Événements
Roy Wagner , professeur d’histoire et philosophie des sciences mathématiques à l’ETH de Zürich: deux conférences dans le cadre de ce séminaire, le 14 et le 16 avril.
Abstract. The purpose of this paper is to analyze the mathematical practices leading to Rafael Bombelli’s L’algebra (1572). The context for the analysis is the Italian algebra practiced by abbacus masters and Renaissance mathematicians of the 14th–16th centuries. We will focus here on the semiotic aspects of algebraic practices and on the organization of knowledge. Our purpose is to show how symbols that stand for underdetermined meanings combine with shifting principles of organisation to change the character of algebra.
I then move on to show that despite Bombelli’s careful adherence to a form of homogeneity, he constructs several diff erent ways of relating algebra and geometry, building on Greek, Arabic, abbacist and original approaches. I further show how Bombelli’s technique of reading diagrams, especially when representing algebraic unknowns, requires a multiple view that makes lines stand for much more than the diagrams present to an untrained eye. This multiplicity reflects an exploratory approach that seeks to integrate the algebraic and geometric strata without reducing one to the other and without suppressing the idiosyncrasies of either stratum.
readings
I will discuss three case studies. The first is the the Sanskrit and Malayalam versions of Citrabhānu’s Twenty-One problems: a discussion of quadratic and cubic problems from 16th-century Kerala. I will review the differences in the approaches of the two versions, highlighting the distinction between the Sanskrit indeterminate integer-remainder arithmetical techniques and the Malayali fixed point iterations, which may reflect aspects of two co-present mathematical cultures.
This second case study is the 16th century Sanskrit mathematical treatise called Kriyākramakarī. This treatise is a commentary on Bhāskara II’s twelfth-century Līlāvatī, one of the most famous mathematical treatises of the Sanskrit mathematical tradition. The Līlāvatī covers all standard areas of Sanskrit arithmetic and geometry, from the most elementary calculations to advanced procedures, up to but excluding algebra and trigonometric tables (which are covered in other works by the same author). I will review the general organization of knowledge of the Kriyākramakarī. Specifically, I will argue that the presentation of justification or proofs is integrative, rather than hierarchical or cumulative. In other words, the purpose of proofs in the Kriyākramakarī is, among other things, to connect various different aspects of mathematics, rather than just to convincingly establish or explain mathematical claims by means of previously known claims.
The third case study is the Kanakkatikāram – a genre of treatises for teaching practical mathematics beyond the level of basic arithmetic that disseminated in late medieval and early modern South India in Kerala and Tamil Nadu. I will focus on the discussion of numbers and units of measurement that opens this treatise, and try to suggest preliminary reconstructions of the practical mathematics culture of the time.
I will also mention some speculations on the possible transmission of algebraic knowledge from Kerala to the west in the context of solving cubic equation, and Jost Burgi’s method of constructing sine tables.
SERAFINA CUOMO, Professeure invitée de l’EHESS
Invitée au CAK par Giovanna Cifoletti du 25 mars au 13 avril 2019 dans le cadre du « Programme professeurs invités » de l’EHESS.
Serafina Cuomo is currently Professor of Ancient History at Durham University, in the North-East of England. Previously, she has worked at Imperial College and Birkbeck, University of London. Her main area of expertise is the history of science in the ancient Greek and Roman world. Her publications have focussed on mathematics and technology. She is working on numeracy in ancient Greece and Rome, and hopes in the future to develop a new project on concepts and views of work and labour.
She will take part of the Visiting Professors Program designed by EHESS, on proposal of Giovanna Cifolletti (Centre Alexandre-Koyré) from March 25th to April 13th 2019. She will be available to meet with students during this period.
