Aristotle's Theory of Bodies (Oxford Aristotle Studies Series) [Illustrated] 9780198779728, 0198779720

Table of contents :
Cover
Aristotle’s Theory of Bodies
Copyright
Contents
Acknowledgements
1: Introduction
PART I: Putting the Theory of Bodies in its Place
2: A Remark on the Notion of Body
3: Body in the Context of Physical Science
3.1. The Importance of an Account of Body for Physical Science
3.2. The Connection Between Physics III–IV and the Study of Body
3.2.1. A remark on Physics III–IV
3.2.2. Body as part of the conceptual underpinnings of physical science
3.2.3. Further evidence from On Generation and Corruption
3.3. A Physical Study of Body
3.3.1. Why the study of bodies belongs to physical science
3.3.2. More exemplary cases
3.3.3. The limits of the study of physical bodies
4: Mathematics and Physical Science
4.1. The Theory of Physics II.2
4.1.1. Studying X but not qua Y
4.1.1.1 a remark on x qua y
4.1.1.2 magnitudes, but not qua physical
4.1.2. Separation and falsity
4.1.3. The snub example
4.2. The Mathematician and the Physicist
4.2.1. The most important results reviewed
4.2.2. Drawing on mathematical results
PART II: The Theory of Bodies
5: Body in Categories 6
5.1. Introduction and Framework
5.2. The Continuous and the Discrete
5.2.1. The definition of the continuous and the discrete
5.2.2. Getting the quantifiers right
5.2.3. Defining the continuous and the discrete
5.2.4. Some limitations of the theory
5.2.5. Is it really the definition of continuity?
5.2.5.1 continuity and connection
5.2.5.2 continuity and divisibility
5.3. Having Parts with Position versus Having Parts without Position
5.3.1. PPos considered
5.3.2. Position, place, and space
5.3.3. Further evidence on thesis
5.3.4. Thesis and the six spatial directions
5.3.5. A classification of bodies
6: A Topological Conception of Bodies
6.1. Introduction
6.2. Bodies are Complete
6.2.1. De Caelo on the completeness of bodies
6.2.1.1. aristotle’s arguments for the priority of bodies
6.2.1.2. what does teleion mean?
6.2.2. Substantiality and the dimensions
6.2.2.1. quantity is not a substance
6.2.2.2. three-dimensionality and priority in substance
6.2.2.3. substance, causes, and dimensionality
6.3. Bodies and Limits
6.3.1. An account of limits
6.3.2. Internal and outer boundaries
6.3.2.1. boundary in and boundary of
6.3.2.2. boundaries and the (topological) form of an object
6.3.3. The relation of bodies and their limits
6.3.3.1. does the question make sense?
6.3.3.2. bodies include their limits
6.3.4. The ontology of boundaries
6.3.4.1. the ontology and dependence of boundaries
6.3.4.2. boundaries and properties
6.3.4.3. boundaries versus parts
6.4. The Matter of Body
6.4.1. Two meanings of ‘matter’ distinguished
6.4.2. An account of extension
6.4.3. Why is the interval indeterminate?
6.4.4. The ontology of extension
6.4.4.1. independent and dependent extension
6.4.4.2. the matter of magnitude
6.4.5. The divisibility of matter and extension
6.4.5.1. the place of the argument
6.4.5.2. the argument against composition fromlower-dimensional entities
6.4.5.3. matter without foundation
7: Contact and Continuity
7.1. Introduction
7.2. Contact and Continuity—the Formal Theory
7.2.1. The first definitions: coincidence, separation, and contact
7.2.1.1. coincidence and separation
7.2.1.2. contact
7.2.2. The second definitions: in succession, contiguity, and continuity
7.2.2.1. being in succession
7.2.2.2. contiguity
7.2.2.3 continuity
7.3. Contact and Continuity—the Metaphysical Theory
7.3.1. The possibility of contact
7.3.1.1. why the boundaries have to be coincident
7.3.1.2. why two limits?
7.3.2. Continuity and the ontology of physical objects
7.3.2.1. why one boundary, rather than two?
7.3.2.2. identical or unified boundaries?
7.3.2.3. do all boundaries have the same status?
7.4. Continuity, Causality, and Unity
7.4.1. Continuity and causality
7.4.2. Continuity, causality, and the instantiation of a form
7.4.2.1. causality and form in metaphysics v.6
7.4.2.2. causality and metaphysical predication in metaph. viii.2
8: Conclusions
PART III: Appendices
Appendix A: Metaphysics V.13
A.1. Metaphysics V.13 on What a Quantity is
A.1.1. Condition [1]: quantity is divisible
A.1.2. Condition [2a] and [2b]: the parts are one and a this
A.2. Pluralities and Magnitudes
A.2.1. The division into magnitudes and numbers
A.2.2. The subdivision of magnitudes according to dimensionality
A.2.3. Limited plurality and magnitude?
A.3. A Comparison of Categories 6 and Metaphysics V.13
A.4. The Definition of Body
Appendix B: A List of the Propositions
Bibliography
Index Locorum
Index Nominum
Index Rerum

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Aristotle’s Theory of Bodies

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OX F O R D A R I S T O T L E S T U D I E S General Editors Julia Annas and Lindsay Judson published in the series Passions and Persuasion in Aristotle’s Rhetoric Jamie Dow How Aristotle gets by in Metaphysics Zeta Frank A. Lewis The Powers of Aristotle’s Soul Thomas Kjeller Johansen Aristotle on the Apparent Good Perception, Phantasia, Thought, and Desire Jessica Moss Teleology, First Principles, and Scientific Method in Aristotle’s Biology Allan Gotthelf Priority in Aristotle’s Metaphysics Michail Peramatzis Doing and Being An Interpretation of Aristotle’s Metaphysics Theta Jonathan Beere Aristotle on the Common Sense Pavel Gregoric Space, Time, Matter, and Form Essays on Aristotle’s Physics David Bostock Aristotle on Teleology Monte Ransome Johnson Time for Aristotle Physics IV. 10–14 Ursula Coope Political Authority and Obligation in Aristotle Andres Rosler

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Aristotle’s Theory of Bodies Christian Pfeiffer

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Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Christian Pfeiffer 2018 The moral rights of the author have been asserted First Edition published in 2018 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2018931634 ISBN 978–0–19–877972–8 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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Contents Acknowledgements

ix

1. Introduction

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Part I. Putting the Theory of Bodies in its Place 2. A Remark on the Notion of Body 3. Body in the Context of Physical Science 3.1. The Importance of an Account of Body for Physical Science 3.2. The Connection Between Physics III–IV and the Study of Body 3.2.1. A remark on Physics III–IV 3.2.2. Body as part of the conceptual underpinnings of physical science 3.2.3. Further evidence from On Generation and Corruption 3.3. A Physical Study of Body 3.3.1. Why the study of bodies belongs to physical science 3.3.2. More exemplary cases 3.3.3. The limits of the study of physical bodies

4. Mathematics and Physical Science 4.1. The Theory of Physics II.2 4.1.1. Studying X but not qua Y 4.1.1.1. A remark on X qua Y 4.1.1.2. Magnitudes, but not qua physical 4.1.2. Separation and falsity 4.1.3. The snub example 4.2. The Mathematician and the Physicist 4.2.1. The most important results reviewed 4.2.2. Drawing on mathematical results

7 12 13 14 14 15 17 19 21 23 24 26 27 30 30 33 34 37 42 42 43

Part II. The Theory of Bodies 5. Body in Categories 6 5.1. Introduction and Framework 5.2. The Continuous and the Discrete 5.2.1. The definition of the continuous and the discrete 5.2.2. Getting the quantifiers right 5.2.3. Defining the continuous and the discrete 5.2.4. Some limitations of the theory 5.2.5. Is it really the definition of continuity? 5.2.5.1. Continuity and connection 5.2.5.2. Continuity and divisibility

53 53 55 56 59 60 61 63 63 64

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vi contents 5.3. Having Parts with Position versus Having Parts without Position 5.3.1. PPos considered 5.3.2. Position, place, and space 5.3.3. Further evidence on thesis 5.3.4. Thesis and the six spatial directions 5.3.5. A classification of bodies

6. A Topological Conception of Bodies 6.1. Introduction 6.2. Bodies are Complete 6.2.1. De Caelo on the completeness of bodies 6.2.1.1. Aristotle’s arguments for the priority of bodies 6.2.1.2. What does teleion mean? 6.2.2. Substantiality and the dimensions 6.2.2.1. Quantity is not a substance 6.2.2.2. Three-dimensionality and priority in substance 6.2.2.3. Substance, causes, and dimensionality 6.3. Bodies and Limits 6.3.1. An account of limits 6.3.2. Internal and outer boundaries 6.3.2.1. Boundary in and boundary of 6.3.2.2. Boundaries and the (topological) form of an object 6.3.3. The relation of bodies and their limits 6.3.3.1. Does the question make sense? 6.3.3.2. Bodies include their limits 6.3.4. The ontology of boundaries 6.3.4.1. The ontology and dependence of boundaries 6.3.4.2. Boundaries and properties 6.3.4.3. Boundaries versus parts 6.4. The Matter of Body 6.4.1. Two meanings of ‘matter’ distinguished 6.4.2. An account of extension 6.4.3. Why is the interval indeterminate? 6.4.4. The ontology of extension 6.4.4.1. Independent and dependent extension 6.4.4.2. The matter of magnitude 6.4.5. The divisibility of matter and extension 6.4.5.1. The place of the argument 6.4.5.2. The argument against composition from lower-dimensional entities 6.4.5.3. Matter without foundation

7. Contact and Continuity 7.1. Introduction 7.2. Contact and Continuity—the Formal Theory

65 67 69 70 72 74 76 76 76 77 78 80 81 82 85 88 89 90 95 95 96 100 101 102 107 107 114 119 121 122 124 125 128 129 130 133 134 138 145 147 147 148

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contents vii 7.2.1. The first definitions: coincidence, separation, and contact 7.2.1.1. Coincidence and separation 7.2.1.2. Contact 7.2.2. The second definitions: in succession, contiguity, and continuity 7.2.2.1. Being in succession 7.2.2.2. Contiguity 7.2.2.3. Continuity 7.3. Contact and Continuity—the Metaphysical Theory 7.3.1. The possibility of contact 7.3.1.1. Why the boundaries have to be coincident 7.3.1.2. Why two limits? 7.3.2. Continuity and the ontology of physical objects 7.3.2.1. Why one boundary, rather than two? 7.3.2.2. Identical or unified boundaries? 7.3.2.3. Do all boundaries have the same status? 7.4. Continuity, Causality, and Unity 7.4.1. Continuity and causality 7.4.2. Continuity, causality, and the instantiation of a form 7.4.2.1. Causality and form in Metaphysics V.6 7.4.2.2. Causality and metaphysical predication in Metaph. VIII.2

8. Conclusions

148 149 154 157 157 158 159 160 161 161 163 171 173 176 178 182 185 188 188 190 193

Part III. Appendices Appendix A. Metaphysics V.13 A.1. Metaphysics V.13 on What a Quantity is A.1.1. Condition [1]: quantity is divisible A.1.2. Condition [2a] and [2b]: the parts are one and a this A.2. Pluralities and Magnitudes A.2.1. The division into magnitudes and numbers A.2.2. The subdivision of magnitudes according to dimensionality A.2.3. Limited plurality and magnitude? A.3. A Comparison of Categories 6 and Metaphysics V.13 A.4. The Definition of Body

197 197 199 199 201 201 203 204 207 209

Appendix B. A List of the Propositions

211

Bibliography Index Locorum Index Nominum Index Rerum

214 222 226 228

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Acknowledgements This book is a revised version of my dissertation, which was accepted at the Humboldt University Berlin in 2012. The book would not exist in its present form, or any form at all, without the support of many people. Above all, I must thank my doctoral supervisors Jonathan Beere, Ben Morison, and Christof Rapp. Jonathan Beere greatly helped to shape my views and his insightful comments improved many of the arguments. His encouragement was essential in many moments of doubt. I am especially grateful to Ben Morison. I have benefited enormously from continually working with him in Berlin, during several months in Princeton, and finally by video conferencing. Our discussions were a constant source of inspiration and motivation. My greatest debt is to Christof Rapp for his unwavering support over many years. He guided me, professionally and intellectually, from my days as an undergraduate student at the HU Berlin to my current position as a member of the Munich School of Ancient Philosophy at the Ludwig Maximilians University (LMU) Munich. I am most thankful to him for our philosophical discussions and collaborations, and, above all, for bringing about perfect conditions for studying ancient philosophy. I wrote my dissertation with the generous support of the Excellence Cluster TOPOI in Berlin. While at Berlin, I was also part of the Graduate Program in Ancient Philosophy where I was fortunate to have had as advisers Marko Malink and Jacob Rosen. They both read substantial parts of this book and their invaluable criticisms and suggestions saved me from many errors. With my fellow students in Berlin I not only enjoyed many philosophical conversations, but they were also responsible for improving my table soccer skills. I revised this book during my years as a member of MUSAPh. Both the quality of the book and the well-being of its author have benefited from its intellectually stimulating atmosphere. During the long time spent working on this manuscript I have incurred a debt to the following people, all of whom read and commented on various stages of the work: Andreas Anagnostopoulos, Laura Castelli, Pieter Sjoerd Hasper, Henry Mendell, Christopher Noble, and Christopher Shields. The same holds true of audiences in Berlin, Cambridge, Edinburgh, Munich, and Zagreb, where I had the opportunity to present parts of my work. The extensive written comments from two anonymous readers for Oxford University Press helped me to greatly improve the book. One of them (now no longer anonymous) was Gábor Betegh from whom I learned about the significance of the notion of body in ancient philosophy more generally and with whom I had the privilege to co-author a paper on this topic.

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x acknowledgements I would also like to thank Joshua Crone for reading and correcting the English and Peter Momtchiloff of OUP for his help in preparing the book for publication. Finally, I must thank my family, Helmut, Karin, and Philipp, for their unfailing support during difficult times. This book is dedicated to them.

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1 Introduction This book explores Aristotle’s conception of body. It aims at a systematic understanding of the concept of three-dimensional magnitude and related notions such as surface, boundary, extension, contact, and continuity. The whole project is guided by two questions: What does Aristotle’s theory of body look like? I answer this question in Part II. Where in Aristotle’s philosophy does such a study belong? This question is answered in Part I. The two questions are, to a certain extent, independent of one another. To put it in the context of modern-day philosophy, Part II can be seen as a systematic elucidation of basic mereotopological concepts in Aristotle. Aristotle was perhaps the first philosopher to develop highly sophisticated views that lie at the heart of contemporary mereotopology. When contemporary philosophers discuss, for example, the nature of boundaries they work with a conception that was shaped by Aristotle. One aim of this book is to analyse the core concepts of Aristotle’s theory of body as fully as possible. It is, however, best to approach philosophical questions from within an author’s philosophical system. It is mandatory to ask not only what Aristotle’s views on these topics are, but also why he was interested in them or what role they play in his philosophy. Putting it in the context of Aristotle’s philosophy, then, I will argue that the treatment of bodies and magnitude is important for Aristotle’s conception of the physical sciences because almost all branches of physical science employ the notion of body and related notions, and hence an elucidation of the notion of threedimensionally extended magnitudes is essential. What holds these notions together is that they are all properties of physical substances insofar as they are physical bodies. More specifically, I believe that Aristotle’s approach to the theory of bodies can be compared—both in importance and scope—to Physics III–IV. In these books Aristotle elucidates and systematizes basic concepts that are crucial to physical science, a discipline that Aristotle himself developed into a science. Why should a study of bodies be crucial for Aristotle’s natural philosophy? As Aristotle repeatedly emphasizes, natural science concerns natural substances, for instance substances that have an internal principle of motion and rest in them. This is why natural philosophy is concerned with animals or plants. However, in virtue of being bodies, physical substances have a place, are in contact with each other, or have parts that are continuous. All of these properties are explained by the body of physical substances. That is to say, a body is

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 aristotle’s theory of bodies a three-dimensional magnitude bounded by surfaces which is a quantitative feature of physical substances. This book is concerned with body as a feature of physical substances.1 Two aspects of my project might be controversial. One is that as a matter of fact we do not find a self-contained treatment of body and magnitude as we do in the case of motion, the infinite, time, or place. This may partly explain why there are many studies dedicated to the latter topics, but no book-length treatise on the former.2 This may even raise the suspicion that the topic is not important for Aristotle.3 Whatever Aristotle’s reasons for not writing such a treatise, it is certainly not true that such a treatise would be misplaced in his physical writings. As I will argue, a proper elucidation of the notion of body is a prerequisite for an understanding of many parts of Aristotle’s natural philosophy such as his theory of place or his theory of action and passion. To put it slightly hyperbolically, Part I can be read as making the case that Aristotle did write this treatise. He announces the need for such a treatise, determines its scope and purposes, discusses problems that might come up for the overall project, and finally lays out step by step the concepts employed in this study. But this treatise, in contrast to Aristotle’s discussion of place or time or the soul, is scattered throughout his works. The other aspect which might be controversial is my claim that Aristotle’s treatment of bodies should be placed in the context of his natural philosophy. It is, as I will argue, part of physical science, rather than part of mathematics. This may initially be surprising, but, as we will shortly see, when Aristotle discusses the notion of body and related notions, as, for example, boundary, extension, contact, and continuity, he has physical bodies in mind. An understanding of the concept of a body and related concepts such as continuity, contact, or boundary is essential to physical science. In the course of her investigation the physicist can draw on results from mathematical science. Yet, the study of physical bodies qua physical bodies is a distinctively physical undertaking. This is, in a compressed form, the argument of the first part of my study. In the second part I will lay out how the study of bodies in the context of Aristotle’s Physics might look. I will give a detailed overview in the introduction to the second part. For now, let me make some general remarks about how Part II should be read. As I have already said, I think that Aristotle’s remarks about bodies should be seen as similar to the discussion of place, time, void, or the infinite. I also said that, just like the discussion of time or place, the study of body is concerned with the conceptual 1

I explain this notion at greater length in Chapter 2. With the possible exception of White 1992. White, however, does not recognize the treatment of bodies as distinct from what he calls a ‘detailed, rigorous analysis of the formal, structural features of motion, time, and spatial magnitude’ (White 1992, 6). Moreover, White’s study is most explicitly concerned with mathematical models in the modern sense, and not with the physical ontology of bodies. I shall say more on this later. 3 Even though I think that the mere lack of a continuous treatment is not reason enough to judge a topic as spurious. There is no continuous discussion in Aristotle on ontological priority or related themes either, yet, it is widely accepted that this is a proper research topic. 2

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introduction  underpinnings of physical science. By speaking of ‘conceptual underpinnings’ I do not mean to imply that the study is a conceptual analysis in the modern sense of the word. It is not an analysis of the meaning of those terms, much less an a priori investigation. Rather, it is an investigation of concepts that are crucial to any branch of natural science.4 For example, there can be no study of animal motion without a grasp of what motion, place, or time is. Thus, for a study of animal motion it is a prerequisite that the concepts of time, place, and motion in general be understood. Understanding the concept of place is, according Aristotle, not a task that is distinct from understanding what place is. Aristotle is not interested in the meaning of the word ‘place’, but in what places are. The same, I claim, holds true of the study of bodies. It aims to elucidate what bodies, boundaries, continuity, and contact are, and this elucidation must be seen in the light of its possible application in physical science. The comparison to Aristotle’s account of place or time also helps to put the account of bodies into context for the modern reader. Today, philosophers discuss the topics I deal with in the second part of my work in the field called ‘mereology’ or, more specifically, ‘mereotopology’. They study concepts such as ‘part’, ‘whole’, ‘being connected’, ‘being continuous’, or ‘having a boundary’. It would be wrong, however, to conclude that Aristotle had a theory of mereotopology—a formal theory of parts and wholes and their spatial relations. Aristotle does not discuss these issues under the heading of mereotopology, nor did he have a formal system in our sense of the words. Modern theories are often presented as axiomatic systems.5 Aristotle’s discussion is not intended as an axiomatic or formal system. It is, to repeat, embedded in his conception of physical science. It is an investigation of the conceptual underpinnings of such a science. Accordingly, I use throughout my work sentences labelled as ‘propositions’. These propositions do not add up to an axiomatic and formal system. Rather, they are meant to indicate important results obtained in my study. I am aware that tastes differ on the use of such labels, especially if they are not part of a formal theory. Personally, I think that they are both helpful in highlighting important results and outlining a fragmentary theory of bodies, even if they do not add up to a formal and axiomatic treatment. Having said this, it nevertheless remains true, I believe, that many aspects of Aristotle’s thought are of interest for those working on these issues within contemporary metaphysics. Aristotle might provide fresh stimulus or even some alternative to current orthodoxy, which could enrich current debates. When we modern-day philosophers define what it is to be whole or what a boundary is, we are working with a conception that has been shaped by Aristotle and that still incorporates a large part of his thinking.

4 That Aristotle’s Physics should be seen as a conceptual work laying out the principles for the science is defended at length in Wieland 1970. 5 For an overview see especially Simons 1987; Casati and Varzi 1999.

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 aristotle’s theory of bodies A final remark on the scope of this book. Obviously, many topics or problems which will be discussed in this work have been discussed by other ancient authors besides Aristotle. Some are inherited from Plato (the status of boundaries, for example) and in later periods we find, especially in Hellenistic philosophy, a continuing discussion about these topics. It certainly would required another book to enter these debates, or even to locate Aristotle’s theory in this wider context. My focus will be Aristotle, but sometimes it might be helpful to be reminded of the fact that there was indeed this larger context. Therefore, I will at some points in the discussion direct the reader to relevant discussions in other authors.

