Sciences, Philosophie, Histoire – UMR 7219, laboratoire SPHERE

Mathematics and philosophy have maintained rich and constant relationships throughout history. As early as Plato and Aristotle, there are closely matched mathematical and philosophical considerations, according to a tradition that is perpetuated through commentators down to at least the classical period. Theory of science, status of demonstrations, axioms and postulates, classification of propositions, analysis and synthesis, treatment of infinity, later place and role of algebra, differential calculus, etc., continuously nourish the reflections of mathematicians And philosophers in a fruitful interaction. This interaction is not only interesting on a historical level : it still allows the present to enrich one another. On the one hand, it can provide the historian with lines of problematization that guide him in his conceptual reconstructions ; On the other, it offers the philosopher the possibility of developing a reflection attentive to the variety of mathematical practices and epistemological frameworks that have developed over time. Naturally, two of the major stakes that the history and philosophy of mathematics now have to face : developing a conceptual history that avoids the danger of artificial reconstructions (in particular by relying on the philosophical orientations of the actors themselves And how they may occasionally serve as standards in their practices) ; Develop in parallel a philosophy of mathematics that can account for their historical evolution and the variation of the conceptual frameworks that accompanies it.
The creation of the SPHERE unit in 2009 allowed a unique gathering of researchers working continuously on the history and philosophy of mathematics from Greek antiquity to the classical age. This research has been structured in the form of working groups (around the Greek and Arab periods, the Renaissance and the Classical Age), which are described below. The researchers participate in the activities of these groups in order to develop a systematic comparative approach, as well as a sensitivity to their specificities, of these different corpuses. Moreover, the fact that the unit develops research on other periods, such as Mesopotamian antiquity, and other cultural areas, such as India and China, has made it possible to confront explicitly Provided by our actors to corpuses where these explanations are often lacking, but where they are no less enlightening.

• « Greek and Arabic Mathematics » 2012–2017
  Knowledge of mathematics written in Arabic has undoubtedly progressed considerably over the past forty years, often calling into question the most commonly accepted historiographic framework. It is this movement that we intend to pursue here, continuing to nourish the research and reflection of newly established texts, whether they are Arabic translations of treatises lost or not in Greek (the first seven books of the Conics of Apollonius, the Apollonius Reporting Section, Euclid’s Data, etc.), or from mathematicians writing in Arabic (Abū Kāmil, Thābit ibn Qurra, al-Siğzī, al-Jayyānī, al-Zanjānī , Etc.). It is thus not only a matter of studying the scientific developments of the Islamic area, but also - and above all - of reinterrogating all the classical mathematics by restoring to the past mathematical activities the epistemic horizon that is theirs. Themes such as the history of curves, the concepts of relationship and proportion, the place and role of algebra in mathematics, the relationships between algebra and geometry, the introduction of geometry, or the practice of analysis and synthesis. All these and other questions are addressed in a dedicated monthly seminar (http://www.sphere.univ-paris-diderot.fr/spip.php?article738&lang=en).

• « Mathematics in the Renaissance » 2012–2017
  In the Renaissance, mathematics asks specific questions : the return to ancient texts, the development of algebra which upsets the frameworks of traditional mathematical disciplines, new fields of application, new relationships to the study of nature. There are few studies on this period, no doubt because, for many historians, mathematics really develops only from the works of Descartes. In doing so, studies of the Renaissance were sometimes carried out in a retrospective way, reducing this period to a prefiguration of the seventeenth century. There is an interest in studying restored mathematics for themselves, by not neglecting the reception of medieval commentaries and the innovations they aroused, nor the contributions of the Renaissance period to the later developments of mathematics. The study of these questions will be carried out during a monthly seminar (http://www.sphere.univ-paris-diderot.fr/spip.php?article681) and study days which will give rise to publications Individual and collective.

• « Mathematics in the Modern Age » 2012–2017
  The Modern age offers a privileged laboratory for the study of the relations between philosophy and mathematics, if only by the richness and diversity of the sources offered. However, these relations proved to be more complex than could be expected from the existence of great figures of "philosophers-mathematicians" such as Descartes, Pascal or Leibniz. We have too often postulated a perfect match between the development of these two aspects, without always seeing, for example, that the history of the philosophy of mathematics constituted a third term, with its own temporality. Thus the modern age is characterized from this point of view by a strong continuity with the earlier periods (see P. Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century, 1996), which philosophy and mathematics do not know not. At the same time, Henk Bos’s study of the transformation of the idea of ​​accuracy in classical geometry has clearly indicated the strong presence of epistemological norms informing mathematical practices without belonging to the history of philosophy stricto sensu (H. Bos, Redefining Geometric Exactness, 2001). These new historiographic options form the basis of the program launched in the seminar on mathematics in the classical age (http://www.sphere.univ-paris-diderot.fr/spip.php?article740). Its wider ambition is to inform a philosophy of mathematics attentive to the historical evolution of this discipline - a program which informs both the renewal of the French tradition of "historical epistemology" and the recent evolution of a part of the so-called "analytical" tradition of philosophy of mathematics.