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Accueil > Séminaires en cours > Séminaire PhilMath Intersem 12. 2022

Axe Histoire et Philosophie des mathématiques

Séminaire PhilMath Intersem 12. 2022



PhilMath Intersem est une coopération entre Notre Dame University et le laboratoire SPHERE (UMR7219), initiée avec le regretté professeur Michael Detlefsen.

Archives : 2022, 2021,
2019, 2018, 2017, 2016, 2015,
2014, 2013, 2012, 2011, 2010


PROGRAMME 2022


Les séances auront lieu du 1er au 14 juin 2022.

Le thème de cette année sera "Logique et géométrie". La langue du séminaire sera l’anglais.


Le séminaire aura lieu en format hybride :
– Pour obtenir le lien Zoom, merci de contacter Emmylou Haffner par email avec comme objet "Zoom-Intersem2022"

– Présentiel : salle Mondrian (646A) du bâtiment Condorcet (10 rue Alice Domon & Léonie Duquet, 75013 Paris), SAUF la séance du 14 juin (salle Rothko, 412BA), sur le campus de l’Université Paris Cité (ex-Diderot)

Planning, accès mercredi 1er vendredi 3 mardi 7 jeudi 9 mardi 14

mercredi 1er, 14h30 - 17h45, salle Mondrian (646A) & Zoom
14h30
16h
Mathieu Anel (Carnegie Mellon University)
Three kinds of interactions between logic and geometry
résumé
16h15
17h45
Chris Miller (Ohio State University)

Collapse of dimensions in the absence of arithmetic

résumé

vendredi 3, 14h30 - 17h45, salle Mondrian (646A) & Zoom
14h30 - 16h Thomas Seiller (CNRS, Laboratoire d’Informatique de Paris Nord)

Linear logic and the geometry of computation

16h15
17h45
David Waszek (AHP-PReST, CNRS) & Nicolas Michel (Universiteit Utrecht)

“Eine prachtvolle Machine” : Notational innovation and the genesis of the Schubert calculus

résumé

mardi 7, 14h30 - 17h45, salle Mondrian (646A) & Zoom
14h30
16h
Alberto Naibo (IHPST, Université Paris 1 Panthéon Sorbonne)

Gentzen, proof theory, and projective geometry

16h15
17h45
Vincenzo de Risi (SPHere, CNRS)

Common Axioms in Euclid and Aristotle

résumé

jeudi 9, 13h30 - 18h30, salle Mondrian (646A) & Zoom
13h30
15h
Marwan Rashed (Centre Léon Robin, Sorbonne Université)

Al-Samaw’al between mathematics, logic and theology

15h15
16h45
Tabea Rohr (AHP-PReST & IHPST)

The distinction between analytic and synthetic geometry from an axiomatic point of view

résumé
17h00
18h30
Jean-Jacques Szczeciniarz (SPHere, Université Paris Cité)
Quelques remarques sur Le concept d’espace à travers les topologies de Grothendieck, et des analyses de Lawvere

mardi 14, 13h30 - 18h30, salle Rothko (412B) & Zoom
13h30
15h
Victor Pambuccian (Arizona State University)

The Parallel Postulate : The View from Logic

résumé
15h15
16h45
Marco Panza (CNRS, IHPST, Université Paris 1 Panthéon Sorbonne)

What Universality Could Have Been for Euclid

17h00
18h30
Jemma Lorenat (Pitzer College)
Russell, Von Staudt, and Hilbert in the Bryn Mawr College Mathematics Journal Club Notebooks
résumé


RÉSUMÉS


mercredi 1er, 14h30 - 17h45, salle Mondrian (646A) & Zoom

  • 14h30 - 16h
    Mathieu Anel (Carnegie Mellon University)
    Three kinds of interactions between logic and geometry
    I will introduce the following notions and explain how they articulate logic and geometry :
    (1) the models of logical theories in spaces, (2) the space of models of a theory, and (3) the underlying space of a logical theory.
  • 16h15 - 17h45
    Chris Miller (Ohio State University)
    Collapse of dimensions in the absence of arithmetic
    One often finds informal definitions of the dimension of a mathematical object to be along the lines of “the minimum number of coordinates needed to describe any arbitrarily given point of the object”. Even assuming for the moment that we understand what is meant by “coordinates”, “mathematical objects” and “points of mathematical objects”, there are still the issues of what it means to describe points of objects and the extent to which these informal definitions capture the notion of dimension in contemporary mathematics. It is now widely accepted in mathematics that dimensions can take on non-integer real values (e.g., the classical middlethirds Cantor set has Hausdorff dimension log3(2)) and that different notions arise even among integer-valued dimensions. However, one easily sees that the standard examples witnessing these differences tend to involve inductive constructions and choice principles. Thus, the question arises as to how “natural” are these differences. In joint work with Philipp Hieronymi [1], we provide at least one reasonably concrete answer via mathematical logic. Our result can be stated heuristically as follows :

