Sciences, Philosophie, Histoire – UMR 7219, laboratoire SPHERE

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 : Marie-José Durand-Richard (Univ. Paris 8, SPHERE).

PROGRAMME 2014-2015 : Mondays, 9:30–17:00,
Room Mondrian (646A), 6th floor, Building Condorcet, Paris Diderot University, 4 rue Elsa Morante, 75013 Paris. Campus map with access.

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Dates : November 17, December 15, January 12, February 9, March 9, April 13, May 11, June 15

October 13, 9:30 – 17:00

The interest of contemporary mathematics in the history of mathematics in objective and subjective sense.

Session (in French) organised by Jean-Jacques Szczeciniarz, on a synthetic text of Alain Connes : ’A view of mathematics’.

How mathematics are constructed and are working throughout history they restore, and what use a historian of mathematics can make of this story. The two issues intersect and engage mathematicians and historians of mathematics. Contemporary means a current state in the making of theories and mathematical results. A text by Alain Connes is the basis for reflection : “A view of mathematics”. One deals with three topics : a architectonic organisation of the mathematical corpus (structural unity), the relationship between algebra and geometry, the relations with physics.

• Jean-Jacques Szczeciniarz (Univ. Paris Diderot, SPHERE)
  Une synthèse sur les mathématiques : quelle unité, quel rapport à la physique. Pour l’histoire et la philosophie des mathématiques.
• Joël Merker (Univ. Paris XI)
  L’ange de la géométrie, le démon de l’algèbre.
• Marc Lachièze-Rey (Univ. Paris Diderot, APC)
  Quelles mathématiques pour quelle physique.

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November 17, 14:00 – 17:30

History of mathematics and mathematics education.

Session (in English) organised by Renaud Chorlay (ESPE Paris IV, SPHERE) in the context of the ERC Project SAW "Mathematical Sciences in the Ancient World".

• Charlotte de Varent (ERC Project SAW)
  Relations between history of mathematics and mathematics education : a case study.
  We will present elements from our PhD work in progress, developed in the context of the SAW - ERC project (Mathematical Sciences in the Ancient World) lead by C. Proust in history of mathematics, and co-tutored by N. Décamp in science education.
  On the case of the area of the square, we will present the construction of a classroom experiment based on a dialogue between ancient texts and contemporary textbooks.
  The ancient texts will be : a clay-tablet from Nippur (paleo-babylonian period) ; an excerpt from the Nine chapters and its commentaries (Han dynasty, written on the basis of texts from before the Qin) ; an excerpt from a VIIth century commentary by Bhāskara, on an Vth century astronomical treatise, the Āryabhaṭīya of Āryabhaṭa.

• Renaud Chorlay (ESPE, SPHERE), et Cécile de Hosson (LDAR)
  History and science and mathematics education : methodological issues.
  The two disciplines have different empirical domains of investigation, different ways of validating claims, and different reference works ; thus, some structural problems need to be addressed before we begin to engage in fruitful dialogue. This talk will be given by two speakers, one working in physics education research, one working in the history of mathematics. We wish to spell out some prerequisites for collaboration, and point to potential avenues for research.

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December 15, 9:30 – 13:00

On the methods of approximation and on questions of discretization.

Session (in French) prepared with the ERC Project SAW "Mathematical Sciences in the Ancient World" organised by Christine Proust, Maarten Bullynck and Marie-José Durand-Richard (SPHERE).

• Nadine de Courtenay (Univ. Paris Diderot, SPHERE) [in English]
  The gift of mistaking. Approximation and uncertainty as modalities of the mathematical translation of physical phenomena.
• Maarten Bullynck (Univ. Paris 8 and SPHERE) and Marie-José Durand-Richard (SPHERE) [powerpoint in English, speak in French]
  Douglas R. Hartree (1891-1958) : les méthodes d’approximation à m’épreuve de l’ordinateur
  D.R. Hartree both witnessed and contributed to the transition from analogue machines (the differential analyser) to digital machines, from desk machines to computers. This influenced both his practices and his thoughts about mathematical physics and calculating machines. We will examine the impact of this transition on approximation methods as Hartree viewed it, both in Calculating Machines (1949) and in Numerical Analysis (1952).

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January 12, 9:30-17:00

The evolution of mathematical proofs as choices by their authors.

