Mercredi 16 octobre
Bâtiment Libération (91, avenue de la Libération), rez-de-chaussée, salle 002
10h30-12h – Chair : Gerhard Heinzmann
Olivier Fouquet (Laboratoire de Mathématiques de Besançon)
Get real! (How $p$-adic numbers came to be).
After Hensel and several important advances in the 1920s and 1930s, it has been understood that to natural, rational, real and complex numbers should be added a new family of numbers for each prime number $p$: the so-called $p$-adic numbers. Discussing this example, I will give in this talk a working mathematician perspective on how and why new mathematical concepts are established and gain existence, at least in the mind of mathematicians.
14h-17h30 – Chair : Francesca Poggiolesi
Iulian Toader (University of Vienna)
Judging Properly: Weyl on Consistency, Quantification, and Correctness
Abstract: Rather than being a philosopher of mathematics who often jumped boat, Weyl was deeply committed to several philosophical ideas that he never gave up (so far as we know). I will argue that in order to understand his criticism of the formalist interest in consistency proofs, one must figure out Weyl's conception of quantification, and in order to do this, one has to appreciate his commitment to a correctness-first epistemology.
Wesley Holliday (U. California - Berkeley)
From Constructive Mathematics and Quantum Mechanics to Fundamental Logic
Non-classical logics have been proposed in a number of domains, including constructive mathematics and quantum mechanics. In this talk, I will identify a base logic beneath some of these non-classical logics that I suggest has a certain fundamental status. I will give an introduction to the proof theory and semantics of this “Fundamental Logic.”
An associated paper is available at https://arxiv.org/abs/2207.06993
Jeudi 17 octobre
Bâtiment Présidence Léopold (34, cours Léopold), amphithéâtre Léopold
9h-12h30 – Chair : Marianna Antonutti
Sandra Bella (AHP, Université de Lorraine)
L’invention de l’osculation par Leibniz. Enjeux techniques et conceptuels
Au début des années 1680, Gottfried Wilhelm Leibniz entreprend des recherches pour donner sens à la notion « d’angle entre deux courbes » et questionne de ce fait la figure ancienne et polémique d’angle de contact. Ses recherches aboutissent à des résultats conceptuels de son propre cru qu’il publie en 1686. Il y généralise de manière originale la notion d’angle de contact par celle d’angle d’osculation, ce qui lui permet de donner des clarifications inédites concernant la mesure de la courbure ainsi que celle de l’angle entre deux courbes. Pourtant Leibniz ne réussit pas à accompagner ses inventions conceptuelles des techniques qui permettraient de représenter ces dernières par le biais d’un calcul.
Des brouillons mathématiques de Leibniz ainsi que certains de ses échanges épistolaires rendent manifeste la tension au cœur des réflexions leibniziennes entre recherche de fondements conceptuels et inventions calculatoires, dont nous souhaiterions rendre compte dans cette intervention.
Guillaume Massas (Scuola Normale Superiore, Pisa)
A New Way Out of Galileo’s Paradox
Galileo asked in his Dialogue of the Two Sciences what relationship existed between the size of the set of all natural numbers and the size of the set of all square natural numbers. Although one is a proper subset of the other, suggesting that there are strictly fewer squares than natural numbers, the existence of a simple one-to-one correspondence between the two sets suggests that they have, in fact, the same size. Cantor famously based the modern notion of cardinality on the second intuition. However, recent advances in mathematical logic have renewed an interest in the question whether Cantor’s way out of Galileo’s paradox was the only possible one.
In this talk, I will present a new solution to Galileo’s paradox and argue that it is the only legitimate alternative that can be based on the “Euclidean” intuition that the whole is always greater than any of its proper parts.
14h30-18h – Chair : Paola Cantù
胡靖凯 (Hu Jingkai) (Pays Germaniques, ÉNS)
Phenomenological Analysis on Visualization and Intuition in Mathematics
In mathematical practice, diagrams are generally regarded as cognitive tools that assist us in understanding and presenting arguments. Therefore, the use and understanding of diagrams depend on a certain mathematical expertise. However, some cases about how finite diagrams visualize the infinite limit in mathematics suggest to us that the visualization of mathematics involves not only the auxiliary function of visible images in arguments but also the question of how mathematical objects are constructed in our consciousness in a way that is different from objects of everyday experience. Therefore, this paper will use sequences as an example and employ a phenomenological analysis method to illustrate a broader conception of mathematical visualization.
Mark Wilson (University of Pittsburgh)
Reasoning Control through Topographic Chart
In this talk I shall attempt to blend Riemann’s structural insights with some recent advances in multiscalar modeling as a means of addressing the vexed problems of crediting puzzling words such as “force” and “cause” with coherent forms of supportive “meaning.” In the course of working out these suggestions, I will also provide a useful gloss on Frederick Waismann’s enigmatic remarks with respect to “language strata” and “open texture.”
