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Axis History and philosophy of mathematics

Mathematics 19th – 21st, History and Philosophy

The seminar on the history and philosophy of modern mathematics is a place of presentation and discussion of mathematical texts produced in the 19th and 20th centuries, in both historical and philosophical perspectives. It intends to serve as a place of exploration (reading, translation, explanation) of mathematical documents little or poorly known, but also to present work in progress on these periods. The emphasis is on proximity to textual sources. The sessions usually take the form of a discussion of the said sources (to which speakers give prior access), preceded by a historical or mathematical exposition.

Organization: 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

We will gather on a monthly basis on Fridays in Room Malevitch (483A) building Condorcet, Université Paris-Cité - 4 rue Elsa Morante, 75013 - Paris. The seminar usually starts at 2.00 pm until 5.30 pm.
The program will be announced progressively throughout the semester.

Friday, 6th October 2023 from 2 pm to 5.30 pm

Program :

  • 2.00 pm - 3.30 pm
    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.

    Abstract :
    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.

  • 3.30 pm – 4.00 pm

  • 4.00 pm – 5.30 pm
    Jan von Plato (University of Helsinki)
    New light on Gödel’s life and work
    Abstract :
    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"

Friday, 17th November 2023 from 2 pm to 5.30 pm

Programme :

  • 2 pm - 3.30 pm
    Michael Friedman (The Cohn Institute for the History and Philosophy of Science and Ideas - Humanities Faculty - Tel Aviv University)
    Title : Models and calculations at the end of the 19th century
    Abstract :
    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.

  • 3.30 - 4 pm

  • 4 pm - 5.30 pm
    Patrick Popescu-Pampu (Laboratoire Paul Painlevé - Université de Lille)
    Title : From infinitely near points to trees and lotuses
    Abstract :
    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.

Friday, 15th December 2023 from 2 pm to 5.30 pm

Homotopy theory

Andrea Gentili (Università di Genova), The homotopical approach to mathematics
Abstract : 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
Abstract : 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.

Practical informations: the session will be held in a hybrid modality. People interested in attending in person are welcome at the Grands Moulins campus of the Université Paris Cité, Bâtiment Condorcet, Salle Malevitch (483A), 4 rue Elsa Morante, 75013 Paris. People interested in attending online may write to with “HPM19-21_15.12.23” as email subject to receive a Zoom link.

Friday, 19th January 2024 from 2 pm to 5.30 pm

Numbers and continuum

Gilles Godefroy (IMJ-PRG, CNRS)
The use of large cardinals in analysis.
Abstract : We’ve known since Cantor that there are different infinities, and the work of twentieth-century set theorists has shown that it’s difficult to compare infinite cardinals, or to assert the existence of some of them. However, such cardinals are useful for deciding quite concrete questions, such as subsets of the real right. We will present a few examples of this type, trying to avoid the technical aspects.

Thomas Berthod (SPHERE, Université Paris Cité)
The real numbers according to Lebesgue: a new construction imbued with philosophical reflections?
Abstract : Henri-Léon Lebesgue, a French mathematician of the early 20th century, is best known for his eponymous integral construction. However, in an article published in L’Enseignement mathématique in 1932 under the title "Sur la mesure des grandeurs", he also proposed a construction for real numbers. After presenting this construction, we’ll raise two questions. The first, of a historical nature, concerns the novelty of Lebesgue’s approach. Indeed, during the second half of the 19th century, various mathematicians had already proposed constructions of these numbers. Does Lebesgue’s construction offer anything new? Does it endorse the point of view of one of his predecessors? Or does it completely break with previous constructions to put forward a new approach? To answer these questions, we compare Lebesgue’s approach with 19th-century constructions presented by mathematicians of his time. Secondly, we’ll show that this mathematical construction is articulated with a particular philosophical reflection, visible notably through Lebesgue’s definition of numbers and through the prism of the epistemological values he advocates in the course of his analysis of real numbers.

Friday, 23rd February 2024 2.00 pm - 5.30 pm

Michael Harris (Columbia University)
Abstract :
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)
Abstract :
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.

Friday, 1st March 2024 from 2 pm to 5.30 pm

Calculation and demonstration
Additional session

Program :

  • 1st session
    general discussion
  • 2nd session
    Marie-José Durand-Richard (SPHERE-UMR 7219-Chercheuse associée, Université Paris8- MCF honoraire)
    Herschel and Babbage 1812-1820: Towards a new calculus of functions: why and how?
    Abstract :
    The first generation of the network of English algebraists - which led to Symbolic Algebra - is well known for having succeeded in imposing the differential notation of infinitesimal calculus at Cambridge, in place of Newtonian notation, essentially since their translation of the Traité Elémentaire du Calcul Différentiel et du Calcul Intégral (1802) by Sylvestre-François Lacroix (1765-1843).
    At the same time, Charles Babbage (1791-1871) published a very important two-part article: "An Essay towards the Calculus of Functions" .
    In fact, these landmark works are the visible part of the abundant publications and exchanges between John F.W. Herschel (1792-1871) and Babbage between 1812 and 1820. Beyond their constant preoccupation with the question of notation, they envisaged using processes of invention, notably analogy, to establish a new calculus on functions, generalizing practices inherited from the Continent. I will examine the advances and failures of such a project.

Friday, 22nd March 2024 from 2 pm to 5.30 pm
Continuous modal analysis

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

Brice Halimi (Univ. Paris Cité & SPHERE)
Continuous modality analysis

Friday, 12th April 2024 from 2 pm to 5.30 pm

Dirk Schlimm (McGill University)
Notations, calculations, and proofs
Abstract :
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
Abstract :
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.

Friday, 26th April 2024 from 2 pm to 5.30 pm

Session on co-algebra concepts
Jeffrey Elawani (University McMaster et Université Paris Cité)
Measurement and numbers in Leibniz. A coalgebraic approach to continuous quantities?
Abstract :
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
Abstract :
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.

Friday, 17th May 2024 (this session will take place from 9.30 am to 1.00 pm in room Rothko 412B Condorcet building Université Paris Cité 4 Rue Elsa Morante Paris 75013)

François Lê (Institut Camille Jordan, Université Claude Bernard Lyon 1)

Nicolas Michel (Bergische Universität Wuppertal)

Friday, 31st May 2024 from 2 pm to 5.30 pm. Room to be defined
Additional session

Friday, 10th June 2024 (this session will take place from 9.30 am to 1.00 pm in room 569 Olympe de Gouges building Université Paris Cité 8 Rue Albert Einstein)

. . . . . . . .


Building Condorcet, University Paris Cité, 4 rue Elsa Morante, 75013 - Paris*.Map
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