The irony of essence

Proclus and Descartes on geometry

Abstract

The aim of this article is to question a common interpretation of Cartesian philosophy of mathematics according to which Descartes is a Platonist. Such a controversial issue is faced by contrasting Cartesian geometry with Proclus’ commentary to Book I of Euclid’s Elements, which in the 16^th century was the main source for a Platonic interpretation of mathematics. Despite many apparently common aspects and concepts, we will see that Proclus’ and Descartes’ accounts are mutually irreconcilable. This is the case because the kinds of mathematics, which they are trying to philosophically explain, are completely different. Actually, Euclidean geometry is structurally based on two irreducible elements: the image and the word. Thus, Proclus is forced to articulate an epistemology that is able to account for a mathematical practice which is intrinsically divided, creating a hierarchy among its elements. On the contrary, Cartesian geometrical calculus is a unitary field where the language of proportions includes the inner duality of ancient geometry. From this basic difference we can show how distant they are on the role of the epistemic faculties in mathematics and especially on that of the imagination. This makes us possible to look differently at Descartes’ usage of Platonic conceptuality in the Fifth Meditation. Concepts like essence or form are no longer notions corresponding to different levels of being, but rather representatives of the different operations of the human mind in producing its own instruments of knowledge.