Descartes's Mathematical Thought (review)

In Descartes's Mathematical Thought, Chikara Sasaki argues that RenéDescartes began a Kuhnian revolution in mathematics — a paradigm shift including nothing less than "a gestalt switch from the geometrical to the algebraic mode of thought in mathematics" (271). The book outlines the development ofDescartes's mathematical projects, his philosophical accounts of mathematics, and the interaction between the two. Among major recent studies of Descartes, the book stands out for lucidly examining the full range of Descartes's mathematical activities and for working to illustrate the centrality of algebra for the development of Descartes's broader philosophical thought.

Sasaki has not produced a narrow monograph focused upon Descartes alone. While underscoring the novelty of Descartes's mathematics, Sasaki surveys the range and limits of premodern algebra, European as well as Arabic, and documents the mathematics and philosophy of mathematics Descartes knew, used, and surpassed. The narrative begins, for example, with an impressive eighty pages on Jesuit mathematical pedagogy, especially the mathematical program of Christopher Clavius and Descartes's training therein.

The second half of the work outlines the development and fate of mathesis universalis from antiquity into the twentieth century. Focusing on interpretations of this "universal mathematics," Sasaki tracks the relationship of mathematics and metaphysics according to Aristotle and his commentators from Alexander ofAphrodisias through ibn Rushd and Aquinas. Turning to the Renaissance, he details both scholastic discussions, notably those of the Jesuit Pedro da Fonseca, and more Platonic ones, notably those of Francesco Barozzi and Adriaan van Roomen, a likely and important influence on Descartes. (The discussion of van Roomen is a highlight of the book.) For most of Descartes's contemporaries, Sasaki persuasively argues, "mathesis universalis" was "primarily not a philosophical concept but a mathematical one" (394). The phrase referred to a more basic mathematical discipline prior to other mathematical disciplines such as geometry. Descartes's immediate predecessors never became "totally free" of the scholastic claim that mathematics depended on metaphysics for its first principles (356). For Descartes, in contrast, mathematical principles were evident in themselves. No longer logically and epistemically subordinate to metaphysics or logic, mathematics could become properly autonomous and a model for other intellectual inquiry. For Sasaki, Descartes spurred the movement toward a new form of mathesis universalis, modeled on algebra, capable of serving as a universal tool of discovery and judgment. Sasaki concludes by examining the attempts of Leibniz and twentieth-century thinkers to perfect such formal tools.

In considering each major example of Descartes's mathematical work, Sasaki argues that Descartes's innovations were algebraic in character, despite their often geometric trappings. "With Descartes," he argues, "the paradigm procedure of mathematics became mechanical calculation with algebraic symbols. His method of mathematical inquiry consisted in an individual manipulating signs, i.e., a non-verbal activity" (422). Much recent scholarship has qualified such claims. Somewhat lost in Sasaki's account are the geometric goals and geometric standards of simplicity and acceptance central for Descartes, which H. J. M. Bos has elegantly described in a series of important articles and his recent monograph Redefining Geometrical Exactness (2001). Also lost at times are Descartes's concerns that the mechanical manipulation of algebraic and other symbols could too easily supplant genuine reasoning using clear and distinct intuitions — discussed by Stephen Gaukroger among others. Although he refers to numerous recent discussions about the relationship of geometry and algebra in Descartes, especially those of Bos and Paolo Mancuso, Sasaki does not weigh these various interpretations of the mathematics with same depth and rigor he brings to many other contentious questions in Descartes's philosophy, such as the difficulties around the term mathesis universalis in the Regulae of the 1620s.

While the book at times loses focus in covering so many topics, its loose structure accommodates numerous compelling discussions, including hitherto untranslated letters concerning the Jesuit mission to Japan, a detailed footnote sketching the introduction of analytical mathematics in China and Japan in the nineteenth century...