderni ad minorem divisionem vocis pervenerunt, scilicet ad crochetam.’ Willelmus and Torkesey (1966, 25). (Therefore, so that I conform with the moderni, I call this a crochet or simpla or minim. Not because a smaller note could not exist, but because with the given measure of the longae and breves a perceptibly smaller note cannot be uttered well by the human voice. And from this the single objection of the moderni is evident. For they argue against the crotchet through this [line of reasoning]: that nothing is smaller than the minim. I respond that Odington did not call this note the minim but rather the minuta, because he posited that a smaller note could exist. Or otherwise, it is said that because the note that was at that time called the minim has been divided, but now I call the crotchet the minim, singers today have attained yet smaller divisions of sound through artifice and practice, namely the crotchet.) 42. This particular variant appears in Handlo and Hanboys (1991, 188–93). 43. Handlo and Hanboys (1991, 189–91). 44. Aristotle (1990, 20). Murdoch (2001, 97). 45. Sylla (1973, 229). 46. Michael (2003, 348). 47. Two different interpretations of such a minimum naturale were commonly proposed; one in which the minimum naturale was viewed as an extensive

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minimum, a ‘minimum quod non,’ and the other in which the minimum naturale was viewed as an intensive minimum, a ‘minimum quod sic.’ For a detailed discussion of this, see Murdoch (2001, 114–18). 48. Desmond (2018b, 178). 49. Murs (1972, 69). 50. Murs (1998, 263) (modified). 51. Murs (1972, 102). 52. For a discussion of des Murs’ interpretation of time as continuous versus Jacobus’ interpretation of time as discrete, see: Desmond (2018b, 175–83), and Seta (1984, 169–219). 53. Tanay has also suggested that des Murs’ minim is a minimum naturale. Tanay (1999, 125). 54. ‘Secundum priores figura quadrilatera, aequilatera, rectiangula, caudata dextrorsum sursum vel deorsum in secundo gradu imperfectum significat pariter et perfectum, hoc est ternarium et binarium. Eadem figura non caudata significat unitatem, sed eadem significans unitatem in secundo gradu, ternarium et binarium significat in tertio. Figura vero quadrilatera, aequilatera, obtusiangula unitatem significat in eodem.’ Murs (1972, 76). (According to the above, a square, equilateral, rectangular figure with an ascending or descending stem to the right in the second gradus signifies either a perfect or an imperfect note, that is ternary or binary. The same figure without a stem signifies the unitas, but the same figure signifying the unitas in the second gradus signifies a ternary and binary note in the third. A square, equilateral, obtuse-angled figure signifies the unitas in the same [gradus].) 55. Gallo (1966, 66). 56. Anagnia (1977, 30). Boethius’ original reads: ‘Numerus est unitatum collectio, vel quantitatis acervus ex unitatibus profusus’ (Number is a collection of unitates, or a multitude of quantity produced from unitates). Boethius (1867, 13). Masi translates this passage as ‘A number is a collection of unities, or a big mass of quantity issuing from unities’. Boethius (1983, 76). 57. ‘Dividitur tamen tempus per annum, menses, hebdomodas, dies, quadrantes, horas, punctos, momenta, uncias et atomos. Atomus vero indivisibilis est . . . Dicendum est quod in quattuor principales quadrantes dividitur . Quadrans habet horas sex. De hora nascuntur puncta quattuor. Punctus habet momenta decem. Momentum habet uncias duodecim. Uncia habet atomos 54.’ Anagnia (1977, 28–9). (Time is divided into the year, months, weeks, days, quarters, hours, points, moments, ounces and atoms. The atom is indivisible . . . It should be said that the day is divided into four principal quarters. Quarters have six hours. Four points are born of the hour. A point contains ten moments. A moment contains twelve ounces. An ounce contains fifty-four atoms.) A similar set of divisions is found in Jacobus’ Speculum musicae, although he does not state the duration of each of the parts of time: ‘Important enim notulae quaelibet determinatas temporis morulas et in hoc inter se distinguuntur, licet in hoc generaliter conveniant quod tempus important ad modum quo annus, mensis, dies, quadrans, hora, momentum, uncia, atomus.’ Ispania (1973, vol. 7, 85). ‘For all notes convey determinate stretches of time, and are distinguished from each other in this respect, yet they generally agree on this point that they convey the tempus in the same way as the year, the month, the day,

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the quarter, the hour, the moment, the twelfth part [ounce], the atom.’ Ispania (2017, 70) (modified). In another part of the Speculum musicae he paraphrases Aristotle’s Physics to reject the atomistic organization of general time. This exemplifies the distinction he makes between general time, which is an infinitely divisible continuum, and the measurement of discrete musical time. Ispania (1973, vol. 1, 78). Seta (1984, 186). 58. Anglicus (1975, 529). 59. Pyle (1995, 195). 60. The imposition of the terms ‘proper’ and ‘improper’ as they relate to Vetulus’ divisions of notes is based upon my translation and commentary to Vetulus’ Liber de musica, which will appear in my dissertation. 61. As far as I am aware, Vetulus is unique in theorizing a layer of atoms below the level of the shortest note within the mensural hierarchy. Both Torkesey and Willelmus view the shortest note of their system – the simpla itself – as an indivisible particle. 62. Anagnia (1977, 96). 63. Three or more divisions of largae, longae, breves and semibreves – the four types of note in Vetulus’ mensural hierarchy – are provided. Largae, breves and semibreves come in three sizes (or divisions) – greater, lesser and least. Longae can be perfect or imperfect (worth three or two breves respectively), duplex or triplex. 64. Tables 12.4 and 12.5 corroborate Lefferts’ assertion that Vetulus’ system is an adaptation of the gradus system to Italian mensural theory. Lefferts (2001, 238–9). 65. Anagnia (1977, 32–3). 66. ‘Interest valor atomorum 54 et particularum vocis 27, quarum quaelibet est indivisibilis quoad vocem sicut atomus quoad tempus.’ Anagnia (1977, 44). (Within [this tempus] there are fifty-four atoms and twenty-seven particles of sound, of which each is indivisible with respect to sound just like the atom is with respect to time.) 67. A hemiola occurs in music when two timespans at a ratio of 3:2 sound in conflict with one another. 68. As Emily Michael has shown, John Wyclif also rationalized an atomic layer overlaid by minima naturalia. See: Michael (2003, 352, 2009, 205). 69. For example, Vetulus’ adoption of both atomism and the latitude of forms thesis would be forbidden if he were either a pure ‘atomist’ or an ‘Aristotelian’, since Aristotle rejected atomism.

REFERENCES Anglicus, B. (1975), On the Properties of Things: John Trevisa’s Translation of Bartholomaeus Anglicus De proprietatibus rerum, trans. John Trevisa, ed. M. C. Seymour, Oxford: Clarendon Press. Apel, W. (1953), The Notation of Polyphonic Music 900–1600, Cambridge: Medieval Academy of America. Aristotle (1990), Physica: Translatio vetus, ed. Fernand Bossier and Jozef Brams, Corpus philosophorum Medii Aevi: Aristoteles Latinus, vol. VII 1.2, Boston and Leiden: Brill.

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Aristotle (1995), Metaphysica Libri I-XIV: Recensio et translatio Guillelmi de Moerbeka, trans. Guillelmus de Moerbeka, ed. Gudrun Vuillemin-Diem, Corpus philosophorum Medii Aevi: Aristoteles Latinus, vol. XXV 3.2, Leiden, New York and Cologne: Brill. Boethius (1867), De institutione arithmetica libri duo, ed. Godofredus Friedlein, Leipzig: B. G. Teubner. Boethius (1983), Boethian Number Theory: A Translation of De institutione arithmetica, ed. and trans. Michael Masi, Amsterdam: Rodopi. Boethius (1989), Fundamentals of Music, trans. Calvin M. Bower, ed. Claude V. Palisca, New Haven: Yale University Press. Cohn, R. (2016), ‘Graph-theoretic and geometric models of music’, in Jordan B. L. Smith, Elaine Chew and Gérard Assayag (eds), Mathemusical Conversations: Mathematics and Computation in Music Performance and Composition, 237–55, Singapore: World Scientific Publishing. Cook, K. (2012), ‘Theoretical treatments of the semiminim in a changing notational world c.1315-c.1440’, PhD diss., Duke University. da Padova, M. (1961), Pomerium, ed. Joseph Vecchi, Corpus scriptorum de musica, vol. 6, Rome: American Institute of Musicology. de Anagnia, J. V. (1977), Liber de musica, ed. Frederick Hammond, Neuhausen and Stuttgart: American Institute of Musicology. de Boer, S. W. (2009), ‘The importance of atomism in the philosophy of Gerard of Odo (O.F.M.)’, in Christophe Grellard and Aurélien Robert (eds), Atomism in Late Medieval Philosophy and Theology, 85–106, Leiden and Boston: Brill. de Colonia, F. (1974), Ars cantus mensurabilis, ed. Gilbert Reaney and André Gilles, Corpus scriptorum de musica, vol. 18, [Rome]: American Institute of Musicology. de Handlo, R. and J. Hanboys (1991), Robertus de Handlo Regule; Johannes Hanboys Summa: A New Critical Text and Translation on Facing Pages, with an Introduction, Annotations, and Indices verborum and Nominum et rerum, ed. Peter M. Lefferts, Greek and Latin Music Theory, vol. 7, Nebraska: University of Nebraska Press. de Ispania, J. (2017), The Mirror of Music Book the Seventh, trans. Rob C. Wegman, Middletown: Lamotte. de Ispania, J. (1973), Speculum musicae, vols. 1–7, ed. Roger Bragard, Corpus scriptorum de musica, vol. 3, Rome: American Institute of Musicology. della Seta, F. (1984), ‘Utrum musica tempore mensuretur continuo, an discreto: Premesse filosofiche ad una disputa del gusto musicale’, Studi musicali 13: 169–219. Desmond, K. (2018a), ‘“One is the loneliest number…”:the semibreve stands alone’, Early Music 46, no. 3: 403–16. Desmond, K. (2018b), Music and the moderni, 1300–1350: The ars nova in Theory and Practice, Cambridge: Cambridge University Press. des Murs, J. (1972), Notitia artis musicae et Compendium musicae practicae; Petri de Sancto Dionysio Tractatus de musica, ed. Ulrich Michels, [Dallas]: Corpus scriptorum de musica, vol. 13, [Dallas]: American Institute of Musicology. des Murs, J. (1998), ‘Notitia artis musicae, Book Two: Musica practica’, in Oliver Strunk (ed.), Source Readings in Music History, 261–9, New York and London: W. W. Norton & Company.

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Gallo, F. A. (1966), La teoria della notazione in Italia dalla fine del XIII all’inizio del XV secolo, Antiquae musicae Italicae subsidia theoretica, Bologna: Tamari. Grellard, C. and Robert, A. (2009), ‘Introduction’, in Christophe Grellard and Aurélien Robert (eds), Atomism in Late Medieval Philosophy and Theology, 1–14, Boston: Brill. Grellard, C. (2004), ‘Les présupposés méthodologiques de l’atomisme: La théorie du continu chez Nicholas d’Aurécourt et Nicholas Bonet’, in Christophe Grellard (ed), Méthodes et statut des sciences à la fin du Moyen Âge, 181–99, Lille: Presses Universitaires du Septentrion. Hicks, A. (2014), ‘Pythagoras and pythagoreanism in Late Antiquity and the Middle Ages’, in C. A. Huffman (ed.), A History of Pythagoreanism, 416–34, Cambridge: Cambridge University Press. Jung, E. and Podkoński, R. (2009), ‘Richard Kilvington on continuity’, in Christophe Grellard and Aurélien Robert (eds), Atomism in Late Medieval Philosophy and Theology, 65–84, Leiden and Boston: Brill. Kappraff, J. (2000), ‘The arithmetic of Nichomachus of Gerasa and its applications to systems of proportion’, Nexus Network Journal II: 41–55. Koehler, L. (1990), Pythagoreisch-Platonisch Proportionen in Weken der ars nova und ars subtilior, Kassel, Basel and London: Bärenreiter. Lefferts, P. M. (2001), ‘An anonymous treatise of the theory of Frater Robertus de Brunham’, in M. Bernhard (ed.), Quellen und Studien zur Musiktheorie des Mittelalters, 217–51, Munich: Verlag der Bayerischen Akademie der Wissenschaften in Kommission bei der C.H. Beck’schen Verlagsbuch. Manzari, F. and Stoessel, J. (2019), ‘The intersection of Anglo-French culture and Angevin illumination in a fourteenth-century ars nova miscellany: A new dating of Biblioteca Apostolica Vaticana, Barb. lat. 307 and Sankt Paul im Lavanttal, Archiv des Benediktinerstiftes, MS. 135/6’, Miscellanea Bibliothecae Apostolicae Vaticanae XXV 25: 283–331. Michael, E. (2003), ‘John Wyclif on body and mind’, Journal of the History of Ideas 64, no. 3: 343–60. Michael, E. (2009), ‘John Wyclif ’s atomism’, in Christophe Grellard and Aurélien Robert (eds), Atomism in Late Medieval Philosophy and Theology, 183–220, Boston and Leiden: Brill. Michels, U. (1970), Die Musiktraktate des Johannes de Muris, Wiesbaden: F. Steiner. Murdoch, J. E. (1987), ‘Thomas Bradwardine: mathematics and continuity in the fourteenth century’, in Edward Grant and John E. Murdoch (eds), Mathematics and Its Applications to Science and Natural Philosophy in the Middle Ages: Essays in Honor of Marshall Clagett, 103–37, Cambridge: Cambridge University Press. Murdoch, J. E. (1974), ‘Naissance et développement de l’atomisme au bas Moyen Âge Latin’, in John E. Murdoch et al. (eds), La science de la nature: Théories et pratiques, 11–32, Montreal: Bellarmin. Murdoch, J. E. (2001), ‘The tradition of minima naturalia’, in Christoph Herbert Lüthy, John Emery Murdoch and William Royall Newman (eds), Late Medieval and Early Modern Corpuscular Matter Theories, 91–132, Leiden: Brill.

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Murdoch, J. E. (2009), ‘Beyond Aristotle: Indivisibles and infinite divisibility in the later Middle Ages’, in Christophe Grellard and Aurélien Robert (eds), Atomism in Late Medieval Philosophy and Theology, 15–38, Boston: Brill. Page, C. (1992), ‘A treatise on musicians from ?c. 1400: The “Tractatulus de differentiis et gradibus cantorum” by Arnulf de St Ghislain’, Journal of the Royal Musical Association 117, no. 1: 1–21. Pyle, A. (1995), Atomism and Its Critics: Problem Areas Associated with the Development of the Atomic Theory of Matter from Democritus to Newton, Bristol: Thoemmes Press. Rico, G. (2005), ‘Music in the arts faculty of Paris in the thirteenth and early fourteenth centuries’, PhD diss., Oxford University. Robert, A. (2012), ‘Le lieu, le vide et l’espace chez quelques atomistes du XIVe siècle’, in Joël Biard and Sabine Rommevaux (eds), La nature et le vide dans la physique médiévale: Études dédiées à Edward Grant, Studia artistarum: Études sur la Faculté des arts dans les Universités médiévales, 67–98, Turnout: Brepolis. Robert, A. (2017), ‘Atomisme pythagoricien et espace géométrique au Moyen Âge’, in Tiziana Suarez-Nani, Olivier Ribordy and Antonio Petagine (eds), Lieu, espace, mouvement: Physique, métaphysique et cosmologie (XIIe-XVIe siècles). Actes du colloque international Université de Fribourg (Suisse) 12-13 mars 2015, Textes et études du Moyen Âge 86, 181–206, Barcelona and Rome: Fédération international des Instituts d’Études Médiévales. Robert, A. (2018), ‘Space, imagination, and numbers in John Wyclif ’s mathematical theology’, in Delphine Bellis and Frederik A. Bakker (eds), Space, Imagination and the Cosmos from Antiquity to the Early Modern Period, 107–31, Cham: Springer International Publishing. Rowell, L. (1979), ‘Aristoxenus on rhythm’, Journal of Music Theory 23, no. 1: 63–79. Stoessel, J. (2002), ‘The captive scribe: The context and culture of scribal and notational process in the music of the ars subtilior’, PhD diss., University of New England. Stone, A. (1996), ‘Che cosa c’è di più sottile riguardo l’ars subtilior?’, Rivista Italiana di Musicologia 31, no. 1: 3–31. Sylla, E. D. (1973), ‘Medieval concepts of the latitude of forms: The Oxford calculators’, Archives d’Histoire Doctrinale et Littéraire du Moyen Âge 40: 223–83. Tanay, D. (1999), Noting Music Marking Culture: The Intellectual Context of Rhythmic Notation, 1250–1400, Holzgerlingen: Musicological Studies & Documents. Willelmus and Johannes Torkesey (1966), Ms. Oxford, Bodley 842 (Willelmus): Breviarium regulare musicae, edidit Gilbert Reaney. Ms. British Museum [sic] Royal 12. C.VI: Tractatus de figuris sive de notis, edidit Gilbert Reaney. Johannes Torkesey, Declaratio trianguli et scuti, ediderunt André Gilles et Gilbert Reaney, ed. Gilbert Reaney and André Gilles, Corpus scriptorum de musica, vol. 12, Rome: American Institute of Musicology. Zayaruznaya, A. (2020), ‘Old, new, and newer still in book 7 of the Speculum musice’, Journal of the American Musicological Society 73, no. 1: 95–148.

CHAPTER 13

Atomism and the Cambridge Platonists ADRIAN MIHAI

The Cambridge Platonists were a most influential seventeenth-century group of philosophers and ‘latitudinarian’ theologians who attempted to unify in a philosophical and theological system freedom, reason, morality and love. Notwithstanding its importance on the seventeenth century and its legacy (Mihai 2020a), this group was barely taken into consideration in the scholarly circles some forty years ago and is little studied even today (Micheletti 1997, 2010, 2011, 265– 84; Beiser 1996, 134–219; Darwall 1995, 109–48; Hutton 2015, 136–59).1 Ralph Cudworth (1617–1688) is the most systematic representative of this movement, Anne Conway (1631–1679), the most radical, while Henry More (1614–1687) is certainly the most inventive and innovative, not to mention the most prolific. Their inquiries mark the early introduction of Cartesianism into England and gave rise to extensive research into the fundamental premises of atomism and materialism, especially by Cudworth. One cannot fully appreciate or properly evaluate their metaphysical views without having a clear picture of their understanding of atomism. Though there are some studies that discuss the theory of atomism in the Cambridge Platonists, particularly in Henry More (Sailor 1962, 1964, 1988; Saveson 1960; Pacchi 1973, 3–48; Gabbey 1982, 171–249; Fallon 1991, 51–78; Webster 1969), the topic has never been thoroughly analysed. Henry More has not only coined the term Cartesianism in English (More 1662, xvii) – five years before its French equivalent, coined by the physician André Graindorge in 1667 (Tolmer 1942) – but was also one of the earliest British philosophers to have a philosophical and scientific correspondence with René Descartes, from December 1648 to October 1649. In his monumental work, The True Intellectual System of the Universe (1678), Ralph Cudworth traces the genesis, development and growth of both atomism and atheism, from antiquity up to the seventeenth century, including discussions on religion, biology, psychology, botany, physics and philosophy. Anne Conway’s Principles of the Most Ancient and Modern Philosophy (written around 1677, but published only in 1690, in a Latin translation) exposes a radical philosophy, in which the theory of atomism plays a central role.

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THE ‘CAMBRIDGE PLATONISTS’ A word must be said about the historiographical label ‘Cambridge Platonism’, which summarize a dense network of positions. The expression and the reality that it embodies have been recently put into doubt by certain historians, holding even that this group has never actually existed, and thus there has never been a group of thinkers sharing some common philosophical and theological views and who would recognize themselves as pertaining to the same intellectual community (Levitin 2015, 16, 126–38). However, considering not only the historical context (Micheletti 2011; Hutton 2005, 340–58; Hickman 2017; Lewis, Secci and Hengstermann 2017, 43–124; Nicolson 1929, 35–53; Lewis 2020) but also, more importantly, the philosophical attitudes and systems of these authors,2 it is rather easy to identify common ideas, shared premises, basic problems and solutions that unify this group, a compound of ideas and methods called by Dieter Henrich as framework of thought (Denkraum).3 The expression ‘Cambridge Platonism’ was used for the first time in the 1800s, around the constellation formed by mostly Cambridge University thinkers like Coleridge, Cattermole, Whewell, Morell, F. D. Maurice and also, outside Cambridge, by the Scottish philosophers Dugald Stewart and John Tulloch (Micheletti 1997 and 2011). It has not been noted until now, and thus not sufficiently explored, that this specific interest in the ‘Cambridge Platonists’ also explains, in part, why England was one of the first countries to open its doors to Kant and to post-Kantian philosophy. I offer here a very plausible reason for this reception: Kantian and post-Kantian philosophy found similar views in England thanks to the Cambridge Platonists. This influence had been noted already in 1801, when Christoph Meiners, a German philosopher influenced by British idealism, remarked that Cudworth had anticipated many of Kant’s ideas (Meiners 1801, 151–2). At the same time, the first interpreters of Kant in England, such as Coleridge, Dugald Stewart and, later, James Mackintosh, had also noticed major similarities between the German idealists and the Cambridge Platonists’ philosophy.4 In addition, John H. Muirhead was the first to note that the Cambridge Platonists, especially Henry More and Ralph Cudworth, were the forefathers of the nineteenth-century British idealist movement (Green, Bradley, Bosanquet, Caird, Royce), who adapted and adopted many ideas of the German Idealists, via the eighteenth-century thinkers like Berkeley, Norris and Collier (Muirhead 1931; Pucelle 1955). Thus, the appearance of the label ‘Cambridge Platonism’ in the nineteenth century is not fortuitous: the expression was used in debates about the introduction of Kant and post-Kantian philosophies into England.5 Notwithstanding the historical occurrence of the term, I shall still use this expression not only for heuristic reasons and convenience’s sake but also, more importantly, in order to characterize certain philosophical and spiritual tendencies that took shape in the 1640s in Cambridge, around Benjamin Whichcote (1609–1683), Henry More, John Smith (1618–1952), Anne Conway and Cudworth, among others.6 And for a good reason: there are certain permanent features amid all variations that define our movement and that indicate a fundamental philosophical orientation.

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Furthermore, the Cambridge Platonists were not just religious moralists, as they are commonly described,7 but true metaphysicians that struggled to produce a complete philosophical system of the universe (Cudworth was the first philosopher to use the word ‘System’ in the title of a philosophical treatise).

ATOMISM AS THE TRUE THEORY OF MATTER In natural philosophy, the Cambridge Platonists hold that atomism, or corpuscularianism, is the true theory of matter. In the words of Cudworth: we conceive this Atomick Physiology, as to the Essentials thereof, to be Unquestion­ ably True, viz. That the onely Principles of Bodies, are Magnitude, Figure, Site, Motion, and Rest; and that the Qualities and Forms of Inanimate Bodies, are Really nothing, but several Combinations of these, Causing several Phancies in us . . . This Atomick Physiology rightly understood, is so far from being either the Mother or Nurse of Atheism, or any ways Favourable thereunto, (as is Vulgarly supposed;) that it is indeed, the most directly Opposite to it of any, and the greatest Defence against the same. (Cudworth 1678, *9r–10v, my emphasis) Atomism not only is the most appropriate theory to explain the physical world but also, through its explanations, refutes any sort of atheism, by showing ‘mathematically’, that thought cannot arise from the power of matter alone. This is one of the reasons that the Cambridge Platonists, especially Smith, More and Cudworth, supported and applauded Descartes’ so-called ‘atomism’,8 which they saw as distinct from the Democritean materialist approach (Gysi 1962). Cudworth writes in one of his posthumously published manuscripts that we can never sufficiently applaud that ancient atomical philosophy, so successfully revived of late by Cartesius [scil. Descartes], in that it shows distinctly that matter is, and what it can amount unto, namely nothing else but what may be produced from mere magnitude, figure, site, local motion, and rest. From whence it is demonstrably evident, and mathematically certain, that no cogitation can possibly arise out of the power of matter. (Hutton 1996, 151, my emphasis)9 However, according to Cudworth, shortly after the Cartesian restitution of this atomism, which does acknowledge incorporeal substance, the Democritean atomism was ‘revived and brought again upon the stage’ too, making senseless and lifeless atoms, to be the only principles of all things in the universe,10 thereby necessarily excluding, besides incorporeal substance and immortality of souls, a Deity and natural morality; as also making all actions and events, materially and mechanically necessary. (Cudworth 1678, 175) Cudworth and More impute this atheistic and materialist atomism to Thomas Hobbes and Pierre Gassendi.

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THEIST ATOMISM The Cambridge Platonists, as has been stated earlier, endorse atomism as the best explanatory model of the physical universe. However, for them, there are two forms of atomist philosophy. First, the most ancient and genuine that was religious, called ‘Moschical or Mosaical’, and Pythagorean. Secondly, the ‘adulterated’ atheist atomism, called Leucippean or Democritean (Cudworth 1678, 174). It is Descartes that revived the first theist atomism, by clearly showing that matter depends upon mind. For Renatus Cartesius first revived and restored the Atomick philosophy, agreeably for the most part, to that ancient Moschical and Pythagorick form; acknowledging besides extended substance and corporeal atoms, another cogitative incorporeal substance, and joyning Metaphysicks or Theology, together with Physiology, to make up one entire System of Philosophy. (Cudworth 1678, 174–5) The second, the atheist atomism, supposes that there is nothing but matter as the only substance in the universe (Cudworth 1678, 7). Cudworth reiterates that atomic physiology, in its Democritean form, is the only true form of atheism or materialism. The essence of atomism, accordingly, is that the only properties of beings are magnitude, figure, site, motion and rest. The Atomick Physiology, [is] the Foundation of the Democritick Fate . . . the onely Form of Atheism, that hath publickly appeared upon the Stage, as an Entire Philosophick System; or hath indeed been much taken notice of in the World, for these Two Thousand years past . . . Democritus and Leucippus . . . were indeed, the First Atheizers of this Ancient Atomick Physiology, or the Inventors and Broachers of the Atomick Atheism . . . were the First, who made Unqualified Atoms, the principles of all things in the Universe without exception; that is, not onely of Inanimate Bodies . . . but also of Soul and Mind. (Cudworth 1678, *8v–*9r, my emphasis) Yet, if rightly understood, atomism not only recognizes and accepts incorporeal substances but also completely refutes atheism and materialism. Thus, atomism is the most ‘effectual Engin’ and ‘Sovereign Antidote’ against atheism (Cudworth 1678, 12). Moreover, More and Cudworth even assert, following other Renaissance thinkers, that the inventor of theist atomism is none other than Moses (Sailor 1964). In his Hypomnemata physica (1636), translated into English by Peter Cole as Thirteen Books of Natural Philosophy (1659), Daniel Sennert (1572–1637) asserts that the theory of atomism could be attributed to Moses: Now amongst other Opinions ascribed to Democritus, Empedocles, and other most noble ancient Philosophers, is this; That they held Atomes or individual Bodies to be the Principles of Natural things, from the various mixture whereof other Bodies have their Original. And this Opinion was a most ancient Opinion, and is now being attributed to one Mochus a Phoenician, who is reputed to have flourished before the destruction of Troy; yea, and that it was the common Opinion of Philosophers before Aristotle. (Cole 1659, 446)

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Taking over Sennert’s judgement, More writes in his Collection of Several Philosophical Writings (1662), that the first Authour of the Atomick Philosophy, or of that Philosophy that gives an account of the Phenomena from the figures and motions of the Particles. Whence there must be no small affinity betwixt this ancient Moschical, or rather Mosaical Physiology, and the Cartesian Philosophy. (More 1662, 112) Similarly, in his System, Cudworth also proposes Moses as the inventor of original, that is, theist, atomism: But besides Reason, we have also good Historical probability for this Opinion, that this Philosophy was a thing of much greater Antiquity than either Democritus or Leucippus: and first, because Posidonius, an Ancient and Learned Philosopher, did (as both Empiricus and Strabo tell us) avouch it for an old Tradition, that the first Inventour of this Atomical Philosophy was one Moschus a Phoenician, who, as Strabo also notes, lived before the Trojan Wars . . . Mochus or Moschus, is plainly a Phoenician name, and there is one Mochus a Phoenician Writer cited in Athenaeus, whom the Latin Translator calls Moschus; and Mr. Selden [scil. John Selden] approves of the Conjecture of Arcerius [scil. Johannes Arcerius], the Publisher of Jamblichus, that this Mochus was no other man than the Celebrated Moses of the Jews, with whose Successors the Jewish Philosophers, Priests, and Prophets, Pythagoras conversed at Sidon. (Cudworth 1678, 12)

RALPH CUDWORTH’S GENEALOGY OF THE SEVENTEENTH-CENTURY ATOMISM(S) Based on this division of atomism, Cudworth offers a genealogy of the various theories of matter that suppose atoms to be the primordial elements of the universe. There can be no little doubt that Cudworth’s assessment of atomism plays an essential part in this chapter. And for two reasons: Cudworth spent most of his True Intellectual System, which fills two volumes in our critical edition (see Mihai 2020c), to discuss the premises of atomism. Secondly, he proposes a systematic account of a new form of atomism, which he denotes as ‘Theist Atomism’.

