Mathematical Impossibility in History and in the Classroom
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Lützen, Jesper
Resumen
Mathematical Impossibility in History and in the Classroom Theorems stating that something is impossible are notoriously difficult to understand for many students and amateur mathematicians. In this talk I shall discuss how the role of such impossibility statements has changed during the history of mathematics. I shall argue that impossibility statements have changed status from a kind of metastatement to true mathematical theorems. I shall also argue that this story is worth telling in the classroom because it will clarify the nature of impossibility theorems and thus of mathematics. In particular it will show to the students how mathematics is able to investigate the limits of its own activity with its own methods
Fecha
2011
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Términos clave
Dificultades | Historia de la Educación Matemática | Reflexión sobre la enseñanza | Teoremas | Tipos de metodología
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Nivel educativo
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Revisado por pares
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Usuario
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Conferencia Interamericana de educación Matemática
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1-9
Referencias
Abel, N. (1824). Mémoire sur les équations algeébriques où on démontre l'impssibilité de la résolution de l'equation générale du cinquième dégré. Christiania : Groendahl. Extended German edition : “Beweiss der Unmöglichkeit algebraische Gleichungen von höheren Graden als dem vierten allgemein aufzulösen”. Journal für die reine und angewandte Mathematik. 1 (1826), 65-85. Abel, N. (1839). “Sur la resolution algébrique des equations”. In: Oeuvres Complètes de N.H. Abel. 1st ed. by Holmboe, B., vol. 2. Christiania: Gröndahl, pp. 185–209; 2nd ed. by Sylow, L. and Lie, S., vol. 2. Christiania: Grøndahl, pp. 217–243. Condorcet, M. (1778). Motivating note with no title and no author. Histoire de l’Académie Royale des Sciences for the year 1775, pp. 61–66. Gauss, C. (1801). Disquisitiones arithmeticae, Braunschweig, 1801. English translation by Clarke, A. A. New Haven: Yale University Press 1966. Goldstein, C. (1995). Un théorème de Fermat et ses lecteurs. Saint-Denis: Presses Universitaires de Vincennes. 9 Mathematical Impossibility in History and in the Classroom Hilbert, D. (1900). “Mathematische Probleme”, Archiv für Mathematik und Physik, 3. Reihe, 1, 44-63 and 213-237. Page references to Gesammelte Abhandlungen. 3, 290–329. Katz, V. (2009). A History of Mtthematics. An Introduction. Boston : Addison-Wesley. Lagrange, J. (1770/71). “Réflexions sur la résolution algébrique des équations”, Nouveaux Mémoires de l’Académie royale des Sciences et Belles-Letters de Berlin. Page references to Lagrange's Œuvres. 3, 205–421. Jesper, L. (1985). Cirklens Kvadratur. Vinklens Tredeling og Terningens Fordobling. Fra Oldtidens Geometri til Moderne Algebra, Herning: Systime. Lützen, J. (1990) Joseph Liouville 1809–1882. Master of Pure and Applied Mathematics, Studies in the History of Mathematics and Physical Sciences No. 15, New York: Springer-Verlag. Lützen, Jesper (2009) “Why was Wantzel overlooked for a century? The changing importance of an impossibility result”. Historia Mathematica 36, 374-394 Lützen, Jesper (2010) “The Algebra of Geometric Impossibility: Descartes and Montucla on the Impossibility of the Duplication of the Cube and the Trisection of the Angle”, Centaurus 52, no. 1, pp. 4-37. Montucla, J. (1754). Histoire des recherches sur la quadrature du cercle (Paris: Jombert). Pappos of Alexandria. (1933). Collection. Translated into French by Ver EeckeP. Pappus d’Alexandrie: La Collection mathématique, I, II, Paris: Desclée de Brouwer. Ruffini, P. (1915). Opere Matematiche, 3 vols., ed. E. Bortolotti. Palermo: Tipografia matematica di Palermo Voelke, J. (2005). Renaissance de la géométrie non euclidienne entre 1860 et 1900, Bern: Peter Lang Wallis, J. (1657). Letter to Kenelm Digby, French translation pp. 427–451 in Oeuvres Complètes de Fermat, vol. III, Paris: Gauthier Villars. Wantzel, P. (1837). “Recherches sur le moyens de reconnaître si un problème de géométrie peut se résoudre avec la règle et le compas”. Journal de Mathématiques pures et appliquées. 2, 366–372. Wantzel, P. (1843). “Pierre Laurent Wantzel, Classification des nombres d’origine algébrique”, Nouvelles Annales de Mathématiques. 2, 117–127.
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9