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PA R T I

Putting the Theory of Bodies in its Place

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2 A Remark on the Notion of Body Since Aristotle uses the word ‘body (σμα)’ in three interrelated, though distinct ways, writing on Aristotle’s theory of bodies can mean different things. Before I begin my discussion let me elaborate further on the notion of body which is crucial to this study. My remarks here are intended as a general orientation for the reader. Body as quantity. The notion of body which is at the heart of this study is the notion of a three-dimensionally extended and continuous magnitude bounded by surfaces. Of quantities some are discrete, others continuous; ( . . . ) continuous are lines, surfaces, bodies.1 (Cat. 6 4b20–24)2 Of magnitude that which is continuous in one dimension is length, in two breadth, in three depth. Of these, limited plurality is number, limited length is a line, limited breadth a surface, limited depth a body.3 (Metaph. V.13 1020a11–14) Body is what has extension in all dimensions.4

(Ph. III.5 204b20)

According to these passages, body is in the category of quantity.5 Thus understood, this is not simply the notion of physical substance. It is true that all physical substances are bodies or, as I shall say, have bodies, but a physical substance is not simply a body in this sense.6 For although a perceptible substance such as Socrates is three-dimensionally extended and bounded by a surface, this is not what Socrates is essentially. The substance and essence of Socrates is his soul. Although Socrates is three-dimensionally extended, this is not what makes him the substance he is. On the other hand, when we refer to the body of Socrates, we do not speak about Socrates as such. Rather we speak about a certain feature of Socrates, namely the threedimensionally extended magnitude of a certain shape which Socrates has. Therefore,

ο δ ποσο τ μ ν στι διωρισμ νον, τ δ συνεχ · (…) συνεχ  δ γραμμ, πιφνεια, σμα. Throughout my work I will use the following conventions. The default translation for Aristotle is Barnes 1984, sometimes modified, and for Plato, Cooper and Hutchinson 1997. Exceptions are noted. 3 μεγ θου δ τ μ ν φ’ ν συνεχ  μκο τ δ’ π δ ο π!το τ δ’ π τρ"α βθο. το των δ π!θο μ ν τ πεπερασμ νον $ριθμ  μκο δ γραμμ% π!το δ πιφνεια βθο δ σμα. 4 σμα μ ν γρ στιν τ πντ& 'χον διστασιν. 5 See especially Categories 6 (discussed in Chapter 5) and Metaphysics V.13 (discussed in Appendix A). 6 For the terminological distinction between what something is and what something has see Code 1986. 1 2

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 aristotle’s theory of bodies I think that we must distinguish between an account of a physical substances as such, for instance an account of what it is to be that kind of substance, and an analysis of what belongs to physical substances insofar as they are bodies. In the latter case, we are interested in characteristics, such as extension, continuity, boundaries, which a physical substance has in virtue of being a body. This study is concerned with such an account.7 Body as Substance. Aristotle also calls the four elements and what is composed out of them, including living beings, ‘bodies’. Bodies seem most of all to be substances, and among these especially natural bodies.8 (de An. II.1 412a11–12) Substance seems to belong most obviously to bodies; therefore we say that both animals and plants and their parts are substances, and also natural bodies such as fire and water and earth and each of the sort, and all things that are parts of these or composed of these, either of some or of all, such as the heaven and its parts, stars and moon and sun.9 (Metaph. VII.2 1028b8–13)

As the quotations suggest, ‘body’ may refer to the whole class of physical substances. Physical substances have according to Aristotle a nature, that is, an internal principle of motion and rest,10 and it is the task of physical science—by this I mean what Aristotle calls ‘( φυσικ’—to deal with these substances.11 This may suggest that mobile body is the proper subject of physical science and, thus, it may come as no surprise that ‘the Aristotelian tradition (broadly understood) places the notion of 7 I may be wrong with my claim that being a body is not part of the essence of a physical substance. Still I think that the distinction between a study of physical substances as such and what belongs to them qua bodies could be upheld. For even if it is part of the essence of a physical substance to be a body, it seems intuitive to distinguish between those characteristics which immediately derive from its being a body and those characteristics which it has in virtue of being a specific kind of substance. For instance, the capacity to perceive is due to the animal having a soul. Animals do not perceive in virtue of being bodies. But animals are continuous in virtue of being bodies. 8 ο)σ"αι δ μ!ιστ’ ε*ναι δοκοσι τ+ σ,ματα, κα το των τ+ φυσικ. 9 -οκε. δ’ ( ο)σ"α /πρχειν φανερ,τατα μ ν το. σ,μασιν (δι τ τε ζ1α κα τ+ φυτ+ κα τ+ μ2ρια α)τν ο)σ"α ε*να" φαμεν, κα τ+ φυσικ+ σ,ματα, ο3ον πρ κα 4δωρ κα γν κα τν τοιο των 5καστον, κα 6σα 7 μ2ρια το των 7 κ το των στ"ν, 7 μορ"ων 7 πντων, ο3ον 6 τε ο)ραν  κα τ+ μ2ρια α)το, 8στρα κα σε!νη κα :!ιο)· 10 Ph. II.1 192b8–14. In this passage Aristotle also mentions the elements and living beings as physical substances. 11 In the course of my study I will use the term ‘physical substances’ in a slightly broader sense. It is meant to apply not only to things that are by nature, but also to things like tables and chairs which are artefacts and not by nature. In this sense, one could use the terms ‘perceptible substances’ or ‘material substances’ to cover both cases. However, I think that for those without constant exposure to Aristotelian philosophy the notion of perceptibility is potentially misleading. Similarly, the notion of matter is, due to its distinct usage in Aristotelian and modern philosophy, also not helpful. Therefore, I will mainly use the term ‘physical’. Moreover, in some places Aristotle denies that the elements or artefacts are substances. Thus, it may be better to use ‘physical beings’ instead. But since this expression is, to my mind, too opaque and for the purposes of this study the question whether the elements or artefacts are substances stricto sensu is not important I will stick to the expression ‘physical substances’. After all, from the perspective of the Categories they should fall under the category of substance.

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a remark on the notion of body  body at the heart of Aristotle’s physics’ (Falcon 2016, 423).12 A study of body, then, would be a general study of physical substances. In the next chapter, I will begin by arguing that an account of body is necessary for doing physical science. With this, however, I do not envisage this general study of physical substances, but I am concerned, more specifically, with the role of quantitative body in physical science. Thus, in claiming that the physicist studies bodies and magnitudes I do not merely want to say that the physicist studies animals, which, of course, she does. I want to make the stronger claim that the notion of body as a threedimensionally extended and continuous entity is central to the project of physical science, as Aristotle conceived of it. The physicist needs an account of what belongs to physical substances qua physical bodies, that is, insofar as they are the body and magnitudes of physical substances. As I already said in the introduction, by placing the study of (quantitative) body in the context of physical science I also oppose the assumption that the study of body is exclusively part of mathematics. Body as matter. Finally, at some places Aristotle uses the term ‘body’ to refer more specifically to the matter of an animal, as opposed to its soul. It follows that every natural body which has life is a substance, and a substance in the sense of a composite. Since it is also a body of such and such a kind, viz. having life, the soul cannot be a body; for the body is not among the things said of a subject, but rather it is the subject or matter, not what is attributed to it. Hence the soul must be a substance in the sense of the form of a natural body having life potentially within it.13 (de An. II.1 412a16–21)

Animals are hylomorphic compounds whose form is their soul and whose matter is their (organic) body. Aristotle’s famous claim that body is a homonymous term is connected to this usage of body. The organic body of an animal is essentially ensouled. If the animals dies, its body is not the same type of entity as it was before.14 I mention this use because one could assume that the bodily features of physical substances, according to the first usage, derive from their matter, in particular their ultimate matter. Animals are bodies because they are composites of soul and body. The body of animals is flesh and bones. Flesh and bones are not defined by being threedimensionally extended, at least not immediately so. But, many commentators have argued, the ultimate matter of physical substances, what traditionally is called ‘prime matter’, is body in the specific sense of three-dimensional extension.15

12

I refer the reader to this article for a discussion of the evidence. ;στε πν κιν(σειε τν μαθηματικν.

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a physical study of body  Notice that the importance of a correct account of the underlying topology of bodies and magnitudes for physical science does not necessarily mean that it is the task of the physicist to work out the proof that bodies are infinitely divisible. In fact, one way to take the quotation is that the proof that bodies are infinitely divisible is probably worked out by the mathematician.19 I am inclined to read the passage differently. It merely says that a mistaken physical account may contradict established mathematical theorems, which, of course, it shouldn’t. Be that as it may, here I wish to emphasize two points: first, the physicist is concerned with the question of infinite divisibility; and second, although the importance of a study of magnitudes for the conceptual underpinnings of physical science does not entail that in this study physical consideration alone play a role, a study of body has a distinctly physical character. This is the topic of Section 3.3.20

3.3. A Physical Study of Body Having placed the study of body within the realm of physical science, one may wonder whether it should not better be seen as a part of mathematics. After all, Aristotle himself suggests that mathematics is the study of lines, surfaces, and bodies.21 It is true that the study of magnitudes is in this respect unlike the study of place, time, or motion. The latter seem obviously to be concepts only dealt with by the physicist.22 The study of magnitudes, however, cannot be so firmly placed in the context of a single science. Aristotle, I believe, is aware of this. The correct answer to the question, ‘who it is that studies magnitudes, the mathematician or the physicist’, is ‘both’. As I will show in Chapter 4, Aristotle explains in Physics II.2 how this is possible. Both the mathematician and the physicist study magnitudes, but not qua the same. The physicist studies magnitudes insofar as they belong to physical substances. In this sense, she studies magnitudes qua physical. The mathematician, on the other hand, studies them not insofar as they are physical. She studies magnitudes as if they were separate. However, since sciences are individuated by their manner of investigation and their respective focus, physical science and mathematical science are distinct

19 Cf. also Cael. III.1 299a1–11 which contains a reference to Physics VI.1 and says that the refutation was part of the analysis of motion. It is unclear whether Physics VI.1 should be seen as a mathematical argument. I think that it should for reasons that will be fully explained only later (part of the reason is that the account of contact differs from the canonical statement in Physics V.3). See Section 6.4.5. 20 In Chapter 4, I will argue that in the account of magnitudes the physicist may draw on mathematics too. But, of course, it remains true that it is first and foremost the task of the physicist to provide a theory of the ontology of bodies. 21 Metaph. XIII.3. 22 I intentionally use the word ‘seem’ here because it is not clear whether this is true. The astronomer, for example, deals with moving things. In the science of astronomy, it seems, motion-predicates occur. It is, however, not clear whether astronomy should be placed under mathematics or physics. Simplicius in his commentary on Physics II.2 argues that astronomy is part of mathematics. For a discussion see Mueller 2006.

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 aristotle’s theory of bodies sciences. Though concerned with the same underlying reality, the way in which the mathematician and the physicist study magnitudes explains why they are engaged in two distinct sciences. To be sure, the physicist may use theorems established in mathematics. The mathematician might show that no line is composed of points, and the physicist could draw on this result and conclude that there are no bodies made up of indivisible atoms.23 It is, however, crucial to see that this does not imply that the study of magnitudes really is part of mathematics or that the part of physics dealing with body and other magnitudes is really a branch of mathematics. It is a mistake to assume that Aristotle’s arguments are mainly mathematical and simply carried over to the realm of physics. The focus on this paradigm has, in my mind, misled commentators. They jump from the fact that Aristotle employs mathematical arguments to the conclusion ‘that [it] is fair to say that Aristotle’s conception of continuous quantity or magnitude is a geometrical conception’ (White 2009, 266). I disagree with this conclusion, if it means that Aristotle’s arguments should be considered as mathematical by his own standards. Some of them might be, but surely not all. It is plausible, I believe, to assume that in giving an account of bodies the physicist can draw on several sources and forms of argument. Some of them are mathematical, but most are distinctively physical arguments. Therefore, I think it is important to outline here a distinct form of argument that the study of magnitudes involves over and above mathematical arguments. This form of argument specifically asks what belongs to body insofar as it is the body of a physical substance.24 It is distinctly physical because it considers the mereotopological properties of body qua physical. 23

This, I believe, is the lesson to be learned from Cael. III.1 and GC I.2. See also Section 4.2.2. One might distinguish a further form of argument that might be called ‘topic-neutral’ or ‘logical’. This form of argument relies on a quite general conception of bodies and magnitudes. As such, the conception is neither part of mathematics nor of physics, but rather of dialectic. In this way it connects to my earlier remarks about the study of magnitudes belonging to the conceptual underpinnings of physics. In calling this form of argument ‘logical’ I want to capture Aristotle’s expression ‘ογικ ’. We already encountered this expression in Aristotle’s discussion of the infinite in Physics III.4–8. Aristotle announces that his question is whether there is an infinite body among physical substances. He then argues that if all bodies are bounded by a surface, it is impossible that there is an infinite body (Ph. III.5 204b4–7). Aristotle calls this a ‘logical’ argument. What type of argument is that? One might assume that it means it is a ‘merely verbal’ argument. Interpreted thus, the expression might have a pejorative connotation. The argument is not really an argument, but rather some form of linguistic trick. I think that this interpretation should be resisted, and I agree with Burnyeat 2001, ch. 5 that it means more than ‘verbal’. Aristotle does not want to say that the use of our words suggests that there cannot be an infinite body. This seems to me to impose an anachronistic reading of Aristotle. I believe that Aristotle intends this as a proper argument. If the argument is successful, it follows that there cannot be an infinite body. The argument is logical because it is topic-neutral. It is not part of any specific science, nor does it hold only of some bodies, be they physical or mathematical. This is indeed suggested by our passage. For Aristotle explicitly says that the argument implies that there can be neither a sensible nor an intelligible body which is infinite. A similar point could be made about the discussion of body in Cael. I.1 (which seems to be the position of Falcon 2005, 38). In this sense it might be connected to the proposal by Owen 1986 that ‘ογικ ’ is connected to conceptual analysis. A similar position is taken by Frede and Patzig 1988, who hold that logical is used by Aristotle in order to characterize investigations as linguistic, conceptual, as opposed to scientific. Nevertheless, Aristotle does not in any way dispute the importance of such formal analyses. One must only be aware that they need to be supplemented with contentful considerations. The last point is important. A logical argument is an argument. For the meaning of ‘logical’ see also Smith 1997b, 90–2. 24

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a physical study of body 

3.3.1. Why the study of bodies belongs to physical science There are two arguments why the study of body that Aristotle carries out is physical most of the time rather than mathematical in nature. First, although parts of physical science can draw on results in mathematics, this does not make the enterprise mathematical. To return to the example of GC I.2: It is the task of the physicist to show that no physical substance can be composed of atoms or be divided into nonmagnitudes. This is a theorem about physical substances insofar as they are physical bodies. The mathematician is not concerned with this theorem. There is no theorem in mathematics stating that physical bodies cannot be composed of atoms or be divided into non-magnitudes. There might be a theorem in mathematics stating that no line can be composed of points. In the course of her argument the physicist can draw on this mathematical theorem, but it is clear that her argument is in the context of physical science. This, however, is not a shortcoming on either side. The physicist should not prove the mathematical theorem, nor should the mathematician apply her theorems to physical reality. Second and more important, the study of physical substances qua physical bodies contains many results that could not possibly be mathematical results. The reason is its intimate connection to ontological considerations about bodies and magnitudes and their relation to the physical substance to which they belong. The physicist studies bodies and magnitudes as bodies and magnitudes of physical substances.25 The mathematician, on the other hand, treats quantities as if they were separate. As I will argue later, the mathematician studies a body only insofar as it is a threedimensionally extended quantity. She does not take into account that there is no such body ontologically separate from physical substances. But she is not concerned with the underlying ontology of these items. In this the study of the physicist differs. The physicist takes into account that it is the body of a physical substance. To illustrate the point consider, for example, the definition of boundary and figure that we find in Euclid: 13.

A boundary is that which is the extremity of something.

14.

A figure is that which is contained by some boundary or boundaries.26 (Euc. I. Def. 13 and 14)27

Equipped with these definitions the mathematician knows what a boundary is and what a figure is. On the basis of these definitions she can then define circles, rectilinear figures, etc. But it would be awkward to ask the mathematician how we should conceive of the ontological relation between the boundary and that which it bounds. The mathematician can do her study quite independent of the question whether the lines which are the boundaries of a rectangle are ontologically dependent on the body.

25 26 27

This is, I believe, also a lesson to be learned from Physics II.2. See Section 4.1. ιγʹ. UQρο )στν, V τιν/ )στι πρα . ιδʹ . Wχ4μ8 )στι τ$ -π/ τινο X τινων Vρων περιεχ/μενον. All translations of Euclid by Fitzpatrick 2007.

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 aristotle’s theory of bodies Those questions, I suggest, do not arise within mathematics. Rather, they arise outside mathematics as part of the foundation of mathematics or a general ontological study of magnitudes. But in proving theorems the mathematician does not need to consider such questions. The physicist, on the other hand, must consider these questions or at least be aware of them. Ontological considerations, like the question whether boundaries depend on their host, must be taken into account by the physicist. The reason is, as I said, that the physicist studies lines, surfaces, or bodies as belonging to a physical substance. Questions about ontological priority of bodies over surfaces naturally occur in a discussion of physical substances and their attributes. Physical substances are threedimensionally extended and are bounded by surfaces. The surfaces, Aristotle argues, are ontologically dependent on the physical substance they bound.28 Thus, if the physicist studies bodies and magnitudes insofar as they are the bodies and magnitudes of a physical substance, the concept of ontological dependence plays a role. None of these insights could be part of mathematics, simply because in mathematics the concept of physical substance is not used at all.29 There may also be structural differences.30 Consider the division of a line. Let us suppose that the line Z is divided at point X into the discrete halves A and B. Does X belong to A or B after the division? Moreover, if X belongs to A, does B have two endpoints? Mathematically, the first question is answered by stipulation. There is no fact of the matter whether X belongs to A or B. Moreover, if X belongs to A, then B does not have two endpoints, but it corresponds to a half-open interval. However, it is doubtful whether these answers make sense in the case of physical lines. If Z is a physical line (assuming that there are one-dimensional lines in the physical world), can it really be a matter of stipulation whether X belongs to A or B? Or does it make sense to say that there is a line without an endpoint? Typically, we assume that a physical line has two endpoints. Thus, there is a real question whether mathematical structures fit the physical structures. Aristotle, as I will argue, distinguishes between the mathematical and the physical case and answers these with respect to physical lines in this way: in cases of division there is no stipulation, but rather two new endpoints come into existence. Also, there are no open or half-open intervals in the physical world, but every line has two endpoints that belong to the line. Commenting on a similar example in a discussion of Zeno’s paradoxes Mark Sainsbury writes:31

28

I will discuss these questions in Section 6.3.4. This is not to say that the boundaries were always sharply demarcated in practice. In many discussions of Platonic or sceptical provenience ontological and mathematical considerations are not duly separated. See, e.g. Sextus Empiricus M3 and M9. My point is that the boundaries are sharply demarcated in Aristotle’s theoretical conception of the sciences. 30 I discuss this case in Section 6.3. 31 I want to thank Joseph Bjelde for bringing this book to my attention. Sainsbury 2009, 15–18 makes exactly the same distinction between a mathematical and a physical approach. 29

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a physical study of body  In these remarks, I have assumed that we have coherent spatial notions, for example, that of (two-ended) length, and that if some mathematical structure does not fit with these notions, then so much the worse for the view that the structure gives a correct account of our spatial notions. (Sainsbury 2009, 18)

I think that this comment is very apt and that the failure to see the potential differences between mathematical and physical structures has been part of the reason why the study of magnitudes as a prerequisite for doing physical science has been unduly neglected. In the case of infinite divisibility, a—by modern standards—mathematical approach might lead to impressive results. But for many of the other concepts involved in the study of magnitudes, concepts such as continuity, boundary, or position, an overly mathematical approach might be detrimental. Surely, this does not yet show that we have, in fact, coherent spatial notions which are applicable to the physical world. But it shows that we cannot undertake this task by relying on mathematical considerations alone. And Aristotle, as I argue, is aware of this.

3.3.2. More exemplary cases Considering two further examples will, I hope, help to illustrate the distinctive nature of the study of body and magnitude and why it is important for physical science. Both examples will be discussed in detail in Part II, Section 6.4 and Chapter 7. First, any particular physical substance has a certain extension, a diastema as Aristotle calls it. The extension can, roughly speaking, be conceived as that which lies within the limits of the magnitude or body of the physical substance. In virtue of having a certain extension the physical substance has a certain length, breadth, and depth. In short, it can be measured. But what is this extension, ontologically speaking? Does the extension belong to the body and the physical substance, or is it independent of it? This question is not irrelevant for the physicist. For, if the extension is independent, this affects the view one might have about void. A void can be conceived of as an independent extension that is unoccupied. If extension is independent of physical substances, it should be possible that there is a void. Moreover, a place according to this conception could be seen as occupied extension. Thus, when the physicist says that a substance has a certain extension, this is explained by the extension that the body of the substance occupies, but the extension itself is separate from the body. However, things radically change if there is no independent extension. In an explanation of the extension of a physical substance one would rely on features of that substance. Extension is not something over and above objects. Rather, it is, as I will argue in detail, an abstract way of considering the (ordinary) matter of an object. Second, the notions of contact and continuity are arguably at the heart of thinking about extended objects. I have already quoted passages from the Parts of Animals and On Generation and Corruption where these notions play a central role.32 As will 32

See Sections 3.1 and 3.2.3.

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 aristotle’s theory of bodies emerge later, the relation of contact holds between two independent objects, whereas the relation of continuity holds between the parts of a single object. The notion of continuity is intimately tied to the idea of a causal factor that unites the parts into a single object. But the idea of a causal connection between the parts of a single object is clearly an idea that has its place in physical considerations. If the physicist employs the notion of continuity to explain the topological integrity or connectedness of an object, she does so by invoking inter alia a certain causal factor that explains this form of unity.33 Thus, the concept of continuity as it is used in the study of bodies and magnitudes that I envisage is metaphysically loaded. It is essentially tied to the concepts of unity and causality. Consequently, in the explanation of the continuity of the body of an animal or of the parts of the body of an animal with each other, the physicist will necessarily rely on considerations about the unity, nature, and form of that animal. Her explanation will without doubt fail if she is not aware of the ontological implications the concept of continuity has. As I said, the details of my interpretation will be made much more precise in Part II of this work. Let me here instead draw attention to a general and important trend in these arguments: all of these arguments are arguments about magnitudes insofar as they are the lines, surfaces, or bodies of physical substances. To provide the conceptual underpinnings of physical science the physicist must include ontological considerations in her account of body. These ontological consideration are not idle speculations that the physicist might dispose of. Rather, they are central to an adequate account of body and magnitudes. This then is the context in which the results of the second part of my work should be seen. In her study the physicist does more than consider ‘the shape of the moon or the sun’ (Ph. II.2 193b29). Of course, this is the task of the physicist, too, but by no means does it fully describe the method in which the physicist studies magnitudes qua physical.

