PRESENTATION

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 2013-2014** :

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.

Oct. 15, Dec. 16, Jan. 13 2014, Feb. 10, March 10, April 7, May 19, June 2, June 30 |

June 30, 9:30–13:15

**Mathematics, History of mathematics, Philology and Linguistics, 18th-20th century**

Session prepared with the ERC Project SAW "Mathematical Sciences in the Ancient World" by Marteen Bullynck (Univ. Paris 8, SPHERE), Pierre Chaigneau (Univ. Paris Diderot, SPHERE), Marie-José Durand-Richard (Univ. Paris 8, SPHERE), Agathe Keller (CNRS, SPHERE & ERC SAW), Pascale Rabault (ENS, CNRS), Ivahn Smadja (Univ. Paris Diderot, SPHERE).

This session continues last year’s session reflexion on the relations in the 18th-20th century of mathematics, the writing of history of mathematics, with the birth of linguistics and philology.

- 9:30 – 10:40

Maarten Bullynck (Univ. Paris 8, SPHERE)

*Transfers and transformations of numerals. The brothers Humboldt at the crossroads of disciplines (1819-1835).*

In 1819 Alexander von Humboldt presented his "Considérations générales sur les signes numériques des peuples’’ before the Paris Academy. Based upon a collection of data he had received from the various ethnographers and linguists in his scientific network, Alexander developed in this fragment a hypothetical history of the genesis of the hindu-arabic numeration system. The fragment would be reworked and updated in 1829 but it would remain but a fragment of a bigger, unfinished project. Nevertheless, the sketch would prove to be an important source of inspiration for the linguistic studies of his brother Wilhem (1830-1835) on the one hand, and for the prudent beginnings of a history of mathematics (Rosen, Woepcke, Nesselmann) on the other hand. Especially the reflections on the transfer and the transformation of numerals proved to be stimulating and led to the more general question of transfer between cultures and of contact between languages, problems that are still highly topical today.

- 10:40 – 11:50

Ivahn Smadja (Univ. Paris Diderot, SPHERE)

*More on Brahmagupta in Germany : Which Quadrilaterals ? Which Proofs ?*

As a result of Colebrooke’s 1817 English translations of Brahmagupta’s and Bhaskara’s mathematical works spreading across Europe among scholarly circles, the French geometer Michel Chasles pointed out the importance of a group of sutras within the mathematical chapters of Brahmagupta’s*Brahma-sphuta-siddhanta*, viz. BSS XII 21-38. He argued that in spite of the sutras being mere statements unsupported by individual proofs, the whole system of them made sense as providing a unified geometrical theory which, in his view, would solve completely, with precision and in all its generality, one single question, viz. how to construct a cyclic quadrilateral whose sides, diagonals, perpendiculars, segments and area, as well as the diameter of the circumscribed circle, may be expressed in rational numbers. A few years later, that interpretive puzzle and Chasles’ purported solution to it aroused rewewed interest in Germany. At about the same time he developed his theory of ideal numbers, E. E. Kummer also fell under the spell of Brahmagupta’s quadrilaterals. In connecting the problem of rational quadrilaterals, duly transposed into an algebraic framework, to Eulerian methods in Diophantine analysis, Kummer established, through reconstructing Brahmagupta’s methods, that Chasles wrongly imputed generality to Brahmagupta’s theory. Eventually, more than two decades later, by virtue of his conjoining philological precision and mathematical expertise, Hermann Hankel outlined an alternative interpretation of BSS XII 21-38, which would do justice to Chasles’ main insight, though in a minor key, owing to Kummer’s rectification. The present paper unfolds this reading against the backdrop of Chasles’ and Kummer’s previous attempts, thereby showing how it also qualifies as the cornerstone of Hankel’s characterization of so-called Indian intuitive ‘proofs’, by contrast to Greek deductive ones.

- 11:50 – 12:05 Break

- 12:05 – 13:15

Pierre Chaigneau (Univ. Paris Diderot, SPHERE)

*Some links between Otto Neugebauer’s work and the history of ancient sciences as made before him in Germany.*

Otto Neugebauer (1899-1990) entered the field of history of mathematics in 1926 with his doctoral dissertation, which focuses on the use of fractions in Old Egypt. For the next decades, he will be known as one of the greatest specialist of the sciences in Antiquity, bringing new methods and theses. This presentation look back at his influences. It is usual to recall his prestigious mathematical entourage at Göttingen, but the focus here will rather be on his references, in his first writings, to previous German historians of sciences, especially Hermann Hankel (1839-1873).

June 2, 9:30–13:00

**On Heinrich Weber**

Session prepared by Emmylou Haffner (Univ. Paris Diderot, SPHERE).

