Presentation

**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)

**PROGRAM 2015-2016 **

Mondays, 9:30–17:00, this year in Room Klimt (366A), 3rd floor,

Building Condorcet, Paris Diderot University, 4 rue Elsa Morante, 75013 Paris. Campus map with access.

Go to : 16-11-2015, 14-12-2015, 11-01-2016, 15-02-2016, 7-03-2016, 4-04-2016, 9-05-2016, 13-06-2016

October 12 (tbc)
**On computations. Some analysis**

Session organised by Jean-Jacques Szczeciniarz (University Paris Diderot, SPHERE and GdR Philmath)

Computations (« le calcul ») have been seen in very opposite ways. On one hand, in the division of mathematical labour, it is placed at the lowest, the least noble of mathematical practices. It is the case with some readings of Plato, which put practices of calculs on the side of merchants or craftsmen. In mathematical practice, while some of the most renowned mathematicians were calculating prodigies (e.g. Gauss, Riemann), computations are put in the background, a mechanical activity to be executed by commoners, a common conception which we will call mathematico-philosophical vulgate. It is opposed either to the conception of structures, which would precisely avoid computations and be the true essence of the mathematical elaboration with its great architectures, or to mathematical reflection itself, which ought to be the realization of conceptual mathematics. What is less explicitly noticed, is that computations have been seen as an activity devoid of meaning *par excellence*, an empty thought so to speak.

On the other hand, the philosophy of mathematics has developed a very different conception of calculus. It has been seen as representing, contrary to what was just said, the essence or at least an essential form of the mathematical thought, of its practice, and even of its theory. Non only because it represents, often in a rather rough way, a mark of objectivity, but also because the computational achievements which paved the history of mathematics make light of very highly elaborated productions.

We wish to begin to show, through these talks, that the first conception often relies on a certain ignorance and a form of normativity deeply rooted in the history of mathematics and of philosophy. In all the mathematical practice, computations play a role, which we will highlight, in mathematical disciplines themselves and in various ways. The descriptive and analytical sketches that we will give will cover pur computations and disciplinary calculus, shared forms and differences. The goal will be to show that mathematical practice is also a freeing of computation. The analyses proposed will be of two kinds : those coming from the practice of computations, and those coming from computations in computer science. The idea is that the differences between these two kinds of analyses are not essential. The last two talks make hypotheses on the nature of calculus which must highlight the first two.

- 9:30 Jean-Jacques Szczeciniarz (University Paris Diderot, SPHERE)
*Introduction. Toward a concept of computations.*

- 10:00 Joël Merker (University Paris-Sud)
*The practice of computation as a demand for mathematical syntheses.*

The first part of the talk shall describe, by means of some accessible elementary examples, the dynamical and mobile features of mathematical computations, first examined in their essence, and then analyzed as they inscribe in an irrevocable temporality which, as a means of knowledge, take them away from any legitimate platonistic desire of immobile internalization.

Next, I will attempt to exhibit the mysteriously irreversible character of invariant mathematical results that can only be attained by means of a targetted computation — Gauss’ Theorema Egregium for instance produces ’the’ formal completed symbolic representation of the concept of curvature —, when the importance of a mathematical concept is not known in advance, the metaphysical issues possibly becoming aporetic, since the yellow line between the ’before’ and the ’after’ traverses every instant of computation.

Lastly, by focalizing on a few contemporary questions (still very open) concerning the nature of certain cohomology groups with values in high rank vector bundles above projective manifolds of large dimension, I will attempt to make clear why and how the idealization of a ’brute force’ computational power encounters limits that are stably due to the resistance of mathematical syntheses, the structured ontology becoming more extrinsic that one would believe, because the exponentiality without concept always runs into impossible totalizations.

- 11:15-12:15 Jean-Jacques Szczeciniarz (University Paris Diderot, SPHERE)
*Computations as an expression of the mathematical form.*

We will examine the effects of computations in the mathematical adventure : the case of the*theorema egregium*, Gauss’ extraordinary theorem, and insist on calculus from the viewpoint of its forms of consubstantiality in the mathematical structures : some examples of computations "à la Cartan"

- 13:30-14:30 David Rabouin (CNRS, SPHERE)
*Logic, mathematics and calculus in Leibniz.*

- 14:45-15:45 Maël Pegny (University Paris 1 Panthéon Sorbonne)
*Quantum calculus, brute force and computational utopia.*

Does quantum calculus rely on brute force ? We will try to clear the vast mathematical stakes which are hidden behind the apparent technicity of this question, by showing what it implies for the foundations of computational complexity.

- 16:00-17:00 Franck Varenne (University of Rouen & GEMASS–UMR 8598)
*Computation as emulation and simulation.*

I would like to show that the omnipresence of the computer in the contemporary practices of modelisation is not only due to the fact that it can be an incredible number cruncher, neither to the sole fact that it can be a flawless logical machine. I will defend the idea that its ever pressing omnipresence is due to the instability and to the ambivalence that it can each time confer to the computations which has been delegated to it, and to the results of this computation. In empirical sciences, this ambivalence is indeed positive because it solidifies the epistemic contribution of the computer instead of weakening it : computation can be considered and treated either as emulation, or as simulation, or both simultaneously albeit from different angles. In the formally more composite computing simulations, the effect of this ambivalence is to combine, enrich, and by this stabilize the epistemic functions of these*simulats*. By doing so, the computation allows spaces in which of computations and reasonings overlap and interact in ways that we consider to be hitherto unknown.- References :

Dowek Gilles,*Les métamorphoses du calcul*(2007), Paris, Le Pommier, 2011.

