Plato's Geometrical Logic

Phronesis, 2002

Late ancient Platonists and Aristotelians describe the method of reasoning to first principles as "analysis." This is a metaphor from geometrical practice. How far back were philosophers taking geometric analysis as a model for philosophy, and what work did they mean this model to do? After giving a logical description of analysis in geometry, and arguing that the standard (not entirely accurate) late ancient logical description of analysis was already familiar in the time of Plato and Aristotle, I argue that Plato, in the second geometrical passage of the Meno (86e4-87b2), is taking analysis as a model for one kind of philosophical reasoning, and I explore the advantages and limits of this model for philosophical discovery, and in particular for how first principles can be discovered, without circularity, by argument.

2003

Socrates’ brief mention of a complex problem in geometrical analysis at Meno (86d-87c) remains today a persistent mystery. The ostensible reason for the reference is to provide an analogy for the method of hypothesis from the use of hypotheses in analytic geometry. Both methods begin by assuming what is to be demonstrated and then show that the assumption leads to a well-founded truth father than something known to be false. But why did Plato pick this particular problem in analysis and why at this particular place in the inquiry? For those of us who view the dialogues as pedagogical puzzles for readers of all time to “scour” out the subtle and complicated details, this is an unquiet mystery that demands further examination. In this paper I will defend the claim that Plato had developed a powerful new heuristic method for the clarification and resolution of a broad range of philosophical problems. This method, based on the techniques of inquiry used in geometry, was a kind of concep...

polar.alaskapacific.edu

2014

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? 18...

Apeiron, 1988

Apeiron, 2015

This paper examines the second geometrical problem in the Meno. Its purpose is to explore the implication of Cook Wilson’s interpretation, which has been most widely accepted by scholars, in relation to the nature of hypothesis. I argue that (a) the geometrical hypothesis in question is a tentative answer to a more basic problem, which could not be solved by available methods at that time, and that (b) despite the temporary nature of a hypothesis, there is a rational process for formulating it. The paper also contains discussion of the method of analysis, problem reduction and a diorism, which have often been ambiguously explained in relation to the geometrical problem in question.

Logica

(please cite that version) Richard Robinson in his classic work Plato's Earlier Dialectic (1953) describes the following difference between dialogues which he takes to represent Plato's 'early period' -and dialogues which he takes to represent Plato's 'middle period': the early gives prominence to method but not to methodology, while the middle gives prominence to methodology but not to method. In other words, theories of method are more obvious in the middle, but examples of it are more obvious in the early. Actual cases of the elenchus follow one another in quick succession in the early works; but when we looked for discussions of the elenchus, we found them few and not very abstract. The middle dialogues, on the other hand, abound in abstract words and proposals concerning method, but it is by no means obvious whether these proposals are being actually followed, or whether any method is being actually followed. (Robinson 1953:61-62) Robinson goes on in what follows to soften this distinction between the two sets of dialogues, but scholarly discussion of Platonic method in the latter set of dialogues has continued to focus more upon Plato's explicit proposals than on Plato's actual practice in those dialogues. No doubt part of the explanation for this tendency is Robinson's suggestion that in the latter dialogues Plato appears not to practice what he preaches. The philosophical method that Plato has Socrates recommend in dialogues like the Meno, Phaedo, and Republic is apparently not the method that Plato has Socrates practice in those dialogues. In this chapter I resist such a conception of Platonic dialectic. I will begin by looking briefly at Plato's explict recommendations of philosophical method in three key middle dialogues -the Meno, the Phaedo, and the Republic. We will see that while differences in the methods recommended in these three dialogues are apparent, certain January 2005 2 core features remain invariant. These core features can be reduced to two processes: a process of identifying and drawing out the consequences of propositions, known as hypotheses, in order to answer the question at hand, and a process of confirming or justifying those hypotheses. I will then maintain that in three pivotal and extended stretches in these three dialogues Plato has Socrates practice one or the other of these processes of the method he has had Socrates recommend. Such a view of Platonic dialectic has two immediate consequences. First, there is more continuity and commonality to Plato's discussion of method, his 'methodology' to use Robinson's word, than has often been supposed. The methods of hypothesis introduced in the Meno and again in the Phaedo and the method of dialectic explicitly introduced in the Republic are versions of a single core method. Second, in order to understand Plato's recommended philosophical method in the so-called middle dialogues we should not restrict ourselves to Plato's explicit discussions of that method. Just as in the so-called early dialogues we look at both Socrates' explicit discussions of method and his actual practice in order to understand the elenchos (SEE YOUNG), so in the so-called middle dialogues we should look at both Socrates' explicit discussions of method and his actual practice in order to understand dialectic. We should, that is, look at both his 'methodology' and his 'method' to use Robinson's words. Nevertheless, we will see that the philosophical method that emerges from both of these sources remains by Plato's own lights in some way inadequate. I will conclude by offering an explanation of this apparent inadequacy -an explanation that points in the direction of further study.

Ancient Philosophy Today: DIALOGOI [proofs], 2023

This paper reconstructs Plato's 'philosophy of geometry' by arguing that he uses a geometrical method of hypothesis in his account of the cosmos' generation in the Timaeus. Commentators on Plato's philosophy of mathematics often start from Aristotle's report in the Metaphysics that Plato admitted the existence of mathematical objects in-between (metaxu) Forms and sensible particulars (Meta. 1.6, 987b14-18). I argue, however, that Plato's interest in mathematics was centred on its methodological usefulness for philosophical inquiry, rather than on questions of mathematical ontology. My key passage of interest is Timaeus' account of the generation of the primary bodies in the cosmos, i.e. fire, air, water and earth (Tim. 48b-c, 53b-56c). Timaeus explains the primary bodies' origin by hypothesising two right-angled triangles as their starting-point (arkhê) and describing their individual geometrical constitution. This hypothetical operation recalls the hypothetical method which Socrates introduces in the Meno (86e-87b), as well as the use of hypotheses by mathematicians which is described in the Republic (510b-c). Throughout the passage, Timaeus is focussed on explicating the bodies in terms of their formal structure, without however considering the ontological status of the triangles in relation to the physical world.