The fourfold division of atomism All through the System, Cudworth offers a genealogy of atheism, or perhaps we should say a genealogy of atomism, since the atheism discussed by Cudworth refers to the early modern philosophy of nature, and thus to the fact that its premises are completely materialistic. Cudworth distinguishes between four kinds or schools of atheism or atomism:

1. Democritean or Epicurean atheism makes unqualified atoms the principles of all things.



2. Anaximandrean or Hylopathian atheism poses matter devoid of all manner of life as the only substance of the universe.11

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3. Stratonical or Hylozoic atheism poses an irrational animated matter.12



4. Stoical or Cosmo-Plastic atheism poses a world-soul (anima mundi), but not a superior divinity.

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The main difference between these four kinds of atheist atomism is their understanding of matter (materia, hyle). On one hand, for the Democriteans and the Hylopathians, the first principle is ‘stupid’ matter, that is, inanimate, lifeless, inert matter. On the other hand, for the Stoics and the Hylozoists, the first principle is ‘vital’ matter. In other words, animate matter, that is, matter containing in itself an originating life-force. In modern times, since Cudworth’s main argument is that modern atheism is nothing but a misreading or distortion of ancient atheism, the representatives of these four types are the following:

1. Atheistic atomism, represented by Democritus and Epicurus, is revived by Pierre Gassendi (1592–1655).



2. Materialist or Hylopathian atomism, represented by Anaximander, has as direct follower Thomas Hobbes (1588–1679), this one being a latter-day Hylopathian.



3. Hylozoic atomism, represented in antiquity by the Aristotelian Strato of Lampsacus, is in the Modern Age represented by William Harvey (1578– 1657), Francis Glisson (1597–1677) and Baruch Spinoza (1632–1677).



4. Stoical or Cosmo-Plastic atomism, represented by the Stoics in antiquity, has as parallel the seventeenth-century Neostoics.

For Cudworth, these forms of atheist atomism are the ‘Kingdom of Darkness Divided’, since they hold, all in different ways, matter to be the fundamental basis of the world. Moreover, since they all miss the most fundamental aspect of atomism, its theistic premise, they contradict and destroy each other.

The twofold division of theories of matter: Atomism and corpuscularianism In addition, Cudworth distinguishes between the four classes of atomism, dividing them further into two pairs: the Democritean and the Anaximandrean, on one side, the Stoical and the Stratonical, on the other. For first, those two Pairs of Atheisms, on the one hand the Anaximandrian and Democritick, on the other the Stoical and Stratonical, do absolutely destroy each other; (1) the Former of them supposing the First Principle of all things to be Stupid Matter devoid of all manner of Life, and contending, that all Life as well as other Qualities is Generable and Corruptible, or a mere Accidental thing, and looking upon the Plastick Life of Nature as a Figment or Phantastick Capritio, a thing almost as formidable and altogether as impossible as a Deity; (2) the other on the contrary, founding all upon this Principle, that there is a Life and Natural Perception Essential to Matter, Ingenerable and Incorruptible, and contending it to be utterly impossible to give any accompt of the Phaenomena of the World, the Original of Motion, the Orderly Frame and Disposition of things, and the Nature of Animals, without this Fundamental Life of Nature. (Cudworth 1678, 142)

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Therefore, we could place Cudworth’s divisions into the two aspects of theories of matter that we find in the sixteenth and seventeenth centuries, atomism and corpuscularianism. Democritean and Hylopathian or Anaximandrean atomism could be placed under the general label of Democritean atomism or, to be more specific, ‘Atomical Physiology’, which holds, according to Cudworth, that ‘Indivisible Particles and Atoms [are] the first Principles of Bodies’ (Cudworth 1678, 17). This definition is close to what is usually understood today by ‘atomism’, that is, the belief in indivisible atoms (Pasnau 2011, 88–9; Pyle 1995, xi). Corpuscularianism, on the other hand, comprises Hylozoic and Stoical atomism, holding that all bodies are composed of very small corpuscles with the following properties: magnitude, figure, site, motion and rest (Cudworth 1678, 7).

THE MIND–BODY PROBLEM The general epistemological perspective of the Cambridge Platonists, especially More, Cudworth and Anne Conway, is all in all ‘Cartesian’, being characterized by the search of an indubitable evidence which procures the basis for all the sciences. The various aspects of this evidence, as for Descartes, are deduced from one first principle, the cogito, or, as Cudworth calls it, ‘Life and Mind, or the Self Active Cogitative Nature, and Inside Being’. We grant indeed that the Evidence of Particular Bodies, existing Hic & Nunc [i.e. here and now], without us, doth necessarily depend upon the Information of Sense: but yet nevertheless the Certainty of this very Evidence, is not from Sense alone, but from a Complication of Reason and Understanding together with it. Were Sense the only Evidence of things, there could be no Absolute Truth and Falsehood, nor Certainty at all of any thing; Sense as such being only Relative to Particular Persons, Seeming and Phantastical, and obnoxious to much Delusion. (Cudworth 1678, 637) Philosophical evidence depends thus upon a complication of reason and sense. Furthermore, the materialists commit a grave antinomy by maintaining that whatever exists must be given through the senses. But, responds Cudworth, atoms, vacuum and empty space are never seen nor felt. Thus, even the materialists themselves must here ‘go beyond the Ken of Sense, and appeal to Reason only for the Existence of these Principles’ (Cudworth 1678, 637). Therefore, Cudworth, as More and Conway, holds that Mind and Understanding is as it were a Diaphanous and Crystalline Globe, or a kind of Notional World, which hath some Reflex Image, and correspondent Ray, or Representation in it, to whatsoever is in the True and Real World of Being. And upon this account may it be said, that whatsoever is in its own Nature Absolutely Unconceivable, is indeed a Non-Entity. (Cudworth 1678, 638) Here, Cudworth argues that the objects of knowledge are modifications of mind. In other words, with this affirmation of the priority of mind over experience, but not overlooking completely experience, Cudworth attempted to discover the system

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which binds together everything in creation and thus to find the first principles or elementary laws that govern everything. In the same way as Descartes, both Cudworth and More pose in nature two distinct substances: bodies and minds. The first, resisting or antitypous extension, is nothing else but body – Descartes’ res extensa: Extension or Magnitude, Really Existing without the Mind, which is a thing, that hath no Self-Unity at all in it, but is Infinite Alterity and Divisibility, as it is also meer Outside and Outwardness, it having nothing Within; nor any other Action belonging to it, but only Locally to Move, when it is Moved. (Cudworth 1678, 832) The second, life, internal energy and self-activity, is mind or thought – Descartes’ res cogitans: Life and Mind, or the Self Active Cogitative Nature, and Inside Being, whose Action is not Local Motion, but an Internal Energy, Within the Substance or Essence of the Thinker himself, or in the Inside of him; which therefore (though Unextended, yet) hath a certain Inward Recess, βάθος, or Essential Profundity. And this is a thing, which can Act all of it Entirely, upon either a Greater or Lesser Quantity of Extended Substance or Body, and its Several Parts, Penetrating into it, and Coexisting in the same Place with it. (Cudworth 1678, 832) However, to avoid Descartes’ mind–body problem, where their natures are radically distinct – if that is the case, How can they act on each other? – Cudworth affirms that there is a kind of vital sympathy ‘by which our Soul is united and tied fast, as it were with a Knot, to the Body’ and of which ‘we have no direct consciousness, but only in its effects’ (Cudworth 1678, 160). Now, Anne Conway, unlike More and Cudworth, will argue that Corporeity (body) and Spirituality (mind) are but modes of one Substance, rather than defining essences. Like them, she holds that something mental (God) is the ultimate foundation of all reality. But she distinguishes herself from them though, by holding that there are only three substances in the universe, God, Christ and Creature: The first reason hereof shall be from the Order of Things, before-mentioned, which I have already proved to be but Three; to wit, God the Supreme or Chiefest, Christ the Medium or Middle, and the Creature the lowest in Order; which Creature is but one Essence or Substance, as to Nature or Essence, as is above demonstrated, so that it only differs secundum modos existendi; or, according to the manners [or modes] of existence. (Coudert and Corse 1996, 84) This premise presupposes that that there is only one substance that extends to both human minds and angels – ‘Creature is but one essence or substance’. Moreover, it is very consentaneous to sound Reason, and so also to the Order of Things, that as God is but One, neither hath he two, three, or more distinct Substances in him; and Christ but one Christ, neither hath in him more distinct Substances, inasmuch as he is the Heavenly Man, and very First Adam; so likewise

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the Creature, or whole Creation, is but one only Substance or Essence in Specie, although it comprehends many Individuals placed in their subordinate Species, and indeed in Manner, but not in Substance or Essence distinct one from another. (Coudert and Corse 1996, 55) To support her assertion, Conway argues in the following way:

1. If substances distinguish themselves from one another by virtue of their essences, then



2. The substances that have different essences would not have anything in common.



3. There would thus exist as many substances as there are essences.



4. Therefore, there must be one Substance in se and a se, i.e. God (I 3) or Christ or Creature (VII 1).



5. A Substance can have more than one mode or attribute, according to the degree of perfection.



6. Mode or Property (i.e. Attribute) and Substance are therefore interchangeable.

Consequently, in contrast to Descartes and her Cambridge fellows, Conway recognizes only one substance, thought and extension being merely modes. These modes are distinguished only through rarefaction and condensation or lightness and thickens or grossness. And indeed every Body is a Spirit, and nothing else, neither differs any thing from a Spirit, but in that it is more dark; therefore by how much the ticker and grosser [crassius] it is become, so much the more remote is it from the degree of a Spirit, so that this distinction is only modal and gradual, not essential or substantial. (Coudert and Corse 1996, 81)

PLASTIC NATURE AS FORMATIVE MOTION From its appearance in the digression of chapter three of the True Intellectual System, much ink has been spilled on the concept of plastic nature (Hunt 1970; Hunter 1950, 197–213; Allen 2013, 337–47; Smith and Phemister 2007, 96–9). In order to save purposiveness in nature and trying to avoid the Cartesian theory of continuous creation, some of the Cambridge Platonists introduce a median force or power, situated between God and matter. Cudworth calls it ‘plastic nature’, while Henry More calls it ‘spirit of nature’ (Greene 1962, 451–74). Cudworth derives his notion of ‘plastic nature’, with considerable latitude, from Plotinus’ notion of the logos (ὁ λόγος), especially from Enneads III 2 and 3, which talk about Providence. In these passages (see, e.g. Cudworth 1678, 163), Cudworth translates Plotinus’ ὁ λόγος as Spermatick Reason or Plastick Nature. For Plotinus, logos, at least in the two treatises on Providence, refers to a formative or formal principle, an activity of the intellect (nous) responsible for the rational order of the cosmos.

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This concept is applied by Cudworth to avoid both materialism and occasionalism or the continuing intervention of God. Since neither ‘all things are produced Fortuitously, or by the Unguided Mechanism of Matter’ – avoiding thus materialism – nor ‘God himself may reasonably be thought to do all things Immediately and Miraculously’ – circumventing thus the Cartesian continuous creation – Cudworth poses the existence of a plastic nature, or a formative principle of motion, that regulates the movement of matter in the universe. [the] Laws of Nature concerning Motion, are Really nothing else, but a Plastick Nature, acting upon the Matter of the whole Corporeal Universe, both Maintaining the Same Quantity of Motion always in it, and also Dispensing it (by Transferring it out of one Body into another) according to such Laws, Fatally Imprest upon it. (Cudworth 1678, 151) In other words, for causation to happen, one must suppose either that the laws of motion execute themselves or that God is the immediate motion of every atom of matter throughout the universe. But the former is ‘a thing plainly absurd and ridiculous’, and the latter is extremely abhorrent. Therefore, there must be something distinct both from matter and from God that governs and orders all motion in the universe, which Cudworth baptizes as ‘plastic nature’.

CONCLUSION: THE FUNDAMENTAL PRINCIPLE OF ATOMISM REFUTED What better way to finish this chapter on the theory of atomism in the Cambridge Platonists than to give a brief description of Cudworth’s refutation of the essential principle of any materialist theory of atomism. For Cudworth, any atomism, well understood, leads only to theism, and the proof that there must exist an incorporeal substance or God. In order to better refute this kind of atomism, which he calls atheist, or materialist, Cudworth identifies the main principle of all these atomistic systems, the maxim ‘Nothing could be made out of Nothing’. The Principle, upon which this Atomology is Founded, and from whence it Sprung, was no other than this, Nothing out of Nothing, in the True Sense thereof; or, That Nothing can be Caused by Nothing: from whence it was concluded, that in Natural Generations, there was no new Real Entity produced, which was not before: the Genuine Consequence whereof was Two-fold; That the Qualities and Forms of Inanimate Bodies are no Entities Really distinct from the Magnitude, Figure, Site and Motion of Parts; and, That Souls are Substances Incorporeal, not Generated out of Matter. (Cudworth 1678, *10v) Little has been written on the reception of Lucretius’ maxim, ex nihilo nihil fit, found in his poem De Rerum Natura (I 44 and II 155-158), much quoted and commented by Cudworth.13 This, I think, is essential for a better understanding of the development of modern philosophy. For Pierre Bayle, for example, the maxim

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ex nihilo nihil fit is characteristic of all the tradition of philosophical rationalism, from Parmenides to Spinoza. Before the enlightenment of the seventeenth century, Pierre de Lostal, in his Les discours philosophiques (1579), had already affirmed that this maxim is in reality a major philosophical axiom: C’est un axiome en la philosophie que de rien nulle chose ne peut estre faite, et mesmes l’experience nous sert de tesmoignage pour l’approbation d’iceluy. (De Lostal 1579, 5–6) Cudworth’s argument to use this Epicurean maxim against the materialists themselves goes as follows: both theists and materialists hold that ‘unquestionably something or other, did exist from all eternity without beginning’. For the former, is God, for the latter, matter. Cudworth then gives three philosophical explanations of the maxim ‘nothing can come from nothing’. The first meaning of nothing is that no being can bring itself into existence by itself. First, That Nothing which was Not, could ever bring it self into Being, or Efficiently Produce it self. Or, That Nothing can possibly be Made, without an Efficient Cause. (Cudworth 1678, 745) The second meaning of nothing is that no being can be brought into existence by a being inferior to it. Secondly, that Nothing which was Not, could be Produced or brought into Being, by any other Efficient Cause, than such, as hath at least, Equal Perfection in it, and a Sufficient Active or Productive Power. For if any thing were made by that, which hath not Equal Perfection, then must so much of the Effect as Transcendeth the Cause, be indeed Made without a Cause [. . .] or be Created by it self, or by Nothing. (Cudworth 1678, 745) The third meaning of nothing is that a pre-existing being will preserve its identity and contents over time. But the Third and Last Sense is this; That Nothing which is Materially Made out of things Prae-Existing, (as some are) can have any other Real Entity, then what was either before contained in, or resulteth from the Things themselves so Modified. (Cudworth 1678, 746) The conclusion of these explanations is that in order for something to be created or generated, there must be an efficient pre-existing cause. Next, Cudworth includes a Neoplatonic premise into his argumentation, to wit the doctrine of the great chain of being and the hierarchical ordering of reality. The Controversie being thus clearly Stated betwixt Theists and Atheists, it may now with great ease [. . .] be determined. It being on the one hand, undenyably evident, that Lesser Perfections may Naturally Descend from Greater [. . .]: but on the other hand utterly Impossible, that Greater Perfections and Higher Degrees of Being, should Rise and Ascend out of Lesser and Lower, so as that which is the

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most Absolutely Imperfect of all things, should be the First Fountain and Original of All. Since no Effect can possibly transcend the Power of its Cause . . . This being undeniably Demonstrable, from that very Principle of Reason, which the Atheists are so fond of, but, misunderstanding abuse, [. . .] that Nothing can come from Nothing. (Cudworth 1678, 728) Thus, Cudworth follows Proclus (Dodds 1963, 7), affirming that the doctrine of the chain of being posits that every productive cause is superior to that which it produces: no finite, imperfect substance [is] able to produce another substance out of nothing. Much less can such a substance, as hath a lower degree of entity and perfection in it, create that, which hath a higher. There is a scale, or ladder of entities and perfections in the universe, one above another, and the production of things cannot possibly be in way of ascent from lower to higher, but must of necessity be in way of descent from higher to lower. Now to produce any one higher rank of being from the lower [. . .] is plainly to invert this order in the scale of the universe from downwards to upwards, and therefore is atheistical; and by the same reason, that one higher rank or degree this scale is thus unnaturally produced from a lower, may all the rest be so produced also. (Cudworth 1678, 862–3) This belief is based especially on Plotinus and the Platonic tradition. Cudworth actually quotes Plotinus in order to demonstrate the rationality and validity of this doctrine. From Ennead V 1, 6.36-40, On the Three Primary Hypostases, Cudworth translates the following passage: Perfect Intellect Generates Soul; and it Being Perfect, must needs Generate, for so great a Power could not remain Steril. But that which is here Begotten also, cannot be greater than its Begetter; but must needs be Inferiour to it, as being the Image thereof. (Cudworth 1678, 580) And from Ennead V 3, 16.5-8: On the Knowing Hypostases, and That Which is Beyond: In the things Generated from Eternity, or Produced by way of natural Emanation, there is no Progress upwards, but all Downwards, and still a Gradual Descent into Greater Multiplicity. (Cudworth 1678, 581) Based on this premise, Cudworth continues his refutation by showing the weakness of the materialist argument: matter, which is ‘Dead and Senseless’ and ‘a mere Passive, Sluggish and Unactive thing’ cannot create or generate mind and understanding (Cudworth 1678, 729 and 756 respectively). Thus, Cudworth uses the same arguments of the atheists against a deity, that nothing can come from nothing, against the atheists themselves, by showing that they themselves hold that there must be something from eternity, and that is matter. But matter cannot create anything, and thus they themselves bring all things out of the nothingness of matter. And thus is that Formidable Argument of the Atheists, that there can be no God, because Nothing can be made out of Nothing; not only proved to be False, but

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also Retorted upon these Atheists themselves, they bringing all things besides Senseless and Unqualified Matter, out of Nothing. (Cudworth 1678, 756) If one understands and accepts this conclusion, then one must of necessity be brought to accept an absolute incorporeal substance as efficient cause of everything. The Intrinsick Constitution of this Atomic Physiology also is such, as that whosoever admits it, and rightly understands it, must needs acknowledge Incorporeal Substance; which is the Absolute Overthrow of Atheism. (Cudworth 1678, *10v)

NOTES 1. See also the following monographs on Anne Conway (Hutton 2004; Orio de Miguel 2004; White 2008), Henry More (Hall 1990; Fouke 1997; Reid 2012; Crocker 2013) and Ralph Cudworth (Bergemann 2012; Mihai 2020c). 2. This essential aspect has been left inadequately studied, or not studied at all, by the holders of the opinion of the non-existence of the group, who consider our thinkers mostly as philologists and theologians. See, for example, Levitin (2015, 16), who refers to Cudworth as a philologist! However, see Cudworth (1678, *15r), where it is written that he uses a ‘Mixture of Philology’, throughout the whole of the True Intellectual System, in order to ‘Sweeten and Allay the Severity of Philosophy’, which is the main goal of his treatise. 3. Henrich (1991). On the application of the constellation method to the seventeenth century, see Hutton (2005, 340–58), Muslow (2008, 109–22), and Lewis (forthcoming). 4. On the influence of Cudworth and Henry More on Kant, see Lovejoy (1908, 270), Baker (1937, 298–306), Schneewind (1998, 207), Darwall (1995, 124), Muirhead (1927, 166). On the legacy of Ralph Cudworth in the eighteenth century, see Mihai (2020a). 5. This philosophical explanation has been completely neglected by the historians who argue for the ‘non-existence’ of the group as a social body, who prefer to affirm that the ‘coherence and importance’ of the group ‘are predicated on the same nineteenth-century Whig story that sought to trace a “rationalist” lineage for “liberal” Anglicanism’ (Levitin 2015, 16). 6. We could also add Peter Sterry (1613–1672), Nathaniel Culverwell (1609–1651) and John Worthington (1618–1671). 7. In his recent book on the Cambridge Platonists, Micheletti (2011) still considers them chiefly as moralists and theologians. An exception to this rule is the book of Bergemann (2012). 8. Descartes has always rejected the attribution of this term to his philosophy. See Alquié (2018, 518–19). 9. See also the manuscript notes of Cudworth, Add. ms 4980, fol. 221, which repeat the judgement expressed here. 10. Here, Cudworth cites Diogenes Laertius, Lives IX 44.1. 11. Hylopathian was coined by Cudworth from hyle, matter and pathos, an affection or property. According to this belief, all bodies are generated by certain qualities from primordial matter.

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12. Hylozoic, hylozoism, hylozoist are terms coined by Cudworth, formed from the Greek words hyle (or matter) and zoe (or life), thus one who holds matter to be animated or alive. 13. Mihai (2020b). See also, on the use of this principle by David Hume, who seems to have taken it from Cudworth and Samuel Clarke, Russell (2008, 113–28).

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Hickman, L. (2017), Eighteenth-Century Dissent and Cambridge Platonism, London: Routledge. Hunt, C. C. (1970), Plastic Nature: A Study of the Cambridge Platonists, Ph.D. Diss., Harvard University. Hunter, W. B. (1950), ‘The Seventeenth Century Doctrine of Plastic Nature’, Harvard Theological Review, 43: 197–213. Hutton, S., ed. (1996), Ralph Cudworth, A Treatise concerning Eternal and Immutable Morality, Cambridge: Cambridge University Press. Hutton, S. (2004), Anne Conway: A Woman Philosopher, Cambridge: Cambridge University Press. Hutton, S. (2005), ‘Eine Cambridge-Konstellation? Perspektiven für eine Konstellationsforschung zu den Platonikern von Cambridge’, in M. Mulsow and M. Stamm (eds), Konstellationsforschung, 340–58, Frankfurt: Suhrkamp. Hutton, S. (2015), British Philosophy in the Seventeenth Century, Oxford: Oxford University Press. Keith Allen, K. (2013), ‘Cudworth on mind, body, and plastic nature’, Philosophy Compass 8: 337–47. Levitin, D. (2015), Ancient Wisdom in the Age of the New Science: Histories of Philosophy in England, c. 1640–1700, Cambridge: Cambridge University Press. Lewis, M. A. (2020), ‘Christ’s college and the Latitude-Men’ revisited: A seminary of Heretics?’ History of Universities 33: 17–68. Lewis, M. A., Secci, D. A. and Hengstermann, C., eds (2017), ‘Origenian platonism in interregnum Cambridge: Three academic texts by George Rust, 1665 and 1658’, History of Universities, 30: 43–124. Lovejoy, A. (1908), ‘Kant and the English Platonists’, in Essays Philosophical and Psychological in Honor of William James, 265–302, New York: Longmans. Meiners, C. (1801), Allgemeine kritische Geschichte der ältern und neuern Ethik, vol. 2, Göttingen: Johann Christian Dieterich. Micheletti, M. (1997), Dai latitudinari a Hume. Saggi sul pensiero religioso britannico dei secoli XVII e XVIII, Perugia: Benucci. Micheletti, M. (2010), ‘John Locke, i platonici di Cambridge e i latitudinari: la critica alla superstizione e al fanatismo e il problema della toleranza religiosa’, in F. Rossi (ed.), Cristianesimo teologia filosofia. Studi in onore di Alberto Siclari, 265–84, Milano: FrancoAngeli. Micheletti, M. (2011), I platonici di Cambridge. Il pensiero etico e religioso, Brescia: Morcelliana. Mihai, A. (2020a), ‘The reception of Ralph Cudworth (1617–1688)’, in D. Hedley and D. Leech (eds), The Cambridge Platonism Sourcebook. http:​/​/www​​.camb​​ridge​​-plat​​onism​​ .divi​​nity.​​cam​.a​​c​.uk/​​about​​-the-​​cambr​​idge-​​plato​​nists​​/rece​​ption​​​/cudw​​orth-​​ralph​. Mihai, A. (2020b), ‘Strato’s Ghost: A Critique of Hylozoick philosophy by Henry More and Ralph Cudworth’, in C. Hengstermann (ed.), Henry More’s ‘Immortality of the Soul’, Münster: Aschendorff, in press. Mihai, A. (2020c), Ralph Cudworth, The True Intellectual System of the Universe, ed. Adrian Mihai, Turnhout: Brepols. More, H. (1662), A Collection of Several Philosophical Writings, London: James Flesher.

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Muirhead, J. H. (1927), ‘The Cambridge Platonists I-II’, Mind, 36: 158–78, 326–41. Muirhead, J. H. (1931), The Platonic Tradition in Anglo-Saxon Philosophy. Studies in the History of Idealism in England and America, London: George Allen & Unwin. Muslow, M. (2008), ‘The third force revisited’, in J. D. Popkin (ed.), The Legacies of Richard Popkin, 109–22, Dordrecht: Springer. Nicolson, M. N. (1929), ‘Christ’s college and the Latitude-Men’, Modern Philology 27: 35–53. Orio de Miguel, B. (2004), La filosofia de Lady Anne Conway, Un Proto-Leibniz, Valencia: Universidad Politecnica de Valencia. Pacchi, A. (1973), Cartesio in Inghilterra da More a Boyle, Bari: Laterza. Pasnau, R. (2011), Metaphysical Themes (1274–1671), Oxford: Oxford University Press. Pucelle, J. (1955), L’idéalisme en Angleterre. De Coleridge à Bradley, Neuchâtel: La Baconnière. Pyle, A. (1995), Atomism and Its Critics, Bristol: Thoemmes. Reid, J. (2012), The Metaphysics of Henry More, Dordrecht: Springer. Russell, P. (2008), The Riddle of Hume’s Treatise: Skepticism, Naturalism, and Irreligion, Oxford: Oxford University Press. Sailor, D. B. (1962), ‘Cudworth and Descartes’, Journal of the History of Ideas 23: 133–40. Sailor, D. B. (1964), ‘Moses and atomism’, Journal of the History of Ideas 17: 3–16. Sailor, D. B. (1988), ‘Newton’s debt to Cudworth’, Journal of the History of Ideas 49: 511–18. Saveson, J. E. (1960), ‘Differing reactions to Descartes among the Cambridge Platonists’, Journal of the History of Ideas 21: 560–7. Schneewind, J. B. (1998), The Invention of Autonomy, Cambridge: Cambridge University Press. Smith, J. E. H. and Phemister, P. (2007), ‘Leibniz and the Cambridge Platonists’, in P. Phemister and S. Brown (eds), Leibniz and the English-Speaking World, 96–9, Dordrecht: Springer. Tolmer, L. (1942), ‘Une page d'histoire des sciences (1661–1669): Vingt-deux lettres inédites d’André de Graindorge à P.-D. Huet publiées et annotées’, Mémoires de l’Académie nationale des sciences arts et belles-lettres de Caen 10: 243–337. White, C. W. (2008), The Legacy of Anne Conway (1631–1679), Albany: State University of New York Press.

CHAPTER 14

Atomism and society in William Petty AKOS SIVADO

INTRODUCTION In the seventeenth century, many mechanically minded natural philosophers came forward with atomistic theories of matter,1 though hardly any of them ventured so far as to extend such metaphysical foundations to the social realm as well. The potential drawbacks of a hypothetical expansion are relatively easy to assess: supposing that atoms are in any way responsible for social matters, one would seemingly need to account for such distinctively human phenomena in terms of war and peace, the fiscal peculiarities of the commonwealth or criminal behaviour. While the task does seem impossible, a handy metaphor could have lent itself to atomists since the dawn of atomistic thinking itself: just as the supposed atoms coalesce to give shape to sensible objects, the commonwealth itself could have been thought of to be comprised of its members qua atoms that come together to form a supraindividual ‘being’ of sorts.2 Other than presenting a potentially powerful image, the metaphor itself could hardly have been able to underpin a thoroughgoing metaphysics of the social – should one wish to extend the boundaries of atomistic thinking, something more would have been needed. There was, however, one such account that took it upon itself to attempt exactly that – Sir William Petty’s political arithmetic.3 This chapter will concentrate on Petty’s view of society from an atomistic perspective, emphasizing the potential consequences of such a view rather than the accuracy of his mathematical machinations. Many have argued that Petty’s treatment of society paved the way towards modern statistics, demographics or epidemiology,4 and while those are all valid arguments to make, the present chapter aims to flesh out his version of atomism within the context of seventeenth-century thinking. The first part will concentrate on articulating the grander atomistic framework of Petty’s thinking and placing societal phenomena within it, arguing that for Petty the treatment of social relations followed naturally from his proposed treatment of phenomena found in the natural world. The second part draws out the implications of Petty’s atomistic view of society, understood not primarily as individuals qua atoms constituting a supraindividual whole, but instead as individuals being constituted by atoms themselves. To extrapolate from such individual but atomistically constituted

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people to account for matters of the state constitutes a further step in Petty’s thinking, one which lends itself to comparisons with modern-day statistics – and the final part will take a closer look at how such a step could have allowed Petty to confess to such seemingly contradictory beliefs as the neutrality and impartiality of numbers and the power of a single person to change these supposedly neutral and impartial numbers in the same time.