Dowek Gilles,*La logique*, Paris, Le Pommier, 2015.

Varenne Franck,*Qu’est-ce-que l’informatique ?*, Paris, Vrin, 2009.

« La surprise comme mesure de l’empiricité des simulations computationnelles », in Claudia Serban et Natalie Depraz (dir.),*La Surprise. A l’épreuve des langues*, Paris, Hermann, 2015, pp. 199-217.

« La reconstruction phénoménologique par simulation : vers une épaisseur du simulat », in Parrochia D. et Tirloni V.,*Formes, systèmes et milieux techniques après Simondon*, Lyon, Jacques André, 2012, p. 107-123.

“Chains of Reference in Computer Simulations”, Tech. Report, FMSH-WP-2013-51, GeWoP-4, 2013, p. 1-29.

- References :

November 16
**History of reviews**

Session organised by Simon Decaens

- Michèle Audin (prev. IRMA, University of Strasbourg, CNRS)
*War and reviews*

Les journaux de recensions d’articles mathématiques,*Zentralblatt*et*Math Reviews*, sont nés dans des contextes politiques particuliers. Celui de l’Allemagne des années 1930 pour le premier, de la guerre en Europe pour le deuxième. J’évoquerai bien sûr ces naissances successives, mais je me concentrerai surtout sur l’étude de l’effet de la guerre sur les recensions d’une célèbre note d’André Weil en 1940 par les différents journaux.

- Barnabé Croizat (University of Lille 1)
*Création et débuts du Bulletin de Darboux : recensions, mélanges et influences.*

Malgré son surnom de*Bulletin de Darboux*, le "Bulletin des Sciences Mathématiques" doit en grande partie sa création à la volonté d’un homme : Michel Chasles. Tant par sa forme de publication que par l’objectif de ses rédacteurs, le*Bulletin*constitue une entreprise novatrice qui s’inscrit dans le contexte de l’émergence d’un sentiment de déclin des mathématiques françaises, avant tout face à la voisine rivale : la Prusse. Notre étude se concentrera sur les 8 années 1868-1875 : nous commencerons par détailler l’émergence du "projet de création d’un Bulletin" dont le lien avec la mise en place d’une institution nouvelle, l’Ecole Pratique des Hautes Etudes (1868), est resté mal appréhendé. Puis nous insisterons sur les buts mis en avant par les rédacteurs quant au rôle d’un tel*Bulletin*, ainsi que sur les nombreuses difficultés qu’ils rencontrent pour mettre en marche leur journal durant les premières années. Les recensions d’ouvrages et de mémoires apparaissent à la fois dans ces buts et dans ces difficultés. Après avoir souligné la place des recensions dans la composition du*Bulletin*, nous présenterons une étude du contenu des 9 premiers tomes (1870-1875) qui visera à analyser la représentation des différentes disciplines mathématiques dans les recensions des mémoires parus dans les périodiques. En comparant avec une section du*Bulletin*non dédiée aux recensions, nous tenterons de décrire l’influence de Darboux sur la composition du journal. Enfin, un regard porté sur la partie du*Bulletin*dédiée aux recensions d’ouvrages parus indépendamment hors des périodiques nous amènera à critiquer les limites d’une analyse centrée sur la statistique à l’image de notre propre étude des recensions du contenu des périodiques.

- Simon Decaens (University Paris Diderot, SPHERE)
*Some reviews of algebra treatises from the 1930s in two american journals.*

In this presentation, I would like to highlight a few uses of reviews for the history of mathematics, presenting in particular the review as a specific reading. To do so, I will rely on examples of reviews of algebra treatises published in the 1930s in two american journals : the*Bulletin of the American Mathematical Society*and the*American Mathematical Monthly*.

- Anne-Sandrine Paumier (IHES)
*Recensions et réceptions de la théorie des distributions de Laurent Schwartz, une étude de cas.*

The reception of Laurent Schwartz’s theory of distribution is a collective phenomena, in which he participates actively. We will focus on one of these actions : the way in which he norms the reception of his theory by the intermediary of reviews in*Mathematical Reviews*between 1947 and 1958.

December 14
**Historical approaches to the editions of complete works of mathematicians in Germany in the 19th century**

Session organised Karine Chemla (CNRS, SPHERE & ERC Project SAW) in the framework of the ERC Project SAW "Mathematical Sciences in the Ancient World".

- 9:30–9:45 Introduction

- 9:45–11:15 Emmylou Haffner (University of Lorraine)
*Some remarks on Dedekind and Weber’s edition of Riemann’s*Gesammelte Werke.

