New perspectives: differentiating cultures in ancient mathematics
Agathe Keller
Fanglei Zheng

This symposium provides reflective case studies on ancient mathematical cultures. Thinking of culture as not determined by language, place, race or religion but produced by work collectives as they share practices and epistemic values related to these practices, and wanting to overcome homogenizing representations of «Chinese», «Egyptian» or «Greek» mathematics, we invite papers which describe a plurality of ways of doing and thinking mathematics in ancient times and within groups writing in a same language, living in a same region or sharing a religion. Recent studies have shown that Akkadian texts could testify to different conceptions of numbers, or different ways of making diagrams, ancient Greek, Latin or Chinese mathematical sources of different modes of proofs or ways of carrying out procedures. Ancient mathematical sources have thus borne witness to previously unsuspected varieties. In which way then did collectives shape specific ways of doing mathematics in ancient times and how can we account for these differences? Can we differentiate how different authors in similar language sources or in a same place thought of numbers and of their relations to measuring units and operations? How they thought differently of objects in geometry? How they thought or used proofs? Can we ascribe different types of sources to different working collectives? Are practices always so easy to differentiate? These are some of the questions we hope our symposium can tackle.

Novelty and Motivation:

The research involved in this symposium rests on a new understanding of mathematical cultures as a way to deeply transform the study of ancient mathematical sources. Emancipated from Eurocentric nationalist historiographies forged in the 19th century- our symposium deals with a great variety of ancient mathematical sources (Latin, Sanskrit, Akkadian, Arabic, Chinese, English, Greek.…)-the texts approached in this way yield new clues to the social and intellectual contexts in which they were produced and used. All sorts of new mathematical concepts and practices are also brought to light: different modes of quantification, a variety of instruments and computational devices, diverse practices of proofs, etc.

Part 1

MIDDEKE-CONLIN Robert (University of Copenhagen)

Variety in a uniform tradition: A comparison of metrology and mathematical education in Old Babylonian sources

Cuneiform mathematics is often perceived as a single, uniform entity. This unity was practiced in the same manner throughout Mesopotamia and beyond. While it’s admitted that there is regional variety in sign use and writing, modern works often treat mathematical activity in Mesopotamia as uniform. This is the case whether, for instance, a text was produced at Nippur in the heart of Babylonia, at Eshnunna of the Diyala region, or at Susa in modern Iran. Challenging this view, the current presentation will explore variety within the cuneiform mathematical tradition. It will propose the examination of three kinds of texts, based on their production source: texts of student practice, texts of erudite practice, and texts of professional practice. These distinct text types are chosen to isolate distinct mathematical practices based on region and profession. As a case study, metrology within the Old Babylonian Kingdom of Larsa (the early second millennium BCE in southern Iraq) will be examined. Metrological values found on lists and tables from three distinct cities (Larsa, Ur, and Nippur) that were memorized early in a scribe’s education are explored and compared to values found on both mathematical and administrative texts to show that there were different, distinct schools within this kingdom with their own distinct metrological traditions and that these traditions had an affect on mathematical practice of the actors produced by these traditions. Thus, the listener will discover a distinct regional variety in cuneiform mathematics as well as the significance of this variety for individual practitioners. 

KELLER Agathe (Sphere, CNRS & Université de Paris Diderot)

Cultures of quantification and computation as testified by the Śulbasūtras

The oldest mathematical texts that have been handed down to us from South Asia, collectively known as the śulbasūtras (Chord treatises), are a group of texts of different authors belonging to separate schools of Vedic ritual. The oldest such texts the Baudhāyana or Apāstamba śulbasūtras might have been composed around 600 BCE while others such as the Hiraṇyakeśi, Mānava and Kātyāyana could have been devised and re-edited after that and as late as the 3rd century CE. These texts contain procedures and definitions related to the construction of ritual spaces, sacrificial altars but also general mathematical considerations. They often have common parts, and sometimes quote one another. The śulbasūtras were first edited and translated into English in the late 19th century and further studied and re-edited through out the 20th and 21st century in the works of historians of mathematics such as B. Datta to more recently J. M. Delire. In contrast with a historiography of mathematics that overall tends to view their mathematical contents as being homogeneous, this paper will look closely at the different ways in which constructions of the areas of sacrificial grounds may rest on different practices and maybe conceptions of the relations of numbers and measuring units to elementary geometrical figures (right triangles, rectangles and trapeziums). As such then we hope to investigate the different practices of quantification in relation to geometrical figures and of computation in geometry as testified by different śulbasūtras.

