Sciences, Philosophie, Histoire – UMR 7219, laboratoire SPHERE

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 ℝ^m (real m-space), and F be a continuous map from X into ℝⁿ, regarded as a subset of ℝ^m+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) : x ∈ X⎬, 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

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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.
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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.