Programme des conférences
Ancient accounting: a view from the bottom
Dans le cadre du séminaire de Giovanna Cifoletti « L’algèbre comme art de penser entre cosmographie et mathématiques du négoce »
Mardi 26 mars 2019, 17h-19h- EHESS, salle AS1_08, 54 bd Raspail, 75006 Paris
This session will explore the practices of accounting in ancient Hellenistic and Roman-era Egypt. We will look at examples of accounts, both from domestic contexts, and from public administrative ones. We will focus on questions such as: how is the account formatted? How are the entries organized? What evidence is there, both explicit and implicit, of calculations? What kind of knowledge does the account imply, and what is the account for? What information can we glean about the people involved, and in particular the author of the account? We will try to draw general features arising from the examination of individual accounts, and ask the key question about generalization: can we generalize on the basis of particular accounts? Can we generalize from the predominantly Egyptian material? Can we generalize in terms of the principles of economic behaviour? The session should be relevant for students of ‘practical’ mathematics focussing on later periods.
The Morals of Measurement
Dans le cadre du séminaire de Giovanna Cifoletti « L’algèbre comme art de penser entre cosmographie et mathématiques du négoce »
Mardi 2 avril 2019, 17h-19h – EHESS, salle A04_47, 54 bd Raspail, 75006 Paris
This session will follow a philosophical thread from Plato to Seneca, to Hero of Alexandria, to Pliny the Elder, exploring the question of whether measurement in particular, and some forms of mathematics in general, are ethically sound. Mathematical practice is often explicitly value-laden. The example of measurement is particularly poignant, because on the one hand, it is represented as a constituent of justice and fair distribution, and conducive to good order, both politically and socially, and cosmically, while on the other hand it is criticized as morally dubious. We will discuss passages in Plato’s Republic, Seneca’s Letters and Pliny the Elder’s Natural History to the effect that some forms of measurement (in Pliny’s case, measurement of the heavens) are reprehensible, or indeed downright deranged. We will focus on questions such as: what arguments are presented on each side? Can we detect intertextuality between the sources, and what can be inferred from these connections? To what extent does the ethical blur into the epistemological, and into the social – are criticisms of a mathematical practice also criticisms of the mathematical practitioners? This session should be useful to students of the later debates around the utility and status of the mathematical disciplines, particularly as they revolve around ancient authors who were widely read in the Renaissance.
Standardization in Antiquity – Measurement, Power, and Trust
Dans le cadre du séminaire de Grégory Chambon et Giovanna Cifoletti « Des histoires de nombres : une approche historique des calculs, mesures, valeurs chiffrées et représentations de grandeurs », et du séminaire de Cecilia D’Ercole « Déplacements et circulations dans la Méditerranée antique »
Jeudi 4 avril 2019, 9h-11h – EHESS, salle 1, 105 bd Raspail 75006 Paris
This session will look at the question of standardization of weights and measures, in particular in what ways it was conceptualized in the ancient Greek and Roman period, and to what extent it was practised or enforced. We will focus on questions such as: what was defined as a ‘standard’? How were standards arrived at? What discourses and rhetorical strategies surrounded the idea of standardization? What evidence do we have of standardization in actual practice? On the one hand, we will analyse some key textual documents, for instance the so-called Athenian standards decree, in the form of a 5th-century-BCE ‘patchwork’ inscription. On the other hand, we will survey the archaeological evidence available for metrological ‘standards’ – cups, measuring tables, weights, and the problems arising from their classification and typology. The session should be relevant for students of metrology and of the mathematics applied to metrology and standards, including those concentrating on different periods and contexts.
Numeracy in Antiquity. Sources, Methods and Problems
Dans le cadre du séminaire de Grégory Chambon « Savoirs et savoir-faire au Proche-Orient ancien »
Vendredi 5 avril 2019, 9h-11h – EHESS, salle 6, 105 bd Raspail, 75006 Paris
This session will explore the general question of how to approach the history of counting, calculating and measuring in ancient Greece and Rome, touching also upon other pre-modern contexts (Egypt, Mesopotamia, China) for comparison. What sources should we draw upon – in particular, given that evidence about numeracy is scarce in literary sources, how should we go about reconstructing it from archaeological, epigraphical, and other ‘material’ sources? What historiographical approaches are available, and what disciplines (anthropology, sociology) could we marshall in our aid? Ultimately, what questions and problems should guide our research? Hopefully the issues raised will be relevant for students of later pre-modern and early-modern periods as well.