THE ROOTS OF PETTY’S ATOMISM Most (if not all) atomistic philosophers have had to suffer being stigmatized as either naturalists or atheists, if not both.5 Petty’s professed version of atomism aimed to counteract such accusations in probably the most self-conscious (and some would say audacious) way possible: he wished to anchor atomism in the teachings of the Bible itself, based on what he took to be textual evidence in support of his claims. Such an anchoring serves at least two possible functions: on the one hand, it is supposed to automatically disqualify objections based on the theory’s inherent atheism, while on the other hand, it enables Petty to ‘transport’ his views on inanimate matter onto the realm of the social. This latter function might not be immediately clear: just because something is based on Biblical teachings, it should not automatically be treated as a universally valid theory to be applied, without reservations, to nature and society alike. To see how Petty’s system might be able to be so comprehensive, it is best to examine the precious few things he says about the universe in The discourse made before the Royal Society the 26. of November, 1674, concerning the use of duplicate proportion in sundry important particulars together with a new hypothesis of springing or elastique motions, for however brief its pertinent parts might be, they may be able to shed some light on his first philosophy.6 Atomism enters into the picture very early on in the Discourse when Petty boldly states in the introductory parts that I suppose all the First Matter of the World to be Atoms; that is, Matter Immutable in Magnitude and Figure. I suppose Corpuscles to be as many Atoms joyned together, as make up a visible or sensible Object, and that all Iuncture of Atomes is made by their Innate motions. Moreover I suppose, That every Atom is like the Earths Globe or Magnet, wherein are three Points considerable, viz. two in the surface, called Poles, and one within the substance, called Center, or rather Byas, because in Atoms we consider neither Magnitude nor Gravity.7 It is clear from the outset then that Petty’s universe is comprised of atoms through and through, atoms that behave very much like invisibly small magnets that attract and repel one another, while their motion and multiplicity ultimately account for the properties of all visible objects: We further say, that the motions of Corpuscles are compounded of the abovementioned motions of Atoms; and the motions of bigger and Tangible Bodies (viz. their qualities) are decompounded out of the Motions, Situation, Figure, and Magnitude of Corpuscles; and that out of, and by, the premisses all Phaenomena in nature must be solved.8

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It will only become clear from the later sections of the text (most importantly the Appendix attached to it) just how inclusive Petty is when he is talking about ‘all Phaenomena’. In the Appendix, he compares the motion of atoms to that of the planets and other heavenly bodies in a Copernican system (they move about their own axes), while their behaviour is also influenced by their magnet-like tendencies (in fact, they have two different motions according to Petty: one of ‘Gravity’ which pulls them towards the centre of the earth, and one of ‘Verticity or Polarity’ that draws them towards one another).9 Petty, however, goes even further than that and lends his atoms some qualities that are truly unique compared to other atomistically minded natural philosophies of his (or any other) time: Lastly, I might suppose (even without a Metaphor) that Atoms are also Male and Female, and the Active and Susceptive Principles of all things; and that the abovenamed Byasses are the Points of Coition: For, that Male and Female extend further than to Animals, is plain enough; the fall of Acorns into the ground, being the Coition of Oaks with the Earth. Nor is it absurd to think, that the words in Genesis, [Male and Female created he them] may begin to take effect, even in the smallest parts of the first Matter. For although the words were spoken onely of Man; yet we see they certainly refer to other Animals, and to Vegetables in manner aforesaid, and consequently not improbably to all other Principles of Generation.10 The most important thing that lets Petty carry atomistic thinking across the natural/ social divide is his willingness to attribute more to atoms than anyone else would: to assign them sexes and see them capable of intercourse not unlike any kinds of animate organisms are. There are three points worth emphasizing about his remarks (however puzzling they may seem). First, he is happy to assert that this language is not metaphorical – that atoms as they actually exist are indeed male or female. Second, that this claim could be backed up by referencing the book of Genesis, hence the atomistic framework could be connected to an intelligent creator of the universe. And third, that atoms are indeed the smallest parts of the ‘first Matter’, meaning that everything else that exists is comprised of them. Such a claim does not exempt society itself from the scope of investigation, in fact it is precisely what is needed to build a social or political philosophy based on the doctrine of atomism.

THE ‘SOCIAL SCIENCE’ OF ATOMS An overarching system built on atoms as foundational building blocks is, however, not something Petty ever developed, nor is it fruitful to attribute something resembling that to him. What might be fruitful instead is to read Petty’s thoughts on social issues (most importantly economic ones) with such a metaphysical background in mind, for his method of political arithmetic does treat social phenomena the same way as if they were natural. In the light of his take on atomism, however, such a treatment seems less likely to be inspired by simply and conveniently disregarding specifically human characteristics that could hardly be fitted into a nature-oriented ‘science’ – it is perhaps more harsh but also more accurate to suppose that he did

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not think there were such characteristics.11 Given so, there is nothing standing in the way of extending the natural philosophical research methods for which he advocates in his Discourse (mostly mathematical operations centred around square roots and duplicate proportions) to the realm of society itself. Petty’s greatest legacy undoubtedly lies in devising the proto-economical method of political arithmetic: a practical way of accounting for a country’s population in terms of their ‘number, weight and measure’. On the surface, it is fairly straightforward: ‘people’ are ‘things’ about which we could generate knowledge, since ‘people’ are part of the world we inhabit, and they are sensible. Since we already use mathematics to calculate and predict the behaviour of inanimate objects (with ever greater precision, as evidenced by the ‘success stories’ of navigation or warfare), we should not shy away from applying the same methods to the social realm as well.12 Some brief examples from Petty’s manuscripts should illustrate the point. First, a remark about how one tenth of England’s population should be able to maintain ten thousand feet, forty thousand horses and forty thousand men at sea per annum: To clear this Point, we are to find out, what is the middle expence of each Head in the Kings Dominions, between the highest and the lowest; to which I say it is not probably less than the expence of a Labourer, who earneth about 8d. a day; for the Wages of such a Man is 4s. per week without Victuals, or 2s. per week, or 5l. 4s. per annum: Now the value of Clothes cannot be less than the Wages given to the poorest Maid-Servant in the Country, which is 30s. per annum, nor can the charge of all other Necessaries, be less than 6s. per annum more; wherefore the whole charge is 7 l. [. . .] Now if the expence of each Man, one with another, be 7 l. per annum, and if the number of the Kings Subjects, be ten Millions, then the tenth part of the whole expence, will be seven Millions: but about five Millions, or a very little more, will amount to one years pay for one hundred thousand Foot, forty thousand Horse, and forty thousand Men at Sea, Winter and Summer; which can rarely be necessary.13 Such an assessment reads like taking stock of inanimate objects – seemingly entirely devoid of humanity, treating its subject (the population of England) as if it were comprised of ‘mere’ matter and nothing more. What Petty is doing in such calculations is, however, nothing unusual for him considering his atomistic background: things comprised of atoms behave in specific ways under specific circumstances. Seeing how people are also ultimately comprised of atoms, individual differences based on character (or on all those ‘mutable’ things Petty does not wish to discuss) could be neglected. The underlying nature is what counts, and that is practically the same in all cases. Or, to quote another example of proto-demography, this time dealing with the population of London: I found that the years 1684 and 1685, being next each other, and both healthfull, did wonderfully agree in their Burials, viz. 1684 they were 23202, and Anno 1685 23222, the Medium whereof is 23212; Moreover that the Christnings 1684 were 14,702, and those Anno 1685 were 14730, wherefore I multiplied the Medium of Burials 23212 by 30, supposing that one dies out of 30 at London, which made the number of People 696,360 Souls.14

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Petty then goes on to give his reasons why ‘one dies out of 30’ in London, and why this number should be arrived at when one takes into account only sensible (in his case, quantifiable) things. Childbirths and burials are definitely quantifiable – the multifaceted circumstances of individual lives are not necessarily so. But a science of the state based on an atomistic world view would do better not to dwell much upon unquantifiable phenomena. Such a motivation is implicit in Petty’s more wellknown and more outrageous schemes (wherein he gives the exact worth of a person in pound sterling, or plans the transportation and forced exchange of multiple thousands of Irish and English women in order to keep Ireland rebellion-free). In such proposals and such calculations, the influence of atomism is twofold. On the one hand, nothing of the sort would ever ring true should one suppose there was a fundamental difference between objects found in nature and persons found in societal settings. The atomistic background, viewed from this perspective, serves as a condition of possibility: shared natures mean the possibility of shared methods of gathering knowledge. The people whom Petty counts (or wishes to count) are every bit as atomistically constituted as bullets or mountains are, making it unnecessary to develop new and untested ways of accounting for them. On the other hand, there is a different argument to be made that shifts the perspective from the individual to the state: just like people are comprised of atoms that have individual qualities (shapes, sizes, motions, even genders), so are states comprised of people with the like individual qualities. Petty, a self-professed admirer of both William Harvey and Thomas Hobbes, extends the conclusions he had arrived at while examining the parts to the whole of the social realm as well. While it is never made explicit, one could hardly miss the same principles at work here: in order to gain insight into the inner workings of a phenomenon, one should attempt to say something about its building blocks. In the case of societies, such building blocks are actual people: societies do not ‘die’, but there is such a thing as ‘mortality’ in them; they do not ‘pay taxes’, but there are people who do. Petty always emphasizes the practical utility of his observations and (often unrequested) counsels – he is primarily interested in state governance, not individual welfare. Political arithmetic itself is often referred to as a tool with which rulers (in most cases the monarch) could ‘perform’ their duties more efficiently – Petty’s friends and foes alike are quick to agree on such an assessment, although with different implications.15 Petty’s own most damning phrase is to be found in the preface to Political Arithmetic as well, where he states that while his numbers might not be accurate (while also professing that he wishes to underpin his argument by virtually nothing but numbers), ‘yet [the numbers] may be made so by the Sovereign Power’.16 Such a wording raises an immediate objection: if numbers should be trusted exactly because they merely reflect the inner nature of things themselves, then why grant anyone access to ‘make them accurate’? Upon first glance, such a statement reads very much like an admission of epistemological guilt: numbers, weights and measures are supposed to be impartial measurements, although all of Petty’s writings aim at changing the impartial measures arrived at, so it is better to say outright that, when everything else is accounted for, the monarch has the right to tinker with

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impartiality in any way he or she sees fit. In the final part of this chapter, I attempt to argue that it is not necessarily the case.

COUNTING AND CHANGING NATURES Petty’s statement about ‘making the numbers right’ reads quite differently once one supplements it with his atomistic world view and his self-professed Baconianism. There could be multiple reasons why one would wish to obtain knowledge about a certain object – and to change that very object (or ‘transmutate’ it, to borrow from the language of alchemy also not foreign to Petty)17 is deemed a reasonable aim among Baconians.18 Observing and predicting the behaviour of phenomena could be extended to include changing those phenomena as well: should we know more about the nature of iron, we could make better use of it when fashioning tools or weapons out of it. In an analogical way, should we know more about the nature of society, we would be better able to change it for the ‘peace and posterity of all’. The atomistic world view lends a helping hand in this endeavour as well: we are already in a better position to reach such a goal the more we are acquainted with the fundamental constituents of our subject matter. And the aforementioned two perspectives can now be reversed: people constitute states, and atoms constitute people. Should either part of such an assertion be questioned, Petty offers a humorous, although nonetheless helpful remark in the final lines of his Appendix to the Discourse, where he mentions how all that he has said ‘is humbly submitted to the Censure of this Society (the Royal Society – AS); whose Atoms or inseparable Members I wish may happily Conglomerate, and Unite themselves into the most fixed and most noble Bodies amongst the Sons of Men’.19 In order to further illustrate how counting something and then arriving at changing that thing by changing the resulting numbers, it is now worth to take a closer look at Petty’s most inhumane proposal concerning the transmutation of the Irish. After Oliver Cromwell’s army had put an end to the Irish Confederate Wars in 1653, Petty, a physician general in said army, began to think about ways one could prevent a similar uprising and the resultant civil war in the future. He was not the first to arrive at a solution that had to do with relocating people, but his version was more elaborate and more drastic than any such scheme put forward before – also, more methodically in line with his main theses concerning numbers, weights and measures. Here are his proposal’s more controversial parts: Whereas there are now 300 M. British, and 800 M. Papists [. . .] If an Exchange was made of but about 200 M. Irish, and the like number of British brought over in their rooms, then the natural strength of the British would be equal to that of the Irish; but their Political and Artificial strength three times as great; and so visible, that the Irish would never stir upon a National or Religious Account. There are among the 600 M. above-mentioned of the poor Irish, not above 20 M. of unmarried marriageable Women; nor would above two thousand per Ann. grow and become such. Wherefore if ½ the said Women were in one year, and ½ the next transported into England, and disposed of one to each Parish, and

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as many English brought back and married to the Irish, as would improve their Dwelling but to an House and Garden of 31. value, the whole Work of natural Transmutation and Union would in 4 or 5 years be accomplished.20 As it can be observed, Petty begins his calculations with the actual numbers of Irish and English people living in a specific time – numbers he arrived at through extrapolations from birth and death certificates, through his method of political arithmetic. What follows is a plan to change those numbers, thereby changing the nature of the ‘things’ counted. (Petty’s rationale for how and why such relocation processes should conclude in the desired result is also rather shaky: he supposes that English women are stronger in character and more virtuous than Irish men, therefore they would be able to instil in them a ‘better nature’ of sorts, while the same holds for Irish women and English men, only in reverse.) Petty himself is, of course, not in any position to change the numbers. However much he writes about the situation he deems ideal, and however elaborately he details the way to bring about such a situation, pamphlets and manuscripts circulated among friends and the nobility are hardly enough to implement the proposed modifications. Comprehensive policy decisions are needed, and the one who has the power to make such decisions is indeed the head of the state – in Petty’s case, the monarch, or the ‘Sovereign Power’. Here, then, is one instance of the possibility for numbers to being made correct (as in ‘ideal’ or ‘desired’) – and it follows from Petty’s mechanical philosophy that the corrections should work both ways. Changing the numbers will amount to changing the natures over time; and of course, changing the natures will further change the numbers (in the aforementioned example, the number of English and Irish people) in the long run. It may seem a bit far-fetched to trace back such overarching practical proposals with potentially disastrous consequences to a version of atomism mentioned in a single Discourse and then never taken up again, although two main reasons point in favour of doing so. First and foremost, it is reasonable to assume that Petty, a man of the state himself and a founding member of the Royal Society, did not want to further elaborate a philosophical theory that, in spite of his precautions, ended up generating accusations of atheism levelled against him.21 Second, no reading of Petty’s texts could be said to apply the principle of charity if it left his proposals and his economic ideas in a vacuum – read that way, they do seem to lack any kind of foundation and seem to be nothing more than disjointed fragments of political scheming. It is somewhat counterintuitive to see someone heralded as the forerunner of economics, demographics, statistics and the like while also maintaining that he had no underlying principles to maintain the coherence of his writings. As this chapter has attempted to argue, there is indeed such an underlying principle, and the practical aspects of Petty’s economic writings are a corollary to such a principle.

CONCLUSION This chapter attempted to show what kind of consequences could follow once one applies atomistic philosophical principles across the board and extends them to the

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realm of society. As William Petty’s case demonstrates, such thinking could provide the foundations for at least two different stages of theorizing. On the individual level, it could prove to be essential in not distinguishing between natural and social phenomena when it comes to methodological issues. Knowledge about one set of entities could be fostered through the same ways that one would use in order to gain knowledge about the other, for they share the same metaphysical background. Placed in the context of seventeenth-century British philosophy, where Baconian induction and empiricism started to take centre stage, such a doctrine could allow social philosophers to measure and take account of phenomena related to people and social life in much the same way as one would write natural histories or design experiments that are supposed to shed light on the inner machinations of inanimate matter. Once the foundational level is homogenized, everything else could be rendered subservient to such a structure. On the level of the state, the thoroughgoing atomistic perspective could allow for a more literal interpretation of the organicist metaphor that compares the body politic to actual human bodies: such as the latter are comprised of invisible atoms, so is the former comprised of its much more visible atom-like constituents. The continued emphasis on atomism with regard to Petty’s economic writings helps explaining why there is no qualitative shift between at least three levels of potential data: the principles of explanation, prediction and manipulation do not change when the scope of our investigation changes from inanimate to animate matter, neither should it do so when it changes from individuals to states. To turn the picture around: Petty’s proposals were fashioned towards the betterment of state governance (third level) while they were based on data gathered about individual birth, death and income rates (second level) – and the whole enterprise seemed feasible to him because those individuals did not differ in their constitution from any other phenomenon found in nature (first level). Histories of economic thinking usually regard Petty as a precursor of political economics, only paying attention to the fiscal/monetary aspects of his thought. While he definitely has his rightful place among the forerunners of modern-day economic sciences, it may be worthwhile to try to reconstruct his account of social phenomena based not on his explicitly economical writings but on his far less famous remarks about natural philosophy and atomism. Much like how without paying attention to said remarks Petty’s proposals would seem to be nothing more than practice without theory, his atomistic world view would also fail to be informative (and would probably remain a curiosity at best on account of his gendered atoms) unless one complemented it with the actual proposals it could have made possible. I attempted to juxtapose the two sides of Petty’s thought with a heavier emphasis on the atomistic underpinnings of his practical suggestions, for this way might allow us to reconsider his schemes under a different light. It would, of course, not redeem the now-apparent cruelty of much of his proposed solutions to socio-economic problems, but it could at least help in their understanding and interpretation, while also drawing attention to those aspects of atomism that could have played a part in making a science of society possible.

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NOTES 1. See Kargon (1966). 2. The infamous image gracing the cover of Leviathan by Thomas Hobbes could serve as an illustration of the idea, although he himself definitely did not intend it to be one – neither was atomism a part of the popular organicist metaphor at the time. 3. The secondary literature on Petty’s work is ever-growing in the twenty-first century, with monographs by McCormick (2009) and Goodacre (2018) as well as numerous articles. 4. See, respectively, Mykkänen (1994), Rusnock (2002), and Banta (1987). 5. See Kargon (1966), esp. chapters 8–11. 6. The first analysis devoted entirely to this aspect of Petty’s thought is to be found in Kargon (1965). 7. Petty (1674, 17–18) (emphasis in original). 8. Petty (1674, 19–20) (emphasis in original). 9. See (Petty 1674, 125–9). 10. Petty (1674, 130–2). 11. As it pertains to visible (sensible) qualities. As he professes in the preface to his manuscript on Political Arithmetic, he does not wish to enlighten his audience about anything regarding the immaterial: the mind, the thoughts or the spirit of human beings. See Petty (1899, vol. 1, 244). 12. It is safe to say that political arithmetic itself could hardly be considered such a ‘success story’: while it did attract followers after Petty’s death, and played a huge part in calculating the costs of England’s union with Scotland, by the middle of the eighteenth century, it was pretty much replaced by Adam Smith’s take on political economy. On the history of the method, see Buck (1977, 1982), and Hoppit (1996). 13. Petty (1899, vol. 1, 305–6) (emphasis in original) 14. Petty (1899, vol. 2, 535) (emphasis in original) 15. Examples of seeing political arithmetic as nothing but a promotional trick with which Petty came up to elevate his own standing include Rothbard (1995) and Mirowski (1989), while more favourable accounts also often treat his method as something that is most importantly a tool for practical purposes. See Poovey (1998), Deringer (2012). On the seventeenth-century context of improving a nation through the generation and dissemination of knowledge, see Slack (2015). 16. Petty (1899, vol. 1, 244). 17. See McCormick (2006). On the interconnections between atomism and alchemy, see Newman (2006). 18. See Rusu (2013). 19. Petty (1674, 134–5). 20. Petty (1899, vol. 1, 157–8). 21. On how this might have led Petty to refrain from pure natural philosophical writing, see the introduction in Lewis (2012).

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REFERENCES Banta, J. E. (1987), ‘Sir William Petty: Modern epidemiologist (1623–1687)’, Journal of Community Health 12, nos. 2–3: 185–98. Buck, P. (1977), ‘Seventeenth-century political arithmetic: Civil strife and vital statistics’, Isis 68, no. 1: 67–84. Buck, P. (1982), ‘People who counted: Political arithmetic in the eighteenth century’, Isis 73, no. 1: 28–45. Deringer, W. P. (2012), Calculated Values: The Politics and Epistemology of Economic Numbers in Britain, 1688–1738. PhD thesis, Princeton University. Goodacre, H. (2018), The Economic Thought of William Petty: Exploring the Colonialist Roots of Economics, London: Routledge. Hoppit, J. (1996), ‘Political arithmetic in eighteenth-century England’, Economic History Review 49, no. 3: 516–40. Kargon, R. (1965), ‘William Petty’s mechanical philosophy’, Isis 56: 63–6. Kargon, R. (1966), Atomism in England from Hariot to Newton, Oxford: Clarendon Press. Lewis, R. (2012), William Petty on the Order of Nature: An Unpublished Manuscript Treatise, Tempe: ACMRS. McCormick, T. (2006), ‘Alchemy in the political arithmetic of Sir William Petty (1623– 1687)’, Studies in History and Philosophy of Science 37: 290–307. McCormick, T. (2009), Sir William Petty and the Ambitions of Political Arithmetic, New York: Oxford University Press. Mirowski, P. (1989), More Heat than Light – Economics as Social Physics: Physics as Nature’s Economics, New York: Cambridge University Press. Mykkänen, J. (1994), ‘“To methodize and regulate them”: William Petty’s governmental science of statistics’, History of the Human Sciences 7, no. 3: 65–88. Newman, W. R. (2006), Atoms and Alchemy: Chymistry and the Experimental Origins of the Scientific Revolution, Chicago and London: The University of Chicago Press. Petty, W. (1674), The Discourse made Before the Royal Society the 26. of November, 1674, Concerning the Use of Duplicate Proportion in Sundry Important Particulars Together with a New Hypothesis of Springing or Elastique Motions, London: J. Martyn. Petty, W. (1899), The Economic Writings of Sir William Petty, ed. John Henry Hull, Cambridge: Cambridge University Press. Poovey, M. (1998), A History of the Modern Fact, Chicago: Chicago University Press. Rothbard, M. N. (1995), Economic Thought Before Adam Smith. Volume I, Auburn: Ludwig von Mises Institute. Rusnock, A. A. (2002), Vital Accounts – Quantifying Health and Population in EighteenthCentury England and France, New York: Cambridge University Press. Rusu, D.-C. (2013), From Natural History to Natural Magic: Francis Bacon’s Sylva Sylvarum. PhD thesis, Universiteit Nijmegen. Slack, P. (2015), The Invention of Improvement. Information and Material Progress in Seventeenth-Century England, New York: Oxford University Press.

CHAPTER 15

Atoms, colours and God in Leibniz ALBERTO ARTOSI

Cependant j’ay changé et réchangé sur des nouvelles lumiéres.1 In his mature years, Leibniz often hinted at his early commitment to atomism. Thus in his Système nouveau de la nature (1695), he writes: ‘In the beginning, when I had freed myself from the yoke of Aristotle, I accepted the void and atoms, for they best satisfy the imagination’ (G IV 478, AG 139, L 454). Two years later, writing to his (and Locke’s) correspondent Thomas Burnett, he recalled his conversion to atomism, which would have taken place when he was less than fifteen years old. He tells of walking in a wood for entire days, trying to decide whether to ‘take part for Aristotle or Democritus’,2 an account that has been judged to be of dubious credibility, as it is unlikely that the conversion could have taken place before 1664, when Leibniz was eighteen years old.3 But this is not the point. The point is that, despite these autobiographical accounts, Leibniz’s first writings have generally been considered to offer no conclusive evidence of an atomist phase in his early mechanistic thinking. This has led Leibnizian scholars to debate whether Leibniz had ever really subscribed to atomism. In this chapter, I will not review the literature on this vexing question,4 as I consider it settled from the start on the basis of the available evidence. In fact, in my view, the existing evidence is strong enough to warrant the conclusion that – even if only for a short time, between 1666 and 1669 – Leibniz effectively endorsed the atoms and the void, and that even if this endorsement appears to be transient, it constitutes a significant chapter not only in Leibniz’s intellectual development but, as I will argue, also in the history of seventeenth-century atomism.

VOID AND ATOMS IN LEIBNIZ’S WRITINGS OF 1664–6 The first mention of atoms we find in Leibniz’s earliest works is in his 1664 dissertation for the title of master of philosophy, Specimen quaestionum philosophicarum ex iure collectarum. In Question VIII, devoted to the discussion of the legal status of animals, Leibniz brings up Pierre Gassendi, the major proponent

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of atomism in the seventeenth century, just to take him to task for having followed Epicurus in construing the rational soul as composed of atoms and in ascribing this soul to brutes as well (A VI/1 84, APS 21–2). No other mention of atoms is to be found, but Leibniz touches on the void in one of the corollaries for the disputation appended at the end of the dissertation, to the effect that no definitive proof of its non-existence has yet been provided (A VI/1 95, APS 39). Atoms burst onto the scene two years later with the Dissertatio de arte combinatoria. Here Leibniz overtly embraces atomism while exalting the prospects of his combinatorial art as a method for investigating nature: No one who sees that all derives from the innermost core of the doctrine of variations (ex intima variationum doctrina) will say that we have departed from the argument – that doctrine which is perhaps the only one which leads through all the infinite the compliant soul to itself, and which embraces at once the harmony of the world and the deepest structures of things and the series of forms, and whose incredible utility will rightly be appreciated once philosophy will be finally accomplished or almost accomplished. In fact, the seventeenth use consists in complicating the geometrical figures, in which matter Johannes Kepler broke the ice in bk 2 of the Harmonice. Thanks to these complications (complicationes), not only can geometry be enriched with infinitely many new theorems, for a new complication produces a new composed figure, by contemplating the properties of which we construct new theorems and new demonstrations, but (if it is indeed true that large things are composed of small ones, whether you call these atoms or molecules) this is also the only way of penetrating into the secrets of nature. For it is said that the more perfectly one knows a thing, the more one perceives the parts of the thing and the parts of the parts, as well as their figures and positions. This reason (ratio) of the figures must first be abstractly investigated in geometry and stereometry: from here you will approach natural history and existence, that is to say, that which is effectively found in bodies; the big portal of physics will open, and we will be amazed by the aspect of the elements and the origin of qualities and the origin of the mixture and of the mixture of mixtures and of anything so far removed in nature. (A VI/1 187–8) Atoms, with their different figures, order and position, resurface towards the end of the dissertation as a fascinating counterpart to the calculus of variatio situs (variation of position). Leibniz eloquently emphasizes the analogy between the generation of worlds from letters and the origination of things from atoms, to this end citing two classic atomistic loci from Aristotle’s On Generation and Corruption and Metaphysics.5 There follow two quotations, one from Lucretius’s De Rerum Natura,6 the other from Lactantius’s Divinae institutions,7 and two further references to two cornerstones of modern atomism such as Gassendi’s Animadversiones in decimum librum Diogenis Laertii8 and Jean Chrysostome Magnen’s Democritus reviviscens,9 after which it becomes difficult to dispute Leibniz’s engagement with atomism at this early stage of his philosophical development. Admittedly, the fourth axiom of the demonstration of God’s existence, premised to the dissertation, states that ‘each

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body whatsoever has infinitely many parts; or, as is commonly said, the continuum is infinitely divisible’ (A VI/1 169, L 74) – an assumption that is evidently incompatible with the existence of indivisible atoms.10 But, contrary to what Richard Arthur has asserted, there is no ‘real enigma’ here.11 At most it could be said of Leibniz that at that early stage he had not formed any coherent idea about the way atomism connects to the problem of the continuum, as in fact he would later concede himself.12 At any rate, my reasons for maintaining that in this period Leibniz was strongly attracted to atomism are also grounded in another Leibnizian writing of that period, to which it is now time to turn.

