Dogmas and the changing images of foundations
Philosophia Scientiae, Fonder autrement les mathématiques, Tome 9 (2005) no. S2, pp. 27-42.

We offer a critical review of several different conceptions of the activity of foundational research, from the time of Gauss to the present. These are (1) the traditional image, guiding Gauss, Dedekind, Frege and others, that sees in the search for more adequate basic systems a logical excavation of a priori structures, (2) the program to find sound formal systems for so-called classical mathematics that can be proved consistent, usually associated with the name of Hilbert, and (3) the historicist alternative, guiding Riemann, Poincaré, Weyl and others, that seeks to perfect available conceptual systems with the aim to avoid conceptual limitations and expand the range of theoretical options. I shall contend that, at times, assumptions about the foundational enterprise emerge from certain dogmas that are frequently inherited from previous, outdated images. To round the discussion, I mention some traits of an alternative program that investigates the epistemology of mathematical knowledge.

Ofrecemos una revisión crítica de varias concepciones de la investigación sobre los fundamentos de la matemática, desde los tiempos de Gauss hasta el presente. Se trata de (1) la imagen tradicional, que guió a Gauss, Dedekind, Frege y otros, y que ve en la búsqueda de sistemas básicos más adecuados una excavación lógica de estructuras a priori, (2) el programa de encontrar sistemas formales correctos para la llamada matemática clásica que puedan demostrarse consistentes, habitualmente asociado al nombre de Hilbert, y (3) la alternativa historicista, que guió a Riemann, Poincaré, Weyl y otros, la cual busca perfeccionar los sistemas conceptuales disponibles a fin de evitar limitaciones conceptuales y ampliar el abanico de opciones teóricas. Defenderé que, en ocasiones, se encuentran supuestos acerca del trabajo sobre fundamentos que emergen de ciertos dogmas, frecuentemente heredados de imágenes previas ya superadas. Para completar la discusión, menciono algunos rasgos de un programa alternativo, que investiga la epistemología del conocimiento matemático.

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Ferreirós, José. Dogmas and the changing images of foundations. Philosophia Scientiae, Fonder autrement les mathématiques, Tome 9 (2005) no. S2, pp. 27-42. http://www.numdam.org/item/PHSC_2005__9_S2_27_0/

[1] Benoit, P., Chemla, K., Ritter, J. eds. 1992.- Histoire de fractions, fractions d'histoire, Basel: Birkhäuser, 1992. | MR | Zbl

[2] Bernays, Paul 1976.- Abhandlungen zur Philosophie der Mathematik, Darmstadt: Wissenschaftliche Buchgesellschaft, 1976. | MR | Zbl

[3] Corry, Leo 1997.- Modern algebra and the rise of mathematical structures, Basel/ Boston: Birkhäuser, 1997. | MR | Zbl

[4] Dedekind, Richard 1888.- Was sind und was sollen die Zahlen?, Braunschweig: Vieweg, 1888; reprinted in Gesammelte mathematische Werke, New York: Chelsea, 1969; eng. trans. in [Ewald 1996, vol. 2]. | MR

[5] Ewald, William, ed. 1996.- From Kant to Hilbert, Oxford: Oxford University Press, 2 vols. | MR

[6] Ferreirós, José forthcoming.- >'O jeoc <'arijmht'izeiu : The rise of pure mathematics as arithmetic with Gauss, in C. Goldstein, N. Schappacher, J. Schwermer, eds., The Shaping of Arithmetic : Number theory after Carl Friedrich Gauss's Disquisitiones Arithmeticae, Berlin : Springer. | MR

[7] Frege, Gottlob 1893.- Grundgesetze der Arithmetik, vol. 1, Jena: Pohle, 1893; reprint: Hildesheim: Olms, 1969. | EuDML

[8] Gauss, Carl F. 1900.- Werke, vol. 8, Göttingen: Dieterich, 1900; reprint Hildesheim: Olms, 1973.

[9] Hallett, Michael 1994.- Hilbert's Axiomatic Method and the Laws of Thought, in A. George, Mathematics and Mind, Oxford: Oxford University Press, 1994, 158-200. | MR | Zbl

[10] Van Heijenoort, Jean 1967.- From Frege to Gödel, Harvard Univ. Press, 1967; reprinted in 2002. | MR

[11] Hilbert, David 1897.- Bericht über die Theorie der algebraischen Zahlen, Jahresbericht der DMV, 4, 1897; reprint in Gesammelte Abhandlungen, Berlin: Springer, vol. 1, 1932.

[12] Hilbert, David 1935.- Gesammelte Abhandlungen, vol. 3, Berlin: Springer, 1935; reprinted in 1970.

[13] Maddy, Penelope 1992.- Realism in mathematics, Oxford: Clarendon Press, 1992. | MR | Zbl

[14] Peckhaus, Volker 1991.- Hilbertprogramm und kritische Philosophie, Göttingen: Vandenhoeck & Ruprecht, 1991. | MR

[15] Quine, Willard Van O. 1940.- Mathematical Logic, New York: Norton, 1940. | JFM | MR | Zbl

[16] Quine, Willard Van O. 1963.- Set theory and its logic, Harvard Univ. Press, 1963. | MR | Zbl

[17] Quine, Willard Van O. 1969.- Epistemology naturalized, in Ontological Relativity and other essays, Columbia Univ. Press, 1969. | MR

[18] Riemann, Bernhard 1854.- Über die Hypothesen, welche der Geometrie zu Grunde liegen, in Gesammelte Werke, Berlin: Springer, 1991; Eng. trans. in Ewald, op. cit., vol. 2. (See also Riemann's Fragmente philosophischen Inhalts, in Gesammelte Werke.)

[19] Sieg, Wilfried 1999.- Hilbert's programs: 1917-1922, The Bulletin of Symbolic Logic, 5, 1-44. | MR | Zbl