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## Séminaires

Combination of several approaches to achieve our aims

Wider societal impact

**SAW : GENERAL AIMS**

The SAW project is devoted to mathematical sources that have come down to us from the ancient world and, more specifically, though not exclusively, to the sources produced in

**Mesopotamia, China**, and the

**Indian subcontinent**. The ambition of SAW is to develop new theoretical approaches in the field of the history of science in order to highlight a motley of mathematical practices within what at the present day are too often presented as homogeneous wholes, that is, “Mesopotamian mathematics”, “Chinese mathematics”, and “Indian mathematics.” SAW also intends to carry out research on the processes that have been at play in the construction of this homogeneity in the historical discourses about the sciences that have been produced worldwide since the 19th century. In broader terms, the impact of the theoretical results obtained on the basis of the research program described will be explored for the historiography of mathematics in general.

**A motley of mathematical practices**

To achieve its goal and introduce an initial decomposition into smaller entities of the aforementioned “homogeneous wholes,” the strategy SAW adopted is systematic. Whether we look at Cuneiform, Sanskrit, or Chinese sources documenting ancient mathematical practices, in each area, some of the sources adhere to spheres of

**astronomical/astrological**activity, whereas others appear to be closer to the workings of administrations and institutions dealing with

**administrative and financial matters**. SAW intends to focus on sources specifically adhering to either domain of activity and to detect, by contrasting them with each other, specificities in the practice of, and knowledge in, mathematics to which they bear witness.

The project aims to highlight how one can identify, in these three geographical areas, different

**mathematical practices**and bodies of

**knowledge**, anchored in specific

**social and professional contexts**. At a theoretical level, the fulfillment of this objective will require a critical reflection on the concept of a “culture of scientific practice”—to be distinguished from more essentializing uses of the word “culture.” This reflection will be carried within the domain of history and philosophy of science at large. Moreover, we shall have to address the issue of how different cultures, in a sense to be determined, connect with each other. We propose rather to scrutinize how they seem to partially “overlap” with each other. We are indeed confronted with local cultures of mathematical practice that are not confined within sealed boundaries, but share common knowledge and ways of working.

**Historiography of science : Two types of uniformity**

Due to the prestige attached to science, and also specifically to mathematics, history of science has provided some of the main cultural artifacts that entered into the making of nations. This also held true for empires, “civilizations,” or other kinds of “communities.” Means have been invented to create

**communities**around science and its history. SAW intends to analyze in detail how the connection between the making of communities and the historiography of science was shaped and how the process is still at work in the modern world. Eurocentrism in the history of mathematics was, at least partly, a product of this phenomenon. This factor has also, most probably, played an important part in maintaining Eurocentrism in its dominant historiographic position until today. All over the world, at the present time, communities claim their value on account of “their contributions” to mathematics, unless they claim their difference on account of the “specific kind of mathematics” that is suited to them.

The main objective SAW set itself was determined by the following observation. The historiographies that are affected by these phenomena present mathematics as a discipline that shows an overwhelming uniformity according to two very different models. Following the

**first**approach the nature of mathematics would not have changed at all in the course of history—this is what we shall call “global uniformity”. In such context the various communities are characterized by feats of being the “first” to introduce a concept or obtain results. According to the

**second model**, the idea has been advanced that mathematics is portrayed as practiced in a specific manner depending on the “nation” or the “civilization” considered. This introduces, then, a “regional uniformity” in which different avatars correspond to what is designated by the expressions “Western,” “Chinese,” “Indian” or “Babylonian” mathematics. The “regional uniformity” is the one that has probably the tightest connection to the making of “communitarian” historiographies, which can be identified at work in some trends in all the societies of our global world. SAW will devote some effort to the history of history and philosophy of science to shed light on the history and the uses of these two types of uniformity.

**COMBINATION OF SEVERAL APPROACHES TO ACHIEVE OUR AIMS**

To fulfill its objectives, SAW will develop several research programs.

