To debate the mathematical thought of René Descartes is to debate modern European mathematics as a whole. The central role Descartes played in forming the tradition of modern European mathematics can be well understood from the facts that the style of contemporary mathematics is basically algebraic and that the degree of mathematization of a science is taken to be a mark of the degree to which it has become a true science. Few contemporary thinkers would question this characterization of mathematics and science. Yet we must not forget that the Cartesian vision of the world is the product of particular circumstances, which figure prominently in the following pages.