Nội dung được dịch bởi AI, chỉ mang tính chất tham khảo
Trực giác trong Toán học: từ Chủ nghĩa Phân biệt Chủng tộc đến Chủ nghĩa Đa dạng
Tóm tắt
Vào thế kỷ XIX và XX, nhiều nhà toán học đã coi trực giác là công cụ nghiên cứu không thể thiếu để đạt được những kết quả mới. Trong bài luận này, chúng tôi sẽ phân tích một nhóm các nhà toán học (Felix Klein, Henri Poincaré, Ludwig Bieberbach, Arend Heyting) đã tương tác với Luitzen Egbertus Jan Brouwer (cha đẻ của trường phái nền tảng trực giác) nhằm so sánh các khái niệm về trực giác của họ. Chúng tôi sẽ thấy rằng từ "trực giác" (trong tiếng Đức là Anschauung) tương ứng với nhiều nghĩa rất khác nhau: chúng dao động từ tầm nhìn hình học, đến cái nhìn tổng thể về một chứng minh, đến nhận thức về thời gian, đến khả năng (được mọi người chia sẻ) để xem xét những khái niệm thường xuyên xuất hiện trong suy nghĩ của chúng ta một cách riêng biệt. Hơn nữa, chúng tôi sẽ khám phá rằng những nghĩa khác nhau này có một mặt đối lập triết học rất quan trọng: chúng đã chuyển từ một sự phân loại chủng tộc trong toán học sang một cái nhìn đa dạng về nó.
Từ khóa
#trực giác #toán học #triết học #phân biệt chủng tộc #chủ nghĩa đa dạngTài liệu tham khảo
Allmendinger H. (2014) Felix Kleins Elementarmathematik vom höheren Standpunkte aus: Eine Analyse aus historischer und mathematikdidaktischer Sicht. Universi – Universitätsverlag Siegen, Siegen.
Bair, J., Laszczyk, P. B., Heinig, P., Katz, M. G., Schafermeyer, J. P., & Sherry, D. (2017). Klein vs. Mehrtens: Restoring the reputation of a great modern. Mathematical Studies, 48(2017), 189–219.
Bieberbach, L. (1934). (1934) Personlichkeitsstruktur und mathematisches Schaffen. Unterrichtsblätter Für Mathematik Und Naturwissenschaften, 40, 236–243.
Biermann K.R. (1988) Die Mathematik und Ihre Dozenten an der berliner Universität. Akademie Verlag, Berlin: 1810-1933.
Boutroux P. (1914) L’œuvre philosophique, in Henri Poincaré. L’œuvre scientifique, L’œuvre philosophique, Alcan, Paris: 205–264.
Brouwer L.E.J. (1905) Leven, kunst en mystiek; Engl. Transl. Life, Art an Mysticism by W.P. van Stigt, Notre Dame Journal of Formal Logic 37 (3), 1996, pp. 391–428.
Brouwer L.E.J. (1975) Collected Works vol. 1 (A. Heyting ed.), North Holland, Amsterdam.
Brouwer, L. E. J. (1981). Brouwer’s Cambridge Lessons on Intuitionism. Cambridge U.P.
Carnap, R. (1934). Die logische Syntax der Sprache. Springer.
van Dalen, D. (1990). (1990) The War of the Frogs and the Mice, or the Crisis of the Mathematische Annalen. The Mathematical Intelligencer, 12(4), 17–31.
van Dalen D. (2005) Mystic, Geometer, and Intuitionist. The Life of L.E.J. Brouwer 1881–1966. Vol II. Clarendon Press, Oxford.
van Dalen D. (2011) The Selected Correspondence of L.E.J. Brouwer. Springer, London.
Detlefsen, M. (1992). Poincaré against the Logicians. Synthese, 90, 349–378.
Detlefsen M. (1993) Logicism and the Nature of Mathematical Reasoning, in Russell and Analytical Philosophy, A. Irvine & G. Wedekind (eds.), Toronto University Press, Toronto: 265–292.
Dunlop, K. (2016). Poincaré on the Foundations of Arithmetic and Geometry, Part 1: Against ‘Dependence-Hierarchy’ Interpretations. HOPOS (The Journal of the International Society for the History of Philosophy of Science), 6(2), 274–308.
