Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-21T23:51:17.575Z Has data issue: false hasContentIssue false
  • English
  • Français

De la logique à l’arithmétique. Pourquoi des logiques et des mathématiques constructivistes?

Published online by Cambridge University Press:  22 March 2018

YVON GAUTHIER
YVON GAUTHIER*
Affiliation:
Université de Montréal
*
*<[email protected]>
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, I wish to discuss in an informal way the motivations and the motifs of the constructivist approach to logic and mathematics and by a natural extension to the general field of science, particularly theoretical physics. Foundational questions in those domains are not ruled by philosophical principles, but a critical philosophy of foundations could be the leitmotiv to the extent that it can be used as a criterion to decide between the theoretical options of scientific practices that are often oblivious to their own doctrinal presuppositions. My objective is to provide the justificatory reasons for a constructivist option or posture in the field of scientific knowledge.

Dans cet essai informel, je veux exposer les motivations et les motifs de l’approche constructiviste en logique et en mathématiques et, au-delà, dans le savoir scientifique en général et la physique théorique en particulier. Les questions fondationnelles dans ces domaines ne relèvent pas de la philosophie, mais la critique philosophique des fondements est ici un motif recteur ou un leitmotiv dans la mesure où elle départage les options théoriques d’une pratique scientifique souvent aveugle à ses propres présupposés doctrinaux. Il s’agit dès lors de la justification d’une option ou d’une posture philosophique critique dans le champ du savoir scientifique.

Keywords

logiquemathématiquesconstructivismearithmétiqueKronecker
Type
Original Article/Article original
Information
Dialogue: Canadian Philosophical Review / Revue canadienne de philosophie , Volume 57 , Issue 1 , March 2018 , pp. 1 - 28
Copyright
Copyright © Canadian Philosophical Association 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références bibliographiques