3.3.3. The limits of the study of physical bodies In the previous section I have argued that there is a distinctly physical study of body insofar as it is the body of a physical substance. How should one delineate such a study? How to distinguish between what belongs to physical substances qua physical bodies and what belongs to physical substances as such? Implicitly, I will delineate the content of the study in Part II.34 Nevertheless, let me outline a more explicit answer here. In its most general form, the study of bodies takes into account only features of physical substances which they have in virtue of being bodies. This, of course, does not enable us to delineate the study because it only invites the further question ‘How do we single

33

Aristotle illustrates this idea with the example of a shoe. Metaph. V.6 1016b11–16. There is one topic which I will not address in this work, although it may justifiably be thought to be part of a theory of bodies. This the question why two bodies cannot be in the same place at the same time. I have discussed this question in detail in Pfeiffer 2016. 34

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a physical study of body  out these features?’ But it provides a heuristic in the sense that each candidate-feature should be tested whether physical substances have it in virtue of being bodies. Since the task of checking each candidate-feature cannot be accomplished here, let me apply this general idea to the particular case of causal efficacy, which, I think, is a prime example.35 Here is how one might reason: action and passion require contact,36 hence, being causally efficacious is something that belongs to physical substances insofar as they are physical bodies. This impression may be reinforced if one thinks, for example, of the Epicurean definition of body as that which is three-dimensionally extended and has antitupia.37 Antitupia distinguishes bodies from void and explains why bodies do not move through each other, but collide when they meet. Thus, according to an Epicurean theory, the causal efficacy of physical substances seems, at least partly, to be a direct consequence of their being bodies. For Aristotle, I suggest, this is not the case. The causal efficacy of physical substances is not due to their being bodies. It does not derive from their being extended, continuous, or bounded that physical substances act upon each other. This is a necessary, but not sufficient condition. Causal efficacy derives from having contrary properties in the same genus.38 At the basic level, these are the pairs of elemental qualities hot– cold and wet–dry. The elemental qualities belong to things insofar as they are physical substances, not insofar as they are bodies. Although all physical substances are, in fact, bodies and thus it is true that only bodies possess the elemental qualities, the elemental qualities are not themselves bodily properties that are comparable to features such as, for example, continuity. Moreover, the comparison with the Epicurus’s view of bodies and their causal efficacy suggests a general observation: whether causal efficacy is part of the conception of physical substances qua bodies or not depends on the broader commitments and specifics of the theory in question. There is no single answer whether a theory of body should include such a discussion. For Epicurus it does, for Aristotle it does not. This divide, I suggest, runs through the history of philosophy. Causal efficacy belongs to the conception of body as such for Locke and Leibniz, for example, but not for Descartes who is closer to Aristotle in that respect.

35 36 38

I want to thank Gábor Betegh for pressing me on this point. 37 Cf. GC I.6 322b21–5, quoted in Section 3.2.3. Cf. S.E. M 1.21. GC I.7 323b29–24a11.

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4 Mathematics and Physical Science In the preceding chapter I have attempted to justify the first two claims that I announced.1 The study of bodies and magnitudes can be seen as a part of the conceptual underpinnings of physical science. It is a study that the physicist cannot do without. Moreover, this study is not purely mathematical in nature. This supposition does not capture the genuine interest the physicist takes in physical bodies. The physicist studies these insofar as they contribute to his understanding of natural substances. Thus, her study is more concerned with ontological issues arising in the study of magnitudes. A full justification of these claims is this study as a whole. In the present chapter I will return to the remaining claims. These revolve around the question of the relation between physical science and mathematics. Of course, the general question of how mathematics and physical science are related cannot be dealt with here. For our purposes it is salient to see that there is no contradiction involved in the assumption that both the physicist and the mathematician study body and magnitudes, but are nonetheless engaged in two distinct sciences (claim 3). And despite being engaged in distinct sciences the physicist may draw on results from mathematical results (claim 4). My view, to put it in a nutshell, is that physical science studies quantities qua physical or insofar as they are the bodies and magnitudes of physical substances. Mathematics, on the other hand, studies bodies, lines, and surfaces insofar as they are separate. The reason why mathematics is a science distinct from physical science is that quantities are logically separable from physical substances. Mathematics studies the quantitative aspect of physical substances in isolation. To be sure, the mathematician is concerned with objects that are ultimately physical substances, but she studies them not insofar as they are the magnitudes of a physical substance. Rather, the mathematician studies them as if they were separate. This is something that the physicist does not do, as I argued in the last chapter. She studies bodies, surfaces, and lines not insofar as they are separate, but rather insofar as they belong to a certain substance.

1

See the beginning of Chapter 3.

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the theory of physics ii. 

4.1. The Theory of Physics II.2 Aristotle, I have argued, not only announces the need for an account of bodies and magnitudes for the purposes of physical science, but also discusses problems that may arise for such a project. One such problem is whether this inquiry should rather be the task of the mathematician. Isn’t it the mathematician who studies magnitudes? And if so, should we say that physics is a branch of mathematics? Aristotle picks up these questions explicitly in Physics II.2. This chapter, I argue, provides a systematic answer to the question how it should be possible that both the mathematician and the physicist study magnitudes. Thus, Aristotle asks at the beginning of Physics II.2: We must consider how the mathematician differs from the physicist; for natural bodies have surfaces and solids, lines and points, which are treated by the mathematician.2 (Ph. II.2 193b23–25)

Aristotle observes that the mathematician and the physicist apparently study the same objects. Since a science is individuated by its subject matter and the attributes belonging to this subject matter, the problem arises how there can be two sciences. One could assume that the physicist studies the moon and the mathematician studies the attributes of the moon, e.g. sphericity. One version of this proposal could be that physical objects have a certain extension and are bounded by surfaces, but the study of the extension and the surfaces is part of mathematics. Thus, there is a division of labour between the mathematician and the physicist. I have already argued against such an interpretation, but Aristotle himself makes clear in this passage that such a proposal won’t work. The physicist clearly studies bodies and their per se attributes (τ συμβεβηκ τα καθ’ α τ).3 As Charlton rightly notes: By ‘things which supervene’ [τ συμβεβηκ τα καθ’ α τ/CP] here Aristotle probably means not just ‘accidents’, things which are affections of natural things, but features about which it is the business of the student of nature to argue and attempt demonstrations. (Charlton 1992, 93)

What are the per se attributes? Following Charlton we should say that they are those attributes about which the scientist makes demonstration. This is indeed a very plausible suggestion. For, consider the questions whether there is an infinite body or how large the place of an object is. These questions cannot be decided without taking into account the extension of a physical object. Extension and sphericity are not properties that the moon just happens to have and that play no role in a scientific account of the moon. The physical scientist may not be concerned with the question who was the first man on the moon. This is something that should not concern the physicist. But the shape and extension of the moon clearly is an altogether different thing. Thus, Aristotle rightly claims that it is obvious that the physicist is 2 θεωρητον τνι διαφρει  μαθηματικ το φυσικο (κα γρ ππεδα κα στερε χει τ φυσικ σ!ματα κα μ"κη κα στιγμ#, περ $ν σκοπε%  μαθηματικ ). 3 Ph. II.2 193b27–28.

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 aristotle’s theory of bodies concerned with the quantitative properties of the moon.4 Of course, this does not mean that the physicist is solely concerned with the quantitative properties of the moon. It is equally important for her to know what the moon is made of, or where the moon will be in two weeks’ time. Be that as it may, it seems generally true that both the mathematician and the physicist study body and magnitudes and questions pertaining to the dimensions and distances of the heavenly bodies was a well-established topic in natural philosophy. Nevertheless, Aristotle equally opposes the suggestion that the mathematician and the physicist do not differ after all. Mathematics is not a branch of physical science, nor is physical science a branch of mathematics. How, then, do they differ? Aristotle explains it in the following lines: Now the mathematician, though [1] he too treats of these things [i.e. shape of the moon and sun], nevertheless [2a] does not treat of them insofar as they are the limits of a natural body; nor [2b] does she consider the attributes insofar as they are the attributes of such bodies. [3] That is why she separates; [4a] for in thought they are separable from motion, and [4b] it makes no difference, nor does any falsity result, if they are separated. [5] The holders of the theory of Forms do the same, though they are not aware of it; for they separate the natural things, although they are less separable than the mathematical objects. [6] This becomes plain if one tries to state in each of the two cases the definitions of the things and of their attributes. [6a] Odd and even, straight and curved, and likewise number, line, and figure, can be defined without reference to motion; [6b] not so flesh and bone and man—these are said to be like a snub nose, not like a curve.5 (Ph. II.2 193b31–194a7)

The point of the passage is to explain the difference between the mathematician and the physicist.6 Let me paraphrase the course of the argument as I understand it. Aristotle first affirms that [1] the science of nature and mathematics do indeed concern the same things.7 But the way in which they study these items is different.

4

Ph. II.2 193b28–30. περ το&των μ'ν ο(ν πραγματε&εται κα  μαθηματικ , )**’ ο+χ , φυσικο σ!ματο πρα -καστον· ο+δ' τ συμβεβηκ τα θεωρε% , τοιο&τοι ο(σι συμββηκεν· δι κα χωρζει· χωριστ γρ τ/ νο"σει κιν"σε! στι, κα ο+δ'ν διαφρει, ο+δ' γγνεται ψεδο χωριζ ντων. *ανθ#νουσι δ' τοτο ποιοντε κα ο1 τ 2δα *γοντε· τ γρ φυσικ χωρζουσιν 3ττον 4ντα χωριστ τ5ν μαθηματικ5ν. γγνοιτο δ’ 6ν τοτο δ7*ον, ε8 τι 9κατρων πειρ:το *γειν το; κα σχ7μα, =νευ κιν"σεω, σρξ δ' κα @στον κα =νθρωπο ο+κτι, )** τατα Aσπερ B σιμ> )**’ ο+χ C τ καμπ&*ον *γεται. 6 In an interesting paper Lennox 2008 argues that the passage has an important function within the context of Physics II. According to Lennox, the point of the passage is to make clear that the natural form of physical objects is not identical to mathematical attributes. Since Aristotle has argued in Physics II.1 that natural objects have matter and form, he wants to fend off a possible misconception which says that physical science has a material part and a formal part, the latter being mathematics. Though I am sympathetic with the attempt at giving a unified reading of the passage and believe that Lennox’s reading may be correct, I remain non-committal with regard to the question, what is the overall function of this passage. It seems to me that the distinction between the mathematician and the physicist can be studied in isolation from its immediate surroundings. 7 I follow Hussey 1991, 108 and use the phrase ‘to be concerned with’ here and elsewhere to be neutral with respect to the question, what is the primary subject matter of a science. In a study of X qua X the 5

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the theory of physics ii.  For, the mathematician studies bodies and their limits, but not [2a] insofar as they are physical and not [2b] insofar as the attributes are attributes of a physical body, i.e. a physical substance. In other words, the mathematician studies bodies and their attributes without reference to their physical nature. She studies them as if they were pure quantities. By implication, the physicist studies bodies and magnitudes insofar as they belong to physical substances. I take this to be the gist of my argument in Section 3.3, where I discuss the specific mode and interest with which the physicist studies magnitudes. If this is right, we see that Aristotle explicitly recognizes the two distinct ways in which body, surfaces, and lines can be studied: either as bodies, surfaces, and lines of physical substances, or as bodies, surfaces, and lines tout court, and not insofar as they belong to a physical substance. Aristotle claims further that [3] the work of the mathematician involves a separation. The reason why the study involves a separation is due to the specific character of the mathematician’s investigation. Because she does not study magnitudes insofar as they belong to a physical substance, she separates. From the sentential connection we can, moreover, read off what is separated. The geometer separates quantities (lines, planes, bodies) and their attributes (straight, curved, etc.). In [4] Aristotle justifies this method with two important points. First, he explains what they are separated from. Quantities are separated from motion. I think that the ‘from what’ and the ‘what’ of separation are simply opposing perspectives on the same method. Either one can ask what the mathematician separates—to this the answer is ‘magnitudes’—or one can ask what the magnitudes are separated from—to this the answer is ‘motion’. By subtracting all properties that imply motion, the mathematician separates magnitudes. Second, Aristotle makes clear that the separation the mathematician performs neither makes a difference nor produces any falsity. Thus, in separating magnitudes from motion or in studying magnitudes, but not insofar as they belong to a physical substance,8 the mathematician does not make false claims. If the mathematician considers what belongs to a physical substance solely insofar as it is extended, she does not get false results. Aristotle compares his method [5] to the method of the Platonists, who unknowingly do the same with regard to the objects of physical science. They separate things which are less separable than mathematical objects. This incidental remark is important, since it shows that Aristotle thinks that there must be a fundamentum in re for separation. There is a reason why there is no falsity when a mathematical object is separated and there is a falsity involved when a physical object is separated. This reason is grounded in the difference of the two classes of entities. It is a special characteristic of quantities that they can be separated. [6] This special characteristic is explained by the simile of the snub nose. Physical things are defined like the snub, primary subject matter is X. In a study of X qua Y it is Y which is the primary subject, but the study is concerned with X. 8

These two phrases should be taken as equivalent.

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 aristotle’s theory of bodies whereas quantities are defined like the concave. Quantities are separable because of what they are. The nature of quantities explains why the mathematician can study them the way she does. This is the argument as I understand it. Of course, this is no more than the sketch of a detailed interpretation and it would be a separate task to flesh out the details. But since our topic is the more limited question how the mathematician and the physicist can both study quantities and yet be engaged in different sciences, I would like to address the following set of questions more specifically: 1. What does it mean that the mathematician studies magnitudes, but not insofar as they are the magnitudes of a physical substance? 2. Why can the physicist draw on results from mathematics? An answer to the first question will separate a mathematical study from a physical study of magnitudes. In showing how these studies differ, we may show what is distinctive about them. An answer to the second question will bring the two studies closer together again. We may show that a focus on the physical ontology of bodies does not mean that we have to neglect mathematical results.

4.1.1. Studying X but not qua Y Both the physicist and the mathematician study magnitudes. However, though the mathematician studies physical bodies and their boundaries, she: [2a] does not treat of them insofar as they are the limits of a natural body; nor [2b] does she consider the attributes insofar as they are the attributes of such bodies.9 (Ph. II.2 193b32–33)

With this sentence Aristotle opens his argument on why the mathematician and the physicist differ. The physicist studies body and magnitudes insofar as they are the bodies and magnitudes of a physical substance. The mathematician studies them, but not insofar as they are the bodies and magnitudes of a physical substance. .... a remark on x qua y Let me begin with a brief general remark about the qua-locution. What is it to investigate Xs qua Y? To prove something of X qua Y means that one takes Y as the proper subject and proves a theorem about Y.10 The theorem holds of X because it is a Y. Take the brazen triangle as an example. If one proves that the brazen triangle has internal angles equal to two right angles (2R), the proof is about the brazen triangle qua triangle. Insofar as the object is a triangle, it has internal angles equal to two right angles. The proper subject for the proof is triangle, not brazen triangle. In this sense )**’ ο+χ , φυσικο σ!ματο πρα -καστον· ο+δ' τ συμβεβηκ τα θεωρε% , τοιο&τοι ο(σι συμββηκεν· I do not agree with Detel, who thinks that logical abstraction and the qua-locution are quite different. Cf. Detel 1993a, 175. It is true that in the special case of mathematics a genus is abstracted. But I think that this is an extension, rather than revision, of the theory. After all, a genus can be seen as a (special kind of) subject. 9

10

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the theory of physics ii.  the property of being brazen is disregarded. The qua-locution is intimately connected to Aristotle’s concept of an universal proof.11 A universal proof proves something of an arbitrary and primitive case.12 That is to say, even though 2R holds of brazen triangle, there is no universal proof of this fact. The reason is that it does not hold primitively of brazen triangle. Insofar as it is a triangle, it is true of the object that it has 2R. 2R holds primitively and universally of triangle. Hence, the proof concerns its being a triangle alone. The method of subtraction thus understood is not a psychological theory.13 To take away or to subtract a property in this context does not refer to a psychological process of imagining the object without the property.14 If one takes away the red colour of a triangle, one does not (and how should one?) imagine a colourless triangle. Rather one considers the red triangle only insofar as it is a triangle. Moreover, the qua-locution describes a manner of investigation.15 This has an implication for the ontological import of sciences. Sciences are individuated by their manner of investigation which does not imply a separate set of objects with which a given science deals.16 In studying X qua Y one does not study an object ‘X qua Y’. In contrast to modern theories about the qua-locution, Aristotle’s use does not introduce a ‘qua-object’ different from the original object.17 Rather, in studying X qua Y one studies X insofar as X is Y.18 In studying physical substances insofar as they are moved, one considers what is true of them in virtue of the fact that they are moved. And in studying physical substances insofar as they are extended, one considers what is true of them in virtue of the fact that they are extended. Following Jonathan Barnes, one can express this by distinguishing between the domain and the focus of the study: In phrases of the form ‘to study Fs qua G,’ the term replacing F fixes the domain of the study, and the term replacing G fixes the aspect or the focus of the study. (Barnes 1995, 70)

The metaphysician studies beings insofar as they are beings. She considers beings and studies what belongs to them in virtue of the fact that they are beings. The domain of

11

See APo I.5 and the analysis of that chapter in Hasper 2006b. APo I.5 73b32–33. 13 Cf. Annas 1976; Philippe 1948. ‘Subtraction’ is my translation of ‘)φαρεσι’. I prefer it over ‘abstraction’ because, as we will see, the method of )φαρεσι relies on disregarding or omitting certain properties of an object in the study of this object. 14 As noted by Henry Mendell: ‘By subtracting properties, we are not depriving the object of them’ (Mendell 1986, 62). 15 As was noted by Barnes 1995, 70. 16 As is emphasized by Netz 2006, 18: ‘The existence of a separate science does not entail the existence of a separate set of objects.’ 17 Therefore, it is misleading to speak of ‘the qua-operator’. See Netz 2006, 11f. For a modern theory in which the qua-locution is treated as an operator on objects cf. Fine 1982. 18 In the words of Netz: ‘An argument about A, QUA G is an argument whose only special presupposition is that A is G’ (Netz 2006, 13). 12

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 aristotle’s theory of bodies metaphysics is beings and the focus is what belongs to them as beings.19 Similarly, Aristotle states that the geometer studies physical magnitudes insofar as they are extended in n-dimensions (n=0,1,2,3).20 Clearly it is possible that there should also be both formulae and demonstrations about perceptible magnitudes, not however insofar as they are perceptible, but insofar as they are of such-and-such a kind.21 (Metaph. XIII.3 1077b20–22)

Thus the domain of the study of the geometer are the bodies and magnitudes of physical substances , but the focus of her study is simply magnitudes. The geometer studies physical substances, but considers only what belongs to them qua being body. In the context of Metaphysics XIII.3, this distinction between the domain and the focus of a study is used to explain why the work of the mathematician does not involve the assumption that she studies a separate domain of objects. Both the mathematician and the physicist are ultimately concerned with the same underlying reality. There is no ontologically separate realm of bodies and magnitudes that the mathematician studies. In doing her science the mathematician is ultimately concerned with the domain of physical substances, though her focus is on them only insofar as they are extended.22 However, it is important to see that the distinctive character of a science, what a science is about, is given by the focus of the study. The mathematician, for example, is concerned with physical magnitudes, but the subject matter of mathematics is not physical magnitudes.23 Rather, the focus of the science of mathematics is simply magnitudes. In the course of a mathematical deduction no terms are used that reveal the physical character of the objects under investigation. It is not a theorem of mathematics that a brazen triangle has internal angles equal to two right angles. The mathematician studies triangles. So when she considers the brazen triangle qua triangle, the proper object of her study is a triangle.24 In general, geometry is about extended and continuous objects (plus points). This is the subject matter of geometry. Geometry is concerned with physical magnitudes, but it is not about them. It is concerned with physical objects because the objects

19 I assume that what Barnes calls ‘domain’ and ‘focus’ corresponds to what Netz calls ‘object’ and ‘manner’. Cf. Netz 2006, 14. 20 I include points which are extended in 0-dimensions, i.e. not extended at all. In what follows I will, for reasons of simplicity, use the term ‘extended objects’ in a way that tacitly includes points. 21 δ7*ον , δ' α2σθητ )**’ , τοιαδ. 22 For a more complete interpretation I refer the reader to Lear 1982; Detel 1993b; Pettigrew 2009; Hussey 1991. 23 Again, in speaking of the mathematician I use the phrase ‘is concerned with physical magnitudes’ to capture the Greek ‘περ τ5ν α2σθητ5ν μεγεθ5ν’ (Metaph. XIII.3 1077b21). In this usage I follow Hussey 1991, 108 fn. 7. If a science is concerned with X, it does not follow that the subject matter of this science is X. Cf. Hussey 1991, 108–9. 24 Cf. Mueller 1970, 164: ‘To say that the mathematician studies man as solid is not to say that he studies man at all. Rather, it is to say that he studies what is quantitative and continuous in three-dimensions.’

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the theory of physics ii.  geometry is about are ultimately the body and magnitudes of physical substances. Note again that, on my interpretation, the distinction between the physicist and the mathematician is not merely that the former studies physical substances whereas the latter studies magnitudes. I am inclined to take the reference to ‘physical magnitudes (περ τ5ν α2σθητ5ν μεγεθ5ν)’ to refer to the study of magnitudes insofar as they are the bodies of physical substances. In other words, the physicist also studies magnitudes, but qua physical. In this way, both the physicist and the geometer are concerned with physical magnitudes, the magnitudes of physical substances, but mathematics is about magnitudes, whereas physical science is about physical magnitudes. .... magnitudes, but not qua physical In the larger context of Aristotle’s philosophy of mathematics, the fact that geometry is ultimately concerned with physical substances is undoubtedly a key doctrine. For our present purposes, however, we are not interested in the question why the existence of a separate science of geometry, which seems to be taken for granted in Metaphysics XIII.3, does not imply a separate domain of objects over and above physical substances.25 Rather, our interest is how both the mathematician and the physicist can study magnitudes and yet be engaged in distinct scientific endeavours. What is the focus or manner of the mathematician’s investigation that justifies there being a separate science of mathematics? It is in this light that the passage in Physics II.2 should be read. Aristotle wants to show the difference between the physicist and the mathematician by noting that the mathematician’s work involves a separation that the physicist’s work does not involve.26 The physicist studies—and as I argued must study—magnitudes. The mathematician, too, studies them, but not insofar as they are physical. The mathematician studies surfaces, but treats these surfaces not insofar as they are limits of physical substances. The qua-locution is combined here with a negation. How should we understand this? The negation applies to the whole qua-locution. Aristotle does not say that the mathematician studies surfaces insofar as they are not boundaries of a physical substances. This would mean that the mathematician explicitly denies that the surfaces she studies are boundaries of a physical substances. If she did this, it would be hard to explain why mathematical theorems should apply to physical substances.27 Rather, Aristotle makes clear that the mathematician studies surfaces that are in fact boundaries of a physical substance, though in the course of her study she does 25 Commentators agree that Aristotle’s major point in Metaphysics XIII.3 is a negative one. Cf. Detel 1993b; Lear 1982; Hussey 1991. Aristotle wants to show that the existence of a science like mathematics does not presuppose a distinct ontological realm of mathematical objects. 26 It has been noted by Mueller that it is difficult to say how the theory of Physics II.2 and Metaphysics XIII.3 go together: ‘There seems to be a significant difference between separating mathematical objects from physical bodies and treating physical bodies as mathematical objects’ (Mueller 1970, 159). For an attempt to reconcile the two passages cf. Detel 1993b, 189–232. 27 This distinction in connection with the applicability of mathematical theorems will be discussed further below (see Section 4.2.2).