LEIBNIZ ON ‘SNOW IS BLACK’ This is a short text that Leibniz wrote in 1666 at the request of his teacher Jacob Thomasius. In this text – which in the Academy Edition appears under the title A Conjecture on Why It Seems That Anaxagoras Could Have Said That Snow Is Black, for the Requester Jacob Thomasius in a Sheet Sent on 16 February 1666 – Leibniz endeavours to provide an explanation of Anaxagoras’s celebrated argument that ‘snow is frozen water; water is black; therefore, snow is black’, from Sextus Empiricus’s Pyrrhoniae Hypotyposes, I, 33 (Diels A97). Since this text has generally gone unnoticed,13 it deserves to be quoted in full:14 Hypothesis



I. All colour is an impression on the sensory, not a certain quality [inhering] in things, but an extrinsic denomination, or, as Thomas Hobbes calls it, a phantasm. II. Therefore, for us unperceiving there is no colour.

III. Blackness is not so much a colour as the privation of colour, namely, we say that we see black when we see nothing.



IV. By hypothesis II together with III, all opaque things in themselves are black. Therefore, so is snow. Anaxagoras, however, in order to make his paradox more startling, assumed in particular what is considered the whitest. V. Colour is nothing other than an impression in the eye, which comes about from luminous atoms impinging from a luminous body on an opaque body, and thence reflected to the eye.

VI. The optical principles are three: fire, whose atoms are pyramidal; water, which, dilated, becomes air, whose [atoms are] spherical; and earth, whose [atoms are] cubical. VII. Fire is the principle of light, water of blackness, earth of colour. In fact, pyramidal atoms are the most subtle – they have the power to pierce, etc. – which are characteristics of fire. Indeed, fire and light are materially the same thing. Cubical atoms can be so joined to one another that no vacuum is left between them. They are therefore the cause for why atoms of fire

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are reflected; that is, by hypothesis V, they are the cause of colour. But between spherical atoms there is a great deal of vacuum; they are therefore the cause of no reflection (indeed where nothing obstructs, [pyramidal atoms] penetrate rather than being reflected), or of no colour, that is, by hypothesis III, of blackness. VIII. Whatever is such when rare, it is more such when condensed. Because force, when combined, is stronger. IX. Snow is condensed water.

X. By hypothesis VII, together with IX, and VIII, snow should therefore also appear as black as possible. Q.E.D. Therefore, this argument is just like Zeno’s argument against movement,15 whether Anaxagoras meant to convince a boastful sophist, parade his ingenuity in proving and defending anything whatsoever or support the sceptics by demonstrating the separation between senses and reason, so that one or the other must fail. On the other hand, if he said that snow looked black to him as well, it seems that he said so in jest, because he knew that no one could refute this paradox.

What emerges from this text, even at first glance, is that here Leibniz is decidedly moving in the stream of the just-revived mechanist/atomist tradition. Falling squarely into this tradition is not only Leibniz’s explanation of colour as a sensible or ‘secondary’ quality – or, as he puts it, an ‘extrinsic denomination’, using the Scholastic term for something that does not intrinsically belong to a thing in an eclectic juxtaposition to Hobbes’s term ‘phantasm’16 – but also his explanation of colour in terms of the mechanical qualities of light-emitting and light-reflecting bodies.17 Hypothesis VI, however, introduces a touch of Leibnizian originality into the picture. For in pursuing the reduction of colours to differences in the shape of atoms, Leibniz draws on Plato. In fact, the pyramidal and cubical shape of the atoms of fire and earth, respectively, is indebted to Plato’s account of the four kinds of geometric figures associated with the corpuscular texture of the four basic elements in Timaeus, 56a–c, a critical exposition of which was available to Leibniz in Magnen’s treatise. As Magnen points out, Plato assigns the cubical shape to earth because of its immobility and the pyramidal shape to fire because of its mobile and penetrating nature (M 134–5). Leibniz clearly follows Plato on both points, rejecting the attribution of the spherical shape to the atoms of fire, a view that Magnen and Gassendi ascribe to Democritus.18 Instead, in keeping with the Democritean tradition, Leibniz assigns the spherical shape to the atoms of water. This enables him to contrive a mechanistic solution to the paradox by explaining blackness as a phenomenon owed to large interatomic voids causing no reflection. Of course, those who support the thesis of a Leibniz who never embraced atomism object that what we are looking at is no more than the exercitation of a young scholar. This is precisely the path that Mercer takes in order to minimize the significance of Leibniz’s endorsement of atomism in this writing: ‘There would be reason to take this position as somehow representative’, writes Mercer (2001, 40), ‘if Leibniz continued to make important use of the same principles. He does not.’ But, as we

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shall see, this argument can hold up only on condition of ignoring – as Mercer does – that Leibniz in fact ‘continued to make important use of ’ atomist principles at least until 1669.19

FROM COLOUR TO GOD As was just being discussed, evidence of such an ‘important use’ of atomist principles can be found in various writings of 1667–9. Thus, for example, in the Nova methodus discendae docendaeque jurisprudentiam, Leibniz’s first jurisprudential treatise written and published in 1667, we find the following definition of body: Whatever has one more sensible quality besides extension and number is called body. Whatever does not have this added quality is called vacuum. Here physics arises. (A VI/1 287, L 90) In defining body as something endowed with a physical quality in addition to mere mathematical extension, Leibniz is clearly reprising the atomists’ ontological dichotomy between corporeal extension – constituted by solid, impenetrable bodies, namely atoms – and incorporeal extension, meaning the void. An even clearer statement of an atomist principle can be found in a letter to Thomasius dated 26 September/6 October 1688 in which Leibniz defines void as unbodied space (‘Once the body is eliminated, there remain space and its dimension, which, if no other body succeeds, is called vacuum’, A II/1 11). This elaboration of atomist themes culminates in Leibniz’s redefinition of Aristotle’s prime matter as extension plus antitypy or impenetrability in his famous letter to Thomasius from 20–30 April 1669,20 in which he traces the first outline of his physical theory in an attempt to reconcile Aristotle with mechanism: Space is a primary extended being or mathematical body, which contains nothing but three dimensions and is the universal locus of all things. Matter is a secondary extended being, or that which has, in addition to extension or mathematical body, also a physical body, that is, resistance, antitypy, thickness, the property of filling space, and impenetrability. (A II/1 21–2, L 100) As the Italian scholar Gianfranco Mormino has argued on a specific textual basis, this definition is strictly indebted to Gassendi, and through him, to the atomist (and in general anti-Aristotelian) tradition; the very term antitypy – denoting the mutual resistance of atoms – refers to the property this tradition identified with the essence of bodies.21 Still, Leibniz’s April 1669 letter to Thomasius is a text whose complexity precludes a straightforward atomist reading. This, however, does not imply that Leibniz had rejected atomism, for it would otherwise prove incomprehensible that in closing the letter he should have referred to a short theological essay of his by the title Confessio naturae contra atheistas, written in 1668 and published in the same year, 1669,22 which more than any other text from this period discusses his reasons for embracing atomism. As mentioned, it was in 1668–9 that Leibniz first outlined his physical theory – a period culminating precisely in his April 1669 letter to Thomasius. The Confessio naturae springs from Leibniz’s concomitant interest

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in championing Christian orthodoxy, and it marks his first, and perhaps boldest, attempt to resort to physics as a ground for demonstrating God’s existence. He begins by acknowledging his mechanistic conviction that natural phenomena should be explained in terms of the primary qualities of bodies: At the beginning I readily admitted that we must agree with those contemporary philosophers who have revived Democritus and Epicurus and whom Robert Boyle aptly calls corpuscular philosophers, such as Galileo, Bacon, Gassendi, Descartes, Hobbes and Digby, that in explaining corporeal phenomena, we must not unnecessarily resort to God or to any incorporeal thing, form, or quality [. . .] but that so far as can be done, everything should be derived from the nature of body and its primary qualities – magnitude, figure, and motion. (A VI/1 489–90, L 110) ‘But’, he hastens to add, what if I should demonstrate that the origin of these very primary qualities themselves cannot be found in the essence of body? Then, indeed, I hope, these naturalists will admit that body is not self-sufficient and cannot subsist without an incorporeal principle. I will prove this without obscurity or detours. (A VI/1 490, L 110)23 In the ensuing demonstration, Leibniz, having first shown ‘that there can be no determinate figure and magnitude, or any motion whatever, in bodies left to themselves’ (A VI/1 491, L 111), goes on to argue that cohesion – the property by which ‘bodies or their parts cohere with each other’ causing such tactile or secondary qualities as ‘solidity, fluidity, hardness and softness [. . .] tenacity and fragility, etc.’ – cannot be explained in terms of resistance and motion, as is suggested by a simple experiment: If I push part of a paper, the part which is pushed gives way: therefore no reaction or motion of resistance can be assumed. But not only does it give way; it also carries with it the remaining parts which adhere to it. (A VI/1 491, L 112) One way to explain this phenomenon is to take the atomists’ point of view: It is indeed truly and with good reason that Democritus, Leucippus, Epicurus, and Lucretius of old, and their modern followers, Pierre Gassendi and JeanChrysostôme Magnen, asserted that the whole cause of cohesion in bodies may be explained naturally through the interweaving of certain shapes such as hooks, crooks, rings, projections, and in short, all the curves and twists of hard bodies inserted into each other. But these interlocking instruments themselves must be hard and tenacious in order to do their work of holding together the parts of bodies. (A VI/1 491–2, L 112) This immediately raises a question: ‘Whence this tenacity? Must we assume hooks on hooks to infinity?’ Given that this would bring us in an infinite regress, [t]here remains only one answer which those most subtle philosophers can make to such difficulties; they must assume certain indivisible corpuscles, which they

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call atoms, as the ultimate elements of bodies, which, by their varied shapes, variously combined, bring about the various qualities of sensible bodies. (A VI/1 492, L 112) On this account, however, the problem is only kicked farther down the field, for ‘no reason for cohesion and indivisibility appears within these ultimate corpuscles’. The ancient atomists’ solution, on which what holds atoms together and makes them indivisible is the absence of an internal void, is absurd and contrary to experience, as it would imply that all atoms should coalesce into one upon coming into contact, with the consequence that all bodies should coalesce in the same way, when colliding, because no void exists between them. Consequently, no possibility remains other than to appeal to God: Rightly therefore, in providing a reason for atoms (in reddenda atomorum ratione), we have at last to resort to God, who endows with firmness these ultimate elements of things. I marvel that neither Gassendi nor any other of these most acute philosophers of our century has noticed this excellent opportunity to demonstrate the divine existence. For through the ultimate analysis of bodies, it becomes clear that nature cannot dispense with the help of God. (A VI/1 492, L 112) To be sure, what we marvel at is that Leibniz ignored that the ‘most acute philosophers of our century’ had not in fact missed that ‘excellent opportunity’.24 But this need not concern us here. What does concern us is that, despite evidence to the contrary, the advocates of the anti-atomist thesis have insisted that the Confessio naturae reveals no Leibnizian engagement with atomism. Quite the opposite, it would testify to Leibniz’s firm rejection of it: ‘In the Confession of nature against atheists’, Mercer (2001, 295–6) writes, ‘Leibniz is adamant about his opposition to the atoms of Gassendi and others. [. . .] In one of his first letters to Oldenburg, Leibniz criticizes Gassendi’s account of the cohesion within material atoms.’25 But, once again, this can be maintained only at the cost of ignoring some crucial facts. True, the Confessio naturae contains Leibniz’s most forceful criticism of atomism, one he would return to and refine in his mature writings.26 But it is equally true that this criticism is placed in the context of what appears to be an unequivocal appreciation of atomism – ‘It is indeed truly and with good reason that Democritus, Leucippus, Epicurus [. . .]’, and so on – and, what is more, atomism seemed at the time to fit perfectly into Leibniz’s theological narrative of an omniscient and omnipotent God, ruler and harmonizer of the universe, as the conclusion of the Confessio naturae clearly implies: Since we have demonstrated that bodies cannot have any determinate figure, quantity, or motion, without assuming an incorporeal being, it readily becomes apparent that this incorporeal being is one for all [bodies] because of the harmony of things among themselves, especially since bodies are not moved individually by their incorporeal being but by each other. But no reason can be given why that incorporeal being chooses this magnitude, figure, and motion rather than another, unless he is intelligent and wise with regard to the beauty of things and powerful with regard to their obedience to his commands. Therefore such an

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incorporeal being will be a mind ruling the whole world, that is, God. (A VI/1 492, L 112) On the other hand, it must be acknowledged that Leibniz’s embrace of atomism in the Confessio naturae was destined to be soon thereafter eclipsed in light of both his apologetic worries and the continuing development of his physical views. Indeed, only a few months after his April 1669 letter to Thomasius, Leibniz would dramatically change his mind about atomism and its theological significance. As we shall see in the next section, it is precisely this change which is reflected in the letter to Oldenburg mentioned by Mercer.

DISENCHANTMENT WITH ATOMISM While sojourning in Bad Schwalbach in the summer of 1669, Leibniz chanced upon Huygens-Wren’s just-published rules of impact.27 ‘When first I saw these’, he wrote to the secretary of the Royal Society Henry Oldenburg in September 1670, ‘I said that those phenomena seemed to me true, but that prime and abstract reasons (rationes) of motion of a very different character were necessary. [. . .] Accordingly I at once took up my pen and having the urge to write I began straightaway to indite my thoughts on this subject.’28 The result of this zestful writing effort was a short essay that Leibniz entitled De rationibus motus, dated August–September 1669.29 In this essay Leibniz staked out a position contrary to the one he himself had argued in the Confessio naturae, in that he now rejected the view that cohesion is a feature of corporeal matter for which there exists no purely mechanical explanation in terms of resistance and motion. The key to understanding such turnabout lies in Leibniz’s opposition to Huygens’s first rule of impact, stating that if a hard body at rest is struck by a hard body of equal mass, the latter will come to a stop and be at rest, and the former will acquire the speed of the striking body. But, as Leibniz objects, rest is the cause of nothing, that is to say, a body at rest imparts neither motion to another body nor rest, direction, or speed. [. . .] It is thus necessary that the body at rest resists; resistance is action; all action of bodies is motion. Therefore, a body at rest, if it resists motion, moves. (A VI/2 161) From this it follows that the sensible qualities resulting from cohesion are to be accounted for in terms of resistance and motion: Indeed, that hardness which is perceived through the senses is nothing but resistance; all resistance is motion; therefore, only those things are hard whose surface parts are moved by a motion so strong that they repel the impetus of an external impinging body. (A VI/2 161) Having reached this conclusion, Leibniz is ready to make explicit the anti-atomist implications of this explanation: This proposition speaks contrary to the followers of Democritus and Epicurus, eminent among them being Gassendi, by whom it is supposed that there exist

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certain corpuscles (which they call atoms) so solid that even when they are at rest, they are not dissolved by the impact of any other body whatsoever. (A VI/2 161) The consequences of this conversion to anti-atomism are clearly stated in Leibniz’s September 1670 letter to Oldenburg. The starting point, for Leibniz, is still the vexed question of cohesion: In providing a reason for the appearances from very precise notions of body, magnitude, figure, and mobility nothing gives me more trouble than the cohesion of parts in a whole, or of several wholes between themselves: the various species of cohesion are hardness, softness, tenacity, flexibility, fragility, friability, and many other of those tactile qualities commonly called secondary. (A II/1 63, H&H 167) Leibniz concedes he is at a loss to come up with an explanation for those phenomena. Even Descartes failed to provide one, and the attempt to explain cohesion as a result of density does not succeed, since a mass at rest is incapable of resisting penetration even if it has the greatest density. Gassendi, Leibniz acknowledges, ‘seems to have seen this difficulty; accordingly, in order to join his atoms together, he has contrived hooks and barbs, but’, he goes on to reason, when the firmness and hardness of the atoms and of the hooks themselves is to be explained, he is forced to take refuge in the will of the Creator. And so, in order for the atoms to be held together, one must resort to a perpetual miracle. (A II/1 63, H&H 167) What Leibniz puts forward here is an argument he will use again and again over the years against any variety of philosophical adversaries, from Descartes and Malebranche to Newton.30 The argument consists in the charge of attributing to a thing features which, although regularly occurring, cannot be explained from the nature of the thing itself, and whose existence would thus require a perpetual miracle. As Leibniz will explain in his much later Demonstratio contra atomos: If anyone at least think that atoms can be made by a decree of God, we admit to him that God can bring them about, but a perpetual miracle would be needed so that they resist separation, since a principle of perfect firmness cannot be understood in body itself. (G VII 288, S 123) Let us now have a second look at the previously quoted passage from the Confessio naturae: Rightly therefore, in providing a reason for atoms, we must resort at last to God, who endows with firmness these ultimate elements of things. I marvel that neither Gassendi nor any other of these most acute philosophers of our century has noticed this excellent opportunity to demonstrate the divine existence. This unexpected reframing of an ‘excellent opportunity to demonstrate the divine existence’ into a specious ‘resort to a perpetual miracle’ raises a number of questions, and it would require far more attention than Leibnizian scholars have devoted to it.

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It is completely overlooked by Christia Mercer, for example, when, in her masterly book on the origin and development of Leibniz’s metaphysics, she accounts for the crucial change which took place in Leibniz’s views in late 1669.31 This change takes us back to Leibniz’s April 1669 letter to Thomasius.

RETROSPECTIVE CHANGES As we saw, in the letter Leibniz writes to his mentor he launches into an ambitious attempt to reconcile Aristotle with mechanism. To this end he defines the basic Aristotelian notions of matter, form and change in terms of magnitude, figure and motion. The point that concerns us here is the consequence he draws from his definition of corporeal matter as extension and antitypy,32 namely that nothing should be supposed in bodies which does not follow from the definition of extension and antitypy. But from them only magnitude, figure, situation, number, mobility, etc., follow. Motion itself does not follow. Hence, properly speaking, there is no motion in bodies as a real entity [ens reale] in them. (A II/1 23, L 102) Though highly problematic, Leibniz’s argument is quite simple: the real features of body should be derived from its nature or essence as captured by its definition; motion cannot be derived from the body defined in terms of extension and antitypy; therefore, motion is not a real feature of bodies, or, as Leibniz says, it is not an ens reale in them. This would seem to cause trouble for Leibniz’s mechanistic/Aristotelian physics, for the unreality of motion in bodies threatens the very reality of moving bodies. But this is not Leibniz’s conclusion. On the contrary, he confidently proclaims to have instead demonstrated that whatever moves is continuously created and that bodies are something at any instant in assignable motion, but are nothing at any intermediate time between instants of motions, a view that has never been heard of until now but which is clearly necessary and will silence the atheists. (A II/1 23–4, L 102) Here Leibniz appears to ignore that, far from having ‘never been heard of until now’, the thesis of God’s continuous creation had already been advanced not only by Descartes but also by his own teacher in Jena, Erhard Weigel.33 Even more importantly, however, he is invoking the same ‘perpetual miracle’ thesis for which he would shortly thereafter fault Gassendi. We are now in a position to see another striking effect of the change in Leibniz’s thought which took place in the months immediately following the letter he wrote to Thomasius. In revising the letter for publication,34 Leibniz struck out the entire passage that runs from ‘Hence, properly speaking’ to ‘silence the atheists’.35 ‘By deleting the passage’, Mercer (2001, 143) sums up, ‘Leibniz retracts two closely related assertions: that God continuously creates bodies and that motion does not belong to bodies as a real thing.’ And, it might be added, he deletes as well an important chapter of his previous intellectual story.36

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CONCLUSION Many years later, returning to the deleted passage from his letter to Thomasius , Leibniz will express a less severe judgement about his youthful work:37 ‘I’ve thought a lot about this argument [i.e. the cohesion of bodies]’, he recalls. In fact, it deals with the innermost corporeal nature. In my youth I believed that this could not be explained except through God’s will, and a little dissertation I wrote on this subject was accepted by Spener and included by a certain Spitzel in his letter to Reiser against atheism. Now, however, I am rather inclined to believe that it can be explained through natural laws, even though these same laws originate from God and do not arise solely from mathematical principles.38 This late account reveals Leibniz’s main reason for accepting as well as rejecting atomism. These reasons can be seen to have been at bottom the same: Leibniz embraced atomism in view of the connection it bears to voluntarism and dismissed it in view of the same connection. Leibniz’s rejection of atomism enabled him to address the tension – already present in the Confessio naturae – between voluntarism and his ‘fundamental commitment to the harmony and intelligibility of the world’ (Mercer 2001, 90), resolving that tension in favour of the latter. At the heart of the matter lies Leibniz’s keen awareness of the problem at the very foundation of the atomic hypothesis. The problem, as Leibniz clearly saw, was that cohesiveness could not be considered a primary property that atoms are endowed with by divine intervention, since, in Wilson’s (1982, 192) words, that would have us ‘allow that in collision God brings it about that something perfectly incomprehensible happens’ (cf. Pyle 2018, 438, 440). From here springs another (even more) important chapter in Leibniz’s intellectual story. But, of course, that chapter falls beyond the scope of this chapter.

NOTES 1. Leibniz to Thomas Burnett, 18/28 May 1697 (A I/14 224). The abbreviations used in this chapter are listed in the reference section. 2. Leibniz to Thomas Burnett, 18/28 May 1697 (A I/14 224). 3. Kabitz (1909, 50–1). 4. Suffice it to say that whereas at the beginning of the twentieth century scholars were happy to acknowledge an atomist phase in Leibniz’s early thinking (see esp. Cassirer 1902, 438–41; Hannequin 1908, 24–40; Kabitz 1909, 51–5), subsequent scholars have revealed themselves to be less and less inclined to acknowledge such a phase. Exceptions are Čapek (1966), Moll (1982), Wilson (1982), and Mormino (1999). More recently, Arthur (2003), while arguing for a youthful commitment to atomism by Leibniz, has substantially denied that the atomism that Leibniz was committed to was the material atomism of the Democritean–Epicurean tradition. 5. On Generation and Corruption, bk. 1, chap. 2, 315b6–15; Metaphysics, bk. 1, chap. 4, 985b14–19. Moll (1982, 91) lays out the textual coincidence between Leibniz’s references and two corresponding references in Gassendi’s Syntagma philosophicum (GO 366).

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6. Bk. 2, vv. 1013–22. Moll (1982, 92) identifies the source of this reference in GO 366–7. 7. Bk. 3, chap. 17. According to Moll (1982, 92), this reference, too, has a Gassendian source in GO 366 (incidentally, Leibniz erroneously refers to chap. 49, whereas Gassendi cites correctly). 8. Leibniz refers in particular to GA I 227, where Gassendi elaborates on the literarum similitudo. 9. Lebniz’s full reference is ‘Disp. 2 de Atomis c. 4 prop. 32 p. 269’, but I suspect that here Leibniz intended to refer to p. 161, where Magnen alludes to the alphabetical metaphor and quotes part of the same verses from the poem by Lucretius that Leibniz also quotes (see note 6 above). Indeed, at M 269 Magnen treats of the nature of colours, which could have attracted Leibniz’s interest because of the contemporary writing of the 16 February 1666 note to Thomasius (see the next section). 10. That the actually infinite division of matter rules out the existence of atoms is something Leibniz would first acknowledge in his 1670–71 note De materia prima (A VI/2 280). 11. ‘The real enigma [. . .] is that this thesis, that the continuum is not just potentially but actually divided into an infinity of parts, is one he seems to have held from as early as 1666, and consistently from then on, even while proposing atoms’ (Arthur 2003, 186). Arthur’s solution to this ‘enigma’ is that Leibniz never subscribed to the material atoms of classical atomists. 12. ‘But not being yet well versed in geometry, I persuaded myself that the continuum is composed of points’ (Phoranomus seu de potentia et legibus naturæ, 1686, P 786). See Cassirer (1902, 439–41). 13. To my knowledge, only Hannequin (1908, 30–1), Kabitz (1909, 51–2), Beeley (1996, 190–3), and De Olaso (1997, 101–2) deal with this writing (the latter is concerned with the way Leibniz’s thought relates to scepticism, not to atomism). Extended mentions of it can be found in Čapek (1966, 250), Mormino (1999, 262), and Mercer (2001, 40). Moll (1982, 46) relies on Kabiz. Wilson (1982, 178) alludes to it. Even in a paper such as Puryear (2013), frontally addressing Leibniz’s view of colour, there is no mention of it. 14. A translation of this text by J. K. McDonough appears in Donald Rutherford’s site G. W. Leibniz: Texts and Translations. The translation offered here is a substantially revised version of it based on the original Latin text (A II/1 4–5). 15. On the alleged analogy between Anaxagoras’s and Zeno’s arguments, see De Olaso (1997, 102). 16. Another clue to Leibniz’s eclecticism lies in his combining the Democritean ‘subjectivist’ account of colour with the Epicurean–Lucretian ‘objectivist’ account, the former stating that the existence of colours depends on the existence of perceivers (see hypothesis II), the latter stating that the existence of colours depends on the existence of light (Rickless 2018, 71). 17. Versions of this explanation can be found in Hobbes, Descartes, Gassendi and Boyle and include both subjectivist accounts (e.g. Hobbes’s) and objectivist ones (e.g. Gassendi’s). See Rickless (2018, 71–3).

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18. M 138–9. According to Magnen, however, the atoms of fire are originally spherical, but they can accidentally (per accidens) assume any shape whatsoever (M 154). In the Syntagma philosophicum, by contrast, Gassendi rejects the attribution of the pyramidal shape to the atoms of fire, emphatically arguing for their spherical shape (GO 394–5, 426–7). 19. Three years later, to be sure, writing to Thomasius, Leibniz did acknowledge a mistake in the solution that in his 1666 note he offered to Anaxagoras’s paradox (‘it is false that snow is condensed water’, A II/1 18, L 96). But the alternative solution advanced in the letter to Thomasius is fully compatible with atomist principles. 20. Leibniz to Jacob Thomasius, 20–30 April 1669 (A II/1 12–24, L 93–103). A revised version of this letter was published the following year by Leibniz as an appendix to the preface to his edition of Mario Nizolio’s Antibarbarus seu de veris principiis et vera ratione philosophandi contra pseudophilosophos. It appeared under the title ‘Letter to a Man of the Most Refined Learning Concerning the Reconciliability of Aristotle and the Moderns’ (A VI/2, 433–44). I shall return to this version in section ‘Retrospective changes’ below. 21. Mormino (1999, 260–3, 269–70). 22. Here is Leibniz’s account of the circumstances in which this tract was written and subsequently published: ‘In a period of leisure, but working in the confusion of an inn, I once wrote about two sheets in which I tried to demonstrate, more accurately than usual, the immortality of the soul and the existence of God. I sent these sheets to a friend who passed them on to the reverend Mr. Spener, a pastor in Frankfurt, with their authorship properly concealed. Spener sent them to Spizel [another Lutheran clergyman], and Spizel recently attached them to the end of his letter to Anton Reiser on the eradication of atheism, with the title Confessio naturae contra atheists’ (A II/1 24, L 102). 23. As Christia Mercer (2001, 71–2) remarks, ‘Leibniz is hopelessly confused about what the mechanists’ position actually is. While they do think that all corporeal features are explicable in terms of the fundamental features of body [. . .] without recourse to anything incorporeal, they do not believe that the fundamental features are themselves wholly derivable from the nature of body (res extensa) taken by itself. [. . .] Descartes and Gassendi [for example] are perfectly happy to let God be the cause of the motion of bodies and see no problem in the fact that the full account of motion does not rest in the nature of body.’ 24. See the previous note 23. 25. Despite these assertions, in her detailed account of the Confessio naturae Mercer completely ignores the atomist argument (see Mercer 2001, 70–82, in particular 79, 81). Arthur (2003, 198) quotes it just to remark, with Mercer, that ‘by 1671 a different conception has emerged, in which every body contains its own principle of activity’. 26. He did so, for example, in the Demonstratio contra atomos sumta ex atomorum contactu, 1690 (G VII 284–8, S 119–23), and especially in the Animadversiones in partem generalem Principiorum Cartesianorum, 1692 (GP IV 386–7, L 405–7). 27. Philosophical Transactions, 43 (11 January 1668–9), 867–8 and 46 (12 April 1669), 925–8. 28. Leibniz to Oldenburg, 28 September 1670 (A II/1 62, H&H 166).