Bishop, Errett 1967 Foundations of Constructive Analysis, New York (NY), McGraw-Hill.Google Scholar
Bishop, Errett 1970 «Mathematics as a Numerical Language», dans Myhill, J., Kino, A. et Vesley, R.E., dir., Intuitionism and Proof Theory, Amsterdam, North-Holland, p. 53–71.Google Scholar
Bourbaki, Nicolas 1970 Théorie des ensembles, Paris, Hermann.Google Scholar
Edwards, Harold M. 1990 Divisor Theory, Boston, Bikhäuser.Google Scholar
Gauthier, Yvon 1989 «Finite Arithmetic with Infinite Descent», Dialectica, vol. 43, no 4, p. 329–337.Google Scholar
Gauthier, Yvon 1991 De la logique interne, Paris, Vrin (coll. «Mathesis»).Google Scholar
Gauthier, Yvon 1997 Logique et fondements des mathématiques, Paris, Diderot.Google Scholar
Gauthier, Yvon 2000 «The Internal Consistency of Arithmetic with Infinite Descent», Modern Logic, vol. 8, nos 1/2, p. 47–87.Google Scholar
Gauthier, Yvon 2002 Internal Logic. Foundations of Mathematics from Kronecker to Hilbert, Dordrecht, Kluwer (coll. «Synthese Library»).Google Scholar
Gauthier, Yvon 2010 Logique arithmétique. L’arithmétisation de la logique, Québec, Presses de l’Université Laval, (coll. «Logique de la science»).CrossRefGoogle Scholar
Gauthier, Yvon 2013a «Kronecker in Contemporary Mathematics. General Arithmetic as a Foundational Programme», Reports on Mathematical Logic, vol. 48, p. 37–65.Google Scholar
Gauthier, Yvon 2013b «A General No-Cloning Theorem for an Infinite Multiverse», Reports on Mathematical Physics, vol. 72, no 2, p. 191–199.CrossRefGoogle Scholar
Gauthier, Yvon 2015 Towards an Arithmetical Logic. Arithmetical Foundations of Logic. Bâle, Birkhäuser-Springer.Google Scholar
Gauthier, Yvon 2017a Nouveaux entretiens sur la pluralité des mondes. Essai de cosmologie sauvage à l’usage des profanes. Québec/Paris, Presses de l’Université Laval/Hermann.Google Scholar
Gauthier, Yvon 2017b «From the Local Observer in QM to the Fixed-Point Observer in GR», Advanced Studies in Theoretical Physics, vol. 11, no 2, p. 687–707.Google Scholar
Gauthier, Yvon 2017c «Arithmetical Logic for AI Deep Learning», International Journal of Soft Computing, vol. 12, no 6 (à paraître).Google Scholar
Gauthier, Yvon 2018 «A Quadratic Reciprocity Theorem for Arithmetical Logic», Logica Universalis, vol. 12, no 2 (à paraître).Google Scholar
Gödel, Kurt 1958 «Über eine noch nicht benütze Erweiterung des finiten Standpunktes», Dialectica, vol. 12, p. 280–287.Google Scholar
Gödel, Kurt 1967 «On Formally Undecidable Propositions of Principia Mathematica and Related Systems I», dans Heijenoort, Jean van, dir., From Frege to Gödel, Cambridge (MA), Harvard University Press, p. 616–617.Google Scholar
Hilbert, David 1926 «Über das Unendliche», Mathematische Annalen, vol. 95, p. 161–190, trad. par André Weil sous le titre «Sur l’infini», dans Acta Mathematica, vol. 48, nos 1-2 (1926), p. 91–122.Google Scholar
Hilbert, David 1935 «Neubegründung der Mathematik», dans Gesammelte Abhandlungen, vol. 3, Berlin, Springer, p. 157–177.Google Scholar
Hilbert, David et Bernays, Paul 1968-1970 Grundlagen der Mathematik, I, II, 2e édition, Berlin/Heidelberg/New York, Springer.Google Scholar
Hodges, Wilfrid 1993 Model Theory, Cambridge (MA), Cambridge University Press Google Scholar
Kronecker, Leopold 1968 «Grundzüge einer arithmetischen Theorie der algebraischen Grössen» [1889], dans Hensel, K., dir., Werke, vol. III, New York (NY), Chelsea, p. 245–387.Google Scholar
Nelson, Edward 1986 Predicative Arithmetic, Princeton (NJ), Princeton University Press (coll. «Mathematical Notes»).Google Scholar
Tarski, Alfred 1933 «Einige Betrachtungen über die Begriffe der ω-Vollständigkeit», Monatshefte für Mathematik und Physik, vol. 40, p. 97–112.Google Scholar
Tarski, Alfred 1951 A Decision Method for Elementary Algebra and Geometry, 2e édition, Berkeley/Los Angeles (CA), University of California Press.Google Scholar
Van den Driess, Lou 1988 «Alfred Tarski’s Elimination Theory for Real Closed Fields», Journal of Symbolic Logic, vol. 53, p. 7–19.Google Scholar
Weil, André 1941 «On the Riemann Hypothesis in Function Fields», Proceedings of the National Academy of Science of the United States of America, vol. 27, no 7, p. 345–347.Google Scholar
Weil, André 1949 «Number of Solutions of Equations in Finite Fields», Bulletin of the American Mathematical Society, vol. 55, p. 497–508.Google Scholar
Weil, André 1979 «Number Theory and Algebraic Geometry», dans Œuvres scientifiques. Collected Papers, vol. III, New York (NY), Springer, p. 442–454.CrossRefGoogle Scholar
Weil, André 1984 Number Theory. An Approach through History. From Hammourabi to Legendre, Bâle, Birkhäuser.Google Scholar
Weil, André 1995 Basic Number Theory, New York (NY), Springer.Google Scholar
Weyl, Hermann 1939 The Classical Groups. Their Invariants and Representations, Princeton (NJ), Princeton University Press.Google Scholar
Weyl, Hermann 1940 Algebraic Theory of Numbers, Princeton (NJ), Princeton University Press.Google Scholar