18h – Réunion du RT Philosophie des mathématiques
Vendredi 18 octobre
Bâtiment Libération (91, avenue de la Libération), rez-de-chaussée, salle 002
9h-12h30 – Chair : Viviane Durand-Guerrier
Jean-Philippe Narboux (CREPhAC, Université de Strasbourg)
Wittgenstein on the Status of the Law of Excluded Middle in Mathematics
In Book 5 of the Remarks on the Foundations of Mathematics, Wittgenstein probes the inclination to hold the question whether there are five consecutive occurrences of ‘7’ in the decimal expansion of π to be already settled, or at any rate meaningful, prior to the provision of a method for settling it. He contends that this inclination rests on sand insofar as it builds upon a picture of the infinite that cannot support its weight. It helps itself to an extensionalist picture of the infinite, a central symptom of which is an insistence on the unconditional validity of the principle of excluded middle. I propose to show that notwithstanding superficial affinities with the intuitionists’s case against the unrestricted validity of the principle of excluded middle, Wittgenstein’s line of thought is ultimately at odds with the intuitionists’.
Andrea Ariotto (Sorbonne Université / Università del Piemonte orientale)
Fécondité et objectivité du formalisme chez Cavaillès et Husserl
Dans cette conférence je vise à discuter les traits essentiels du « formalisme modifié » qui définit la position propre à Jean Cavaillès. Premièrement, à partir d’une lecture de Méthode axiomatique et formalisme, j’entends aborder deux thèmes, celui de la « fécondité propre », et celui de la « portée objective » du formalisme. Afin d’expliquer le premier, il faut remonter aux considérations de Cavaillès sur la théorie de la généralisation, qu’il expose au début de son ouvrage et pour laquelle les travaux de Dedekind offrent un exemple paradigmatique. En revanche, la deuxième question fait référence à la philosophie hilbertienne du signe, que Cavaillès réinterprète en développant de manière originale l’inspiration kantienne à laquelle Hilbert fait lui-même allusion.
Deuxièmement, le positionnement de Cavaillès vis-à-vis du formalisme hilbertien permet de comprendre la critique qu’il formule à l’égard des conceptions propre au « formalisme radical » et au logicisme, qui impliquent une réduction des mathématiques à un jeu combinatoire et aveugle de symboles et, corrélativement, empêchent une compréhension de l’historicité qui caractérise la pensée mathématique à travers l’imposition d’une syntaxe générale. Dans Sur la logique et la théorie de la science, les difficultés que Cavaillès relève dans la Syntaxe logique du langage de Carnap l’amènent à prendre en compte la phénoménologie husserlienne.
Dans la troisième partie de ma conférence, j’en viens alors à la position de Husserl pour développer essentiellement deux points. D’abord, je m’arrête sur la critique que Husserl formule, à son tour, à l’égard des « mathématiques purement formelles » (notamment dans Logique formelle et logique transcendantale et dans la Krisis). Ensuite, je remonte aux premiers travaux husserliens où il est question de justifier, dans un cadre que l’on peut définir comme pré-phénoménologique, ce que Cavaillès appelle la « fécondité propre » des mathématiques formelles. Le but de mon intervention est de montrer, d’une part, le fil conducteur qui conduit Cavaillès dans l’analyse du logicisme et de la phénoménologie, d’autre part, de souligner que, malgré les critiques adressées à Husserl, les deux positions demeurent plutôt complémentaires dans le cadre d’une compréhension alternative des conséquences philosophiques du formalisme.
14h-15h30 – Chair : Emmylou Haffner
Valérie Debuiche (CGGG, Aix-Marseille Université)
Leibniz's work on perspective: from representation to projection
It is well known that Leibniz was a great philosopher and a creative mathematician. More precisely, he was an audacious geometer and a convinced metaphysician. One of the most famous Leibnizian topic linking geometry and metaphysics, present both in mathematical notes and drafts and in more achieved philosophical texts, is the one of perspective: each substance is a point of view on universe, a kind of perspective representation of it. Quite recently Leibniz’s manuscripts on geometrical perspective have been published and shed some light on his own conception of perspective. The main manuscript, intitled Scientia perspectiva, is a masterpiece of genuine inventiveness. It reveals Leibniz’s idea of a new kind of geometry, that is of a pure geometry of spatial relations (and not of magnitudes). My purpose is to present how, in this text, he went from a very consensual formulation of perspective relations to an unexpected projective geometry of space and points, consistent with his metaphysics of centers of perception and deeply innovative for the mathematics of 17th century.