CAO Jingbo (Shanghai Jiao Tong University)

An Analysis of the Double-Fourteenth Book in Billingsley's Translation of Euclid’s Elements

Henry Billingsley published the first English translation of Euclid’s Elements in 1570. His work included a lot of additional “corollaries”, “demonstrations”, “assumpts”, “propositions”, and notes from forty scholars. Billingsley’s translation contained Books XIV and XV, which Hypsicles of Alexandria is supposed to have added to the Elements, and which discuss the ratio between Platonic Solids when one is inscribed or circumscribed in the other. It is worth noting that Billingsley put two different Books XIV in his translation, which are called 14A (Hypsicles, Zamberti and Flussates) and 14B (Hypsicles, Flussates and Campanus) for convenience. The double-fourteen book sheds light on two different aspects of Billingsley’s translation, on which this talk will focus. Firstly, Billingsley paid a lot of attention to the research on Euclid’s Elements carried out by previous translators. His edition attempted to unify the various traditions of the study of the Elements that existed at the time. These included the tradition embodied by the Arabic works written during the Abbasid period, the Latin tradition in the Renaissance and Hermetism. Secondly, the revival of Platonism had a noticeable influence on Billingsley’s attitude towards the interpretation of solid geometry. For example, he proposed that "number" and "quantity" could work together to interpret the significance of solid geometry in natural philosophy, as well as the status of solid geometry in the structure of mathematics.

PAN Shuyuan, (Institute for the History of Natural Sciences, Chinese Academy of Sciences)

What is “Multiplying by the Different and Dividing by the Same”? Differentiating Two Practices and the underlying Epistemic Principles in the “Rule of Three” Procedures in China

When the Jesuit Matteo Ricci and his Chinese collaborators introduced the “rule of three” from Europe into China in early 17th century, they identified this proportional procedure with “Multiplying by the different and dividing by the same” (yicheng tongchu 異乘同除, hereafter MDDS), a method which had been discussed in Chinese mathematical texts for about four centuries. What does this name exactly mean? The procedures that correspond to this name in those Chinese mathematical texts present differences and this fact suggests that there existed two approaches to determine “the different” and “the same”. Moreover, a procedure named “suppose” (jinyou) in the ancient Chinese mathematical canon from about the beginning of the common era, Mathematical Procedures in Nine Chapters (Jiuzhang Suanshu), was also a version of the “rule of three”, from which the method MDDS derived. Philological evidence from the original text and the commentaries of this canon shows that there also were two different practices of operating and employing the “suppose” procedure, and two corresponding considerations on the relationship of those involved objects. Drawing on these investigations, we will highlight the two practices of the “rule of three” procedures in China and show that their underlying epistemic principles enjoyed some continuity from scholarly writings to popularized texts, and from ancient to medieval times. Further, this study will cast light on why and how Chinese scholars in the 17th and 18th centuries integrated the introduced mathematics into their knowledge tradition — through a shared mathematical culture.

Part II

Lu Peng

Using the Square or Using the Circle?

Different Proofs on the “Broken Bamboo” Problem

Many parallel problems are found in ancient Chinese and Sanskrit mathematical texts. One typical case of them is the so-called “Broken Bamboo” or “Sinking Lotus” problems. The descriptions of the problems and the rules attached to them are extremely similar in Chinese and Sanskrit sources. This has been well noted in the historiography, and it has been assumed that Chinese and Sanskrit sources may have shared similar circulating mathematical practices. However, if we check the verifications/proofs of the algorithms carefully, we find that not all authors proceed in the same way. Liu Hui, the 3rd century commentator of the Nine Chapters, uses a right triangle which he then changes to a “square”. Bhāskara I, the 7th century commentator of the Āryabhaṭīya, explained the rule by using the relationships of arc, arrow and radius of “one’s own circle”. Moreover, later the same problems and rules are found in other Sanskrit works.  It is interesting that the 9th century Pṛthūdaka explains the rule in a way similar to Bhāskara I while the 12th century Bhāskara II uses the three sides of the right triangle, which "seems like" Liu Hui. In the presentation, we re-examine a tradition which considers a culture of Chinese and Indian mathematics, to rather focus on the different practices shared or not by different authors. 

