Perfectly Understood Problems

Abstract

AbstractChapter 9 shows how Descartes’s requirement that the material simple natures be intuited by the intellect aided by the imagination determines the activity of the Cartesian scientific habitus in mathematics. Descartes introduces several problem-solving techniques in Rules 13–21. Sections 9.2–9.3 discuss the reduction of problems about subject-matters to problems about relations between pure magnitudes (Rules 13–14). Section 9.4 discusses the representation of these magnitudes by means of the material simple nature of figure alone via the unit (Rule 15). Section 9.5 discusses Descartes’s algebraic notation, which symbolically represents relations between pure magnitudes (Rule 16). Sections 9.6–9.7 discusses Descartes’s geometrical interpretation of addition, subtraction, multiplication, division, and root extraction (the “geometrical calculus” in Rule 18). In mathematics, intuition requires the construction of a geometrical figure that embodies the relevant mathematical operations and satisfies the solution criteria expressed in the problem. Section 9.8 discusses how Descartes’s method in mathematics encounters a number of problems that lead him to end the treatise prematurely at Rule 21.