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 aristotle’s theory of bodies not consider them qua boundaries of a physical substance. Thus, Aristotle does not deny that the mathematician considers surfaces of a physical body. In mathematics this simply plays no role. There are no theorems involved that refer to the physical nature of magnitudes. Whether or not these magnitudes are physical magnitudes is not in the focus of the mathematician’s study. This might be better expressed in terms of ‘omitting’. The mathematician omits that the surfaces are in fact surfaces of a physical substance. However, in omitting that the surfaces are in fact surfaces of a physical substance she does not neglect her duty as a mathematician. As a geometer she should not be studying surfaces insofar as they belong to physical substances. It simply is not her field of study.

4.1.2. Separation and falsity Aristotle expresses this idea in terms of a separation that the mathematician performs. In studying surfaces but not qua limits of a natural body the mathematician separates them as follows: [3] That is why she separates; [4a] for in thought they are separable from motion, and [4b] it makes no difference, nor does any falsity result, if they are separated. [5] The holders of the theory of Forms do the same, though they are not aware of it; for they separate the natural things, although they are less separable than the mathematical objects. [6] This becomes plain if one tries to state in each of the two cases the definitions of the things and of their attributes. [6a] Odd and even, straight and curved, and likewise number, line, and figure, can be defined without reference to motion; [6b] not so flesh and bone and man—these are said to be like a snub nose, not like a curve.28 (Ph. II.2 193b33–194a7)

Aristotle presents it as an inference: because the mathematician studies surfaces, but not qua limits of a natural body, she separates. The separation points again to the distinctive method of the mathematician. She treats of surfaces, but—in a sense to be explained in a moment—she separates them. This is, I believe, also the thought behind Aristotle’s remark: Each thing will be best investigated in this way—by supposing what is not separate to be separate, and this is exactly what the arithmetician and the geometer do.29 (Metaph. XIII.3 1078a21–22)

The mathematician supposes that surfaces constitute an independent field of study. Again, this neither means that surfaces are in fact part of a realm ontologically distinct

28 δι κα χωρζει· χωριστ γρ τ/ νο"σει κιν"σε! στι, κα ο+δ'ν διαφρει, ο+δ' γγνεται ψεδο χωριζ ντων. *ανθ#νουσι δ' τοτο ποιοντε κα ο1 τ 2δα *γοντε· τ γρ φυσικ χωρζουσιν 3ττον 4ντα χωριστ τ5ν μαθηματικ5ν. γγνοιτο δ’ 6ν τοτο δ7*ον, ε8 τι 9κατρων πειρ:το *γειν το; κα σχ7μα, =νευ κιν"σεω, σρξ δ' κα @στον κα =νθρωπο ο+κτι, )** τατα Aσπερ B σιμ> )**’ ο+χ C τ καμπ&*ον *γεται. 29 =ριστα δ’ 6ν οEτω θεωρηθεη -καστον, ε8 τι τ μ> κεχωρισμνον θεη χωρσα, κα Hρμονικ> κα )στρο*ογα· )ν#πα*ιν γρ τρ πον τιν’ χουσιν τ/ γεωμετρI. F μ'ν γρ γεωμετρα περ γραμμ7 φυσικ7 σκοπε%, )**’ ο+χ , φυσικ", F δ’ @πτικ> μαθηματικ>ν μ'ν γραμμ"ν, )**’ ο+χ , μαθηματικ> )**’ , φυσικ". 61 On this I agree with Mueller 2006.

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 aristotle’s theory of bodies be part of physical science and not merely be principles of a mixed science, such as optics or astronomy? The answer lies, I believe, in the way these principles are linked to the ontology of physical substances. To use an example from Part II of this work:62 if two objects are in contact, there are two boundaries in the same place. The reason for there being two boundaries, rather than one, is due to the fact that boundaries are dependent particulars. In the case of continuity, on the other hand, the parts of an object share a boundary. There is, thus, a topological distinction between contact and continuity. However, this difference is grounded in the ontology of physical objects. It is the nature or form of a physical object which ultimately explains why some parts are continuous and constitute a whole.63 My claim, therefore, is not that the principles concerning bodies which I discuss in this work cannot be applied in the mixed sciences. The difference between the purely mathematical and the optical treatment lies, as I suggested, in the fact that lines are treated as light rays, or a point as an eye. This is, I take it, behind Aristotle’s remark that ‘optics investigates mathematical lines, but qua natural, not qua mathematical’ (Ph. II.2 194a10–11). Insofar as the optician takes the mathematical lines in a diagram to be a representation of actual physical lines, she will have to say that, if the lines are in contact, there are two boundaries in the same place. Crucially, however, the mathematician or the optician are in no position to justify the claim that physical objects are continuous due to their nature or that, if two physical objects are in contact, there are two boundaries in the same place. An explanation of why physical substances have the topological structure they in fact have requires an analysis of them qua physical bodies. A principle which relies on the nature of an object falls outside the realm of mathematical considerations. Thus, my point is that the principles of a theory of bodies are set up in such a way that their main application as well as their justification relies on considerations which are, since they rely on the notion of a nature, genuinely physical. Related to this discussion is another point I wish to address briefly. In the case of mixed sciences, Aristotle not only claims that the subordinate science can draw on truths of the higher science, but also claims that the subordinate science states the fact, whereas the higher science states the reason why: For here it is for the empirical scientists to know the fact and for the mathematical to know the reason why; for the latter have the demonstrations of the explanations, and often they do

62 I will discuss these issues at length in Sections 7.3 and 7.4. Cf. also Propositions 31 in Section 7.3.2.1, Proposition 32 in Section 7.4.1, and Proposition 33 in Section 7.4.2.1. 63 Of course, these considerations may not always be relevant to an issue at hand. Most of the time, referring to the topological distinctions is enough as an explanation. As I said in the Introduction, depending on the level of generality, Aristotle may or may not adduce certain considerations or explanations. The theory of natural places is discussed in Physics IV, but it is absent from Physics VI. It is again very important in De Caelo. Does this imply that the notion of place has changed in Physics VI? Not necessarily; it may simply be due to the fact that, on the level of generality the discussion in Physics VI proceeds, these considerations are not relevant.

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the mathematician and the physicist  not know the fact, just as those who consider the universal often do not know some of the particulars through lack of observation.64 (APo. I.13 79a2–6)

Consider a work such as Ps.-Euclid’s Catoptrics: the optician knows that light rays are reflected in a certain way when they hit a mirror, but the mathematician knows why they are reflected in a certain way. The mathematician may not know that rays of light are reflected in this way. Nothing in her study says anything about light rays. Importantly, however, once this evidence is presented to her, she will be in a position to explain why this is the case. For the laws of geometry explain this law of optics. It is precisely at this point where a genuine physical science differs from optics or the other mixed sciences. For the explanatory ground is not provided by any higher science. An act of seeing, for example, has to be explained by a causal story which involves an account of the reception of a perceptible form. This explanation lies outside the realm of optics. In this sense, a philosophical discussion of the principles of physical science and the demonstrations within that science are closely related: in both cases, the fundamental explanatory grounds rely on physical considerations.65 Why can the physicist draw on mathematical results? Although the theory of bodies belongs to the foundations of physical science, it remains the case that in a physical study of magnitudes one may, and presumably must, draw on mathematical results. How is that possible? As I said, I am inclined to believe that there is a quite harmless version of subordination in Aristotle, but, since this is quite controversial, I will develop my argument without relying on the notion of subordination. Instead I will rely on the terminology of the focus and domain of a science, which I developed earlier. In short, I will argue that the focus of the physicist and the focus of the mathematician overlap. The physicist studies physical substances insofar as they have a nature. The study of nature, in turn, requires a study of magnitudes. The mathematician, on the other hand, studies physical substances insofar as they are magnitudes. This constitutes an overlap between the two sciences. To put it simply, the study of geometry is concerned with physical substances qua extended and part of the study of the physicist is concerned with physical substances qua extended and movable magnitudes. The reason why the physicist can rely on mathematical truths is based on the thought that what holds of an object insofar as it is a body also holds of the object insofar as it is a movable body. In more general terms, what holds of X qua Y also holds of X qua Z, if being Z implies being Y. A simplified model of what I have in mind can be found in APo. I.5.66 Aristotle argues that to prove that brazen isosceles triangles have interior angles equal to two

64 νταθα γρ τ μ'ν δ γραμμ8 συνεχ στιν· (στι γ#ρ αβεν κοιν ν -ρον πρ  ?ν τ# μρια ατ4 συν&πτει, στιγμ+ν· κα τ4 πιφανε α γραμμ+ν, – τ# γ#ρ το πιπδου μρια πρ τινα κοιν ν -ρον συν&πτει. – @σα6τω δ κα π το σ/ματο (χοι 1ν αβεν κοιν ν -ρον, γραμμ8ν : πιφ&νειαν, πρ  Aν τ# το σ/ματο μρια συν&πτει.

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the continuous and the discrete  they are connected.26 The boundary is at least one dimension less than the thing whose boundary it is.27 As stated above, it is reasonable to assume that the adjacent parts of a line or a body must be connected by a boundary.28 a

•

•

b

•

c

•

Clearly, this line is continuous, though part a and part c do not join at a common boundary. It is not a counterexample because a and c are not next to each other.

5.2.2. Getting the quantifiers right Whether an object is discrete or continuous is determined by the absence or presence of a common boundary for the adjacent parts. If X is discrete, its adjacent parts lack a common boundary. If X is continuous, its adjacent parts join at a common boundary. Let us ask ourselves whether all the adjacent parts of a continuous object should be connected, or only some? It seems to me that all the adjacent parts must be connected. Consider the following example: X:•

a

•

b

•

•

c

•

Does this represent a continuous line? It seems not to, because parts b and c are not connected. There are two lines, namely the line consisting of parts a and b and another line c. Therefore, in the case of continuity we should say, it seems, that all the adjacent parts must be connected. It is not sufficient for continuity that only some adjacent parts are connected. In our example it is not sufficient that a and b are connected because the further connection to c fails. This, however, makes it difficult to classify X at all, because Aristotle seemingly also holds that no adjacent parts of an object are connected when he says that: nor could you ever in the case of a number find a common boundary of its parts, but they are always separate.

This is the only instance where Aristotle makes the quantifier explicit, so one should not take it lightly. However, if we follow Aristotle’s assertion that the parts of something discrete never have a common boundary, line X is not discrete. Since it is also not continuous, it is neither continuous nor discrete. Of course, one could conclude that division into the continuous and the discrete is not exhaustive. But in my opinion, 26 Though I leave it to the next chapter to explain precisely what object means. For the time being it is sufficient to think of an individual item such as a triangle rather than the property triangularity. 27 Though it will be discussed later, I wish to say that boundaries are not fictional items. A body is bounded by a surface. The surface is a dependent individual, not a fiat object. The equator is a fiat object. The surface of a cube has for Aristotle a different ontological status. For more on boundaries see Section 6.3. For a modern discussion see Varzi 2008, 1997; Casati and Varzi 1999; Smith 1997a. 28 Setting aside several topological relations where an object may (intuitively) be said to have two parts which are connected by two boundaries. A torus could be an example: its two halves are connected by two boundaries.

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 aristotle’s theory of bodies this assumption goes against the spirit of these classifications. I believe that Aristotle wants to classify every quantity as either continuous or discrete.29 So, what shall we say about X? I suggest that X is discrete. I have two reasons for this assumption. The first is simply that it seems to me that X cannot be classified as continuous. I do not think that Aristotle would be willing to classify X as continuous. Hence, it must be discrete. This leads to my second reason. When Aristotle says that discrete objects have no boundaries at which their parts join, he is speaking about numbers. With respect to numbers it is true that all the parts are unconnected. The problem is that our object X is in no straightforward sense a number. A number is, roughly, an aggregate of things which are treated as indivisible units.30 In the case of X, I think, we should say that it represents the number 2. If we think that X represents the number 2, there are two entities a+b and c (rather than one single continuous entity). In this case, the question whether there is a boundary at which part a and part b are connected simply does not arise. It is not the case that X has two continuous parts a+b and another part c that is discrete from a+b. Rather there are only two parts a+b as an indivisible unit and c as another indivisible unit. In general, an object should be regarded as discrete if the connection between its parts fails once. If my analysis of objects such as X is correct, an object is discrete if at least some of its adjacent parts are not connected.31

5.2.3. Defining the continuous and the discrete The criterion determining whether something is continuous or not is the connection of its parts. A body, for example, is continuous because it is connected. How should we understand this ‘because’? Though Aristotle does not say so explicitly I think we might reasonably interpret it as a definition of the continuous. To be continuous is defined by having parts that join at a boundary, and to be discrete is defined by the lack of such a connection. I think that Aristotle not only mentions a necessary condition for continuity,32 but in fact spells out what it is to be continuous. To be connected in the right way is necessary and sufficient for being continuous. Proposition 2. X is continuous (CONT) = connected by a boundary.

def

All the adjacent parts of x are

Proposition 3. X is discrete (DISC) = def Some adjacent parts of x are not connected by a boundary.

29 Notice that this is also the case if these distinctions are drawn for dialectical purposes. For it seems plausible that winning the dialectical ‘quantity-game’ gets difficult if the classifications of quantities are not exhaustive. 30 Cf. Metaph. XIII.3 1077b30. 31 I am not saying that Aristotle explicitly considered such an example. My suggestion is that examples like this are not foreign to Aristotelian texts and all the tools we need for an answer can be extracted from the texts. 32 In logical terms: X is continuous only if the parts of X are connected.

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the continuous and the discrete 

5.2.4. Some limitations of the theory Aristotle’s discussion in the Categories, however, has certain limitations that should not escape attention. These limitations are, I believe, mostly due to the status of the Categories and the resulting caveat I discussed in Section 5.1. In the context of the present discussion the most important limitation concerns the absence of the distinction between continuity and contact. This distinction is crucial in his discussion in Physics V.3.33 In the Categories Aristotle even uses the word ‘being in contact (συν&πτειν)’ in his elucidation of continuity. This can be misleading because in other contexts ‘contact (Bφη)’ is explicitly contrasted with ‘continuity (συνχεια)’.34 The absence of this distinction means that on the basis of the theory of the Categories one cannot distinguish between the case of several things that touch each other and several parts of one thing that join. Take Fig. 5.2 as an example: a

b Fig. 5.2.

In the Categories Aristotle gives us no guidelines to decide the question whether the picture shows two objects a and b that touch each other or one object that has a and b as parts. Or consider the following two objects in Fig. 5.3:35

Fig. 5.3.

Do they have the same topological arrangement? Both are continuous objects and both look like discs. In both cases there are no ‘holes’ in the object. Yet there seems 33

See Chapter 7 for a detailed discussion. Cf. Ph. V.3 and Metaph. III.5 1002a34–b5. 35 Except the spiral which is from wikipedia (http://en.wikipedia.org/wiki/Archimedean_spiral) all other graphics in this work were kindly provided by Mareike Witt. 34

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 aristotle’s theory of bodies to be a difference between those objects.36 If one thinks of the second object as a snake, one can see the difference. The spiral line can be seen as the skin of the snake at which the parts are not continuous, but merely in contact. The difference between continuity and contact, however, is not explicitly drawn in the Categories. We learn that something is continuous if and only if all its adjacent parts join at a common boundary. But whether and in what respect ‘joining at a common boundary’ is different from ‘being in contact’ is not stated in the Categories.37 As a consequence, in the Categories CONT and DISC are not fully explained because the notion of being connected by a boundary is not fully explained. In other words, Aristotle states both in the Categories and in Physics V.3 that the parts of something continuous join at a ‘common boundary’.38 I believe that it is plausible to think that he means the same. But from the Categories we cannot tell whether this is so. And the reason is that only in the Physics Aristotle explicitly contrasts ‘being continuous’ with ‘being in contact’. The theory from the Categories is in this respect not specific enough. A similar limitation might hold of the notion of part that is employed in the Categories. Aristotle does not explicitly define what counts as a part, but I think it is plausible that the notion of part employed here corresponds to the first sense of part distinguished in Metaphysics V.25: We call a part that into which a quantity can in any way be divided; for that which is taken from a quantity qua quantity is always called a part of it, e.g. two is called in a sense a part of three.39 (Metaph. V.25 1023b12–15)

The parts into which a quantity, insofar as it is a quantity, can be divided are themselves quantities. The parts into which a surface can be divided are themselves surfaces, as opposed to the colour of the surface which is a part in another sense. This, however, does not yet tell us whether, for example, points are parts of the line. In chapter six of the Categories Aristotle does not inform us.40 However, I think that these limitations should not be seen as a serious shortcoming. As I said earlier, in the Categories Aristotle is not concerned with laying out a complete theory. This is not the purpose of the Categories. However, I think Mann is right in emphasizing the ‘potential availability of explanations, even if none are provided’ (Mann 2000, 4 fn. 4). The account of the Categories can be consistently extended by accounts of quantities from the Metaphysics or Physics. The necessary distinctions can be drawn using the Categories’ account as a basis which is then further augmented. Although, admittedly, many questions are left open, the question can be answered by taking in more material and not by revising or rejecting the account of the Categories.

36

Or to be more precise: given the expressive power of Aristotle’s theory we can draw a distinction between those objects. 37 I address this question in Chapter 7. 38 Contra Cattanei 2011, 137 who thinks that the Categories and the Physics disagree on this point. 39 Cρο γεται Dνα μν τρπον ε% ? διαιρεθε η 1ν τ ποσ ν 3πωσον (*ε γ#ρ τ *φαιρο6μενον το ποσο E ποσ ν μρο γεται κε νου, ο)ον τ=ν τρι=ν τ# δ6ο μρο γετα πω). 40 For a discussion see Sections 6.3 and 6.4.

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the continuous and the discrete  Nevertheless, there are two questions pertaining to Aristotle’s conception of continuity and discreteness that need to be addressed at this stage. First, is this the correct explication of continuity? Second, is the division of quantities into continuous and discrete successful? These questions need to be addressed here because a failure to answer them threatens the usefulness of the Categories for a general account of bodies and magnitudes.

5.2.5. Is it really the definition of continuity? I presented CONT and DISC as definitions. I think that they indeed capture the essence of the continuous and the discrete. In particular, I think that this is the same definition that Aristotle uses in other works as well. My argument comes from looking at non-technical as well as technical uses of the word ‘τ συνεχ’. .... continuity and connection In a non-technical sense τ συνεχ translates best as ‘that which holds together’, which captures the composition of the word from σ6ν and (χω.41 Moreover, Aristotle himself uses that derivation to explain the meaning of the word. In his entry on ‘(χειν’ in Metaphysics Delta he remarks: That which hinders a thing from moving or acting according to its own impulse is said to have it, as pillars have the incumbent weights, and as the poets make Atlas have the heavens, implying that otherwise they would collapse on the earth, as some of the natural philosophers also say. In this way that which holds things together is said to have the things it holds together, since they would otherwise separate, each according to its own impulse.42 (Metaph. V.23 1023a17–24)

Aristotle emphasizes the active role which something that holds together exerts. Of course, it is going too far to assume that continuous things are generally held together by active forces. Nonetheless this passage makes an important point. Continuous objects are connected and hold together. They are ‘ones’, as Aristotle says in Metaphysics V.6.43 Correspondingly, Aristotle defines the continuous explicitly as a kind of connection. Now if the terms ‘continuous’, ‘in contact’, and ‘in succession’ are understood as defined above—things being continuous if their extremities are one, in contact if their extremities are together, and in succession if there is nothing of their own kind between them, . . .44 (Ph. VI.1 231a21–23) 41

Cf. Liddell et al. 1996, ad loc. (τι τ κωον κατ# τ8ν αFτο 3ρμ+ν τι κινεσθαι : πρ&ττειν (χειν γεται τοτο ατ, ο)ον κα οG κ ονε τ# πικε μενα β&ρη, κα @ οG ποιητα τ ν HIJταντα ποιοσι τ ν οραν ν (χειν @ συμπεσντ’ 1ν π τ8ν γ4ν, 2σπερ κα τ=ν φυσιογων τιν φασιν· τοτον δ τ ν τρπον κα τ συνχον γεται K συνχει (χειν, @ διαχωρισθντα 1ν κατ# τ8ν αFτο 3ρμ8ν Dκαστον. 43 Metaph. V.6 1015b35–16a1. 44 L% δ’ στ συνεχ κα Mπτμενον κα φεξ4, @ δι/ρισται πρτερον, συνεχ4 μν Nν τ# (σχατα Dν, Mπτμενα δ’ Nν Bσμα, φεξ4 δ’ Nν μηδν μεταξO συγγεν, . . . 42

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 aristotle’s theory of bodies The reference goes back to Physics V.3, where we find the same definition: The continuous is a certain kind of the contiguous: I call things ‘continuous’ when the limits of each at which they are in contact become one and the same and, as the word implies, hold together.45 (Ph. V.3 227a10–12)

I think that this definition of the continuous is a further elucidation of the definition we find in the Categories.46 But regardless of the question how the connection should be precisely spelled out it seems clear to me that the continuous is in Aristotle’s writings defined as a kind of connected entity. .... continuity and divisibility The reader may suspect that my account contradicts the position that continuity is defined by infinite divisibility. In this vein Wieland thinks that the definition of the continuous as a connected entity is preliminary and is later replaced by the proper definition of the continuous as something that is infinitely divisible.47 Of course, I do not wish to deny that according to Aristotle continuous things are infinitely divisible. But I believe that this is a theorem or inference about the continuous, rather than a matter of definition. I think that Aristotle argues in Physics VI.1 that because a continuous entity is defined by having parts that are connected by a boundary, it cannot consist of indivisible parts, nor can it be divided into indivisibles.48 It is admittedly hard to tell whether the argument begins with an intuitive understanding of continuity and leads to a proper definition, as Wieland believes, or whether the argument begins with the definition of the continuous and deduces what follows from it, as I claim. But I think that the following passage might tip the scales in favour of my reading: All magnitudes, then, which are divisible are also continuous. Whether whatever is continuous is also divisible is not yet, on our present grounds, clear.49 (Cael. I.1 268a28–30)

This question would obviously be senseless if the continuous were defined outright as being infinitely divisible. This passage suggests therefore that Aristotle kept the notion of divisibility and the notion of continuity apart. Continuity is not defined in terms of divisibility. Rather, infinite divisibility is a consequence of the account of continuity. Moreover, it seems plausible to assume that the question Aristotle raises in De Caelo

τ δ συνεχ (στι μν -περ χμενν τι, γω δ’ εPναι συνεχ -ταν τατ γνηται κα Qν τ 0κατρου πρα ο) Bπτονται, κα 2σπερ σημα νει τοRνομα, συνχηται. 46 I discuss the definition from the Physics at length in Chapter 7. 47 Cf. Wieland 1975, 257–8. 48 Cf. Ph. VI.1 231a21–b18. For an analysis see Section 6.4.5 and the literature mentioned there. 49 .ISσα μν οTν διαιρετ# τ=ν μεγεθ=ν, κα συνεχ4 τατα· ε% δ κα τ# συνεχ4 π&ντα διαιρετ&, οRπω δ4ον

κ τ=ν νν. 45

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having parts with position vs without position  is in fact answered in Physics VI.1.50 Accordingly, I believe that the logic of Aristotle’s argument in Physics VI.1 has to be understood thus: Moreover, it is plain that [1] everything continuous is divisible into divisibles that are always divisible; for [2] if it were divisible into indivisibles, we should have an indivisible in contact with an indivisible, since [2a] the extremities of things that are continuous with one another are one and are in contact.51 (Ph. VI.1 231b15–18)

Aristotle shows that something continuous cannot be made up by indivisibles because they cannot touch each other in the way that is requisite for continuous entities. Aristotle states in [1] that something continuous is divisible ad infinitum. This is a theorem about the continuous. But it is not built into the definition of the continuous. It is not the case that someone who supposes that the continuous is composed of indivisibles has failed to understand what continuity means. Therefore, Aristotle supports his claim in [2] by a reductio argument. If it were the case that the continuous is composed of indivisibles, those indivisibles should be in contact. It is in [2a] that Aristotle relies on the definition of the continuous. Since the extremities of the parts of the continuous are in contact and one, or, as we might also put it, have a common boundary, and since indivisibles cannot be in contact with each other,52 it is impossible that the continuous is composed of indivisibles. But if it cannot be composed of indivisibles, it must be composed of divisibles. Thus, we see that Aristotle provides a full-blown argument to the effect that everything continuous is divisible ad infinitum.53 All these passages point towards the fact that the notion of continuity that is employed in the Categories is not deviant, but compatible with other central texts on continuity.