In 1876, the collected works of Bernhard Riemann were published, with additional manuscripts from his Nachlass, by Richard Dedekind and Heinrich Weber. The edition of Riemann’s works was a long and difficult work, which took Dedekind and Weber more than two years to complete. Indeed, as the letters exchanged by Dedekind and Weber tell us, the state of Riemann’s manuscripts made it necessary for them to thoroughly work through the manuscripts, sometimes struggling to understand what Riemann was doing.

We will consider some elements of their correspondence and highlight key points to understand the process of edition through which Dedekind and Weber went, such as the steps followed to unfold Riemann’s texts until publishable versions could be obtained. We will suggest that it is important, here, to elucidate to what extent Weber and Dedekind’s very thorough editing work led to publish adapted or even rewritten versions of Riemann’s texts, and to clarify whether Dedekind and Weber’s reading of Riemann could have had an influence on our own reading, through their re-appropriation of Riemann’s texts that accompanied the edition of the manuscripts.

- 11:30–13:00 Maarten Bullynck (University Paris 8)
*The edition of Gauss’ works and Gœttingen’s self-portrayal.*

The talk wants to explore some aspects of the edition of Gauss Werke (1863-1933). This edition was first under Ernst Schering’s direction, later under Felix Klein’s supervision. We will look at the scholarly apparatus developed during this edition and its evolution. The focus will be especially on the interaction with contemporary mathematics and the development and promotion of a proper Goettingen school of mathematics.

- 14:00–15:30 Christophe Eckes (University of Lorraine)
*On the publication of the complete works of Hermann Minkowski by David Hilbert, Andreas Speiser and Hermann Weyl.*

We will first describe the close links between Minkowski, Adolf Hurwitz et Hilbert, as can be seen from the letters from Minkowski to Hilbert which were written between 1885 and 1908. These letters have been published in 1973 by Lily Rüdenberg and Hans Zassenhaus. Then we will examine the*Gedächtnisrede*given by Hilbert at the beginning of may 1909 in tribute to Minkowski. This*Gedächtnisrede*is reproduced at the very beginning of Minkowski’s collected papers. More generally, we will study the overall structure of Minkowski’s Gesammelte Abhandlungen. In particular, we will try to identify the choices made by the publishers in the composition of Minkowski’s collected papers before studying their early reception. Finally, we will try to measure the impact of Minkowski’s work on Hermann Weyl, who was one of the publishers of Minkowski’s collected papers.

Round table with Karine Chemla and (Pierre Chaigneau (Univ. Paris Diderot, SPHERE, & SAW Project)

January 11
**History of the historiography of algebra**

Session organised Karine Chemla (CNRS, SPHERE & SAW Project) in the framework of the ERC Project SAW "Mathematical Sciences in the Ancient World".

- Jens Hoyrup (Roskilde University)
*What is "geometric algebra", and what has it been in historiography ?*

This title is adapted from that of Hans Freudenthal’s contribution to a famous debate dealing not least with the interpretation of ancient Greek geometry through the concept of "geometric algebra". Freudenthal argued that through this concept it was possible to point to an underlying unity of algebra, in spite of historical transformations. The present contribution, looking at the main creators and users of the notion of a "geometric algebra" since the 1880s, that their explanations of what they intend by the term and their use of it changes so fundamentally from one of them to the other that it is difficult to argue for even an underlying unity, and that the use of shared words means little more than, exactly, shared words.

- TIAN Miao (Institute for the history of natural sciences, CAS, Beijing)
*Western and Chinese Algebraic Methods in 19th Century Chinese Historical Works.*

At the beginning of 18th century, European algebra, Jie genfang, was transmitted into China by Jesuit missionaries. It was soon accepted by Chinese mathematicians. It is based on the study of European algebraic method, the old Chinese Tianyuan method was rediscovered, and the two methods were regarded as identical at the beginning. In the second half of the 18th century, in the context of research and publication of ancient works in China, most of old Chinese mathematical books became accessible to mathematicians and scholars. Aim at the fully understanding of ancient knowledge and texts, some Chinese mathematicians focus on the study of 13th century Chinese mathematical texts concerning Tianyuan algebra, even they fully understand the advantage of European algebra. However, they reach the result that Chinese methods have more advantage than the ones from Europe. at the beginning of the 19th century, there was a hot debate among Chinese mathematicians about the advantage and disadvantage of Western and Chinese algebraic methods. In this paper, I will focus on the following problems :

1) The context of the study of Algebra method in the 19th century China.

2)The attitude of Chinese mathematicians toward the algebraic method from Europe and China in historical works.

3) The attitude of Chinese mathematicians toward the theoretical knowledge unveiled in their study concerning algebra.

- Marie-José Durand-Richard (SPHERE)
*The English algebraists’ view of historiography of algebra.*

English algebraists of the first half of the 19th century, notably George Peacock (1791-1858) and Augustus de Morgan (1806-1871), initiated a large program restructuring algebra, for both pedagogical, epistemological and even political purposes. Their conception of algebra was founded on an empiricist philosophy and on a view of mathematical language as the culmination of a symbolizing process. So, their views on the history of algebra is articulated on the history of arithmetic. This talk will focus on examining how they considered in a same process the development of algebra in India, in the Arabic world and in Italy.

