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Mathematical Reasoning in Plato’s Epistemology Jane Orton PhD Classics, Edinburgh University 2013 1 Declaration (a) this thesis is my own work (b) it has been composed by me (c) it has not been submitted for any other degree or professional qualification except as specified. Jane Orton 05/09/2013 102,000 words approx., excluding bibliography and appendix. 2 Contents Abstract 4 Introduction 5 Aims of the thesis 7 Statement of Terminology 9 Section One: The Republic 31 Chapter One: Introducing the Sun, Line and Cave 33 Chapter Two: Readings of the Allegories in Context 44 i. Knowledge, Belief and Gail Fine 44 ii. Propositions or Objects: Gonzalez on Fine 56 Chapter Three: My Reading 68 i. How Seriously Should We Take the Allegories? 68 ii. Ascending the Scale 78 iii. Noēsis and the Role Definition in Plato’s Epistemology 109 Chapter Four: A Closer Look at dianoia 112 i. The Dianoectic Image 113 ii. The Hypothesis 134 iii. Theaetetus: Hypothesis and Image in the Search for a Definition 146 Section Two: The Meno 152 Chapter One: Definition in the Meno 155 a. Meno’s Definitions 156 b. Socrates’ Definitions 159 c. What is Plato’s preferred answer? 162 ii. What role do definitions play in Plato’s epistemological scale? 169 a. Definition and Essence 170 b. Gail Fine and the Meno 171 Chapter Two: Aporia and the Psychology of Mathematics 181 i. What is aporia? 182 ii. What good is aporia? 183 iii. What is distinct about aporia in Meno? 184 iv. Aporia and hypothesis 186 Chapter Three: The Hypothetical Passage and Mathematical Method in the Meno 191 i. Mathematics as a model for philosophical investigation 192 ii. The Hypothetical Method in the Meno 194 Chapter Four: The Hypothetical Passage in the Context of the Inquiry 214 Chapter Five: Knowledge and True Belief in the Meno 226 Section Three: The Phaedo 229 Chapter One: Dianoia and the Objections of Simmias and Cebes 232 i. Plato Wishes to Illustrate Mathematical Reasoning in the Phaedo 233 ii. The Hypothetical Method as a Second Best Method 235 Second Sailing as Second Best? 242 iii. More Textual Evidence for Dianoia in Phaedo 245 iv. The Objections of Simmias and Cebes and the Divided Line 246 Chapter Two: Socrates’ ‘Intellectual Autobiography’ 257 i Early Enthusiasm and Disappointment 258 ii. Socrates’ ‘Secondary Approach’ 265 Chapter Three: The Hypothetical Passage 271 Conclusion 284 3 Abstract According to Plato, we live in a substitute world. The things we see around us are shadows of reality, imperfect imitations of perfect originals. Beyond the world of the senses, there is another, changeless world, more real and more beautiful than our own. But how can we get at this world, or attain knowledge of it, when our senses are unreliable and the perfect philosophical method remains out of reach? In the Divided Line passage of the Republic, Plato is clear that mathematics has a role to play, but the debate about the exact nature of that role remains unresolved. My reading of the Divided Line might provide the answer. I propose that the ‘mathematical’ passages of the Meno and Phaedo contain evidence that we can use to construct the method by which Plato means us to ascend to knowledge of the Forms. In this dissertation, I shall set out my reading of Plato’s Divided Line, and show how Plato’s use of mathematics in the Meno and Phaedo supports this view. The mathematical method, adapted to philosophy, is a central part of the Line’s ‘way up’ to the definitions of Forms that pure philosophy requires. I shall argue that this method is not, as some scholars think, the geometric method of analysis and synthesis, but apagōgē, or reduction. On this reading, mathematics is pivotal on our journey into the world of the Forms. 