**A theoretical program in history and philosophy of science : “cultures of mathematical practices”**
The conceptual attempts to grapple with the diversity of scientific practices in a systematic way were conceived in the course of researching the history, sociology or philosophy of sciences bearing mainly on contemporary, or modern at the earliest, knowledge. They overwhelmingly consider only European or North-American source material. Mathematics plays a minor part, if a part at all, in all three. In the last years, and through an engagement with ancient Chinese sources, K. Chemla has shaped a first set of tools suited to approach ancient mathematical sources as follows. She has suggested to decompose “knowledge machineries” which these documents attest to into elements (problems, algorithms, diagrams, kinds of text and inscriptions used, and so on). Moreover, she has collected evidence which permits the historian to reconstruct the way in which the actors worked with these elements (“practices with these elements” or “elementary practices”). In this respect, as E. Fox Keller had suggested, epistemological factors proved to be essential : what epistemological values did actors prize ? What were the objectives actors assigned to, for instance, mathematical proof ? It also appears as essential to introduce the specific practices that actors have shaped with these “epistemological elements.” In fact, their epistemological practices permeate the practices with other material elements. For example, in ancient China, the way generality is practiced echoes into how problems like algorithms are written as well as how diagrams are designed. Raising these kinds of questions enables a description of the nexus of practices with elements, into which a given way of doing mathematics can be analyzed. Such an analysis is crucial to compare and distinguish between different ways of doing mathematics. A network of relations links these "elementary practices" with each other, the whole set thereby forming what K. Chemla suggests should be understood with the expression "culture of mathematical practice. These tools are discriminating enough to approach more extensively different corpuses of ancient mathematical sources worldwide and identify different practices.

However, these concepts still need theoretical refinement. This research will be conducted within history and philosophy of science at large to benefit from the insights that different sets of sources can provide. We will devote a first critical effort to bear on the concept of “practice” itself. Even though it is now quite widespread in history and philosophy of science, there is neither consensus on the meaning of the term nor on how to describe a given scientific practice. We shall develop a collective research to analyze in a critical way various concrete uses of the concept that appear meaningful and determine what they can contribute to a theory of “practice.” A second effort will be devoted to the question of how different practices connect to each other. The hypothesis which SAW will test is that, far from forming disparate entities, different socially situated cultures of mathematical practices formed, rather, a continuum with specific poles and common regions. We therefore hope to forge new theoretical tools in order to elaborate an approach to “cultures without culturalism.

Seminar Practices

**Research in history and philosophy of mathematics : Quantities and operations**

The mathematical sources on which SAW focuses give quantities and operations a central role. It is thus essential for the project to concentrate on these elements in order to develop tools enabling us to identify differences among sources. We shall examine the actual quantities on which the sources describe or prescribe operating. The project will analyze the sets of operations to which distinct sources bear witness and explore the organization of these sets to which the writings testify. We shall concentrate on the ways in which these operations were executed and especially on the layouts shaped for the calculation. We shall attempt to capture the theoretical questions that were raised about these operations : Can one, for instance, identify an interest in fundamental operations or in elementary operations in our sources ? How can one account for a shift in the organization of sets of operations ? Lastly, how was the correctness of the algorithms executing these operations addressed ? These are some of the questions that will lie at the center of the inquiry, which we shall carry out on a broad set of sources selected among documents written anywhere and at any time period. The objective will be to develop tools to allow us to understand the differences and evolutions in these respects.

Seminar History and Philosophy of Mathematics

**Mathematics, economy and finance : A joint approach between historians of mathematics and historians of economy and finance**

This research program has several goals. On the one hand, it aims to identify in detail the mathematical work and the kinds of mathematics involved in actions taken by institutions in charge of financial or economic matters. For instance, one of the key operations needed by such an administration is the definition of measuring units. This operation requires mathematical work which is relatively well documented in our sources. SAW will systematically explore this type of information. On the other hand, mathematical sources regularly evoke questions that can be correlated to those addressed by practitioners in charge of economic and financial operations. What is the nature of this parallel ? Does the correlation indicate an actual connection between these practitioners and those who composed mathematical writings ? Or should other hypotheses be adopted ? In fact, we know that in some cases, mathematical sources do testify to a strong connection with milieus of practitioners of economic and financial matters. Further, these sources sometimes provide evidence that complements aspects on which administrative sources remain silent. We point out, for example, the way in which in the ancient world one measured quantities of grain, assessed the surface area of land or the capacity of storage silos ; all operations so essential to the management of the state. This will be the first range of issues we shall address before turning to the mathematical practices that developed in relation to the astral sciences.

