Nick Winters, Northwestern University
Whenever two groups working in the same field (be it mathematics or any other science) seem to contrast sharply with one another, an investigation into points of similarity is merited. In this paper, I will undertake an investigation of one such point in the history of Greek mathematics: the hybrid systematist-heurist texts. It has been shown in a recent dissertation (Winters 2020) that two distinct schools of Greek theoretical mathematicians, dubbed “systematists” and “heurists,” can be observed in the textual record. These schools, in addition to behaving as synchronic and diachronic professional networks, used different methodologies, vocabularies, and stylistic conventions in their work.
The two schools coalesced during the Hellenistic period and remained distinct and indeed occasionally hostile to one another down into late antiquity. Certain authors, however, produced texts that I describe as “hybrid,” which contain within the same body of material some passages in the systematist style and others in the heurist. This paper will attempt to explain why they did so and what the hybrid texts reveal about social tensions between the two schools regarding issues of professional legitimacy and authority.
Remarkably few hybrid texts exist. Excluding elementary introductions, handbooks, commentaries, compilations, and artificially composite works by multiple authors (all of which have obvious reasons for including both systematist and heurist passages, and are not therefore interesting as hybrid texts), I will survey the only three that I consider to be genuine systematist-heurist hybrids: Archimedes’ Quadrature of the Parabola (3rd c. BCE), Hypsicles’ Anaphoricus (2nd c. BCE), and Ptolemy’s Almagest (2nd c. CE). Archimedes develops a highly original heurist method of geometric problem solving, then re-demonstrates his most important results independently in systematist style. Hypsicles begins in the systematist mode with three arithmetical demonstrations, then switches abruptly to the heurist style to exposit method for calculating zodiacal rising times. Ptolemy lays the groundwork for his table of chords by interspersing systematist propositions with heurist-style calculations that illustrate each proof. These texts deploy both schools’ practices in different ways, but all apparently to a similar end: the legitimization of heurist innovations through an appeal to established systematist norms.
Systematist mathematicians seem to have understood themselves as an authoritative elite. They show little indication of interest in heurist methods, and no need to validate their own practices in heurist terms. On the other hand, the hybrid texts show that as early as Archimedes, but consistently throughout antiquity, heurist mathematicians were positioning themselves as innovators against a systematist establishment. Drawing on the scholarship of Acerbi, Berrey, Horky, Netz, Taub, and others, I will argue that this asymmetrical rivalry between the mathematical schools is comparable to patterns of professional agonism in other areas of Greek intellectual life, such as philosophy and medicine. The broader goal of this line of inquiry is to show that theoretical mathematics was not constrained within an isolated circle of disinterested researchers but was fully integrated into the social (and indeed economic and political) dynamics of Greek intellectual culture.