Document sans nom

From China to the Mediterranean

Inaugural worshop

November 24 & 25, 2009

November 24, 2009

Karine CHEMLA (REHSEIS, CNRS)
Introduction

Christine PROUST (CNRS-NYU & REHSEIS CNRS Université Paris 7)
Structure of series texts: a new approach of cuneiform mathematical corpus

Scholarly texts from Mesopotamia are most often presented in list form. These lists gather and organize data related to various fields as writing, lexicography, grammar, law, mathematics, divination, astronomy and so on. Revealing specific meanings conveyed by lists structures is thus an important issue so as to understand scholarly cuneiform texts. Mathematical texts are generally organized as a chain of problems. Each problem is inscribed in a section, which is defined by horizontal lines. The majority of mathematical texts can thus be considered as lists of sections, each one containing a problem. In most cases, a problem includes the following elements: a statement, a question, a detailed resolution procedure, and the solution. However, the resolution procedure is often not indicated. In some texts, problems are restricted to statements, without question or solution. Such statements lists are generally very long, built on a sophisticated structure, and can cover several numbered tablets, then called series. Mathematical series are probably the most accomplished production of the Mesopotamian scholars in the art of list making.
In my talk, I focused on the series texts, notably on two unpublished series texts now kept in Musée du Louvre. I underlined some issues raised by series texts and the questions that I would like to study in the near future:
1) Why did the scribes write the series texts?
2) How can we describe the general architecture of these texts?
3) What was the language used to write the series texts?
4) How did the scribes express arithmetical operations? What kind of terminology and what kind of syntax were in use for these operations?
5) Do series texts have a consistent mathematical content?
6) What is the scholarly tradition to which these series texts belong?
7) It is also interesting to approach these texts from a historiographical perspective: why did the series texts only play a minor role in the study of cuneiform mathematics?

John STEELE (Brown University)
Shadows in Babylonian Astronomy

Several Babylonian cuneiform texts are known which present schemes for the length of the shadow cast by a gnomon at different times of the year. In this presentation, I reviewed the three sources that have been published and proposed new interpretations for the schemes. In addition I presented a new source: BM 33564, a cuneiform tablet in the British Museum. The first sources discussed was the early compendium MUL.APIN. This source contains a description of a scheme for the length of the shadow cast by a gnomon at different times during the day in the first, fourth, seventh and tenth month of the ideal year (the months of the equinoxes and solstices). It is well known that this scheme is constructed mathematically so that the length of the shadow in cubit multiplied by the time in US since sunrise is equal to a constant in each month. In my presentation I showed that the scheme was constructed using the 2:1 ratio for the longest to the shortest daylength found elsewhere in MUL.APIN and a ratio of 3:2 as has generally been accepted (this ratio only being found in much later cuneiform sources). Furthermore, I showed that the noon-shadow implied by the scheme is in good accord with reality. I then discussed the text SpTU IV 172, a metrological list which includes a noon-shadow table at the end. Again, I showed that the noon shadows were in good accord with reality. I then discussed the catchline at the end of the tablet and showed that it referred to a tablet describing a shadow scheme that is also known from BM 29371. I proposed a new reading for this tablet based upon the catchline of SpTU IV 172, and showed that the scheme was reasonable accord with nature. Finally, I presented the tablet BM 33564, an unpublished astronomical tablet from the British Museum. This tablet, which discusses shadow lengths, the length of daylight, and the calendar, is interesting because it can be linked with a number of other important astronomical tablet through the scribe named in the colophon.

Agathe KELLER (REHSEIS, CNRS-Université Paris 7)
Reflecting on the different social groups that produced mathematical knowledge and texts in ancient India: different research perspectives, with a special emphasis on the history of versified problems and the perspective they open.

