Plato\u27s Geometrical Logic

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 conceptual analysis using geometry as an interpretive model to elaborate the allowable structures of logic. I will argue that Plato adopted geometrical constructions to navigate the precarious passage between the syntactical and semantic structures in philosophical arguments, and that this procedure amounts to a kind of logic of discovery. My approach in this inquiry will be twofold. I will first attempt to elaborate the precise philosophical question that the method of analysis is meant to resolve. I will pay particular attention to any critical hurdles that remain intractable within the limits of the conversation itself. I will then look closely at the details of the geometrical problem to determine just how it could possibly reflect any light back onto the original inquiry. There have been many other interesting attempts to figure out just what Plato meant by “analysis” or the method of hypothesis. They have all focused completely on the interpretation textual references in the dialogues and the commentaries. And they have achieved almost no common ground of assent as to the nature of this process. By examining at close quarters the specific problem in geometrical analysis that Plato wished to use as his example, I hope to overcome some of the contention that originates from the ambiguity of relying merely on the translation of terms