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29. The story of this essay, bound to remain unknown, is briefly told by Leibniz in the same letter to Oldenburg (A II/1, 62–3, H&H 166). 30. See Piro (2017). 31. See note 25 above. 32. See section ‘From colour to God’ above. 33. Cassirer (1902, 443). 34. See note 20 above. 35. Compare A II/1 23–4 with A VI/2 443. 36. In the published version Leibniz essentially disowns the Confessio naturae: ‘I discussed this topic in a little extemporaneous paper’, he writes, ‘which the Illustrious Teoph. Spizel, having received it in in his hands, although it did not deserve to, stitched to his recently published letter to the Illustrious Reiser on the eradication of atheism like a tattered cloth to a purple robe, under the title Confessio naturae contra atheistas.’ 37. See the previous note 36. 38. Leibniz to Gerhard Maier, 17/27 July 1696 (A I/12 736–7).

REFERENCES a) Primary Sources A = G. W. Leibniz, Sämtliche Schriften und Briefe. Ed. German (formerly Prussian) Academy of Sciences at Berlin, Darmstadt: Reichl, 1923– (reprint Hildesheim: Olms, 1972–), cited by series, volume and page number. AG = G. W. Leibniz, Philosophical Essays. Ed. and trans. R. Ariew and D. Garber, Indianapolis: Hackett, 1989. APS = Leibniz: Philosophical Puzzles in the Law. Ed. and trans. A. Artosi, B. Pieri, G. Sartor, Dordrecht: Springer, 2013. G = Die philosophischen Schriften von Gottfried Wilhelm Leibniz. Ed. C. I Gerhardt, 7 vols, Berlin: Weidmann, 1875–1890 (reprint Hildesheim: Olms, 1965), cited by volume and page number. GA = P. Gassendi, Animadversiones in decimum librum Diogenis Laertii, qui est de vita, moribus, placitisque Epicuri, Tomi I–III, Lugduni, Apud Guillelmum Barbier, 1649 (reprint New York: Garland, 1987), cited by volume and page number. GO = P. Gassendi, Opera omnia in sex tomos divisa, Tomus Primus quo continentur Syntagmatis Philosophici Pars Prima, sive Logica itemque Partis Secundae, seu Physicae Sectiones duae priores, Lugduni, Sumptibus Laurentii Anisson & Ioan. Bapt. Devenet, 1658 (reprint Stuttgart-Bad Cannstatt: Frommann-Holzboog, 1964). H&H = Correspondence of Henry Oldenburg, Volume VII 1670–1671. Ed. and trans. A. Rupert Hall and M. Boas Hall, Madison: The University of Wisconsin Press, 1970. L = G. W. Leibniz, Philosophical Papers and Letters. Ed. and trans. L. E. Loemker, 2nd edn, Dordrecht: Reidel, 1976. M = J.-C. Magnen, Democritus reviviscens, sive de atomis, Papiae, Apud Io. Andream Magrium, 1646.

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P = G. W. Leibniz, Dialoghi filosofici e scientifici. Ed. and trans. F. Piro, Milan: Bompiani, 2007. S = L. Strickland, The Shorter Leibniz Texts, London: Continuum, 2006.

b) Secondary Sources Arthur, R. (2003), ‘The Enigma of Leibniz’s atomism’, in D. Garber and S. Nadler (eds), Oxford Studies in Early Modern Philosophy, I, 183–227, Oxford: Oxford University Press. Beeley, P. (1996), Kontinuität und Mechanismus, Stuttgart: Franz Steiner. Čapek, M. (1966), ‘Leibniz’s thought prior to the year 1670: From atomism to a Geometrical Kinetism’, Revue Internationale de Philosophie 20: 249–56. Cassirer, E. (1902), Leibniz System in seinen wissenshaftlichen Grundlagen. Hamburg: Meiner, 1998. De Olaso, E. (1997), ‘Leibniz and scepticism’, in R. H. Popkin et al. (eds), Scepticism in the Enlightenment, 99–130, Dordrecht: Kluwer. Hannequin, A. (1908), ‘La première philosophie de Leibniz’, in Études d’histoire de sciences et d’histoire de la philosophie, Tome II, 24–224, Paris: Alcan. Kabitz, W. (1909), Die Philosophie des jungen Leibniz, Heidelberg: C. Winter (reprint Hildesheim: Olms, 1974). Mercer, C. (2001), Leibniz’s Metaphysics: Its Origins and Development, Cambridge: Cambridge University Press. Moll, K. (1982), Der junge Leibniz II, Stuttgart-Bad Cannstatt: Frommann-Holzboog. Mormino, G. (1999), ‘Atomismo e volontà divina nei primi scritti leibniziani (1663– 1671)’, Rivista di storia della filosofia 2: 255–81. Piro, F. (2017), ‘L’argomento del “miracolo perpetuo” e i suoi sottintesi teologici: Ancora sui rapporti Leibniz-Malebranches’, Laboratorio dell’ISPF XIV, no. 5: 1–19. Puryear, S. (2013), ‘Leibniz on the metaphysics of color’, Philosophy and Phenomenological Research 86: 319–34. Pyle, A. (2018), ‘The theory of matter’, in D. Kaufman (ed.), The Routledge Companion to Seventeenth Century Philosophy, 410–46, London and New York: Routledge. Rickless, S. C. (2018), ‘Qualities’, in D. Kaufman (ed.), The Routledge Companion to Seventeenth Century Philosophy, 60–86. London and New York: Routledge. Wilson, C. (1982), ‘Leibniz and atomism’, Studies in the History and Philosophy of Science 13: 175–99.

PART III

Atomism in contemporary thought

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Philosophy

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CHAPTER 16

Logical atomism and Wittgenstein ANNALISA COLIVA

Notoriously, Wittgenstein endorsed a form of logical atomism in the Tractatus LogicoPhilosophicus. Key to his defence of atomism was the idea that simple objects (T2.02) must exist in order for names to have meaning (T3.202, 3.21-3.22, 3.203, 4.0311). Names mean objects, which, in their turn, are related to one another to form states of affairs (T2.01, 2.0121, 2.0272-2.032). Propositions,1 in the Tractatus, are ultimately juxtapositions, or concatenations of names (T3.1432, 4.22), which manage to depict states of affairs (T4.01, 4.021). They do so thanks to the fact that their component parts stand for objects (T3.22, 4.0311) and that they share with reality the same logical form (T4.12-4.1211). That is to say, the combinatorial possibilities of objects among themselves, and of their names, match one another (T4.04). According to the Tractatus, the existence of simple objects is thus necessary to guarantee the possibility of propositions – that is, the possibility of there being pictures of states of affairs (T2.1-2.225) – whose truth or falsity can only be determined by comparing them with reality (T4.05-4.06). Hence, the existence of simple objects is required to guarantee the possibility of sense (Sinn) within the framework of the picture theory that Wittgenstein endorsed in the Tractatus. As he put it: Objects make up the substance of the world. That is why they cannot be composite. If the world had no substance, then whether a proposition had sense would depend on whether another proposition was true. In that case we could not sketch any picture of the world (true or false). (T 2.0212.0212, emphasis added) That is, for Wittgenstein, logical atomism secured the possibility of their being propositions capable of representing reality, which could be made true or false by the latter. The opposite picture – that is, that propositions were not anchored to reality directly, but only to one another – was simply rejected as failing at objectivity and therefore at representationality. While the existence of simple elements that could guarantee names’ meaning, and ultimately the very possibility of sense, was clearly required by the semantics of the Tractatus, their nature was left in the dark. That spurred a heated discussion

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among interpreters concerning their identity.2 Some scholars defended the idea that they were sense-data,3 or objects of direct experience (thus likening Wittgenstein’s and Russell’s forms of atomism),4 while others defended the view that they were ultimately physical entities – if not mid-size physical objects, at least the building blocks of reality identified by physics.5 Still other scholars opted for the idea that the notion of object in the Tractatus is actually formal and hence that it escaped any ontological determination.6 Reaching no consensus, the controversy was later abandoned. Thus, scholars at least implicitly agreed with David Pears’ injunction to follow the ‘golden rule to treat as peripheral the questions that [Wittgenstein] himself treats as peripheral’.7 For all Wittgenstein cared about, in the Tractatus, was the overall function simple objects played with respect to the possibility of meaning, not the ontological issue of determining their identity.8 In the Philosophical Investigations Wittgenstein revisits the issue. This time, however, he is critical of atomism and of the key thought behind it in the Tractatus. In fact, he thinks that his early atomism was the outcome of a fatal mistake concerning the meaning of names. Namely, the identification of a name’s meaning with its bearer (T3.203), such that if the bearer did not exist, then the name would lack a meaning (and in fact would no longer be a symbol in the language but merely a sign). Wittgenstein’s critique starts at Philosophical Investigations §38 and occupies about forty passages.9 He starts by actually criticizing Russell’s idea that, logically speaking, the only real proper name is ‘this’.10 Wittgenstein asks why we are tempted to think that only ‘this’ could be a genuine proper name, when in fact ‘we call very different things “names”; the word “name” is used to characterize many different kinds of use of a word, related to one another in many different ways; – but the kind of use that “this” has is not among them’ (PI §38). Wittgenstein then continues by analysing the differences between ‘this’ and a name. They both occupy the same position in a sentence. However, only names are defined by means of the demonstrative expression ‘That is N’ or ‘That is called “N”’. Still, we do not give definitions such as ‘That is called “this”’ or ‘This is called “this”’. More importantly, we are tempted by the idea that ‘a name ought really to signify a simple’ (PI §39, cf. also §55). If it did not signify a simple, it could lose its meaning, if the complex object that constitutes its meaning was destroyed. Yet, in that case, the sentence ‘Excalibur has a sharp blade’ ‘would contain a word that would have no meaning, and hence the sentence would be nonsense’ (ibid. Cf. PI §55). Since this is not the case, we are tempted to think that ‘there must always be something corresponding to the words of which it [i.e. the sentence] consists’ (ibid.). Thus, Wittgenstein concludes that when the sense of the word ‘Excalibur’ is analysed, the word must actually disappear and its place must be taken by words that name simples. This, in fact, is the conception of names that, as we briefly saw, Wittgenstein himself had proposed in the Tractatus Logico-Philosophicus. Russell too, in The Philosophy of Logical Atomism, advanced a similar view and ended up proposing the idea that ordinary proper names do not refer to, but actually denote an object by means of some property that only that object should have.11

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But why should one think that a word has no meaning if nothing corresponds to it? According to Wittgenstein, this idea depends on the confusion between the meaning of a name with the bearer of the name (PI §40). To expose the confusion, he famously quips ‘when Mr. N. N. dies one says that the bearer of the name dies, not that the meaning dies. And it would be nonsensical to say that, for if the name ceases to have meaning it would make no sense to say Mr. N. N. is dead’ (ibid. Cf. PI §55). Thus, the confusion between the meaning of a name and its bearer is at the origins of the mistake of thinking that only ‘this’ would be a genuine proper name. Now, clearly, the early Wittgenstein or Russell would have been unmoved by such a remark. For they would have acknowledged that in ordinary language we have proper names that continue to have a meaning even if their bearers do not exist, or no longer do so. Indeed, as we just saw, Wittgenstein thought that ordinary propositions should be completely analysed in their constituent elements and only then would we encounter genuine names; and Russell too endorsed a similar idea, together with the further claim about the fact that ordinary proper names do not refer to but actually denote individuals through some of their characteristic properties. Thus, ‘N.N.’ would not have counted as a genuine or logically proper name for either of them. What did change in the meanwhile to make Wittgenstein think that the example of N.N. being deceased, in Philosophical Investigations §40, could actually be a criticism of his earlier views? It seems safe to say that his overall perspective on language had changed. According to his new perspective, it is indeed our ordinary use of words in our real language games that has pride of place and that should be investigated to clarify the nature of language. Furthermore, instead of pursuing a quest into the essence of the world and meaning, Wittgenstein is now interested in providing a ‘perspicuous representation’ (PI §122) of linguistic use capable of dispelling philosophical problems or, in fact, misconceptions. Key to this new perspective onto language is the idea that words can have a multiplicity of functions (PI §§11-15, 23-24), varying from one context of their use to the next. Similarly, the representational function is only one function among many that words, and indeed propositions, can have. Moreover, our concepts, including central philosophical ones such as the concepts of name, meaning, truth and proposition, work by family resemblance (PI §§65-78): they resist definitions in terms of necessary and sufficient conditions and do not pick out entities that share a common essence. It is only by keeping in mind this larger background that one can make sense of Wittgenstein’s remarks against logical atomism – let it be his own or Russell’s version of it. That is why, at the beginning, I have suggested that the criticism of logical atomism occupies about forty paragraphs of the Philosophical Investigations, and does not stop at §64, as is more commonly held among interpreters. For, as I read it, it comprises the sections on family resemblance and culminates in PI §79 with the discussion of the proper name ‘Moses’. Once it is clear that the meaning of a name is not its bearer but rather its use or function in the language game, as Wittgenstein famously states in Philosophical Investigations §43, then it may still be possible for the name to have meaning even if its bearer does not exist, or if it no longer exists. If it is possible for a name to retain

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meaning even if its bearer does not exist (any longer), then the temptation to think that only ‘this’ is, logically speaking, a genuine proper name vanishes. That the criticism is of Russell’s individuals and of the notion of object in the Tractatus Logico-Philosophicus becomes clear in Philosophical Investigations §46, where Wittgenstein discusses Socrates’ conception of the primary elements in the Theaetetus and connects his discussion to Russell and the Tractatus. As Wittgenstein explains, while rehearsing his Tractarian theses, primary elements are what everything else consists of. They cannot be defined because any definition would make them composite. If I define red as ‘a primary color with such-and-such a wave length’, I would be decomposing red into some more fundamental elements such as wavelengths and it would no longer be a simple element but a complex one. Hence, on pain of contradiction, primary elements cannot be defined, but only named. Sentences, and therefore descriptions, are simply connections of names, in their turn. The problem arises, however, of understanding what the simple constituent parts of which reality is composed are. This is a problem that, as we mentioned, the Tractatus Logico-Philosophicus had actually left open. For in it, Wittgenstein demonstrated that there must be simple objects for meaning to be possible at all (under the aegis of the picture theory of meaning), but he had not explicitly said what they are. Here he returns to the issue, but to show how in fact it cannot be resolved and therefore why the very philosophical question, whether this or that is a simple element of reality, is in fact nonsensical. First, he asks whether the simple constituent parts of a chair are the bits of wood of which it is made, or the molecules or the atoms, and then he argues that there is no answer to that question (cf. also PI §59). For, he claims, the answer to that question depends on the language game we are playing. Furthermore, Wittgenstein holds that there are many different ways in which we can describe an entity, such that it can turn out to be simple or complex depending on the description we give of it. Writes Wittgenstein: We use the word ‘composite’ (and therefore the word ‘simple’) in an enormous number of different and differently related ways. (Is the colour of a square on a chess board simple, or does it consist of pure white and pure yellow?) And is white simple or does it consist of the colours of the rainbow?12 Is this length of 2 cm simple, or does it consist of two parts each 1 cm long? But why not of one bit 3 cm long, and 1 bit 1 cm long measured in the opposite direction? To the philosophical question: ‘is the visual image of this tree composite and what are its component parts?’ the correct answer is: ‘That depends on what you understand by “composite”.’ (And that is of course not an answer but the rejection of the question). (Philosophical Investigations §47) Wittgenstein multiplies examples and presents a case in which there are primary colours arranged like in a chess board with names for each of them, such that, by juxtaposing one to the other, an instruction can be given to someone – for example, ‘Bring me 2 red squares, 1 black square, 3 green ones, and 2 white ones’. He then asks us to consider it as a series of simple objects with their names. Then

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he notices that ‘under other circumstances I should call a monochrome square “composite” consisting perhaps of two rectangles or of the elements colour and shape. But the concept of complexity might also be so extended that a smaller area was said to be “composed” of a greater area and another one subtracted from it’ (PI §48). The moral, once more, is that there is nothing which, in and of itself, is simple or complex. Rather, being simple and being complex are a function of the role these various elements play in our descriptions and language games. It should be noted, however, that this is no rejection of the very idea that we can carve up reality in simple and complex elements. Rather, it is a rejection of the idea that this distinction is metaphysically grounded – that it carves nature at its joints, as it were. On the contrary, it is a function of our (partly arbitrary) descriptions. Thus, the distinction can still be drawn, but the ‘metaphysical’ status of simples (and complexes) must be clear: they are not the ultimate elements of reality itself but the ultimate elements of our descriptions of reality, from which complex entities are said to be composed of, within our descriptions.13 In PI §60 Wittgenstein presents a different objection. Namely, that there is no need to analyse the sentence ‘My broom is in the corner’ to clarify or to understand what it means, as opposed to what he himself had maintained in the Tractatus. Indeed, substituting that sentence with ‘There is a broomstick and a brush in the corner’ (perhaps with the addition of a specification of their arrangement) could actually hinder our understanding. For, if we asked subjects whether this is what they meant or understood, they would probably say that they ‘had not thought specially of the broomstick or specially of the brush at all’ (ibid.). At any rate, the two sentences seem to belong to two different language games (PI §64), which may be related, and yet have different functions. Once again, the author of the Tractatus would have been unmoved by such a consideration. For, in that work, the complete analysis of a proposition was never thought to be directly accessible to consciousness. It was rather a requirement imposed, as we saw, in order to guarantee the possibility for our ordinary language could play a representational function. Yet, this overall perspective on language has changed and now Wittgenstein thinks that it is only by paying attention to how we actually use words in our everyday interactions that we can make sense of their meaning and of what it means to understand them. Furthermore, it is only by paying attention to their actual use in our everyday interactions that we can dispel misguided philosophical attempts at determining the nature of meaning and understanding, including his own endeavours in the Tractatus. Following the text once again, Wittgenstein then returns to the issue of whether the simple, according to the Tractatus, can be defined. He claims that it cannot, because definitions consist of parts and simples do not have any. Therefore, simples can only be named. In some limiting case, however, a complex that consists only of one element could be described by a name. Hence, whether a symbol is a name or a description also depends on the context of its use. Writes Wittgenstein: ‘This was what Frege meant too, when he said that a word had meaning only as part of the sentence’ (PI §49).14

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Nor can existence be predicated of simple elements because non-existence cannot be predicated of them either. For non-existence would consist in the separation of elements, but simples have no elements. Hence, they cannot be destroyed, and therefore they cannot but exist. Yet, if their existence is necessary, it cannot be meaningfully predicated. For only bipolar propositions – that is, propositions that can be both true and false, and with respect to which, to know whether they are true or false, we need to compare them with reality – are meaningful, for Wittgenstein in the Tractatus. This is indeed a point that Wittgenstein had made in the Tractatus too, where the existence of objects could not be said, but showed itself in the fact that there are meaningful sentences and therefore names endowed with meaning.15 The metaphysical argument for the existence of simples, however, seems flawed. For why cannot something indivisible stop existing all at once? Why cannot something pop in and out of existence without depending on the prior existence of its component parts, or without being decomposed into its components? If simples were not actually identified with physical entities, for which the argument would go through, but with sense-data, or other phenomenological entities such as after images, they could pop in and out of existence at once, without either ontologically depending on or being decomposed in more elementary elements. Whether this is an argument that would ultimately lead to the identification of the objects in the Tractatus with this kind of entities, rather than physical ones, is something we need not settle here. Yet, it speaks in favour of such an interpretation. The consequence, however, would then be that the Tractatus would be very close to Russell’s view, in The Philosophy of Logical Atomism, that the only proper names, from a logical point of view, would be ‘this’ and ‘that’ as used to name sense-data, at least as long as these entities exist. Yet, the reason why this is ultimately irrelevant, in the context of the Philosophical Investigations, is that the master thought behind the whole quest of simple elements in the Tractatus has been exposed as flawed, due to the confusion between a name’s meaning and its bearer. That is, for Wittgenstein, in the Philosophical Investigations, there is no need to identify simples, which could confer meaning to (logically proper or genuine) names, for the meaning of a name is not its bearer but the use we make of it in our language. Sometimes we explain the meaning of a name by indicating its bearer, but sometimes we do not, for instance, if the bearer does not exist, or does not exist any longer. As we noted, the interest of the author of the Tractatus has shifted and now the key intuition is that there is a multiplicity of functions words and names among them can play within language, while still be seeing as names (or as propositions, etc.). The master thought that occupies centre stage is now the idea that words function through family resemblance, as Wittgenstein insists between Philosophical Investigations §§66–76, before returning to the issue of the meaning of a proper name like ‘Moses’ in PI §79. Of simples, Wittgenstein insists, we cannot predicate either their existence or their non-existence (PI §50). For they must exist for the words that would name them in the corresponding sentence to have a meaning. Thus, ‘x does not exist’ would be meaningless. If so, however, ‘x exists’ would be meaningless too, since it would have to be necessarily true and its negation impossible (for the reason just

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seen). Hence, of a simple we cannot meaningfully predicate either its existence or its non-existence. Interestingly, this is what, by analogy, leads Wittgenstein to discuss the role of paradigms, like the standard metre in Paris. For of it, for Wittgenstein, it does not makes sense to say that it is one metre long or that it is not. Here is his argument: if that stick is what is used to ostensively define what ‘being one meter long’ means, then it cannot fail to be one metre long. But if it cannot meaningfully be said of that stick that it is not a metre long, then, because of bipolarity, it cannot meaningfully be said that it is one metre long either. We may be tempted to think otherwise if we were oblivious to the role that that stick (like a sample of colour (like sepia or red, respectively discussed in PI §50 and §57) or any other paradigm (PI§55), even a mental one (PI§56)) plays in our language. A similar confusion, by Wittgenstein’s lights, is what would later afflict Kripke’s reading of this example.16 For, on that reading, the stick would be used only to fix the reference of ‘being one meter long’ such that it could actually turn out to be the case that that very stick is not one metre long, after all. While this could obviously happen to the stick, which like any physical object may be subject to alternations and can therefore change its length, or a different stick could be taken as the standard metre, so that that stick would not be one metre long not even in this world, it could not happen to it as used as the standard metre. If used as such, whatever its physical length could be, across possible worlds, it would necessarily be one metre long. Leaving aside this discussion, Wittgenstein points out that when we say ‘red exists’ and ‘red does not exist’, all we are actually trying to say is either ‘The word “red” has a meaning’ or ‘The word “red” does not have a meaning’ (PI §58). The sample is only what we sometimes use to ostensively define or explain the meaning of that word. The key thought, then, is that the bearer is something we use in connection with a name only in certain cases – to repeat, when we ostensively define or explain its meaning. Yet, there are many more ways in which we use names and explain their meaning, such that, even when there is no (longer a) bearer, a name can retain a meaning. The take-home message of Wittgenstein’s discussion of the name ‘Moses’ in PI §79. Before delving into it, however, it is crucial to recall Wittgenstein’s own defence of his new perspective onto language and philosophy, to which the remarks in Philosophical Investigations §§66-78 are devoted. The defence is introduced after the objection, in PI §65, ‘You take the easy way out! You talk about all sorts of language games, but have nowhere said what the essence of a language game, and hence of language, is.’ Wittgenstein acknowledges that he is not trying to define what language is, or to get at its essence, contrary to his previous project in the Tractatus. That project was deeply misguided – a vestige of a metaphysical way of looking at things, which Wittgenstein is completely rejecting in Philosophical Investigations. For there is no common essence shared by all entities called the same or subsumed under one word or concept. Famously, Wittgenstein exemplifies the idea with different games that share no common set of necessary and sufficient conditions, which could then be used to define the word ‘game’ or the very concept of a game. Rather, we subsume under that concept activities that

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resemble other ones, already comprised within it, in at least one varying respect (PI §66), like ‘the various resemblances between members of a family’ (PI §67) – whence the name of ‘family resemblance’ (ibid.) for the way in which Wittgenstein thinks of the workings of our concepts. For instance, in patience, like in tennis and unlike playing with dolls, there is winning and losing, but patience, unlike tennis, and like playing with dolls, can be played alone. Tennis and playing with dolls, then, do not have anything in common. Yet they are both games because they each resemble in at least one varying respect something else that is considered to be a game too. Nor can we say that they are all games because they share the disjunction (ibid.) of all these features. Of course we could close the frontiers of concepts in this way, if we needed to, but we typically don’t and that is why they have, to put it with Friedrich Waismann (1945), an ‘open texture’, which allows us to extend them to new and unforeseen cases (PI §68).17 Pace Frege, then, concepts can function perfectly well even if they have no rigid boundaries and even if, as it happens, they admit of vagueness (PI §§ 69-71, 76-77). The same is true of philosophical key concepts such as language, meaning, proposition, name and so on. In particular, as we saw, a name, for Wittgenstein, can have a meaning even if it has no (longer a) bearer. The discussion of the name ‘Moses’, just after the remarks on family resemblance, exemplifies his point. In it, Wittgenstein notices that we do not use that name with a fixed meaning, for different definite descriptions can be used to explain its meaning. We use each of them in turn as ‘props . . . we lean on . . . if another should be taken’ away from us. Nor there is any prior decision made as to how many definite descriptions have to turn out to be false before saying that Moses did not exist or that ‘Moses’ does not have a meaning. Now, clearly, the latter two claims are not on par. For fictional proper names, like ‘Santa Claus’, retain a meaning even if the definite descriptions we use to explain it are in fact empty – that is, even if they are not satisfied by any individual. Conversely, Moses – that is, the individual named thus – could have existed even if he actually did none of the things the Bible attributes to him. Maybe he was not saved from the Nile by the pharaoh’s daughter, or did not lead Israelites outside of Egypt, or any other thing the Bible says Moses did. Yet he did exist at some point in history. Again, the account Wittgenstein proposes of the meaning of proper names is not entirely plausible. For, as Kripke has shown in Naming and Necessity, it does not explain the modal behaviour of proper names. Moreover, as we have briefly mentioned, it does not seem plausible of fictional proper names either, since the falsity of the definite descriptions we use to explain their meaning, due to the non-existence of their alleged referents, does not prevent those names from having a meaning.18 Yet, it clearly marks the departure from his earlier views and is made possible by his new outlook onto language. As Wittgenstein concludes in Philosophical Investigations §81, we use words in everyday language without a fixed meaning or ‘according to definite rules’. That, however, is perfectly all right, and we do not need to construct an ideal language to explain how ours works. That is what Wittgenstein himself did in the Tractatus, and that is what led him to embrace a form of logical atomism. That is what he is

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now rejecting as unnecessary and as driven by an entirely philosophical or even metaphysical way of looking at language, instead of looking at its actual use. Hence, by enjoining ‘Don’t think, but look!’ (PI §66), Wittgenstein is dismissing, at once, that outlook and one of its most notable consequences.

NOTES 1. Let them be the elementary propositions resulting from logical analysis or the ordinary propositions which we normally traffic in, as Pears (1987, 76) points out. 2. Copi (1958) is considered to be at the origin of the controversy. For an overview of this controversy, see Pears (1987), vol. I, chapter 4 and 5, and Dionigi (2001), chapter 3. The controversy has been taken to bear on the vexed issue of the realism of the Tractatus. That is, whether, for Wittgenstein, it is the nature of the object – its inherent combinatorial properties in states of affairs – that determines which uses of its name within a proposition make sense; or else, whether it is the meaningful use of the name within a proposition that determines the possible occurrence of the object in states of affairs. Notoriously, Pears has been the chief support of the realist interpretation and Ishiguro and McGuinness of the anti-realist (or constructivist) one. 3. Anscombe (1959). 4. Most notably, Hintikka and Hintikka (1986). 5. Griffin (1964). 6. Ishiguro (1969) and McGuinness (1981). 7. Pears (1987, 89). 8. Indeed, he said to Malcolm (1958, 99–100) that this was none of his business, since it would ultimately be an empirical, not a logical, issue. Also, in the Notebooks 1914–1916, Wittgenstein claims that for his purposes it is actually expedient to consider ordinary proper names and their relations to the objects they name to exemplify and reflect on their function, even if at the end of the analysis they would be considered neither as names nor as simples (Notebooks 14.6.15, 21.6.15 and 30.5.15). 9. See in the following for the rationale of this claim. 10. Ishiguro 1969 denies that anything like Russell’s idea of logically proper names can be found in the Tractatus. 11. This in turn originated a debate about the (in)determinacy and possible variability of names’ meaning which occupied philosophers of language till the late 1970s–early 1980s. 12. Here the reference is to the Newton–Goethe debate over the nature of white light. The issue was extremely important to Wittgenstein who grappled with it until the very end of his career (see his Remarks on Color, written during the last eighteen months of his life). For a discussion, see Coliva (2020). 13. This indirectly could be evidence in favour of the realist interpretation of the Tractatus, which Pears endorsed. For here Wittgenstein seems to be criticizing his earlier views.

