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Axe Histoire et philosophie des mathématiques

Mathématiques 19e-21e, histoire et philosophie

Le séminaire d’histoire et de philosophie des mathématiques modernes est un lieu de présentation et de discussion de textes mathématiques produits aux 19e, 20e et 21e siècles, dans des perspectives aussi bien historiques que philosophiques. Il entend servir de lieu d’exploration (lecture, traduction, explication) de documents mathématiques peu ou mal connus, mais également de présentation de travaux en cours sur ces périodes. L’accent est mis sur la proximité avec les sources textuelles. Les séances prennent, le plus souvent, la forme d’une discussion desdites sources (auxquelles les orateurs donnent accès au préalable), précédée d’un exposé historique ou mathématique.

Organisation : Thomas Berthod, Paul-Emmanuel Timotei et Clément Bonvoisin

2011-2012, 2012-2013, 2013-2014,
2014-2015, 2015-2016, 2017-2018,
2018-2019, 2019-2020, 2020-2021,
2021-2022, 2022-2023

PROGRAMME 2023-2024

Nous nous réunirons un vendredi par mois en Salle Malevitch (483A) bâtiment Condorcet, Université Paris-Cité - 4 rue Elsa Morante, 75013 - Paris. Le séminaire a lieu habituellement de 14h00 à 17h30.
Le programme sera annoncé progressivement au cours du semestre.

Vendredi, 6 octobre 2023 de 14h00 à 17h30

Programme :

  • 14h00 - 15h30
    Leo Corry (Tel Aviv University)
    Two Views of Excellence in Research, Two Views of Zionist Nation-Building : Pure Mathematics at the Hebrew University, Applied Mathematics at the Weizmann Institute.

    Résumé :
    I will present a comparative analysis of the early years of two world-class centers of mathematical research in Mandatory Palestine, and then in the recently created State of Israel. They pursued different ideals of mathematical excellence which were strongly associated with two different views of Zionism and of the role that science institutions should play in the national project envisioned by each.

    The Einstein Institute of Mathematics was established in 1925 at the Hebrew University in Jerusalem (HUJI). Edmond Landau came in 1937 as the first professor and scientific leader, and was succeeded by Avraham Halevy Fraenkel. The neo-humanistic, conceptual spirit of German pure mathematics dominated activities in Jerusalem and it was very much in accordance with a view of Zionism that sought to establish a leading intellectual and spiritual center for the Jewish people in Palestine with a Hebrew University as its flagship.

    The Weizmann Institute of Science (WIS) was established about twenty years later in the rural town of Rehovot. It had a thoroughly practical and applied orientation meant to serve the aims of political Zionism in its most activist version, which saw in the creation of a Jewish State in Palestine the most urgent and significant task. A Department of Applied Mathematics was established at WIS in 1948 under the leadership of Chaim Leib Pekeris, whose mathematical views consolidated against the background of his wartime activities at MIT and Columbia, and under the marked influence of John von Neumann. His purpose when joining WIS was to build a high-speed electronic computer and to implement a wide-ranging program of research in various fields of applied mathematics based on computing-intensive methods.

  • 15h30 – 16h00

  • 16:00 – 17:30
    Jan von Plato (University of Helsinki)
    New light on Gödel’s life and work
    Résumé :
    Kurt Gödel (1906-1978) was a secretive character who published very little. His foremost result, the incompleteness theorem, revolutionized the foundations of mathematics in 1931. By 1937, he had come half-way through with the solution of Hilbert’s famous first problem about the cardinality of the set of real numbers. After this success, Gödel’s only new published results were about the strange circular-time solutions to the field equations of Einstein’s theory of general relativity he had found in 1949.