The talk was an attempt at confronting ethnomathematical field work with questions linked to how different groups practiced mathematics and produced mathematical texts in ancient and medieval India.
The ethnomathematical field work considered, was the one conducted by Senthil Babu, at the time associated with the French Institute of Pondicherry, in the Nagai area of Nagapattinam district in Tamil Nadu (South India).
The aim of S. Babu’s investigation was to list all different mathematical activities that were conducted in that specific rural area of south India. Here we concentrate on three unpublished features of this field work: the great diversity of measuring units used, the specific computational features of some activities and a dying tradition of mathematical riddles.

Before, to prepare a close comparaison, some brief facts on ancient scholarly tradition of mathematics in the Indian subcontinent were presented. Focussing on Sanskrit texts before the XIIth century, one can distinguish the mathematical chapters of astronomical texts transmitted through a tradition of copying, and those found in unique recensions, sometimes excavated by chance, of the pâti tradition (see power point for a list of precise texts).

For one and the other tradition we have very little information on who produced these texts, who copied them, how they were used and read. For instance, they all insist that knowledge is made to be transmited orally: why then do we have written copies of them? They are written in Sanskrit: does this mean that they testify only of Brahmin highclass practices? Pâtî texts (whose general structure are given in the powerpoint) are quite understudied in the secondary literature. We do not yet have a global view of how each text is structured, what is the deep inner coherence of its different chapters. In particular how does the paribhâsa part, which lists measuring units, articulates to other mathematical chapters? Mathematial texts in astronomical works always include pâtî problems. Whatever the mathematical tradition in Sanskrit, it always testifies of several different mathematical cultures.

A great diversity of measuring units are still in use in the Nagai area. A “traditional tamil system” used by all co-exists with English measuring units and the official decimal ones. The “traditional tamil system” itself testifies of a long history. Indeed, from at least the XVIIth century onwards, changing rulers would each impose new measuring units in the region. Those that survive today articulate well with the English system but are difficult to convert in decimal measuring units. Numerical literacy in computation with fractions and rule of threes seems thus of a crucial importance. Traditional house hold measures bear the memory of attested ancient tamil numerical systems to state numbers smaller than one, using partly 1/320 as a base, and 1/4th as an other. Finally, all use the pala, a measure also found in all Sanskrit texts. First then, wondering if Sanskrit texts are far from rural realities, the paribhâsa parts could account for a real diverse reality. T. Hayashi has shown that Sanskrit mathematical texts testify not only of a variety of regional measures, but also that measures used through out the sanskrit world such as the yava, had a converting ratio that varied from text to text. These facts then point to the fact that there was some kind of interaction between the abstract world of sanskrit mathematical texts and a diverse, profuse measuring reality. We have also seen that measuring units could testify of traditions of numeration that existed alongside the Sanskrit decimal system. Indeed, folk and non-scholarly numerations were amply scrutinized in the XIXth century in search of the origins of the decimal place value notation. After the rediscovery of Sanskrit scholarly texts that give definitions of such a notation, the study of other system of numerations was set aside. They are subject of a renewed interest today. Many acts of measuring as seen on the field, associated exact measuring procedures and units with approximative ones. Those who composed mathematical manuals reflected on the oppostion drawn between vyavahâra (practical, commercial) and sûksma (exact) mathematics.

Today as trained historians of mathematics we are not familiar with daily computations in rural or “traditional” societies. Some computations appear as crucial for certain occupational activities: thus for the carpenter, krakaca types of computations involving an evaluation of the number of planks contained in a tree trunk of a given circumference and length are crucial, for the masons and the brick maker, those evaluating heaps of bricks…
Many political tensions rise from different modes of measuring and computing. Turning from striving field questions to look closely at pâtî texts could maybe then tell us more on the issues not only mathematical that such texts were striving to solve? Finally, S. Babu has collected twenty “mathematical” riddles, most of which were learned at the beginning of the XXth century by those who went to village (tinnai, pyal) schools. A selection is given in the hand out: in some we recognize traditional problems known in many different area of the world: rules of three with a subtraction, time of meeting, but also chineese remainder theorem problems and “pure” mathematical ones. Not all are trivial. As a study of problem 1 or 4 may show, they reflect the existence of a “culture” of problems, in which one may disguise a problem into another, and slightly change a solution.

