Quadrivial Pursuits: Case Studies in the Conceptual Foundation of the Mathematical Arts in the Late Middle Ages.

摘要

The quadrivium, the four mathematical disciplines of the Middle Ages, described the structure of the medieval cosmos, both macrocosm and microcosm. Arithmetic and music were the mathematics of Platonic counting numbers. Geometry and astronomy were the mathematics of continuous magnitude. All four disciplines worked in concert to describe a cohesive and harmonious universe, which in the late Middle Ages incorporated everything from Aristotelian elemental theory to astrology. This dissertation describes the early philosophical formulation of these disciplines from Pythagorean and Platonic roots and the foundation of the quadrivium itself in the mathematical writings of Boethius in the early sixth century. This dissertation then examines the mathematical philosophy of three late medieval authors who were proficient in the quadrivial arts: Nicole Oresme (ca. 1320--1382), Prosdocimo de' Beldomandi (ca. 1375--1428), and Leon Battista Alberti (1404--1472). All three demonstrate that the Boethian quadrivial philosophy continued to be relevant in the late 14th and early 15th centuries, but all three studies point to a significant fault line in the metaphysical structure of the quadrivium itself---the distinction between discrete and continuous, the quadrivial distinction between arithmetic and geometry.