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SAW Report Summary

Project ID: 269804
Funded under: FP7-IDEAS-ERC
Country: France

Final Report Summary - SAW (Mathematical Sciences in the Ancient World)

The general objective of the SAW project was to develop historiographic tools enabling the perception of diversity in the mathematical practices and bodies of knowledge to which the available Chinese, Sanskrit and cuneiform sources from the ancient world attest. To achieve this aim, we have first focused on documents related to administrative and economic activities (Phase 1). We have then explored the context of astral sciences (Phase 2). Finally, we have dealt with the modern history of historiographies of ancient mathematics worldwide (Phase 3). In parallel with case studies, we have developed methodological and historiographic reflections, in particular on the notions of “practice” and “culture”, now widely used in the history and philosophy of science.
During Phase 1, we have uncovered sets of sources as yet unexplored in the history of mathematics, thereby shedding light on cultures of quantification and computation so far unknown. This conclusion derives, in particular, from analyses of the various ways of working with numbers / quantities and carrying out basic arithmetical operations on them to which these documents attest. Notably the emergence, spread and diversity of place-value number systems and their uses has been documented in a new way.
Concerning China, new approaches to ancient mathematical sources, and Zhu Yiwen’s study of commentaries on Confucian canons brought to light previously unstudied mathematical practices different from those to which mathematical canons attest. Some of the new approaches adopted have an impact on the field of history in general. For instance, Morgan’s handwriting analysis of manuscripts belonging to a single tomb enabled the identification of a network of hands, changing the approach to the entire corpus and the interpretation of its mathematical manuscript. We also explored the tight relationships between administrative regulation texts and the earliest extant mathematical sources, hence providing a new interpretation of the state’s grain management.
The study of new corpuses of economic documents from Southern Babylonia, in the early 2nd millennium BCE, notably in Middeke-Conlin’s dissertation, has enabled the description of mathematical practices at different levels of the bureaucracy. More largely, we have undertaken studies of the mathematical knowledge and practices involved in economic transactions and bureaucratic activities. We clarified how metrological tables established relationships between three types of quantification of spatial extension: volume, capacity and bricks quantities. Specific weighing and surface quantification practices in 2nd millenium Assyria and in Early Dynastic tablets from southern Mesopotamia were also brought to light.
For Sanskrit sources, new studies based on legal and administrative texts have started to unravel and complexify historiographies that had created homogenous metrological systems. We established that testimonies on operations in medieval Sanskrit sources showed variations in the execution of multiplication on integers, opening a larger venue into the description of operations in these texts.
Phase 2 of the SAW project (the study of mathematical practices in the context of astral sciences) also made new corpuses of sources available (Hirose’s critical edition, translation and analysis of the Sanskrit Goladipika, composed by Parameśvara around 1430-1450; an edition of cuneiform astronomical and mathematical tablets from Hellenistic Uruk). Issues addressed to perceive the diversity of mathematical cultures, and so far overlooked in the history of the ancient astral sciences owing in particular to widespread historiographic practices of converting texts into modern counterparts, included the shaping of operations, quantities and measurements units. Progressions and ordered lists of numbers proved essential mathematical tools in the astral sciences of ancient China and Mesopotamia, and could help connect sources in astral sciences and mathematical writings. Mathematical reasonings, in the context of the astral sciences, and how they put space and time into play; instruments, and the mathematical practices shaped to use them or work with them; and the contrasts between and syntheses of mathematical cultures were shown to be crucial, notably in Chinese, Sanskrit and Latin sources.
During Phase 3, theoretical work on mathematical (and more generally scientific) “practices” and “cultures,” showed the fruitfulness of concentrating on epistemological values and their historiography, on the variety of types of texts produced in the context of different scholarly cultures, and on the issue of how documents reflect the milieus in which they were produced and their mathematical practices. We have collectively developed critical approaches to the sources of ancient mathematics.
Phase 3 also systematically explored 19th and 20th century historiographies of mathematical sciences in the ancient worlds. It proved essential to pay attention to the diversity of milieus in which, from the 18th century onwards, an interest in the history of mathematics of the ancient world emerged. This diversity enabled us to account for specificities of different historiographies, and also to describe how knowledge circulated and was transformed from one milieu to another. Major differences appeared in the history of historiographies of mathematics in the Indian subcontinent, China, and Mesopotamia (Chaigneau, Chen, Dessagnes, Schneider, Zhou).
The research conducted in the context of the SAW project shaped a renewed vision of the field, reflected in articles and chapters of books that members composed for more general audiences. This vision has also inspired de Varent’s PhD dissertation, which explored the uses of ancient mathematical documents coming from everywhere in the world for the classroom.

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