    Let m and n be positive integers, X be a closed (in the usual topology) subset of ℝm (real m-space), and F be a continuous map from X into ℝn, regarded as a subset of ℝm+n. If the first-order structure (ℝ,+, ·, F) does not define ℤ (the set of all integers), then all notions of topological or metric dimension commonly encountered in fractal geometry, geometric measure theory and analysis on metric spaces agree on F(X) := ⎨F(x) : xX⎬, the image of the set X under F. Moreover, they all agree on F(X) with what is arguably the most naive notion of dimension that could ever arise in dealing with subsets of finite cartesian powers of ℝ.

    It follows immediately from Gödel’s Incompleteness Theorem that if the set F(X) is an example witnessing the inequality of certain notions of dimension, then the complete theory of (ℝ,+, ·, F) is undecidable. But much more is true : The definable (allowing real parameters) sets of (ℝ,+, ·,ℤ constitute the real projective hierarchy, and so F must somehow encode enough information so that every real Borel set is first-order definable from F over the field of real numbers (allowing all real numbers as constants). I shall attempt to explain the result and its significance without assuming knowledge of dimension theory or descriptive set theory.
    – Référence :
    [1] Philipp Hieronymi and Chris Miller, Metric dimensions and tameness in expansions of the
    real field
    , Trans. Amer. Math. Soc. 373 (2020), no. 2, 849–874, DOI 10.1090/tran/7691.
    MR4068252



vendredi 3, 14h30 - 17h45, salle Mondrian (646A) & Zoom

  • 14h30 - 16h
    Thomas Seiller (CNRS, Laboratoire d’Informatique de Paris Nord)
    Linear logic and the geometry of computation
  • 16h15 - 17h45
    David Waszek (AHP-PReST, CNRS) & Nicolas Michel (Universiteit Utrecht)
    “Eine prachtvolle Machine” : Notational innovation and the genesis of the Schubert calculus
    New geometric calculi (i.e., new ways of obtaining geometric results by means of symbolic computations) are often seen as transformative achievements in the history of mathematics—analytic geometry being the best-known example. From afar, such successes seem unmysterious : they appear like straightforward applications of some kind of algebraic device to geometry. Looking more closely, however, reveals little-studied complexities to how and why such calculi succeed.
    As an example, we shall explore the genesis of Hermann Schubert’s 1879 enumerative calculus, the purpose of which is to count how many geometric figures of a certain kind satisfy some conditions (for instance, how many plane conics are tangent to five given lines). This calculus struck its contemporaries by its remarkable, though rather mysterious, computational power ; in 1900, Hilbert included its rigorous justification as the 15th of his famous list of open problems. While it looks like an application of Boole and Schröder’s algebra of logic to geometry, and has regularly been described as such —both by contemporaries, like Cayley or Peirce, and by later historians— we shall argue that it instead grew out of a circuitous sequence of computationally effective notational innovations rooted in classical projective geometry. We shall then draw broader lessons about the making of symbolic calculi.



