From Intermediates through Eidetic Numbers: Plato on the Limits of Counting

Many have argued that Plato’s intermediates are not independent entities. Rather, they exemplify the incapacity of discursive thought (διάνοια) to cognizing Forms. But just what does this incapacity consist in? Any successful answer will require going beyond the intermediates themselves to another aspect of Plato’s mathematical thought - his attribution of a quasi-numerical structure to Forms (the ‘eidetic numbers’). For our purposes, the most penetrating account of eidetic numbers is Jacob Klein’s, who saw clearly that eidetic numbers are part of Plato’s inquiry into the ontological basis for all counting: the existence of a plurality of formal elements, distinct yet combinable into internally articulate unities. However, Klein’s study of the Sophist reveals such articulate unities as imperfectly countable and therefore opaque to διάνοια. And only this opacity, I argue, successfully explains the relationship of intermediates to Forms.