Franchella, M. (1994a) Brouwer and Griss on Intuitionistic Negation. Modern Logic, 4(3), 256–265.
Franchella, M. (1994b). Heyting’s Contribution to the Change in research into the Foundations of Mathematics. History and Philosophy of Logic, 15(1994), 149–172.
Franchella M. (1995) Negation in the Work of Griss, in Perspectives on Negation, (H.C.M. de Swart e L.J.M. Bergman eds.) Tilburg University Press, Tilburg 1995: 29–40.
Franchella M. (2019) Shaping the Enemy: Foundational Labelling by L.E.J. Brouwer and A. Heyting, History and Philosophy of Logic, 40:2, 152–181.
Von Franz M-.L. (1964) Science and the unconscious, in Man and his Symbols, (Jung, C.G., von Franz, M-.L.eds.), Aldus, London: 301–307.
Franzini E. (2009) Elogio dell’Illuminismo, Bruno Mondadori: Milano.
Goldfarb W. (1985) Poincaré against the Logicists, in History and Philosophy of Modern Mathematics, (W. Aspray & Kitcher P. eds), Minnesota Press, Minneapolis: 61–81.
Griss G. F. C. (1946) Idealistische filosofie. Van Loghum Slaterus, Arnhem.
Griss G.F.C. (1948) Sur la négation (dans les mathématiques et la logique), Synthese VII: 71–74.
Heinzmann G. (1995) Zwischen Objektkonstruktion und Strukturanalyse. Zur Philosophie der Mathematik bei Henri Poincaré, Vandenhoek & Ruprecht, Göttingen.
Heinzmann G. and Nabonnand P. (2008) Poincaré, Intuitionism, Intuition and Convention in One Hundred Years of Intuitionism (1907–2007), M. v. Atten, P. Boldini, M. Bordeau & G. Heinzmann eds., Birkhäuser, Basel/Boston/Berlin: 163–177.
Heinzmann G. and D. Stump (2017) Henri Poincaré, The Stanford Encyclopedia of Philosophy (Winter 2017 Edition), Zalta E.W. (ed.), URL = <https://plato.stanford.edu/archives/win2017/entries/poincare/>.
Heyting A. (1930) Die formalen Regeln der intuitionistischen Logik, Sitzungsbericht der preussischen Akademie der Wissenschaften, physikalische-mathematische Klasse: 42–56, 57–71, 158–169.
Heyting A. (1934) Mathematische Grundlagenforschung. Intuitionismus. Beweistheorie, Springer, Berlin.
Heyting, A. (1935). Intuitionistische wiskunde. Mathematica B, 5, 123–128.
Heyting, A. (1956). Intuitionism: An Introduction. North-Holland.
Heyting, A. (1958a). Blick von der intuitionistischen Warte. Dialectica, 12, 332–345.
Heyting A. (1958b) lntuitionism in Mathematics, in Philosophy in the Mid-century. A Survey. (R. Klibansky ed.) La Nuova Italia, Firenze: 101–115.
Heyting, A. (1962). After thirty years, in Logic (E. Nagel, P. Suppes, & A. Tarski Eds.), Stanford University Press: Stanford, 194–197
Heyting A. (1980) Collected papers (A. Troelstra, J. Niekus, H. van Riemsdijk eds.), Mathematisch Instituut, Amsterdam.
Hilbert D. (2013) David Hilbert's Lectures on the Foundations of Arithmetic and Logic 1917–1933. (W. Evald and W. Sieg eds.), Springer, Berlin.
Jaensch, E. R. (1931). Ȕber die Grundlagen der menschlichen Erkenntnis. Johann Ambrosius Barth: Leipzig.
Jaensch, E. R. (1938). Der Gegentypus. Johann Ambrosius Barth: Leipzig.
Jaensch, E. R., & Althoff, F. (1939). Mathematisches Denken und Seelenformen. Johann Ambrosius Barth: Leipzig.
Kant I. (1838) Critique of Pure Reason. William Pickering, London; available online at: https://www.gutenberg.org/cache/epub/4280/pg4280-images.html#chap13
Klein F. (1892) Nicht-Euklidische Geometrie I. Vorlesung gehalten während des Wintersemesters 1889–90. Göttingen. Available at: http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN51697307X|LOG_0003&physid=PHYS_0003
Klein F. (1894) Lectures on Mathematics delivered from 28 august to 9 september 1893 before members of the congress of mathematics held in connection with the world fair in Chicago at Northwestern University. Reported by Alexander Ziwet. Macmillan, New York and London.