5.3. Having Parts with Position versus Having Parts without Position Besides the division into the continuous and the discrete Aristotle offers another classification as well. 50 Commentators of De Caelo agree with me that this is the place where the question is addressed. Cf. Leggatt 1995; Jori 2009, ad loc. 51 φανερ ν δ κα -τι πUν συνεχ διαιρετ ν ε% α%ε διαιρετ&· ε% γ#ρ ε% *δια ρετα, (σται *δια ρετον *διαιρτου Mπτμενον· Qν γ#ρ τ (σχατον κα Bπτεται τ=ν συνεχ=ν. 52 As was argued before by Aristotle. Cf. Ph. VI.1 231a18–b10. 53 As I said, these arguments will be reviewed in greater detail in Section 6.4.5. Of course, it is an intricate question what a defining feature is in contrast to something that follows from this feature. Aristotle clearly believes that it is metaphysically (in our sense of the word) impossible that the continuous is composed out of indivisibles. Neither it is an empirical question. Hence one might question on what grounds one can distinguish between the defining properties of the continuous and those properties it has as a consequence of that definition. Yet, I think that the test I offered points in the right direction. To argue about the continuous at all one has to grasp what it is. That it is ever-divisible is not part of what it is to be continuous. Otherwise Aristotle would hardly feel the need for an elaborate argument.

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 aristotle’s theory of bodies Further, some quantities are composed of parts which have position in relation to one another, others are not composed of parts which have position.54 (Cat. 6 5a15–16)

Aristotle does not say how he thinks that this classification corresponds to classification into the continuous and the discrete. Yet his discussion makes it clear that the quantities are arranged thus (see Fig. 5.4):55 quantity ~ PPos

PPos

line

surface

body

place

time

speech

number

Fig. 5.4.

The classification suggests that the extension of having parts with position is a subclass of being continuous, because everything that has parts with position is continuous, but not vice versa. In logical terms we may say that having parts with position implies being continuous. The difference between the two classifications is that time does not have parts with position, but is nonetheless continuous. Besides the extensional difference regarding the classification of time, we should ask in what way the notions of continuity and having parts with position differ intensionally. Is there a definitional nexus between the two or are they definitionally independent from each other? For example, the parts of a line have position in relation to one another. [PPos] For each of them is situated somewhere, and you could distinguish them and say where each is situated in the plane and [CONT] which one of the other parts it joins on to. Similarly, the parts of a plane have some position. For one could say [PPos] where each is situated and [CONT] which join on to one another. So, too, with the parts of a solid and the parts of a place.56 (Cat. 6 5a17–23)

Aristotle apparently justifies (note the γ&ρ) his claim that the parts of a line have position in relation to one another by invoking two criteria. First, the parts are situated somewhere, and more precisely they are distinguishable such that one could name their position. Second, all the parts join to one another. The second condition is, of course, what defined being continuous. The occurrence of CONT is probably due to the aforementioned example of the line. For, if one considers the positional relations HILτι τ# μν κ θσιν χντων πρ  ηα τ=ν ν ατο μορ ων συνστηκεν, τ# δ οκ ξ χντων θσιν. Cf. Cat. 6 5a15–37; Studtmann 2002, 2004. 56 ο)ον τ# μν τ4 γραμμ4 μρια θσιν (χει πρ  ηα, – Dκαστον γ#ρ ατ=ν κετα που, κα (χοι 1ν διααβεν κα *ποδοναι ο7 Dκαστον κεται ν τ πιπδ< κα πρ  ποον μριον τ=ν οιπ=ν συν&πτει· – @σα6τω δ κα τ# το πιπδου μρια θσιν (χει τιν&, – 3μο ω γ#ρ 1ν *ποδοθε η Dκαστον ο7 κεται, κα ποα συν&πτει πρ  ηα. – κα τ# το στερεο δ @σα6τω κα τ# το τπου. 54 55

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having parts with position vs without position  between the parts of a line it is natural to assume that it is possible to say where the line parts join to each other. But this is true in virtue of the line being continuous, not in virtue of its parts having a certain position. Therefore, one should not assume that θσι is necessarily tied to continuity.57 It is, of course, perfectly conceivable for two non-continuous objects to stand in a positional relation. Aristotle uses in the Categories the example of the density of an aggregate. An aggregate is dense if and only if its parts are close to each other.58 This is, as Aristotle himself says, a positional relation. Moreover, as we will learn later, the exclusion of time is due to the fact that the parts of time do not exist at the same time.59 If my analysis is correct so far, the definition of having parts with position (PPos) should be stated like this: Proposition 4. x has parts with position (PPos) = def x has parts which [1] are (definitely) spatially situated and [2] exist at the same time.

In what follows I shall first spell out what I understand by the two conditions and then present some more evidence on the notion of position by drawing on other related texts.

5.3.1. PPos considered For an object to have parts with position, it must satisfy two criteria. Its parts must be [1] definitely situated and [2] exist at the same time. To be sure, the two conditions are not independent of each other. The parts of an object cannot be definitely situated (=[1]), if they do not exist at the same time (=[2]). Yet, for expository reasons, it is useful to discuss the two conditions separately. At some points, Aristotle seems to suggest that being situated somewhere (κεσθα που) and having a position (θσι (χειν) are roughly equivalent. With a number, on the other hand, one could not observe that the parts have some position in relation to one another or are situated somewhere.60 (Cat. 6 5a24–25)

This quotation may suggest that being situated somewhere is a mere stylistic variant of having parts with position. Though it surely is difficult to keep these two apart, I am inclined to take being situated somewhere as an explication of having a position. First, this is suggested by what seems to me the more authoritative passage that I quote above and which reads: For example, the parts of a line have position in relation to one another. [PPos] For each of them is situated somewhere, and you could distinguish them and say where each is situated in the plane. (Cat. 6 5a17–19)

57 58 59 60

Contra Cattanei 2011, 148. She thinks that being continuous is part of the definition of PPos. Cat. 8 10a19–21. This is obviously close to a definition of density as mass per volume. Cat. 6 5a26–28.

π δ γε το *ριθμο οκ 1ν (χοι τι πιβψαι @ τ# μρια θσιν τιν# (χει πρ  ηα : κετα που.

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 aristotle’s theory of bodies The notion of being situated somewhere is here used as an explication of the notion of position, as is signalled by the word ‘for (γ&ρ)’. Second, ‘being situated somewhere’ is, in terms of the concepts employed, richer. Its very words make the idea of some sort of spatiality (που) explicit.61 Therefore, I assume that ‘being situated somewhere’ is more basic and an explanation of ‘having a position’. As Aristotle makes clear, the first criterion [1] not only requires that the parts are somewhere in an indefinite sense, but that they have a precise position. For each of them [i.e. parts of a line] is situated somewhere, and you could distinguish them and say where each is situated in the plane. (Cat. 6 5a18–19)

I do not think that Aristotle has some sort of epistemological argument in mind here. He does not mean to say that a mathematician or anyone else has a certain ability to distinguish the parts of a line. Rather, he says that the parts of a line not only lie somewhere on a plane, but have a specific and determinate position. They are, as we will say, definitely situated. Consider the following line, which is composed of two parts. •

a

• b

• Part a and part b not only have a common boundary, but stand in a positional relation to each other. They are connected in a 90 degree angle, and one part is below the other. Having a position thus implies that the parts have a certain kind of spatial orientation towards each other.62 Therefore, the division of quantities into those whose parts have position and those whose parts do not is a division into what we might call spatial entities and non-spatial entities. By ‘spatial entity’ I mean any object that has these positional properties. In this sense, position (θσι) is a relation between the parts of an object.63 In the Categories it is a relation between the parts of an object which assigns a certain position to them. The key concept for assigning a position to the parts is the notion of being somewhere. This notion is central to the notion of spatiality I wish to describe. Moreover, things that have parts with position are coextensive with things that have extension.64 In this sense, only extended things have parts with position which suggests the appropriateness of the term ‘spatiality’.

61 Note that the term ‘spatiality’ is used as a translation of what it means to be που. It is distinct from the related notions of ‘space’ and ‘place’. 62 63 Cf. Ph. IV.1 208b12–25. I discuss the passage below. Cf. Cat. 7 6b12. 64 Cf. the reference to extension in the definition of quantities in Metaph. V.13: ‘In magnitude, that which is continuous in one dimension is length, in two breadth, in three depth. Of these, limited plurality is number, limited length is a line, breadth a surface, depth a body’ (Metaph. V.13 1020a11–14).

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having parts with position vs without position  We may further clarify this by turning to the existence condition (=[2]). Thus, the grounds on which Aristotle denies position to the parts of time is that they do not endure (Fπομνει). Nor with the parts of a time either; for none of the parts of a time endures, and how could what is not enduring have any position? Rather might you say that they have a certain order in that one part of a time is before and another after.65 (Cat. 6 5a26–30)

The parts of something have position only if they endure (Fπομνει). The parts of time are continuous, they join at a common boundary. They have a certain order insofar as the parts of time imply an ordering by priority.66 Perhaps we can even say that the parts of time exist insofar as the past has existed, the present exists now, and the future will exist. But the parts do not stand in a positional relation in a stricter sense, because for Aristotle things which have position must exist at the same time. This, I submit, is another reason for thinking that θσι is a spatial relation. It seems generally true that things standing in spatial relations have to exist at the same time. Let us return to the example of the line. a

•

b

•

•

The line-part a is to the left of part b and the line-part b is to the right of part a. But now let us imagine that the parts do not exist simultaneously: •

a

t1

•

•

b

•

t2

At t2 line-part a has no position with respect to part b. It is neither left nor right of b. Part a does not exist at all. It seems nonsensical to say that part a is left of part b. This sets spatial objects apart from other entities. Only spatial objects have parts with position, which presupposes the simultaneous existence of the parts. Aristotle’s rhetorical question: ‘how could what is not enduring have any position?’ (Cat. 6 5a27– 28), is thus certainly not superfluous, as Ackrill maintains.67 Quite to the contrary: it is (among other things) what distinguishes a spatial order from other types of order, which is precisely what Aristotle does when he says that the parts of time and number have order (τ&ξι), but not position (θσι). The kernel of truth that lies in Ackrill’s remark is that θσι indeed brings in the idea of spatiality.

5.3.2. Position, place, and space It is important, though, that one keeps the notions of position and spatiality distinct from the notion of space or the notion of place. To begin with, one cannot equate the former with space. Wσι is a spatial or topological relation between the parts of an 65 οδ τ# το χρνου· Fπομνει γ#ρ οδν τ=ν το χρνου μορ ων, ? δ μ+ στιν Fπομνον, π= 1ν τοτο θσιν τιν# (χοι; *# μUον τ&ξιν τιν# ε;ποι 1ν (χειν τ τ μν πρτερον εPναι το χρνου τ δ’ 5στερον. 66 67 Cf. Metaph. V.11 1018b14–19. Cf. Ackrill 1963, 94.

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 aristotle’s theory of bodies object, and Aristotle explicitly says that the parts are situated somewhere (κετα που) which seems to presuppose some kind of space. However, one should not presuppose that it involves a reference to an independently existing space. Speaking of spatial relations does not commit us to the view that Aristotle believed in a space that is independent from those objects. More likely, it is an abstract way of describing the spatial configuration of an object. It describes the spatial arrangements of the parts, an arrangement which presupposes the existence of the parts and which brings in the idea of spatiality. This, to repeat, does not require the actual existence of a space, but only the applicability of concepts like extension, being apart from and being close to, above, below, etc. We might freely speak of one part being above the other or a line being extended in one dimension without implying an ontological correlate. On the other hand, the notion of position clearly is connected to the six spatial directions.68 If the parts of a body have a certain position, one can apply notions such as up, down, left, and right. Moreover, all the things that have parts with position are extended in one, two, or three dimensions. This warrants the title of ‘spatial entities’. Second, the notion of spatiality is independent from the concept of place (τπο) as we find it in the Physics.69 Having a position or, as we might say, being somewhere is distinct from being in a place. Two examples may show this. First, only movable bodies have a place, but mathematical items which are unmoved and have no place nevertheless have position.70 Second, the parts of an extended object are positionally related, but they do not have a place. Thus, being in a place and having a position are distinct.71 Moreover, it even seems that the notion of position is more basic than the notion of place because in the explication of the concept of place Aristotle makes use of positional relation, for example the coincidence of the boundaries of the surrounded and the surrounder. The notion of surrounding is a positional relation.72 All of the above shows, it seems to me, that Aristotle has a notion of position in mind which differs both from having a place and an (abstract) space.

5.3.3. Further evidence on thesis I have argued that the notion of position is a topological or spatial relation between the parts of an object. Though not identical to having a place, it clearly is the case that only spatially extended objects have parts with position. That the idea of a determinate

68

Cf. Ph. IV.1 208b12–25 and my discussion in Section 5.3.4. For a comprehensive interpretation of Aristotle’s account of place in the Physics see Morison 2002. 71 Ph. IV.1 208b22–25. Ph. IV.1 211a29–31. 72 I am, of course, referring to the notion of place as it is expounded in the Physics. It is still debated how the notion of place in the Categories relates to the notion of place in the Physics. Cf. Morison 2002; Mendell 1987, 2005. But whatever the precise relation it seems that the Categories account of place also makes place explanatorily posterior to the concept of position. The reason is that in the Categories place is a species of things that have parts with position. In this sense, explaining what place is relies on an elucidation of what position is. 69 70

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having parts with position vs without position  spatial position is important is implicit in the way in which Aristotle distinguishes between points and units. That which is in no way divisible in quantity is a point or a unit—that which has no position is a unit, that which has a position is a point.73 (Metaph. V.6 1016b29–31)

Both a unit and a point are indivisible. But they are distinct types of entities. A unit is a measure of numbers and it belongs to discrete quantity and lacks a position. The point, on the other hand, belongs to continuous quantity, though it is not itself a quantity.74 A point is, to pick up our characterization, definitely situated on a line; each point has a precise position in relation to all the other points that are on the line.75 The second passage comes from On Generation and Corruption. Nevertheless contact in the strict sense belongs only to things which have position. And position belongs only to those things which also have a place; for in so far as we attribute contact to the mathematical things, we must also attribute place to them, whether they exist in separation or in some other fashion.76 (GC I.6 322b32–323a3)

Aristotle defines the notion of contact in this passage and maintains that the proper sense of contact applies only to things which have a position.77 He then draws the further inference that things that have position have place too. Again, the importance of this passage comes from the explicit connection between having a position and having a place. Having a position is a spatial property which is closely connected to, but still conceptually distinct from, having a place. In On Generation and Corruption Aristotle restricts contact in the proper sense to physical bodies which are the per se occupants of places. In the same vein he restricts having a position to things that are in a place. This seemingly contradicts the account of the Categories as well as the

τ δ μηδαμX διαιρετ ν κατ# τ ποσ ν στιγμ8 κα μον&, > μν θετο μον# > δ θετ  στιγμ+. Because a point has no parts that can be connected nor is a point measurable as quantities. Cf. Metaph. V.13. 75 The example with points and units is problematic for two reasons. First, according to the Categories points are not quantities. In the Categories points are not mentioned as quantities and they do not satisfy either classification since they are neither continuous nor do they have parts with position. Hence, I use the weaker expression that points belong to continuous quantity (since they are the boundary of lines) and units to discrete quantity (since they are the measure by which numbers are counted). Second and related, position is a relation between the parts of an object, but, strictly speaking, points are not parts, but boundaries of a line (see Section 6.3). Thus, it may seem that points cannot have a position at all. However, it should be remembered that in Categories 6 Aristotle glosses over this distinction, or does not explain what criteria are set on parthood. Moreover, if we allow for a slightly weaker notion of part which includes items that belong to the magnitude in question, as limits surely do, the notion of position is applicable to points. This also makes intuitive sense because, if, e.g. we take two points A and B on a line, A could be said to be positioned to the right of B. In this respect, there seems to be no difference to parts. What is important for our purposes is that position marks a distinction between entities with spatial properties and other objects. A point belongs to spatial entities because it has a certain position whereas units do not. 76 .ISμω δ τ κυρ ω εγμενον Fπ&ρχει το (χουσι θσιν, θσι δ’ ο)σπερ κα τπο· κα γ#ρ το μαθηματικο 3μο ω *ποδοτον Mφ8ν κα τπον, ε;τ’ στ κεχωρισμνον Dκαστον ατ=ν ε;τ’ ον τρπον. 77 For a detailed interpretation of GC I.6 see Natali 2004; Williams 1982; Joachim 1926; Buchheim 2010. Buchheim, however, constructs the logic of the sentence quite differently. 73 74

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 aristotle’s theory of bodies Physics, where mathematical entities have a position but no place. I am inclined to explain the discrepancy by distinguishing between a narrow sense of θσι and a broad sense. In On Generation and Corruption Aristotle maintains that the things which have a position in the primary sense are only physical bodies. Mathematical entities have position only in a secondary sense. The reason for this is, I believe, the different ways in which θσι involves the six spatial directions. If spatial directions are understood objectively, then things having a position must have a place, that is, the relation of θσι enables the applicability of the six spatial directions, up, down, left, right, etc. In On Generation and Corruption Aristotle understands these as objectively given. Thesis implies the six spatial directions which in turn imply place, if one understands the directions objectively.

5.3.4. Thesis and the six spatial directions This brings us to the interesting question of the precise way in which position relates to the six spatial directions. In the Categories position is a relation between the parts of an object. Or perhaps more generally, it is a relation between objects. Since it is a spatial relation it seemed appropriate to say that one part lies to the left of another part. Yet, the assignment of left and right is in many cases not objectively determined, as Aristotle points out in the Physics, where he contrasts having a position with having a place by demarcating a difference in the way things relate to the six spatial directions. Now these are parts and kinds of place—up and down and the rest of the six dimensions. Nor do these (up and down and right and left) hold only in relation to us. Relative to us they are not always the same but depend on the position in which we are turned: that is why the same thing is often both right and left, up and down, before and behind. But in nature each is distinct and separate. Above is not anything you like, but where fire and what is light are carried; similarly, too, down is not anything you like but where what has weight and what is made of earth are carried—the implication being that they do not differ merely in position, but also in power. This is made plain also by mathematical objects. Though they have no place, they nevertheless, in respect of their position relatively to us, have a right and left as these are spoken of merely in respect of position, not having by nature these various characteristics.78 (Ph. IV.1 208b12–25)

Aristotle points out that ‘up’ and ‘down’ are objectively determined in the natural world. Up is where fire is or naturally tends to. Mathematical items do not have a place, and yet it is possible to ascribe the six spatial directions to them. One can characterize

78 τατα δ’ στ τπου μρη κα ε;δη, τ τε νω κα τ κ&τω κα αG οιπα τ=ν Qξ διαστ&σεων. (στι δ τ# τοιατα ο μνον πρ  >μU, τ νω κα κ&τω κα δεξι ν κα *ριστερν· >μν μν γ#ρ οκ *ε τ ατ, *# κατ# τ8ν θσιν, -πω 1ν στραφ=μεν, γ γνεται (δι κα τατ πο&κι δεξι ν κα *ριστερ ν κα νω κα κ&τω κα πρσθεν κα 9πισθεν), ν δ τX φ6σει δι/ρισται χωρ Dκαστον. ο γ#ρ - τι (τυχν στι τ νω, *’ -που φρεται τ πρ κα τ κοφον· 3μο ω δ κα τ κ&τω οχ - τι (τυχεν, *’ -που τ# (χοντα β&ρο κα τ# γεηρ&, @ ο τX θσει διαφροντα μνον *# κα τX δυν&μει. δηο δ κα τ# μαθηματικ&· οκ 9ντα γ#ρ ν τπ< -μω κατ# τ8ν θσιν τ8ν πρ  >μU (χει δεξι# κα *ριστερ# @ τ# μνον εγμενα δι# θσιν, οκ (χοντα φ6σει το6των Dκαστον.