    The study of Gödel’s tens of thousands of pages of notebooks since 2017, written in an abandoned shorthand, gives a picture of his achievements, as well as of the aims of his life, that is quite different from the one suggested by his publications. As to achievements, there is a plethora of results on logic and foundations of mathematics he revealed to no one. As to the aims, these reflect a vision of science and philosophy he had formed early on in his life, while still a high-school student. Said vision contained that "the world and everything in it has a meaning and makes sense, and it is a good and doubtless meaning.
    So, the talk would be in part about the results Gödel had achieved from 1940 on, when he ceased to publish, in part about his grand program as dictated by his youthful philosophy"

Vendredi, 17 novembre 2023 de 14h00 à 17h30

Programme :

  • 14h-15h30
    Michael Friedman (The Cohn Institute for the History and Philosophy of Science and Ideas - Humanities Faculty - Tel Aviv University)
    Titre : Models and calculations at the end of the 19th century
    Résumé :
    On November 7th, 1886 Alexander Brill gave a lecture on the collection of material mathematical models at the university of Tübingen. These models usually modelled various algebraic surfaces and curves of various degrees, and were wide spread in Germany during the last third of the 19 th century. In this lecture, after describing the preparation of models from various materials (plaster, strings, cardboard) usually by students, he notes that these students could “write a paper on this [subject], the publication of which […] played no small part in encouraging one to carry out the often-arduous calculations […],” implicitly noting that calculations precede the formulation and the proving of theorems. Brill then continues, claiming that the reverse direction also occurred : “the model often prompted subsequent investigations into the specific features of the represented structure.” [Alexander Brill, “Über die Modellsammlung des mathematischen Seminars der Universität Tübingen (Vortrag vom 7. November 1886),” Mathematisch naturwissenschaftliche Mitteilungen 2 (1887), 69–80.] Thus, according to Brill, mathematical models did not merely serve to visualize lengthy calculations, they were also an object of research, and prompted not only the discovery of new theorems but also ways to prove them.
    My talk will ask whether this statement was only a rhetorical one, meant to further support such model collection (and perhaps others collections), or whether indeed Brill or other mathematicians considered calculations, modeling and proving mathematical claims as supporting and essential to each other. I will also attempt to characterize the changing role of calculation, when the tradition of material models declined and when the term “model” acquired new meaning during the 1920s and the 1930s, signifying procedures of abstraction and mathematical representation of certain well-selected processes.

  • 15h30-16h

  • 16h-17h30
    Patrick Popescu-Pampu (Laboratoire Paul Painlevé - Université de Lille)
    Titre : From infinitely near points to trees and lotuses
    Résumé :
    I will explain what are the constellations of infinitely near points, how they were codified by various kinds of trees, and how these trees may be compared by embedding them in lotuses.

Vendredi, 15 décembre 2023 de 14h00 à 17h30

Théorie de l’homotopie

Andrea Gentili (Università di Genova), The homotopical approach to mathematics
Résumé : After introducing with some motivations the homotopical point of view and some of the models for it, I will compare it with the classical one. I will explain how certain classical constructions translate in the homotopical framework and, on the other hand, how some classical categorical constructions can be easily interpreted in the homotopical language.

Jean-Pierre Marquis (Université de Montréal), Why homotopy theory matters to philosophy of mathematics
Résumé : With roots in 19th century mathematics, homotopy theory slowly rose in the 20th century to become a fundamental theory in the 1950s and 1960s. Nowadays, it is central to algebraic topology, its methods are relevant to many other fields and it gave rise to a global foundational framework, namely homotopy type theory. In this talk, I want to focus on the rise of abstract homotopy theory and its significance for philosophy of mathematics.

Informations pratiques : le séminaire se tiendra dans une modalité hybride. Les personnes souhaitant assister en personne au séminaire sont bienvenues au campus des Grands Moulins de l’Université Paris Cité, Bâtiment Condorcet, Salle Malevitch (483A), 4 rue Elsa Morante, 75013 Paris. Les personnes souhaitant assister en ligne au séminaire sont invitées à écrire à avec pour objet du mail « HPM19-21_15.12.23 ».