- Simon Decaens (University Paris Diderot, SPHERE)
*The history of American abstract algebra, two accounts by E. T. Bell and G. Birkhoff.*

In 1938, E. T. Bell presented « modern abstract algebra » as a very recent trend in America. Although « practically all German » in its history, it was carried by a american « vigorous young school ». In this talk, I would like to discuss two different images of this history of an American abstract algebra. First, we will focus on Bell’s history of algebra in the 1930s. We will see how his specific way of presenting the develoment of mathematics by « abstraction » is linked to his definition of abstract algebra as a theory of structures. In a second part, we will use historical papers by G. Birkhoff from the 1970s to look at the same issue from an other point of view. In particular, in both historical presentations, we will question the use of national categories and their link to the history of algebra.

February 15
**The transfer of mathematical knowledge between Europe and East Asia : case studies from the end of the Ming dynasty (China), to the Taisho era (Japan)**

Session organised by Marion Cousin (University Paris Diderot, SPHERE)

The transfer of science from the West to East Asia as well as its dynamics have often been studied by historians of mathematics, especially the transfer to China. During this session, we will highlight four specific approaches to this subject by presenting four well-defined case studies concerning China and Japan. This shall allow us to have a broad but precise understanding of the dynamics and to underline the contrast between the Chinese and the Japanese integration of Western mathematics. ZHOU X. examines the first contacts between East Asia and Jesuit mathematics and discusses the relationship between imported and traditional knowledge by a case study on a mathematical procedure in a work from the early 17th century. CHEN Z. addresses the acceptance/rejection of the Western astronomical and mathematical knowledge among the Confucian academic circle between late 18th century and early 19th century. Then CHEN Z.and M. Cousin discuss about the introduction of Western symbolism during this period, both in Chinese and Japanese case, before M. Cousin talks about the evolution of Japanese mathematical language in the 19th century. H. Kümmerle closes the session with an institutional approach, analyzing the institutionalization of mathematical research in the Japanese universities during the Taishō era (1912-1926).

- 9:30-9:40 Introduction

- 9:40–10:40 ZHOU Xiaohan (Univ. Paris Diderot, SPHERE, & SAW Project)
*How Did Chinese scholars treat the Western mathematical procedures and their similar traditional methods : A case Study on the double false position method in the early 17th century*

In*The Nine Chapters on Mathematical Procedures*which date from of the first century CE, a chapter titled “Excess and Deficit” (Ying Buzu) containing 20 questions and the detailed procedures for treating them is included. Thereafter many scholars made commentaries, subcommentaries, “detailed drafts” (annotations on procedures and their former commentaries), explanations, etc. to this book. In the Ming Dynasty (1368-1644), the method of “excess and deficit” still existed as a principal subject in mathematical works which was in the framework of*The Nine Chapters*and was supplemented with vast analogous questions. In the early 17th century, Christopher Clavius’s (1538-1612) Epitome*Arithmeticae Practicae*was introduced into China by Li Zhizao (1565-1630) and Matteo Ricci (1552-1610). Nevertheless, this is not only a simply literal translation, considering Li compiled some questions of other Chinese mathematical works into it. So the question that how Chinese scholars treated the western mathematical methods can be enquired into from this point of view, especially when the introducers were faced with a procedure which has a counterpart in Chinese mathematics context, like the method of double false position. Li considered the traditional method of double false position was somewhat not as ingenious as the western method. He applied the western method to give an alternative solution to the traditional question. However both the two approaches were given in the traditional canon. Did not Li know this or he just ignored the traditional method because he preferred the western one ? What real new knowledge concerning this method, which has not been found in the tradition, did Li and Ricci bring into China ? Did Li and Ricci use the Jiu Zhang’s classical classification to reconstruct Clavius’s work ? And considering the manual calculation with pen was introduced for the first time by the translation of this book, what difference of practice with the traditional counting rod calculation, alongside the continuity the traditional practice remained, were represented in the method of double false position ? These are questions I want to investigate in this talk.

- 11:00–12:00 CHEN Zhihui (CNRS, SAW Project)
*Confucian scolars’ attitudes towards the Western astral-mathematical knowledge in the early 19th century — a study of on the civil examination and correspondence resources.*

For the acceptance of the newly-imported astral-mathematical knowledge from the West by the scholars in the early-middle Qing dynasty, there are many previous studies on their treatises. In this presentation, I mainly focus on the civil examination and correspondence resources. On one hand, I will study on the correspondences on discussing precession among scholars during Qianlong (1736-1795) and Jiaqing (1796-1820) eras (also abbreviated as Qian-Jia), such as Jiang Sheng (1721-1799), Qian Daxin (1728-1804), Sun Xingyan (1753-1818) and Ling Tingkan (1757-1809). On the other hand, I will take the provincial examination of Jiangnan Province in 1804 as an example, in order to examine why and how the examiner Dai Junyuan (1746-1840) set some questions on the astral-mathematical knowledge in a rigid examination system which was previously thought it should be confined in the neo-Confucianism theories, and to analyse examinees’ answers and the selection criteria. Instead of basing on the treatises materials, these cases can show Qian-Jia scholars’ attitudes towards the Western astral-mathematical knowledge in a different perspective.

