Chihon Ley Polanco

Aristotle’s report in Metaphysics A.6, 987b14-18, says that, apart from the Forms and the sensible things, Plato posited a third kind of entity: the objects of mathematics or mathematicals (τὰ μαθηματικά). This latter entity would share properties with the former two, but not fully belonging to either of them, belonging to a place in between Forms and sensible particulars. Hence its name: intermediates. This report generated an important controversy for ancient philosophy scholars, since nowhere in Plato’s dialogues it is explicitly stated that he advocated for this third kind of entity. Most efforts to make sense of this report have been based on an exegesis of Plato’s works, looking for hints that would indicate that he advocated for objects of mathematics, or, conversely, to dismiss Aristotle’s report as spurious. My proposal here is to freshen this discussion by focusing on Greek mathematics and the way it worked. Indeed, most of the literature that deals with intermediates tends to focus on Plato’s ontology, but lacks a more detailed account of the mathematical side of things. Moreover, Plato’s Academy was one of the main places in Antiquity to pursue the study of mathematics. My contention is that, by looking at Plato’s dialogues through the lenses of mathematics (i.e. by having a clear account of how it worked and what makes it different from the mathematics we nowadays use), we would be able to find new insights in Plato’s mathematical passages in the middle dialogues. The upshot of this analysis is that Plato not only posited these mathematicals, but he also incorporated this notion of third kind of entity into his account of the soul in the Phaedo and the ideal city in the Republic.