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Home > Seminars > History and Philosophy of Mathematics

Axis History and Philosophy of Mathematics

History and Philosophy of Mathematics

The seminar is the meeting point between different SPHere teams that are interested in mathematics. It fosters dialogue between philosophers and historians of mathematics while focusing on textual sources. Speakers are encouraged to make their sources available to the participants.

Coordination : Thomas BERTHOD, Clément BONVOISIN, Simon GENTIL (Université Paris Cité, SPHERE)


Thanks to Charlotte de Varent, Simon Decaens, Marie-José Durand-Richard, Emmylou Haffner, Arilès Remaki, Adeline Reynaud, Eleonora Sammarchi, and Alexis Trouillot, who proudly provided the coordination of the seminar for the previous years.

Archives
2009-2010, 2010-2011, 2011-2012,
2012-2013, 2013-2014, 2014-2015,
2015-2016, 2016-2017, 2017-2018,
2018-2019, 2019-2020, 2020-2021,
2021-2022, 2022-2023

PROGRAM 2023-2024
Seminars are held from 9:00 am to 5:00 pm in Room 628 (6th floor-Olympes de Gouge building) - Université Paris Cité 8 rue Albert Einstein Paris 75013

To join us online, please see the login details at the top of each session’s developer page.

The detailed program will be posted online at a later date.


Monday, 9th October 2023
This session will take place in room 569 (5th floor) Olympe de Gouges building.

Axis History and Philosophy of mathematics’s activities presentation


Program

  • 9.00 am - 9.15 am
    Welcoming participants and Introduction
  • 9.15 am - 10.15 am
    Presentation : Mathematics from the 19th to the 21st century sub-axis
  • 10.15 am - 11.15 am
    Presentation : Mathematics from antiquity and modern age sub-axis
  • 11.15 am - 11.30 am
    Break
  • 11.30 am - 12h30 pm
    Presentation : Practical mathematics sub-axis
  • 12.30 pm - 14.00 pm
    Lunch break
  • 14.00 pm - 14.45 pm
    Mathesis seminar
  • 14.45 pm - 15.30 pm
    Cuneiform mathematical texts reading seminar
  • 15.30 pm - 15.45 pm
    Break
  • 15.45 pm - 16.30 pm
    Reading ancient mathematical texts seminar
  • 16.30 pm - 17.15 pm
    Presentation : History and Philosophy of mathematics workshops
  • 17.15 pm - 17.30 pm
    Conclusion




Monday, 18th December 2023

Curve algebraization
Organization : Paul-Emmanuel Timotei
Simon Gentil (SPHERE/UPC), Brief overview of the use of algebra for a theory of curves between 1650 and 1750.
Abstract :
In this communication, we propose to look at the use of algebra in geometry during the early modern period, particularly with the aim of establishing a theory of curves. We will take Descartes and the publication of his "Geometry," in French in 1637, then in Latin in 1649, and in 1659-1661 as a starting point. We will demonstrate that Descartes’ algebraic manipulations radically transform the geometric landscape of the time while following a certain tradition from the ancients. We will focus on how Descartes legitimizes and organizes the entire set of curves, rendered infinite, through his various classifications. We will also briefly discuss the popularization of the idea of a "curve in general". In the continuation of the presentation, we will comment Leibniz’s work on the "Conic Section" to highlight some issues in Descartes’ algebraic approach, as well as Newton’s work on the classification of third-order lines to pose some epistemological questions related to the use of algebra in the context of a general discourse on curves. In particular, we will address issues of unity, link between curve, equation and coordinates, handling of specific cases, consideration of infinite elements, etc. Finally, we will look at Euler’s work, in particular his method of identifying a curve with an equation, and we will comment the distance between Descartes’ descriptive algebra and Euler’s representative algebra. It will become apparent that algebra does not play the same role in the works of the second half of the 17th century and those of the following century. Understanding this change in status is crucial for comprehending how algebra and geometry intersect, especially in the case of studies on curves.

Claire Schwartz (Institut de Recherche Philosophique, Université Paris Nanterre), The correspondance between curves and equations in Reyneau’s Analyse démontrée
Abstract :
L’Analyse démontrée, written in 1708 by C. Reyneau, a close collaborator of N. Malebranche, is one of the first textbooks including both Cartesian algebra and infinitesimal calculus. Famous geometers like A. Clairaut and J.L-R d’Alembert read it and used it to learn and practice the differential and the integral calculus.
If it is one of the first-generation textbooks about the Leibnizian calculus, it can also be considered as a second-generation treatise on Cartesian algebra that it expanded upon: two of its main features consist of a generalization of the concept of equation that is not restricted anymore to polynomial equations, and of a systematic use of Cartesian coordinates. Reyneau can rely on these two elements to develop a program that the Cartesian Geometry of 1637 started but did not fully accomplish: a systematic study of curves by their equations.
We will therefore examine the goals set by this program, its accomplishments, and the relationship between geometry and algebra it presupposes.

