Sciences, Philosophie, Histoire – UMR 7219, laboratoire SPHERE

The research developed in this theme aims, in part, to reconstruct and analyze the various mathematical practices that we can observe by relying on our sources and, on the other hand, to develop a philosophical and anthropological reflection. about these practices as such. We propose a study of the practices which breaks them up in a systematic way, before examining the modalities of the recomposition of their elements in various contexts.
1) Some of these elements have been the subject of particular attention in the past, for example, numeric tables, instruments, operations, notation of numbers and quantities. They will continue to be, but new elements will be articulated with the previous ones, for example the diagrams, the curves, the reasoning, the algorithms, the approximations. The research focuses more specifically on the following elements :

• Numbers - Numbers as tools shaped during the practice of quantification, calculation, or storage of information, will continue to be a major focus of researchers working on this topic . Part of this work, because of its interdisciplinary nature, will be integrated into the thematic ’Numbers, measurement and measurability ...’ of axis 5 link. A collective work on the various forms taken in history by what we call ’positive numbers’ and ’negative numbers’ is being prepared.

• Mathematical Reasoning - Work is underway on mathematical reasoning in a variety of contexts, including economics and astral sciences. This work intends to show the diversity of reasoning tools, whether discursive or non-discursive. For this, the role of diagrams, spatial organizations of texts, or the serialization of problems in the reasoning will be analyzed. A collective work, Practices of Mathematical Reasoning, under the dir. Husson, Keller & Proust, is under construction.

• Instruments - The study of the interaction between instruments and knowledge has a long tradition in the history of physics where experimentation relies on measurement and calculation practices that mobilize a constant interaction between theory and instruments. Researchers in this area are particularly interested in measurement and calculation instruments for elementary operations, boards and calculators, planimeters and machines such as harmonic and differential analyzers, up to the computer. They wish to conduct this research in coordination with the historians of physics.

• Machines and approximation - The use of analog and digital machines to obtain elaborate mathematical results (like integrals for example) raises the question of approximation in mathematics. Thematic researchers are interested in how these questions were raised and treated by mathematicians and physicists who used these machines.

• Calculation - The examination of the above elements leads to paying particular attention to calculation and operations, particularly in connection with the research carried out on mathematics in the nineteenth and twentieth centuries. Link

2) Mathematical practices, and in particular the modalities of the recomposition of their elements, are studied in the context of the environments or communities that produced them. As in the past, we will focus on communities of scientists, engineers or astronomers, administrative and financial communities, merchant groups, and village collectives in communities. societies without writing. Research will focus more specifically on the following environments :

• Astral Sciences - The mathematical practices developed in relation to the astral sciences have been part of the SAW European Project. In the continuity of these researches, a work is in progress on how were treated the operations, the quantities, the units of measurement, the arithmetic and geometric progressions and the reasonings by the practitioners of the astral sciences in the old worlds. A collective work, Mathematical Practices in relation to the Astral Sciences, under the dir. Chemla, Husson & Keller, is already under construction.

• Ethnomathematics - Research in ethnomathematics is developing in the frame (1) of the ANR project ETKnoS (2016-2020, coordinated by E. Vandendriessche) which associates ethnomathemati- cians, anthropologists and linguists, to lead in particular a reflection on the mathematical rationality involved in the practice of string games in various societies oral, (2) doctoral work "Ethnomathematics of drawings on the sand of Vanuatu", (Alban Da Silva, director Eric Vandendriessche), and (3), through a cooperation with the University of Reunion which aims to study mathematical activities practiced in the islands of the South-West Indian Ocean, notably Madagascar and Mayotte (current thesis on the seeding games, L. Tiennot, D. Tournès). The work in this theme is closely coordinated with the one developed in the theme ’Ethnographic Approaches in the History of Science’ of the transverse axis (axis 5).