La generalización es necesaria o incluso inevitable

• Autores: Michael Otte, Tania Maria Mendonça Campos , Luiz de Barros
• Localización: PNA: Revista de investigación en didáctica de la matemática, ISSN-e 1887-3987, Vol. 9, Nº. 3, 2015, págs. 143-164
• Idioma: español
• DOI: 10.30827/pna.v9i3.6101
• Títulos paralelos:
  • Generalizing is necessary or even unavoidable
• Enlaces
  • Texto completo (pdf)
  • Texto completo  1    2  
• Resumen
  • español

    Los problemas de geometría y mecánica han motivado la generalización de los conceptos de número y función. Esto muestra cómo la aplicación y la generalización previenen que las matemáticas sean un mero formalismo. Los pensamientos son signos y los signos tienen un significado dentro de un cierto contexto. El significado es una función de un término: esta función produce un patrón. El álgebra o la moderna axiomática vienen a la mente como ejemplos. Sin embargo, las matemáticas estrictamente formales no prestaron suficiente atención al hecho de que las teorías axiomáticas modernas requieren un elemento complementario, en términos de aplicaciones intencionadas o modelos, para no terminar en un juego meramente formal.

  • English

    The problems of geometry and mechanics have driven forward the generalization of the concepts of number and function. This shows how application and generalization together prevent that mathematics becomes a mere formalism. Thoughts are signs and signs have meaning within a certain context. Meaning is a function of a term: This function produces a pattern. Algebra or modern axiomatic come to mind, as examples. However, strictly formalistic mathematics did not pay sufficient attention to the fact that modern axiomatic theories require a complementary element, in terms of intended applications or models, not to end up in a merely formal game.

• Referencias bibliográficas
  • Agdestein, S. (2013). How Magnus Carlsen became the Youngest Chess Grandmaster in the World. Alkmaar, The Netherlands: New Chess.
  • Aristotle (1960). Posterior Analytics. Topics. (Loeb Classical Library No. 391.) Cambridge, MA: Harvard University Press.
  • Belhoste, B. (1991). Augustin-Louis Cauchy. New York, NY: Springer.
  • Bochner, S. (1974). Mathematical reflections. The American Mathematical Monthly, 81, 827-852.
  • Carrol, L. (1895). What the tortoise said to Achilles. Mind, 4, 278-80.
  • Cassirer, E. (1910). Substanzbegriff und funktionsbegriff [Substance and function]. Berlin, Germany: Bruno Cassirer.
  • Cavell, S. (2008). Must we mean what we say? Cambridge, United Kingdom: UP.
  • Chatelet, F. (1992). Une histoire de la raison [History of the reason]. Paris, France: Editions du Seuil.
  • Dehn M. (1983). Mentality of the mathematician. Mathematical Intelligencer, 5(2), 18-26.
  • Durkheim, E. (1995). The elementary forms of the religious life. New York, NY: The Free Press.
  • Eisenstein, E. (2005). The printing revolution in early modern Europe. Cambridge, United Kingdom: University Press.
  • Gödel, K. (1944). Russell’s mathematical logic. In P. A. Schilpp (Ed.), The philosophy of Bertrand Russell (pp. 123-154). La Salle, France:...
  • Gödel, K. (2001). Collected works: Volume III unpublished essays and lectures. New York, NY: Oxford University Press.
  • Goodman, N. (1965). Fact, fiction and forecast. Indianapolis, IN: Bobbs-Merril.
  • Hilbert, D. (1964). Über das unendliche [About the infinite]. Darmstadt, Germany: Hilbertiana.
  • Hustvedt, S. (2012). Living, thinking, looking. London, United Kingdom: Sceptre.
  • Kant, I. (1787). Critique of pure reason [English translation by Norman Kemp Smith, 1929]. London, United Kingdom: Macmillan.
  • Klein, J. (1985). Lectures and essays. Annapolis, MD: St John’s College Press.
  • Klein, S. B. (2014). The two selves. New York, NY: Oxford University Press.
  • Koetsier, T. (1991). Lakatos’ philosophy of mathematics. Amsterdam, The Nether lands: North-Holland.
  • Lovejoy, A. O. (1964). The great chain of being. Cambridge, MA: Harvard University Press.
  • Mueller, I. (1969). Euclid’s elements and the axiomatic method. The British Journal for the Philosophy of Science, 20, 289-309.
  • Otte, M. (2003a). Complementary, sets and numbers. Educational Study in Mathemat ics, 53(3), 203-228.
  • Otte, M. (2003b). Does mathematics have objects? What sense? Synthese, 134, 181-216.
  • Parmentier, R. (1994). Signs in society. Bloomington, IN: Indiana University Press.
  • Peirce, C. S. (1857-1886). Writings of Charles S. Peirce: A chronological edition. Peirce Edition Project (Eds.). Indiana, IN: Indiana University...
  • Peirce, C. S. (1892). The law of mind. The Monist, 2(4), 533-559.
  • Peirce, C. S. (1931-1935). Collected Papers of Charles Sanders Peirce (Vols. 1-6. edited by C. Hartshorne & P. Weiss; Vols. 7-8 edited...
  • Peirce, C. S. (1966). How to make our ideas clear. New York, NY: Dover Publications.
  • Quine, W. V. (1990). The roots of reference. La Salle, IL: Open Court.
  • Rorty, R. (1979). Philosophy and the mirror of nature (PMN). Princeton, United Kingdom: Princeton University Press.
  • Rorty, R. (1989). Contingency, irony, solidarity (CIS). Cambridge, United Kingdom: Cambridge University Press.
  • Russell, B. (1967). Introduction to mathematical philosophy. London, United Kingdom: Routledge.
  • Spengler, O. (1918). Der untergang des abendlandes [The decline of the west]. Darmstadt, Germany: Bibliographisches Institut.
  • Stolzenberg, G. (1984). Can an inquiry into the foundations of Mathematics tell us anything interesting about the mind. In P. Watzlawik (Ed.),...
  • Valéry, P. (1998). Leonardo da Vinci. Frankfurt, Germany: Suhrkamp.
  • Wertheimer, M. (1945). Productive thinking. New York, NY: Harper.