Contenido principal del artículo

Michael F. Otte
University of Bielefeld, Alemania Universidade Bandeirantes de São Paulo, Brasil
Tânia M. Mendonça
Universidade Bandeirantes de São Paulo, Brasil
Luiz de Barros
Universidade Bandeirantes de São Paulo, Brasil
Vol. 9 Núm. 3: Número monográfico en Generalización (Marzo, 2015), Artículos, Páginas 143-164
Recibido: jun 24, 2017 Aceptado: jun 24, 2017 Publicado: mar 1, 2015


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.

La generalización es necesaria o incluso inevitable

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.



Nº de citas en WOS (2017): 1 (Citas de 2º orden, 1)


La descarga de datos todavía no está disponible.

Detalles del artículo


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: Open Court.

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 Netherlands: 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 Mathematics, 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 Press.

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 by A. W. Burks). Cambridge, MA: Belknap Press of Harvard University Press.

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.), Invented reality (pp. 257-308). Nueva York, NY: W. W. Norton Inc.

Valéry, P. (1998). Leonardo da Vinci. Frankfurt, Germany: Suhrkamp.

Wertheimer, M. (1945). Productive thinking. New York, NY: Harper.