Vendredi, 19 janvier 2024 de 14h00 à 17h30

Nombres et continu

Gilles Godefroy (IMJ-PRG, CNRS)
The use of large cardinals in analysis.
Résumé : Nous savons depuis Cantor qu’il y a différents infinis, et les travaux des théoriciens des ensembles du vingtième siècle ont montré qu’il était difficile de comparer des cardinaux infinis, ou d’affirmer l’existence de certains d’entre eux. Il se trouve pourtant que ces cardinaux sont utiles pour décider de questions assez concrètes, portant par exemple sur des sous-ensembles de la droite réelle. Nous présenterons quelques exemples de ce type, en nous efforçant d’éviter les aspects techniques.

Thomas Berthod (SPHERE, Université Paris Cité)
The real numbers according to Lebesgue : a new construction imbued with philosophical reflections ?
Résumé : Henri-Léon Lebesgue, un mathématicien français du début du XXème siècle, est principalement connu pour sa construction d’une intégrale éponyme. Dans un article de L’Enseignement mathématique publié en 1932 sous le nom « Sur la mesure des grandeurs », il propose pourtant aussi une construction des nombres réels. Après avoir présenté cette construction, nous soulèverons deux questions. La première de nature historique concerne la nouveauté de l’approche de Lebesgue. En effet, durant la seconde moitié du XIXème siècle, différents mathématiciens ont déjà proposé des constructions de ces nombres. La construction de Lebesgue apporte-t-elle de nouvelles choses ? Rejoint-il le point de vue d’un de ses prédécesseurs ? Rompt-il totalement avec les constructions précédentes pour mettre en avant une nouvelle approche ? Pour tenter de répondre à ces questions, nous comparons l’approche de Lebesgue avec les constructions du XIXème siècle présentées par des mathématiciens de son époque. Dans un deuxième temps, nous montrerons que cette construction mathématique s’articule avec une réflexion philosophique particulière, visible notamment à travers la définition que donne Lebesgue des nombres et par le prisme des valeurs épistémologiques qu’il prône au cours de son analyse sur les nombres réels.

Vendredi, 23 février 2024 de 14h00 à 17h30

Michael Harris (Columbia University)
Mechanical understanding of proofs ?
Résumé :
The drive to incorporate methods of artificial intelligence in mathematics raises the possibility of the separation of proofs from understanding. A formal proof can be formally validated but not understood by anyone : not by humans, either because the proof is written in impenetrable code or simply because it is too complex ; not by machines, unless the term "understanding" is radically redefined. It may be objected that many existing proofs are too long and intricate for any one person to understand. Nevertheless, it is a common practice to invite the authors to explain such a proof in seminar talks, or for study groups to break down a proof into comprehensible chunks, so that one may speak of a collective (human) understanding. My talk will try to imagine machine-made proofs that cannot be understood even in this way. In the process, I will examine strategies by which mathematicians seek to make proofs understandable ; how one might determine whether or not these strategies have been successful with a human audience ; and what it would mean to test a machine’s understanding of a proof.

The talk will be presented from the standpoint of a practicing mathematician and will take the form of questions of a philosophical nature arising from the practice. If time permits there will also be reports on attempts by large language models to demonstrate understanding of relatively simple proofs.

Clément Bonvoisin (SPHERE, Université Paris Cité)
Computations made proof. On the reassessment of an equation a decade after its writing at RAND Corporation (1948-1961)
Résumé :
In this talk, I wish to discuss the relationship between computations and proofs in contemporary mathematics. I intend to do so by focusing on an equation written by US mathematician Magnus Hestenes (1908–1991). The equation lies in a research memorandum produced by Hestenes in 1950 for the RAND Corporation, a think tank funded by the US Air Forces. The memorandum itself was part of a broader research work on the computation of minimum time flight paths for aircrafts and missiles, which lasted from 1948 through 1951. However, in an article published in 1961, US mathematician Leonard Berkovitz (1924–2009) cited the equation, and framed it as an early version of a result obtained in 1958 by a group of Soviet mathematicians—namely, Pontryagin’s maximum principle.
This situation calls to mind a set of issues for the history of mathematics. How did Berkovitz come to know of this equation, buried in the middle of a report with poor circulation outside of RAND ? What status did the equation play in the initial memorandum, and what role did it play over the 1950’s ? How similar was it to Pontryagin’s maximum principle, and how did it differ from it ? How can we account for the likening of the equation to a result derived some eight years after the memorandum was written ? And what historiographical consequences did such a likening have ? To answer these questions, I will trace the circulations, uses and changes in status of the material contained in Hestenes’ memorandum at RAND over the 1950’s. In doing so, I will show the importance of the initial purpose of the project Hestenes took part in—computations. This, I will argue, shaped the initial status of the equation as a mere intermediate step, rather than as an original result—a status Berkovitz misrepresented in his 1961 article. Overall, I hope to provide insights on the relationship between computations and proofs in mathematics, and on its historiographical consequences.