Session organised by Eleonora Sammarchi.
This session deals with types of mathematical proofs. Continuing some considerations about the notion of demonstration for second degree equations which were developed during last year’s session Algèbres arabes et gréco-latines du IXe au XVIe siècles, we will focus now on some questions such as : how do demonstrations evolve in time and space ? Which reasons can influence mathematicians in choosing a certain type of proof rather than another one ? What does rigour mean in mathematical proofs ? Two case studies will be given as a contribution for formulating a satisfactory answer for these kind of questions.

• Eleonora Sammarchi (Univ. Paris Diderot, SPHERE)
  Short reminder on thought elements ht raised last year concerning "new demonstrations" of the resolution algorithms quadratic equations.
• Wang Xiaofei (Univ. Paris Diderot, SPHERE)
  The demonstrations in a purely analytic way of Lagrange.
  A the turn of the 19th century, Lagrange undertook to give the infinitesimal calculus a basis on algebra with two important works, the Théorie des fonctions analytiques and the Leçons sur le calcul des fonctions, of which the first editions appeared respectively in 1797 and 1801. Both of them were published as a result of his teaching on analysis at the École Polytechnique from 1794 to 1799. And the latter was indicated as a supplement to the former. Through a detailed study of the two works and a comparison between the demonstrations, in this présentation, I would try to answer the questions raised in today’s session.
• Alberto Cogliati (Università degli Studi di Milano)
  Algebra vs Geometry in the domain of mathematical proof : a case study from the theory of Lie groups.
  For their own nature, Lie groups position themselves at the crossroad of Algebra and Geometry. Consequently, some of their properties can equally well be characterized by means of algebraic and geometrical tools. Their history reects this twofold nature and somehow testifies a great variety of attitudes concerning technical choices and proof strategies. In my talk, I will present two examples of this fact both taken from Cartan’s work on the theory of Lie groups. The first one deals with the classification problem of real Lie groups (algebras), the other deals with Cartan’s collaboration with J. A. Schouten on the geometry of Lie groups.
• Marion Cousin (Univ. Paris Diderot, SPHERE)
  Mathematical Proof in American and English Textbooks on Elementary Geometry during the second part of XIXth century, and its Introduction in Japanese Education during Meiji Era (1868-1912).
  During the Meiji era, the political context in East Asia led the Japanese authorities to embark on a nationwide modernization program. This resulted in the introduction of Western mathematics, and especially elementary geometry into Japanese education. However, as traditional mathematics (wasan 和算) were very successful, there were no Japanese translations of texts dealing with this new geometry available.
  In this session on demonstrative methods, I will analyze the proofs presented in various textbooks of the XIXth century that contributed to the introduction of Western elementary geometry in Japanese education. I will focus on commentaries of procedures extract from Japanese was an textbooks of the beginning of XIXth century, on proofs of theorems presented in English and American textbooks translated in Japan during the Meiji era, and, finally, on the translations of these proofs in Japanese textbooks of the end of XIXth century.

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February 9, 9:30-17:00

Session prepared with the ERC Project SAW "Mathematical Sciences in the Ancient World"

1rst part, follow-up to "On the methods of approximation and on questions of discretization" (December 15, 2014).
Session prepared with the ERC Project SAW "Mathematical Sciences in the Ancient World" organised by Christine Proust, Maarten Bullynck and Marie-José Durand-Richard (SPHERE).

• 9:30-10:45 Nadine de Courtenay (Univ. Paris Diderot, SPHERE)
  The gift of mistaking. Approximation and uncertainty as modalities of the mathematical translation of physical phenomena. (in English)

2 nd part, 11:00–17:00, Mathematics, Astral science and Trigonometry.
Session prepared with the ERC Project SAW "Mathematical Sciences in the Ancient World" organised by Agathe Keller (CNRS, SPHERE & ERC Project SAW)

• 11:00-12:15 Clemency Montelle (Un. Canterbury, N.-Z., and researcher invited by the ERC SAW project)
  Some reflections on trigonometry in the Sanskrit astral sciences. (in English)
  Solar declination is a very important concept in astronomy and relies heavily on trigonometry. A twelfth century Indian scholar, Āmarāja, gives us insight into the various nuances surrounding its role and reckoning in his commentary to the Khaṇḍakhādyaka by noted seventh century CE Indian astronomer Brahmagupta. Āmarāja devotes a large part of his commentary to providing the background computations and underlying mathematical techniques to declinations presented in this work. His exposition includes detailed mathematical derivations, second order interpolation of sines, basic operations of arithmetic, and allusions to the geometry of the sphere. We explore these mathematical details and their role in this astronomical topic and explore the language invoked in these procedures.