Klein F. (1895) Ueber die Beziehung der neueren Mathematik zu den Anwendungen, Zeitschr. f. math.-naturw. Unterricht Bd.XXVI (1895): 535–540. Available at http://digital.slub-dresden.de/id404375774-18950260. Engl. Transl. Felix Klein’s Erlanger Antrittsrede in Rowe 1985.
Klein F. (1892). Nicht-Euklidische Geometrie I. Vorlesung gehalten während des Wintersemesters 1889–90. Göttingen: Autograph. Available at: http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN51697307X|LOG_0003&physid=PHYS_0003
Klein F. (1896) Über Arithmetisierung der Mathematik. Zeitschrift für mathematischen und naturwissen-schaftlichen Unterricht, 27, 143–149. Available at http://digital.slub-dresden.de/id404375774-18960270
Klein F. (1908) Elementarmathematik vom höheren Standpunkte aus. Teil I: Arithmetik, Algebra, Analysis. Leipzig ; Engl. Transl. Elementary mathematics from an advanced standpoint. Vol. I. Arithmetic, Algebra, Analysis. Translation of the 4th German edition, Springer, Berlin 2016.
Klein F. (1909) Elementarmathematik vom höheren Standpunkte aus. Zweiter Band. Geometrie. Ausgearbeitet von E. Hellinger. Teubner, Leipzig; Engl. Transl. Elementary Mathematics from a Higher Standpoint: Vol. II. Geometry. Translation of the 4th German edition, Springer, Berlin, 2016.
Klein F. (1923) Göttinger Professoren. Lebensbilder von eigener Hand. Mitteilungen Universitätsbund Göttingen 5, I, I–36.
Klein F. (1926) Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, Für den Druck bearbeitet von R. Courant und O. Neugebauer. Springer. Reprint of the Berlin edition of 1926–1927 by Chelsea: New York, 1967.
Klein F. (1977) Felix Klein: Handschriftlicher Nachlass, K. Jacobs, Erlangen.
Kronfeldner M. E. (2011) Darwinian Creativity and Memetics, Acumen, Durham.
Kuiper J. (2004) Ideas and Explorations: Brouwer's Road to Intuitionism. Ph. D. Thesis. Amsterdam. Available at: https://www.cs.ru.nl/~freek/brouwer/phdthesis.pdf
Luciano E. – Roero C. (2005) Introduzione, in Giuseppe Peano – Louis Couturat. Carteggio (1896–1904). Olschki: Firenze, 2005, V-LXIX.
Manegold K. (1970) Universität, Technische Hochschule, und Industrie. Ein Beitrag zur Emanzipation der Technik im 19. Jahrhundert unter besonderer Berűcksichtigung der Bestrebungen Felix Kleins. Dunker & Humblot: Berlin.
Manger, E. (1934). (1934) Felix Klein im Semi-Kürschner! JDMV, 2, 4–11.
Mattheis, M. (2019). Aspects of “Anschauung” in the Work of Felix Klein. In The Legacy of Felix Klein, H. G. Weigand, W. McCallum, M. Menghini, M. Neubrand, & G. Schubring eds., 93–106, Springer: Cham.
Mehrtens, H. (1987) Ludwig Bieberbach and ‘Deutsche Mathematik’ in Studies in the History of Mathematics (Phillips E. ed.), Mathematical Association of America: 195–241.
Menghini, M., Neubrand, M., & Schibring, G. (Eds.). (2019). The Legacy of Felix Klein. Springer: Cham.
Miller, A. I. (1984). Imagery in Scientific Thought: Creating 20th Century Physics. Birkhäuser: Basel.
Miller, A. I. (1992). Scientific Creativity: A Comparative Study of Henri Poincaré and Albert Einstein. Creativity Research Journal, 5, 385–418.
Miller, A. I. (1996). Insights of genius: Imagery and creativity in science and art. Copernicus: Goettingen.
Miller, A. I. (1997). Cultures of creativity: Mathematics and physics. Diogenes, 45(177), 53–72.
