DESCARTES ON THE ROLE OF DIAGRAMS AND SYMBOLS IN MATHEMATICS

Descartes seems to have held to the end of his life that imagination plays a prominent role in mathematics. In distinguishing sharply between images and ideas in the Meditations, he is adamant in stating that we do not need images in order to have mathematical ideas, and that there is a purely intellectual conception of extension and even presumably extended objects, existing or not. However, he seems to advocate that imagination does make these ideas better. The main goal of the present paper is to analyze, specially by means of Rules XIV-XVI and the Discourse on Method, how this auxiliary role played by imagination can be found in the use of diagrams and mathematical symbolism as an aid for mathematical invention. As I will argue, while Descartes seems to have thought that (ordinary) language is a source of error, prejudice, and obscurity, he also thought that a good use of signs in mathematics provides our ideas with distinctness, as also would the remote possibility of an idealized universal philosophical language.