Vendredi, 1er mars de 14h00 à 17h30

Calcul et Démonstration
Séance supplémentaire

Programme :

  • 1ère séance
    Discussion générale
  • 2ème séance
    Marie-José Durand-Richard (SPHERE-UMR 7219-Chercheuse associée, Université Paris8- MCF honoraire)
    Herschel et Babbage 1812-1820 : Vers un nouveau calcul sur les fonctions : pourquoi et comment ?
    Résumé :
    La première génération du réseau des algébristes anglais – qui a conduit à l’Algèbre Symbolique – est bien connue pour avoir réussi à imposer la notation différentielle du calcul infinitésimal à Cambridge, en lieu et place de la notation newtonienne, essentiellement depuis leur traduction du Traité Elémentaire du Calcul Différentiel et du Calcul Intégral (1802) de Sylvestre-François Lacroix (1765-1843).
    À la même époque, Charles Babbage (1791-1871) a publié un article très important en deux parties : “An Essay towards the Calculus of Functions” .
    En fait, ces travaux marquants sont la partie visible de publications et d’échanges très abondants entre John F.W. Herschel (1792-1871) et Babbage entre 1812 et 1820. Au-delà de leur préoccupation constante pour la question de la notation, ils envisagent de s’appuyer sur des processus d’invention, notamment l’analogie, pour établir un nouveau calcul sur les fonctions, généralisant des pratiques héritées du Continent. J’examinerai les avancées et les échecs d’un tel projet.

Vendredi, 22 mars 2024 de 14h00 à 17h30

Analyse modale du continu

Filippo Costantini (ERC Philiumm, SPHERE)
The continuum from Hellman & Shapiro to Leibniz

Brice Halimi (Univ. Paris Cité & SPHERE)
Analyse continue des modalités

Vendredi, 12 avril de 14h00 à 17h30

Dirk Schlimm (McGill University)
Notations, calculations, and proofs
Résumé :
Notations are essential in mathematical practice, serving both representational and operational roles. On the one hand, they provide names for mathematical entities, and, on the other hand, we gain insights into the intended subject matter by manipulating them. These manipulations depend in part on the structure of the notations, but also on the computational resources employed and the algorithms used. Using examples from arithmetic and logic, I discuss various aspects of the assessment of the operational role of notations and argue for the importance of comparative studies.

Mary Louise Elworth (Aarhus University)
Computation or mutation ? Cluster algebras as a case study
Résumé :
Studying the role of signs (e.g. notations or figures) in mathematical practice can be a way to gain insight into how human computational constraints shape our mathematical theories (see Waszek 2023). I investigate the role of signs—also called ‘representations’—in the context of the recently developed theory of cluster algebras. Within this theory, one finds different forms of representations which are used to encode the ‘same’ information. While such representations are considered to be equivalent, I argue that they differ in terms of the support they provide for visualizing both the relations which they encode and the transformations which are defined on them. Moreover, we can examine the specific contexts in which these representations appear and the aims which they are intended to be used for in order to better understand their epistemic and pragmatic roles. This case study also offers opportunities for considering which transformations or operations may count as computations.

