Aristotle on the metaphysical status of mathematical entities Vangelis Pappas Faculty of Classics University of Cambridge This dissertation is submitted for the degree of Doctor of Philosophy Pembroke College September 2018 2 Preface This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration. I state that no substantial part of my dissertation has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge or any other University or similar institution. It does not exceed the prescribed word limit for the Classics Degree Committee. 3 Στην μνήμη της γιαγιάς μου 4 Acknowledgements My greatest thanks go to my supervisor, Dr Robert Wardy. I am very grateful for his unwavering support over my years here in Cambridge. Dr Wardy is an ideal supervisor: not only did he encourage me to explore modern debates in the philosophy of mathematics, he also provided deeply insightful comments on the (sometimes very rough) drafts I gave him, always accompanied with helpful advice. Under his patient and wise guidance and tutelage I was able to rectify the many deficiencies of my writings and, more importantly, I learned a great amount about how to do philosophy in a critical, scholarly manner, and how to properly engage with problematic texts. I would also like to thank wholeheartedly my secondary supervisor, Mr Nicholas Denyer, for his constructive criticism of my drafts, and for his warm encouragement and advice. His extensive, acute, and eminently useful annotations on my drafts as well as the conversations with him proved to be an invalueable source of philosophical stimulation. I would like to express my gratitude to Professor Vassilis Karasmanis for sparking my interest in philosophy and for a great many discussions about Greek mathematics and philosophy in Athens and in Oxford. I wish to express my thanks to my good friend Vangelis Triantafyllou for our many discussions on a variety of topics in the philosophy of mathematics. I would like to acknowledge the generous support received for the fruition of my research project from the Arts and Humanities Research Council in the form of a Doctoral Training Partnership award and from Pembroke College in the form of a graduate studentship. I would also like to thank the Cambridge Trust and especially the George and Marie Vergottis Foundation for providing me the necessary means to cover a significant part of the living costs and thereby boosting my capacity to conduct research at the highest academic level without being financially distracted; I am very grateful and thankful for this unique opportunity. 5 My heartfelt thanks go to my parents for their unconditional support, encouragement and love over the years, and especially to my brother Giannis for being there for me through good times and bad times. Finally, I would like to thank my partner, Stefania, for making me extraordinarily happy. V.P. 6 Contents Synopsis .................................................................................................................................................. 8 Chapter 1: Aristotle’s realism about mathematicals in Physics B.2 ........................... 10 [1.1] Introduction ....................................................................................................................... 10 [1.2] A discussion of the Physics B.2 passage .................................................................. 12 [1.3] The question of idealisation ........................................................................................ 20 Chapter 2: Aristotle on the metaphysical status of geometricals .............................. 22 [2.1] (Not really) a tetrachotomy ......................................................................................... 22 [2.2] The notion of priority and its role in the M.2 discussion ................................ 25 [2.2.1] The multifaceted notion of priority ................................................................. 25 [2.2.2] Aristotle’s arguments against the metaphysical priority of lower- dimensional entities .............................................................................................................. 29 [2.2.3] Mathematicals as prior in definition ............................................................... 34 [2.3] Mathematicals in sensible things. A fanciful doctrine? .................................... 37 [2.3.1] Arguments against the existence of intermediate mathematicals in the sensibles ............................................................................................................................. 37 [2.3.2] Arguments against a conception of lower-dimensional mathematicals as constitutive parts of bodies .......................................................................................... 42 [2.4] Metaphysics M.3 analysis and related excursus .................................................. 47 [2.4.1] The analogy from the universal propositions in mathematics and the related discussion–part one .............................................................................................. 47 [2.4.2] The analogy from the universal propositions in mathematics and the related discussion– part two ............................................................................................. 61 [2.4.3] The analogy from physics/astronomy ........................................................... 68 [2.4.4] Conclusions and further discussion ................................................................ 73 [2.5] The crucial M.3 passage (1078a17-31) .................................................................. 80 [2.5.1] Untangling a highly compressed text .............................................................. 80 [2.5.2] The metaphysical priority of bodies ............................................................... 90 [2.5.3] Ian Mueller’s interpretation of the M.3 passage and its background 95 [2.6] The potential being of geometricals ...................................................................... 103 [2.6.1] The meaning of ὑλικῶς ........................................................................................ 103 [2.6.2] A normal understanding of potentiality ...................................................... 105 [2.7] Towards a new interpretation of the M.3 passage .......................................... 110 [2.7.1] A radical interpretation ...................................................................................... 110 [2.7.2] A more ‘exotic’ potentiality ............................................................................... 111 [2.7.3] Aristotle’s ‘intuitive concept of a limit’ ........................................................ 117 [2.7.4] Menn’s interpretation and its problems ...................................................... 120 [2.7.5] My interpretation and a modern analogue ................................................ 124 [2.7.6] Further suggestions .............................................................................................. 129 Chapter 3: Aristotle on the metaphysical status of numbers: An exploration .... 132 [3.1] Introduction: A Fregean Aristotle? ......................................................................... 132 [3.2] Frege’s arguments against numbers as properties of perceptibles ......... 135 7 [3.2.1] A first argument ..................................................................................................... 135 [3.2.2] A second argument ............................................................................................... 137 [3.3] A Fregean reading of Phys. Δ.12, 220b10-12 ...................................................... 139 [3.4] Two senses of number ................................................................................................. 141 [3.4.1] A first sense of number ....................................................................................... 141 [3.4.2] An Aristotelian response to Frege .................................................................. 143 [3.4.3] Aristotle’s criticisms of Platonic Form numbers in Meta. M.6-8 ....... 144 [3.4.4] A second sense of number ................................................................................. 150 [3.5] The beginnings of a non-Fregean account .......................................................... 154 [3.6] A surprising Platonic account ................................................................................... 158 [3.7] Platonic complications and interim conclusions .............................................. 160 [3.8] The thesis that composition is identity ................................................................ 165 [3.8.1] Frege against Mill .................................................................................................. 165 [3.8.2] Frege and Plato ....................................................................................................... 170 [3.8.3] Plato and Aristotle on composition as identity ........................................ 171 [3.9] Aristotelian complications ......................................................................................... 177 [3.9.1] A hylomorphic account of number? .............................................................. 177 [3.9.2] Discussing Meta. H.3 ............................................................................................