- 13:15–14:15 Marion Cousin (SPHERE & SAW Project)
*Writing the Western mathematics : mathematical language and symbolism in Japanese textbooks during the Meiji era (1868-1912).*

In Japan, the introduction of Western mathematics began later than in China, as part of the general movement of modernization during the Meiji era (1868-1912). With the Decree on Education (gakusei 学制), the exclusive teaching Western mathematics and the abandon of traditional practices were ordered. Giving that there was no Japanese book on this new subject, authors had to write, in Japanese, mathematics textbooks based on European and American works.

In this presentation, I will examine the language used in these mathematics textbooks. We will see that a methodic analyze of the evolution of the several aspects of Japanese mathematical language (on the cultural level, on the formulation level, and with purely linguistic analysis – that is to say on syntax and terminology) reveals that the integration of Western knowledge in Japanese mathematical culture was a complex process. It involved several steps, during which we can see, for example, the aspects of the modern mathematical language got fixed one after the other. We will also see that these analyses show the contrasts between the various domains of mathematics on the one hand, and between the Chinese and the Japanese case on the other hand.

- 14:30–15:30 Harald Kümmerle (Martin-Luther-Universität Halle-Wittenberg)
*The institutionalization of higher mathematics in Taishô-era (1912-1926) Japan : the organizational viewpoint.*

The beginning of the Taishō era (1912-1926) coincided with a flourishing of mathematical research in Japan, which is both well known in the secondary literature and can easily be verified by an inspection of quantitative indicators (for example the number of research articles per year). On the other hand, the further development has not been investigated sufficiently so that the narrative still abounds with discontinuities and conflicting explanations.

In this presentation, an analysis of the institutions in which mathematical research took place will be carried out with a focus on the organizational structures. While all of the important institutions - the three Imperial Universities in Tōkyō, Kyōto and Tōhoku, the Mathematico-Physical Society and the Imperial Academy of Sciences – had been established in the Meiji era (1868-1912) and no other independent unit of this size emerged until the early Shōwa era (1926-1989), their relationship and importance for the Japanese mathematical research community as a whole changed dramatically during the years in between. In tracing these changes, I will give an overview of the history of mathematics in Japan against a rapidly changing societal and economic background in the Taishō period. Furthermore, the contributions of key figures like Fujisawa Rikitarō (1861-1933), Hayashi Tsuruichi (1873-1935) and Takagi Teiji (1875-1960) are isolated from each other and set into context.

March 7
**Historical approaches to the history of indeterminate analysis**

Session organised by I. Smadja, A. Keller, K. Chemla (CNRS, SPHERE & ERC SAW Project) in the fremework of the ERC Project SAW "Mathematical Sciences in the Ancient World".

- Ivahn Smadja (University Paris Diderot, SPHERE & SAW Project)
*On Two Conjectures That Shaped the Historiography of Indeterminate Analysis : Strachey and Chasles on Sanskrit Sources.*

This paper is part of a research project on the historiography of mathematical proof in ancient traditions. Its purpose is to shed light on the various ways in which nineteenth-century European scholars attempted to make sense of Sanskrit mathematical sources dealing with indeterminate analysis. Attention will be paid to the historical processes by which these different strands interwove into a cumulative historiography of the field. The focus is on two interpretive conjectures that shaped alternative readings of an evolving corpus of texts, with significantly different emphases and viewpoints.

The British scholar and EIC servant Edward Strachey first identified a consistent algebraic theory in Bhāskara’s Bīja-gaṇita, which he translated from a seventeenth-century Persian manuscript. While reading his sources through the lens of the Euler-Lagrange theory of periodic continued fraction expansions for quadratic irrationals, he offered an insightful interpretation of the so-called cakravāla, or « cyclic method ». Two decades later, in the context of his investigations on the historiography of geometry, the French geometer Michel Chasles delved into Henry Thomas Colebrooke’s translations of Bhāskara and Brahmagupta, from the Sanskrit original, which had become authoritative all over Europe in the meantime. While working out an overall interpretation of Brahmagupta’s theory of quadrilaterals, Chasles incidentally spotted a geometrical construction which opened the way to a geometrical solution of the indeterminate équation Cx^{2}±A=y^{2}. He conjectured that this geometrical way may have been the Sanskrit path to indeterminate analysis. Furthermore, on the basis of textual reconstruction, he supplemented his rigorous interpretive conjecture with a more sweeping historical assumption about a possible transmission of this geometrical approach to algebra, from Sanskrit to European mathematics, through the Arabs and Fibonacci. Owing to further scholarship by Boncompagni, Woepcke and others, the wheat would be sorted out from the chaff.

