THE THEORY OF MATHEMATICAL SUBTRACTION IN ARISTOTLE by Ksenia Romashova Submitted in partial fulfilment of the requirements for the degree of Master of Arts at Dalhousie University Halifax, Nova Scotia August 2019 © Copyright by Ksenia Romashova, 2019 DEDICATION PAGE To my Mother, Vera Romashova Мама, Спасибо тебе за твою бесконечную поддержку, дорогая! ii TABLE OF CONTENTS LIST OF FIGURES .................................................................................................................. iv ABSTRACT ............................................................................................................................... v LIST OF ABBREVIATIONS USED ....................................................................................... vi ACKNOWLEDGEMENTS ..................................................................................................... vii CHAPTER 1 INTRODUCTION ............................................................................................... 1 CHAPTER 2 THE MEANING OF ABSTRACTION .............................................................. 6 2.1 Etymology and Evolution of the Term ................................................................... 7 2.2 The Standard Phrase τὰ ἐξ ἀφαιρέσεως or ‘Abstract Objects’ ............................ 19 CHAPTER 3 GENERAL APPLICATION OF ABSTRACTION .......................................... 24 3.1 The Instances of Aphairein in Plato’s Dialogues ................................................. 24 3.2 The Use of Aphairein in Aristotle’s Topics II, III, V, VI, VIII, and Metaphysics VII .................................................................................................... 32 CHAPTER 4 TECHNICAL APPLICATION OF ABSTRACTION IN ARISTOTLE .......... 39 4.1 The Use of Abstraction Outside Mathematics ...................................................... 40 4.1.1 Two Non-mathematical Uses of τὰ ἐξ ἀφαιρέσεως ................................ 40 4.1.2 Aphairesis and the ‘Qua’ Method Outside Mathematics ........................ 51 4.2 The Use of Abstraction Within Mathematics........................................................ 56 4.2.1 Seven Mathematical Uses of τὰ ἐξ ἀφαιρέσεως ..................................... 56 4.2.2 Aphairesis and the ‘Qua’ Method in Mathematics ................................. 77 4.3 Potential Existence of Intelligible Matter in Sensible Substances ........................ 81 CHAPTER 5 MODERN INTERPRETERS OF ARISTOTLE’S SO-CALLED “THEORY OF ABSTRACTION” ........................................................................................... 98 5.1 Charles De Koninck ............................................................................................... 99 5.2 Auguste Mansion ................................................................................................ 103 5.3 Allan Bäck ........................................................................................................... 106 5.4 Julia Annas ........................................................................................................... 114 CHAPTER 6 CONCLUSION ............................................................................................... 124 BIBLIOGRAPHY .................................................................................................................. 129 iii LIST OF FIGURES Figure 1 Additional lines EC and CD prove the truth of the theorem ‘the angles of the triangle equal to two right angles’ ................................................................. 88 Figure 2 Lines and surfaces BED and DEC demonstrate that the angle in a semicircle in all cases is a right angle ............................................................................. 88 iv ABSTRACT The problem with most modern accounts of Aristotle’s so-called ‘theory of mathematical abstraction’ or aphairesis (ἀφαίρεσις) is that it is interpreted primarily through the scope of the epistemological process of an immediate reception of mathematical forms by the soul without matter from which the former are said to be ‘abstracted’. However, this interpretation is not present in Aristotle’s texts. Instead, aphairesis presents itself as a method by which the mode of being of the objects of mathematics is explained: it elucidates the location and place of the category of quantity within a particular sensible substance, but not some kind of an abstracting activity of drawing mathematical forms or universals from matter. This latter type of the epistemological abstraction of mathematicals found in most modern commentaries was developed by later commentators of Aristotle. The analysis of the modern scholarship shows a clear trace of influence of the ancient tradition. v LIST OF ABBREVIATIONS USED