Vendredi, 26 avril 2024 de 14h00 à 17h30

- Séminaire est annulé
Séance sur la notion de co-algèbre

Jeffrey Elawani (University McMaster et Université Paris Cité)
Measurement and numbers in Leibniz. A coalgebraic approach to continuous quantities ?
Résumé :
In his important works on the evolution of the concept of ratio in Renaissance and early modern time, Antoni Malet points out that many early modern scholars assumed a concept of continuous numbers that is at best ill-grounded relative to the traditional Euclidean conception of ratios and at worst plainly circular. At first sight, the conception of numbers and ratios found in Leibniz is open to these same critiques.
In this talk, to the contrary, I argue that Leibniz developed a conception of number that is free of the circularity criticized by Malet. Moreover, although it is not without difficulties, I hint that the conception is based on valuable and (for the time) commendable foundational insights. I end the talk with a discussion of some aspects of contemporary coalgebraic characterizations of the continuum that, I believe, are relevant for the appreciation of Leibniz’s own conception.
The Leibnizian concept of number I discuss is based on the measurement of continuous quantities. In Leibniz, this measurement consists in a procedure of iterative division that yields a sequence of repetitions of parts. Following (Costantini, F. and Elawani, J. (2024)), I argue that this procedure involves no notion of abstract numbers. Abstract numbers are rather the common feature (the common form of comparison) shared by all the different measurement procedures, be they measurement of lines, surfaces, forces etc.

Graham Leach-Krouse (Software engineer et auparavant à Kansas University)
Coalgebras for philosophers
Résumé :
In this talk, I introduce F-algebras and F-coalgebras, with some familiar examples of each. I make the case for both the intrinsic philosophical interest of coalgebras (for example, as potential sources of insight into the intuitive concept of the continuum) and for the significance of the algebra/coalgebra duality.

Vendredi, 17 mai 2024 (exceptionnellement, la séance se tiendra de 9h30 à 13h00 en salle Rothko 412B bâtiment Condorcet Université Paris Cité 4 Rue Elsa Morante Paris 75013)

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.

Lundi, 10 juin 2024 (exceptionnellement, la séance se tiendra de 9h30 à 13h00 en salle 569 bâtiment Olympe de Gouges Université Paris Cité 8 Rue Albert Einstein)

Jean-Jacques Szczecziniarz (SPHERE, Université Paris Cité)
About Penrose Transform the rôle played by the computation
Résumé :
This presentation is all about showing you how to install a calculation in multiple variables. It’s all about embracing the power of plurality ! In particular, we’ll be exploring how matrices can be treated as variables, and how points can be represented by matrices.

Paul-Emmanuel Timotei (SPHERE, Université Paris Cité)
Georges-Henri Halphen’s approaches to the reduction of singularities
Résumé :
In the second half of the 19th century, in Germany, England, Italy and France, mathematicians became interested in certain geometric invariants. These invariants enabled certain classifications to be made. The development and study of these invariants led mathematicians to study the singularities of geometric objects, insofar as they have an influence on the determination of these invariants. Once knowledge of the influence of "simple" singularities on invariants was established, mathematicians turned their attention to more complex singularities. One way of approaching this problem is to reduce these high singularities to simpler ones.
Georges-Henri Halphen (1844–1889) presented two methods for the reduction of singularities of plane algebraic curves. He developed his methods in reaction to the method that Max Noether (1844–1921) had published, which Halphen considered to be unclear with regard to the result.
In two later texts, Halphen qualified his methods in two different ways. He describes the first one as being more geometric than Noether’s method, while he comments on the second one, stating that it proceeds using of truly geometric transformations. We notice that Halphen always compares his methods to Noether’s and he uses the term geometric to qualify them twice, but with different meanings.
What meaning does Halphen give to the word geometric in the context of the reduction of singularities ?
To explore this issue, I will present Noether’s and Halphen’s methods of the reduction of singularities and their knowledge about this notion. Then I compare the three methods and I explore the different meanings Halphen gives to the word geometric.

. . . . . . . . . . . .

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