The Culture of the Method

Abstract

Abstract Chapter 4 explains the role played by mathesis universalis in the formation of the Cartesian scientific habitus. The principal function of mathesis universalis is to shore up and intensify perspicacity in intuition and sagacity in deduction. Only after one has acquired sufficient practice in mathesis universalis can one apply the method to more complex problems. Section 4.2 discusses Descartes’s definition of mathesis universalis as the science of “order and measure,” and argues that, contrary to a widespread tendency to interpret mathesis universalis in expansive terms as either identical to Cartesian mathematics or (even more expansively) to Descartes’s method, mathesis universalis refers to one of the simplest mathematical sciences, well-known since antiquity: the theory of proportions. Section 4.3 argues that reflection on the operations needed to solve problems about continuous and mean proportionals yields Descartes’s theory of relatives and absolutes in Rule 6 of Rules for the Direction of the Mind. Section 4.4 argues that mathesis universalis also yields a classification of different ways in which perfectly understood problems may be solved. Section 4.5 shows how mathesis universalis builds on the abilities acquired via practice in recreational mathematics (discussed in Chapter 3, Section 3.6). Section 4.6 shows how mathesis universalis unifies mathematics, and Section 4.7–4.7.1 delves into the debate about the relation between mathesis universalis, Descartes’s mathematics, and the method. These sections draw on the Cambridge manuscript in order to bolster the principal arguments developed in Sections 4.2–4.5 and propose a revision in standard accounts of the controversial and rather complex chronological genesis of Rule 4.