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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 : Simon Decaens (Univ. Paris Diderot & SPHERE), Emmylou Haffner (Centre A. Koyré, Univ. de Lorraine, & SPHERE), Eleonora Sammarchi (Univ. Paris Diderot & SPHERE)

SCHEDULE 2017-2018 : Details and abstracts soon online

Sessions as usual on Mondays, 9:30–17:00, in Room Klimt (366A), 3rd floor,
Building Condorcet, Paris Diderot University, 4 rue Elsa Morante, 75013 Paris. Campus map with access.

Date Thema Organisation
9/10/2017 Trigonometry and maths related to astronomy K. Chemla & A. Keller
13/11 Algebra and geometry. Guests : J. Gray, P. Nabonnand, A. Brigaglia E. Haffner & N. Michel
4 to 7/12 Practices of Mathematical Reasoning C. Proust & al.
15/01/2018 Teaching and rigor C. Proust
12/02/2018 Duality S.Decaens
19/03 On the works of Legendre K. Chemla
9/04 Algebra and arithmetics M. Houg
14/05 Proportions and Reports S. Rommevaux-Tani, A. Malet
4/06 Geometry and computation N. Michel

October 9
Mathematical Practices in Astronomy


  • A. J. Misra (Tamas, Observatoire de Paris)
    The computational procedures in reconstructing the Sine tables of Amṛtalaharī of Nityānanda.
    The Amṛtalaharī (or perhaps, the Kheṭakṛti) of Nityānanda is an undated Sanskrit manuscript found in the collection of the University of Tokyo discovered by Prof. Pingree. This manuscript contains astronomical tables for computing calendrical elements (like tithis, nakṣatras, yogas), right and oblique ascensions tables of the zodiacal signs, the planetary equation tables and mean motion tables. In addition to this, it also includes a Sine table (sinus totus of 60), solar declination table (maximum obliquity of 24 ;0,0), lunar latitude table, and three shadow-length tables for gnomons of height 7, 12, and 60 digits. This last set of tables are collectively presented for arguments one to ninety degrees or arc over three folia. In this talk, I discuss the mathematical techniques and generative algorithms in recomputing the Sine table within the context of its relation with the other tables in this collection. .

10:45-11:00 Break


  • Jiang-Ping “Jeff” Chen (StCloud State University)
    The “First” Chinese Treatise in Trigonometry, Its Latin Sources, and the Algebraic Techniques.
    The calendar reform at the end of Ming China (1368-1644) produced many “firsts” in the efforts. The Jesuits involved and their Chinese collaborators faced many challenges to adapt/collate/translate certain existing European treatises into Chinese : coining new terms, introducing foreign concepts, and working with extreme time constraints, to name a few. Included among the products is the “first” trigonometric treatise in China, 大測 Dace (Grand Measure), which was presented to the court in 1631. This work establishes the paradigm of the principles for constructing trigonometric tables in China and inspires many Chinese trigonometric treatises in the centuries that followed.