4ème atelier Séries de Problèmes
Mercredi 19 décembre, 14h-16h, Centre A. Koyré.
Dans le prolongement d’une journée d’étude organisé en 2012 dans le cadre du projet, il s’agit d’étudier le principe de sériation dans les documents à caractère juridique, écrit en cunéiforme, afin d’identifier des « pratiques intellectuelles » cohérentes dans le système de pensée des Mésopotamiens, ainsi que leurs formes expression. Il s’agit à terme d’établir méthodologiquement un cadre comparatiste avec d’autres productions de cette culture, qui s’organisent aussi en série, avec une terminologie spécifique et une mise en ordre sous forme de récapitulatifs : les textes comptables établis dans les institutions (palais, temples). Ce dernier genre de document, qui par son caractère laconique et stéréotypé en lien avec le quotidien ne semble a priori ne pas se prêter à l’étude des pratiques intellectuelles, peut dévoiler, grâce à cette approche comparatiste, une forme de rationalité qui va au-delà de la simple réalisation mécanique de calculs ou le report de données effectives et répond à des phénomènes de normalisation et de sériation, par et pour l’écriture, qui se réfère à des modèles et représentations sociales particulières. Cette comparaison des formes de rhétorique sera en particulier mise en parallèle lors des discussions avec le cas des juristes de la Renaissance, qui offre un terrain d’étude privilégié entre les normes du domaine du droit et les normes du domaine des mathématiques.
Mise à disposition des textes: Des traductions en français des textes cunéiformes (juridiques et comptables) seront données, ainsi que leur transcription en langues anciennes (akkadien ou sumérien), l’objectif n’étant pas d’effectuer une analyse philologique pure mais une réflexion sur le rôle de la terminologie dans la structuration des textes et les processus de normalisation.
Journée d’études
Mathesis Universalis:
historical and systematic perspectives
Dans le cadre du séminaire de Giovanna Cifoletti (EHESS, CAK) « L’algèbre comme art de penser entre cosmographie et mathématiques du négoce ».
9h30-9h50. Introduction by Giovanna Cifoletti (EHESS, CAK)
10h-11h10. Mark van Atten (CNRS, IHPST)
Definite multiplicities and the incompleteness theorems
11h20 – 12h30. Michael Roubach (Hebrew University, Israel)
Fine’s notion of arbitrary objects and Husserl’s phenomenology
12h30-14h30. Lunch
14h30 – 15h40. David Rabouin (CNRS, SPHERE)
Mathesis universalis in Leibniz
15h50 -17h. Giovanna Cifoletti (EHESS, CAK)
Arguing with letters and numbers: notiones communes in Ciceronian dialectic and in mathematics
17h10 – 18h20. Jean-Marie Coquard (EHESS, doctorant)
Number and unit in Stevin
Discussants :
Burt Hopkins (Université de Lille, STL)
– Carlos Lobo (Collège International de Philosophie, Paris / Centro de Filosofia das Ciências da Universidade de Lisboa, Portugal)
– Fabrizio Palombi (Università della Calabria)
Conférences
Jacqueline FEKE
« Ptolemy’s philosophy »
Dans le cadre du séminaire « L’algèbre comme art de penser entre cosmographie et mathématiques du négoce », animé par Giovanna Cifoletti
This talk will provide an introduction to Ptolemy’s philosophy : its scope, major claims, and appearance across Ptolemy’s corpus in his texts on astronomy, astrology, harmonics, cosmology, and psychology.
