The Mixed Mathematical Intermediates

In Metaphysics B.2 and M.2, Aristotle gives a series of arguments against Platonic mathematical objects. On the view he targets, mathematicals are substances somehow intermediate between Platonic forms and sensible substances. I consider two closely related passages in B2 and M.2 in which he argues that Platonists will need intermediates not only for geometry and arithmetic, but also for the so-called mixed mathematical sciences, and ultimately for all sciences of sensibles. While this has been dismissed as mere polemics, I show that the argument is given in earnest, as Aristotle is committed to its key premises. Further, the argument reveals that Annas’ uniqueness problem is not the only reason a Platonic ontology needs intermediates. Finally, since Aristotle’s objection to intermediates for the mixed mathematical sciences is one he takes seriously, so that it is unlikely that his own account of mathematical objects would fall prey to it, the argument casts doubt on a common interpretation of his philosophy of mathematics.