Conférences

Mercredi 11 mars 2015
‘Platonism’ in the Philosophy of Mathematics after Jacob Klein.

Klein’s account of Plato’s philosophy of mathematics is presented as the basis for reassessing the standard interpretation of “Platonism” in mathematics. Specifically, Klein’s presentation of the mathematical and philosophical context of Plato’s “unwritten doctrine” of eidetic numbers, which he reconstructs primarily on the basis of Aristotle’s critique of it, is considered. On the basis of the results of this consideration, it is argued that the standard view of Platonism in mathematics, as the positing of mind independent mathematical objects, has no basis in Plato’s thought. Once this is established, Klein’s account of the problems in the philosophy of mathematics found in Plato is presented with a focus on the foundational problem of the unity of mathematical and eidetic numbers.

Dans le cadre du séminaire de Giovanna Cifoletti « L'algèbre comme art de penser entre cosmographie et mathématiques du négoce » 105 bd Raspail, salle 2, 17-19h

vendredi 13 mars
Jacob Klein on François Viète and the Birth of the Modern Symbolic Concept of ‘Number’

Klein’s mathmatico-philosophical account of the birth of the modern symbolic concept of number in Viéte’s analytic art is presented. Klein’s thesis is unpacked, that the logistice speciosa of Viéte’s analytic art extends Diophantus’ arithmetical eidos (species) concept to geometrical magnitudes, thereby effectively formalizing magnitude and generating for the first time “general number” in the medium of the species as an “object” in itself. Special attention is paid to Klein’s employment of the mediaeval distinction between first and second intentions and their respective intentional objects to articulate the symbolic constitution of “general number.” Finally, Klein’s account of the role the reinterpretation of the ancient arithmos concept of number (i.e., the determinate unity of a determinate multitude of monads) from the point of view of their symbolic representation plays in our understanding of both ancient “arithmetic” and “logistic” and of “ordinary numbers” as coinciding with the number sign as such is discussed.

Dans le cadre de la journée « Séries de problèmes au carrefour des cultures. La Renaissance »
Centre A. Koyré, 27, rue Damesme, 9h30 -11h30

mardi 17 mars
Jacob Klein on the Philosophy of the History of Exact Sciences

In his 1934 letter to Strauss, Klein succinctly assesses the significance of his “Die griechische Logistik und die Entstehung der Algebra” as follows: “So far as I am competent to judge, I can only say: this work is the first attempt to develop a fundamentally different approach to the history of the exact sciences and philosophy as it is ordinarily practiced.” Six years later, Klein supplements Edmund Husserl’s “intentional-historical” account of the development of modern mathematics with his own account, based on the results of his research in “Die griechische Logistik und die Entstehung der Algebra.” This paper will argue that the significance of Klein’s approach to the history of the exact sciences and philosophy lies in the originality of his philosophy of the history of mathematics. This originality is elaborated in terms of Klein’s unprecedented and underappreciated account of the shift in the basic mode of concept formation behind the ancient and modern epistemology of the most basic concept of mathematics, number.

Dans le cadre du séminaire de Giovanna Cifoletti « L'algèbre comme art de penser en te cosmographie et mathématiques du négoce » Bât. Le France, salle 1, 17-19h

mercredi 1er avril
Philosophical Problems in the Foundation of Arithmetic: Ancient and Modern.

Building on the seminal mathematico-philosophical researches of Jacob Klein, the philosophical problem “foundation” as it applies to ancient and modern mathematics is presented. The problem of the foundation of “number” in Pythagorean, Platonic, Aristotelian, and contemporary philosophy is discussed within the context of the shift in the conceptuality of number from its ancient, non-formalized meaning, to its modern, symbolically formalized meaning. This discussion highlights the peculiar “historicity” of a priori mathematical concepts and the paradoxes inherent in that historicity.

Dans le cadre du séminaire collectif « Mathématiques avec et sans discipline. Ethnomathématique, anthropologie, histoire »
bât. Le France, 190-198 av de France 75013 Paris), 13 h à 17 h (salle 3, RdC)