The enthusiastic reaction of all, has given me confidence in pursuing this improbable confrontation of contemporary practices with those existing in ancient texts. I have made explicit my “fantasy” of mathematical texts solving social problems. K. Plofker has underlined how on the contrary looking at pâtî texts from after the XIIth centuries, the “real problems” are just disguises for mathematical problems. Bref, she thinks that traditions of riddles might come from a more scholarly background. All have corrected a false reasoning I had in the rule of three problem, which underlined how these riddles might in fact not be so mathematical. Others, consequently, have asked questions on the riddle culture of Tamil Nadu, of which I am ignorant. However, mathematical riddles seem to exist today in Asia, Central Asia and Europe, in pastoral and agricultural communities. The study of such traditions may give us ideas on how such riddles circulated, were transformed and how they articulate to larger riddle traditions.

Markus Asper’s presentation echoed into my study of riddles, opening onto the narrative and linguistic choices that maybe can mark different (mathematical) cultures. The meeting has also enabled K. Plofker, T. Knudsen and me to agree in publishing yearly a review of the field, both in institutional terms (who has what job, what students are studying what text) and intellectual trends. Furthermore, I hope to follow the work of Kevin T. van Bladel whose work and culture discovered, very impressed, his culture and work.

Toke KNUDSEN (SUNY)
The Direction of Down and Adhesive Antipodeans:
Tradition and Innovation in Medieval Indian Astronomy

The Siddhanta-sundara is a treatise on cosmology, astronomy, and mathematics composed in Sanskrit by Jnanaraja in around 1500 CE. The treatise was the first major text of this type to have been composed in India since Bhaskara II composed the Siddhanta-siromani in 1150 CE. While the text is important for our understanding of the history and development of Indian astronomy, it has hitherto remained unstudied. During the presentation, an overview of the contents of the work will be given, focusing on the main contributions of Jnanaraja. While a capable mathematician and astronomer, the sections on mathematical astronomy and mathematics offer little that is not found in earlier Indian treatises. In other words, when it comes to these subjects, Jnanaraja follows the Indian tradition of astronomy and mathematics closely. However, when it comes to cosmology, his presentation differs significantly from that of his predecessors. Jnanaraja’s main concern is that the cosmological model of the Indian astronomical tradition differs from that expounded in the sacred texts of India. Being a practicing astronomer, he cannot abandon the scientific tradition, but on the other hand he wants to assert the correctness of the cosmology of the sacred texts. He thus attempts to create a synthesis between the two, leading to novel interpretations of both traditions and new ideas. This approached greatly influenced the Indian astronomy in the period following him and helped shape the indigenous response to Islamic (and later Western) astronomy.

Michio YANO (Waseda University, Japan)
Buddhist Astronomy and Astrology. An Aspect of Cultural Propagation

Buddhism was born in India during the time of a new intellectual movement when people began to doubt the orthodox view of priests (Brahmanas or Brahmins) that taught that performing sacriﬁcial rites was the only way to reach heaven (svarga). The main teaching of Buddha, as represented by Four Noble Truths and the Noble Eightfold Path, was nothing more than the search for the right way to achieve nirvana (ultimate tranquillity). Buddha himself was a person of insight and in a sense he was a “scientist”, as the XIVth Dalai Lama said in an interview. The Brahmins, however, regarded Buddhism as a heterodoxy and were unwilling to share with Buddhists the traditional Indian culture, especially intel- lectual activities which we now call ‘sciences’. However, the two social groups, Hindus (this is a modern word) and Buddhists were not totally separate from each other. I would like to talk about some aspects of their relationship with a special interest in scientiﬁc activities of Buddhists as well as the transmission of science from India to East Asia through Buddhism. From a viewpoint of the history of science, the long history of Buddhism is divided into three periods. (1) Early Buddhism (2) Mahayana Buddhism (3) Tantric Buddhism I will give a brief overview from the aspects of astronomy and astrology in these three periods. The original sources that I use were written in Sanskrit, Pali, and Chinese.

