THEORETICAL PRESUPPOSITIONS OF DESCARTES’ “GEOMETRY”

Abstract

Descartes’ mathematical thought, given to us in his “Geometry”, is based on several conceptual presuppositions which separate him from his predecessors. The underlying idea of mathesis universalis as the source of all sciences enables him to adequately synthesize algebra and geometry, which is not a one-sided reduction of geometry to algebra, but rather their unification. The analytical method that he applies to problems is truly mathematical, as it consists of finding roots of equations. These equations are, in turn, possible due to another concept, that of a mathematical variable. Equations can be used to describe geometrical figures, as they can express the relations between certain points on the figure and certain points on the coordinate reference system. Qualities of the figure are thereby translated into magnitudes, and the figures themselves are at the same time associated with certain ideal motions. The importance of these presuppositions is evident from Descartes’ ability to solve mathematical problems that were previously deemed unsolvable.