A Relfection On Descartes’ Mathematical Thought

MY discussion of Rene Descartes’ mathematical thoughts is mainly ontwo perspectives.1. Mathematical angel. Based on the review of Descartes’ discussion onthe Pappus’ Problem in Geometry, I discuss the differences of the uses ofarithmetical phrases and geometrical phrases between Descartes andancient mathematicians. I think the differences are rooted in theirdifferent attitudes to what geometry should study.Descartes thought that a geometrical problem was solved unlessmathematicians found an approach to constructive the target objects. And,in analytical geometry, he transferred the constructions geometricalapproaches to arithmetical approaches.2. Mathematics-natural science relation. I give a review of the relationsbetween Descartes’ mathematical thoughts with his natural philosophy.Descartes thought that the objects in physics theories and in mathematicsare both inherent. He would recognize mathematics knowledge to be prio,but would not acknowledge physics knowledge to be prio.At the end of my discussion, I discuss the mapping relationship betweencoordinate space with the numbers Descartes would use. It shows that thecareful considerations the foundations of analytic geometry woulddefinitely lead to the development of real number theory.