Karine CHEMLA (REHSEIS, CNRS)
Writing down texts for algorithms:
views from ancient China

The presentation focused on the question of the texts by means of which ancient practitioners of mathematics wrote down algorithms. Like many other historians, I regularly described the texts of algorithms, which constitute, with problems, the core of ancient Chinese mathematical writings, as sequences of operations leading from the data of a given problem to the value of the unknowns sought-for. The presentation called this common description into question, by showing that the reality we face when we read ancient sources is much more complex and in fact much more interesting. This phenomenon is in my view an aspect of a much more general one. The textual elements that compose mathematical writings display, when examined closely, technical features. Moreover, these features relate to the fact that professional cultures shaped them so as to be able to use them in their scholarly activities. Practitioners that were to use them received training to handle such textual elements and to work with them in a proper way. Further, in the course of their activities, they happened sometimes to modify their technical features. Studying these elements from a textual point of view and in their technical dimensions is thus a means of bringing to light the scholarly cultures to which they adhere. Moreover, it is, I argued, an essential task if we are to interpret our sources in a rational way. The presentation illustrated these theses on the case of texts for algorithms. With one central example, chosen for its capacity to illustrate the points I wanted to make, I highlighted the problematic character of some of the texts with which algorithms are given and thereby disclosed their adherence to a professional culture. Moreover, I pointed out the problem raised by the interpretation of such texts and how it could be solved by attending to the technical features of the texts. The example chosen was taken from the Book of mathematical procedures 算數書, which was excavated 20 years ago from a tomb sealed around 186 B.C.E. Placed on the bamboo slips that the editor Peng Hao numbered 74 and 75, the section examined bears the title “lü-ing on the basis of the dan 石率。”I showed that the way in which the practitioner read the text of the algorithm presented in this section and turned it into actions was far from obvious to |restore. I indicated why it was impossible to simply follow one after the other the terms for operations listed in the text. Moreover, I brought to light that the actions to which some of these terms corresponded changed, depending on where these terms occurred in the text.
On the one hand, I concluded that these remarks compelled us to introduce the hypothesis that the sequence of actions to be derived from the text was determined on the basis of a specific circulation within the text, and not a circulation that simply followed the sentences from beginning to end. Moreover, I showed that circulations of exactly the same kind accounted for how texts for algorithms were written and used in China until at least the 7th century. On the other hand, I suggested that some terms of operations designated the actions to be carried out by means of the reasons to carry them out. The practitioner thus needed to know how to derive the appropriate actions from the prescription by means of the reasons. It was on this feature that I indicated diachronic changes, pointing out that the way of stating the reasons evolved from what we find in the Book of mathematical procedures to what later sources attest to.

November 25, 2009

Emilienne BANETH-NOUAILHETAS (CNRS, Univ. Rennes 2),
Edward BERENSON (NYU),
co-Directors, Transitions UMI 3199,
Christophe J. GODDARD (CNRS, Univ. Reims),
Deputy Director, Transitions UMI 3199 (CNRS-NYU)
Foreword

Alexander JONES (ISAW, NYU)
Introduction

Markus ASPER (NYU)
Narratives in Greek Mathematics?

At first glance, both ancient Greek and modern mathematics appear to be narrative-free. Nonetheless, there has been recent debate about the relative merits of looking for narratives in modern mathematics (see Senechal 2006). Two aspects of this discussion have been discussed in this talk:
(1) How do narratives about Greek mathematics relate to mathematical practices? One finds such narratives occasionally in introductory letters, anecdotes, and ancient commentaries. Research on modern mathematics suggests that such stories do contribute in certain ways to the practices that they are about. I have looked at two anecdotes about Euclid in this talk and found that the tales situate the theoretical texts within their social, dogmatic, historical, and ideological contexts.
(2) Mathematical argument and narrative: My argument tackled the problem from a structural-logical and from an aesthetic perspective: (1) I have tried to show with some examples ranging from Euclid to Andrew Wiles that mathematical proof itself follows narrative structures: On the one hand, one can understand proof-as-discovered as a narrative. But even deductive proof-as-published the logical structure of which is non-narrative, translates into a narrative when describes as a process of understanding (that is, mostly, from a reader’s perspective).
(2) A closer look at the mathematical texts themselves reveals that they occasionally they adopt aesthetic norms that one also finds in literary narratives (suspense, surprise, sequential organization, closure). Notions of ‘beauty’ and ‘elegance’ come up. Especially rewarding would be a closer look at how Greek mathematical aesthetic relates to literary aesthetic of the time.
Conclusion: Perhaps one could understand (mathematical) proofs and solutions of problems as specialized stories.

