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.
In this article, I tackle an heretofore unnoticed difficulty with the application of Pyrrhonian scepticism to scientific definitions. Sceptics can suspend belief regarding a dogmatic proposition only by setting up opposing arguments or considerations for and against that proposition. Since Sextus provides arguments exclusively against particular geometrical definitions in Adversus MathematicosIII, commentators have argued that Sextus’ method is not scepticism, but negative dogmatism. However, commentators have overlooked the fact that arguments or considerations in favour of particular geometrical definitions were absent in ancient geometry, and hence unavailable to Sextus. While this might explain why they are also absent from Sextus’ text, I survey and evaluate various strategies to supply arguments in support of particular geometrical definitions.