The Transformation of Mathematics in the Early Mediterranean World

Abstract

The transformation of mathematics from ancient Greece to the medieval Arab-speaking world is here approached by focusing on a single problem proposed by Archimedes and the many solutions offered. In this trajectory Reviel Netz follows the change in the task from solving a geometrical problem to its expression as an equation, still formulated geometrically, and then on to an algebraic problem, now handled by procedures that are more like rules of manipulation. From a practice of mathematics based on the localized solution (and grounded in the polemical practices of early Greek science) we see a transition to a practice of mathematics based on the systematic approach (and grounded in the deuteronomic practices of Late Antiquity and the Middle Ages). With three chapters ranging chronologically from Hellenistic mathematics, through late Antiquity, to the medieval world, Reviel Netz offers an alternate interpretation of the historical journey of pre-modern mathematics.

The Unattainable Attempt to Avoid the Casus Irreducibilis for Cubic Equations

“…80 Moreover and in any case, no explanation on the different status of (AMA XXXIX.6) is given. But he does not explain in which sense a complete equation "depends" on, or "is equivalent" to, or "is composed" by two other equations.…”

“…80 Moreover and in any case, no explanation on the different status of (AMA XXXIX.6) is given. 80 We cannot always pass from a complete equation to three-term equations by transformations, since also quadratic equations are involved. 77 See above, after (AMA XXII.iv).…”

A Historically and Philosophically Informed Approach to Mathematical Metaphors

“…16-18), and are quickly set aside. When the algebraic problems are interpreted geometrically, the reasoning is highly classical, using Euclidean, Archimedean and Apollonian inferences (except for the conflation of numbers and lines; see Netz 2004 for an analysis). Moreover, in the solutions, division into cases is based on geometric, rather than algebraic considerations, and even numerical examples undergo a highly geometric analysis.…”

Medieval Arabic Algebra as an Artificial Language

“…75 On the Sphere and Cylinder, Proposition II.4. See Netz (2004) for a discussion of the history of this problem. 76 Woepcke (1851, p. 2;7).…”