Colloquium 3: Plato’s Geometry of Being: Mathematical Objects and the Theory of Forms

This paper contributes to the historical discussion of how to think about Aristotle’s report in Metaphysics A 6, what Plato said about mathematical objects, that they are between, µεταξύ, Forms and sensible objects (Metaph. A 6, 987b15–20). We don’t see Plato have Socrates in the dialogues directly mention these mathematical objects. Whenever Socrates offers an ontological distinction, the divide is into what is intelligible and what is sensible. The paper shows how mathematical intermediates in the dialogues bind the two different and distinct objects together, using arithmetic and geometric relations. Arithmetic relations are the set-member or one-many relation, using comparable units. Geometric relations are more complex arithmetic relations using comparable and non-comparable units, in linear and non-linear planes and dimensions. These mathematical relations are then used to analyze the final proof for the immortality of the soul, two geometry problems in the Meno, and Parmenides’ questions about the Forms. The analysis of mathematical relations in Plato’s Meno, Phaedo, and Parmenides offers a charitable answer to Aristotle’s claim in A 6 that Plato certainly could have held true (and probably did) the mathematical intermediates.