Sciences, Philosophie, Histoire – UMR 7219, laboratoire SPHERE

W 2, 16:00 – 18:30, Room Malevitch (483A) & Zoom

• 4pm Opening

• 4:15pm – 5:15pm
  Marco Panza (CNRS, IHPST, University Paris 1 Panthéon Sorbonne)
  Michael Detlefsen and Mark Luker on the Proof of the Four-Colors Theorem : Empiricism, Platonism, or both ?

• 5:30pm – 6:30pm
  Andrew Arana (CNRS, Archives Henri Poincaré - PReST, University of Lorraine)
  A vectorial conception of problem-solving
  In our paper “Purity of Methods” (Philosophers’Imprint, 2011), Mic & I characterized one epistemic value of purity as better ensuring the stability of problem-solutions through changes in the epistemic attitudes of the problem-solver than impurity. In our account we took what we called to each other informally at least, a “vectorial conception” of problem-solving. On this view, a mathematical problem represents a particular ignorance, and a solution to that problem can be more or less directed at relieving that ignorance, rather than some other ignorance. In this talk, I would like to say some more about this vectorial conception, and about its relation to what we might roughly call the “constructivist” epistemologies of Poincaré and Brouwer in particular.

Th 3, 3pm – 6:30pm, Room Malevitch (483A) & Zoom

• 3pm – 4pm
  Andrei Rodin (Saint Petersburg State University) (Saint Petersburg State University)
  Mic Detlefsen on Frege-Hilbert Controversy
  The exchange of letters between Frege and Hilbert took place during an early stage of Hilbert’s work in foundations of mathematics and stopped before Hilbert shaped what is commonly known today as Hilbert‘s Program. As Mic Detlefsen convincingly shows in his 1986 monograph ‘Hilbert’s Program’, Frege’s challenges remain fully relevant to Hilbert’s foundational project at its later mature stage. In this new context Detlefsen reconstructs Hilbert’s argument in support of his notion of formal aka “ideal” proof as follows. While on Frege’s view a formal derivation of formulas from other formulas does not constitute a proof unless all involved formulas are semantically interpreted in an appropriate way, on Hilbert’s view at least a part of these formulas may remain uninterpreted being instead “metamathematically evaluated”. This opens a possibility to build metamathematical proofs of mathematical theorems ; the needed join between mathematics and metamathematics is provided by what Detlefsen calls the Equivalency Principle. This principle, which Detlefsen attributes to Hilbert on the basis of available textual evidence, postulates that metamathematical reasoning about symbolic formulas has the same contentual character as the elementary contentual arithmetical reasoning and thus equally belongs to the intuitively trusted core of mathematics.
  In my paper I describe how the method of formal ‘ideal’ proof as conceptualised by Detlefsen after Hilbert, i.e., the method of proof by a ‘metamathematical replacement’, performs in today’s mathematical practice, and demonstrate that it still remains highly problematic and, generally, doesn’t lead to a stable consensus in the mathematical community. For this end I briefly review the cases of Continuum Hypothesis, Consistency of Arithmetic, Four Colour Theorem, and Kepler Conjecture. On the basis of these examples I identify some general problems of formal methods and, finally, I show how these problems can be partly resolved with the homotopy-theoretic rendering of formal proofs used in the Univalent foundations of mathematics.