Poincaré, H. (1902). La Science et l’hypothése. Flammarion: Paris.
Poincaré, H. (1905). La valeur de la science. Flammarion: Paris.
Poincaré, H. (1906). M. Poincaré’s Science et hypothése. Mind, 15, 141–143.
Pyenson L. (1979) Mathematics, education, and the Göttingen approach to physical reality. 1890–1914 . Europa: A Journal of Interdisciplinary Studies 2(2), 91–127.
Poincaré, H. (1908). Science et méthode. Flammarion: Paris.
Poincaré H. (1913) Derniéres pensées, Flammarion: Paris; Engl. Trans, Mathematics and Science: Last Essay, Dover, New York, 1963.
Poincaré, H. (2014). The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method. Cambridge U.P: Cambridge.
Rowe, D. E. (1985). Felix Klein’s “Erlanger Antrittsrede”. A transcription with English translation and commentary. Historia Mathematica, 12, 123–141. https://doi.org/10.1016/0315-0860(85)90003-5
Rowe, D. (1986). (1986) ‘Jewish mathematics’ at Göttingen in the era of Felix Klein. Isis, 77(3), 422–449.
Rowe D. (1994) The philosophical views of Klein and Hilbert. In The intersection of history and mathematics, Ch., Sasaki, Ch., Sugiura, M., Dauben, D.W. eds., Birkhäuser, Basel, 187–202.
Russell, B. (1967). The Autobiography of Bertrand Russell. Allen and Unwin: Crows Nest.
Ruvidotti, L. (2011). Il concetto di numero nella fondazione dell'aritmetica in Kant e Dedekind. Rivista Italiana di Filosofia Analitica Junior. 2. https://doi.org/10.13130/2037-4445/1513.
Segal, S. (1986). Mathematics and German Politics: The National Socialist Experience. Historia Mathematica, 1986, 118–135.
Segal, S. (2003). Mathematicians under the Nazis. Princeton University Press: Princeton.
Shabel, Lisa, Kant’s Philosophy of Mathematics, The Stanford Encyclopedia of Philosophy (Fall 2021 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/fall2021/entries/kant-mathematics/>.
Siegmund-Schultze R. (2008) Antisemitismus in der Weimarer Republik und die Lage jüdischer Mathematiker: Thesen und Dokumente zu einem wenig erforschten Thema, Sudhoffs Archiv Bd. 92, H. 1 (2008): 20–34
Siegmund-Schultze, R. (2016). (2016) Mathematics Knows NoRaces : A Political Speech that David Hilbert Planned for the ICM in Bologna in 1928. Math Intelligencer, 38, 56–66. https://doi.org/10.1007/s00283-015-9559-4
Stauff P. and Ekkehard E. (1929) Sigilla veri (Ph. Stauff's und Semi-Kürschner): Lexikon der Juden,-Genossen und-Gegner aller Zeiten und Zonen, insbesondere Deutschlands, der Lehren, Gebräuche, Kunstgriffe und Statistiken der Juden sowie ihrer Gaunersprache, Trugnamen, Geheimbünde, usw. Bd. 1 Bodung: Erfurt.
Thiele R. (2018) Felix Klein in Leipzig: Mit F. Kleins Antrittsrede, Leipzig 1880, Edition am Gutenbergerplatz, Leipzig, 2.te erweiterte Aufl.
Tobies R. (2019) Felix Klein, Springer Spektrum: Heidelberg.
Todorov T. (1982) La conquête de l’Amerique. La question de l’autre. Seuil, Paris; Engl. Transl. The Conquest of America: The Question of The Other. University of Oklahoma Press: Norman, 1999.
Vahlen T. (1923) Wert und Wesen der Mathematik. Greifswalder Universitätsreden Nr. 9, Greifswald.
van Atten, Mark, The Development of Intuitionistic Logic, The Stanford Encyclopedia of Philosophy (Winter 2017 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2017/entries/intuitionistic-logic-development/>.
van Atten, Mark, Luitzen Egbertus Jan Brouwer, The Stanford Encyclopedia of Philosophy (Spring 2020 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/spr2020/entries/brouwer/>.
van Stigt, W. P. (1990). Brouwer’s Intuitionism. North-Holland.