- Karine Chemla (CNRS, SPHERE & ERC SAW Project)
*Remarks on the historiography of the Chinese remainder theorem in the 19th and 20th century.*

In*Jottings on the Science of the Chinese*(hereafter abbreviated to*Jottings*), which the protestant missionary Alexander Wylie published in 1852, he presented for the first time to a European audience elements of a history of indeterminate analysis in China. His publication also included an outline of a comparison with Sanskrit sources. Ulrich Libbrecht has analyzed the reception of this part of Wylie’s book, showing how it had been mistranslated and thus misinterpreted. Ulrich Libbrecht has further examined the most significant contribution to this field in China, in the 13th century mathematical book by Qin Jiushao. The talk intends to examine the historiographic principles that have presided over the analysis of Qin Jiushao’s work in these various publications, and the part the topic of indeterminate analysis has played in the European historiographies of mathematics in ancient China and of indeterminate analysis.

- Agathe Keller (CNRS, SPHERE & SAW Project)
*Unravelling Libbrecht’s*kuṭṭaka

In Chapter 14 of his 1973*Chinese Mathematics in the Thirteenth Century. The Shu-shu chiu-chang of Ch’in chiu-shao*, Ulrich Libbrecht discusses a reconstruction of what is both a procedure of resolution and the name of the problem to be solved in Sanskrit sources : the*kuṭṭaka*(pulverizer). He then proceeds in Chapter 18 to compare the algorithm he has reconstructed with what he calls the Ta-yen rule. This presentation will look at the basis and sources with which Libbrecht made his reconstructions, hoping to also shed light on how he constructed his comparaisons, and how he understood the arithmetical and algebraical objects of Sanskrit sources he was dealing with.

April 4
**Mathematical work on measurement units**

Session organised in the framework of the ERC Project SAW "Mathematical Sciences in the Ancient World" by Christine Proust (CNRS, SPHERE & SAW Project) and the members of SAW.

This session presents several case studies in very different contexts and periods, showing how mathematical work on the measurement units of is an integral, and sometimes critical, part of the mathematical elaboration. In particular, case studies explore how taking in consideration the measurement units changes the understanding of the nature of numbers and fractions used in treaties and manuals, ancient or modern. The presentations also show that, by focusing on mathematical elements often overlooked such as measurement units, one can open new opportunities for transversal discussions on mathematics practices specific to some crafts, quantification methods developed in preliterate societies, or mathematical contents taught today.

- 9:30-11:00 Marc Moyon (University of Limoges) & Maryvonne Spiesser (Univ. Paul Sabatier de Toulouse)
*Rectangular Surfaces and Units of Measurement with Leonardo of Pisa (13th century*

We focus our work on the first chapter of the Practica Geometriae of Fibonacci, in which the author proposes to calculate the area of rectangles given their dimensions. Although the algorithms used are elementary and well-known, their implementation is not as simple as it may seem. Indeed, the purpose of Fibonacci is to calculate these surfaces regardless of the units and subunits of measurement used. We detail here a ’set of problems’ reflected from units to demonstrate the use of various mathematical tools, including fractions, by the author from Pisa.

- 11:30-13:00 Thomas Morel (Laboratoire de Mathématiques de Lens, Université d’Artois)
*Subterranean geometry and its measurement units (16th—18th centuries).*

Subterranean geometry belonged to the practical mathematical disciplines and aimed at improving and planning mining operations. It developed from the 16th century onwards in the mining states of central Europe. These regions had distinctive juridical and technological systems, which implies, amongst others, that they had specific measurement units. Firstly, I will give an overview of the measuring units that were used and described both in G. Agricola’s*De Re Metallica*and in various mining laws of central Europe. Secondly I will explain how the subterranean geometers (*Markscheider*) would use these units for the resolution of various problems (setting property limits, calculating directions). Thirdly, I will focus on two case studies for the normalization that occurred in late 18th century Saxony : the determination of the fathom (*Lachter*) and the standardization of volume and shape of the mining hoppits (*Kübel*).

- 14:00-15:30 Eric Vandendriessche (CNRS, SPHERE)
*Estimating quantities of yams in the Trobriand Islands.*

As is generally the case in Melanesia (South Pacific), making gardens is the main livelihood of the Trobriand Islanders from Papua New Guinea. Among the tubers grown there, yams are of fundamental importance, and individual performance for cultivating them is socially valued. After the harvest, all gardeners pile up their crops of yams in cone-shaped stacks, in preparation for the ritual of measuring and comparing the sizes of each crop. First, I will describe the Trobrianders’ method for counting yams, which is based on a traditional system of numbers and measurement. Secondly, we will see that body measurement units enable Trobrianders to estimate—before the yam-counting ritual—the quantity of yams contained in each stack, allowing them to make bets on the basis of these estimations.

- 16:00-17:30 Charlotte de Varent (Univ. Paris Diderot, SPHERE & SAW Project)
*Contributions of ancient metrological systems’ diversity to the teaching of today : comparative reflections with elementary textbooks from the fifth grade.*

I intend to investigate the status of measuring units in teaching sequences, introducing the area of the rectangle in the fifth grade. I compare this approach of measuring units with other metrological systems from sanskrit, chinese and cuneiform sources. I use this geometrical example to interrogate the introduction of metrology in elementary teaching of mathematics.