- Mercredi 9 mai 2018, 17h-19h – EHESS (salle AS1-08), 54 boulevard Raspail 75006 Paris
« Ptolemy on theoretical and practical philosophy »
Dans le cadre du séminaire « L’algèbre comme art de penser entre cosmographie et mathématiques du négoce », animé par Giovanna Cifoletti
In the very first lines of the Almagest, Ptolemy responds to a contemporary debate over the relationship between the theoretical and practical parts of philosophy. In this talk, I will examine Ptolemy’s accounts of theoretical and practical philosophy and address what their respective roles are in Ptolemy’s virtue ethics.
- Mardi 15 mai 2018, 17h-19h – EHESS (salle A04_47), 54 boulevard Raspail 75006 Paris
« Ptolemy’s harmonics »
Dans le cadre du séminaire « Pratiques, artefacts et mathématiques. Ethnomathématique, anthropologie, histoire », animé par Marc Chemillier, Giovanna Cifoletti et Sophie Desrosiers
This talk will examine the epistemology and ethics of Ptolemy’s harmonics with a focus on the scientific method Ptolemy prescribes for harmonics and the application of harmonics in his study of astrology, astronomy, and psychology.
- Mercredi 16 mai 2018, 15h-17h – EHESS (salle 13), 105 boulevard Raspail 75006 Paris
« Ptolemy on mathematics and categories »
Dans le cadre du séminaire « L’algèbre comme art de penser entre cosmographie et mathématiques du négoce », animé par Giovanna Cifoletti
Ptolemy defines mathematics in Almagest 1.1, and he investigates the causes of mathematics and mathematical objects in Harmonics 3.3. In this talk, I will examine the nature of mathematics and mathematical objects within the larger context of Ptolemy’s account of the three theoretical sciences : physics, mathematics, and theology.
- Mardi 22 mai 2018, 17h-19h – EHESS (salle A04_47), 54 boulevard Raspail 75006 Paris
Conférences de Massimmo MAZZOTTI
« Shifting control: mathematical writing in the age of revolutions »
Dans le cadre du séminaire de S. Desrosiers, M. Chemillier, G. Cifoletti, E. Vandendriessche « Pratiques, artefacts et mathématiques. Ethnomathématique, anthropologie, histoire
Is mathematics primarily about intellectual intuition and the visual, or is it primarily about writing, and portable algorithmic procedures? This age-old question assumes new and urgent meaning in an age of political turmoil and redistribution of authority.
- Mercredi 13 juin 2018, 15h-17h – EHESS, salle 4, 105 bd Raspail, 75006 Paris
« Calculus as spiritual exercise »
Dans le cadre du séminaire de Grégory Chambon et Giovanna Cifoletti « De la pratique à la théorie. Comptes et mesures des marchands en Mésopotamie et en Europe de la Renaissance »
An intepretation of Maria Gaetana Agnesi’s Instituzioni Analitiche (1748), the first systematic presentations of the techniques of differential and integral calculus.
- Jeudi 14 juin 2018,13h-15h – EHESS, salle 3, 105 bd Raspail 75006 Paris
« History of mathematics as archeology »
Dans le cadre du séminaire de Giovanna Cifoletti « L’algèbre comme art de penser entre cosmographie et mathématiques du négoce »
The first journal of the history of mathematics was founded in Rome, in 1868. That the scientific study of mathematical documents should be supported by the last pope-king is not a coincidence.
- Mardi 19 juin 2018, 17h-19h – EHESS, salle A04_47, 54 bd Raspail 75006 Paris
« Algorithmic Life »
Dans le cadre du séminaire « L’algèbre comme art de penser entre cosmographie et mathématiques du négoce », animé par Giovanna Cifoletti
Historians of science and mathematics can offer precious and much needed contributions to contemporary discussions on digital algorithms.
- Mardi 26 juin de 17h à 19h (salle A04_47, 54 bd Raspail 75006 Paris
