Abstract

In Metaphysics M.3, Aristotle puts forward two main claims. He argues (i) that geometry is about sensible objects, but not qua sensible, and (ii) that geometers can posit as separate what is not separate, and consider it as such. I believe that the two statements are not two reformulations of the same idea, but two distinct claims, aimed at solving two different problems: the Grounding Question (what makes geometrical statements true?) and the Reference Question (what do the singular terms in geometrical statements refer to?). I submit that, contrary to our expectations, in Aristotle’s account the answers to these two questions come apart. This is because the ontology that seems to be implicit in geometrical practice and language is at odds with Aristotle’s metaphysical commitments.

The qua-theory is offered as an account of geometrical truth, while the doctrine of separation is supposed to account for geometrical practice and for the way in which geometers speak of the objects they study. I give a radically anti-Platonist account of geometrical truth, according to which Aristotle does not need to introduce any kind of object that is specific to geometry (not even objects that depend for their existence on sensible substances): geometry is true in virtue of sensible substances and their properties. However, I believe that Aristotle relies on a moderate form of fictionalism (which only concerns the mode of being of the entities in question) in order to account for geometrical practice.