Séminaire – Mathématiques 19e-21e, histoire et philosophie

François Lê (Institut Camille Jordan, Université Claude Bernard Lyon 1)
On Charles Hermite’s style
Abstract :
This presentation takes a historiographical approach to the tension between demonstration and computation. It aims to discuss the notion of style of writing, understood as « the set of expressive traits that denote the author of a piece of writing », by studying the case of Charles Hermite (1822-1901). To be more precise, my point is to propose a way to account precisely for the overall impression of reading that one has when reading the latter’s prose. I focus on Hermite’s writing peculiarities that can be detected through the use of words which are not associated a priori with the mathematical lexicon : non-technical nouns, verbs, adjectives, and adverbs, as well as functionals words such as conjunctions and pronouns. The analysis of all these words is quantitative, using tools from textometry, i.e. statistical analysis of textual data. It is also comparative, Hermite’s corpus being contrasted with Camille Jordan’s one. Among other results, I will show that Hermite’s prose is characterized by a higher lexical diversity than Jordan’s one, and that it corresponds to a lively mathematical narration where the first person and other words which describe the mathematical processes are of great importance.

Nicolas Michel (Bergische Universität Wuppertal)
Computations and exactness in G.-H. Halphen’s enumerative geometry
Abstract :
In a letter to his friend and colleague Hieronymus Zeuthen, the mathematician Georges-Henri Halphen credited computations (le calcul) with « putting him back on the right track » with respect to enumerative geometry, and to the theory of conic sections in particular. Indeed, his latest publications on the subject make extensive use of the patient and painstaking description of systems of curves and geometric conditions afforded by a variety of complex algebraic tools. The analysis of the singularities of such systems, conducted by analogy with the study of the singularities of curves (a subject in which Halphen was also a recognized specialist), occupies a particularly important place in them ; an analysis which, Halphen would retrospectively diagnose, had only been made possible by the use and adaptation for geometry of the « theory of algebraic functions ».

In this talk, I will examine the changing role of algebraic computations and description in Halphen’s geometrical practice. To do so, I will draw on a succession of Halphen’s writings on the enumeration of conics, from his manuscripts preserved at the Institut de France to his memoirs for the Journal de l’École Polytechnique and the Mathematische Annalen.

Lieu : salle Rothko 412B bâtiment Condorcet Université Paris Cité