Descartes ’ Philosophical Grounds for Algebra and Geometry

Cathay Ming Lih Liu: Descartes' Philosophical Grounds for Algebra and Geometry (Under the direction of Alan Nelson) Though neither Descartes nor Fermat developed analytic geometry as we understand the mathematical discipline today, they each developed a different central feature of it. But history of mathematics has not explained their differences in mathematical aims and practice in terms of philosophical commitments. Generally, Descartes' metaphysics has not played a role in the explication of his mathematical emphasis. While it is undisputed that Descartes placed a lot of emphasis on the geometrical aspect of analytic geometry over the algebraic component, I explain his mathematical practice by appealing to his philosophical views. I argue that Descartes views geometry to be more basic than algebra. Algebra, and its significance, depended on geometrical objects. I have defend this view by appealing to Descartes metaphysical and epistemological commitments, but I also defend this view against a possible alternative interpretation of Descartes' philosophy that would allow algebra not to depend on geometry.