Progress in mathematics: Descartes' 'Geometry'

This dissertation gives an account of the goals of inquiry of Descartes's 1637 Geometry. This account of the goals of inquiry is based on historical study of what is actually achieved by the mathematical work done in the Geometry. This account of the Geometry is a contribution toward a naturalistic understanding of mathematical inquiry in general.; This dissertation first examines an account by Putnam of goals of mathematical inquiry. This dissertation argues that this extension of a realist view of mathematics results in a contrived interpretation of the achievement of the Geometry and is rejected. Instead this dissertation argues that Kuhn's view of the historical development of inquiry can be interpreted as an account of the goals of scientific inquiry and this account can be extended to the mathematical inquiry of the Geometry.; This dissertation explains this extension of Kuhn's account of inquiry by examining the variety of solutions to a single long-standing and relatively simple mathematical problem. With respect to the most general goals of mathematical inquiry, Descartes's solution of the cube duplication problem is at the midpoint in a development extending from the ancient Greeks to a contemporary abstract view.; This account of mathematical inquiry is also extended to a study of the applied mathematics of Descartes's geometrical optics. This account of applied mathematics circumvents the problem of understanding the ontology of the mathematics involved.