Inventing Intermediates: Mathematical Discourse and Its Objects in Republic VII

Plato is commonly taken to be committed to the existence of intermediates, ideal mathematical particulars distinct from Forms. But the main argument by which he is thought to arrive at this commitment makes a problematic assumption about mathematical discourse. By examining the role of mathematics in the philosopher’s education of Republic Bk. VII, I show that Plato does not accept this assumption, and therefore does not endorse the argument for intermediates attributed to him. I then offer a new account on which Plato regards the objects of mathematics as theoretical fictions, objects invented by mathematicians to further their theoretical aims.