• 12:30-13:15 / 14:15–14:45 Nathan Sidoli
  (Waseda University, Japan, and researcher invited by the ERC SAW project)
  Episodes in Greek Trigonometry. (in English)
  In this talk I will look at a number of calculations using trigonometric methods in Greek sources. I will begin with some proto-trigonometric work found in Aristarchus and Archimedes, then look at the way that Ptolemy handles trigonometry in his astronomical writings, and finally look at discussions of the computational techniques in late ancient authors.

• 15:00-16:30 Philippe Abgrall (CNRS, CEPERC)
  Le recours au théorème des sinus pour résoudre certains problèmes liés à l’astrolabe, par Ibn ‘Irāq. (talk in French, documents in English)
  In some of his treatises, Ibn ‘Irāq (second half of the 10th century) deals with the astrolabe’s construction or with some of its uses. The problem of constructing height circles (or azimuth circles), the great circles which are orthogonal to the horizon circle on the sphere, is integrated into treatises on the astrolabe only from the 9th century onwards. It is the most difficult problem when constructing an astrolabe. We shall examine some situations which Ibn ‘Irāq solves by applying the sines’ theorem which he had elaborated in his works on spherical trigonometry. In another problem concerning the use of the astrolabe this time, Ibn ‘Irāq will resort to this theorem to demonstrate al-Ṣaghānī’s solution which was criticized by al-Harawī.

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March 9, 9:30–17:00

Operations and Objects.

Session (in French) organised by Emmylou Haffner (Univ. Paris Diderot, SPHERE)

• Frédéric Jaëck (Univ. Paris Diderot)
  Some examples of the use of the word ’operation’ in the shaping of functional analysis.
  Starting with symbolical algebra in the first part of the 19th century we will trace through a couple of examples the use of the word ’operation’ in some key steps in the shaping of functional analysis. We shall show the links with arithmetical operations and move torwards some abstract use of the idea of operation.

• Catherine Morice-Singh (Univ. Paris 3 et SPHERE)
  Operations and classification of numbers in ancient texts of the Jaina tradition (India - before the 10th century A.D.)
  The Jaina tradition shows a deep commitment to using quantification to describe the different components of the cosmos. Some of the computations Jaina scholars made for this purpose led them to express very large numbers going beyond the reach of ordinary numeration, and to confront the notions of the innumerable (asamkyeya) and the infinite (ananta).
  In this presentation, we intend to show that the theories Jaina scholars devised for handling such notions were neither vague nor naïve. On the contrary, they were based on precise definitions and a coherent mathematical treatment. We will present these operations and algorithms (as they appear in ancient texts), which allowed the authors to establish a unique classification of numbers.

• David Rabouin (CNRS, SPHERE)
  Operations and Numbers in John Wallis’s works. [talk in English]
  Since Jakob Klein’s Greek Mathematical Thought and the Origin of Algebra, John Wallis has been regularly presented as a turning point in the conception of numbers in mathematics. For the first time, claimed Klein, numbers were seen as apurely symbolic entity, characterized by and in a system of operations. This would correspond to a radical change in the very concept of what a mathematical object is and a culminating point in the development of the so called “symbolic algebra”. By transforming mathematical objects into symbolic entities, a path was opened for a new form of reflexivity in mathematics, operations being themselves symbols upon which one can operate through increasing levels of abstraction. In this talk, I would like to challenge this reading by coming back to Wallis’s texts, paying particular attention to the way in which he described what he called the “foundations of algebra” and the role played in it by the structure of symbolic writing.

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April 13, 9:30-17:00,

: : Morning 9:30–13:00 : Historiography : Comparing in History of Mathematics
Organized within ERC Project SAW

• Karine Chemla (CNRS, SPHERE & SAW) & Agathe Keller (CNRS, SPHERE & SAW)
  Introduction [in English]

• Pierre Chaigneau (Univ. Paris Diderot, SPHERE & SAW)
  Comparisons around Egyptian mathematics in Otto Neugebauer and Kurt Vogel’s dissertations. [in French]
  In 1926, then in 1929, that is, after an interval of only two years, Otto Neugebauer and Kurt Vogel (respectively) publish a dissertation about the history of mathematics in ancient Egypt. Both works rely widely on an analysis of the Rhind mathematical papyrus, especially on the so-called 2/n table. The proximity of both texts, in time, themes, in the situation of their author, makes them ideal sources for an inquiry on the historiographical trends in Germany at the end of the twenties. We propose here to contrast the ways in which the authors use comparison. What is compared ? What for ? Is it possible to deduce from that some differences between the approach of the Munich group, to which Vogel belongs, and these of the Göttingen group, to which Neugebauer belongs ? Such are the questions we aim to explore.