Thierry Joffredo (AHP-PReST at Archives Henri-Poincaré - UMR 7117 and IMJ-PRG UMR 7586), Singular points of algebraic curves: rediscoveries of Newton’s parallelogram method in the second half of the 19th century.
Abstract:
After 1850, in England, Germany or France, some of the mathematiciens who are interested in algebraic curves and their singular points rediscover the Newton’s parallelogram method, which seems then largely neglected, even forgotten, since the past century. "How completely it has dropped out of sight will appear from the uses which can be made of it, and which, it seems to me, must have been most obvious to any writer on curves, or on the theory of
equations, who had really obtained possession of it.", said Augustus de Morgan, obviously surprised, in a lecture read in front of the members of the Cambridge Philosophical Society in 1855 and later published in the Philosophical Transactions under the title „On the Singular Points of Curves, and on Newton’s Method of Coordinated Exponents“. In this talk, we will shortly expose some of the works of these 19th century geometers on algebraic curves putting into action the Newton’s parallelogram. We will therefore show that these new uses are mostly based on new readings of Gabriel Cramer’s Introduction à l’analyse des lignes courbes algébriques, printed in Geneva in 1750, in which is made extensive use of this method to study infinite branches and singular points of curves, thus illustrating the continuities that exist between the 18th and 19th centuries in geometry.

Les présentations seront suivies d’une table ronde avec les intervenant·es et l’audience du séminaire, animée par Karine Chemla, David Rabouin et Paul-Emmanuel Timotei.

Practical informations : The seminar will take place in Salle 628, Bâtiment Olympe de Gouges (Place Paul Ricœur, 75013 Paris) from 9:30 am to 5:00 pm. If you wish to attend, you will need to request an access badge on the 6th floor at the building reception desk. If you wish to attend the session online, a Zoom link can be provided on request - please write an email to bonvoisin.clement@gmail.com, with "HPM18-12-2023" in the subject line. The link will be sent to you the day before the seminar.



Monday, 8th January 2024

Presentations session
Organization : Clément BONVOISIN
Frederike Lieven (Sorbonne Université)

The "New Math" reform in France and in Germany: the challenges of a comparative analysis
Abstract:
In the 1960s, the “New Math” was a movement that promoted deep changes in the teaching of mathematics. In many countries, it led to reforms of mathematical instruction, in its content and methods. In the 1970s, after a strong public contest, the reforms were gradually abandoned.
The reform movement was international, whereas the reforms took place in a national context. Indeed, in France and Germany, the curricula were elaborated on a national or regional level. This entanglement of international and national factors makes it interesting, but also challenging, to study the “New Math” reform in a comparative way.
First of all, what does the “New Math” stand for? Since the 1950s, the need for reform has become generally accepted among mathematics educators. At that time, the actors spoke about a modernisation of mathematics teaching. However, sources show that there was no consensus on the content of modernisation, and that the term was used with diverse meanings. International meetings and organisations contributed to shaping a more homogeneous understanding of the aims of the reform.
With regard to the institutional aspects, both in France and in Germany, the reform is carried out by a commission. However, there are important differences in the composition and the tasks of these commissions. Differences in the organisation between the two countries raise the question of the sources that the historian can use and of the possibility of making a comparison using different types of sources.
As a final point, I would like to analyse the public debate sparked by the reform, in order to show that the actors involved in the debate, as well as the issues mobilised for the contest, were by no means the same in France and in Germany. This raises the question of the interaction between an international movement and different national contexts.

Elisa Dalgalarrondo (SPHERE, Université Paris Cité)
Gendered representations in the mathematics of the Ladies’ Diary (1760-1784)
Abstract
The Ladies’ Diary is a British almanach published annually between 1704 and 1840. In this periodical, we can find different sections containing enigmas, charades, rebuses and mathematical questions to which members of the readership could directly participate by submitting their questions and solutions to the editor. Few of the contributors could read their own contributions with their name in the magazine issues, while the great majority of them could just see their names among lists of contributors following the published solutions. Initially adressed to « the Ladies », the Diary soon attracted a male readership. The period between around 1760 and 1790 tends to appear in the historiography as a time during which no women contributed to the mathematics of the Ladies’ Diary. Actually, two female names can be found among the frequent contributors of the mathematical questions. The first one is a man’s pseudonym from which we can find published participations. The second one is indeed a woman’s name, but just appeared among the unpublished contributors. In this talk, I will focus on these two cases in order to try to show how studying the mathematics of the Ladies’ Diary can raise some questions in terms of gendered representations.

