The Trouble With Touching: A Problem In Aristotle's Continuity Theory by Sammy T. Jakubowicz Graduate Program In Philosophy Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy Faculty of Graduate Studies The University of Western Ontario London, Ontario July 1999 O Sammy T. Jakubowicz 2999 National Library Bibliotheque nationale 1*1 of Canada du Canada Acquisitions and Acquisitions et Bibliographic Services services bibliographiques 395 Wellington Street 395, rue Wellington Ottawa ON K1A ON4 Ottawa ON KIA ON4 Canada Canada Your Me Vofre referame Our fire Notre reference The author has granted a non- L7auteura accorde une licence non exclusive licence allowing the exclusive pennettant a la National Library of Canada to Bibliotheque nationale du Canada de reproduce, loan, dstribute or sell reproduire, preter, distribuer ou copies of this thesis in microform, vendre des copies de cette these sous paper or electronic formats. la forme de microfiche/film, de reproduction sur papier ou sur format eiectronique. The author retains ownerslup of the L'auteur conserve la propriete du copyright in this thesis. Neither the droit d'auteur qui protege cette these. thesis nor substantial extracts fi-om it Ni la these ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent Ctre imprimes reproduced without the author's ou autrement reproduits sans son permission. autorisation. Abstract Aristotle's physics rests upon the foundation of the continuum. This entails understanding space to be infinitely divisible. However, when space is understood in this way we find a number of problems. In particular, this work deals with the difficulties posed by the problem of touching in Aristotle's continuum. It would seem that, given his understanding of space, it would not be possible for any two objects to come into contact. And this would apply not only to physical situations, but to geometricaf ones as well. In order to solve this difficulty, or, more accurately, to see how it might have been possible for Aristotle himself to have solved it, a number of related topics are examined. The early stages of the examination focus primarily on the geometry of the situation and explore the concepts of points and boundaries. After acquiring a better understanding of these and of what they entail, we are able to move to the physical side of the problem and examine such concepts as intelligible matter, mixing, and actuality. When taken together, these allow us to do two things. First, we are more easily able to see the relation between the physical and the geometrical concepts and difficulties. More importantly, we are able to see how the initial problem of touching, as well as the variations on the problem, could have been solved by Aristotle within his own framework. Keywords: Aristotle, touching, point, boundary, limit, geometry, intelligible matter, mixing, actuality, potentiality, place, continuum, space, unit, tangent, history of mathematics, history of physics Appropriately enough, given Aristotle's views, a thesis is not written in a vacuum. There are a great many people who should be acknowledged for the parts they played in bringing this work to completion. My supervisor, John Thorp, deserves a great deal of credit for guiding me through the process of writing this thesis. My finding this topic is a direct result of having taken, and having enjoyed, his courses. Without his advice and constructive criticism this would have been a far different - and probably far less interesting - work. I would also like to thank the members of my advisory and defence committees - John Bell, Lome Falkenstein, Paul Potter, hing Block, and John Magee - for their comments on the thesis. Thanks are also due to John Scott who, at the 1997 meeting of the CPA, commented on a very early draft of the chapter on points. Their comments were helpful and appreciated. Since my arrival in this department I have felt welcome, and this has made my work all the easier. This is in no small part due to the people who REALLY run the show - Andrea Purvis, Julia Mcdonald, Glenda Ogilvie, and Ella Young. Credit should also be given to the friends I have made while living in London. Thank you all for putting up with my idiosyncrasies. And last, though by no means least, are the members of my family. None of this work would have been possible had it not been for the support and encouragement they gave me from an early age. I place the blame for my love of learning squarely on your shoulders, and I thank you for giving me the blame to place. Table of Contents Page .. Certificate of Examination ll .. Abstract 111 Ac know Iedgements iv Table of Contents v A Note on Texts vii Chapter 1 - Introduction 1 The Problems 2 The Method 8 Chapter 2 - Points 11 The Traditional Interpretation of Points: The EuclideanKartesian View 13 What Is a Unit? - A Digression Into How Units Relate to Points 16 Problems With the Traditional Interpretation of Points I. The Place of Points 26 2. The Size of Points 29 3. Points in Contact 3 1 A Different Interpretation: Ideal Versus Real 35 The Advantages of this Interpretation 47 Chapter 3 - Boundaries 53 What is a Boundary? 