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having parts with position vs without position  their position as being to the left or being to the right, which are also the differentia of place.79 It is, however, not an objective position, but rather a position that holds in relation to us. This is why mathematical objects have no place, since place implies absolute positioning. In his explanation Aristotle seemingly relies on the idea that in the case of mathematics position is a relation between us and the objects. They have ‘a position relative to us’, as he says. This is evidently a slightly different sense of θσι than the one in the Categories. Still, I think that these two senses—position as a relation between parts of an object and position as a relation between us and mathematical objects—do not contradict each other. Consider again the two parts of the following (mathematical) line: •

a

•

b

•

The two parts have a positional relation to each other. This relation, the configuration of the line, is not dependent on us. What is, however, dependent on us is which part is said to be on the left and which to the right. It depends on us that a is to the left of b. And this, I believe, is what Aristotle points out in the Physics passage. Of course, it is not up to us that the two parts of a line have a positional relation. It is, for example, not up to us that the parts are side by side. Nor is Aristotle committed to the view that the positional relations between the parts of an object change in relation to us. What Aristotle is committed to is the view that the application of the six spatial directions, left, right, up, down, and so on, is not objectively determined in the case of mathematical objects. We do not change the configuration by first saying that a is on the left and then saying that b is on the left. We merely state that in relation to us the configuration has changed (by having turned the book, for instance). And there is no fact of the matter which of the two ascriptions is correct. Thus it remains true that the notion of position involves the six spatial directions in all cases, be it mathematical or physical objects. But in the case of mathematical objects the way in which the six directions are applied depends on us. The reason why there is a different emphasis in these passages can again be explained by the different projects pursued by the treatises. In the Categories the project is classificatory. The classification of quantities into those whose parts have position and those whose parts do not is a neat division. It is, we argued, what distinguishes spatial entities from others. In the Physics and On Generation and Corruption Aristotle ultimately wants to establish a contrast between spatial directions and places that are by nature and those that are not by nature, but depend on us. Therefore, it is important whether the position applies to the whole object in relation to us or in relation to nature. We want to say that fire moves upwards not in relation to us, but in accordance with nature. Given the different types of project Aristotle pursues in the Categories and in the Physics such differences are only to be expected. 79

Ph. IV.2 208b12–13.

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 aristotle’s theory of bodies However, the underlying thought is the same in all of the texts that I discussed: θσι is a relational spatial property.

5.3.5. A classification of bodies We have now arrived at an interpretation of the two classificatory schemes according to which quantities are categorized (see Fig. 5.5): quantity

continuous line

surface

body

discrete place

time

speech

number

quantity ~ PPos

PPos

line

surface

body

place

time

speech

number

Fig. 5.5.

Bodies are continuous (CONT) and have parts with position (PPOS).80 A body is thus a spatial entity characterized by its topological properties. All its parts are spatially related and have a position with respect to each other; for instance some parts are above, others below, and so on. Moreover, all of the adjacent parts of a body are connected with each other. Body as presented in the category of quantity is a connected, three-dimensional spatial entity whose parts are definitely situated.81 We can put it as follows: Proposition 5. x is a body in the category of quantity (‘quantitative body’) = def [1] x is extended in three dimensions and [2] all adjacent parts of x are connected by a boundary and [3] the parts of x have position.

This is a formal description of how body is presented in the Categories. The content of Proposition 5 will guide us through pretty much the whole remaining treatise. This guidance is to be understood as a framework. For example, in the next chapter I will review Aristotle’s argument that bodies are prior to other magnitudes in virtue of having three dimensions (Section 6.2). Following this, I will analyse in what way 80 81

For the exact formulation see Proposition 2 in Section 5.2.3 and Proposition 4 in Section 5.3. This is not explicitly stated in Categories 6. It is made explicit in Metaphysics V.13. See Appendix A.

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having parts with position vs without position  quantitative bodies can be seen as composites of form and matter. The form of the body will be identified with its limit (Section 6.3) and the matter with its extension (Section 6.4). In Chapter 7, I will discuss the crucial distinction between continuity and contact and the implication this has for the ontology of bodies. In this sense, we will investigate the metaphysical underpinnings as well as many other questions that are left open, such as for example the relation of bodies to their limits. Nonetheless, the Categories can be seen as providing the background for the later discussion as well as being a guide for deciding what questions need to be addressed.

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6 A Topological Conception of Bodies 6.1. Introduction Categories 6 yields a systematic theory and classification of body.1 Aristotle introduces fundamental properties such as continuity, extension, position. These concepts are the building blocks for a theory of extended objects. Thus, I believe that a study of this text is fundamental for any study of quantities and extended objects. But the theory presented there is, as I said, not complete. It provides important background for understanding Aristotle’s theory of extended objects, but it does not answer all of the questions. It is never the case, of course, that all of the questions are answered. So let me be more specific on the argument of this chapter: first, Aristotle believes that threedimensional objects are complete and perfect in virtue of having three dimensions. The Categories allow for a classification of continuous quantities and body, but there is no indication that bodies are privileged. Aristotle claims, however, that they are. I will discuss Aristotle’s claim that bodies are complete in Section 6.2. Second, Aristotle suggests in the Categories that lower-dimensional entities are boundaries of higherdimensional objects. The boundaries at which the parts of a body are connected are planes. But what are boundaries? How should we conceptualize them? Is there a distinction between the outer limits and the internal limits of an entity? I will address the topic of limits in Section 6.3. Third and closely connected to the previous point, a body can be analysed into its limit and its extension, but what is the latter? What is the ontological status of extension? This is the task of Section 6.4.

6.2. Bodies are Complete In Part I of this study I claimed that the study of bodies is a part of the conceptual underpinnings of physical science. It investigates what belongs to them insofar as they are bodies of physical substances. In this section we will consider a concrete example. Aristotle argues that bodies are complete and perfect in virtue of being 1 The same is true of Metaphysics V.13, which is another text that is rarely studied. I discuss this text in Appendix A.

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bodies are complete  three-dimensional. If an item is three-dimensional, it is implied that it is prior to an item that is two-dimensional. Additionally, Aristotle claims that because bodies are complete, there cannot be a four-dimensional magnitude. How can this be explained? I think that the best explanation is that certain topological properties are linked to and determined by the nature of the object in question. Even though the content of this claim is applicable to any three-dimensionally extended object, the justification of the claim is based on considerations regarding the nature of bodies. The claim about topological and quantitative features is based on the ontology and nature of physical bodies. This, I propose, is argued for by Aristotle in De Caelo I.1.2

6.2.1. De Caelo on the completeness of bodies Bodies are exceptional among extended objects. Bodies are complete and perfect (τειον)3 because they have all possible dimensions. Accordingly, lines and surfaces are incomplete. They do not have all possible dimensions and are posterior to bodies. Aristotle defends this claim in the first chapter of De Caelo. He begins the chapter with a specification of the proper subject matter of physical science. Aristotle claims that magnitudes and especially bodies are the primary objects of physical science.4 They are the primary objects because these bodies and magnitudes are constituted by nature.5 This preamble, I suggest, puts the following discussion in the context of physical science. Aristotle then goes on to say: Of magnitude, that which is extended in one dimension is a line, that which is extended in two is a surface and that which is extended in three dimensions is a body. There is no other magnitude beyond these, since the three (dimensions) are all and the thrice is in every way.6 (Cael. I.1 268a7–10)

In the first sentence Aristotle mentions the previously discussed definition of magnitudes according to their dimensions. But in the next sentence, Aristotle makes a stronger claim. He claims that there cannot be other magnitudes besides these. There can be no other magnitude because body, by being extended in three dimensions, is extended in all dimensions. Aristotle grounds his claim on the shaky evidence that being three implies being an ‘all’.7 As it stands, the argument seems deeply flawed 2 In my discussion I rely on Betegh et al. 2013. The paper presents a full analysis of the difficult and puzzling first chapter of De Caelo. Since I cannot do justice here to the complexities of the chapter, especially the striking allusions to the Pythagoreans, I refer the interested reader to this paper. 3 ‘Complete’ or ‘perfect’ are my translations of τειον. Cf. Section 6.2.1.2 in this chapter for a discussion of the precise meaning of the term. 4 Cael. I.1 268a1–4. 5 Cael. I.1 268a4–6. 6 εγθου δ τ μν φ’ ν γραμμ, τ δ’ π δο ππεδον, τ δ’ π τρα σμα· κα παρ τατα ο κ !στιν "ο μγεθο δι τ τ τρα π#ντα ε$ναι κα τ τρ π#ντ%. 7 ‘For, as the Pythagoreans say, the world and all that is in it is determined by the number three, since beginning and middle and end give the number of an “all”, and the number they give is the triad. And so, having taken these three from nature as (so to speak) laws of it, we make further use of the number three

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 aristotle’s theory of bodies because, if we grant that being three implies being an all and being complete, it is hard to explain why there are more than three people in the world. I do not see how Aristotle’s argument could be saved from these objections. But perhaps it can be made more understandable by noting that his argument stands in a dialectical context and he seems to rely on premisses that might be shared by him and his opponents. Be that as it may, Aristotle’s discussion of the relation between the nature of physical substances and three-dimensionality is fine-grained and well worth studying, even if the connection between being three and being all proves to be unsatisfactory. It is worth studying because Aristotle employs a model of how three-dimensionality, though not being part of the essence of physical substances, is nonetheless caused by and due to the essence of physical substances. Before entering the discussion I must emphasize that it is crucial to bear in mind that the whole discussion is set in the framework of physical science. Aristotle even calls the bodies he discusses ‘substances’ (Cael. I.1 268a3). Thus it should be clear that Aristotle’s arguments are not independent of physical considerations. The argument is underpinned by considerations about the nature of the bodies in question. It is not a mathematical argument. It is not an argument about three-dimensionally extended objects in general. Rather it is an argument about physical bodies. That is to say, it is an argument that concerns the question why physical substances are necessarily threedimensional. Part of the answer is that by being three-dimensionally extended they have a body that is complete. But the completeness of body in the sense of a threedimensionally extended quantity can only be understood in the context of that body being the body of a physical substance. Having said that, let us turn to Aristotle’s two major claims in the chapter: first, body is exceptional among the existing magnitudes. Second, there can be no other continuous magnitude besides lines, surfaces, and bodies. .... aristotle’s arguments for the priority of bodies The first claim, the priority of bodies over lower-dimensional magnitudes, is vindicated thus: Therefore, since ‘every’ and ‘all’ and ‘complete’ do not differ from one another in respect of form, but only, if at all, in their matter and in that of which they are said, body alone among magnitudes can be complete. For it alone is determined by the three (dimensions), that is, is an ‘all’.8 (Cael. I.1 268a20–24)

in the worship of the Gods. Further, we use the terms in practice in this way. Of two things, or men, we say “both”, but not “all”: three is the first number to which the term “all” has been appropriated. And in this, as we have said, we do but follow the lead which nature gives’ (Cael. I.1 268a10–20). For an interpretation see again Betegh et al. (2013). 8 &'(στ’ πε τ π#ντα κα τ π)ν κα τ τειον ο κατ τ*ν +δαν διαφρουσιν ,ων, ,’ ε.περ, ν τ/ 0% κα φ’ 1ν γονται, τ σμα μ2νον 3ν ε.η τν μεγεθν τειον· μ2νον γρ 5ρισται το6 τρισν, τοτο δ’ στ π)ν.

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bodies are complete  Aristotle’s thought is hard to grasp, but I suggest taking the passage in the following way:9 Aristotle takes himself to have shown that being extended in three dimensions is being extended in all dimensions because the number three implies being an all. Now he introduces the further thought that being an ‘all’ also implies a certain kind of completeness and perfection. For, Aristotle argues, the terms ‘every’, ‘all’, and ‘complete/perfect’ do not differ in form. Thus, since body is an all, it must be complete and perfect, too. To justify this inference it is, I think, plausible to assume that ‘having the same form’ is here used as an equivalent of ‘synonymous’.10 All these terms have the same meaning, and therefore it is permitted to predicate ‘completeness’ of an item, if it is permitted to predicate ‘all’ of the same item.11 Therefore, we can conclude that body is complete, whereas lines and surfaces are incomplete. The latter cannot be complete because they are not extended in all dimensions, but only in some dimensions. Having shown that body is unique among the existing magnitudes, Aristotle goes on to argue that that the transition to another genus of magnitude (in this case the fourth dimension) is impossible. This can be seen as the flip side of the coin in Aristotle’s argumentation. The completeness and perfection of bodies not only accounts for the priority of bodies over lines and surfaces, but also for the fact that bodies are posterior to nothing. One thing, however, is clear. There is no transition to another kind of magnitude, as we passed from length to surface, and from surface to body. For if we could, it would cease to be true that body is complete magnitude.12 We could pass beyond it only in virtue of a defect in it; and that which is complete cannot be defective, since it is in all ways.13 (Cael. I.1 268a30–b5) 9

See Betegh et al. for the details. This is also suggested by the following two parallel passages. The first is Cael. I.8 276a32–b4: ‘Moreover each of the bodies, fire, I mean, and earth and their intermediates, must have the same power as in our world. For if those elements are named homonymously and not in virtue of having the same form (μ* κατ τ*ν α τ*ν +δαν) as ours, then the whole to which they belong can only be called a world homonymously.’ The second is EN V.1 1129a27–b1: ‘Now “justice” and “injustice” seem to be ambiguous, but because the homonymy is close, it escapes notice and is not obvious as it is, comparatively, when the meanings are far apart. For here the difference in form is great (7 γρ διαφορ πο* 7 κατ τ*ν +δαν). E.g. as the homonymy in the use of kleis for the collar-bone of an animal and for that with which we lock a door.’ See also Wildberg 1988, 22, who quotes Cael. 11 What does difference in matter amount to? I suggest assuming that for two terms to differ in their matter is for those terms to be predicated of different items. Alexander (according to Simplicius, in Cael. 9.5–8) had a similar interpretation. He maintains that ‘every’, ‘all’, and ‘complete’ are the same in form but not with reference to their objects (8ποκεμενα) because ‘every’ is predicated of a determinate quality, ‘all’ of continuity, and both of the ‘complete’. Thus, ‘all’ is predicated of masses, like water, whereas ‘every’ of countable items like horses. In this sense, the meaning of ‘all’ in the sentence ‘She poured all the water out’ and the meaning of ‘every’ in the sentence ‘Every person in the room drank a martini’ is the same. 12 This is not the only possible reading of the sentence. Leggatt translates: ‘For magnitude of such a kind would no longer be complete’ (Leggatt 1995, 49). Thus, he takes τοιοτον to refer to the hypothetical fourdimensional entity. But τοιοτον has the same referent as τειον in line b9 and the latter refers back to ‘body’. The sentence does not just mean that something complete cannot be deficient, but that body being complete cannot be deficient. Aristotle’s main objective is to argue that bodies are complete, and not that a four-dimensional object would not be complete. If we follow Leggatt’s reading, this essential connection is lost, because if there were a fourth dimension, a four-dimensional object would be complete (assuming that filling every possible dimension is a mark of completeness). 13 9:’ κε6νο μν δ;ον, < ο κ !στιν ε+ "ο γνο μετ#βασι , 5σπερ κ μκου ε+ πιφ#νειαν, ε+ δ σμα ξ πιφανεα · ο γρ 3ν !τι τ τοιοτον τειον ε.η μγεθο · ,ν#γκη γρ γγνεσθαι τ*ν !κβασιν κατ τ*ν !ειψιν, ο χ οA2ν τε δ τ τειον επειν· π#ντ% γ#ρ στιν. 10

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 aristotle’s theory of bodies A transition is possible, if and only if the entity in question is deficient. Bodies, however, are complete and, hence, cannot be deficient. Something complete cannot be surpassed, for the possibility of being surpassed depends on a deficiency of the object in question. Since body is complete, it cannot be surpassed. But if there were another magnitude beyond body, body could be surpassed. Since we have seen that this is not the case, we can conclude that there is no other magnitude. Aristotle thus holds that bodies are complete, whereas other magnitudes are not. And he believes that because bodies are complete there cannot possibly be another magnitude beyond body. Bodies are unique and singular among magnitudes. They are complete. This, however, raises two questions. First, what precisely does the claim that bodies are complete or perfect (τειον) mean? Second and more importantly, how does the notion of dimensional completeness relate to the notion of priority in nature? .... what does teleion mean? Aristotle says that bodies are τειον. I have translated the word as ‘complete’ or ‘perfect’. Aristotle’s claim is that bodies, insofar as they are extended in three dimensions, are complete and perfect. The completeness of bodies is due to and grounded in their being three-dimensional. Due to this connection it is natural to assume that being teleion means in its first and foremost sense that body is complete because it fills all the dimensions that exist. To be teleion means being dimensionally complete. However, I believe that there is more to it. By saying that body is teleion Aristotle means more than just stating the dimensional completeness of body. Aristotle is making a normative claim. This normativity is better captured by the word ‘perfect’. Body is prior to the other magnitudes. This priority is, I believe, suggested by the semantics of the word ‘teleion’. For Aristotle distinguishes several senses in Metaph. V.16: Something is teleion if it includes all its parts (that is, is something complete) and if it cannot be surpassed with respect to the excellence proper to its kind (that is, is something perfect).14 Both senses are relevant in De Caelo I.1. Insofar as body is extended in all the dimensions in which a magnitude can be extended, body is a complete magnitude. Insofar as no further magnitude can surpass body, body is the perfect magnitude according to the second meaning of teleion.15 There is also a third sense of teleion which is connected to final causation. Insofar as something has reached a (good) end, it is teleion.16 Wildberg, by contrast, thinks that the attribution of a normative meaning to teleion in this context is ‘philosophically absurd’ because it is ‘simply false to say that a body qua body is perfect’ and ‘Aristotle never wanted to claim this’.17 Wildberg does not specify why it should be philosophically absurd to assume this view. I cannot see the absurdity involved. For perfection means that body cannot be surpassed in the 14 Cf. Metaph. V.16 1021b12–22a3. See also Cael. II.4 286b18–19, Ph. II.6 207a8–14, Metaph. X.4 1055a10–16. Cf. the discussion in Betegh et al. 2013, 44ff. 15 16 17 See also Falcon 2005, 35. Metaph. V.16 1021b23–24. Wildberg 1988, 22.

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bodies are complete  relevant sense. This goes beyond the statement that as a matter of fact there are only three dimensions. It is impossible that there should be something that is extended in more than three dimensions. Bodies are perfect magnitudes because they cannot be surpassed. And this is why Aristotle later says that lines and surfaces are deficient. Lines and surfaces can be surpassed. Thus, they are posterior to bodies. Proposition 6. Bodies are complete and perfect in the following way: Body is the only magnitude that is dimensionally complete because it is extended in all possible dimensions. In virtue of being extended in all possible dimensions body is prior to lower-dimensional magnitudes and it is not possible that there should be another magnitude that surpasses bodies.

In the following section I will elaborate on the claim of priority. Body is prior to lowerdimensional magnitudes because it is extended in more dimensions than they are. This, I have argued, is a claim about the perfection of bodies. But there is an objection to this account. Speaking of perfection implies a certain normativity. But the modal claim that body is complete could be understood in a weaker sense. It could simply mean that body is three-dimensional and therefore is extended in more dimensions than, for example, a surface, and additionally that it is impossible that there is another magnitude with more than three dimensions. It may seem off the mark to speak of normativity here. If we understand ‘normativity’ in the sense of (morally) good or bad, this criticism is indeed justified. But this is not what perfection expresses. Perfection, as I understand it, is due to the systematic connection between dimensionality and substantiality. Extended substances are three-dimensional due to their substance and nature. At this point the third sense of teleion distinguished above becomes important. The normativity of three-dimensional extension is due to the fact that bodies can be seen as the endpoint or goal of a teleological quasi-process. This is the claim of the next section.

6.2.2. Substantiality and the dimensions The connection between substantiality and the dimensions is not at all obvious. It is a topic that is rarely discussed either by Aristotle or by his commentators. Accordingly, it is difficult to reconstruct Aristotle’s view on these matters. I offer here what I think is a possible and plausible interpretation. Other interpretations might be possible, too, and my interpretation is tentative. Having said that, the line of my argument runs thus: I will first argue that Aristotle does not conceive of the extension as a substance. This conclusion, I propose, can be drawn from the famous ‘stripping’ argument in Metaphysics VII.3. Being three-dimensional is not part of the definition of physical substances. Even though all physical substances are bodies, being a body, that is, being three-dimensionally extended, is not part of their essence.18 18 As I said in Chapter 2, I disagree with Studtmann 2002 who thinks that body is a genus in the category of substance and a genus in the category of quantity. Because it is not part of the essence of a physical substance to be extended in three dimensions (though it is extended in three dimensions out of necessity),

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 aristotle’s theory of bodies However, even if extension in three dimensions is not the essence of a physical substance, it is connected to its essence. I will attempt to spell this out in terms of final causation. The three-dimensional extension of physical substances is due to and flows from their nature. .... quantity is not a substance In a first step I wish to argue that being extended is not part of the essence of a physical substance. If so, it cannot be a substantial feature or substance of a substance. In the course of the famous ‘stripping’ argument in Metaphysics VII.3 Aristotle remarks the following:19 If it (i.e. matter) is not a substance, it is hard to see what else could be; for when all else is taken off, nothing apparent remains. For while other things are attributes, products, and capacities of bodies, length, breadth, and depth are quantities and not substances (for a quantity is not a substance). Rather, the substance is that primary thing to which these quantities belong. And yet when length, breadth, and depth are taken away, we see nothing left behind unless there be something which is determined by these, so on this view it must appear that matter alone is substance.20 (Metaph. VII.3 1029a10–19)21

This passage occurs in a notoriously difficult and opaque discussion of substance.22 One criterion for being a substance is, Aristotle says, not to be predicated of a subject, while other things are predicated of it.23 In the course of this discussion Aristotle presents the ‘stripping’ argument. I am not going to join the debate on what ‘matter’ means here, or on whether the whole of the argument represents Aristotle’s opinion, and whether this passage either affirms or denies prime matter (but see fn. 26 in this chapter). For our purposes the important question is how quantities and magnitudes occur in the course of the argument. This is to say, the focus lies on lines 12–16 where Aristotle apparently allows quantities to have a peculiar status. For as it stands, the argument introduces an asymmetry between quantities and other properties. Roughly, the asymmetry is that other properties, like colours, are there is, according to my interpretation, no ‘body problem’ in Aristotle. For the body problem to arise it is not sufficient that physical substances are bodies, even necessarily so. They must have part of their essence in common with body in the category of quantity. But, as Studtmann 2002, 215 himself notes, Aristotle never makes this claim. And I will argue that Aristotle has good reasons for not making this claim. 19 I want to thank Alan Code for pointing out the importance of this passage for the present context and discussing it with me. 20 ε+ γρ μ* α0τη ο σα, τ στιν "η διαφεγει· περιαιρουμνων γρ τν "ων ο φανεται ο δν 8πομνον· τ μν γρ "α τν σωμ#των π#θη κα ποιματα κα δυν#μει , τ δ μ;κο κα π#το κα β#θο ποσ2τητ τινε ,’ ο κ ο σαι (τ γρ ποσν ο κ ο σα), , μ)ον B 8π#ρχει τατα πρCτD, κε6ν2 στιν ο σα. , μ*ν ,φαιρουμνου μκου κα π#του κα β#θου ο δν Eρμεν 8ποειπ2μενον, π*ν ε. τ στι τ Eριζ2μενον 8π τοτων, 5στε τ*ν 0ην ,ν#γκη φανεσθαι μ2νην ο σαν ο0τω σκοπουμνοι . 21 Translation is by Morison 2002, 111. The translation is a modification of Bostock 1994. 22 The literature on this passage is vast. A first orientation can be provided by the commentaries of Bostock 1994; Frede and Patzig 1988; Detel 2009. For two recent interpretations see Green 2014; Lewis 2013. 23 Metaph. VII.3 1029a7–8.