Workshop SAW 2013 : Cultures of Computation and Quantification

Seminar SAW 2011-2012 : History of Mathematics, History of Economical and Financial Practices

**Mathematics and Astral Sciences : A joint approach between historians of mathematics and historians of astral sciences**

In addition to looking at the practice of mathematics associated with the description and determination of astronomical phenomena — the motion of the planets, computations of specific moments such as eclipses, risings and settings—we intend to examine how mathematics was involved in the production and use of calendars and almanacs. In relation to these themes, we shall also devote equal attention to the mathematical practices that can be identified in various aspects of musical theory. We will concentrate on this aspect of the project from May 2013 on. However, we will before that time occasionally organize events on this theme.

Seminar 2013-2014 Mathematical practices in the context of the astral sciences

**History of Science, History of Text**

The goal of this research project is twofold. On the one hand, by a minute attention to our sources, we want to be able to grasp the distinct milieus that produced them, on account of material features of these documents. Among these features, we shall explore the terminologies used, the syntax employed for technical statements, and the kinds of abbreviations used. When applicable, we shall naturally place due emphasis on the layout and other physical features of the documents themselves. We shall focus equally on the structure of the texts, on the parts of which they are composed and, when available, to the names given to these parts. All these factors can furnish evidence of the setting in which writings were produced. Further, attention will be paid to how sources bear witness to the material environment in relation to which they are composed (“epitext”) and to the practices with which the elements of this environment were put into play. In particular, we shall examine how sources relate to oral activities to which they adhere. On the other hand, we want to develop working tools that will help the historian of science interpret his or her sources in the most accurate and fruitful way possible. Thus, we will study the different compilation practices which can be identified in the various traditions. The practice of quoting or of making fragments will receive attention. We shall scrutinize the temporal thickness of documents, in our investigation of, for example, the history and analysis of their critical editions. In both respects, observing ancient readers and their practices with these sources will provide an essential tool for our work. In this part of our program too, for a better apprehension of these phenomena, research on textual and critical aspects will be developed in history of science and not simply mathematics

Seminar History of Science, History of Text

**Reading ancient sources linked to mathematics**

In parallel with the research programs described above, we plan to organize a seminar in the framework of which we shall read ancient sources together and shape common tools for their description.

Seminar Reading Mathematical Texts

**WIDER SOCIETAL IMPACT**

The goal SAW has set itself is to produce documentary resources so as to promote new representations of mathematics and its history. The wider societal impact at which the project aims is to define an alternative to widespread visions of mathematics that have made it possible for this field and its history to become tools playing a part in the shaping of communities worldwide. SAW will work to make the outcome of the research available to secondary school teachers, in order to contribute to bridging the widening gap between young people and mathematical sciences.

### 2015-2016 : Seminar Exploring 19th and 20th centuries historiographies of mathematics in the ancient world

ERC (European Research Council Advanced Grant 2010) Project : Mathematical Sciences in the Ancient World (SAW)

### 2015-2016 : Seminar Exploring 19th and 20th centuries historiographies of mathematics in the ancient world

ERC (European Research Council Advanced Grant 2010) Project : Mathematical Sciences in the Ancient World (SAW)

### Seminar Theoretical approaches to uses of the concepts of scientific “practice” and “culture” in History and Philosophy of Science

ERC (European Research Council Advanced Grant 2010) Project : Mathematical Sciences in the Ancient World (SAW)