Laura Winters
Ancient Greek mathematics has been associated for centuries with the systematic, deductive style of Euclid’s Elements. The strength of this association has tended to obscure the variety of ancient methodological and stylistic practices. In this paper I shall argue that this variety is not a happenstance result of individual authorial preferences, but that there were two distinct intellectual traditions within ancient Greek theoretical mathematics, each with its own stylistic conventions, problem-solving methods, epistemological priorities, and even preferred vocabulary. Despite their long history and broad cultural penetration, these intellectual traditions have not been recognized as such, largely because of the impact of late ancient philosophy on the transmission of mathematical texts and the telling of mathematical history.
We can see the contours of the two intellectual traditions in a pair of telling examples. Two mathematicians of the third century B.C., Apollonius of Perga and Archimedes, each wrote a work detailing a method for making calculations with very large numbers (i.e. numbers greater than one hundred million). Archimedes’ Sand Reckoner is known for its preservation of a heliocentric astronomical model of Aristarchus, but Apollonius’ unnamed work is preserved only as a fragment in Book II of Pappus’ Synagoge, and has garnered relatively little attention. Both use a verbal shorthand to express numbers too large for existing notation, but in every other way they are dissimilar. Apollonius works in the same tradition as Euclid: he presents a series of distinct propositions without specific numbers, proving general arithmetical principles (Pappus clarifies these with numerical examples). At the end of the work, Apollonius illustrates his method through the calculation of the numerical value of a hexameter verse in which each letter represents a number. Archimedes, on the other hand, presents his material in a continuous narrative that is framed by a specific problem, that of calculating the number of grains of sand required to fill the universe. Demonstrations of general principles are rare, and are interspersed with specific calculations and methodological discourse. Aside from the differences of structure, the two works use differing vocabulary, methods of calculation, and styles of presentation.
Works of the tradition that Apollonius is a part of are characterized by systematic structuring of the text, formulaic presentation of the material, strong concern for generalizing principles and exhaustive treatment of a subject area, and an idealized, non-physical conceptualization of mathematical objects. This tradition was heavily informed by Pythagorean and Platonist philosophy. In fact, Apollonius’ calculation of the hexameter is an example of the Pythagorean practice of number mysticism and prophecy through arithmetic. Archimedes, by contrast, is part of a tradition that is more flexible in structure and presentation. They are characterized by a problem-based approach to mathematics rather than a proof-based approach, preferring clever and efficient solutions to the formulation of general principles. They tend to conceptualize mathematical objects more physically, subjecting them to measurement, motion, and other types of specification.
These two traditions sharply differentiated Greek schools of thought on mathematics from the classical period into late antiquity. Their cultural influence extended far beyond the boundaries of mathematical treatises, into poetry, music, religion, and the other sciences. During late antiquity the homogenizing tendencies of encyclopedic compilation and neo-Platonist philosophy among scholars blurred the distinctions between the schools, and the tradition of Euclid and Apollonius came to eclipse that of Archimedes. Modern scholars have inherited a history of Greek mathematics that is in many ways more reflective of late antique scholarly activity than of ancient categories of thought. In recognizing the two traditions, we have the opportunity to examine our own narratives about mathematics and its history with a clearer understanding of their origins, to reassess the impact of philosophy and scholarship on mathematical conventions and methodology, and to illuminate the true variety of ancient mathematical practice.