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Accueil > Archives > Séminaires des années précédentes > Séminaires 2015-2016 : archives > Séminaire PhilMath Intersem 7 2016

Axe Histoire et philosophie des mathématiques

Séminaire PhilMath Intersem 7 2016

PhilMath Intersem est une coopération entre Notre Dame University, SPHERE, le Département d’Histoire et Philosophy des Sciences de l’Université Paris Diderot et le laboratoire SPHERE (UMR7219).

La septième édition du PhilMath Intersem (PhilMath Intersem 7) aura lieu au moins de juin 2016.
Le thème sera la méthode axiomatique et ses utilisations en histoire des mathématiques. Nous considérerons des questions historiques, philosophiques et logiques.
La langue du séminaire sera l’anglais.

Les résumés, conseils de lectures et informations pratiques sont disponibles sur le site :
Contact : Emmylou Haffner (Université de Lorraine & SPHERE)

Toutes les séances auront lieu salle Klimt (366A) du bâtiment Condorcet (10 rue Alice Domon & Léonie Duquet, 75013 Paris), sur le campus de l’Université Paris Diderot


Thème 2016 : Méthode axiomatique et ses utilisations en histoire des mathématiques

Séances du 2, 7, 9, 14, 16, 21, 23, 28

Jeudi 2 juin

  • 14:00
 Pascal Crozet (CNRS, SPHERE)
    Thābit ibn Qurra and the fifth postulate

    Thabit ibn Qurra (826-901) was not the first to propose a demonstration of the parallels postulate. However, his use of movement as a primitive notion of geometry, combined with his use of Archimedes’ and Pasch’s axioms, provided the starting point for a research tradition that lasted several centuries.
  • 16:00 Vincenzo De Risi (Max Planck Institute for the History of Science, Berlin)
    The axiomatization of space in the modern editions of Euclid’s Elements.
    The talk deals with the development of the system of axioms grounding elementary geometry in the editions of Euclid’s Elements in the Early Modern Age. Particular importance was given to the axiomatization of purely spatial relations : something that was absent in ancient axiomatics and which reflects a new conception of the object and methods of geometry.

Mardi 7 juin

  • 14:00
 Victor Pambuccian (Mathematics, Arizona State University)
    Why should one follow the axiomatic method in geometry ? An apology.

    In AD 2016 there are fewer than ten people in the world actively working with the axiomatic method in geometry. On a Hegelian reading, the historic demise of the axiomatic method in geometry, where analytical methods have carried the day for more than a century now, is a sure sign of its intrinsic inferiority, and proof that better approaches have replaced it. In this talk, we will try to make the case for the advantages sub specie aeternitatis of the axiomatic method, which opens up a whole new view of truth in geometry, and which presents us with results that have no equivalent whatsoever in algebraic, differential, or topological flavors of geometry.
  • 16:00 Volker Halbach (Philosophy, New College, University of Oxford)
    The axiomatic approach to semantics.

Jeudi 9 juin

  • 14:00
 Volker Peckhaus (Wissenschaftstheorie und Philosophie der Technik, Universität Paderborn)
    Zermelo’s Set Theory and Hilbert’s Philosophy of Axiomatics.

Mardi 14 juin

  • 14:00
 David Rabouin (CNRS, SPHERE)
    "En ne laissant passer aucun axiome sans preuve". On the need to demonstrate axioms in Leibniz.
  • 16:00 Alain Prouté (Mathématiques, Université Paris Diderot)
    The axiom of choice from an algorithmic viewpoint.
    The analysis we propose for the axiom of choice provides a clear definition of what "choosing" means (in contrast with "computing"), so explaining at the same time why some instances of this schema of axioms are non constructive and why the choice by itself remains nevertheless always effective. This pragmatic approach is likely to largely demystify the axiom of choice and was motivated by the realization of a proof assistant software based on topos theory. The subject is highly linked to the question of the indiscernibility of proofs, which currently causes heated discussions within the microcosm of logicians and computer scientists. It is likely that our arguments are already known by some proof assistant experts, but the talk will be the occasion to exchange points of view on this question of indiscernibility of proofs, which is to our opinion the most salient feature of mathematics, compared to programming.