• 4:15pm – 5:15pm
  Paola Cantù (CNRS, Centre Gilles Gaston Granger, Aix-Marseille Université)
  Peano’s philosophical views between structuralism and logicism
  This paper is part of a larger project that aims at a comprehensive historical and philosophical evaluation of the contributions of the Peano School to mathematics, logic, and the foundation of mathematics based on an investigation of the school’s axiomatic practices. A detailed analysis of several mathematical practices that can be considered as markers of logicism is a fruitful way to reconstruct Peano’s philosophical views : the link between functions and relations, the role of metatheoretical investigations, the kind of semantics, the use of definitions by abstraction, and the foundational or non-foundational value of axiomatics. This paper will focus on Peano’s semantics, coping with an apparent contradiction in the Peano School approach to concept script. On the one hand, symbols stand for ideas. On the other hand, the linguistic expressions associated to the symbols are sometimes considered as meanings, sometimes as a way to read the symbols and sometimes as a way to indicate the adopted interpretation. Besides, they are used in the analysis of the independence of the axioms of arithmetic, where they sometimes occur together with the notion of interpretation. Focusing on this aspect and by means of a comparative investigation of Peano’s logico-mathematical, linguistic and notational practices we would like to assess 1) whether Peano’s symbolic language is a concept script and 2) whether Peano’s symbolization is a formalization in modern sense. Peano’s view is characterized as a form of structural algebraism, which differs from both the algebra of logic tradition using mathematical symbols to express logical calculi, and from Frege’s logical investigation centred on the effort to understand the functional nature of predication.

M 7, 3pm – 6:30pm, Room Malevitch (483A) & Zoom

• 3pm – 4pm
  Walt Dean (University of Warwick)
  A royal road to incompleteness ?
  One of the goals of Detlefsen 1990 ("On an alleged refutation of Hilbert’s program using Gödel’s first incompleteness theorem") was to refute an argument popularized by Kreisel, Smorynski, and Simpson that Gödel’s first theorem "effectively kills Hilbert’s programme". The first goal of this talk will be to offer a reading of the second volume of Grundlagen der Mathematik (1939) which supports a key premise in Detlefsen’s argument — i.e. that Hilbert should not be understood as committed to the finitary decidability of real propositions or that his commitment to "real soundness" (i.e. that no sentence provable by an ideal theory can be refutable by real means) should be understood as engendering a commitment to "real conservativity". The second goal of the talk will be to trace the reception of the incompleteness theorems in Grundlagen der Mathematik —e.g. in regard to Hilbert & Bernays’s use of the Liar paradox as a framework for presenting Gödel’s results, the details of their formalization of the second incompleteness theorem, and its interaction with their formal truth definition for first-order arithmetic. Building on this and subsequent work on arithmetical incompleteness, I will finally illustrate a means by which these considerations connect with the second goal of Detlefsen 1990 —i.e. highlighting the potential aptness of "consistency-minded" definitions of provability (e.g. in the manner of Feferman 1960) in the formulation of ideal theories.

• 4:15pm – 5:15pm
  Emmylou Haffner (Institut de Mathématique d’Orsay, Université Paris-Saclay)
  Reassessing Dedekind’s ideal of rigor ?
  Ideals of rigor have been an important interest of Mic Detlefsen’s. In this talk, I will start from Mic’s analysis of Dedekind’s ideal of rigor and question to which extent this ideal is relevant to rigor in the making, as observed in Dedekind’s mathematical drafts. Indeed, considerations about rigor in mathematics often rely on questions of justification and/or verification of results, rather than how they were found. In this talk, I will propose to shift the focus and look behind the scene to consider the shaping of rigorous mathematics. This will raise an additional consideration : To what extent does rigor support or guide mathematical research in its various phases ? What are some consequences of such an ideal of rigor, if any, on mathematical research ? To do so, I will use two examples. Firstly, I will consider the genesis of his late Dualgruppe theory (equivalent to our modern lattice). Focusing on a specific law of Dualgruppe theory, I will show that the elaboration of a rigorous work can be the outcome of a process that is not necessarily so. I will put forward the trial-and-error and inductive aspects of Dedekind’s research practices. Secondly, I will consider the genesis of Was sind und was sollen die Zahlen ?, Dedekind’s famous essay on the natural numbers. Dedekind wrote several versions of this text, from the 1870s to 1888. I will particularly be interested in what seems to be an important step of mathematical writing, in Dedekind’s drafts, namely arranging the order of propositions in a deductive hierarchy.

• 5:30pm – 6:30pm
  Graham Leach-Krouse (Kansas State University)
  Coabstraction and the Continuum