Laura E. Winters
Contained in Book XIV of the Palatine Anthology, among riddles, enigmas, and oracles, are forty-four epigrams that pose mathematical problems. While they may appear at first to be frivolous or even trivial both as poetry and as mathematics, these fascinating pieces lie at a crossroads of Greek intellectual culture, where theoretical and practical mathematics, elementary education and elite intellectual recreation all intersect. Similar epigrams in the works of Archimedes, Diophantus, and others provide a glimpse into the origins of these poems and the cultural orientation of Greek mathematics.
In this paper, I will briefly review the literary and mathematical features of the epigrams in the Palatine anthology, before discussing the significance of the genre to practicing mathematicians. I will argue that the engagement of Archimedes and Diophantus with arithmetical epigrams is evidence of a tradition in which recreational mathematics actively interfaced with the work of serious practitioners.
The most salient feature of the epigrams in mathematical practice is that, although they span from the third century BCE to the fourth century CE, they are only ever associated with one mathematical school. It has recently been demonstrated that there were two primary traditions of Greek theoretical mathematics: the systematist school, epitomized by Euclid, whose methodology was proof-based, generalizing, and highly idealized; and the heurist school, whose methodology focused on flexible and iterative problem-solving and transferrable specifics (Winters 2020). Of the several mathematicians who engaged with epigrammatic poetry, not one worked in the systematist tradition.
Why should only the heurist school take an interest in mathematical epigrams? It is not the case that the systematists were unconcerned with arithmetical theory. Both Euclid and Apollonius of Perga wrote on the subject, and Nicomachus’ Introduction to Arithmetic is a systematist text in outlook, though methodologically rudimentary according to its intended audience. Nor is it the case that the systematists had no interest in poetry. But when systematist mathematics does involve poetry, it is in the style of epic (such as Apollonius’ hexameter lines in his untitled arithmetical work in Pappus' Synagoge) or didactic (such as Aratus’ poetic adaptation of Eudoxus’ astronomical work, the Phaenomena).
I will argue that the reason for this division is that the mathematical epigrams align specifically with the methodological interests of the heurist school, but not the systematist. That is, these poems invite the reader to experiment with a style of problem-solving that requires both literary and mathematical acumen. The problems, when solved, do not prove universal principles as the systematists do in their work—rather, the process of problem-solving itself becomes a form of transferrable knowledge and an algorithm for generating new methods of solution. This is the core of heurist mathematics.
The mathematical epigrams show that the interests of practicing mathematicians were in conversation for centuries with areas of intellectual life such as education and recreation. In short, mathematical theory was not isolated from the rest of Greek intellectual culture, but was part of its literary and social traditions.