Sciences, Philosophie, Histoire – UMR 7219, laboratoire SPHERE

The seminar on the history and philosophy of modern mathematics is a place of presentation and discussion of mathematical texts produced in the 19th and 20th centuries, in both historical and philosophical perspectives. It intends to serve as a place of exploration (reading, translation, explanation) of mathematical documents little or poorly known, but also to present work in progress on these periods. The emphasis is on proximity to textual sources. The sessions usually take the form of a discussion of the said sources (to which speakers give prior access), preceded by a historical or mathematical exposition.

ABSTRACTS

Tuesday January 17, 2pm–5pm, Room Rothko, 412B, hybrid

• Claudio Bartocci (DIMA, Università di Genova) et Jean-Jacques Szczeciniarz (Université Paris Cité, HPS, SPHere)
  The existence of the non-existent, a new problematic, that of the one-element body [in Fr]
  Three points will be addressed :
  1. History and presentation of the problem since the appearance of FUN in the work of Jacques Tits (1958) through the results of some mathematicians, Manin, Kapranov, Soulé, Deitmar.
  2. A new presentation through the works (summaries) of Toën Vaquié, and our understanding (CB AG JJS) of the problem. What the categorical point of view brings.
  3. Philosophy of mathematics, how to non-exist ? Quick comparison with root minus 1. The synthetic power of object distortion.

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Tuesday February 14, 9:30am-12:30am, Room Rothko, 412B, hybrid

• Peter Ullrich (Universität Koblenz-Landau)
  The theory of analytic factorials - a quarrel between proofs and computations
  Already at the beginning of the 19th century it was known that the factorial can be interpolated as an analytical function, at least on the positive real numbers, by means of the Gamma integral. However, some mathematicians tried to end interpolations in other ways, by setting up functional equations that these analytic factorials should satisfy and then trying to compute analytic expressions for them.
  A. L. Crelle, for example, published a whole book on the subject in 1823, which led to a sharp response from M. Ohm in 1829. K. Weierstraÿ brought clarification here, who clearly recognized that the interpolation problem has no unique solution and was thus able to correct Crelle and refute Ohm. Regarding the publication of his results, Weierstraÿ declined Crelle's o er of the Journal für die reine und angewandte Mathematik for this purpose, choosing instead his school’s 1843 program. It was not until 1856 that he published an expanded version in Crelle’s Journal.

• Valeriya Chasova (PLUS, GW Fakultät, Fachbereich Philosophie ; Archives Henri-Poincaré, UniStra/ULorraine/CNRS ; CEFISES, UCLouvain)
  Physical significance versus mathematical surplus in philosophy and history
  Not all there is in physical theory has physical significance (in the sense of contributing to predictions or to ontology of the physical world). There are also elements coming from mathematics and usually interpreted as idle. Separating this mathematical surplus from physical content is an important and non-trivial task. I will tell more (where possible in simple terms) about ways to accomplish it and related topics in recent philosophy and in XX century history.
  A still common solution in philosophy of physics, going back to Leibniz in his debate with Clarke (1717), is to count as mathematical surplus what varies under theoretical symmetry transformations. However, it has been defended from Kosso (2000) on that symmetry-variant elements have physical significance provided the theoretical symmetries concerned are matched with symmetries in the world or with conservation laws.
  Historically, Klein (1917) argued that energy-momentum conservation law is unphysical in Einstein’s just formulated general relativity (1915) because of not being linked appropriately with the field equations. However, Hilbert (Klein, 1917) took this to be a hallmark of that theory, while Noether (1918) showed that this feature is characteristic of a class of theories with certain symmetries. But her solution was itself mathematical, and its physical import only got better clarified later on.
  • text 1 | text 2

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Monday March 6, !! 1:30pm - 4:30pm, Room Gris, 734A !!, hybrid

• Walter Dean
  On models and computation in geometric consistency proofs
  The focus of the talk will be a method for transforming several of Hilbert’s (1899) model-theoretic consistency proofs for Euclidean and non-Euclidean geometries into syntactic demonstrations originally exposited by Paul Bernays (1935, 1939). Bernays’ method is of historical interest due its apparent relation to the Frege-Hilbert controversy. But it is also of mathematical interest because of the specific combination of proof-theoretic, analytic, and algorithmic techniques which it employs. Time permitting, I will also discuss how this method anticipates contemporary developments in reverse mathematics and automated theorem proving.

