In this paper I propose a notion of propria inspired by Aristotle, on which propria are non-essential, necessary properties explained by the essence of a thing. My proposal differs from the characterization of propria by Kit Fine and Kathrin Koslicki: unlike Fine, the relation of explanation on my account can’t be assimilated to a notion of logical entailment. In disagreement with Koslicki, I suggest that the explanatory relation at issue needs not be necessary. My account of essence is conceptually parsimonious: it illuminates the contribution of essence to explanation without relying on obscure notions such as Aristotelian form or identity.
Relying upon a very close reading of all of the definitions given in Euclid’s Elements, I argue that this mathematical treatise contains a philosophical treatment of mathematical objects. Specifically, I show that Euclid draws elaborate metaphysical distinctions between (i) substances and non-substantial attributes of substances, (ii) different kinds of substance, and (iii) different kinds of non-substance. While the general metaphysical theory adopted in the Elements resembles that of Aristotle in many respects, Euclid does not employ Aristotle’s terminology, or indeed, any philosophical terminology at all. Instead, Euclid systematically uses different types of definition to distinguish between metaphysically different kinds of mathematical object.