The history of Greek mathematics, like most ancient histories, has always suffered from a tendency to over-generalize the practices of a given society based on its most prominent or well-preserved remnants. The Euclidean style of deductive proof has loomed so large in the textual record as to obscure the significant variations in mathematical method, conceptualization, standards of verification, and style of presentation among the Greek authors. My research aims to illuminate these variations and to trace the networks of schools and traditions which cultivated them. In this paper, I will present an illustrative example of one such tradition: the Pythagorean school.

Though relatively famous for their interest in mathematics, the early Pythagoreans left little to no written evidence of their practices. However, certain terms and concepts fossilized in later works (neo-Pythagorean treatises and commentaries) can help us to reconstruct a highly localized and specialized tradition of thought. I will show that the Pythagorean school’s early association with Platonism and subsequent entanglement with Hermetic mysticism can be seen in the isolated, in-group usage of certain number-names and a particular method of expressing and calculating ratios and multiples. On a broader scale, this type of reconstruction consistently reveals that ancient mathematical practices were localized within social networks to the point of being nearly proprietary, and that there is a great deal of intellectual history to be learned when generalizing ideas of mathematical content are not permitted to obscure the particularities of method and form.

Cuneiform culture is often spoken of as a monolithic system. This is especially true of its education system, and in many ways it is true: Lexical lists, proverbs, and contracts can seem similar, if not the same from place to place. At the same time the numerous metrological and numerical systems used to assess value and perform complex calculations renders the impression of uniformity. During the Old Babylonian period, roughly between 2000 and 1600 BCE, the cuneiform knowledge system is witnessed by numerous, seemingly similar clay tablets, reinforcing the impression of monotony. If you’ve seen one tablet, you’ve seen them all. We wonder, how uniform is this culture and system? Do we see variety between time and place?

Inspired by the theme of this year’s conference, this presentation explores the local expressions of knowledge in a seemingly globalized system. It focuses on the education institution of the day – the eduba – in cities throughout Babylonia (modern southern Iraq). It examines variety in three ways: first by focusing on the document itself, the tablet, and then the two pillars of cuneiform education, translation and numeracy. It will mine texts from the different iterations of this institution to isolate different scholastic identities. Cuneiform culture will be shown for what it is: a tapestry of local knowledge informed by and informing a global system. The attendee will see variety of voices in a very uniform system.

Mullā Muḥammad Bāqir Zayn al-ʽAbidīn al-Yazdī, a noted Iranian mathematician, was contemporary to the Safavid period. Yazdī in his Sharḥ al-Maqālat al-ʽAshirat min Uṣūl Uqlīdus, has compiled a commentary on 67 propositions of Euclid's Elements. Yazdī, aside from propositions 1, 2, 3, 4, 12, and 14, provides commentary up to proposition 62; thereafter, he only discusses propositions 67, 70, 74, 76, 77, 78, 79, 80, and 81.
In general, it can be said that Yazdī's commentary includes the following:
1. The presentation of many new propositions related to the propositions of the tenth book or in connection with Ṭūsī's explanations.
2. Critiques of Ṭūsī's sentence structure to clarify the meaning.
3. An explanation regarding the naming of certain lines.
4. Mathematical explanations are very useful throughout propositions like: Proposition 2, 6 and 10.
5. An important mathematical discussion in some propositions like: Proposition 1, 9, 23, 25.
Yazdī follows the same tradition as Euclid. Although he provides numerical examples for the various types of irrational magnitudes discussed, he does not mention irrational numbers and he does not introduce compound irrationals composed of more than two terms. Except for proposition 1, where he raises an important challenging discussion and also in proposition 23, he introduces the higher-order irrational magnitudes, in none of the main discussions of the 10th book, Yazdī does not employ any innovation of his own and only provides very useful explanations. In this presentation I’m going to talk about his commentary in detail.