ZHENG, Fanglei (Department of History of Science, Tsinghua University)

How many mathematical cultures are there in the works of Fibonacci?

An alternative perspective on differentiating cultures in mathematical practices

Fibonacci’s works, especially his Liber Abaci, inherited mathematical knowledge from different traditions. Although in most cases the source could not be identified as specific books, our forerunners (Folkerts, Miura, Høyrup, etc.) have convincingly demonstrated the connections between certain parts in the work of Fibonacci to different cultures in a sense that is similar to civilizations: Greek, Indian, Arabic, Byzantine, etc. However, in the present paper, mathematical culture is conceived as a way of doing mathematics. Taking propositions from Fibonacci’s works as examples, I will not only enumerate the different ways betrayed by the texts from different sources but also analyze his various treatments on particular same (would-be from the viewpoint of modern mathematics) pieces of mathematical knowledge. Furthermore, some conjectures will be made, on the one hand, concerning the milieus for the different ways of practice we saw, and on the other hand, concerning the groups of addressees that Fibonacci was aiming at with different types of presentations.

Karine Chemla SPHERE (CNRS & Université de Paris)

Mathematical cultures according to observers and to actors

The historiography of number systems and arithmetic

The historiography of number systems and arithmetic has been for its greatest part organized according to nations and linguistic groups. This holds true, for example, for Geneviève Guitel’s Histoire Comparée des Numérations Ecrites (1975), despite the fact that her focus lies elsewhere: she aims to show that all written number systems can be described and classified according to a few general principles. By concentrating on the example of Guitel (1975), this talk intends to identify recurring problems in the historiography of number systems that have more broadly underpinned the view that national or linguistic groups constituted the relevant frame of analysis. By contrast to observers’ approach to number systems, such as Guitel’s, I will show that this is not the way in which, in his Kitab al-Fusul fi al-Hisab al-Hindi (Saidan 1978), the 10th century practitioner of mathematics, al-Uqlidisi perceived the divide between different number systems and their collectives of users. Drawing on his account, the talk suggests that we derive a research program about number systems and arithmetic that might allow us to highlight the different cultures of computation that coexisted in ancient societies, as well probably as in modern ones.

References

Guitel, Geneviève, 1975. Histoire Comparée des Numérations Ecrites (Paris: Flammarion, 1975).

Saidan, Ahmad S., 1978. The Arithmetic of al-Uqlidisi. The Story of Hindu-Arabic Arithmetic as told in Kitab al-Fusul fi al-Hisab al-Hindi, by Abu al-Hasan Ahmad ibn Ibrahim al-Uqlidisi, written in Damascus in the year 341 (A.D. 952/3) (Dordrecht: D. Reidel, 1978).

WU Yan, Zhihui CHEN (Institute for History of Science and Technology, Inner Mongolia Normal University, Hohhot, China)

19th Century French Sinologists's observations on the Chinese abacus and its cultural background

Paul Perny, Terrien de Lacouperie, Arnold Vissière, and Léon Rodet, who were French sinologists in the 19th century, investigated the notation and operation for the Chinese abacus and made some observations. They praised the Chinese abacus as an ingenious calculating device which had the merit of allowing practitioners to operate quickly in comparison to the pre-modern European abacus. However, they also suggested that the slower calculating instruments in Europe facilitated the development or adoption of a set of notations for calculating and were more useful for arithmetic researches and demonstrations. In this paper, we are examining how their various mathematical cultures, anchored in various 19th century's contexts -- defined by the ways of thinking and doing mathematics, especially by its preferences in evaluating mathematics -- affected their observations. We will also discuss briefly how historians re-considered the textual evidence about ancient mathematics, that is, how they approached the influence of a material instrument -- the abacus, in this case -- on a mathematical textual practice, as well as the thoughts connecting to it.

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