12:15-13:30 Lunch Break


  • Yiwen Zhu (Zongshan Univ.)
    Writing Mathematics in Thirteenth Century China——The Case Study of Qin Jiushao’s Writing on the Procedure for great inference.
    By contrast with early mathematical documents, a conspicuous feature of writing mathematics in thirteenth century China was the use of diagrams, in which mathematical procedures carried out with counting rods were written down. However, the reasons behind this historical phenomenon are still not clear. This talk aims at shedding light on this issue based on the case study of Qin Jiushao’s秦九韶(1208-1261) writing on the Procedure for great inference with all numbers 大衍總數術, i.e. the so-called Chinese Remainder Theorem. In Qin Jiushao’s famous monograph, Mathematical Book in Nine Chapters (Shushu jiuzhang, 數書九章, completed in 1247), the Procedure for great inference with all numbers was used in the first nine problems. But among them only the first problem about Book of Changes contained the graphic explanation of the Procedure for great inference looking for one 大衍求一術, i.e a method for solving linear congruences, that is the core of Procedure for great inference with all numbers. On the other hand, in the twelfth problem about calendric calculations there was no Procedure for great inference with all numbers, but the Procedure for great inference looking for one was written down with diagrams. Hence, I will focus on the first and the twelfth problems. By analyzing these two problems, I find that their modes of using diagrams to write the procedure Procedure for great inference looking for one are different : in the first problem, diagrams are written being intertwined with texts, and in the twelfth problem diagrams are written with lines. Relying on the study of modern historian of mathematics Li Jimin李繼閔(1938-1993), we know the twelfth problem could be real, and the first problem was artificial. Therefore, the writing mode in the twelfth problem could imply Qin Jiushao borrowed the procedure from others, and the writing mode in the first problem implies it was Qin’s own procedure.
    In Qin Jiushao’s preface and the twelfth problem, he stated the Procedure for great inference looking for one came from calendric calculations, but astronomers used it as Fangcheng procedure, i.e. a method for solving liner equations. We know that a Fangcheng procedure was recorded in mathematical books as early as in the Nine Chapters on Mathematical Procedures (Jiuzhang suanshu九章筭術). In order to reveal how Procedure for great inference looking for one and Fangcheng procedure are connected, I make an assumption about the astronomical Fangcheng procedure. From this point of view, we will see how Qin Jiushao improved this old Fangcheng and made it into Procedure for great inference looking for one.
    The necessity of using diagrams could be understood in Qin Jiushao’s classification of mathematical knowledge. According to this classification, mathematics was divided into two parts : inner procedures and outer procedures. Outer procedures, like in the Nine Chapters on Mathematical Procedures, were written down in Chinese characters, and carried out with counting rods. Inner procedures were not recorded. Therefore, when an inner procedure was written, the best way was to use counting diagrams and present the whole process using counting rods. This was exactly what happened in Procedure for great inference as presented in the second problem.

13 novembre 2017
Algèbre et Géométrie

  • Jeremy Gray (Open University & Warwick University)
    Brill and Noether : Cayley—Bacharach, Riemann—Roch.
    In the second half of the 19th century mathematicians began to look for theorems about algebraic curves with complicated singularities. Part of this activity was intended to extend the work of Plücker, part of it was meant to rederive Riemann’s ideas in the setting of plane algebraic curves. As this talk will explain : what is geometry, what is algebra, what is assumed, and what is proved was never fully agreed.
  • Philippe Nabonnand (LHSP - AHP, Univ. de Lorraine)
    Titre et abstract à venir
  • Aldo Brigaglia
    The young Segre and the introduction of hyperspatial geometry in Italian algebraic geometry.
    Thanks to Klein and Lie, the concept of abstract (i. e. multidimensional) geometry made great progress, and (after Segre) it became an ordinary tool for the contemporary Italian geometers. Indeed nothing is more fertile than the multiplication of our intuitive powers operated by this principle : it is as if besides the mortal eyes with which we examine a figure, we have thousands of spiritual eyes to observe its manifold transfigurations ; all this while the unity of the object shines in our enriched reason, and it enables us to easily go from one form to another. These words of Federigo Enriques (written in 1922, only two years before Segre’s death) are important to understand the role played by Segre in introducing a more abstract (but not less intuitive) point of view in algebraic geometry. In my talk I will try to give an overview of Segre’s first papers and how, starting with his dissertation, he faced the problem of developing a complete theory of the projective geometry of hyperspaces by using some of the most important algebraic results that Weierstrass, Kronecker and Frobenius had obtained and applying them to the purely geometric interests favoured by the Italian school.

December 4 -> 7 2017
Practices of Mathematical Reasoning

January 15 2018
Enseignement et rigueur

February 12 2018

Guests : Ralf Krömer (Bergische Universität Wuppertal), Emmylou Haffner (Bergische Universität Wuppertal & SPHERE). À confirmer.

March 19 2018
Autour des travaux de Legendre

April 9 2018
Algèbre et arithmétique
Invités : Kenneth Manders (University of Pittsburgh), Sébastien Maronne (IMT, Toulouse), Erwan Penchèvre (SPHERE)

May 14 2018
Proportions et rapports

June 4 2018
Géométrie et calcul


Building Condorcet, University Paris Diderot, 4 rue Elsa Morante, 75013 - Paris*.Map
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Metro : lines 14 and RER C, stop : Bibliothèque François Mitterrand or line 6, stop : Quai de la gare. Bus : 62 and 89 (stop : Bibliothèque rue Mann), 325 (stop : Watt), 64 (stop : Tolbiac-Bibliothèque François Mitterrand)

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