- Katrin Scheel (TU Braunschweig)

*Heinrich Weber and Richard Dedekind - The story of a great and long-lasting friendship.*

They started their collaboration to achieve their common purpose : publishing the collected works of Bernhard Riemann. Rapidly it became more than a collaboration, it became a close friendship which lasted nearly 40 years until the death of Heinrich Weber in 1913. How did this friendship influenced the life and the scientific work of Heinrich Weber ? Which topics did they discuss ? What did they do together ? Did Heinrich Weber learn form Richard Dedekind or vice versa ? We will try to answer these questions by analysing the extensive but not complete extant exchange of letters between Richard Dedekind and Heinrich Weber."

- Cédric Vergnerie (Univ. Paris Diderot, SPHERE)

*Weber’s Lehrbuch der Algebra and Kronecker’s Vorlesungen über die Theorie der algebraischen Gleichungen : similarities and differences.*

Algebra was marked during the 19th century by Heinrich Weber’s and Kronecker’s research and teaching, in particular the on the theory of equations, which we will here examine. To this end, we will compare, both in their structure and contents, Weber’s Lehrbuch der Algebra and Kronecker’s Vorlesungen über die Theorie der algebraischen Gleichungen. We will then try to show how this study permits us to shed light on their conception of mathematics.

May 19 , 9:30-17:00

Morning : **The Zīj al-Sindhind of al-Khwarizmi**

Session prepared by Matthieu Husson (SPHERE), linked to ERC Project SAW "Mathematical Sciences in the Ancient World"

*Zīj al-Sindhind*is an essential source documenting the circulatin of astronomical parameters’ and theories’ between the IX and XII centuries from Persia and India to the Arabic and Latin Occidents. We wish to sum up what can be said on this important question, but beyond the transmission of astronomical parameters and theories we would like to asses the potential transmission of mathematical objects and practices, for instance in the way to write and operate on numbers or in the treatment of the Sine in "simple" instances like Sun’s declination or Sun’s equation tables.

- José Chabás (Universitat Pompeu Fabra)

*Equations and velocities of the Sun and the Moon in the*Zij al-Sindhind*by al-Khwarizmi.*

The astronomical tables ascribed to al-Khwārizmī (early 9th century) reached al-Andalus quite soon after their compilation in the East, and had a considerable success in the Iberian Peninsula, where they were used for several centuries. This so-called Indian tradition coexisted in the Peninsula with the astronomical practice stemming from Ptolemy’s works, also referred to as the Greek tradition, which was predominant both in the West and the East during the Middle Ages.

The solar and lunar equations in the Zij al-Sindhind present some basic features that make them differ from those compiled by Ptolemy and his followers. In turn, al-Khwarizmi’s tables for the solar and lunar velocities depend on the equations of the corresponding luminaries, and thus gave different results than those in the Greek tradition.

In this presentation we will explain the different models underlying the tables for the equations and the velocities of both luminaries, highlight the different parameters involved in them, and present the methods used in their compilation.

- Michio Yano (Université de Kyoto)

*Indian elements of al-Khwarizmi’s astronomy.*

al-Khwārizmī’s astronomical work*Zīj al-Sindhind*(early ninth century) is very important because there survived only a few Indian texts on mathematical astronomy before him. Only ͞Aryabhata, Varāhamihira, Brahmagupta, and Bhāskara I predate him. So-called modern*S͞uryasiddhānta*was barely formed in the time of al-Khwārizmī. We know no original Sanskrit text(s) on which al-Khwārizmī ’s Arabic work is based, nor his Arabic translation is known to us. What we have is only the Latin translation of Adelad of Bath (early twelfth century).

The Latin text was published by Suter in 1914. Neugebauer published English translation with commentary in 1962. This is an indispensable work for further study. I am very much indebted to Neugebauer’s work. Although I have no knowledge of Latin and I am poorly informed of Islamic-Arabic astronomy, I would like to give some comments from the viewpoint of the history of Indian astronomy. Especially I want to compare al-Khwārizmī’s planetary theory with that of India and raise some questions.

- Afternoon :

–**Presentation of works of Sabine Rommevaux**

*New theories of mathematical proportions from the XIVth to the XVIth century.*

In my talk I would like to expound the conclusions of my book published with Brepols. I showed that, from the fourteenth century to the sixteenth, in West Europa, new mathematical theories were established, in which the main subject is the ratio (or logos in Greek). Medieval mathematicians questioned its nature (relation or quantity) and of its denomination (that is the manner to name it). They developed a new theory on ratios of ratios. For Nicole Oresme the ratio is even an object of calculation in his Algorismus propmortionum (that I edited and translated in French in an appendix of my book). Finally, these new theories will circulate during the Renaissance and the seventeenth century.

