Mathematics

Summary

According to many of its past and present readers, and maybe the author himself, the Geometry forms the core of Descartes’ mathematical practice. From the point of view of its posterity, this judgment seems uncontroversial. By identifying the “geometrical” and the “algebraic,” the treatise from 1637 opened a new era in mathematics. It is well known that other (“infinitesimal”) techniques were also used by the philosopher in his correspondence when dealing with the quadrature of the cycloid or “Debeaune's problem” (Houzel 1997). But even when one acknowledges these other techniques (dealt with only outside of the published treatise), the comparison with the Geometry seems the proper background to assess Descartes’ practice. Yet this comparison raises the following questions: Is the Geometry the ending point of a continuous and self-conscious evolution? Is it part of a homogeneous and coherent practice characterizing Descartes’ mathematics as a whole?

These questions are of interest not only for historians of mathematics. The Discourse on Method presents in effect the study of mathematics as a privileged field where the famous “method” is developed. Just after the statement of the four “rules,” Descartes mentions as an origin for his methodology “the long chains of reasonings, every one simple and easy, which geometers habitually employ to reach their most difficult proofs” (AT VI 19, CSM I 120). Symmetrically, it is with mathematics – the most simple and certain science – that one has to start the inquiry. However, he then makes clear that one need not learn the whole of mathematics but only what forms its core: the study of “ratios and proportions,” which he proposes to represent by lines, which in turn are represented by symbols (AT VI 20, CSM I 121). The program exposed by Descartes in Discourse II seems to have been achieved: starting from the three disciplines he learned when he was young (logic, geometrical analysis, and algebra), Descartes finally succeeds in finding a method that “retained the advantages of all three but was free from their defects” (AT VI 17, CSM I 19).