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14. This, in contrast, could be indirect evidence in favour of the anti-realist interpretation of the Tractatus, championed by Ishiguro and McGuinness. For the passage occurs in the midst of the rehearsal of some of his earlier views and clearly echoes T3.3. 15. Hintikka and Hintikka (1986, 102) insist on the ineffability of the existence of objects. They think Russell too had a similar view and maintain that Wittgenstein’s objects in the Tractatus are phenomenological entities. 16. See Kripke (1980, 201–2). 17. Notice that some disjuncts would actually be contradictory and, if that finding could be extended to all potential disjuncts figuring in this alleged definition of the concept, the disjunction would end up being a tautology, which could not be used to tell games apart from non-games. 18. Dionigi (2001), chapter 4 defends the idea that Wittgenstein is interested only in the existence of objects within the discourse or the story, rather than in their worldly existence. Hence, within the story, Santa Claus would exist. For within it, it is true that one and no more than one individual has the properties typically attributed to Santa Claus. As a result, of distinguishing these two different notions of existence, it would also be possible to reconcile Wittgenstein and Kripke’s different accounts of proper names. For Moses could have existed even if none of the definite descriptions we can derive from the Bible were (uniquely) true of him. Consequently, in that case the name ‘Moses’ could still have a meaning and would behave, modally, as Kripke argues all proper names work. According to Dionigi, Wittgenstein would be merely objecting to the idea that names’ meaning is given by a definition – let it be ostensive or otherwise – and would not be putting forward an account of what their meaning is – let it be their bearer, or a (set of) definite description(s). Rather, their bearer or that (set of) definite description(s) is merely what we avail ourselves to explain their meaning. The discussion of this ingenious interpretation cannot be taken up here.

REFERENCES Anscombe, G. E. M. (1959), An Introduction to Wittgenstein’s Tractatus, New York: Harper & Row. Coliva, A. (2020), ‘Which philosophy after philosophy? Rorty and Wittgenstein’, ms. Copi, I. M. (1958), ‘Objects, properties and relations in the Tractatus’, Mind 67: 145–65. Dionigi, R. (2001), La fatica di descrivere. Itinerario di Wittgenstein nel linguaggio della filosofia, Macerata: Quodlibet. Griffin, J. (1964), Wittgenstein’s Logical Atomism, Oxford: Oxford University Press. Hintikka, J. and Hintikka, M. (1986), Investigating Wittgenstein, Oxford: Blackwell. Ishiguro, I. (1969), ‘Use and reference of names’, in P. Winch (ed.), Studies in the Philosophy of Wittgenstein, 20–50, New York: Humanities Press. Kripke, S. (1980), Naming and Necessity, Boston: Harvard University Press. Malcolm, N. (1958), Ludwig Wittgenstein: A Memoir, Oxford: Oxford University Press. McGuinness, B. (1981), ‘The so-called realism in Wittgenstein’s Tractatus’, in I. Block (ed.), Perspectives on the Philosophy of Wittgenstein, 60–73, Oxford: Blackwell.

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Pears, D. (1987), The False Prison, 2 vols., Oxford: Clarendon Press. Russell, B. (1918), ‘The philosophy of logical atomism’, in Logic and Knowledge, 177–281, London: Allen & Unwin, 1956. Waismann, F. (1945), ‘Verifiability’, Proceedings of the Aristotelian Society Supplementary Volume XIX: 119–50. Wittgenstein, L. (1922), Tractatus Logico-Philosophicus, London and New York: Routledge. Wittgenstein, L. (1953), Philosophical Investigations, Oxford: Blackwell. Wittgenstein, L. (1961), Notebooks 1914–1916, Oxford: Blackwell. Wittgenstein, L. (1977), Remarks on Color, Oxford: Blackwell.

CHAPTER 17

Atomism and semantics in the philosophy of Jerrold Katz1 KEITH BEGLEY

Jerrold J. Katz often explained his semantic theory by way of an analogy with physical atomism and an attendant analogy with chemistry. In this chapter, I track the origin and uses of these analogies by Katz, both in explaining and in defending his decompositional semantic theory, through the various phases of his work throughout his career.

THE ANALOGY WITH PHYSICAL ATOMISM The first instance of the analogy with physical atomism is to be found in Katz and Fodor’s seminal paper, ‘The Structure of a Semantic Theory’ (Katz and Fodor 1963). There they refer to hypothesized ‘atomic concepts’ into which meanings may be decomposed, which are represented in the theory by way of semantic markers. The semantic markers and distinguishers are used as the means by which we can decompose the meaning of a lexical item (on one sense) into its atomic concepts, thus enabling us to exhibit the semantic structure in a dictionary entry and the semantic relations between dictionary entries. That is, the semantic relations among the various senses of a lexical item and among the various senses of different lexical items are represented by formal relations between markers and distinguishers. (Katz and Fodor 1964, 496)2 Later in the same article they point out that semantic markers are merely theoretical constructs that are used in an analogous way to that in which theoretical constructs have been used in other sciences. a semantic marker is simply a theoretical construct which receives its interpretation in the semantic metatheory and is on a par with such scientific constructs as atom, gene, valence, and noun phrase. A marker such as (Human) or (Color) is, then, not an English word, but a construct represented by one. (Katz and Fodor 1964, 517) A similar notion of ‘atomic concept’ is also employed in Katz and Postal’s 1964 work, An Integrated Theory of Linguistic Descriptions:

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The meaning of a lexical item is not an undifferentiated whole. Rather, it is analyzable into atomic conceptual elements related to each other in certain ways. Semantic markers and distinguishers are intended as the symbolic devices which represent the atomic concepts out of which the sense of a lexical item is synthesized. Readings represent such synthesizations of atomic concepts. (Katz and Postal 1964, 14) In an endnote to the first sentence of this passage, they point out that, although componential analysis of the meanings of lexical items was already employed by an approach to the anthropological study of kinship terms in the 1950s, this approach had not been extended to very many other sets of lexical items nor to linguistic description more generally, and had not been integrated into the new generative linguistic descriptions that allow for the interpretation of an infinity of sentences (Katz and Postal 1964, 28, n. 7). Thus, it is clear that they saw the extension of this method to lexical items more generally, and its relation to the new Chomskyan grammar, as constituting part of the novelty of the theory.

THE FREGEAN ANALOGY WITH CHEMISTRY In an article from December 1964, ‘Semantic Theory and the Meaning of “Good”’, Katz extended the analogy between conceptual and physical atoms, to include an analogy with chemistry. This is an analogy between a reading, that is, a theoretical representation of the sense of an expression, composed of semantic markers, and a chemical formula for a molecule, in respect of the manner in which they represent the structures of their respective objects: [Semantic markers] are to be regarded as constructs of a linguistic theory in just the sense in which terms like ‘force’, ‘mass’, ‘molecule’, etc. are accepted as labels for scientific constructs in physical theory. There is here a strong analogy between the manner in which a reading for a sense of a word or expression represents its structure of concepts and their interrelations and the manner in which a chemical formula for a molecule of a substance represents its structure of atoms and the bonds between them. Both employ theoretical constructs of a scientific theory and a schema of representation to exhibit the elements and relations out of which a compound entity is formed. (Katz 1964, 744) Katz later makes similar statements in his book The Philosophy of Language (1966) and provides there in addition the following example of such a formula, a diagram for a molecule of ethyl alcohol:

H H | | H — C— C — O— H | | H H (Katz 1966, 156; cf. Katz [1966] 1970, 187)

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The analogy with chemistry was quite important for Katz, and it continued to appear in various forms in publications throughout his career, the latest usage being in an article from 1992.3 Although Katz does not say so explicitly, I find it likely that the analogy was inspired by a similar analogy used by Gottlob Frege in ‘On Concept and Object’ (1892). There is, I think, good circumstantial evidence for this claim in that the analogy with chemistry certainly began to become important for Katz at just the time that he was beginning to read Frege closely. We should first note that later, in the preface to Semantic Theory (1972), Katz acknowledges a general debt to Frege, but also his and Fodor’s earlier ignorance of the details in Frege’s work: the more our approach was worked out, the more it was found to embody versions of Frege’s principles. Fodor and I were originally unaware of the extent to which doctrines of ours were often replicas of Frege’s. In the present book, I have tried to indicate some of the debt that empirical semantics (in our sense) owes to him. (Katz 1972, xxiv) The first of these indications, following the statement just quoted, is in regard to Frege’s approach to the question ‘What is Meaning?’ (Katz 1972, 3n), a question that we will return to presently. The second indication is in regard to the principle of effability (Katz 1972, 19).4 A third is in regard to the notion of a primitive semantic marker, which is the theoretical representation of an atomic concept. Katz makes the point here that the inventory of such markers or the atomic concepts that they stand for cannot be set out in advance of the theory of which they are a part, but only as ‘required in the formation of dictionary entries’ (Katz 1972, 38). To this, Katz provides a footnote in which he quotes from Frege’s ‘On Concept and Object’ as follows: my explanation is not meant as a proper definition. One cannot require that everything shall be defined, any more than one can require that a chemist shall decompose every substance. What is simple cannot be decomposed, and what is logically simple cannot have a proper definition. Now something logically simple is no more given us at the outset than most of the chemical elements are; it is reached only by means of scientific work. If something has been discovered that is simple, or at least must count as simple for the time being, we shall have to coin a term for it, since language will not originally contain an expression that exactly answers. On the introduction of a name for something logically simple, a definition is not possible; there is nothing for it but to lead the reader or hearer, by means of hints, to understand the words as is intended [Frege 1952, 42–3]. (Katz 1972, 38n) Katz’ first published reference to the source of this quotation and, indeed, his first direct references to Frege5 are given in his book The Philosophy of Language (Katz 1966, 47n), which was in press before October 1965 (Katz 1965, 600n). Moreover, it seems likely that he was asked by Arthur Danto, before the autumn of 1964, to begin writing an essay for inclusion in The Harper Guide to Philosophy, which he probably first submitted by the end of the summer of 1965, if not earlier.6 In that essay, Katz further develops the analogy with atomism, which we will discuss presently. However,

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due to approximately a five-year delay, it is not until 1971 that this essay is published in the form of a book, The Underlying Reality of Language and its Philosophical Import. This short introductory book turns out to be quite programmatic, and it came just a few months before Katz published his main technical work, Semantic Theory (1972), to which the former work already contained many references added in footnotes.7 The 1971 book also contains a short fragment of the same quotation from Frege’s ‘On Concept and Object’ as that which is quoted in Semantic Theory, and in the same connexion regarding primitive semantic markers: ‘As Frege once observed, “. . . something logically simple is no more given us at the outset than most of the chemical elements are; it is reached only by means of scientific work.”’ (Katz 1971, 101). This indicates that Katz certainly had read ‘On Concept and Object’ at least five years before June 1971, and probably before October 1965 since he refers to the source in The Philosophy of Language, and perhaps as early as the summer of 1964, given that he had been recruited to write the essay in which he quotes Frege’s analogy. That is, early enough for it to have inspired the analogy with chemistry that appears in Katz’ article published in December of that year (Katz 1964).

THE DEMOCRITEAN APPROACH Beginning with his short book in 1971, Katz adds a further element to the analogy through conceiving of his semantic theory as being part of a Democritean linguistic theory. This element of the analogy with atomistic theory serves to emphasize the appearance–reality distinction that is integral to the theory, in contrast to some other linguistic theories. Here, Katz takes the Democritean concept of matter to be a paradigm case: The Democritean concept of matter originated as a purely hypothetical postulation. Initially, it could only have seemed the most extravagant of fancies. It proposed to populate the universe with unbelievably many new objects. Such objects were, moreover, supposed to be invisible and yet to provide the true understanding of visible phenomena. Finally, to add insult to injury, the concept flew in the face of the plain testimony of sense experience. But when it proved to yield better predictions and explanations of the observable behavior of physical objects and substances than the concept of continuity, it received scientific acceptance. The continuity hypothesis, which once must have seemed the last word in sober science, became relegated to the status of a depiction of appearance. (Katz 1971, 3) Katz is careful to show how the Democritean approach contrasts with other philosophical approaches. For example, he says that it contrasts with the early Wittgenstein’s philosophy in respect of the latter’s ‘assumption that logical form is [inaccessible]’ (Katz 1971, 11). He quotes Wittgenstein’s famous clothing metaphor for this point from Tractatus Logico-Philosophicus, 4.002: Language disguises thought. So much so, that from the outward form of the clothing it is impossible to infer the form of the thought beneath it, because the

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outward form of the clothing is not designed to reveal the form of the body, but for entirely different purposes. (Wittgenstein 1922, 61–3, quoted in Katz 1971, 9–10) Katz replies to this by tailoring the clothing metaphor to his own Democritean approach as follows: To vary Wittgenstein’s metaphor somewhat: While it is true that language disguises thought, the disguise fits in such a way as to enable us to frame for ourselves a facsimile of the form of the body hidden beneath if we are willing to penetrate the disguise in the way physicists penetrated the disguise in which nature presents matter to us in sense experience. Thus, a Democritean theory of language also contrasts with the later philosophy of Wittgenstein. Within the framework of a Democritean theory, we conceive the problem of understanding the logical features of language as a problem of theory construction. (Katz 1971, 12) This variation would appear to be cut from similar cloth to that of Frege’s clothing metaphor in ‘The Thought’ (1918): The thought, in itself immaterial, clothes itself in the material garment of a sentence and thereby becomes comprehensible to us. We say a sentence expresses a thought. (Frege [1918] 1968, 511)8 Katz takes somewhat of a middle position. That is, although the material garment is what enables us to frame a ‘facsimile’ of what it clothes, it is nonetheless a disguise that must be penetrated by constructing a Democritean theory of what lies beneath its covering. Katz does provide a reference to a translation of Frege’s article, but only at a much later point in the book, where he agrees with Frege’s non-psychological use of the word Gedanke (thought) (Katz 1971, 121n). He quotes the passage from the Tractatus in a number of other publications,9 but never quotes the concomitant passage from Frege’s article.10 So, it is uncertain whether Katz ever noticed a similarity with the passage from Frege. Katz takes the rationalist position, inspired by the work of Noam Chomsky, that since users of a language do indeed often manage to penetrate the orthographical and phonological disguises, we can frame a ‘facsimile’ of the underlying form through providing a theory of their competence to do so. This is also spelt out in terms of a contrast with what Katz calls non-Democritean theories. These theories, such as the pre-Chomskyan taxonomic theories of grammar, naively rely merely upon the manifest features of utterance-tokens when forming theories of sentence types (Katz 1971, 19). As Katz put it: the non-Democritean assumption about the nature of language leads to grammars whose rules only permit descriptions of linguistic features observably manifest in utterances of sentences, while the Democritean assumption leads to grammars whose rules also permit descriptions of linguistic features that are not observably manifest. (Katz 1971, 31) Katz took Chomsky’s transformational model of grammar to be the embodiment of such a Democritean method and to be part of a tradition stretching back to

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the universal grammar of the seventeenth-century Port-Royal grammarians (Katz 1971, 46). Katz also contrasts the Democritean approach with the non-Democritean approach to the question ‘What is Meaning?’, which he calls ‘the big question of semantics’ (Katz 1971, 84). Non-Democritean theories, he says, tend to misconstrue the question as a request for a direct and straightforward answer (Katz 1971, 85–6; Katz 1972, 3). He lists some notable examples of such answers to the question: These include the [Platonic] answer that meanings are eternal archetypes; the Lockean answer that meanings are the mental ideas for which words stand as external signs; the answer that meanings are the things in the world to which words refer; the Wittgensteinian answer that meaning is use; the behaviorist answer that meanings are the stimuli that elicit verbal responses; the introspectionalist answer that meanings are mental images associated with verbal behavior; and so on. (Katz 1971, 85) In contrast, the Democritean approach begins by reversing the order of investigation. That is, not by first attempting directly to state what meaning is in terms of something else that is better or more explicitly known to us antecedent to providing a theory of meaning, rather, by seeking to construct a theory of meaning before subsequently attempting to say more about what meaning must be in order to satisfy this theory as its object. Katz motivates this approach through a comparison with the history of other sciences: Imagine what would have happened if ancient astronomers had insisted on knowing what sorts of things planets are before trying to describe their movements. Or, better yet, suppose mathematicians had insisted on a direct answer to the question ‘What are numbers?’ before trying to explain arithmetic properties and relations like ‘is the sum of ’, ‘is the square root of ’, and ‘is a prime number’. We would now be without a theory of arithmetic (i.e., number theory). (Katz 1971, 94)11 A corollary of this is that there is no room in this approach for philosophical scepticism about planets and numbers within astronomy and mathematics; these disciplines seek merely to provide adequate theories of what appears to us, that is, the phenomena. Analogously, Katz seeks to silence the sceptic of semantics by doing semantics; to refute them thus (cf. Katz 1972, 2). Katz is led by this diagnosis ‘to begin with the assumption that “What is meaning?” is a request for a semantic theory’ (Katz 1972, 3). This is the point from which he sets off in his main technical work Semantic Theory (1972). Here he follows again the example of the Democritean theory of matter, which he explains as follows: Physicists were in no position to say what matter was until they identified a wide range of phenomena exhibited in the behavior of matter and ascertained many of the significant empirical facts in each case. These various phenomena (diffusion, expansion, interpenetration, conduction, etc.) demarcated the domain for a theory of matter by bringing together the properties that such a theory had to explain. It then became possible to compare different conceptions of the

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nature of matter in terms of their explanatory adequacy within this circumscribed domain. We can say, therefore, that the Democritean theory was arrived at by the construal of the questions ‘What is diffusion (of matter)?’ ‘What is expansion (of matter)?’ ‘What is interpenetration (of matter)?’ and so on as components of the question ‘What is matter?’ This reflects the common practice in science of breaking down a large, general question into narrower, more specific ones. The answers to these questions then form integral parts of the answer to the general question because the phenomena about which they ask are within the appropriate explanatory domain. (Katz 1972, 4) Analogously to this approach to matter, Katz breaks the big question of semantics ‘What is meaning?’ into a non-exhaustive list of fifteen sub-questions that identify the most prominent sense properties and relations pertinent to a theory of meaning: synonymy and paraphrase, similarity and difference, antonymy, superordination, meaningfulness and anomaly, ambiguity, redundancy, analyticity and metalinguistic truth, contradiction and metalinguistic falsehood, syntheticity, inconsistency, entailment, presupposition, possible answerhood and self-answerhood (Katz 1972, 4–6). For Katz, providing a semantic theory is to provide an integrated theory of these sense properties and relations, which the theory represents in terms of relations between semantic markers.

THE ANALOGICAL DEFENCES OF THE THEORY These analogies with atomism and chemistry later become important for Katz’ defence of the theoretical approach. For example, in an article by Chomsky and Katz from 1974, the authors reply to what they take to be a Quinean misinterpretation of linguistics that was put forward by Stephen Stich. Stich complained that Katz seemed not to have sufficient evidence for his conclusions given that Stich himself was not able to derive as much from the same evidence using his own (non-Democritean) method. Chomsky and Katz reply to this by stating that there are always at least two ways to react to such a poverty of evidence: we can point out that some people may well be interested in making hypothetical inferences about underlying causes on the basis of certain evidence, while others with different interests and outlook may choose to restrict their attention more narrowly to the evidence. We can certainly imagine that some early physicists might have been quite happy to accept diffusion and similar phenomena at face value, chiding their Democritean colleagues for ‘flamboyant portraits’ of atoms. (Chomsky and Katz 1974, 365) The point here being that, unless one goes beyond the evidence and hypothesizes underlying linguistic form, one will be stuck at the naïve and non-Democritean level of manifest utterances. The analogy with chemistry later forms an important part of Katz’ defence against David Lewis’ (in)famous criticism of his semantic theory, which Lewis puts forward at the beginning of his paper ‘General Semantics’ (1970). There

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Lewis refers pejoratively to the symbolism of the theory as ‘Semantic Markerese’, a characterization that has been parroted for decades following.12 Indeed, his argument relies upon this erroneous characterization of Katz’ theoretical symbolism as being a mere language: Semantic markers are symbols: items in the vocabulary of an artificial language we may call Semantic Markerese. Semantic interpretation by means of them amounts merely to a translation algorithm from the object language to the auxiliary language Markerese. But we can know the Markerese translation of an English sentence without knowing the first thing about the meaning of the English sentence: namely, the conditions under which it would be true. Semantics with no treatment of truth conditions is not semantics. (Lewis 1970, 18) Indeed, as Lewis goes on to say, if this were the case, one might as well have translated the English sentence into Latin for all the good it would do, because we would then be relying on ‘our tacit competence as speakers’ of this language, or our ability to do so-called ‘real semantics’ for the same by providing truth conditions (ibid., 18–19). Katz pointed out in his responses that Lewis completely overlooked or ignored the fact that the semantic theory is a theory and, as such, is distinct from a mere artificial language. Apart from this necessary emendation in the spirit of charity, Katz grants that, equipped with a semantic theory, the reading of an English sentence can be known without knowing the meaning of the English sentence. However, he points out that this is merely because it can be known, in a purely formal and procedural manner, via certain projection rules, that the theory maps a certain reading R onto a certain sentence S. However, this does not show what Lewis intended it to, because, as Katz explains: a person who does not know semantic theory cannot know what claims R makes about the meaning of S. Since semantic theory provides the intended interpretation for the semantic portion of the grammar of a natural language, the reading R is merely a concatenation of meaningless symbols unless interpreted under an appropriate semantic theory. Similarly, someone may know that a chemical description of some covalent bond represents it as

H .. H: C : H .. H without knowing what this symbolism asserts about the nature of the bond. (Katz 1975, 108–9)13 So, it is important to stress, the readings produced by the semantic theory make claims when interpreted within the theory. That is, they are hypotheses about underlying structures, not mere translations. Lewis’ argument failed to show what he intended it to because, as Katz elucidates by way of the analogy with chemistry, ‘the relation between a sentence of a natural language and its reading(s) is no more translation

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than the relation of [a chemical description] to a covalent bond is translation’ (Katz 1975, 110). For one matter, as Katz points out, in contrast to the theoretical ‘reading of ’ relation, the ‘translation of ’ relation is symmetrical. That is, ‘if S1 is a translation of S2, then S2 is a translation of S1 but the latter is asymmetrical, i.e., if R is a reading of S, then S is not a reading of R’ (Katz 1975, 110). So, Lewis would have needed to prove, at the very least, that the ‘reading of ’ relation is symmetrical. However, since a reading is part of a hypothesis generated by a theory of semantic competence, it contains or codifies information that is not directly contained in the sentence onto which it is mapped. So, it follows that such a proof is not possible, or at least not possible in most instances. Lewis’ further and somewhat glib remark that ‘Semantics with no treatment of truth conditions is not semantics’ is no objection to Katz at all, because it relies on an ambiguous antecedent usage of the word ‘semantics’ (cf. Katz 1972, 182), which Lewis himself acknowledged existed in the literature of the time. Katz’ theory is simply not aimed at explaining the same phenomena. Hence, Lewis merely begs the question (Katz 1975, 114).14 That is, he essentially says nothing more than that ‘The theories of meaning with no treatment of truth conditions are not the truthconditional theories of meaning’, which does not even have the virtue of being false.15

SWIMMING IN THE ‘PLATONIST WHIRLPOOL’ In 1977, a major change takes place in Katz’ approach. Previously, he had adopted an anti-sceptical and non-reductionist method, as part of the Democritean approach, regarding the objects and posits of semantic theory. However, he had also expressed difficulties with, among others, some realist approaches such as the Platonist approach, calling it on various occasions ‘wholly uninformative’ (Katz and Fodor 1962, 212),16 ‘empirically inadequate’ (1964, 762), ‘too vague and speculative’ (1971, 85) and even described the later Wittgenstein as attempting ‘to steer a safe course between the formalist rock and the Platonist whirlpool’ (1971, 11). Despite these earlier attitudes, in his article ‘The Real Status of Semantic Representations’ (1977), Katz broaches the topic of Platonism once again and reveals that he is preparing a work in which he will ‘try to develop a tenable version of the Platonist position’ (1977, 564n).17 This is what eventually becomes his book Language and Other Abstract Objects (1981) in which he develops a realist distinction between knowledge of language, in the sense of competence, and the language of which it is knowledge, which he considered to be an abstract object. A Chomskyan conceptualist theory of language is a theory of the former, a Platonist theory of language is a theory of the latter. The explicit analogies with atomism and chemistry are notably absent during this transitional period, but soon return. First, in his article ‘Common sense in semantics’ (1982), the analogy with chemistry remains a means of explaining the decomposition of senses: we have to go beyond a notational scheme consisting, in effect, of numericals functioning as bare names of senses. We require a scheme that describes the structure of senses in the decompositional way that chemical diagrams describe

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the molecular structure of compounds. Our first approximation is inadequate because it only provides names for senses taken as unanalyzed wholes. In going beyond this first approximation, we are going beyond both Fregean and standard predicate calculi theories of the logical structure of natural languages. (Katz 1982, 198) Here, Frege’s analogy with chemistry (if, as I suggested earlier, it was originally his) is turned against him. The ‘numericals’ mentioned here (later called ‘numerals’, cf. Katz 1990, 65) are intended to illustrate elements of a Fregean theory of sense. The point being that such unary designators standing for unanalysed whole senses are unable to describe the decompositional structure sufficient for subordination, antonymy and the other sense properties and relations. These designators are only capable of representing bare sameness and difference of sense, for example, {1} = {1} and {1} ≠ {2} and so on.18 While this was enough to enable Frege to provide solutions to his famous problems regarding co-referential names and opaque contexts, it is not sufficient to describe the other sense relations.19 To see why this is, consider the following simple examples, which Katz uses in a later work. If one attempts to represent the synonymy between ‘sister’ and ‘female sibling’, their senses will be designated by different numerals and hence they will be marked as non-synonymous. If one attempts to represent the antonymy between ‘open’ and ‘closed’ in this manner, one will not be able to mark the difference between this kind of sense relation and that between the merely non-synonymous ‘open’ and ‘destroy’ (Katz 1990, 65). It is clear from this that Katz’ basic decompositional approach remained largely the same but that he was also beginning to distinguish his theory more carefully from its earlier Fregean influences. In his book Cogitations (1986), the analogy with the Democritean theory of matter is also put to work against the later Wittgenstein’s criticisms of the notion of analysis; an account of these criticisms is to be found in the previous chapter of the present volume (cf. Coliva 2020, 304ff.), including PI §47 to which Katz responded as follows: Everything that Wittgenstein says about the analysis of language can also be said about the analysis of physical substances. Physical analysis cannot employ absolute notions of simplicity and compositeness, either, and for the same logical reasons. But physicists from Democritus to Dalton employed a notion of the components of matter that was relativized to the scientific aim of uncovering the truth about its behavior. As a consequence, what physicists said about the nature of matter on the basis of scientific investigation enjoyed a derivative privileged status with respect to counter claims made on other bases. One couldn’t sensibly reply to the Democritean theory, ‘Matter? Well, it’s composed of atoms and molecules, relative to findings based on investigations aiming at the truth, but, of course, it isn’t relative to findings based on investigations with other aims.’ (Katz 1986, 153) That is, in the case of semantic theory, the analysis or, rather, the decomposition, is carried out just as far as is required to define the various sense properties and relations over the relevant atomic concepts, and thereby ‘to state true laws about

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the properties of sentences’ (ibid.). Any less and the theory would lack explanatory power, any more and the theory would contain redundant elements. This is what ‘defines the enquiry itself ’ (Katz 1986, 153–4). In The Metaphysics of Meaning (1990), Katz seems to take a slightly broader view of the application of the term ‘Democritean’ (cf. Katz 1990, 53–7). Here he talks about Democritean approaches in both grammar and logic and reflects on their shortcomings when developing his Democritean approach to semantics. In earlier works, he had already applied the term to the Chomskyan rationalist approach to grammar. Here he says that the approaches of Frege and Russell could be seen as ‘a corresponding ‘Democritean’ tradition in modern logic which advocates posits of underlying logical structure to overcome the insufficiency of surface grammar to account for certain logical inferences’ (Katz 1990, 53). Here ‘Democritean’ means merely the recognition that there must be a distinction between surface and deep structure in the various areas in which it is applied. Katz’ approach to semantic theory draws on both traditions.

