Philosophy in Euclid’s Definitions

What is it about?

Euclid’s Elements is the first treatise of deductive mathematics in history, and it is still regarded as a paradigm of scientific rigour. While the Elements seems to be a purely mathematical work on the surface, I argue that Euclid in fact encrypts a metaphysical theory in his text. Specifically, Euclid systematically uses different types of definition to distinguish between metaphysically different kinds of mathematical object (such as genera and species on the one hand, and differentiae on the other).

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Photo by Andy Art on Unsplash

Photo by Andy Art on Unsplash

Why is it important?

My paper provides the first self-contained study of Euclid’s method of definition. The analysis of the Elements' definitions yields unexpected consequences for interpreting Euclid’s overall project. The fact that Euclid uses different types of definition to encode traditional metaphysical distinctions makes Euclid’s Elements an early instance of the history of ancient philosophy in the Aristotelian and Platonic tradition. Euclid’s distinctions also have a mathematical value: his distinction of the differentiae of a genus from the genera themselves serves to distinguish demonstrable from indemonstrable attributes. Moreover, the fact that Euclid draws metaphysical distinctions by way of employing different types of definition sheds interesting new light on the literary techniques of ancient scientific treatises. Furthermore, my analysis of the linguistic regularities found in the Elements' definitions even warrants a revision of Heiberg’s standard edition (1873–1876), as I show with the case of Euclid’s famous definition of parallel lines.

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