Joe DAUBEN (CUNY)
Archimedes and Liu Hui on Circles and Spheres

Considerable interest in the Archimedes palimpsest has been generated by the careful conservation efforts and powerful scientific tools that have made it possible to reveal previously unobservable details of the original text now being studied at the Walters Art Museum (Baltimore) under the direction of William Noel. This has recently led the historian of ancient mathematics Reviel Netz (Stanford University) to offer important new insights about Archimedes’ Method and the methods he used to determine the volume of the sphere by applying methods involving infinitary arguments. Archimedes is also well-known for his application of the so-called method of exhaustion to approximate the value of . Similar interest in the mathematics of circles and spheres also inspired mathematicians in ancient China to consider the same problems of determining the ratio of the circumference to the diameter of a circle, and the volume of a sphere. Liu Hui’s commentary on the 3rd-century Chinese classic text, 九章算術 Jiuzhang suanshu (Nine Chapters on Mathematical Procedures), addresses these problems in ways that invite comparison with the results and methods of Archimedes. This lecture will address similarities as well as differences between the achievements of these two great mathematicians, and suggest some significant aspects that may serve to distinguish the character of mathematics in both oriental and occidental contexts.

Alexander JONES (ISAW, NYU)
Parapegma puzzles: reconstructing Greek documents on stellar risings and settings

The paper discusses problems motivated by my current work on inscriptions on the Antikythera Mechanism (a hellenistic bronze gearwork device for calculating apparent motions of heavenly bodies and various astronomical and chronological cycles). One inscription under current study is a parapegma, a chronologically ordered list of annually repeating phenomena relating to the stars and constellations. Many parapegmata are at least partially preserved from about 300 B.C. on to late antiquity. The media vary, including stone inscriptions (with peg holes for tracking the current day in the year), bronze inscription (on the Antikythera Mechanism), papyri, and texts transmitted through the medieval tradition. Incompletely preserved parapegmata create interesting problems of restoration: for example one may know that on a particular date a constellation is supposed to become visible, but the name of the constellation may be lost.

For some parapegmata (including the Antikythera Mechanism’s) the problem’s are reduced somewhat by the use of a standard list of constellations, which can be traced back to Euctemon (5th century B.C.). Moreover, texts may be demonstrated to derive data from earlier, lost texts, though the process of transmission was not free of distortions. Ultimately, the data may have been observed or generated from a visibility theory, which before Ptolemy would probably not have reflected the full complexity of stellar visibility conditions. Moreover, observations may have been made in any of a wide range of localities in the Greco-Roman world; Ptolemy gives only a sketchy and inadequate list of places of observation for certain authorities. Our best control of parapegma phenomena, and the best opportunity for restoring incomplete texts, exists when the phenomena refer to single bright stars such as Sirius and Arcturus. Unfortunately none of the parapegmata before Ptolemy’s (second century A.D.) restrict themselves to individual bright stars; the rest indiscriminately employ individual stars, «fuzzy» objects (Pleiades), constellations with several bright stars (Orion), and constellations with no bright stars (Pegasus). Particularly for the latter, the uncertainties involved in ancient criteria for visibility appear to make it unfeasible to reconstruct imperfectly transmitted parapegma texts.

Richard FOLEY,
Dean of Faculty of Arts and Science, NYU
Conclusion