• Chen Zihui (CNRS, SAW)
  Wylie’s Comparative Study on Sino-Western Transmission of Astral Sciences in Ancient China : The Case of the Synthesis Study on the Weekly Calendrical System. [in English]
  Alexander Wylie’s (1815-1887) study on the Weekly Calendrical System in ancient China is mainly embodied in his “On the Knowledge of a Weekly Sabbath in China” (1871, hereafter “Sabbath in China”). This work was a subsequent research, which originated from the study on the Nestorian Tablet which was carved in the 7th century and unearthed in the 17th century (“The Nestorian Tablet in Se-gan-Foo” in North-China Herald, 1854-1855). At the same time (1855), Li Shanlan (1810-1882), a mathematician and one of the closest collaborators of Wylie in China, calculated by using the method of calendrical system in 7th century China, and confirmed that the obscure “day of da yaosenwen” was exactly the Sunday. Besides quoting Li Shanlan’s work in the “Sabbath in China”, Wylie made use of historical texts and the method of comparative philology, and synthesized the works by Alexander von Humboldt (1769－1859), Édouard Biot (1803-1850), Augustus De Morgan (1806-1871) etc., historicizing the knowledge of a weekly calendrical system in ancient China. The “Sabbath in China” also had an influence on some Sinologists later such as Édouard Chavannes (1865-1918) and Paul Pelliot (1878-1945).

: : Afternoon, 14:00–17:00, HPM, with the seminar Mathematics and Philosophy, 19th and 20eth Centuries

• David Rowe (Gutenberg Univ., Mayence)
  On editing Mathematische Annalen, 1872-1928.
  The founding of Mathematische Annalen by Alfred Clebsch (Göttingen) and Carl Neumann (Leipzig) created an important forum for mathematicians in Germany who stood outside the dominant centre in Berlin. Beginning in 1876, Felix Klein and Adolf Mayer served as its principal editors, complemented by Walther von Dyck in 1888. Later, in 1901, David Hilbert joined the Hauptredaktion, supported by Otto Blumenthal from 1905 onward. During these different periods, editorial policies differed greatly, though the Annalen had already by 1880 emerged as one of the leading journals under Klein’s guidance. Both Klein and Hilbert took a strong international orientation, a stance that was gradually undermined by the political activities of L.E.J. Brouwer, who joined the editorial board during the early war years. After describing various aspects of the editorial work behind the scenes, I will take up the tumultuous events of the 1920s that ultimately led to Brouwer’s ouster from the editorial board. My analysis suggests that Hilbert’s motivation for this action had little to do with the famous foundations debate that pitted Hilbert’s formalism against Brouwer’s intuitionism but rather that Hilbert felt compelled to act in order to counteract his rival’s political influence on the future direction of the journal.

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May 11, 9 :30–17 :00

Topological Genesis

Session organised by Pascal Bertin and Ramzi Kebaili, (Univ. Paris Diderot, SPHERE)

• Pascal Bertin, (Univ. Paris Diderot, SPHERE)
  In the neighbourhood of topology : the Hausdorff mystery. [in French]
  In this talk we his talk will start from the observation of the relatively sudden appearance of topological theme in Hausdorff’s writings, and will try to account for this by tracing various milestones that led the mathematician to his axiomatization of topological spaces.
• Jean-Pierre Marquis (Université de Montréal)
  About the devlopment of pointless topology. [in French]
  This talk will examine the history, or prehistory, of pointless topology and will particularly focus on the little known work of Nöbeling.

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June 15, 9:30 – 13:00

Morning : 9:30 – 13:00
Session organised by Pascal Crozet (SPHERE)
: : Les éditions et réécritures des Eléments d’Euclide et la question des fondements

• Pascal Crozet
  al-Qûhî et le quatrième postulat

• Sabine Rommevaux-Tani
  Considérations sur le travail de Campanus (XIIIe siècle) concernant les principes des Eléments d’Euclide et quelques exemples de sa diffusion au XVIe siècle.

• Vicenzo de Risi (Max Planck Institute for the History of Science)
  The Development of Euclidean Axiomatics from Antiquity to the Early Modern Age.

Afternoon, 14:00 – 17:00
Discussion about the programme 2015-16

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