mardi 7, 14h30 - 17h45, salle Mondrian (646A) & Zoom

  • 14h30 - 16h
    Alberto Naibo (IHPST, Université Paris 1 Panthéon Sorbonne)
    Gentzen, proof theory, and projective geometry
  • 16h15 - 17h45
    Vincenzo de Risi (SPHere, CNRS)
    Common Axioms in Euclid and Aristotle
    There is a strong relation between Euclid’s common notions in the Elements and the common axioms that Aristotle mentions in the Metaphysics and the Posterior Analytics. It is not likely, however, that Euclid may have been directly influenced by Aristotle’s work. This makes the relations between Euclid’s and Aristotle’s conceptions of common axioms even more interesting, since it shows how two very different personalities, a philosopher and a mathematician, may have read and interpreted the same set of principles. In this talk, I argue that Euclid’s first three common notions together provide a neat axiomatization of equality and additivity of measure. While they may have first been conceived in geometry, they are easily and flawlessly generalized to numbers and magnitudes in general, thus fitting in with the Aristotelian remarks about common axioms. They are entirely propositional and their application does not rely on any diagrammatic inference. By contrast, the fourth and fifth common notions listed in the Elements were external to this theory and responded to different epistemic needs. In particular, I claim that these common notions are employed in diagrammatic reasoning and display a different underlying epistemology than the other purely propositional principles. I complement these results on ancient mathematics with a discussion on Aristotle’s very different conception of the same axioms, and advance an “inferential” interpretation of Aristotle’s views on axioms that is at odds with the standard reading of them as schemata of principles. I conclude by showing the fruitful interplay between Aristotle’s philosophical speculation and Euclid’s mathematical practices.

jeudi 9, 14h30 - 18h30, salle Mondrian (646A) & Zoom

  • 13h30 - 15h
    Marwan Rashed (Centre Léon Robin, Sorbonne Université)
    Al-Samaw’al between mathematics, logic and theology
  • 15h15 - 16h45
    Tabea Rohr (AHP-PReST & IHPST)
    The distinction between analytic and synthetic geometry from an axiomatic point of view
    19th-century geometry was shaped by the methodological debate between analytic and synthetic geometers. In this talk, it will be shown how these methodological concerns reappear in Hilbert’s axiomatic setting, for example, in the question at which stage continuity principles are introduced. It will be argued that Hilbert favored a more synthetic approach. Nonetheless, Hilbert thought his axiomatic approach was superior to both analytic and synthetic geometry. For this purpose, Hilbert’s writings, including unpublished lectures, of the 1890s and the early 20th century, will be taken into account, as well as writings from Hilbert’s intellectual environment like Hertz and Helmholtz.
  • 17h - 18h30
    Jean-Jacques Szczeciniarz (SPHere, Université Paris Cité)
    Quelques remarques sur Le concept d’espace à travers les topologies de Grothendieck, et des analyses de Lawvere



mardi 14, 13h30 - 18h30, salle Rothko (412B) & Zoom

  • 13h30 - 15h
    Victor Pambuccian (Arizona State University)
    The Parallel Postulate : The View from Logic
    The parallel postulate has several weakenings. Among these are : The rectangle axiom, stating that there exists a rectangle, the Lotschnittaxiom, stating that the perpendiculars raised on the sides of a right angle intersect, and Aristotle’s axiom, stating that the distances from one side of an angle to the other side grow indefinitely. The relations between these axioms, as well as purely incidence-geometric versions of the Lotschnittaxiom and Aristotle’s axiom, with an analysis of their syntactical simplicity. The unexplained fact that, whenever someone comes up with an "of course true" statement weaker than the parallel postulate, one that does not sound like a theorem but rather an "obviously true fact of experience", it turns out that the statement is equivalent to the Lotschnittaxiom. This entire world of weakenings of the parallel postulate collapses in the presence of the Archimedean axiom, given that, in its presence, the Lotschnittaxiom is equivalent to the parallel postulate.
  • 15h15 - 16h45
    Marco Panza (CNRS, IHPST, Université Paris 1 Panthéon Sorbonne)
    What Universality Could Have Been for Euclid
  • 17h - 18h30
    Jemma Lorenat (Pitzer College)
    Russell, Von Staudt, and Hilbert in the Bryn Mawr College Mathematics Journal Club Notebooks
    As mathematics departments in the United States began to shift toward standards of original research at the end of the nineteenth century, many adopted journal clubs to stay abreast of new periodical literature. The Bryn Mawr Mathematics Journal Club, maintained episodically between 1896 and 1924, began as a supplement to the graduate course offerings. The Notebooks document the process of becoming a professional mathematician by recording ways in which graduate students engaged with contemporary literature, formulated research questions, and assessed tools and techniques for potential solutions. This talk will consider how Scott adapted the foundations of geometry in three case studies represented by journal club entries : non-Euclidean geometry in preparation for Bertrand Russell’s lectures on the foundations of geometry at Bryn Mawr (1896), Von Staudt’s treatment of imaginary elements (1898), and Hilbert’s Foundations of Geometry (1902). By situating well-known texts in this pedagogical environment, I will emphasize how they modeled the “local knowledge traditions” of Bryn Mawr.