55 The Problem of Coincident Boundaries 66 The Problem of a Shared Boundary 70 Aquinas' Solution - Touching Boundaries vs. No Boundaries 76 Another Solution - Boundaries in Contact vs. Mixed Boundaries 83 The Missing Piece of the Puzzle: Mixing 87 Chapter 4 - Geometrical Objects and Intelligible Matter 9 1 Why the Subject of Mixing Is Important to the Project 92 Aristotle's Views on Mixing 94 Difficulties with Aristotle's View 1. The Problem of Coincident Bodies 97 2. The Problem of Instantaneous Change 100 3. The Problem of Generation 100 4. The Problem of Geometricd Mixing 102 Intelligible Matter - Solving the Problem of Geometrical Mixing 103 The Difficulty of Making the Transition from Physical to Geometrical Objects 1 15 Solving the Other Difficulties 120 Chapter 5 - Actuality and Potentiality Connecting Intelligible Matter to the Actual/Potential Distinction Explaining Actuality and Potentiality Actualization and Geometry Actualizing Potentials Solving the Problems with Boundaries Solving the Problems of Coincident Boundaries, Instantaneous Change, and Generation Chapter 6 - Conclusion Review: What Has Been Uncovered Thus Far Solving the Original Problems 1. Basic Tangency 2. Touching at Boundaries 3. Physical Tangency and Abstraction The Significance of the Findings Bibliography Vita A Note on Texts References are made throughout this dissertation to the works of Aristotle. Unless otherwise indicated, all English translations of the works of AristotIe have been taken from Aristotle. The Complete Works of Aristotle. Ed. Jonathan Barnes. 2 vols. Princeton: Princeton University Press, 1995. The information about the Greek texts I have used can be found in the bibliography. vii Cha~ter1 - Introduction The Problems There are a number of important problems which arise when we Look closely at the concept of continuity as it is presented in Aristotle's writings. This section introduces four difficulties related to the problem of touching in a continuum: the problems of geometrical and mathematical tangency, the problem of contact at boundaries, and the problem of abstraction. These need to be addressed in order to understand how Aristotle understood the idea of contact in both a geometrical and a physical context. The Method This section serves as a rough guide to the manner in which the project will proceed. In particular, the objectives of the ensuing chapters are mentioned in order better to place them in the broader context of the project at large. 2 The Problems Much of Aristotle's physics rests on the foundation of what is called 'continuum theory'. This theory consists in seeing matter, space, and time as divisible ad infiniturn. However, there is a difficulty with this theory in that it cannot coherently deal with the idea of two objects touching one another. It may be that Aristotle himself was aware of this problem and this can be seen through his various attempts in his works to solve related problems. These attempts, though, often do as much to complicate matters as they do to solve them. In fact, at times the entire exercise seems like a neurotic effort on Aristotle's part to save his theory. Consider the following diagram. If two circles touch at a single point, call this point C, they are understood as being tangent to each other at that point; we standardly anaiyse this mathematical situation as one in which the two circles share a single point, and this analysis gives us little trouble. This problem becomes more complicated when we try to see how it would work in a physical setting. Assume that the circles represent two material spheres which touch one another and which exist as continuous bodies in a continuous space of the type for which Aristotle argues so vehemently. What analysis can be made of the point at which the spheres 3 come into contact? When the problem dealt only with nonmaterial elements such as Iines and circles we were not troubled by saying that two shapes shared a point. However, unlike the situation where the circles are mere geometrical entities, in the material version of the problem we would not want to say that this point is in both spheres. If it were, then the two objects would intermingle and lose their discreteness; we would have a single object, not two objects touching one another. We might say that the spheres could be divided at the point of contact, but this would not work either. To do so would be to say that points are divisible and, therefore, have size.