- 16:00 – 18:00

**Sho HIROSE (SAW ERC doctoral student) will present his research theme in the presence of his codirector YANO Michio**:

*The dual illumination of the sphere : Comparing the two versions of the*Goladīpikā*by Parameśvara.*

The south Indian astronomer Parameśvara (c.1360-1460) wrote two treatises on spherical astronomy under the same title*Goladīpikā*. They differ greatly in structure, and each of them has contents that do not appear in the other. I shall compare the two texts in an attempt to determine their chronological order and to infer about Parameśvara’s intention.

April 7, 9:30 – 17:00

** Arabic and graeco-latin Algebras from IX to XVI centuries**

Session organised by Sabine Rommevaux (CNRS, SPHERE)

- Eleonora Sammarchi (SPHERE)

*Une lecture algébrique des*Éléments*au XIIIe siècle : le texte euclidien mentionné par al-Zanjanī dans son*Livre d’algèbre.

Dans le chapitre VII de son*Livre d’algèbre*, al-Zanjanī développe l’algèbre des polynômes. En s’appuyant sur la théorie éuclidienne des nombres en proportion et sur certains théorèmes d’al-Karajī, il collecte une quarantaine de propositions, parmi lesquelles figurent presque toutes les propositions du Livre II des*Eléments*, interprétées ici algébriquement et expliquées par des exemples purement numériques. Cette lecture algébrique du texte euclidien influencera in primis son exposé de l’algorithme de résolution des équations quadratiques.

- Sabine Rommevaux (CNRS, SPHERE)

*Démonstrations géométriques des algorithmes de résolution des équations du second degré : al-Khwārizmī, Abū Kāmil, Pedro Nuñez.*

Dans son*Libro de algebra*, Pedro Nuñez propose des "démonstrations nouvelles" des algorithmes de résolution des équations du second degré, en plus de celles plus traditionnelles, produites à la suite d’al-Khwārizmī. Nous comparerons ces démonstrations avec celles d’Abū Kāmil pour les équations du type ax2+c=px, qui ont la particularité d’avoir, selon les cas, deux racines, une seule racine ou aucune racine.

- Odile Kouteynikoff (SPHERE)

*La réception des*Arithmétiques*de Diophante par Guillaume Gosselin*.

Les*Arithmétiques*du mathématicien grec Diophante d’Alexandrie ( iiie siècle) ont été traduites en arabe, sous le titre*L’Art de l’algèbre de Diophante*, par Qustā ibn Lūqā à la fin du IXe siècle. Gosselin les découvre dans la traduction latine des six livres connus du texte grec, que Xylander donne à Bâle en 1575. La lecture que fait Gosselin des Arithmétiques est au cœur du questionnement incontournable sur les liens entre algèbre et arithmétique qui étayent son œuvre.

March 10, 9:30-17:00

**Ancient and mediaval mathematics – Means of paiement**

Session organised by Karine Chemla (CNRS, SPHERE & ERC SAW), linked to ERC Project SAW "Mathematical Sciences in the Ancient World"

- CAO Jin 曹晉 (Tübingen University)

*Weighing, Counting, and Calculating Coins in Ancient China.*

In China much as in most other parts of the world, coins have been the most-used means of monetary transaction over the last two millennia. What makes the Chinese case unique is that generally speaking only coins made of copper in varying alloys with inferior metals were in use. While larger and smaller issues did exist, the majority of coins were of one size, stringing them together in bundles rather than changing their denominations was the common praxis to carry out larger transactions. Under these circumstances the coin as an object itself represented the standard of value par excellence. This status was also reflected in its contours with the circular rim outside standing for heaven and the quadrangular hole inside for the earth.

Thus my talk will focus on coins in the context of their relations with weights and measures as a part of China’s monetary policy, their daily use for transactions and calculations, as well as their cultural connotations and meanings. Besides, it will also be observed that coins were used as a means to retrieve ancient weighing units and their values. In the end, a comparison between the “countable” coins and “weighable” silver ingots, which were another important currency, will be presented.

- Matthieu Husson (SPHERE et ERC SAW)

*Remarks on John of Murs’s*De monetis.

As a part of his complex and manifold*Quadripartitum numerorum*completed in 1343 John of Murs wrote a treatise on money. The general theme of his treatise is the making of money with a prescribed proportion of silver and copper starting with other types of money or with a certain amount of the two metals. The relation of this treatise to Fibonacci’s*Liber abbaci*will be acknowledged and its situation in the frame of the*Quadripartitum numerorum*will be examined. Then we will look at a few key passages, comparing them with the Liber abbaci and other parts of the*Quadripartitum numerorum*in order to understand how this problem of money was tackled mathematically and how it was presented to various potential readers of the work.