Clément Bonvoisin (SPHERE, Université Paris Cité)
Failing to apply applied mathematics? On Donald Bushaw’s Ph.D. dissertation (1949 – 1955)
Abstract
In this talk, I will discuss issues pertaining to the application of mathematics. How are mathematical problems and results shaped by technical issues? Conversely, how do hypotheses made by mathematicians raise challenges for the design of instruments? And to what extent can this impede the application of mathematical knowledge, so to speak? In order to address these questions, I will build on the research carried out by US mathematician Donald Bushaw (1926 – 2012) for his Ph.D. thesis at Princeton University.
Back then, Bushaw worked as a consultant in a military-funded project. I will begin by discussing this broader project, what may be said of its practical aims, and the relations it had to the mathematical problem tackled by Bushaw. I will then turn to the mathematical content of the dissertation itself. Here, my focus will be on the differences between Bushaw’s work, and pre-existing research on similar practical problems by German engineer Irmgard Flügge-Lotz (1903–1974). I will show that these differences are threefold: in the relation between mathematics and technics; in the mathematical problem at hand; and in the generality of mathematical hypotheses made on solutions to the problem.
This last aspect will allow me to discuss the application of Bushaw’s results in the context of his project. As I will show, the mathematical generality of Bushaw’s work raised technical difficulties. In turn, these technical difficulties brought social and institutional issues, due to the military context in which Bushaw’s work was carried out. From this perspective, I will discuss intrications between mathematical, technical and social aspects in the application of mathematics, and how this may well cause failures in such applications.



Monday, 5th February 2024

“The Mathematizazion of Nature in the 16th and 17th Centuries” Day
Organization : Vincenzo de Risi

Program

  • 9:30-10:30
    David M. Miller (Auburn University), Catena’s ‘Third Kind of Thing’ in the Quaestio de Certitudine
  • 10:45-11:45
    Álvaro Bo (Academy Vivarium Novum, Rome), Mathematising the unmathematical: Alessandro Piccolomini on the (im)possibility of a quantified science of nature
  • 12:00-13:00
    Michela Malpangotto (Centre Jean Papin, Paris), A revolution before Copernicus. The precopernican astronomy issued from Georg Peurbach’s Theoricae novae planetarum (15th-17th c.)
  • 14:30-15:30
    Sophie Roux (École Normale Supérieure, Paris), Mathematics and Natural Philosophy in Descartes and in the Reception of Descartes
  • 15:45-16:45
    Robert DiSalle (University of Western Ontario), The metaphysics and method of Newton’s Mathematical Principles
    Abstract :
    Newton’s Principia was among the first examples, and eventually an ideal model, of what we now mean by “mathematical physics.” It is a challenge to understand, therefore, the view of some of his contemporaries that this work was only mathematics, and not physics at all. The explanation that is nearest to hand is that Newton’s approach set aside the pursuit of causal explanations: his contemporaries, dedicated to one or another form of the “mechanical philosophy,” would not recognize as a physical theory any mathematical account that, like Newton’s, provided no causal mechanism to explain its mathematical principles. Certainly some of Newton’s own remarks about the mathematical character of his work, and his disavowal of inquiry into the cause of gravity, seem to invite such an interpretation. But this interpretation falsely suggests an abandonment, at the very commencement of mathematical physics, of the very idea of physical understanding in favour of mathematical precision and predictive success. This is very far from Newton’s view of his work, its relation to the mechanical philosophy, and its actual and potential achievements. On the contrary, Newton considered his use of mathematics in physics to be a new and indispensable method for identifying, and understanding, the action of physical causes.
    Few of Newton’s contemporaries grasped the radical novelty of his mathematical method, and this explains some of the difficulties they had in grasping the metaphysical implications of his method. Huygens, in particular— arguably the most accomplished representative of the mechanical philosophy— articulated an ideal of physical explanation and the ways in which Newton’s theory fell short of the ideal. His own theory of gravity was meant to show the conceptual clarity and explanatory power of mechanistic principles. By comparing Huygens’ mechanical method with Newton’s mathematical method, we can see why Newton’s method providing insights into the metaphysics of causation that were hidden from the mechanistic view. In the subsequent evolution of physics, it was the Newtonian conception, properly understood, that illuminated the power of mathematical physics as a means of understanding causes as well as a tool for predicting effects. Newton’s method also shed light on fundamental philosophical problems concerning the applicability of mathematics to the world, including questions about the nature of empiricism and realism regarding mathematical theories, that have preoccupied philosophers even in the realm of post-Newtonian physics.