May 9
**The foundations of infinitesimal calculus XVII- XVIII century**

Session organised by David Rabouin

- João Cortese (Univ. Paris Diderot, SPHERE)
*Pascal and the nature of indivisibles.*

In the text*De l’esprit géométrique*, Pascal considers the nature of "indivisibles". Following the definitions of the fifth book of Euclid’s*Elements*, he considers as homogeneous those magnitudes which, being sufficiently multiplied, can surpass one another. This is not the case for indivisibles with regards to their magnitudes, therefore they are heterogeneous. On the other hand, in the works on the cycloid Pascal used a "method of indivisibles" in which one finds quantities "smaller than any given one", that are in fact … divisibles ! Should we agree with L. Carnot that seventeenth century hypothesis about indivisibles are "absurd", but that we shall consider them as "moyens d’abréviation" (*Réflexions sur la métaphysique du calcul infinitésimal*, 1813) ? Modern scholars have argued that Pascal’s indivisibles are not contradictory, provided we don’t consider them literally (Descotes 2001, 2015), or that Pascal’s "indivisibles" are in fact infinitesimals (Malet 1996). In this context, I will discuss the relations between Pascal’s reflection on "indivisibles" and his mathematical practice.

- R. Arthur (Mc Master University, Canada)
*Leibniz’s infinitesimals and their interpretation.*

In this paper I endeavour to give a historically accurate presentation of how Leibniz understood his infinitesimals, and how he justified their use. He called them fictional or "syncategorematic" terms, by which he meant that such terms could be used to express truths even though there are no infinitesimals, if these are understood as actually infinitely small parts of the continuum. I argue that Leibniz’s justification of their use based on the Archimedean axiom is surprisingly rigorous, and not susceptible to the usual criticisms ; but that their status as fictions does not make them methodologically dispensable. I defend this reading against criticisms of the syncategorematic interpretation given in papers by Katz, Sherry and numerous co-authors, who see Leibniz’s methods rather as partial anticipations of modern infinitesimal methods achieved using Robinson’s Nonstandard Analysis.

- Sandra Bella (Univ. of Nantes)
*L’appropriation du calcul infinitésimal leibnizien à l’Académie des Sciences.*

En 1684, Leibniz rend publiques les règles de son calcul dans un célèbre article, "Nova Methodus pro maximis et minimis". Le calcul tarde à se diffuser, et ce ne sont que quelques savants (essentiellement Leibniz et les frères Bernoulli) qui publient des articles dans lesquels ils utilisent le nouveau calcul pour résoudre des problèmes concrets dits « physico-mathématiques ». En France, la réception du calcul débute vers 1691. Un groupe de savants, dont les acteurs les plus significatifs sont Guillaume de l’Hospital (1661-1704) et Pierre Varignon (1654-1722), va s’initier à ce nouveau calcul. Le début de cet apprentissage n’est qu’une affaire pour l’essentiel privée entre ces savants. En 1693, le Marquis de l’Hospital devient membre de l’Académie Royale des Sciences et lit les premiers mémoires concernant des problèmes mathématiques résolus à l’aide du calcul différentiel. Il introduit ainsi officiellement l’algorithme au sein de l’institution. Quelques académiciens sont plutôt hostiles à ce nouveau calcul, mais ce n’est qu’à partir de 1700 que Michel Rolle (1652-1719) lance le débat sur le bien-fondé de l’utilisation des infiniment petits. Si l’histoire de ce débat a déjà été analysé par des historiens tels que Michel Blay et Paolo Mancosu, nous souhaitons cependant y revenir pour cerner quels ont été les enjeux de sa genèse et son déroulement, qu’ils soient épistémologiques ou politiques.

- WANG Xiaofei (University Paris Diderot, SPHERE)
*Lagrange’s thoughts on the foundation of analysis.*

At the turn of the 18th century, Lagrange published two important works on analysis, or rather, as he called it, on the new calculus, and its applications to geometry and mechanics, but these works bore different titles. Respectively, they are the*Théorie des fonctions analytiques*in 1797, and the*Leçons du calcul des fonctions*in 1801. The latter was conceived as commentary and supplement to the former. In the*Théorie*, Lagrange rejected the conceptualizations of all his predecessors because of their use of infinitely small quantities or vanishing quantities. By contrast, he relied on the method of developing the functions into series, where forming the derived functions. As a result, all calculations are executed on functions instead of infinitely small quantities. He also applied these functions to the problems of geometry and mechanics. The last step he planned is to show the identity of this « calcul des fonctions » with the differential calculus, although this plan was abandoned for the reason of the length of the book. According to my study on these works, I suggest that Lagrange, not only offered a new method for exposing the principles of the differential calculus, but proposed an new "theory of functions", which comprehended the differential calculus and the integral calculus as parts, so to join them together with algebra. In my talk, I will show how Lagrange reduced the differential calculus to algebra, so as to connect all parts of "analysis" as a whole.

June 13, !! 9:30-13:00 !!
**On algebraic equations** (tbc)

Session organised by Sara Confalonieri (Universität Wuppertal) and Emmylou Haffner

- Sara Confalonieri (Bergische Universität Wuppertal)

Titre à confirmer

- Massimo Galuzzi (Università di Milano)
*The theorem of Alexandre Joseph Hidulphe Vincent.*

The*Eléments d’algèbre*of Pierre Louis Marie Bourdon, a famous author of textbooks, are an important manual that has had countless editions and translations since the first edition of 1817 ([7]).