- Marc Bompaire (EPHE)

*Calculations on currencies. Mathematicians’ approaches, Practitionners’ approaches : the case of French money-changers books from the XIVth and XVth centuries.*

The authors of French money-changers books - when one can identify or define their profile - may exhibit originality in the selection of problems and the ways of solving them when compared to models proposed by the books or abacus schools. Some examples will be discussed and the investigation will focus more particularly on the presentation of the monetary formula (title, weight and currency rates) and units used to express it.

Feb. 10 , 9:30-17:00

**What are the links between intuitions and constructions in mathematics ?**

Final discussion moderated by Ramzi Kebaili (SPHERE)

- 9:30 - 11:00 Guillaume Loizelet (SPHERE)

*L’intuition mathématique au 11e siècle : le*Traité pour aplanir les voies en vues de déterminer les propositions géométriques*d’al-Sijzi*

- 11:15 - 12:45 Jeremy Gray (Open University, London)

*Hermann Weyl on mathematical intuition and constructions.*

- 14:30 - 16:00 Sébastien Maronne (Univ. de Toulouse)

*Intutitions et constructions chez Descartes*

Jan. 13, 9:30-17:00

**Approximations and errors in various contexts**

Session prepared by Nadine De Courtenay (Univ. Paris Diderot, SPHERE) and Christine Proust (CNRS, SPHERE & ERC SAW), linked to ERC Project SAW "Mathematical Sciences in the Ancient World"

By comparing the practices of ancient and contemporary approximations, we will attempt to better understand the complexity of the concept of error. We will analyze in particular how the different facets of this concept are related to metrological context (measuring instruments, standards, units, and modes of representation of systems of units).

- Giora Hon (Un. Haïfa)

*Accuracy and Precision : Two Aspects of Approximation.*

Accuracy and precision corresponds to the dichotomy between systematic and random errors. Accuracy refers to the closeness of the measurements to the ‘true’ value, whereas precision indicates the closeness with which the measurements agree with one another. The paper seeks to shed light on the concept of approximation by analyzing these two different kinds of error.

- Robert Middeke-Conlin (SPHERE & ERC SAW)

*Error and Uncertainty in Ancient Texts.*

This presentation examines approximations found in economic texts of the Old Babylonian period Southern Iraq (between 1900 and 1700 BCE) in which there is a clear deviation from an actual total quantity, the real value, and the total quantity represented on the tablet, the represented value.

- Fabien Grégis (SPHERE)

*Error and uncertainty in contemporary metrology.*

In contemporary science, "error" has become a scientific notion, integrated within a theory of experimentation which enables us to deal with it quantitatively. This presentation will describe how error, and its epistemic counterpart, uncertainty, are taken into account in metrology.

- Jonathan Regier, (Univ. Paris Diderot, SPHERE)

*Le statut de la mesure chez Képler.*

Beginning with Philipp Melanchthon’s epistemology and the sixteenth century art of cosmography, we will try to define the meaning of measurement in Kepler’s mundus. Why must one observe, that is, measure precisely ? The answer to this question will allow us to see how Kepler understands necessity and contingence as essentially mathematical.

Dec. 16, 9 :30–13 :00 !!

**Formal-Formalism**

Session prepared by Ramzy Kebaili (SPHERE)

- Frédéric Jaëck (SPHERE)

*Le développement de l’Algèbre Symbolique chez Duncan Gregory.*

- Pascal Bertin (SPHERE)

*La Théorie des Ensembles de F. Hausdorff, une approche formaliste mais pas fondationnaliste.*

Oct. 15

**Tabular Layout of Calculations : Theory and Practice**

Session prepared by Robert Middeke-Conlin (SPHERE & ERC SAW), linked to ERC Project SAW "Mathematical Sciences in the Ancient World"

- Odile Kouteynikoff (SPHERE) and Sabine Rommevaux (CNRS, SPHERE)

*Sur les signes posés dans les opérations et les algorithmes dans les travaux de Stifel, Stevin et Clavius.*

- David Rabouin (CNRS, SPHERE)

*Tableaux, algorithmes et algèbre dans le*Commentaire à Al-Khwarizmi*d’Adriaan Van Roomen.*

- Eleonora Sammarchi (SPHERE)

*Les tableaux et leur règles de formation dans la tradition arithmético-algébrique. Quelques exemples selon la perspective d’al-Zanjani.*

- Robert Middeke-Conlin (SPHERE & ERC SAW)

*Tabular Mathematic Texts from Ur and Elsewhere in the Old Babylonian Period.*