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Tuesday April 18, 1:30pm - 4:30pm, Room Rothko, 412B, hybrid

• Karine Chemla (CNRS, SPHere)
  From computation to a practice of proof through the introduction of a concept : Poncelet’s ideal elements in geometry
  In a first part of this presentation, I intend to return to the nature of the “ideal elements” that Jean-Victor Poncelet introduced into geometry, notably in his Traité des Propriétés Projectives des Figures (1822). This analysis will require that I discuss some features of the diagrams used by Poncelet. Moreover, I will examine the use of elements of this kind in the practice of proof to which the Traité des Propriétés Projectives des Figures attests. In a second part, I will turn to the notebooks that Poncelet wrote in Saratov, as a prisoner of war, between March 1813 and June 1814, to shed light on the part played by Poncelet’s analytical approaches at the time in the genesis of the concept of ideal elements. This talk is based on joint work with Bruno Belhoste.

• Clément Bonvoisin (Université Paris Cité, ED 623, SPHere)
  Des Schwarz-Weiß-Steuereungen aux bang-bang controls. Traductions mathématiques d’un problème d’ingénierie dans la thèse de Donald Bushaw (1952)

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Tuesday May 16, 9:30 am - 12:30 pm, Room Rothko 412B - Université Paris Cité, Condorcet building. 4, rue Elsa Morante, 75013 - Paris

• Patrick Popescu-Pampu, Université de Lille and Paul Painlevé laboratory
  What does it mean to solve the singularities of a plane algebraic curve ?
  I will examine several meanings of the notion of solving singularities of plane algebraic curves, as well as several methods of resolution.
  
• Paul-Emmanuel Timotei, Université Paris Cité and SPHERE laboratory
  Reduction of singularities of a plane algebraic curve by correspondence by Georges-Henri Halphen.
  I will present a technique for reducing the singularities of plane algebraic curves found in the appendix of George Salmon’s Traité de géométrie analytique (courbes planes) (French edition of 1884) : "Etude sur les points singuliers des courbes algébriques planes" by Georges-Henri Halphen.
  
  
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Friday June 2 juin, 9:30am - 12:30am,Room Rothko, 412B, hybrid

• Klaus Volkert Wuppertal/Luxemburg
  The early history of duality
  In my talk I will study three strands of the early history of duality : the theory of ploygons and polyhedra, the case of spherical geometry and the classic polar reciprocity. We will look for the base and the function of duality particular to the different fields, and we will analyze its common features. What is duality, why is there duality, and why it is so useful ?
  
  Lit. Etwein, F./Voelke, J.-D./Volkert, K. : Dualität als Archetypus mathematischen Denkens (Göttingen : Cuvillier, 2019).

• Tinne Hoff Kjeldsen (University of Copenhagen )
  From Mathematical Programming to Convex Analysis : Duality as a driving force in history of mathematics
  The presentation will focus on the emergence of convex analysis in the 20th century in the context of mathematical programming with special attention to the significance of duality. More specific, we will look at duality in the history of mathematical programming from von Neumann’s work in game theory to Fenchel’s duality theorem in nonlinear programming and the role it played for the development of convex analysis. How did ideas of duality emerge in linear programming ? What role did they play for the development of nonlinear programming ? How did Fenchel introduce ideas of duality in nonlinear programming and how did his duality function as a driving force for the development of convex analysis ?