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L’algèbre comme art de penser entre cosmographie et mathématiques du négoce

  • Giovanna C. Cifoletti, directrice d'études de l'EHESS (TH) ( CAK )

    Cet enseignant est référent pour cette UE

S'il s'agit de l'enseignement principal d'un enseignant, le nom de celui-ci est indiqué en gras.

1er et 3e mardis du mois de 17 h à 19 h (salle 1, RdC, bât. Le France, 190-198 av de France 75013 Paris), du 15 novembre 2016 au 6 juin 2017. La séance du 15 novembre est annulée. Séances supplémentaires les 23 et 30 mai et 13 juin 2017

Les séances des 18 avril, 2 et 16 mai et 6 juin 2017 se dérouleront en salle AS1_23 (1er sous-sol) 54 bd Raspail 75006 Paris ; celles des 23 et 30 mai se dérouleront en salle AS1_24 (1er sous-sol) 54 bd Raspail 75006 Paris ; celle du 13 juin se déroulera de 13 h à 15 h en salle AS1_08 (1er sous-sol) 54 bd Raspail 75006 Paris.

L’algèbre prit sa nouvelle forme d’algèbre symbolique dans le contexte français du XVIe siècle. Cette nouvelle forme était accompagnée d’un nouveau sens : d’art du calcul, propre au monde du négoce, elle se transformait en art de penser et en clé de la compréhension cosmographique du monde.

L’enquête historique doit tenir compte des aspects culturels, anthropologiques et sociaux qui ont favorisé cette double transformation. L’histoire de l’algèbre est étudiée en relation avec l’histoire des mathématiques anciennes et modernes, du livre, de la philosophie, des institutions et des techniques à l’époque moderne.

Burt Hopkins (Università Ca' Foscari Venezia, Department of Philosophy and Cultural Heritage), professeur invité à l'EHESS donnera 4 conférences :

Jacob Klein’s Philosophical-Historical-Mathematical Method

23 mai : “Jacob Klein on Simon Stevin’s Symbolic Breakdown of the Ancient Distinction between Discrete and Continuous Magnitude”

I examine Klein’s argument that the new symbolic number concept [Zahl-Begriff] permits Stevin to address the consequences of the schism between the actual understanding of number [Zahl] and the understanding clinging to the traditional number concept [Anzahl-Begriff], as a definite amount of definite objects. According to Klein, these consequences include Stevin’s rejection of the ancient distinction between discrete and continuous magnitude and the traditional appellations of absurd or surd or irrational (i.e., un-speakable) numbers [Zahlen]. According to Klein, Stevin attacks these traditional appellations with the thesis that there are no absurd, irrational, irregular, inexplicable, or surd numbers, which supports the claim that incommensurability does not cause the incommensurable terms to be surds.

30 mai :  “Jacob Klein’s on John Wallis’s Symbolic Realization Number as Dimensionless Magnitude”

I examine Klein’s argument that it is only in terms of a symbolic reinterpretation of the mode of being of ancient numbers, understood as definite amounts of definite objects [Anzahlen], that Wallis’s account of the universality of arithmetic as a general theory of ratios, which depends on the homogeneity of all numbers, can be understood. A major consequence of Wallis’s reinterpretation is that the object of arithmetic and logistic in their algebraic expansion is now determined as number [Zahl], and this means as a symbolically conceived ratio. Not only is this conception of number, as ratio, consonant with that of algebra as a general theory of proportions and ratios, but, also, as the material of arithmetic and logistic, its being no longer presents any problem, since, unlike the problematical mode of being of the ancient material (ὕλη), its mode of being is immediately graspable in the notation. Thus Klein argues that unlike the pure units that make up the material of ancient number (ἀριθμός), the mode of being of which can be subject to dispute because it can be conceived of as formations that are either independent or obtained by abstraction (ἀφαίρεσις), in Wallis’s Mathesis Universalis the being of numbers [Zahlen]—its symbolic character as dimensionless—is indisputable.

6 juin :  “The Phenomenological Context of Jacob Klein’s Philosophy of Mathematics”

Leo Strauss’s famous remark, that there’s barely a sentence in Klein’s work whose meaning wasn’t influenced by his association with Martin Heidegger and Edmund Husserl, is as true as it is misleading. It’s true, because the phenomenological philosophy of Husserl and Heidegger informed both the content and the context of Klein’s philosophical development. It’s misleading, however, because Klein’s work on the origin of modern algebra can only be understood as a fundamental critique of major presuppositions guiding both Heidegger’s and Husserl’s phenomenologies. In the case of Heidegger, the presupposition is that ancient Greek philosophy was basically ontology, the inquiry into the intelligibility and nature of the things that exist, beings. Klein argues that this view of ancient Greek philosophy is basically Aristotelian and therefore elides ancient Greek philosophy’s more original origin in Plato’s and the Pre-Socratics’ preoccupation with the problem of the one and many in nature and number. In the case of Husserl, Klein’s work criticizes the presupposition behind his theory of intentionality, that general ontology provides the answer to the question of the meaning of being. Klein’s work does so by showing that the meaning of general ontology is inseparable from historically dated presuppositions stemming from 16th and 17th century philosophy and mathematics, presuppositions that occlude the intelligibility of non-formalized structures of being.

13 juin :  “Jacob Klein on the Philosophy of the History of the Exact Sciences”

In this lecture I will discuss Klein’s argument that mathematics as a science lacks the conceptual resources to investigate its own history, because the formality of its concepts are not historically datable within the context of mathematical science. The conclusion he draws from this will also investigated, that the history in question only emerges as a problem on the basis of philosophical investigations into the foundations of mathematics, which are historically datable. Finally, the implications of Klein’s conclusion, that the proper discipline of the history of mathematics is the history of the philosophy of the foundations of mathematics, will be discussed and critically assessed.

Aires culturelles : Europe,

Suivi et validation pour le master : Bi/mensuel annuel (24 h = 6 ECTS)

Mentions & spécialités :

Domaine de l'affiche : Histoire - Histoire des sciences

Intitulés généraux :

  • Giovanna C. Cifoletti- Savoirs mathématiques et arts de penser à l'époque moderne
  • Renseignements :

    contact Giovanna Cifoletti, Mathématiques et histoire, Centre Alexandre-Koyré, 27 rue Damesme 75013 Paris.

    Direction de travaux d'étudiants :

    rendez-vous par écrit, adresse ci-dessus.

    Réception :

    sur rendez-vous, demandé par écrit, adresse ci-dessus.

    Niveau requis :

    séminaire ouvert au master.

    Site web : http://mathshistoire.ehess.fr/

    Adresse(s) électronique(s) de contact : cifolet(at)gmail.com

    Dernière modification de cette fiche par le service des enseignements (sg12@ehess.fr) : 24 mai 2017.

    Contact : service des enseignements ✉ sg12@ehess.fr ☎ 01 49 54 23 17 ou 01 49 54 23 28
    Réalisation : Direction des Systèmes d'Information
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