WORKS OF ARISTOTLE APo. Posterior Analytics Cat. Categories DA De Anima Meta. Metaphysics NE Nicomachean Ethics PA Prior Analytics Phys. Physics Pol. Politics Top. Topics WORKS OF PLATO Rep. Republic Symp. Symposium Tim. Timaeus vi ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my thesis advisor, Dr. Eli Diamond, for his support during my years of study here at Dalhousie University. I am thankful, first, for the inspirational classes I have taken from him at the Department of Classics; Ancient Greek, Plato, and Aristotle were always exciting. Secondly, I am grateful for his ability to recognize the interests of a student and place him or her on the correct path of philosophical development. He was the one who recognized and shaped my interests in such a way that they turned into a thesis on Aristotle’s philosophy of mathematics. The process by which he sculpted my interests included several stages. Firstly, I am thankful to Dr. Diamond for encouraging me to take the class on Aristotle’s Metaphysics, which I was hesitant about at first - look at those fourteen dialectical aporiai in book Beta! However, the inspirational side of Dr. Diamond helped us all untie those knots. Secondly, I am thankful to him for including a question on Aristotle’s astronomy as one of our in-class presentation topics. The presentation I gave on this question later turned into a paper that I presented at a graduate conference at the University of Toronto. Thirdly, I am thankful to Dr. Diamond for including a question on Aristotle’s mathematics (books XIII and XIV) on our second take- home exam - he said he included this question particularly for me in case I was interested. My response to this question later grew into a thesis with a specific interest in Aristotle’s theory of subtraction. Fourthly, and most importantly, I would like to thank Dr. Diamond for his guidance throughout my thesis work, his encouragement, constructive criticism, patience, and sympathy for my struggles with writing in a second language. I also wish to express my gratitude to Drs. Michael Fournier and Ian Stewart for their valuable comments on my thesis and for their time they spent reading it. I would like to thank Dr. Alexander Treiger who introduced me to the Classics program and guided me through the entire application process. His presence at the Department of Classics made the first months of my linguistic and academic adaptation pass by in comfort. I am also thankful to Dwight Crowell for his encouragement throughout all these months I spent researching and writing. His desire for knowledge and his pursuit for intellectual development in both philosophy and science has served as a great example for me to follow. Finally, and most importantly, I would like to thank my most loving and most caring mother for her infinite mental support during the years of my study and for her never-ending belief in my goals and interests. She is the primary reason I have been able to succeed academically and complete my thesis. vii CHAPTER 1 INTRODUCTION According to Heraclitus, “πάντα χωρεῖ καὶ οὐδὲν μένει, καὶ ποταμοῦ ῥοῇ ἀπεικάζων τὰ ὄντα λέγει ὡς ‘δὶς ἐς τὸν αὐτὸν ποταμὸν οὐκ ἂν ἐμβαίης” (Plato, Cratylus 402a). This means that everything gives way and nothing remains still just like a flowing river. It is impossible to step in the same stream twice, and what is more, it is impossible to step in it even once (Meta. 1010a14-15). Since everything is in flux and since the matter of physical objects is constantly changing and moving, mathematics cannot be true of the sensible triangles and spheres made of bronze, wood or any other materials, as well as of those drawn on paper or in sand. It follows that when we say that this triangle is of such and such a size or that this triangle is equilateral whose internal angles are each 60°, these truths will almost never hold true of any sensible triangle. Thus, to affirm something will always result in a false statement irrelevant to its physical representative. Even if we suppose the matter of some mathematical shape to be motionless, it would still be impossible to build or draw a perfectly triangular shape due to our limited human abilities. To save the objectivity and precision necessary for mathematics, Plato posited eternal and unchanging intermediates (τὰ μεταξύ) which later became the main focus of Aristotle’s criticisms found primarily in the last two books of his Metaphysics, XIII and XIV. Aristotle describes Plato’s intermediates in the following way: “besides sensible things and Forms he says there are the objects of mathematics, which occupy an intermediate position, differing from sensible things in being eternal and unchangeable, from Forms in that there are many alike, while the Form itself is in each case