Abstract
* Doutor em Lógica e Filosofia da Ciência pela Universidade Estadual de Campinas. Professor Adjunto do Instituto de Filosofia da Universidade Federal de Uberlândia (UFU).Paradoxos , o infinito e a intuição geométricaResumo: Analisam-se neste artigo alguns resultados matemáticos que já foram assinalados como contrários à intuição e se realiza uma pesquisa das possíveis causas desse caráter contraintuitivo. Nesse sentido, são estudados o Paradoxo de Galileu, a demonstração de Cantor de que o segmento tem a mesma quantidade de pontos que o quadrado e o Paradoxo de Tarski- Banach. Primeiro é discutido o papel que o conceito de infinito e princípios como “o todo é maior que a parte” têm nesses paradoxos. Em segundo lugar, é estudada a influência que as intuições geométricas têm em alguns paradoxos e a relação delas com a concepção de geometria do Erlanger Programm de Felix Klein.Palavras-chave: Paradoxos. Intuição. Erlanger Programm.Abstract: This paper studies some mathematical results formerly characterized as opposed to the intuition and also carries out a query for the possible causes of this counter-intuitive feature. In this sense, both Tarski-Banach and Galileo’s paradoxes are discussed and Cantor’s proof that the segment and the square have the same number of points is analyzed. In particular, it is examined the role that the concept of infinity and such principles as “the whole is greater than the part” have in these paradoxes. Furthermore, the effect that geometrical intuitions have on some paradoxes as well as the relationship between these intuitions and the concept of geometry of the Erlanger Programm of Felix Klein are discussed.Keywords: Paradox. Intuition. Erlanger Programm.Referências:ARISTÓTELES. Aristotle’s Metaphysics. A Revised Text with Introduction and Commentary by W. D. Ross, Oxford: Clarendon Press, 1975.BIRKHOFF, G.; BENNETT, M. K. Felix Klein and his “Erlanger Programm”. In: ASPRAY, W.; KITCHER, P. (Ed.). History and philosophy of modern mathematics. Minneapolis: The University of Minnesota Press, 1988. p. 145–176.BOI, L.; FLAMENT, D.; SALANSKIS, J.-M. (Ed.). 1830–1930: A Century of Geometry. Epistemology, History and Mathematics. Berlin: Springer, 1992. (Lecture Notes in Physics, 402).BOYER, C. B.; MERZBACH, U C. A History Of Mathematics. 2 ed. New York: John Wiley & Son, 1991.CANTOR, G. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Berlin: Springer, 1932.CASSIRER, E. The concept of group and the theory of perception. Philosophy and Phenomenological Research, Buffalo, N. Y., v.5, n.1, p. 1–36, Sept. 1944.CLARK, M. Paradoxes from A to Z. 2. ed. London: Routledge, 2007.DAUBEN, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton: Princeton University Press, 1990.DEDEKIND, R. Gesammelte mathematische Werke. Braunschweig: F. Vieweg & Sohn, 1932.FRAENKEL, A.; BAR-HILLELL, Y.; LEVY, A. Foundations Of Set Theory. Amsterdam: Elsevier, 1973.GALILEI, G. Discorsi e dimostrazioni matematiche intorno a due nuove scienze attenenti alla mecanica, & i movimenti locali. In: ______. Milano: Società tipografica de’classici italiani, 1811. (Opere di Galileo Galilei, v. 8).GRAY, J. Poincare and Klein – groups and geometries. In: BOI, L.; FLAMENT, D.; SALANSKIS, J.-M. (Ed.). 1830 – 1930: A Century of Geometry. Epistemology, History and Mathematics. Berlin: Springer, 1992. p. 35–44. (Lecture Notes in Physics, 402).GREENBERG, M. J. Euclidean and Non-Euclidean Geometries. Development and History. 3. ed. New York: W. H. Freeman, 1994.HALMOS, P. R. Measure Theory. Berlin: Springer, 1970.HEATH, S.; HEIBERG, J. The thirteen books of Euclid’s Elements. Translated with Introduction e Commentary by Sir Thomas L. Heath from the Greek Text of J. Heiberg. Cambridge: Cambridge University Press, 1908.HILBERT, D. Grundlagen der Geometrie. 4. ed. Leipzig: B. G. Teubner, 1913.______. Ueber das Unendliche. Mathematische Annalen, Leipzig, v. 95, n. 1, p. 161–190. 1926.JECH, T. The axiom of Choice. Amsterdam: Elsevier, 1973.KLEIN, F. Vergleichende Betrachtungen uber neuere geometrische Forschungen. Programm zum Eintritt in die philosophische Facultat und den Senat der k. Friedrich-Alexanders – Universitat zu Erlangen. Erlangen: Deichert, 1872.______. Gesammelte mathematische Abhandlungen. Ersten Band. Berlin: J. Springer, 1921.MOORE, A. W. The infinite. London: Routledge, 2001.ROWE, D. E. Klein, Lie, and the “Erlanger Programm”. In: BOI, L.; FLAMENT, D.; SALANSKIS, J.-M. (Ed.). 1830 – 1930: A Century of Geometry. Epistemology, History and Mathematics. Berlin: Springer, 1992. p. 45–54. (Lecture Notes in Physics, 402).Data de registro: 21/07/2010Data de aceite: 15/09/2010Date
2011-07-22Type
info:eu-repo/semantics/articleIdentifier
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