Proof and explanation from a semiotical point of View
Tipo de documento
Autores
Lista de autores
Otte, Michael
Resumen
A distinction between proofs that prove and proofs that explain has over and again played an important role within recent discussions in epistemology and mathematics education. The distinction goes back to scholars who, like Bolzano or Dedekind, have tried to reestablish pure mathematics as a purely conceptual and analytical science. These endeavors did in particular argue in favor of a complete elimination of intuitive or perceptual aspects from mathematical activity, arguing that one has to rigorously distinguish between a concept and its representations. Using a semiotical approach which negates such a separation between idea and symbol, we shall argue that mathematics has no explanations in a foundational sense. To explain amounts to exhibiting the meaning of something. Mathematics has, however, as we shall try to show, no definite meanings, neither in the structural intra-theoretical sense nor with respect to intuitive objectivity. Signs and meanings are processes, as we shall argue along with Peirce.
Fecha
2006
Tipo de fecha
Estado publicación
Términos clave
Epistemología | Modalidades de evaluación | Pruebas | Representaciones
Enfoque
Nivel educativo
Idioma
Revisado por pares
Formato del archivo
Volumen
9
Número
Extraordinario 1
Rango páginas (artículo)
23-43
ISSN
16652436
Referencias
Bar-Hillel, Y. (1967). Bernard Bolzano, In Paul Edwards (ed.), The Encyclopedia of Philosophy, vol. II, p.337f Beth, E. (1968). The Foundations of Mathematics: a Study in the Philosophy. of Science. - 2., rev. ed., 2. print . - Amsterdam : North-Holland Publ. Co. Bochner, S. (1974). Mathematical Reflections, American Mathematical Monthly, 830- 859. Bolzano, B. (1810/1926). Beiträge zu einer begründeteren Darstellung der Mathematik, Paderborn: Schöningh. Bolzano, B. (1817/1980). Rein analytischer Beweis des Lehrsatzes, dass zwischen zwei Werten, die ein entgegengesetztes Resultat gewaehren, wenigstens eine reelle Wurzel der Gleichung liege, Translation into English by S. Russ, Historia Mathematica, 7, 156- 185. Bolzano, B (1837/1981).Wissenschaftslehre, 4 vol. (Sulzbach). Ed. by Wolfgang Schultz. Reprint Scientia Verlag Aalen. Boulignand, G. (1933). L’idée de causalité en mathematiques et dans quelques theories physiques, Revue Scientifique, 71, 257-267. Boutroux, P. (1920). L’Idéal Scientifique des Mathématiciens. Paris: F. Alcan. Cantor, G. (1966). Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Hildesheim : Olms. Carroll, L. (1905). «What the Tortoise said to Achilles,»’Mind, N. S. vol. 4, p. 278; reprinted in: D. Hofstadter, Gödel, Escher, Bach, Vintage N.Y. Cauchy, A.L. (1821). Analyse algebrique, Paris. Dedekind, R. (1912). Stetigkeit und irrationale Zahlen, Braunschweig, Vieweg & Sohn, 4th edition. de Jong, W.R. (2003). Bernard Bolzano, Analyticity and the Aristotelian Model of Science, Kant Studien, 92, 328-349. Dilthey, W. (1910/1981). Der Aufbau der geschichtlichen Welt in den Geisteswissenschaften, Frankfurt: Suhrkamp. Fish, S. (1980). Is there a Text in this Class?, Harvard UP, Cambridge USA Hanna, G. (1989). Proofs That Prove and Proofs That Explain. In: G. Vergnaud, J. Rogalski, and M. Artigue (Eds.), Proceedings of the International Group for the Psychology of Mathematics Education, Paris, Vol II, pp. 45-51. Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, Special issue on «Proof in Dynamic Geometry Environments», 44, 5- 23. Hersh, R. (1993). Proving is Convincing and Explaining, Educational Studies in Mathematics, 24, 389-99. Hirsch, E.D. (1967). Validity in Interpretation, Yale UP, Princeton. Jakobson, R. (1985). Selected Writings, vol. VII, Berlin: Mouton. Lenhard, J. y Otte, M. (2005). Grenzen der Mathematisierung – Von der grundlegenden Bedeutung der Anwendung [Limits of Mathematization: the Constitutive Role of Application], Philosophia naturalis, 42(1), 15-47. Mancosu, P. (2000). On Mathematical Explanation, in: E. Grosholz/H. Breger (eds.) The Growth of Mathematical Knowledge, Kluwer, 103-119. Mancosu, P. (2001). Mathematical Explanation, Topoi 20: 97-117. Marx, K. (1906). Capital. Edited by Frederick Engels. Revised and Amplified According to the Fourth German Edition by Ernest Untermann. Translated by Samuel Moore and Edward Aveling, from the Third German Edition (of Das Kapital). Published: Chicago: Charles H. Kerr and Co. First published: 1867. Newton-Smith, W.H. (ed.) (2000). A Companion to the Philosophy of Science, Blackwell Oxford. Otte, M. (2003). Complementarity, Sets and Numbers, Educational Studies in Mathematics, 53, 203-228. Otte, M. (2005). Mathematics, Sign and Activity. In M. Hoffmann et.al. (eds). Activity and Sign (pp. 9-22). N.Y.: Springer. Peirce, Ch. S.: CP = Collected Papers of Charles Sanders Peirce, Volumes I-VI, ed. by Charles Hartshorne and Paul Weiß, Cambridge, Mass. (Harvard UP) 1931-1935, Volumes VII-VIII, ed. by Arthur W. Burks, Cambridge, Mass. (Harvard UP) 1958 (followed by volume and paragraph) NEM = Carolyn Eisele (ed.), The New Elements of Mathematics by Charles S. Peirce, Vol. I-IV, The Hague-Paris/Atlantic Highlands, N.J. (Mouton/Humanities Press) W = Writings of Charles S. Peirce. Ed. by Peirce Edition Project. Bloomington: Indiana University Press (1982-2000) (followed by volume and page). Resnik, M./D. Kushner (1987). Explanation, Independence and Realism in Mathematics, British Journal for the Philosophy of Science, 38, 141-158. Rota, G.-C. (1997). The Phenomenology of Mathematical Proof, Synthese, 111, 183- 196. Snow, C.P. (1993). The Two Cultures, Harvard UP, Cambridge/USA. Steiner, M. (1978). Mathematical Explanation, Philosophical Studies, 34, 135-151. Valery(no year).Leonardo, Insel Verlag Frankfurt. Volkert, K. (1986). Krise der Anschauung, V.+R. Göttingen. Wertheimer, M. (1945). Productive Thinking. New York: Harper. Weyl, H. (1995). Topology and abstract Algebra as two Roads of Mathematical Comprehension, American Math. Monthly, 453-460.