Thematics of research / Members / 2018–2022
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.
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 »
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 »
The Latin Middle Ages inherited Greek and Arabic mathematics thanks to the influx of translations in the 12th century. After a period of assimilation, the fourteenth century sees the appearance of original developments, particularly in the context of reflections on Aristotelian physics and in particular on the movement, which lead to an extension of the Euclidean theory of proportions, but also, in the theological framework, discussions on the movement of angels, which lead to reflections on the composition of the continuum and the status of indivisibles, or reflections on the increase of charity that induce reasoning using infinite summations. Most of the original mathematical and physical treatises produced in connection with these and other questions were published in the Renaissance and nourished the reflections of mathematicians until the seventeenth century, even though often medieval sources are obscured.
In the Renaissance, moreover, a new attention paid to the Greek manuscripts, the interest for mathematical texts a little unnoticed in the Middle Ages like the Archimedean treatises, or the rediscovery of new authors like Proclus, Pappus or Apollonius, of even as the developments of astronomy and the extension of the fields of application of mathematics fuel the reflections of mathematicians. Algebra is also taking off and it would be reductive to focus only on the original work of Italian mathematicians on third-degree equations. Algebra raises new questions, especially on the status of numbers, the links between disciplines, algebra, geometry, arithmetic.
All these questions feed the work of some members of SPHERE and are the subject of presentations during study days or seminars.
- « Mathematics in the Modern Age »
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. Too often, we have postulated a perfect match between the development of these two aspects. Thus the Modern age is characterized from this point of view by a strong continuity with earlier periods (see P. Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century, 1996), which philosophy and mathematics do not know. At the same time, Henk Bos’s study of the transformation of the idea of accuracy into classical geometry clearly indicated the strong presence of epistemological norms informing mathematical practices without falling into the history of philosophy stricto sensu (H. Bos , Redefining Geometrical Exactness, 2001). These new historiographical options form the basis of our work. Our broader ambition is to inform a philosophy of mathematics attentive to the historical evolution of this discipline - a program that informs both the renewal of the French tradition of "historical epistemology" and the recent evolution of part of the so-called "analytical" tradition of philosophy of mathematics.
- Mathematics from Ancient to Modern Age, organized by Pascal Crozet, Vincenzo de Risi, Sabine Rommevaux-Tani, (CNRS, SPHERE)
Members / Thematics /
Organisers |
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CROZET Pascal |
RABOUIN David |
ROMMEVAUX Sabine |
Researchers – Phds students – Post–Docs |
BANCEL Faïza |
BELLA Sandra |
CONFALONIERI Sara |
CORTESE João |
CRIPPA Davide |
DE RISI Vincenzo |
DEBROISE Philippe |
DECORPS-FOULQUIER Micheline |
GROSHOLZ Emily |
HAFFNER Emmylou |
HOUZEL Christian |
LEVY Tony |
KOUTEYNIKOFF Odile |
LOIZELET Guillaume |
MALET Antoni |
MORELON Regis |
PENCHEVRE Erwan |
RASHED Roshdi |
REMAKI Ariles |
ROBERT Aurélien |
SAMMARCHI Eleonora |
SCHWARTZ Claire |
SMADJA Ivahn |
SZCZECINIARZ Jean-Jacques |
TIMMERMANS Benoît |
VAHABZADEH Bijan |