Abstract
As I am not at all a specialist on Gottlob Frege’s work, my comments intended initially to be focused on an aspect that emerges in the last part of Peter Clark’s paper “Frege, neo-logicism and applied mathematics” 1, where he treats the question of “applied mathematics” — an aspect that appealed to me and that was triggered by Frege’s relationship between numbers and concepts, and reasoning. Starting with this concern, I have been led by my subject to propose some considerations about foundations and rationality which will go — briefly — in two directions. The first direction is that of a distinction between logical and rational foundations, whilst the second direction is that of taking into account as a fact the historical development of mathematics among the sciences, which modifies the terms of any foundational program. In delineating these considerations, I found that what I had in mind could apply to mathematics itself as well as to “applied mathematics”, thus deviating somewhat from my first explicit intention. In conclusion I shall consider the possibility of a rational foundation programme for mathematical and physico-mathematical sciences which would take into account the changes in the scientific contents and the widenings of the forms of rationality that, in my view, make these changes possible. Such foundations for knowledge would not be any more static, but dynamical and would be possibly considered only retrospectively: they would be “forward foundations”, in a sense that will be discussed in detail elsewhere.
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References
Peter Clark, “Frege, neo-logicism and applied mathematics”, in this volume.
Michel Paty, “Des fondements vers l’avant. Sur la rationalité des mathématiques et des sciences formalisées”, Contribution to the Colloque International “Aperçus philosophiques en logique et en mathématiques. Histoire et actualité des théories sémantiques et syntaxiques alternatives”, Nancy, 30 Sept.-4 Oct. 2002 (forthcoming).
Gottlob Frege, Die Grundlagen der Arithmetik. Eine logisch-mathematische Untersuchung iiber den Begriff der Zahl, Breslau, W. Koebner, 1884 (Engl. transi. by J.L. Austin, The Foundations of Arithmetic, Oxford, Blackwell, 1950. French transi. by C. Imbert, Les Fondements de l’arithmétique, Paris, Seuil, 1970); Gottlob Frege, Ecrits logiques et philosophiques, transi. in French from German, Paris, Seuil, 1971; J. van Heijenoort, From Frege to Gödel. A source Book in Mathematical Logic (1879–1931) Cambridge (Mass.) Harvard University ssPre 1967
Peter Clark, ibid.
It is true that Frege himself considered a more general symbolic notation than pure numbers (ideography).
Michel Paty, ”The concept of quantum state: new views on old phenomena”, in A. Ashtekar, R.S. Cohen, D. Howard, J. Renn, S. Sarkar & A. Shimony (eds.), Revisiting the Foundations of Relativistic Physics: Festschrift in Honor of John Stachel, Boston Studies in the Philosophy and History of Science, Dordrecht Kluwer forthcoming
Henri Poincaré, “La logique et l’intuition dans la science mathématique et dans l’enseignement”, L’Enseignement mathématique, 1, 1889, 157–162 (Repr. in Henri Poincaré, Œuvres, Paris, Gauthier-Villars (11 vols., 1913–1965), vol. 11. pp. 129–133.
David Hilbert, “Neubegründung der Mathematik” (1922), in David Hilbert, Gesammelte Abhandlungen, Berlin, B. 3, 1935.
Jean Cavaillès (1903–1944), prematurely carried off to death by the Nazis in occupied France, where he was involved in Resistance activities. was a philosonher and Ingi ian
Jean Cavaillès Méthode axiomatique et formalisme, Essai sur le problème du fondement des mathématiques (Thesis, 1937, 1rst ed, 1938), Introduction by Jean-Toussaint Desanti, Preface by Henri Cartan. Paris. Hermann. 1981(My emphasis MP)
Jean Cavaillès, Remarques sur la formation de la théorie abstraite des ensembles (Complementary Thesis, 1937, lrsst ed, 1938), in Jean Cavaillès, Philosophie mathématique, Preface by Raymond Aron, Introduction by Roger Martin. Paris. Hermann. 1962. pn. 23–174
Jean-Toussaint Desanti, “Souvenir de Jean Cavaillès”, in Jean Cavaillès, Méthode axiomatique et formalisme. Essai sur le problème du fondement des mathématiques (1981 ed.). Introduction by Jean-Toussaint Desanti, Preface by Henri Cartan Paris H-ermann 1981
Gilles-Gaston Granger, Science, langage, philosophie, Collection “Penser avec les sciences”, EDP-Sciences, Paris, 2003, chapter on “Jean Cavaillès et l’histoire”, pp. 76–84.
Jean Cavaillès, Sur la logique et la théorie de la science (written in 1942, 1r st ed., 1946), 3rd ed., Paris, Vrin, 1976.
By the expression “working thought” I mean what I would call in French: la pensée au travail. The idea of scientific or rational thought as a working action has been developed by GillesGaston Granger, notably in his Essai d’une philosophie du style, Paris, Armand Colin, 1968; reed., Paris, Odile Jacob, 1988.
Michel Paty, “Intelligibilité et historicité (Science, rationalité, histoire)”, in J.J. Saldaña (ed.), Science and Cultural Diversity. Filling a Gap in the History of Science, Cadernos de Quipu 5, México, 2001, pp. 59–95; “Les concepts de la physique: contenus rationnels et constructions dans l’histoire”, Principia (Florianopolis, Br), 5, nº1–2, junho-dezembro 2001, 209–240 (English version: “The concepts of physics: rational contents and constructions in history”, in J. Margolis and T. Rockmore (eds.), forthcoming.
On the last one, see in particular: Michel Paty, “La physique quantique ou l’entraînement de la forme mathématique sur la pensée physique”, in C. Mataix y A. Rivadulla (eds.), Física cuantica y realidad. Quantum physics and reality, Madrid, Editorial Complutense, 2002, pp. 97134; “ The concept of quantum state: new views on old phenomena”, op. cit.
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Paty, M. (2004). Remarks about a “General Science of Reasoning”. In: Stadler, F. (eds) Induction and Deduction in the Sciences. Vienna Circle Institute Yearbook, vol 11. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-2196-1_12
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