Monday, 4th March 2024

Algorithms and software
Organization : Agathe Keller

Clément Cartier and Eric Vandendriessche
"Algorithms and software: a dialogue between computer theory and ethnomathematics"

Baptiste Mélès
"Programming in classical Chinese: the case of Wenyan software"

Maarten Bullynck (IDHE.S, Paris 8)
"Un algorithme n’est pas l’autre" Forms and materialities of an algorithm in the 1960s"

Clément Bonvoisin
"Algorithm adaptation and computer implementation: cooperation between mathematicians and engineers at the RAND Corporation (1946 - 1952)"

Abstract :
In this presentation, I will discuss the treatment of an aeronautical problem using computers at the RAND Corporation. Within the framework of this think tank funded by the U.S. Air Force, mathematicians and engineers worked, between 1946 and 1952, to minimize the travel time of airplanes and missiles between take-off and goal. In particular, I want to show how the implementation of optimal trajectory calculation was shaped by the nature of the mathematical model posed, the relationships between mathematicians and engineers, and the computational tools available (a REAC analog computer and a digital computer produced by IBM).

Maarten Bullynck (IDHE.S, Paris 8)
""One algorithm is not the other" Forms and materialities of an algorithm in the 1960s"
Abstract :
Far from being an invariant, an algorithm can have many forms and faces once implemented on a computer. This observation will be illustrated by a specific case: the implementation of an arithmetic screen on a variety of software systems from the 1960s. Depending on the machine and the system used (Multics, Illiac IV or other), the screen will take on other, mutually incommensurable forms, reflecting the conceptual, material and social aspects of its implementation. The question arises as to whether these observations can be generalized to other places and times.



Monday, 22nd April 2024

Generality
Organization : Organization : Simon Gentil
Program

  • 9.30 am - 10.00 am
    Welcoming participants
  • 10.00 am - 11.15 am
    David Waszek (ENS, ITEM)
    Expressive Means and Generality: Some reflections on the use of algebra in geometry in the early seventeenth century
    Abstract : The use of algebra is often said to make geometrical problem-solving more “general”: problems that, in traditional diagram-based geometry, would have required separate treatments then become susceptible of a unified solution. This talk looks at how such issues actually played out in early-seventeenth-century applications of algebra to geometrical problems, with a special focus on thorny issues regarding problem individuation and the geometrical interpretation of negative results. This talk is partly based on work by, and joint work with, Ken Manders.
  • 11.15 am - 11.30 am
    Pause
  • 11.30 am - 12.45 pm
    Agathe Keller (CNRS, SPHERE)
    Commenting on general statements: some working reflections on practices of generality in treatises and commentaries dealing with mathematics in Sanskrit.
    Abstract : What tools can we use to try to describe an actor’s understanding of “generality”, when such an actor does not explicitly state much about this topic, while as historians we are tempted to interpret his or her mathematical statements as reflecting such an aim? This presentation will look at three sets of early mathematical texts in Sanskrit, investigating how authors and commentators understand the grouping of things that might be considered distinct. The two first case studies- one on Āryabhaṭa’s mathematical sūtras (499) and Bhāskara’s commentary on them (629) on the one hand, and Brahmagupta’s sūtras on mathematics and mathematical astronomy (628) and Pṛthūdhaka’s commentary (ca 850)- deals with mathematical objects and processes. The practices of specification (uddiś-) and variation (udahṛ-) of commentators will be a way to reflect on what then characterizes for them the statements that elicit their mathematical work. The third case is a reflection on what appears as an enduring standard of the representation of mathematical computations on a working surface in manuscripts: I suggest to consider them as representing a generic situation of computation, representing many possible computational media and instruments. In other words, this presentation will describe a variety of practices of generality in Sanskrit mathematical sources, a fact worth noting in itself as it goes against homogenizing conceptions of mathematical practices from South Asia.
  • 12.45 pm - 2.30 pm
    Lunch
  • 2.30 pm - 3.45 pm
    Marco Panza (Chapman University and IHPST)
    What generality might have been for Euclid ?
    Abstract : In this talk, I’d delineate a conception of generality, which does not depend on any sort of universal generalization of geometrical objects, that seems compatible with Euclid’s conception of geometry and might explain the way he reaches universal results, tough working on particular instanciations.
  • 3.45 pm - 4.00 pm
    Pause
  • 4.00 pm - 5.15 pm
    Filippo Constantini (CNRS, SPHERE)
    Absolute Generality without a Universal Domain
    Abstract : In this talk, after a brief introduction to the absolute generality debate, I introduce the notion of indefinite extensibility and argue that we can have absolutely general statements about an indefinitely extensible sequence of domains. As a consequence, absolutely general statements do not require the existence of a universal domain of everything.


Monday, 6th May 2024

Classification
Thomas Berthod



Monday, 3rd June 2024

History and philosophy of mathematicians
Organization : Karine Chemla



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