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bodies are complete  predicated of quantities, but not vice versa. The subject of colour are surfaces, but the subject of surfaces are not colours. In this sense, magnitudes underlie other properties.24 Because of this asymmetry one might believe—and some philosophers, presumably Platonists, have in fact believed25 —that quantities, like mathematical bodies, surfaces, lines, or points, are substances. One might believe this because if substance is characterized by an is-said-of relation it seems that all other properties are said of magnitudes, but magnitudes are not said of them. Thus, in stripping properties from an object one arrives at some point at the idea of pure quantities. That is to say, there seems to be a stage where we get to objects whose only essential properties are quantitative ones. However, Aristotle is explicit in stating that magnitudes are not substances. For he immediately adds that the length, breadth, or depth of a thing is not its substance. Even if in the course of this ‘stripping’ process we reach magnitudes after all other properties are taken away, the magnitudes themselves must belong to something. And, as Aristotle continues, matter—and if matter qualifies as substance, the substance—is that to which length, breadth, and depth belong. If we take this idea seriously, we see that three-dimensional extension cannot be, wholly or partly, the essence of a substance.26 24 Cf. Metaph. III.5 1001b32–1002a4; Metaph. V.18 1022a16–17. The same point is made by Morison 2002, 110. 25 Cf. Metaph. III.5 and XIII.2. 26 The passage, of course, is less famous for what it implies about the metaphysical status of quantities, than for what it implies for the metaphysical status of substance and matter. It is, alongside GC II.1, the main passage that is discussed under the heading ‘Did Aristotle believe in prime matter?’ Though Aristotle’s theory of substance or matter is not our topic, there is a certain complication we must address in this footnote. For I have just argued that Aristotle believes that extension (in whatever dimension) is not a substance. However, to say that the extension of a body is not its substance contradicts the view of those who believe that extension is prime matter. The controversy revolves around the immediately following sentence: ‘And yet when length, breadth, and depth are taken away, we see nothing left behind unless there be something which is determined by these, so on this view it must appear that matter alone is substance’ (Metaph. VII.3 1029a16–19). The sentence can be understood in several ways. First, it might mean that a determinate length, breadth, or depth is taken away. According to this interpretation, if I take away the length of a line, I take away the specific length of the line. If the line is one metre long, I take away the length of one metre. What I am left with is, however, still a one-dimensional extension, but an indefinite one. It is extended, but not in a determinate length. This interpretation, which ultimately goes back to Simplicius, is put forward, e.g. by Sorabji 1985. Second, it might mean that by taking away length, breadth, and depth extension in one, two, or three dimensions is taken away. What is left, according to the second interpretation, is something that is not extended in any dimensions. This interpretation goes back to Philoponus, but has been developed in different directions. Schofield 1972, e.g. believes that literally nothing remains. The problem with this is, as Bostock 1994, 77 remarks, that it becomes a mystery why Aristotle should think that something, i.e. matter, remains. Third, one could also suppose that what remains is ordinary matter, like, for example, bronze. It is the bronze that is bounded by length, breadth, and depth. Cf. Morison 2002, 111. In the present context only the first of these alternatives is important, since it is the only interpretation that identifies extension with matter. I think that this interpretation is not correct. I cannot go into a detailed criticism, but simply note some points that speak against Sorabji’s interpretation. First, in Aristotle’s regular use, the words ‘length, breadth, and depth’ designate one-, two-, or three-dimensional extension in general and not a specific size of an object. Cf. Metaph. V.13 1020a11–12. Thus, to take away length, breadth, and depth is to take the dimensions of an object and not only its specific size. Second, it is quite plausible that the

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 aristotle’s theory of bodies It is a feature of physical substances, but it is not part of their essence. Humans, dogs, or cats do not have three-dimensional extension as part of their essence. Essentially they are something else, namely living beings. On the other hand, physical substances are not extended in three dimensions by chance. It is predicated of them qua being physical substances:27 Proposition 7. x is a physical substance if and only if x is extended in three dimensions.28 Proposition 8. If x is a physical substance, being extended in three dimensions is not part of the essence of x.

These two propositions give us a necessary and sufficient connection between being a physical substance and being extended in three dimensions, but they deny that it is part of the essence of physical substances to be extended in three dimensions. However, it seems to me a plausible assumption that it is not entirely detached from their essence either. It is not a brute fact that these two are found in constant conjunction. There is a connection between the essence of a physical substance and its being a body. To use a metaphor, it flows from its essence that a physical substance

passage is directed against a Platonic view. This would explain the obvious parallel to Metaph. III.5. In this case the ultimate aim of the passage would be a reductio of a rival theory, not a presentation of Aristotle’s own theory. Third, the argument is entirely based on the logic of predication, as Aristotle himself emphasizes. The account of predication that Aristotle uses comes from the Organon. There being a subject is identified with being a substance by way of the is-said-of relation. But one should not suppose that one can, as it were, read off the metaphysical status of objects by focusing solely on that criterion. From the standpoint of predication, quantities underlie other properties. But it is not an ontological point because a physical substance is not made out of a quantity or extension. Thus, insofar as the concept of matter is connected to that out of which something is made of (Metaph. IX.7), this passage has nothing to contribute. Matter is defined by its potential to be the substance, not by its being a quantity of stuff (Metaph. VIII.1 1042a27–28; GC I.3 317b16–18). Even if Sorabji were correct and indefinite extension remained, I can see no reason to assume that indefinite extension plays a role analogous to the role of bronze in a brazen statue. On this account I also disagree with Studtmann 2006 who argues that extension ‘exists in material composites as the matter for substantial forms’ (Studtmann 2006, 182). Extension is not literally the matter. Extension is a feature of the matter. Studtmann thinks that in studying physical or mobile bodies one studies the matter of material substances; I believe that one studies body (a quantity) insofar as it belongs to physical substances. For further discussion of my view on extension see Section 6.4. 27

For an interpretation of the qua-locution see Section 4.1.1.1 and the literature mentioned there. One may object that this proposition is false because mathematical bodies are extended in three dimensions, but are not physical substances. But one should note that this proposition tells us nothing about the respective definitions, but makes an existence claim. And for Aristotle it is true, I assume, that there are no three-dimensional extended objects which exist independently of physical substances (with the possible exception of the whole cosmos whose status as a substance is unclear. But it is clearly a physical object. See Matthen and Hankinson 1993 for discussion). Yet, since the focus of this section is on Proposition 8 as well as on the question of how three-dimensionality flows from the essence of physical substances, we may also use the weaker proposition 28

Proposition 7*. x is a physical substance only if x is extended in three dimensions. Those who are in serious doubt about Proposition 7 can read the starred version instead.

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bodies are complete  is extended in three dimensions.29 Aristotle, I propose, explains this connection between three-dimensionality and the essence by a comparison to a quasi-natural process. .... three-dimensionality and priority in substance Though three-dimensionality is not the essence of a physical substance, it is, I will argue, tied to the essence of it. It is not a constituent, but a concomitant of the essence of physical substances. Being three-dimensional follows from the essence of physical substances. Aristotle shows this by relying on the principle that what is prior in generation is later in substance.30 Bodies represent the endpoint of a quasi-natural process. That is to say, there is a quasi-generation from lines and surfaces to bodies, which, being the end point of such a process, become alive. Three-dimensionality is prior in nature to lower-dimensional entities because it can be seen as the result of a quasi-natural process of generation and is tied to the concept of a living substance, or, more generally, to the concept of the nature of a thing. Again, the generations show that we are right. First it comes to be in length, then breadth, lastly depth, and it is complete. If, then, that which is posterior in generation is prior in substance, the body should be prior to plane and length. It is more complete and whole in the following way also—it becomes animate. How, on the other hand, could there be an animate line or a plane? The supposition passes the power of our senses.31 (Metaph. XIII.2 1077a24–31)32

The context of the passage is an argument against the claim that lines, planes, and surfaces are prior to perceptible substances. Aristotle considers two considerations against this claim. First, it is a general truth that what is prior in generation is later in substance. Since lines and surfaces are prior to bodies in generation, they are, according to this principle, posterior in substance. Second, a body becomes animate. All living things are three-dimensional, but it is impossible to see how there could be a living surface. This argument has puzzled commentators. And it is indeed puzzling that Aristotle should assert that there is a generation of bodies from lines. Julia Annas, for example,

29 In regard to this metaphor, it is interesting to note that some scholastic authors took such metaphors quite seriously and argued that substances bring about their attributes in an efficient causal way. Suárez, e.g. argues that ‘accidental properties, especially those that follow upon or are owed [to a substance] by reason of its form, are caused by the substance not only as a material cause or final cause, but also as an efficient cause through a natural resulting ([nimirum] proprietates accidentales, praesertim illas quae consequuntur aut debentur rei ratione formae, causari a substantia non solum materialiter et finaliter, sed etiam effective per naturalem resultantiam)’ (DM 18.3.4 [25, 616a]). The thought I will ascribe to Aristotle is, however, closer to final causation. 30 Cf. Metaph. IX.8 1050a4–10, Ph. VIII.7 261a13–21, GA II.6 742a18–22. 31 !τι αG γενσει δηοσιν. πρτον μν γρ π μ;κο γγνεται, ε$τα π π#το , τεευτα6ον δ’ ε+ β#θο , κα το !σχεν. ε+ οHν τ τ/ γενσει 0στερον τ/ ο σI πρ2τερον, τ σμα πρ2τερον 3ν ε.η πιπδου κα μκου · κα τατ% κα τειον κα Jον μ)ον, Jτι !μψυχον γγνεται· γραμμ* δ !μψυχο K ππεδον π 3ν ε.η; 8πρ γρ τ α+σθσει τ 7μετρα 3ν ε.η τ ,ξωμα. 32 Translation adapted from Annas 1976, 94.

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 aristotle’s theory of bodies thinks that there is a confusion on Aristotle’s part. Aristotle, she believes, fails to distinguish between mathematical and physical objects.33 Additionally, one might object that the term ‘generation’ is used in a homonymous sense. For though it might be true that in the generation of physical bodies what is prior in being comes later in generation, this isn’t the case with mathematical entities, where ‘generation’ must mean ‘construction’. There is no generation, no process that leads from surfaces to three-dimensional bodies. The man is prior to the boy in substance, but the boy is prior in generation to the man, as Aristotle explains in Metaph. IX.8 1050a5. And we may agree with Aristotle that this is the case. But whether or not we agree, our agreement would be conditional on the fact that there is a natural process leading from the boy to the man. Since there is no natural process from lines to bodies, why should we believe that bodies are prior in nature? Aristotle’s argument, we may say, is simply irrelevant. This ambiguity [between the two senses of generation] deprives the argument of whatever value it might otherwise have possessed. (Ross 1924a, 414)

And Ross seems right. It seems all too easy to charge Aristotle with being confused and to treat the passage as an isolated and in the final analysis unintelligible piece of writing. I think that these objections are not entirely justified. First, the claim that there is a transition from lower-dimensional to higher-dimensional magnitudes is not restricted to Metaphysics XIII.2, but appears in the De Caelo as well. Second, the context of both passages is dialectical.34 That is to say, the key element of transition between extended objects is a common presupposition shared by Aristotle’s opponents. Beginning with the first remark, the view that there is a transition is also endorsed in the previously quoted passage from the first chapter of De Caelo.35 One thing, however, is clear. There is no transition to another kind of magnitude, as we passed from length to surface, and from surface to body. For if we could, it would cease to be true that body is complete magnitude. We could pass beyond it only in virtue of a defect in it; and that which is complete cannot be defective, since it is in all ways. (Cael. I.1 268a30–b5)

When one closely reads the passage, it becomes obvious that Aristotle not only denies that there is a transition from body to another genus, but he apparently assumes that there is a transition from line to surface and surface to body. This obviously connects this passage to the passage in Metaphysics XIII.2. Moreover, the choice of language betrays that the transition Aristotle has in mind is not a logical construction, but rather a quasi-natural generation. The word ekbasis occurs only here in the corpus aristotelicum and metabasis usually describes the elemental transformations.36 33 36

34 Cf. Annas 1976, 146. Cf. Cleary 1995; Betegh et al. 2013. Cf. Cael. III.1 298b1, III.7 306a32. See also Bonitz 1870 and the TLG.

35

See p. 79.

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bodies are complete  Hence, in their usual meaning these words describe a physical transition. This makes it unlikely that Aristotle wants to restrict the processes referred to here to a mathematical transition.37 In this sense, the passage from Metaphysics XIII.2 cannot be treated as an isolated passage. This is especially apparent when we consider that the conclusion that there is no further transition is central to Aristotle’s own theory. Aristotle endorses the conclusion. This, however, does not mean that the conclusion cannot be situated in a dialectical context. Aristotle endorses the conclusion, but his premisses may partly come from a dialectical context. What could this context be? What could Aristotle have in mind here? One possible candidate is a passage in Plato’s Laws: What happens when the generation of all things occur? Clearly, an archê takes up growth, and reaches a second stage and then the next one out of this second, so that as soon as it reaches the third, there is something for percipient things to perceive.38 (Pl. Lg. X 894a1–5)

This passage is crucial for my interpretation in several respects.39 First, metabasis refers here as well as in De Caelo I.1 and Metaphysics XIII.2 to a transition from n dimension to n+1 dimension. Second, the phrase ‘there is something for percipient things to perceive’ unambiguously shows that it is a generation of physical, perceptible bodies. Finally, the parallel with Laws X also shows that Aristotle is referring here to a doctrine which he may or may not endorse, but which certainly does not originate with him. Rather it is a presupposition Aristotle’s opponents subscribe to and, therefore, Aristotle is justified in drawing on this presupposition in the context of an argument against his opponents. The dialectical context, I propose, can be described as follows: anyone who believes that there is a transition between magnitudes that finally leads to physical substances must at least agree that the transition stops at the third dimension. If so, one must further agree that bodies are complete and perfect. The process of a generation is completed when three-dimensional magnitudes are reached. Since, and this is Aristotle’s own premiss, what is prior in generation is later in substance, bodies are prior in substance to two-dimensional objects. However, once we have reached this conclusion, we could, in a modification of Wittgenstein’s famous saying, throw away the ladder, after we have climbed up on it.40 That is to say, we can draw the conclusion even if we deny that there literally is a transition from n to n+1 dimensions. We can establish the conclusion that bodies are prior in substance to lower-dimensional magnitudes by relying on the

37

This conclusion is reinforced by the fact that the context is physical science. γγνεται δ* π#ντων γνεσι , 7νκ’ 3ν τ π#θο L; δ;ον < Eπ2ταν ,ρχ* αβοσα αMξην ε+ τ*ν δευτραν !θ% μετ#βασιν κα ,π τατη ε+ τ*ν πησον, κα μχρι τριν θοσα α.σθησιν σχ/ το6 α+σθανομνοι . 39 On this passage see also Betegh et al. 2013. 40 The original says: ‘He must so to speak throw away the ladder, after he has climbed up on it’ (Wittgenstein 1961, 6.54). We, on the other hand, need not throw away the ladder, although we are free to do so. 38

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 aristotle’s theory of bodies model of natural generation. For Aristotle this model may not be literally true, since there is no transition between the dimensions. But even if not literally true, this model provides an insight into the relations of priority between dimensional objects. In other words, Aristotle can claim that in the case of magnitudes an analogue to the priority in substance and generation is true. The priority of body over lowerdimensional magnitudes can be made transparent, if we compare it to the case of a natural generation. This comparison is especially apt since Aristotle’s opponents in fact believe that there is a natural generation. This interpretation faces the following objection.41 If we detach the dialectical context, the conclusion does not follow. If there is no generation, then we cannot apply the rule that what is posterior in generation is prior in substance. You cannot climb a false ladder, so to speak. This objection has a point. Surely, one cannot establish a conclusion by means of a false premiss. But this is not the way I would construe the argument. As I said, Aristotle uses the priority in nature as an analogue. This analogue is especially fitting since some philosophers, presumably Platonists, in fact believe in generation of planes from lines. The analogue works insofar as it gives a suitable reason to assume that three-dimensional objects are perfect. This reason is, of course, defeasible and Aristotle has not given sufficient support for proving the claim. But it makes the claim more credible. It shows that his opponents are committed to this conclusion by their own standards. It also provides, through the analogue of priority in substance, a way to understand what it means to claim that bodies are perfect. .... substance, causes, and dimensionality My claim is that we should interpret Aristotle’s remarks in the following way: Though Aristotle did not believe that there is literally a transition from n-1 to n-dimensions, he did believe that bodies are complete and perfect. He engages in a dialectical argument that presupposes that there is a transition from n-1 to n-dimension. If one accepts that there is a transition, one has to admit that there cannot be more than three dimensions. But even if one does not believe that there is a transition, one can establish a link between the essence of a physical substance and its being three-dimensional. Aristotle alludes to this link when he says that only bodies become animate. All extended living substances are three-dimensional bodies.42 This, however, is not a mere coincidence. Substances are three-dimensional in virtue of their essence. This is, I suggest, the crucial connection between the idea of a natural process of generation depicted in Metaphysics XIII.2 and De Caelo I.1 and our question how dimensionality and essence connect. Bodies are perfect and complete because they are three-dimensional. The priority of three-dimensionality, on the

41

This was suggested to me by Ben Morison. Of course, Plato and Aristotle believe that there are non-extended living beings, e.g. the demiurge in Plato’s Timaeus or the unmoved mover in Aristotle’s Metaphysics. The point is that among extended things (including points) only three-dimensional entities can be substances. 42

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bodies and limits  other hand, cannot be established without taking into account the nature of physical substances. The argument for the priority of bodies is grounded in considerations about the nature of physical substances to which the bodies belong. Proposition 9. If x is a physical substance, x is a three-dimensional body in virtue of its essence and nature.

As I said earlier, this line of interpretation is not without its problems. It is difficult to explain the notion of priority in nature with regard to magnitudes once the assumption of a literal transition between objects of different dimensionality is given up. I chose to employ the notions of ‘being due to’ or ‘in virtue of ’. The thought is that, although three-dimensionality is not part of the essence of physical substances, it is still connected to their essence. The essence or nature of physical substances is responsible for their being three-dimensional. Of course, to a certain extent these notions are metaphors. But I believe that sense can be made of those metaphors. This also provides the background for the discussion of the next section where the status of limits will be discussed. Since bodies are complete and ontologically prior to lowerdimensional items, the being of those items is grounded in the being of bodies.

6.3. Bodies and Limits One intriguing topic in any discussion—be it modern or ancient—of extended objects is the nature of limits. How should we think of limits? Consider the case of a body and the surface that is its boundary. Intuitively, the world contains surfaces. A golden sphere has a surface as its limit. But what is this surface? There seem to be two radically distinct ways to think of this surface. Recently, Galton has argued that the existence of these two ways leads to what he calls the ‘paradox of surfaces’ (Galton 2007, 379).43 Galton argues that one way to think of the surface of a golden sphere is as made of gold.44 If you scratch a golden sphere at its surface, you scratch a golden surface. The surface is golden. It is a surface that is made of gold. On the other hand, it belongs to our concept of a surface that a surface is two-dimensional. But you cannot scratch something two-dimensional, nor can anything two-dimensional be made of gold. Moreover, if we accept the second conception of a surface, more questions come up. Shall we say that there are really two-dimensional layers in the world? Or shall we treat them as abstractions, as Whitehead did? If we accept two-dimensional layers in our ontology, do they belong to the objects in question? Obviously, this introduces a whole battery of problems concerning open and closed bodies. Aristotle, no less than other

43 The two distinct ways of conceptualizing a surface are already mentioned by Stroll 1999, who calls them the ‘Somorjai’ and the ‘Leonardo’ conception. 44 Galton 2007, 379.

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 aristotle’s theory of bodies ancient or modern day philosophers, was concerned with these questions.45 He had, I shall argue, an elaborate theory of boundaries. According to Aristotle boundaries are dependent particulars. There are points, lines, and surfaces in the physical world. But these items all ontologically depend on bodies. This is the claim I shall defend.