Jeudi 16 juin

  • 14:00
 Akihiro Kanamori (Mathematics, Boston University)
    Axioms as procedure and infinity as method, both in ancient Greek geometry and modern set theory.
    Axiomatics have become de rigueur in modern mathematics, especially with the development of mathematical logic and formalization. But separate from this, what about axiomatics arising in mathematical practice ? This was in ancient Greek geometry and modern set theory, which act as parentheses for mathematics in several senses. For both, it is corroborated, by looking at the history and practice, that axioms (and definitions) serve to warrant procedures (e.g. constructions) and to regulate infinity as method. Axioms in practice are thus less involved with mathematical truth or metaphysical existence than with the instrumental properties of concepts. For Greek geometry, this view provides counterweight to the doxography (Plato, Aristotle, Proclus) and historiography (Zeuthen, Unguru controversy), which lean to interpreting constructions as establishing existence. For set theory, this view coheres with extensions of set theory through large cardinal and forcing axioms, and arms the eld of mathematics not as invested in some search for truth but as a study of well-foundedness through the transfinite.
  • 16:00 Juliet Floyd (Philosophy, Boston University)
    Gödel on Russell and Axiomatic Method 1942-43
    Gödel’s “MaxPhil” Gabelsburger notebooks IX-X (1942-3), begun on the day he accepted the invitation to write for Russell’s Schlipp volume, reveal a fascinating attempt on Gödel’s part to come to grips with Russell’s overall philosophy : not only the mathematical and logical aspects of Principia, but more purely philosophical ones.
    The distance between Gödel’s manner of working in these notebooks and his published tribute to Russell (1944) is considerable, evincing a more sophisticated and longstanding grappling with Russell than has been thought. In particular, Gödel did not uncritically hold that we can literally “see” sets, or that axioms “force themselves upon us” as physical objects do. Instead, he aimed in 1942-3 to rigorize Russell’s Principia idea of truth-as-correspondence, the “multiple relation theory” of judgment, rooted at the atomic level in (what Russell called) “judgments of perception”.
    Aware of Wittgenstein’s impact on Russell after 1918, and having read Russell’s main philosophical works through Inquiry into Meaning and Truth, Gödel holds out for an infinitary version of the multiple relation view, one that takes order, as well as our capacity for other-than-step-by-step, finistic thought, to be basic.
    Every axiomatization leaves some interpretive residue behind, the trail where the human serpent brings philosophy and knowledge into the garden of analysis. Gödel’s picture of the rigorizations of truth in axiomatic systems, expressed in the following quote, will be analyzed : A board game is something purely formal, but in order to play well, one must grasp the corresponding content [the opposite of combinatorically]. On the other hand the formalization is necessary for control [note : for only it is objectively exact], therefore knowledge is an interplay between form and content. [Max Phil IX 16].

Mardi 21 juin

  • 14:00
 Jean Petitot (CAMS, EHESS, École Polytechnique)
    The intertwining of structures in complex proofs.
  • 16:00 Michael Hallett (Philosophy, McGill University)
    Hilbert’s Axiomatic Method : Logic and Set Theory.

Jeudi 23 juin

  • 16:00 Sébastien Gandon (Philosophie, Université Blaise Pascal)
    Axiomatization as Analysis
    Russell’s conception of axiomatization is puzzling. On the one hand, axiomatization plays a central role in logicism, since the whole of mathematics (and not only arithmetic as in Frege) is said to be deducible from a few logical principles ; on the other, Russell’s view of axiomatization is markedly different from the contemporary Hilbertian approach. In this talk, my aim is to put Russell’s conception in its historical context. Discussing some writings of Moore and Sidgwick, I will suggest that Russell’s view of axiomatization takes its roots in the debate opposing utilitarians and intuitionists in XIXth Century moral philosophy.

Mardi 28 juin

  • 16:00 Gilles Dowek (INRIA)

Directors : Pr. Michael Detlefsen (University of Notre Dame) and Pr. Jean-Jacques Szczeciniarz (Université de Paris 7-Diderot)