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Home > Archives > Axes de recherche : 2012–2017 > Axis 2012-2017: History and philosophy of mathematics > Mathematical Practices: Writings and Instruments, Calculations and Diagrams / 2012–2017

Axis History and Philosophy of Mathematics 2012–2017

Mathematical Practices: Writings and Instruments, Calculations and Diagrams / 2012–2017

-An Historical, Anthropological and Philosophical Approaches

- Thematics
- Members

Thematics 2012–2017

The work carried out within the framework of this project is based on the results of the research already undertaken since 2006 in two previous projects of REHSEIS and then of SPHERE, within the framework of the axis History and Philosophy of Mathematics: "Mathematical Practices" and "Algorithms, instruments, operations, algebra ".
This research aims, in part, to restore and analyze the various mathematical practices that we can observe based on our sources and, on the other hand, to develop a philosophical reflection on these practices as such. The term "practices" here requires a clarification. The current use of this term refers to mathematics as the product of an activity, not a fixed knowledge. The researchers of SPHERE engaged in this project work on various mathematical circles, some in Europe, others in China, India, Mesopotamia and beyond. They naturally subscribe to the idea that mathematical practice is not one and that it must be studied in its variations of fact. The colleagues who have invested in the work on ethnomathematics focus precisely on the question of what criteria can be called a "mathematical practice". They give more broadly to our work the lighting of an anthropological approach.

We believe that mathematical practice can and should be systematically analyzed. This activity is characterized by the implementation of a certain number of artefacts. At various levels, problems, numbers, algorithms, theorems, figures, demonstrations, calculation and drawing instruments can be distinguished. The hypothesis that we wish to test is that these artefacts are the object of singular practices. The English language has a distinct term when it speaks of "engagement with" these artefacts, but it seemed difficult to render this expression in French other than by repeating the word "pratique" again. So we need to study these "commitments" with problems, algorithms, figures, computational instruments, numerical tables, etc., before we worry about the ways in which they are articulated in the In a mathematical practice, understood this time in the global sense. Thus, we propose a study of practices, which breaks them down in a systematic way, before examining the modalities of the reconstruction of their elements. One of the elements entering into the mathematical practices building will continue to be the subject of collective work (Projects under the previous contract: ANR History of Digital Tables, ERC Mathematical Sciences in the Ancient World). This is the practice of calculations. One objective of this research will be to capture the mathematical contribution of computational practices developed in different environments. Particular attention will be paid to the instruments, tables, diagrams, notations of measurements and numbers, and operations, in the continuity of the research already undertaken on these elements. The following list of questions is not exhaustive; It only indicates our working horizon.

  • 1. Numbers and units of measure
    How have human collectives shaped the expression of the quantities on which they operated? How did measurement problems affect the nature and representation of the numbers used? What were the relations between the abstract numbers and the measured numbers? What role did the practice of operations play in the emergence of different forms of abstract numbers? What interactions can be identified between the oral and written representations of numbers on the one hand and their incarnation in material objects on the other?
  • 2. Operations
    What were the theoretical problems raised in connection with the practice of arithmetic operations? What are the possibly contrasted stories of the "four operations" of arithmetic, and were they always only four? (These two first questions related to operations are also present in the study of the historiography of arithmetic developed within the framework of the project on "Scriptures and use of the past of sciences" - see axis "World History of Science" . How to interpret the highlighting of elementary operations and fundamental operations that mathematical documents of different epochs, produced in different environments, testify to? What were the relations between numbers and practice of operations in different contexts? All these questions are developed both within the ERC Project (Mathematical Sciences in the Ancient World -> Research on algebraic symbolism in the 19th century.
  • 3. Numerical tables - an aspect of calculation practice
    How can we synthesize and account for all the situations that lead to the organization of numbers in one-, two- or three-dimensional spatial arrangements? What ties can we perceive in these various situations between tables and other material instruments of calculation? To what extent has the tabular organization of calculations conditioned the developed mathematical theories? What methods can be used to reconstruct the algorithms used to calculate old tables in the event that any trace of these algorithms has disappeared? These questions are at the center of the ANR History of digital tables project (2009-2013). In connection with this latter project, two books and a set of articles on digital tables are being prepared (special issues for East Asian Science, Technology and Medicine journals, Journal of the History of Astronomy and Suhayl).
  • 4. A mutation in the practice of computation: from automation to automation
    Long before the intervention of instruments or machines, parts of computation were automated, that is to say they were subjected to rigid procedures executed automatically. It is often these "repeatable" parts that were then mechanized. In what context did this mutation from automation to automation take place? How much of it was given in the whole process? What were their motivations? Was it valued? What tools did she use? When and how did the transition from the instrument to the machine take place? On what basis: technical or theoretical? What were its implications for the contents of knowledge and the processes of memorization?
  • 5. Shapes
    What is the role of dispositions or textual forms in the understanding of a theorem or a procedure? What are the links between the invented or reworked form and the discovery or implementation of a theoretical result?
  • 6. The social organization of calculations
    Who was doing the calculations? How were the computers recruited, trained, considered, or symbolically valued? How were complex calculations organized and distributed among several individuals? What were the economic and social factors that led to the mechanization of calculations? What is the status of calculations within the theories of knowledge in general, and mathematics in particular? What are they supposed to represent?
  • 7. The practice of mathematics outside academic circles
    What is the relationship between calculation and technique, whether it be manual techniques or mechanical techniques, depending on the contexts? How have mathematical practices developed outside academic circles contributed to the creation of mathematical knowledge? How did mathematicians develop new concepts from algorithms and instruments created in practice in the practice of different trades? How were mathematical theories assimilated and transferred to practice?

These computational lines of research illustrate how we will more generally study other forms of practice with different types of elements as well as the articulation between distinct practices. For example, we will look at the relationship between computation and demonstration.

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In charge
PROUST Christine
Researchers – Phds students – Post–Phds students
BARBIN Évelyne
GAC Philippe
GANDON Sébastien
HUSSON Matthieu
IDABOUK Ghislaine
JAECK Frédéric
LI Liang
MALET Antoni
MOSCA Antonio
PATY Michel
POLLET Charlotte
TOURNÈS Dominique
WANG Xiaofei
ZHENG Fanglei-Félix
ZHU Yiwen

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