**Thematics of research** / Members /

- On the one hand, research on the history of mathematics which focuses more specifically on the study of mathematics in the 19th and 20th centuries in the context of various unifying projects or research groups.

- on the other hand, research in the philosophy of mathematics covering a broad spectrum from the history of the philosophy of mathematics of the 19th and 20th centuries to contemporary issues related to 20th and 21st century mathematics (foundations and theory categories, topoi, theories of abstraction, philosophy of mathematical practices, logic and formalization).

Although relatively independent in their own methods and forms of questioning, these two research orientations are strongly linked by their common grounding in mathematics as they have developed since the 19th century. If from the point of view of the history of mathematics proper, the gap between the 18th and the 19th century appears very largely artificial, and if, in this perspective, it must obviously be itself questioned, the choice we make of to bring together all these researches in the same direction is nevertheless justified by the fruitfulness of the dialogues crossed between these multiple approaches.This theme brings together a body of research on the history and philosophy of mathematics in the nineteenth and twentieth centuries, according to two distinct and complementary orientations:

**1.** With regard to the first orientation, the research is divided over the five-year period 2018-2023 into several new projects:

- A project "
*Demonstration and calculation*" in which we propose to adopt a new perspective on the history of demonstrations in mathematics of the 19th and 20th centuries, by exploring the different forms of relations between demonstration and calculation. It is thus a question of developing an alternative to this form of historiography of mathematical demonstration, very largely dominant in the 20th century, which made the value of axiomatico-deductive rigor prevail over the value of calculus.

- A project "
*Geometry and Calculation*", related to the previous one, but distinct as to its guiding questions, insofar as the accent is placed on the various modalities of parallelism, entanglement and competition according to which geometry enter the calculation, or the calculus graft on the geometry, etc.

- A project "
*Mechanization and calculation: 19th and 20th centuries*", in continuity of previous works of some members of the axis.

- A "
*Legendre*" project, around the work of the mathematician Adrien-Marie Legendre (1752-1833) to which several members of the team devoted previous works that they would like to extend as part of a collective project larger scale, covering all aspects of the work (number theory, geometry, elliptic functions, etc.)

- A project "
*Mathematics and historiography of mathematics*" which would gather, coordinate and extend, in the context of this sub-axis, the work of certain members of the team started notably in the framework of the ERC project "Sciences in Ancient World", led by Karine Chemla. We would like to be able to extend and generalize the type of problematization of the relationship between historiography and mathematical practices in the 19th and 20th centuries.

- A
*Curricula*project

A joint working group is set up which is to be used in the start-up phase of the platform to launch the different projects.

19th-20th century working group:

Organizers: Frédéric Jaeck, Nicolas Michel, Ivahn Smadja

**2.** With regard to the second "philosophy of mathematics" orientation, the work carried out within the SPHERE team is organized around different poles:

- An INTERSEM seminar in the philosophy of mathematics, co-organized each year by Michael Detlfsen and several members of the team.

- A seminar on mathematical abstraction co-organized by Professor Jean-Pierre Marquis of the University of Montreal and Professor Jean-Jacques Szczeciniarz.

- A research program in line with the ANR "Géométrie et physique à la charnière des XIXe et XXe siècles", directed by Jean-Jacques Szczeciniarz (2006-2010), which brings together work around the following themes :

- a) the relationship between geometry and physics, in the continuity of the book*Geometrie au XXe siècle*co-edited by J.-J. Szczeciniarz; in particular the question of the status of complex numbers in mathematical physics;

- b) the status of cosmological science and the role of mathematical conceptual constructs in the constitution of cosmological models (working group bringing together researchers from the team and the APC team - Paris Diderot University);

- c) the mathematical and philosophical question of the unity of mathematics, under the various forms of unification by integral transformations, unification by Galois theory, unification by theory of categories. In this context, a working group on category theory brings together several of the team’s researchers, teacher-researchers and PhD students: Michel De Glas, André Rodin, Jean-Jacques Szczeciniarz and Sylvain Cabanacq.

**Members** / Thematics /

In Charge |
---|

SMADJA Ivahn |

SZCZECINIARZ Jean-Jacques |

Researchers - PhD students - Post-docs |

ANEL Matthieu |

BERTIN Pascal |

BULLYNCK Marteen |

CATREN Gabriel |

CHEMLA Karine |

DECAENS Simon |

DURAND-RICHARD Marie-José |

FERREIROS José |

GASTALDI Juan Luis |

GROSHOLZ Emily |

HAFFNER Emmylou |

HALIMI Brice |

JAECK Frédéric |

KELLER Agathe |

MICHEL Nicolas |

PENCHEVRE Erwan |

PETROLO Mattia |

RABOUIN David |

TIMMERMANS Benoît |

VERGNERIE Cédric |

WANG Xiaofei |