4 Introduction Throughout history, mathematics has influenced those at the forefront of other fields, and philosophy is no exception. Writing about the great philosopher Thomas Hobbes, Aubrey captures the massive impact that the achievements of mathematics can have on a thinker: He was 40 years old before he looked on Geometry; which happened accidentally. Being in a Gentleman’s Library, Euclid’s Elements lay open, and twas the 47 EL. Libri I. He read the Proposition. By G- sayd he (he would now and then swear an emphaticall Oath by way of emphasis) this is impossible! So he reads the Demonstration of it, which referred him back to such a proposition; which Proposition he read. That referred him back to another, which he also read…that at last he was demonstrably convinced of that trueth. This made him in love with Geometry (Cited in Stillwell [1989] p 13). Consider also the words of Bertrand Russell: At the age of eleven I began Euclid…This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world (Cited in Stillwell [1989] p 26). This sentiment is not limited to philosophers: writing of himself, here is what Abraham Lincoln has to say about the same subject: He studied and nearly mastered the six books of Euclid since he was a member of Congress. He began a course of rigid mental discipline with the intent to improve his faculties, 5 especially his powers of logic and language. Hence his fondness for Euclid, which he carried with him on the circuit till he could demonstrate with ease all the six books (Cited in Stillwell [1989] p 26). No doubt Plato would have approved of the study of geometry by a statesman; much of his Republic is taken up with a plea for the rulers of his ideal state to undergo rigorous mathematical training. Plato has his own love affair with mathematics, and, as I shall try to argue, his dialogues can tell us about the effect it had on his philosophy. Plato’s relationship with mathematics, it has already been argued,1 changed his entire philosophical approach. This dissertation explores one aspect of this change: the hypothetical method in the so-called ‘middle dialogues,’ with particular reference to the use of imagery. Plato is famously critical of images2 and although much work has been done to rehabilitate the status of the image in Plato’s political thought (Nehamas [1999] Ch. 12-13), it remains the black sheep of Platonic epistemology. Yet the dialogues are full of images, as Plato himself is not unaware.3 Indeed, the use of images, along with the use of hypothesis, is a distinguishing mark in the science that Plato so much admires: the mathematics of his time. In the divided line passage of the Republic, Plato notes that the use of hypotheses and the use of diagrams are both characteristics of dianoia, or mathematical reasoning. This project will explore the influence of mathematics on Plato’s philosophy. 1 See especially Vlastos (1991), pp. 107-131 for a discussion of the influence of mathematics on Plato. 2 See books IX and X of Republic for Plato’s criticism of images. 3 Socrates refers to such issues at Phaedo 100a. 6 Aims of the Thesis Plato’s theory of Forms is one of the most distinctive aspects of his philosophy. Forms are central to his metaphysics, epistemology and moral and political theory. The Philosopher Kings of the Republic are to spend years in training to attain knowledge of the Forms before they are to rule. In the divided line passage of the Republic (509d-511e), Plato speaks of there being two realms: the intelligible (containing the Forms) and the visible, and each realm has two states of mind associated with it. Dianoia is the state of mind that Plato links to mathematics: I shall argue that it is a kind of reasoning that Plato himself uses in the middle dialogues, and it is the second highest kind of reasoning in Plato’s scheme. The ultimate aim for the philosopher is to attain the highest state of mind, noēsis, but the problem for Plato is how to do that when most of us spend our lives contemplating the visible world. In fact, as Plato notes in the Phaedo (66cd) we are tied to the physical world, so how can we release ourselves from these tethers to contemplate the intelligible? The primary aim of the thesis is to show how Plato can use the mathematical method to provide a solution. I shall suggest that Plato’s divided line can be seen as an epistemological scale to be ascended by the philosopher, and that the mathematical method provides a pivotal role in ascending this scale. In fact, it is by using this method that we may escape the prison of the senses, and begin to contemplate the intelligible. By examining Plato’s remarks about mathematics, and the structure of some of the important arguments in the middle dialogues, we shall see that Plato himself uses the mathematical method to do exactly this: when Socrates’ friends are stuck in pistis, the state of mind that uses physical objects as tools of inquiry, Socrates sometimes uses dianoia to ascend the epistemological scale and reach for knowledge of the Forms.