Abstract

‘The Place of Geometry’ discusses the excursus on mathematics from Heidegger's 1924–25 lecture course on Platonic dialogues, which has been published as Volume 19 of the Gesamtausgabe as Plato's Sophist, as a starting point for an examination of geometry in Euclid, Aristotle and Descartes. One of the crucial points Heidegger makes is that in Aristotle there is a fundamental difference between arithmetic and geometry, because the mode of their connection is different. The units of geometry are positioned, the units of arithmetic unpositioned. Following Heidegger's claim that the Greeks had no word for space, and David Lachterman's assertion that there is no term corresponding to or translatable as ‘space’ in Euclid's Elements, I examine when the term ‘space’ was introduced into Western thought. Descartes is central to understanding this shift, because his understanding of extension based in terms of mathematical co‐ordinates is a radical break with Greek thought. Not only does this introduce this word ‘space’ but, by conceiving of geometrical lines and shapes in terms of numerical co‐ordinates, which can be divided, it turns something that is positioned into unpositioned. Geometric problems can be reduced to equations, the length of lines: a problem of number. The continuum of geometry is transformed into a form of arithmetic. Geometry loses position just as the Greek notion of ‘place’ is transformed into the modern notion of space.