Title: Unity and Priority in Descartes’s Algebra and Geometry

Is Descartes’ geometrical emphasis found in his mathematical practice a consequence of his inability to escape from the classical geometric tradition and their unsolved problems? Or does he instead make a great mathematical advance towards a pure mathematics by allowing algebraic numeric entities that are independent of our special intuitions?  The answer, I argue, is neither.  Taking the issues in reverse, I argue that, Descartes does not free numbers from spatial intuitions because his algebraic entities must be represented geometrically.  To show why this is true, I explain how Descartes conceives the relationship between algebra and geometry.  This reading, based largely on the Regulae, also offers an account of how Descartes’ mathematical practices are a consequence of his metaphysical and epistemological views concerning the nature of mathematics.  Hence, it would be a mistake to think Descartes’ mathematical views are orthogonal to his philosophical views, or that his mathematical practices are merely due to vestiges of the classical tradition.