Dogmas and the changing images of foundations
Philosophia Scientiae, Volume 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assumptions about the foundational enterprise emerge from certain dogmas that are frequently inherited from previous, outdated images.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, Volume 9 (2005) no. S2, pp. 27-42. http://archive.numdam.org/item/PHSC_2005__9_S2_27_0/

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