BETTER SEMANTICS THROUGH CHEMISTRY The final mention of both the analogies with chemistry and atomism occurs briefly in Katz’ article ‘The New Intensionalism’ (1992): Having no conception of analysis on which syntactically simple words in sentences like [‘The spot is blue’, ‘The spot is red’, ‘Bachelors are unmarried’, ‘Red is a colour’, ‘Squares are rectangles’, ‘John is a bachelor’, ‘John is unmarried’] can have complex sense structure, Tractarian semantics had no access to the structure which actually determines the inferential powers of those sentences. Tractarian semantics is like chemistry prior to the period of the atomic theory; decompositional semantics is like chemistry afterward. (Katz 1992, 702) The difficulty to which Katz is referring here is usually referred to as the colour incompatibility problem or the colour exclusion problem.20 That is, for example, the theoretical problem of marking the incompatibility relations between sentences such as ‘The spot is blue’ and ‘The spot is red’, given that their conjunction does not have the form of a logical contradiction, for example, ‘p and not-p’. Katz’ solution manages to link the senses of ‘red’ and ‘blue’ in such a way that avoids positing that the senses of the colour terms are explicitly contained in each other in isolation. Instead, it is in so far as such terms are involved in sentences that such relations hold. As Katz put it: ‘Compositionality makes the difference’ (Katz 1998, 572–3). The theory represents such structures through four interlinked formal devices, which Katz calls the antonymy operator, antonymous n-tuples, categorized variables and selection restrictions. I will now provide a brief exposition of the interlinked formal devices that Katz employs to implement his solution to the problem. The first is the antonymy operator ‘A/. . .’, which is analogous to the negation operator in logical calculi; the difference is that negation is an external operator that toggles the truth value of its argument, that is, a proposition, from true to false or false to true, whereas the antonymy

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operator in Katz’ semantic theory is an internal operator that toggles from one semantic marker to others within the same antonymous n-tuple.21 An antonymous n-tuple is simply a set of markers that all contain the same superordinate marker; so, the antonymous n-tuple comprising the complex markers that contain the marker for the sense of ‘colour’, includes the markers for the senses of ‘red’, ‘blue’, ‘green’ and so on. The markers are automatically grouped in this way in virtue of the fact that each of them contains the superordinate marker ‘(colour)’. Katz states the general form of an antonymous n-tuple as: ‘(M(α1)), (M(α2)), . . . , (M(αn))’ (Katz 1972, 52), where the semantic marker ‘M’ (in this context) represents the superordinate sense component and each α represents a different subordinate sense component, and he defines the notion of an antonymous n-tuple as follows: Two semantic markers belong to the same antonymous n-tuple of semantic markers if and only if one has the form (M(αi)) and the other has the form (M(αj)), where i ≠ j and 1 ≤ i ≤ n and 1 ≤ j ≤ n. (Katz 1972, 52)22 The next piece in the solution is the use that Katz makes of categorized variables and selection restrictions. A categorized variable awaits a semantic value given by another part of the reading of a sentence; they are categorized according to the functional notation of the syntactic theory being used. The variable is also given a selection restriction, for example, ‘’, which constrains the selection of the marker that will replace the variable. So, any semantic marker that replaces the categorized variable must include the superordinate marker ‘(colour)’. That is, the marker must be one that is taken from the antonymous n-tuple comprising the markers for the colour senses. Categorized variables are notated in the following fashion: [  ] & K X

The square brackets include the functional syntactic notation that specifies a particular node of the underlying phrase marker for a sentence, for example, ‘subject of ’ is represented as ‘[NP, S]’, the function from a noun phrase to a sentence. It is the semantic reading (or a component thereof)23 of this node of the underlying phrase marker, represented as structured semantic markers, which will be substituted for the variable in a derived reading for the sentence (or higher-level constituent) as a whole. The angled brackets include the semantic selection restriction for the variable, for example, ‘’ (cf. Katz 1972, 104ff.). Although the sense of ‘red’ does not contain the sense of any other specific colour term, it nonetheless excludes the sense of any antonymous colour term. The reading for ‘red’ is given in the form of the complex semantic marker ‘(((red), (A/(X))) (colour))’,24 where ‘X’ is a categorized variable with ‘’ as its selection restriction, and the function ‘[F]’ as its syntactic function. The ‘[F]’ function picks out the semantic markers of the colour term in the predicate of a verb phrase, for example, the reading for ‘blue’. For instance, only the reading for some other colour term, for example, ‘blue’, could replace ‘X’ when ‘red’ is involved in a sentence

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such as ‘Red is not blue’. The rules of Katz’ semantic theory would then mark this sentence as being analytic, and a sentence like ‘The spot is red and blue’ as being contradictory on a sense.25 Thus, the theory provides a theoretical explanation of such properties. Katz’ Democritean semantic theory allows for the decompositional representation of what were considered by the early Wittgenstein to be independent, non-complex simples, as instead having a minimal internal complexity consummate with the relations into which the senses of such terms enter. As I have shown in this chapter, this approach to semantic theory was both explained and defended by Katz, throughout its various phases of development over more than thirty years, often by way of analogies with physical atomism and chemistry.

NOTES 1. A penultimate draft of this chapter was read at the work-in-progress árd-seimineár at the Trinity Plato Centre, Trinity College Dublin, in October 2019. I am especially grateful for the helpful discussion and for comments that I received from Vasilis Politis, Peter Larsen, Margaret Hampson and Simone Nota. 2. I quote instead from Katz and Fodor’s revised version of their paper included in their edited volume, The Structure of Language, published in 1964. Although they note that the article has been reprinted from the 1963 original, they made quite a number of changes and corrections throughout but make no note of this. Importantly, here the infelicitous phrase ‘the meaning of one ense [sic] of a lexical item’ (1963, 185–6) is replaced with ‘the meaning of a lexical item (on one sense)’ (1964, 496), because, of course, senses themselves do not have meanings, rather, they are meanings. 3. The analogy with chemistry has been noticed by Arnold M. Zwicky, who quotes from Katz (1966, 156) and pursues a related thesis (Zwicky 1973, 476). It was also noticed by Keith Allan ([1986] 2014, 317), who mentions Katz’ article (1964, 744), but grossly misinterprets the analogy there: ‘His semantic markerese is like chemical formulae in that it translates into English’ (Allan [1986] 2014, 317). 4. D. Terence Langendoen has recently claimed that it was instead Edward Sapir’s principle of formal completeness that was ‘reinterpreted’ by Katz as the principle of effability (Langendoen 2018, 261). However, this is not what Katz himself says in print. See my review of the edited volume that contains Langendoen’s article (Begley 2019). It appears that Langendoen (2010, 141n) overinterpreted a claim made by Von Fintel and Matthewson (2008, 142–3), whom he subsequently neglected to cite in 2018. Their claim is in fact much weaker: ‘The effability idea has been around since at least Sapir [1924]’ (2008, 143, emphasis added). Nevertheless, they also overlook an attribution by Katz of the principle to Frege in 1923 (Katz 1976, 36). 5. ‘Fregeans’ are mentioned once before this, but merely as one group in a list of the proponents of various positions (Katz and Fodor 1962, 197). Frege’s work is, of course, mentioned by other authors in Fodor and Katz (1964). 6. In the preface of the book that resulted from this project, The Underlying Reality of Language and its Philosophical Import, dated June 1971, Katz reports that he originally wrote the book as an essay ‘about five years’ previous for inclusion in The

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Harper Guide to Philosophy (Katz 1971, vii), edited by Arthur Danto. This was one of a number of guidebooks conceived by Fred Wieck, an editor at Harper. Robert Paul Wolff, the author of another of the essays, reports that he was the last to be recruited by Danto for the project, when he arrived at the Columbia philosophy department in the autumn of 1964, after Danto had been turned down by Isaiah Berlin. Danto requested that Wolff submit his essay for the end of the summer of 1965, which he did (Wolff 1998, x–xi; Wolff 2010; Wolff 2019). So, it is likely that Katz submitted his essay at the same time or before, given that he was recruited before Wolff. Although Katz noted the guide as being ‘in press’ in 1968 (Katz 1968a, 493), delays ensued. Another section author, Bernard Williams, later reported that he and some of the other authors even took to calling the guide ‘Harper’s Bazaar’ (Williams 1993, xi). In 1970, it was suggested by Wolff that each essay instead appear as a short book. He lists the titles of the other books that came out of the project, apart from his own In Defense of Anarchism and Katz’ book, as: ‘What Philosophy Is, by Arthur Danto; Observation and Explanation, by Norwood Hanson; [. . .] Problems of Mind, by Norman Malcolm; What is Knowledge, by David Pears; The Philosophy of Logic, by Hilary Putnam; Morality: An Introduction to Ethics, by Bernard Williams; and Art and Its Objects; by Richard Wollheim’ (Wolff 1998, xii). Some of these were republished in the UK by Allen & Unwin, with Katz’ book retaining its title as a subtitle and taking the new title Linguistic Philosophy. 7. In a letter to Fritz Staal at the end of August 1968, Katz reported that he had worked on his book over the summer, completing chapters 1–4, and had written but not edited a further five (!) chapters, which he planned to finish over the next semester and then send a resulting ‘800-page opus’ to Staal (Katz 1968b). Semantic Theory (1972) has eight chapters. 8. Translated elsewhere as: ‘The thought, in itself imperceptible by the senses, gets clothed in the perceptible garb of a sentence, and thereby we are enabled to grasp it. We say a sentence expresses a thought’ (Frege 1997, 328). 9. For example, Katz (1981, 163; 1990, 67); and partially in (1992, 693). 10. It is uncertain what connection, if any, there is between the two passages. One might be tempted to think that because the Tractatus was published after Frege’s article, it includes a response to the latter. However, Wittgenstein’s above quoted remarks from 4.002 appear with minor differences in 4.0014 and 4.00141 of an earlier manuscript, MS104 on page 36 (Wittgenstein 2015). Also included there is another relevant remark that is not preserved in the Tractatus: ‘Thus the outward aspect of ordinary language makes every kind of illusion and confusion possible’ (Wittgenstein 1971, 4.0015). On some views, this material may originally date to Wittgenstein’s time in Norway, between October 1913 and June 1914 (Kang 2005, 6), and, on others, at least between October 1915 and March 1916 when added to MS104 (McGuinness 2002, 266), although there is still much disagreement. I do not intend to adjudicate on the matter here. Frege and Wittgenstein met and engaged in correspondence but, from what remains of this, the clothing metaphor is not apparent. However, the metaphor was indeed used much earlier by Frege and so may have been mentioned in conversation with Wittgenstein during the intervening period. The following rather clear and pertinent example comes from a posthumously published paper, entitled ‘Logic’, dated to 1897, in which Frege notes a ‘difficulty’:

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grammar, which has a significance for language analogous to that which logic has for judgement, is a mixture of the logical and the psychological. If this were not so, all languages would necessarily have the same grammar. It is true that we can express the same thought in different languages; but the psychological trappings, the clothing of the thought, will often be different. (Frege [1897] 1979, 142; cf. 135, 185; also printed in Frege 1997, 243; cf. 361) Frege had earlier used a similar clothing metaphor regarding concepts, in The Foundations of Arithmetic (Die Grundlagen der Arithmetik, 1884): ‘Often it is only through enormous intellectual work, which can last for hundreds of years, that knowledge of a concept in its purity is achieved, by peeling off the alien clothing that conceals it from the mind’s eye’ (Frege 1997, 88). Wittgenstein is known to have had a copy of the Grundlagen at Cambridge (now preserved in the Russell Archives), which he abandoned there when he returned to Norway in October 1913 (Kienzler 2011, 81–2). Another possible influence upon Wittgenstein may have been Heinrich Hertz. This is also noted by Hacker (1972, 12). In the closing remark of Hertz’ introduction to a collection of papers he says: scientific accuracy requires of us that we should in no wise confuse the simple and homely figure, as it is presented to us by nature, with the gay garment which we use to clothe it. Of our own free will we can make no change whatever in the form of the one, but the cut and colour of the other we can choose as we please. (Hertz [1892] 1893, 28) 11. Katz here repurposes without citation Quine’s example of ancient astronomy (1953, 47), which he later quotes for the same purpose (Katz 1972, 9). 12. For example, when Keith Allan says: ‘His semantic markerese is like chemical formulae in that it translates into English’ (Allan [1986] 2014, 317), cf. note 3 above. 13. The diagram’s numbering has been removed. 14. Further, as Gilbert Harman makes clear in ‘Meaning and Semantics’ (1974), Lewis’ argument can also be applied to truth-conditional theories of meaning: Similarly, there is a sense in which we can know the truth conditions of an English sentence without knowing the first thing about the meaning of the English sentence. To borrow David Wiggins’s example, we might know that the sentence “All mimsy were the borogroves” is true if and only if all mimsy were the borogroves. However, in knowing this we would not know the first thing about the meaning of the sentence, “All mimsy were the borogroves.” (Harman 1974, 6) Wiggins’ example is to be found in a footnote in his ‘On Sentence-Sense, WordSense, and Differences of Word-Sense’ (1971), which comments on the Vienna Circle’s verifiability criterion of meaning: I surely cannot say or explain what All mimsy were the borogroves means by saying that this sentence will be true if and only if everything satisfies the open sentence if X is a borogrove then X is mimsy. And it is certainly a part of what would still be lacking in this explanation that it gives no idea at all of what investigations with what outcome would count for or against the assertion. (Wiggins 1971, 19n)

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It is somewhat ironic that both Wiggins and, following him, Harman misquoted from the poem ‘Jabberwocky’ by Lewis Carroll. The word is instead ‘borogoves’. Although the borogroves sounds like a miserable place to be, perhaps near the wabe and where one might find a mimsy borogove, it is uncertain whether the borogroves could really be mimsy at all. Perhaps more ironically, the word first appears in Carroll’s earlier ‘Stanza of Anglo-Saxon Poetry’ (1855), where he provided a literal English translation for the ‘nonsense’ sentence containing it: ‘all unhappy were the parrots’ (Carroll [1855] 1932, 141). 15. Admittedly, Lewis seems to rely upon his impression of the early Katz and Postal version of the theory from 1964, and it is not until the Democritean approach is spelt out by Katz in 1971 that its differences with other, non-Democritean approaches generally would have become clear. However, this kind of misunderstanding of Katz has continued for quite some time. 16. Here Platonism is referred to under the banner of ‘theories of meaning based upon notions of “real essence”’ and is called uninformative, together with theories based upon notions of ‘mental idea’, due to a lack of a criterion for two expressions having the same semantic property, in particular synonymy (Katz and Fodor 1962, 212). 17. Keith Allan claimed, rather proleptically, that the change had instead already happened in 1972: K’s Platonism can be traced back to his Semantic theory (1972). Earlier (1967, 129), he had described the conceptual content of a semantic marker as ‘what is common to our individual ideas’. However, five years later he was saying that the concepts represented by semantic markers are not something that people have in mind on any one or any number of occasions: ‘Concepts . . . are abstract entities. They do not belong to the conscious experience of anyone’ (1972, 38). What K seems to intend here is that the content of a semantic marker is something like a Platonic Form (eîdos) or Idea (idéa). (Allan 1983, 678; cf. [1986] 2014, 89–90) Allan overlooked the fact that the quote continues immediately: ‘though they may be thought about’ (Katz 1972, 38). The passage was probably written not five years but one year later (cf. note 7). Further, Allan failed to notice that Katz’ early Democritean approach in 1972 ruled out condensed answers such as ‘Platonic archetypes’ (1972, 3, 7), and that, on the very next page, Katz states that ‘the question of what the ontological status of concepts and propositions is [. . .] will be left here without a final answer’ (1972, 39). 18. The symbols for identity and non-identity are used here redundantly. 19. This distinction, between what he called expressional and non-expressional relations, first appears in Katz and Katz 1977, but is not spelt out in detail there. 20. Apart from Katz’ own accounts, especially Katz (1998), I recommend Jacquette 1990 as an accessible account of the early Wittgenstein’s problem. 21. I exclude here several further clauses to the definition of this operator, which are superfluous for our present purposes (cf. Katz 1972, 160–8). 22. Katz later extends this device to infinite such collections of markers which he calls ‘antonymy sets’ (Katz 1972, 312).

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23. ‘K’ is an optional function that specifies that the variable should be filled by a more precisely determined semantic marker (Katz 1972, 104ff., 258ff.). 24. This formalism can be given an equivalent representation as a tree structure, with ‘(colour)’ dominating ‘(red)’ and ‘A/(X)’ (cf. Katz 1998, 567). 25. Full details can be found in Katz (1998 and 2004).

REFERENCES Allan, K. (1983), ‘Review of Language and Other Abstract Objects by Jerrold J. Katz’, Language 59, no. 3: 678–83. Allan, K. ([1986] 2014), Linguistic Meaning, Vol. 1, London: Routledge. Begley, K. (2019), ‘Review of Essays on Linguistic Realism’, The Linguist List, 30.1644. Available Online: https​:/​/li​​nguis​​tlist​​.org/​​issue​​s​/30/​​30​-1​6​​44​.ht​​ml Carroll, L. ([1855] 1932), The Rectory Umbrella and Mischmasch, London: Cassell & Co. Chomsky, N. and Katz, J. J. (1974), ‘What the linguist is talking about’, The Journal of Philosophy 71, no. 12: 347–67. Coliva, A. (2020), ‘Logical atomism and Wittgenstein’, in U. Zilioli (ed.), Atomism in Philosophy: A History from Antiquity to the Present, 301–11, London: Bloomsbury Academic. Frege, G. ([1892] 1952), ‘On concept and object’, trans. P. T. Geach, in P. Geach and M. Black (eds), Translations from the Philosophical Writings of Gottlob Frege, 42–55, Oxford: Basil Blackwell & Mott. Frege, G. ([1918] 1968), ‘The thought: A logical inquiry,’ trans. A. M. and M. Quinton, in E. D. Klemke (ed.), Essays on Frege, 507–36, Urbana: University of Illinois Press. Frege, G. ([1897] 1979), ‘Logic’, in H. Hermes, F. Kambartel and F. Kaulbach (eds), Gottlob Frege: Posthumous Writings, 126–51, Oxford: Basil Blackwell. Frege, G. (1997), The Frege Reader, M. Beaney (ed.), Oxford: Blackwell. Hacker, P. M. S. (1972), Insight and Illusion: Wittgenstein on Philosophy and the Metaphysics of Experience, Oxford: Oxford University Press. Harman, G. (1974), ‘Meaning and semantics’, in M. K. Munitz and P. K. Unger (eds), Semantics and Philosophy, 1–16, New York: New York University Press. Hertz, H. R. ([1892] 1893), Electric Waves: Being re-searches on the propagation of electric action with finite velocity through space, trans. D. E. Jones, London: MacMillan & Co. Jacquette, D. (1990), ‘Wittgenstein and the color incompatibility problem’, History of Philosophy Quarterly 7, no. 3: 353–65. Kang, J. (2005), ‘On the composition of the prototractatus’, The Philosophical Quarterly (1950–), 55, no. 218: 1–20. Katz, F. M. and Katz, J. J. (1977), ‘Is necessity the mother of intension?’, The Philosophical Review 86, no. 1: 70–96. Katz, J. J. (1964), ‘Semantic theory and the meaning of “Good”’, The Journal of Philosophy 61, no. 23: 739–66. Katz, J. J. (1965), ‘The relevance of linguistics to philosophy’, The Journal of Philosophy 62, no. 20: 590–602.

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Katz, J. J. (1966), The Philosophy of Language, New York: Harper & Row. Katz, J. J. (1967), ‘Recent issues in semantic theory’, Foundations of Language 3, no. 2: 124–94. Katz, J. J. (1968a), ‘The logic of questions’, in B. Van Rootselaar and J. F. Staal (eds), Logic, Methodology and Philosophy of Science III, 463–93, Amsterdam: NorthHolland. Katz, J. J. (1968b), Letter to Frits Staal, 28th of August, 1968. [In the possession of this author.] Katz, J. J. ([1966] 1970), ‘The semantic component of a linguistic description’, in A. and K. Lehrer (eds), Theory of Meaning, 176–98, Englewood Cliffs: Prentice-Hall. Katz, J. J. (1971), The Underlying Reality of Language and its Philosophical Import, New York: Harper & Row. Katz, J. J. (1972), Semantic Theory, New York: Harper & Row. Katz, J. J. (1975), ‘Logic and language: An examination of recent criticisms of intensionalism’, in K. Gunderson (ed.), Language, Mind and Knowledge, Minneapolis: University of Minnesota Press. Katz, J. J. (1976), ‘A hypothesis about the uniqueness of natural language’, Annals of the New York Academy of Sciences 280: 33–41. Katz, J. J. (1977), ‘The real status of semantic representations’, Linguistic Inquiry 8, no. 3: 559–84. Katz, J. J. (1981), Language and Other Abstract Objects, Totowa: Rowman and Littlefield. Katz, J. J. (1982), ‘Common sense in semantics’, Notre Dame Journal of Formal Logic 23, no. 2: 174–218. Katz, J. J. (1986), Cogitations, Oxford: Oxford University Press. Katz, J. J. (1990), The Metaphysics of Meaning, Cambridge, MA: The MIT Press. Katz, J. J. (1992), ‘The new intensionalism’, Mind, New Series, 101, no. 404: 689–719. Katz, J. J. (1998), ‘The problem in twentieth-century philosophy’, The Journal of Philosophy 95, no. 11: 547–75. Katz, J. J. (2004), Sense, Reference, and Philosophy, New York: Oxford University Press. Katz, J. J. and Fodor, J. A. (1962), ‘What’s wrong with the philosophy of language?’, Inquiry 5, nos. 1–4: 197–237. Katz, J. J. and Fodor, J. A. (1963), ‘The structure of a semantic theory’, Language 39, no. 2: 170–210. Katz, J. J. and Fodor, J. A. (1964), ‘The structure of a semantic theory’, in J. A. Fodor and J. J. Katz (eds), The Structure of Language: Readings in the Philosophy of Language, 479–518, Englewood Cliffs: Prentice-Hall. Katz, J. J. and Postal, P. M. (1964), An Integrated Theory of Linguistic Descriptions, Cambridge, MA: The MIT Press. Kienzler, W. (2011), ‘Wittgenstein and Frege’, in O. Kuusela and M. McGinn (eds), The Oxford Handbook of Wittgenstein, 79–104, Oxford: Oxford University Press. Langendoen, D. T. (2010), ‘Just how big are natural languages?’, in H. van der Hulst (ed.), Recursion and Human Language, 139–46, Berlin: De Gruyter. Langendoen, D. T. (2018), ‘Languages as complete and distinct systems of reference’, in C. Behme and M. Neef (eds), Essays on Linguistic Realism, 255–70, Amsterdam: John Benjamins.

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Lewis, D. (1970), ‘General semantics’, Synthese 22: 18–67. McGuinness, B. (2002), ‘Some pre-Tractatus Manuscripts’, in Approaches to Wittgenstein, 259–69, London: Routledge. Quine, W. V. O. (1953), ‘The problem of meaning in linguistics’, in From a Logical Point of View, 47–64, Cambridge, MA: Harvard University Press. von Fintel, K. and Matthewson, L. (2008), ‘Universals in semantics’, The Linguistic Review 25: 139–201. Wiggins, D. (1971), ‘On sentence-sense, word-sense, and differences of word-sense: Towards a philosophical theory of dictionaries’, in D. D. Steinberg and L. A. Jakobovits (eds), Semantics: An Interdisciplinary Reader in Philosophy, Linguistics and Psychology, 14–34, London: Cambridge University Press. Williams, B. ([1972] 1993), Morality: An Introduction to Ethics, Cambridge: Cambridge University Press. Wittgenstein, L. (1922), Tractatus Logico-Philosophicus, trans. F. P. Ramsey and C. K. Ogden, London: Routledge & Kegan Paul. Wittgenstein, L. (1971), ‘Prototractatus: 4.0015’, trans. D. F. Pears and B. F. McGuinness, University of Iowa Tractatus Map. Available Online: http://tractatus​.lib​.uiowa​.edu Wittgenstein, L. (2015), ‘MS-104, 36 Facsimile’, Bergen Nachlass Edition, Bodleian Libraries, Oxford, Wittgenstein Source. Available Online: http:​/​/www​​.witt​​genst​​einso​​ urce.​​org​/B​​FE​/Ms​​​-104​,36_f Wolff, R. P. ([1970] 1998), In Defense of Anarchism, Berkeley: University of California Press. Wolff, R. P. (2010), ‘The philosopher’s stone’, 30 April 2010, Blog. Available Online: http:​/​/rob​​ertpa​​ulwol​​ff​.bl​​ogspo​​t​.com​​/2010​​/04​/m​​emoir​​-volu​​me​-tw​​o​-cha​​pter-​​​three​​-seco​​ nd​.ht​​ml Wolff, R. P. (2019), ‘The philosopher’s stone’, 12 April 2019, Blog. Available Online: http:​/​/rob​​ertpa​​ulwol​​ff​.bl​​ogspo​​t​.com​​/2019​​/04​/c​​redit​​-wher​​e​-cre​​​dit​-i​​s​-due​​.html​ Zwicky, A. M. (1973), ‘Linguistics as chemistry: The substance theory of semantic primes’, in S. R. Anderson and P. Kiparsky (eds), A Festschrift for Morris Halle, 467–85, New York: Holt, Rinehart and Winston.

CHAPTER 18

Atoms and knowledge NICK TREANOR

This is a chapter about knowledge of atoms. It is not, however, about knowledge concerning atomic things, whether those be atoms of chemistry, physics or metaphysics. It is instead about knowledge of what is true of things, atomic or otherwise, and about whether this is or amounts to knowledge of atomic truths. My aim will not be to answer this question, but to trace out the appeal of this picture and unearth and explore a central ambiguity in our thinking about it. The atoms that concern me are atoms of truth not of being. I am not a historian of philosophy, but the roots of the idea probably reach into the early soil of our discipline. Just as it is very natural to think that the objects of ordinary life are made of smaller things, which are made of smaller things, which are made of smaller things and so on, it seems natural to think that the truths of everyday life have a mereological structure of some sort. Suppose it is true, for instance, that most cars on the road today are diesel. That truth (that most cars on the road today are diesel) seems to be made up, in some uncertain way, of various other truths – that there are cars on the road, that being diesel is a way for a car to be, that diesel is a kind of fuel and so on. This may seem an outlandish thing to say. These days we are more accustomed to speak of truths entailing other truths than of truths containing or having other truths as their more basic material (except in cases where words like ‘and’ and ‘but’ do the knitting that a quick pull would unravel). I grant this, but think that talk of entailment rather than of containment or inclusion is a theoretical move rather than an expression of a natural view or starting place. Suppose it is true that most cars on the road today are diesel. Part of what’s true when that is true, surely, is that diesel is a kind of fuel, that there are cars on the road today, the being diesel is a way that cars can be and so on. Or to put it another way, for it to be true that most cars on the road today are diesel, all these other truths have to be true – it has to be true that there are cars, that diesel is a fuel, that being diesel is a way for a car to be, that there is some period of time that is today, that there’s such a thing as time such that there can be periods of it and so on. Moreover, the truth that most cars on the road are diesel does not seem to merely entail these other truths, as it does the truth that 2 + 2 = 4. It is rather that it seems to involve them in some constitutive way. Take all these other truths and fuse them into a mass – that is the truth that most cars on the road today are diesel. This claim is intolerably picturesque, to be sure, but the starting place in philosophy always is.