6.3.1. An account of limits Aristotle mainly defines a body as that which is extended in three dimensions.46 However, he sometimes suggests that a definition of bodies includes surfaces as boundaries.47 In this case, it follows from the definition of bodies that every body has a limit. Aristotle mentions this way of defining bodies in his discussion of the infinite in Physics III.5: For those who investigate logically it appears that there is no such thing [i.e. an infinite body]: If ‘bounded by a surface’ is the definition of body there cannot be an infinite body either intelligible or sensible.48 (Ph. III.5 204b4–7)

Whether or not we want to see this as a proper definition of body, it seems clear that every body has a bounding surface. So the overarching questions are what are boundaries in general and how do they relate to the entity of which they are boundaries? Let us approach these questions by asking what the intuitive notion behind the concept of a limit might be. I believe that Aristotle shares the same intuitive and prototopological notion of limits that underlies our conception of limits as well. What is this idea? I think that this idea can be best elucidated with the help of an example. Consider a certain body. The body has a certain size. It has a certain extension. This body is, we might say, separate from its environment. To put it metaphorically, it stops somewhere. Where it stops, there is something that separates the body from its environment. This ‘something’ is such that everything that belongs to the body lies 45 Many of these issues in Aristotle I am going to discuss are also discussed in the following articles: Galton 2007; Chisholm 1999; Stroll 1999; Smith 1997a. In ancient philosophy, the nature of limits was extensively discussed in Hellenistic Philosophy. A good starting point is S.E. M9, discussed in Betegh 2015. For a recent assessment of the Stoic position see Rashed 2016. Much of the discussion in this chapter should be seen before the background of Plato’s Timaeus or, more generally, discussions in the Academy. I will occasionally point the reader to these discussions, but a comparative study lies outside the scope of my discussion. 46 Cf. Metaph. V.13 1020a11–14; Ph. III.5 204b20. Note that ‘body’ is used here in its quantitative sense. Physical substances are bodies, since they are three-dimensionally extended, but they are not defined by being three-dimensionally extended. 47 I believe that the first definition should be seen as more authoritative. It is, however, interesting to note that we find a similar phenomenon in Euclid. In the eleventh book, the first two definitions read:

1. A solid is a (figure) having length and breadth and depth. 2. The extremity of a solid (is) a surface. (Euc. XI Def. 1 and 2) These two definitions obviously echo Aristotle’s definitions. For a justification of the inference from the claim that body has a limit to the claim that it cannot be infinite see Nawar 2015. For our present purposes the validity of this inference does not make a difference. 48 ογικ μν οNν σκοπουμνοι κ τν τοινδε δ2ξειεν 3ν ο κ ε$ναι· ε+ γ#ρ στι σCματο 2γο τ πιπδD τι (γ.ται κα τα3τα μρη εναι το3 ποσο3· ο&δ#ν γ%ρ τ"ν συμβεβηκτων ο?τω ν τι @ τδε τι κα τ. α&τ. φ/σεω τ* ποκειμν+ εναι. 15 Notice that this does not imply that quantities are infinitely divisible. If we divide the two into its units, these units are no longer divisible. Yet, I think it is plausible to maintain that the units have the same metaphysical nature as the two. 16 This was suggested by Gábor Betegh. 17 Cf. Metaph. V.6 1015b36–16a17; Metaph. X.1 1052a15–21. 18 The suggested link between being one, being a this, and being undivided is also made in Metaphysics X: ‘for that which is either divided or divisible is called a plurality, and that which is indivisible or not divided is called one’ (Metaph. X.3 1054a22–23) and ‘for this reason to be one is to be indivisible (being essentially a ‘this’ and capable of existing apart either in place or in form or thought); or perhaps to be

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appendix a  A.2. Pluralities and Magnitudes After his general description of quantities Aristotle turns to a classification of them. [1] A quantity is a plurality if it is numerable, [2] a magnitude if it is measurable. [1a] That which is potentially divisible into non-continuous parts is called a plurality, [2a] into continuous parts a magnitude.19 We call a plurality that which is divisible potentially into non-continuous parts, [2a] a magnitude that which is divisible potentially into continuous parts; [2ai] in magnitude, that which is continuous in one dimension is length, [2aii] in two breadth, [2aiii] in three depth.20 (Metaph. V.13 1020a8–12)

In the first sentence Aristotle mentions distinguishing marks of pluralities and magnitudes. Quantities are either [1] pluralities which are numerable or [2] magnitudes which are measurable. The idea of measuring or counting a quantity is, I suggest, connected to the description of the parts as ones and thises. For insofar as the parts are ones and thises, they can be seen as a measure of the whole. The number five can be counted because it consists of five units which are ones.21 A.2.1. The division into magnitudes and numbers I think that the additional claim that pluralities are numerable and magnitudes are measurable is not intended to exclude actual infinities (which is, as Alexander remarks, indeed a consequence22 ), but rather to highlight the distinct natures of pluralities and magnitudes. As I have already argued with regard to Categories 6, a plurality is an aggregate of several things. That is to say, typically the number four is represented by an aggregate of four distinct things or, as Aristotle remarks elsewhere, it is a collection of ‘ones’.23 ‘Number on the other hand is whole and indivisible’ (Metaph. X.1 1052b15–17). Admittedly, the second passage is difficult to interpret and the text is problematic, but I believe that it should be clear on any interpretation that Aristotle’s usage of being a one and a ‘this’ is comparable to the usage we find in Metaphysics V.13. For a detailed discussion cf. Castelli 2018, ad loc. In general, it seems to me that in the context of the Metaphysics as a whole chapter V.13 is closely linked to Book X. For the treatment of unity and oneness in Book X relies on the fact that the one is primarily found in the category of quantity. Obviously, it is beyond the scope of this book to spell out this idea. 19 Ross, in his translation in Barnes 1984, connects δυνμει only with plurality. In contrast to this, I believe it modifies the divisibility both of plurality and of magnitude. I follow here the translation of Kirwan 1993. Since ‘being divisible’ is already a modal notion, one could think that the addition of δυνμει is superfluous in either case. Perhaps the thought behind the addition is to emphasize the possibility of repeated divisions (which would connect it to my interpretation of ν κτερον  καστον. Cf. fn. 1 in this Appendix). A line is divisible, but the resulting line parts are themselves further divisible. The same holds typically true of number, although, of course, the possibility of further division comes eventually to a stop. 20 π.θο μ#ν οAν ποσν τι %ν :ριθμητν B, μγεθο δ# 7ν μετρητν B. γεται δ# π.θο μ#ν τ διαιρετν δυνμει ε μ) συνεχ., μγεθο δ# τ ε συνεχ.· μεγθου δ# τ μ#ν φ’ Cν συνεχ# μ.κο τ δ’ π δ/ο πτο τ δ’ π τρ!α βθο. 21 This connection is made explicit in Metaph. X: ‘For measure is that by which quantity is known; and quantity qua quantity is known either by a “one” or by a number, and all number is known by a “one”. Therefore all quantity qua quantity is known by the one, and that by which quantities are primarily known is the one itself; and so the one is the starting-point of number qua number’ (Metaph. X.1 1052b31–35). In general, Aristotle suggests that the parts of a thing measure the whole. Ph. IV.10 218a6–8; Metaph. V.25 1023b12–17. 22 Al. in Metaph. 396. 14–15. It is a consequence for Aristotle because he contrasts being numerable and being measurable with being infinite. Insofar as something is infinite it cannot be counted or measured. Cf. Ph. III.5 204b7–10; Metaph. II.2 994b20–31; Pr. 955b12–13. 23 Which is, of course, again reminiscent of Aristotle’s remark that quantities are divided into ‘ones’ and ‘thises’.

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 appendix a a plurality of ‘ones’ and a certain quantity of them.’24 (Ph. III.7 207b7–8). A magnitude, by contrast, is measured. The idea is probably that ‘to measure’ is to measure one thing.25 The result of measuring a line is that this line is five foot long. Of course, Aristotle’s criteria may seem dubious since one could equally maintain that the line consists of five one-foot-long lines which can be counted.26 Similarly, to say that something is an aggregate of five units can be seen as a measurement, insofar as the units are the measure of the number, as Aristotle himself suggests in places other than Metaphysics V.13.27 Although the terminological distinction may seem ad hoc, the difference between measuring and counting can be grounded in the fact that magnitudes and pluralities are distinct genera in the category of quantity. A plurality is counted because it involves several things; a magnitude is measured and involves only one thing that is measured. The difference between a plurality or a magnitude is also reflected in the predicates which apply to them. ‘The many and few [are affections] of number, and the great and small of magnitude’28 (Metaph. XIV.1 1088a18–19). It would be a mistake to say that a magnitude is many or few. It is great or small. And similarly it is not correct to speak of a small number or plurality. This would be a metaphorical use of ‘small’ in Aristotle’s view because strictly speaking what is meant is that the plurality has few parts. Be that as it may, the distinction between counting and measuring and the assertion that pluralities are counted and magnitudes are measured is not an explanation of the nature of magnitudes and pluralities. It does not tell us what a magnitude or a plurality is. Rather in the context of Metaphysics V.13 it names a per se accident of magnitudes, something that belongs to them exclusively. But it does not explain the essential features of pluralities and magnitudes. This is explained in the next sentence: [1a] We call a plurality that which is divisible potentially into non-continuous parts, [2a] a magnitude that which is divisible potentially into continuous parts.

Aristotle employs here the same criterion as in the Categories. Pluralities have parts that are non-continuous, whereas magnitudes have parts that are continuous. In both works the first differentia of quantity is the continuity of parts.29 This is the important feature for 8 δ’ :ριθμ στιν να πε!ω κα πσ’ 4ττα. A suggestion that is made by Studtmann 2004, 81. 26 Still there is a difference: There is a single correct answer to how long is that line. There is no single correct answer to how many parts has that line because it is divisible into ever-divisibles. Does the line consist of five one-foot-long parts or of two two-and-a-half-foot-long parts? Even if it were possible to count several magnitudes, an aggregate of several magnitudes is not itself a magnitude. In general, one must say that the same problems we encountered in the Categories arise here as well. Especially the question why one and the same item cannot be a plurality and a magnitude. Kirwan notes that ‘Aristotle does not acknowledge, but neither does he deny, that the same thing may be both a plurality and a magnitude’ (Kirwan 1993, 160). Under a strict reading of ‘the same’ this must be wrong, since it surely is metaphysically impossible that the same thing has both continuous and non-continuous parts. Part of the solution to this problem must lie in the fact that the same thing may be viewed under different aspects. A hand can be studied qua continuous object, in which case it is measured. Or it can be studied qua having five fingers, in which case it is counted. However, Aristotle does not address this problem here. 27 E.g. Metaph. X.1 1052b20–24. 28 τ πο0 κα 1!γον :ριθμο3, κα μγα κα μικρν μεγθου. 29 Note that the continuity of parts can be understood in two ways: it could mean either that the parts are continuous with each other or each part is itself continuous. See my discussion in Appendix A.4. 24 25

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appendix a  a classification of quantities. The reason why a plurality is countable and a magnitude is measurable is, as I said, based on the fact that the former is discontinuous and the latter is continuous. The former is an aggregate of various things, the latter a fusion of connected parts and one single thing. A.2.2. The subdivision of magnitudes according to dimensionality Aristotle further divides magnitudes according to their dimensionality [2ai–2aiii]. Thus, we get the following tree-structure (see Fig. A.1.):

quantity magnitude

plurality length

breath

depth

Fig. A.1. Quantity is first divided into plurality and magnitudes. The latter are further divided into length, breadth, and depth. It is important to note that length, breadth, and depth do not indicate directions here. Aristotle does not employ a Cartesian coordinate system with length, breadth, and depth as orthogonal vectors. Rather, length is simply one-dimensional extension, breadth two-dimensional extension, and depth three-dimensional extension.30 A body, in this sense, is not characterized by having length, breadth, and depth, but simply by having depth, that is, three-dimensional extension. Depth is not the third-dimension on top of length and breadth. Depth is extension in three-dimensions.31 ‘Breadth’ is close to what one might call an ‘area’, and ‘depth’ to ‘volume’.32 It is the next sentence [3] which poses an intriguing question. [3] Of these, limited plurality is number, limited length is a line, breadth a surface, depth a body.33 (Metaph. V.13 1020a13–14)

Apparently, Aristotle is here introducing a further division among quantities. A limited plurality is a number and a limited depth is a body. Ross 1924a, who is followed by Studtmann 2004, argues that ‘τ πεπερασμνον’ goes not only with plurality, but with length, breadth, and depth, too.34 The tree is thus to be completed in the following way (see Fig. A.2.):35

30

On this point see Waschkies 1977, 289. Cf. de An. II.11 423a22–23 where ‘depth’ is used in the same way. This contrasts with Ph. IV.1 209a4–6 where body is defined as having length, breadth, and depth. 32 Cf. Kirwan 1993, 160. 33 το/των δ# π.θο μ#ν τ πεπερασμνον :ριθμ μ.κο δ# γραμμ) πτο δ# πιφνεια βθο δ# σ"μα. 34 Kirwan 1993 mentions the reading of Ross 1924a in his commentary and discusses several ways in which the phrase can be understood. But in his translation he connects τ πεπερασμνον only with π.θο. 35 The superscript ‘L’ indicates that the differentia is ‘limited’. 31

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 appendix a quantity plurality

magnitude

L number

length

breadth

L

L

line

surface

depth L body

Fig. A.2. A.2.3. Limited plurality and magnitude? Grammatically, it is plausible that Aristotle is introducing a further division here. The word ‘το/των’ refers to length, breadth, and depth and the phrase ‘τ πεπερασμνον’ apparently functions as a differentia. Since Aristotle has a general tendency to introduce a division by employing a positive and a privative term, the lack of a privative term corresponding to ‘τ πεπερασμνον’ needs an explanation.36 An explanation of this peculiarity is offered by Studtmann 2004. He explains the absence of the privative term as a consequence of Aristotle’s belief that there are no unlimited, for instance, infinite, numbers or magnitudes. Since for Aristotle there are no unlimited magnitudes or pluralities in this sense, Studtmann’s interpretation makes sense on that account. Nevertheless, the interpretation faces two problems. First, it is not clear why one should admit that this brings in a new differentia. Aristotle has already said that a plurality is countable and a magnitude is measurable (Metaph. V.13 1020a8–9). It is already built into the notions of a plurality and a magnitude that they cannot be unlimited in the sense of being infinite. In the Physics Aristotle uses the criterion of countability to argue that there cannot be an infinite number.37 Hence, if to be countable and to be measurable belongs to plurality and magnitude as such, it is nonsense to introduce it as a division of plurality and magnitude. That is to say, at the juncture of the text where being limited is introduced it is already possible to infer that there are no unlimited magnitudes. Thus, the force of ‘τ πεπερασμνον’ must be different. Limited plurality. Moreover, the evidence that is available in the case of plurality and number does not support Studtmann’s interpretation. In Metaphysics X Aristotle states that plurality is a genus of number. Plurality is as it were the genus to which number belongs; for number is plurality measurable by one.38 (Metaph. X.6 1057a2–4)

This is the most explicit statement I am aware of in which Aristotle classifies number as a kind in the genus plurality. The context of this passage, however, is not concerned with being unlimited

36 38

37 Cf. Studtmann 2004, 75–6. Cf. Ph. III.5 204b7–10. τ δ# π.θο ο'ον γνο στ το3 :ριθμο3· Dστι γ%ρ :ριθμ π.θο ν μετρητν.

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appendix a  in the sense of being infinite,39 but rather with the fact that number is measured by one. An interpretation of the immensely difficult chapter six of Metaphysics X would lead too far away from our topic. Still, it seems safe to assert that the puzzle in that chapter is to find the correct opposition between one and many.40 In this sense, Aristotle’s thought seems to be that one has to assign a certain definite or determinate measure if one counts things. The same plurality may be two humans or four legs depending on whether the unit is ‘human’ or ‘leg’. The difference between an unlimited plurality and limited plurality, that is, number, is further reflected by the kinds of predicates that apply to them. A plurality, as long as it is not measured by the one, may simply be said to be ‘few’. In this sense, the plurality is indefinite or indeterminate. A limited plurality, by contrast, is determinate: what counts as a unit is determined and, hence, a numerical value can be assigned to it.41 If this is along the right lines, we should take ‘limited (τ πεπερασμνον)’ in the sense of being determinate. Limited magnitude. There is no further evidence I am aware of for the view that Aristotle conceives of line, plane, and body as kinds of length, breadth, and depth in contrast to unlimited length, breadth, and depth, which he views as other kinds. Of course, Aristotle is unambiguous in his view that magnitudes are limited.42 The question, however, is whether he classifies

39 Studtmann does not distinguish between these two meanings. Studtmann 2004, 76: ‘Uninformed indeterminate extension would be limitless quantity and hence would be an instance of the actual infinite Aristotle denies could exist.’ 40 The connection to Metaph. X.6 is also made by Kirwan 1993, 161. The chapter begins with a series of puzzles among which are the following:

For if the many are absolutely opposed to the one, certain impossible results follow. One will then be few; for the many are opposed also to the few. Further, two will be many, since the double is multiple, and double derives from two; therefore one will be few. (Metaph. X.6 1056b4–8) Aristotle’s solution depends on a distinction within pluralities. But ‘many’ is applied to the things that are divisible; in one sense it means a plurality which is excessive either absolutely or relatively (while ‘few’ is similarly a plurality which is deficient), and in another sense it means number, in which sense alone it is opposed to the one. (Metaph. X.6 1056b16–20)

For example, a plurality might be said to be a ‘few’, because it has only two members and is measured against a group of ten which is many. However, as a number the two is many, because numbers are always measured against the one. Thus the many has in a certain sense two opposites: the few and the one. These are identified with two different ways in which a plurality may be seen. As I said, to fill in the details is difficult, but it is plain that none of this shows much concern with infinities. An alternative is the interpretation of Ross 1924a who relies on Cornford 1923 and interprets the passage as incorporating the Pythagorean doctrine that number is generated ‘by the union of the Limit and the Unlimited’ (Cornford 1923, 8). That would fit the only other occurrence of the phrase. Cornford, however, does not explicitly mention Metaph. V.13 so his interpretation is not tailored to fit the passage as Ross wants to have it. For a recent assessment of this chapter see Castelli 2018. Alexander in his commentary ad loc. interprets the word ‘limited’ as excluding infinities. But he does not assume that a new point is made. 41 This distinction is also present in the Categories: ‘There is nothing contrary to four-foot or to ten or to anything of this kind—unless someone were to say that many is contrary to few or large to small; but still there is nothing contrary to any definite quantity (:φωρισμνων ποσ"ν)’ (Cat.5 3b29–32). For further discussion of this distinction, and how the ancient commentators elaborated on it, see Barnes 2011, 347–53. 42 There are various discussions found in the Corpus Aristotelicum. Most notably, Ph. III.4–8, esp. III.5 204b4–205a7.

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 appendix a body as a kind of depth whose differentiae are limited vs. unlimited. For this claim there is no evidence. Following the account of pluralities, we should say that a limited magnitude is a magnitude which is determinate, rather than merely finite. Once it is made determinate, we can assign a definite measure to it. For instance, breadth is two-dimensional extension. Viewed simply as some extension, it can be said to be small or large, and in principle it can be measured. However, only when we actually ‘limit’ it, and thereby assign a measure, the breadth is a surface. As in the case of pluralities, the measurement of a magnitude is closely linked to its being determinate. At this point, however, a complication occurs. For, as I have argued in Sections 6.3 and 6.4, lines, surfaces, and bodies are composites of extension and a boundary. In this sense, they are matter-form composites.43 Length, breadth, and depth, on the other hand, are simply extensions. The identity criteria of breadth solely depend on the area. In this sense, every breadth can be measured—by its area. But a surface is, in contrast to breadth, not defined by its area. Rather, a surface is defined by its form, its topological limit. A breadth, on the other hand, can be enclosed by various limits. Consider again the two rectangles. They are the same breadth or area, but are nevertheless different surfaces (see Fig. A.3.):

5 4 3

A B

2 1 0

0

1

2

3

4

5

6

7

Fig. A.3.

43 With this, I follow Studtmann 2004 in assuming that in Metaphysics V.13 Aristotle introduces some sort of matter/form contrast. According to this interpretation, this chapter yields an important addition to the account of the Categories. This point has also been made by Cleary 1995, 147. The difference between unlimited magnitude and limited magnitude is not a distinction between two kinds of objects, but rather the distinction between an object considered qua matter and an object considered qua having a form. I disagree with Studtmann’s analysis of the matter/form contrast. I do not see Aristotle as introducing the idea of prime matter that has to be limited by a form to yield an individual.

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appendix a  Some breadth is measurable insofar as it is, for example, an area of four feet.44 However, this does not yet yield a surface because the surfaces are identified by their specific shape or form. In this way, a breadth of four feet is still indeterminate or ‘unlimited’ because no particular surface is associated with it, even though it may seem to satisfy the constraints on measurability Aristotle set at the beginning of chapter 13 of Metaphysics V. Therefore, I suggest that for some breadth to be fully determinate, that is, to be a surface, it is not sufficient to have its area measured. It must also be identified as a specific figure. That is to say, for the two figures in Fig. A.3. to be determinate, one has to say that one is a rectangular surface whose side lines are one foot and four foot long and the other a square whose side lines are two foot long. This determination enables the measurement of the area. What gets measured is the area of specific geometrical figures, not areas as such. Notice also that, according to this picture, for some some breadth to be determinate, it is not sufficient to identify it as a square as opposed to a rectangle. The size is part of what it is to be a determinate breadth: and again, if the formula of its primary substance is one, e.g. equal straight lines are the same, and so are equal and equal-angled quadrilaterals—there are many such, but in these equality constitutes unity. Things are like if, not being absolutely the same, nor without difference in their compound substance, they are the same in form, e.g. the larger square is like the smaller, and unequal straight lines are like; they are like, but not absolutely the same.45 (Metaph. X.3 1054a35–b7)

The determination of magnitudes is along two dimensions, so to speak. They must be determined both with respect to their size and their kind of geometrical figure. Two squares of unequal size are merely like each other. Insofar as they are both squares, they have the same form. Sameness of form is, I suggest, to be understood as sameness in kind of geometrical figure. However, an account of what kind of geometrical figure they are is not an account of their primary substance. This account—which reveals the full essence and what it is to be a determinate magnitude—should include their size. A difference along one dimension already accounts for a difference what these things are according to their primary substance. If we read Metaph. V.13 in the light of the quotation from Metaph. X.3, we can say that a magnitude is fully determined by its primary substance. A.3. A Comparison of Categories 6 and Metaphysics V.13 This then enables us to compare the classifications of Categories 6 and Metaphysics V.13. In my comparison I will focus on body. I shall argue that the two texts are not only compatible but in fact present the same theory, though with a different emphasis.

44 To be sure, there are no areas that are not surfaces. Thus there is a sense in which if one could speak, as Studtmann does, of the metaphysical necessity that every breadth is bounded by limits. However, this is obviously different from the way in which prime matter, if there were any, must have a form. 45 Dτι δ’ %ν 8 γο 8 τ. πρEτη ο&σ!α ε' B, ο'ον αF Gσαι γραμμα ε&θε