In the Avertissement at the beginning of the sixth edition Bourdon observes

that. . . j’y ai fait plusieurs am´eliorations de d´etail et quelques additions dont quelque-unes sont assez importantes, er que je dois en grande partie à M. Vincent, mon gendre et mon ami : je vais

indiquer les principales.^{1}Among the additions and improvements there is ". . . une note fort importante sur la resolution des équations numériques", written by Vincent, who is not mentioned explicitly as the author, which is aimed to “perfectionner la méthode de Lagrange, au moyen des principes dus à M. Budan des Boislaurent."

This note, which will then be published by the same Vincent with some modifications in [13] and [14], will no longer appear in the successive editions of Bourdon’s manual, substituted by the famous result of Sturm, announced in 1829 and given in complete form in 1835.^{2}

Of course Sturm’s theorem has an incomparable elegance and has the possibility to be reformulated and generalized in many very different mathematical contexts.^{3}

3 But from the computational point of view the result of Vincent is comparable to it or even better. Consequently after the rediscovery of Uspensky ([12]), mainly thanks to Akritas, it has become an important tool of modern computer algebra.^{4}And the story of his survival and of the rediscovery of its value is not uninteresting. A good example of the fact that

the development of mathematics can give new life to what seemed destined to oblivion.

–^{1}See [8, p. v].

–^{2}See [10] for the notes of Sturm. To look at the difference in Bourdon successive manuals, see for example [9, p. vj], where he claims to have exposed "avec tout le soin possible le théorême de M. Sturm, et ses applications à la résolution d’une équation mumérique quelconque."

–^{3}See for example [11].

–^{4}Among the many works of Akritas, I just mention [1] and [2].*Références :*

[1] A. G. Akritas.*Elements of computer algebra with applications*. John Wiley and Sons, New York ecc., 1989.

[2] A. G. Akritas. Vincent’s theorem of 1836 : overview and future research.*Journal of mathematical sciences*, 168(3):309–325, 2010.

[3] A. Alesina and M. Galuzzi. A new proof of Vincent’s theorem.*L’Enseignement mathématique*, 44:219–256, 1998.

[4] A. Alesina and M. Galuzzi. Addendum to the paper ”A new proof of Vincent’s theorem”.*L’Enseignement mathématique*, 45:379–380, 1999.

[5] A. Alesina and M. Galuzzi. Vincent’s theorem from a modern point of view. In*[6]*, pages 179–191, 2000.

[6] R. Betti and F. W. Lawvere, editors.*Categorical Studies in Italy*, Palermo, 2000. Supplemento ai Rendiconti del Circolo Matematico di

Palermo, serie II - n. 64.

[7] P. L. M. Bourdon.*Eléments d’Algèbre*. Mme Ve Courcier, Imprimeur-Libraire, Paris, 1817.

[8] P. L. M. Bourdon.*Eléments d’Algèbre*. Bachelier père et fils, Paris, sixième edition, 1831.

[9] P. L. M. Bourdon.*Eléments d’Algèbre*. Bachelier, Paris, septième edition, 1834.

[10] J. C. Pont (in collaboration with F. Padovani), editor.*Collected Works of Charles François Sturm*. Birkäuser, Basel-Boston-Berlin, 2009.

[11] H. Synaceur.*Corps et Modèles. Essai sur l’histoire de l’algèbre réelle*. Vrin, Paris, 1991.

[12] J. V. Uspensky.*Theory of Equations*. Mc Graw-Hill, New York, 1948.

[13] A. J. H. Vincent. Sur la résoluton des équations numériques.*Mémoires de la Société Royale des sciences, de l’agriculture et des arts, de Lille*,

pages 1–34, 1834.

[14] A. J. H. Vincent. Sur la résoluton des équations numériques.*Journal de mathématiques pures et appliquées*, 1:341–372, 1836.

- François Lê (Université d’Artois)
*L’équation aux neuf points : de Hesse à Weber*

Dans les années 1840, Otto Hesse publie une série d’articles sur les courbes cubiques, montrant en particulier que toutes ces courbes contiennent neuf points d’inflexion. Suite à une suggestion de Jacobi, Hesse étudie dans la foulée une certaine équation algébrique de degré 9, appelée "équation aux neuf points." Ce n’est qu’à la toute fin des années 1860 que cette équation sera reprise par d’autres mathématiciens, en tant qu’exemple privilégié des "équations de la géométrie", famille d’équations toutes liées à des configurations géométriques particulières, étudiées en particulier par Camille Jordan, Alfred Clebsch et Felix Klein, et donnant lieu à des techniques de résolution bien particulières liées à une certaine compréhension intuitive de la théorie des substitutions. Le but de l’exposé est de décrire les activités liées aux équations de la géométrie ; l’accent sera surtout porté sur l’équation aux neuf points, dont je chercherai à décrire le rôle particulier qu’elle a occupé dans la deuxième moitié du 19ème siècle, depuis les écrits de Hesse jusqu’à sa place dans le*Lehrbuch der Algebra*de Heinrich Weber.

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