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The ‘and so on’ in the previous paragraph echoes the ‘and so on’ when we say that physical things are made of smaller things, which are made of smaller things and so on. In both cases there is the suggestion that the direction of travel continues to a place we cannot see or cannot say. In both cases we move from what we can see (that a house is made of bricks, and bricks of clay and straw, and clay and straw of finer things in turn) or what we can say (that most cars on the road are made of diesel, that diesel is a fuel, that being diesel is a way a car can be) to things we cannot see or say but conjecture must be there.1 I have tried to motivate the idea that it is natural to think that truth has a mereological structure by appeal to the idea that when something is true, or at least when the truths of ordinary life are true, they seem to consist in other, smaller truths being true, where smaller here is not a spatial notion but a containment notion. But there are other ways to motivate the idea if these remarks do not get purchase with the reader. Think of what is true – all of it. Does it not seem, prima facie at least, that what is true (the totality) is built out of, or made up of, other truths or collections of truths? (The ‘build up’ relation may be no more than conjunction or concatenation, but that is all we’d need.) Or to put it another way: There is what’s true, right? (Here the reader is asked to assent.) Now, is that just what’s true, end of story, or is what’s true made up, in some way, of all the things that are true? I think it is hard to imagine how it could be otherwise. The focus in the paragraph immediately preceding paragraph is on all the truth, or on what’s true without any domain restriction, but a parallel line of thought can be generated for any more narrow subject matter. Think of what’s true concerning continental drift, that is, what’s true concerning the process whereby continents move, migrate, break apart and conjoin. Let’s call that the truth about continental drift. Does that truth not seem to consist in other truths? Again, I find it hard to understand how it could not. Or consider what’s true concerning whether I am writing this chapter on a laptop. If it’s true that the truth about continental drift consists in or contains other truths, it’s not easy to see how the truth about whether I’m writing this chapter on a laptop doesn’t. The truth is that I am writing it on a laptop. And part of that is the truth about what a laptop is, what writing is, what it is to write on a laptop and so on. If there is resistance to these remarks, one source of it could be that there is an important difference between how we think of parthood or containment for physical things compared to for truths. If x is a physical object and y is a proper part of x, then y is smaller than x in a relatively straightforward sense. Moreover, if x is not identical to z, but both x and z have y as a proper part, then x and z must overlap, again in a relatively straightforward sense. Neither of these things seem the case when we think of truths containing other truths. Most cars on the road are diesel, we suppose. Part of what’s true when that is true is that diesel is a fuel – that diesel is a fuel is part of the truth that most cars on the road are diesel. But there is no straightforward or obvious sense in which the truth that diesel is a fuel is smaller than the truth that most cars on the road are diesel.2 Moreover, the first rocket that flew to the moon (let us suppose) ran on diesel. That diesel is a fuel would be part of that truth, too. But the truth that most cars today are

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diesel and the truth that the first rocket that flew to the moon ran on diesel don’t overlap in any straightforward or obvious way, despite them having a part in common.3 In light of these differences, we might find it quite strange to think that truths contain other truths or have other truths as parts. I think this response is perceptive, but misdescribes what it notices. It is strange to think that truths contain other truths as parts. This could be a sign that truths don’t have other truths as parts. But concluding that is rash and two possibilities strike me as more likely. First, we could construe the puzzle as generated by one’s having attached to the general notion of containment or parthood aspects that are particular to spatial containment or parthood. Physical parts are smaller than what they are parts of, and overlap makes sense with physical things, not because of parthood itself but because of the nature of physical objects as extended. Alternatively, we could think the lesson is that work needs to be done to make clear what ‘smaller’ and ‘overlap’ mean in the realm of truth; the lesson on this interpretation is only that there is a gap in theory, not that talk of parthood with regard to truth is unintelligible.4 What I have tried to offer so far is some reason to think it is natural to suppose that truths, at least the ones of everyday life, are made of or contain other truths, however unclear the exact nature of this is. This alone does not get us to atomism about truth. It could be, for instance, that all truths are made of other truths, that the structure of truth is one of infinite descent. As in the case of atomism in the domain of physical things, however, where it is natural to move from observing a direction of travel to imagining there must be an unseen place it ends, in the domain of what is true it is perhaps natural to think that if ordinary truths are made of or contain other truths, this all has to stop somewhere. I am not saying this is correct or, certainly, that to the degree the atomic picture is a natural one that this aspect of it is normally in view. The suggestion is rather that, if pushed on where the travel goes, it is perhaps natural to think it must stop somewhere.5 Wittgenstein and Russell describe the most austere version of this picture, each with his own distinctive eloquence: Wittgenstein: If all true elementary propositions are given, the result is a complete description of the world. (Tractatus, 4.26) Russell: If the world is composed of simples – i.e., of things, qualities and relations that are devoid of structure – then not only all our knowledge but all that of Omniscience could be expressed by means of words denoting these simples. We could distinguish in the world a stuff (to use William James’s word) and a structure. The stuff would consist of all the simples denoted by names, while the structure would depend on relations and qualities for which our minimum vocabulary would have words. (259) For both of them, as expressed here in these passages, it is not just what is true is made up of other truths, nor that there is, way down at the bottom, an atomic level where all the truths bottom out. It is that this set of atomic truths is all the truth there is, at least in the sense that they describe reality completely and one who knew them would be omniscient. I describe this as austere because it holds that just as one

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might think that non-atomic physical objects are no addition to being, non-atomic truths are no addition to truth. I hope the aforementioned remarks capture one way in which it is natural or common to have a picture of truth or of what is true on which it is ultimately atomistic. Much of it will sound at least broadly familiar. My own view is that this picture is ultimately unintelligible, although I will argue for something more modest here. To get to this, however, it will help to briefly address two different ways we might think of atomism in the realm of being rather than truth. Consider what Travis Dumsday (this volume, pg 400–419) calls atomism version 1 and atomism version 2. Version 1 takes nature to bottom out at indivisible nonextended point-particles; things are made of smaller and smaller things, and so on, until you reach things that have no size and are not divisible. Version 2 also takes nature to bottom out, but at indivisible extended objects; things are made of smaller and smaller things, and so on, until you reach things that have a size but aren’t, for whatever reason, divisible. Both versions face thorny problems. Version 1 struggles to explain how things that have no size can, together, have a size, as they seem to when they compose any ordinary physical object. Version 2 struggles to explain how something can be extended but not be divisible, even in principle, given that it seems that to be extended is to be such that a bit of you is here and a bit of you is there. The relevant thing to notice for present purposes is this: if atoms are as version 1 construes them, then they have number but not extent or volume; how much of them you have can be answered with a number but not with a volume. Version 2, in contrast, is different: on that picture, there are some number of atoms, and each has a size because each is extended (perhaps they’re all the same size, perhaps they differ in size). The answer to how much of them you have could, depending on context or speaker intention, properly be given by a number or properly be given by their collective volume. I have briefly looked at this distinction in kinds of atomism about things because a similar distinction can be drawn about atomism in the domain of truth. One sort of atomism about truth would be a version 1 picture, wherein what is true is given by some number of truths and how much is true, or how much truth this person knows or that book expresses, and so on, is only (ever) answered with a number – the only dimension truth has is cardinality. A quite different version would be a version 2 picture wherein there is some proper quantitative dimension in the domain of truth beyond this, something akin to the notion of volume in the domain of atomic objects. On this version of atomism about truth, there would be some number of truths and each truth would have a size, although of course this is analogous to rather than the very same thing as size with regard to physical objects. Set aside for the moment, please, the question of why one would ever endorse this sort of atomism – we will get to that. The point is only to see that there could be such a version of atomism in the domain of truth. The modality in that sentence is meant, at this point, to be only epistemic: for all we know, not yet having thought about it, there could be a size or size-esque dimension to truth beyond cardinality. In my view, this is not something that we have thought enough about, and there are concomitant, deeply interesting questions (to me anyway!) that lie in the same

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region.6 Part of the explanation of this is that we tend not to think about atomism about truth in a serious way; hence it is not surprising that we tend not to think about whether an atomism in which the quantity of truth dissolves into number is or is not to be preferred to one where at the bottom level there is number plus extentof-truth. That said, though, we do think, at least obliquely, about the number versus number plus extent question when we talk about truth not at the bottom level, so to speak. I will try to give a sense of what I mean. First, there are times that we seem to think of truths (ordinary, everyday truths, not atomic truths) as having both number and extent. Consider how common it is to talk, loosely and metaphorically, of adding some truths to one’s stockpile. I don’t think those who employ this metaphor have likely thought much about, much less mean to commit, to what is embedded in that metaphor: that truths have both number and an extent just as the durable goods and materials that are found in literal stockpiles have both number and volume. If ‘stockpile of truths’ were the only such example, it would best be thought of as an idiom rather than suggestive of an ambiguity in thought concerning whether truths (of the ordinary rather than the atomic kind) have only number or number plus truth-extent. But similar examples abound. Philosophers talk of piling up truths or of truths being heaped up, and there is a pervasive use of mass and spatial terms to talk of truth and of content (that which is true or false).7 Sometimes, this is taken as far as the claim that truth has extent only rather than number, a kind of stuff ontology of truth. For instance, in discussing the principle of charity, Davidson notes it is usually characterized as the idea that we should interpret a person such that most of what she believes is true, and then says: This way of stating the position can at best be taken as a hint, since there is no useful way to count beliefs, and so no clear meaning to the idea that most of a person’s beliefs are true. A somewhat better way to put the point is to say there is a presumption in favor of the truth of a belief that coheres with a significant mass of belief. (Davidson 2001, 138–9) Here Davidson is talking about belief, but it is clear the point is as much about truth or true content. There’s no clear meaning to the idea that most of a person’s beliefs are true because what a person believes (true and false propositions) can’t be counted, he thinks. Doubts about the intelligibility of the idea of counting shift him towards a picture on which the plural count vocabulary (most beliefs, most truths) is replaced with massy vocabulary (a significant mass of belief/truth). We need not endorse that more radical conclusion to recognize that in ordinary thought and talk, we sometimes seem to think of truth as both having number and as having extent, albeit in some unresolved, unclear fashion. We speak of the truth, of truths, of many truths, of more truth, of much truth and so on. I take pains to emphasize that the aforementioned talk of truth is suggestive but no more. I think if we look carefully we see that the opposite conception is also latent, that it is also latent that what is true has number but no other dimension of size. One way to see this is to note that many philosophers seem to think the questions ‘how much truth does a person believe’ and ‘how many truths does a

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person believe?’ are synonymous. Consider, for example, the following remark by Foley and Fumerton (1982): It is no doubt true that most people would be inclined to agree with Lehrer’s and Rescher’s suggestion that we should try to believe rationally as many truths as possible. They would be inclined to agree, that is, that we ought to be curious about the world; we ought to find out as much about it as we can. (55) Regardless of whether Foley and Fumerton are right that most people would agree with the suggestion they mention, their phrasing makes clear they think that finding out as much about the world as we can is the very same thing as, or is just another way of talking about, believing as many truths as possible. This is presented not as a substantive picture or theory of the relation between more truth and more truths, but as (and the reader naturally takes it to be, I think) an alternative locution for the very same thing.8 We can also see the grip of the conception of truth on which ordinary everyday truths have number but not extent by looking at what is known as the trivial truths objection to the claim that the goal of inquiry is to acquire more truth. This could not be the goal of inquiry, the objection insists, because some truths are significant and others are trivial and inquiry properly ought to aim for the significant truths over the trivial ones. Inquiry aims for more truth, perhaps, but not just more truth – it aims for some kinds of truth more than other kinds. I have discussed this argument at length elsewhere9 so will only point out here that those who give the objection, or find it compelling, have simply assumed without argument that so-called trivial truths and so-called significant truths are the same amount of truth – one truth’s worth of truth, if you will. Strictly speaking, that is compatible with the idea that ordinary truths have both number and extent, as long as the extent each has is always the same. But it is more reasonable to think that background assumption is simpler and is just that the measure of truth is cardinality alone. At this point the reader may well be confused. I seem to have argued both that we have a latent conception of truth on which ordinary, everyday truths have both cardinality and extent and that we (or many of us) have a latent conception of truth on which, when it comes to ordinary truths, cardinality is everything. That sounds like I am contradicting myself. I think the proper read of the situation, however, is that what I’ve said is true and it points to a deep ambiguity in how we think about the measure of truth. The problem with attributing the numbers-are-everything view as a settled view rather than as something people endorse in temporary or fleeting contexts is that if it is a stable feature of how people think about the measure of truth, then they would have to either (i) maintain that line even in the face of patent counterexamples or (ii) endorse some version of genuine atomism about truth, which few would be willing to do. Consider a philosopher who endorses the trivial truths argument and who alleges they think the questions ‘how much truth does S believe?’ and ‘how many truths does S believe?’ are the same questions worded differently. What do they say when confronted with this pair of ordinary, everyday truths:

1. I am wearing a blue jacket.



2. I am wearing a blue jacket and black shoes.

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Each of those is one truth, but it would be very hard to deny that the second is more truth than the first. If a person really thinks how much truth and how many truths are the same, though, then they either have to affirm that 1 and 2 aforementioned are the same amount of truth or they have to affirm that the second is more than one truth (e.g. two truths). In this example, this latter option is the obvious one. Denying 2 is more truth than 1 seems to be a non-starter, whereas taking 2 to be two truths rather than merely one is an easy thing to endorse and the natural dialectical move. The problem, however, is that it is very hard to see how to stop this process once it starts, or to make sense of where it stops. For consider:

3. I am wearing a blue jacket.



4. I am wearing a jacket.



5. I am wearing something.



6. I stand in some spatiotemporal relation to a jacket.



7. I stand in some spatiotemporal relation to something that isn’t me or a part of me.



8. Something stands in the wearing relation to something else.



9. Something stands in some spatiotemporal relation to something else.

We wanted to solve the original puzzle by saying that 2 was really two truths, the truth that I am wearing a blue jacket and the truth that I am wearing black shoes. As a first step that seems the right thing to say. But it’s very hard to see that things stop there. The truth that I am wearing a blue jacket seems to tell us more about the world, and therefore to be more truth (more truth not more true) than that I am wearing something, that I stand in some spatiotemporal relation (of which wearing is just one option) to a jacket and so on. The person who wants to maintain that numbers are everything can’t stop at explaining that 2 is more truth than 1 by saying it is two truths rather than one truth. They need to find a place where the possibility of further division stops and therefore genuine, non-arbitrary counting can begin. That would be genuinely atomic level – by definition. Let me step back for a moment and address where we are in the discussion. I started by saying that some sort of decompositional picture of truth seems both natural and rooted deeply in our discipline. I then distinguished two versions of this decompositional picture. Both agree that decomposition continues to an atomic level such that there are genuinely atomic, discrete truths, which have a cardinality. But they disagree on whether there is, quantitatively, more than that to the measure of truth (such a thing as the extent of truth, illustrated by comparison to volume in the domain of the physical). I then said that, for the most part, philosophers don’t think about this issue with regard to truth atoms but do think about it, at least in an oblique and indirect way, with regard to ordinary truths, truths that are expressed by the everyday sentences of natural language. ‘Think about’ might be too strong a way to put it, the basic idea being that our thought and talk is largely silent on the question with regard to atomic truths but does reflect something on the question with regard to ordinary truths. What it reflects, however, is a deep ambiguity. We use

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metaphors on which truths have both number and extent, and we naturally move to massy talk when count talk starts to baffle (or find such moves intelligible rather than incoherent when other people make them). But we also employ arguments, and claim to subscribe to views, that collapse quantity into mere number, even when talking about ordinary truths. In the space remaining, what I’d like to do is turn from ordinary truths to atomic truths and put pressure on the possibility that a version 1 atomism is true. My goal will not be to defend version 2 atomism – I have separate doubts about that. It will be rather to say that if the options are version 1 atomism or version 2 atomism, then version 2 atomism, specifically a version 2 atomism in which truths have extent but not all the same extent, must be true. Version 1 atomism holds that there are atomic truths and that there is no dimension to size other than cardinality. To show this sort of atomism is false, what we need is a case where there are two bodies (that is intended to be ontologically uncommitting) of truth, each with the same number of atomic truths, yet where one is more truth than the other. In one sense, it is difficult to imagine how to construct such a case since we have no idea what atomic truths would look like; to paraphrase Russell, as quoted earlier in the chapter, atomic truths are not experienced as such but known only inferentially as the limit of analysis. Thus, it seems impossible to collect 500 of them here and 500 of them there and compare which collection, if either, is more truth than the other. But in another sense, we can do this, for there is another way for the cardinality of two sets to be the same other than by each set containing the same finite number of truths. We need only construct an example where each body of truths has the same infinite cardinality, say the cardinality of the continuum, but where one collection of truths is, or seems to be with whatever clarity our intuitions can establish, more truth than the other. Here is such a case: Start by thinking of the Encyclopaedia Britannica and of a brief pamphlet for tourists that describes a local attraction such as a historic castle. Assume that each contains nothing but the truth. It seems pretty clear that the encyclopaedia contains more truth than the brief pamphlet, but this isn’t the case we’re looking for since it’s far from clear that each contains the same number of truths (finite or otherwise). However, now consider what we can call the alethic complements of each: The alethic complement of the tourist brochure is something that contains or expresses every truth that it does not. The alethic complement of the Encyclopaedia Britannica contains or expresses every truth that it does not. The relation of being an alethic complement is symmetric, and a pair of alethic complements is a complete description of the world. (Treanor 2018, 1062) In the next step, consider which alethic complement seems to contain or express more truth: the one that contains everything that’s true save what’s in the Encyclopaedia Britannica, or the one that contains everything that’s true save what’s in the local tourist pamphlet: The alethic complement of the tourist pamphlet contains a vast amount of truth – just think of all the truth, about any topic whatsoever, that the tourist brochure

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leaves out. The truth contained in the alethic complement of the Encyclopaedia Britannica is also vast, as well-researched and comprehensive an encyclopaedia as it is. Yet the truth contained in the alethic complement of the Encyclopaedia Britannica is not quite so vast as the truth contained in the alethic complement of the tourist pamphlet. This is more or less the claim we started with, given what an alethic complement is: There is more truth in the Encyclopaedia Britannica than in the tourist pamphlet. (Treanor 2018, 1062) For the final step of the argument, notice that the case we are describing is one wherein what’s being compared are two bodies of truth where, if any version of atomism about truth is true, each body of truth has the same cardinality: [I]f there is any number of truths at all, then it is an infinite number, presumably a very large infinity. So the two alethic complements each contain infinitely many truths – and importantly and most plausibly, the same order of infinity. So we have a difference in how much truth each contains without any difference in the cardinality of the truths that each contains. (Treanor 2018, 1062) If this argument is sound, then if atomism is true then there must be something, beyond cardinality, that contributes to the measure of truth. The truths at the bottom level, whatever they are, would not be version 1 atomic truths. Here we can circle back and see the parallel with atomism version 1 and atomism version 2. Take any circle and imagine it cut in four, like a pie with four large slices. Any single slice has exactly as many points as the other three put together – the cardinality of the continuum. Put another way, the number of points in a given plane figure doesn’t tell us what size it is (in fact, it tells us nothing at all about its size). Nonetheless, it is intelligible that any three slices together are bigger than any one slice alone, since they take up more space or extend further in a two-dimensional plane. If there are atomic truths, then there must be something akin in the domain of truth.

NOTES 1. Compare Russell in The Philosophy of Logical Atomism: ‘When I speak of “simples”, I ought to explain that I am speaking of something not experienced as such, but known only inferentially as the limit of analysis. It is quite possible that, by greater logical skill, the need for assuming them could be avoided’ (2010, 143). 2. More is true when it’s true that most cars are on the road are diesel than when it’s true that diesel is a fuel, since when it’s true that most cars on the road are diesel it’s true that diesel is a fuel and more besides. So there is some sense in which the one truth is smaller, or less truth, than the other. My point is not that this isn’t the case, but that it is not straightforward how to understand it. 3. Again, the idea is not that they fail to overlap, but that if they do, it is not in a readily understood way. We could say they overlap by having a part in common. But that is empty in the absence of a way of understanding them as having a part that is in the same place; that is what we are missing. We could talk of propositional space and say that the truth they share as a part occupies such and such position in propositional

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space, and that that is how or why they overlap. But that redescribes the problem rather than solves it. 4. See Yablo 2015 for a discussion of the generality of parthood. As he puts it there, ‘To learn that x is or has a part, however, tells you nothing about the sort of thing it might be, considered in itself. Philosophers have discovered some strange entities over the years, but nothing so ontologically outre as not to stand in mereological relations.’ Yablo’s discussion there clearly suggests that truths have other truths as parts. See also Yablo 2014 and Lewis 1988. 5. Russell seemed to have thought, at least at times, that this was a straightforward matter, as he remarks in his lectures on logical atomism ‘I confess it seems obvious to me (as it did Leibniz) that what is complex must be composed of simples, though the number of constituents may be infinite.’ (2010, 143) An alternative picture is offered by Eugene Bronstein, writing in the newly founded journal Analysis in 1934. He insists: ‘[I]t is nothing but a risky inference from a directional analysis to basic facts; and that as termini ad quos of the analysis basic facts may or may not in fact exist . . . . I wish very humbly to suggest that . . . though we can have several things that are simpler, we can never have anything that is simple’ (1934, 11–14). 6. For instance: How much is true? Does how much is true vary from time to time or world to world, or is it necessarily fixed and unchanging? Is there more true of that city than there is of this apple, or is just as much true of one as of the other? I can change what is true, but can I change how much is true? 7. Some examples: ‘[O]ur basic cognitive aim is to come into possession of as much truth as possible and to avoid false beliefs’ (Alston 1982, 7); ‘A very plausible set of [cognitive] goals are the oft-cited aims of believing the truth – as much truth as possible – and avoiding error’ (Goldman 1980, 32); ‘I have suggested that epistemic justification is essentially related to the cognitive goal of truth . . . . We aim both to avoid as much error as we can and to obtain as much truth as we can’ (Moser 1985, 5). It is also common to talk of propositional space, of truths being regions of propositional space, of propositional space being divided into regions believed or known and regions not believed or known and so on. 8. I discuss this example and the point I’m making here at greater length in Treanor 2018. 9. See Treanor (2013, 598–9), Treanor (2014), and Treanor (2018, 1055–6).

REFERENCES Alston, W. P. (1982), ‘Religious experience and religious belief ’, Noûs 16, no. 1: 3–12. Bronstein, E. D. (1934), ‘Miss Stebbing’s directional analysis and basic facts’, Analysis 2, nos. 1–2: 10–14. Davidson, D. (2001), ‘A coherence theory of truth and knowledge’, in Subjective, Intersubjective, Objective: Philosophical Essays, vol. 3, 137–57. Oxford: Clarendon Press. Foley, R. and Fumerton, R. (1982), ‘Epistemic indolence’, Mind 91, no. 361: 38–56. Goldman, A. (1980), ‘The internalist conception of justification’, Midwest Studies in Philosophy 5: 27–51.

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Lewis, D. (1988), ‘Statements partly about observation’, Philosophical Papers 27, no. 1: 1–31. Moser, P. K. (1985), Empirical Justification, Dordrecht: D. Reidel Publishing Company. Russell, B. (1948), Human Knowledge: Its Scope and Limits, London: Allen & Unwin. Russell, B. (2010), The Philosophy of Logical Atomism. New York: Routledge Classics. Treanor, N. (2013), ‘The measure of knowledge’, Noûs 47, no. 3: 577–601. Treanor, N. (2014), ‘Trivial truths and the aim of inquiry’, Philosophy and Phenomenological Research 84, no. 3: 552–9. Treanor, N. (2018), ‘Truth and epistemic value’, European Journal of Philosophy 26, no. 3: 1057–68. Yablo, S. (2014), Aboutness, Princeton: Princeton University Press. Yablo, S. (2015), ‘Parts and differences’, Philosophical Studies 173, no. 1: 141–57.

CHAPTER 19

Atoms and time II1 MAURO DORATO

THE CURRENT PHILOSOPHICAL RESEARCH FOR THE DISCRETE NATURE OF TIME: SOME MOTIVATIONS There is little doubt that the greatest change in the theoretical pillars on which classical physics was based was not caused by relativity theory (in both its special relativistic and generalist versions), but rather by quantum mechanics. And within quantum mechanics, possibly the best confirmed physical theory ever conceived, the quantization of some physical magnitudes, from action to spin and from energy to angular moment, is of crucial importance. In order to provide some evidence to the claim that quantum mechanics reintroduced a form of ‘atomism’ linked to the discrete character of some of these physical magnitudes – and therefore only partially similar to that defended in ancient philosophy by Democritus, Epicurus and Lucretius – an extremely brief overview of its history will suffice. At the end of the nineteenth century, there were two rival theories about the nature of the physical world, energetism and atomism. The former was based on the idea that reality is made of continuous, viscous stuff (energy), the latter, defended by Boltzmann and others, on the hypothesis that the world is made of atoms (Baggott 2011, 7). Continuity versus discreteness. Already Kirchkoff (1824–1887) had realized that ‘the rate of the radiation emitted and absorbed in a cavity depended only on temperature and frequency and not on the shape and the material of the cavity’ (Baggott 2011, 10). As Baggott notes, this was a sign that he was dealing with something fundamental, something, we could add, like the force of gravity, which does not depend on the particular material of which the mass of a body is made. The challenge was picked up by Planck, who in 1900 speculated that in order to fit already known equations with data, one had to assume that energy comes in packets, or integer multiples of hv, where v is the frequency of the oscillators that he used for his equation and h is a constant. As is well known, in 1905, Einstein used the discrete character of energy to explain the photoelectric effect, and Bohr tried to explain the stability of the atom by assigning the orbiting electrons discrete energy levels. In Bohr’s semi-classical model, when an atom was hit by a photon (or when emitting photons), it gained or lost energy in discrete multiples of hv, so that any electron jumping to another energy level did not go through all possible,

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intermediate values of energy. The fact that there was an orbit at which an electron had the lowest possible value of energy explained why electrons don’t spiral down to the nucleus and therefore also the stability of the atom. When De Broglie later postulated the dual nature of matter, his equation λ = h/p still referred to the discreteness of action as expressed by h, whose presence characterizes the whole theoretical structure of quantum physics. It is important to specify that Planck’s constant h has the dimension of [energy] x [time] or [momentum] x [length]. In this sense, quantum mechanics goes back to ancient atomism, but rather than vindicating an ontology of indivisibly particles moving in the void, it is based on a different kind of atomization, the atomization of action,2 or on the fact that the action h ‘cannot be further divided’. The discreteness of action has also important conceptual consequences. In particular according to Bohr’s theory of measurement, the fact that h is indivisible and ‘atomistic’ limits the very predictability of any finite interaction between a quantum system and a measurement apparatus and therefore of any measurement of a system that has not already been measured. To be more precise, such an interaction establishes a peculiar kind of holism or non-separability, so that it is the whole experimental situation that allows us to make precise predictions (Howard 1994; Dorato 2017). The same ‘atomistic’ behaviour of quantum entities (with or without mass) is evident from many simple experiments: when a photon polarized at 45 degrees interacts with a vertically oriented polarizer (90 degrees), it is either fully absorbed or goes through the polarizer with a probability of ½, which means that it is not partly absorbed and partly transmitted. The same holds for electrons going through two slits, as we must assume in order to explain interference: if we try to detect which slits the electron went through by using two detectors behind both slits, the electron is revealed either by one or the other detector, but not by both. In other words, the electron does not divide into two parts by hitting the two detectors at once. More abstractly, also the non-commutativity of conjugate variables expressed by Heisenberg’s indeterminacy principle (which holds for positions and momentum as well as for energy and time, despite the fact that these are continuous magnitudes) calls into play the discrete nature of h. In short, the omnipresence of this new ‘atomistic’, discrete constant of nature h is the litmus test of quantum physics. Given the emphasis on the discrete/atomistic character that quantum theory brought to light, it is only natural to try to extend this character to the two other fundamental magnitudes of the physical world, namely space and time and therefore to the best theory that we have to describe them, that is, general relativity. Merging general relativity with quantum mechanics via some quantum theory of gravity is proving very hard, however, and all attempts so far are devoid of straightforward empirical confirmation. In particular, we still don’t know whether space–time is quantized, in the sense that there are minimal spatial intervals (Planck length) and temporal intervals (the so-called Planck time on which we will return later).3 Despite and maybe because of this (hopefully temporary) lack of evidence, it seems in any case interesting as well as important to inquire into the conceptual consequences arising from the postulation of something similar to what the ancient philosophers called a ‘chronon’, or later philosopher, the tempusculus, namely

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a minimal atom of time. Considering the inseparability of time from space (the irrevocable conquest of the special theory of relativity), in what follows we will have to keep in mind that any argument about the possible existence of a discrete or atomic time should be discussed within the conceptual possibility to extend this atomization to space as well (and conversely), a problem that we will discuss briefly only at the end of the chapter. In any case, in this chapter our discussion will be limited to atomic time. This restriction is in part justified by the fact that progress in understanding what postulating an atomic discrete time really amounts to may shed light on the conceptual implications of postulating a discrete spatiotemporal structure. With this aim in mind, in what follows I will offer a brief review of the various possible senses in which we could talk about ‘atoms of time’ or ‘chronons’ (§2) and will then ask whether chronons can play a role in contemporary science and philosophy of time.4 My conclusion will be sceptical: not only is it conceptually unclear what the postulation of chronons really means, but also, even if this problem were clarified, there are no convincing philosophical arguments that can show that physics (or other empirical sciences) requires, or can take practical advantageous by, postulating a discrete or atomic time.

THE MATHEMATICAL MODEL OF DISCRETE TIME We are used to representing time as isomorphic to a set of real numbers. But there is no a priori reason why time should not have the structure of the integers. To the extent that time is just the order of succession of physical events, as Leibniz had it, not only the set of real numbers but also the set of integers exemplify the ordering relation ‘smaller than’ or its converse ‘greater than’, which isomorphic to the earlier or later than relations among different instants of time (simultaneity can be defined in terms of these relations). Furthermore, as noted by Whithrow (1984, 201), from the fact that for the formulation of its theories and construction of its mathematical models physics presupposes real or complex analysis and differential manifolds, we cannot infer that physical time is not atomic and therefore infinitely divisible. As we know, the defining characteristic of the integers is that for any integer n there is one and only one integer m = n+1 (the ‘next one’). This aspect is reflected in the axiomatization of arithmetic given by Peano, which relies on the primitive function S of ‘being a successor’: given S and given any integer n, there is only one number S(n) which is the successor of n and such that S(n) = n+1. More formally, take a structure R = {N,