Dogmas and the changing images of foundations

José Ferreirós

Philosophia Scientiae (2005)

  • Volume: 9, Issue: S2, page 27-42
  • ISSN: 1281-2463

Abstract

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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assumptions about the foundational enterprise emerge from certain dogmas that are frequently inherited from previous, outdated images.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.

How to cite

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Ferreirós, José. "Dogmas and the changing images of foundations." Philosophia Scientiae 9.S2 (2005): 27-42. <http://eudml.org/doc/103772>.

@article{Ferreirós2005,
abstract = {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assumptions about the foundational enterprise emerge from certain dogmas that are frequently inherited from previous, outdated images.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.},
author = {Ferreirós, José},
journal = {Philosophia Scientiae},
language = {eng},
number = {S2},
pages = {27-42},
publisher = {Éditions Kimé},
title = {Dogmas and the changing images of foundations},
url = {http://eudml.org/doc/103772},
volume = {9},
year = {2005},
}

TY - JOUR
AU - Ferreirós, José
TI - Dogmas and the changing images of foundations
JO - Philosophia Scientiae
PY - 2005
PB - Éditions Kimé
VL - 9
IS - S2
SP - 27
EP - 42
AB - 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.
LA - eng
UR - http://eudml.org/doc/103772
ER -

References

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  1. [1] Benoit, P., Chemla, K., Ritter, J. eds. 1992.— Histoire de fractions, fractions d’histoire, Basel: Birkhäuser, 1992. Zbl0827.01001MR1278489
  2. [2] Bernays, Paul1976.— Abhandlungen zur Philosophie der Mathematik, Darmstadt: Wissenschaftliche Buchgesellschaft, 1976. Zbl0335.02002MR444417
  3. [3] Corry, Leo1997.— Modern algebra and the rise of mathematical structures, Basel/ Boston: Birkhäuser, 1997. Zbl0858.01022MR1391720
  4. [4] Dedekind, Richard1888.— 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]. Zbl25.0115.05MR106846
  5. [5] Ewald, William, ed. 1996.— From Kant to Hilbert, Oxford: Oxford University Press, 2 vols. MR1465678
  6. [6] Ferreirós, José forthcoming.— &gt;’O jeoc &lt;’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. MR2308276
  7. [7] Frege, Gottlob1893.— Grundgesetze der Arithmetik, vol. 1, Jena: Pohle, 1893; reprint: Hildesheim: Olms, 1969. 
  8. [8] Gauss, Carl F.1900.— Werke, vol. 8, Göttingen: Dieterich, 1900; reprint Hildesheim: Olms, 1973. Zbl0924.01032
  9. [9] Hallett, Michael1994.— Hilbert’s Axiomatic Method and the Laws of Thought, in A. George, Mathematics and Mind, Oxford: Oxford University Press, 1994, 158–200. Zbl0816.00002MR1373897
  10. [10] Van Heijenoort, Jean 1967.— From Frege to Gödel, Harvard Univ. Press, 1967; reprinted in 2002. MR1890980
  11. [11] Hilbert, David1897.— Bericht über die Theorie der algebraischen Zahlen, Jahresbericht der DMV, 4, 1897; reprint in Gesammelte Abhandlungen, Berlin: Springer, vol. 1, 1932. 
  12. [12] Hilbert, David1935.— Gesammelte Abhandlungen, vol. 3, Berlin: Springer, 1935; reprinted in 1970. Zbl61.0022.02
  13. [13] Maddy, Penelope1992.— Realism in mathematics, Oxford: Clarendon Press, 1992. Zbl0762.00001MR1075998
  14. [14] Peckhaus, Volker1991.— Hilbertprogramm und kritische Philosophie, Göttingen: Vandenhoeck & Ruprecht, 1991. MR1116994
  15. [15] Quine, Willard Van O.1940.— Mathematical Logic, New York: Norton, 1940. Zbl0063.06360MR2508JFM66.0027.03
  16. [16] Quine, Willard Van O.1963.— Set theory and its logic, Harvard Univ. Press, 1963. Zbl0122.24601MR274272
  17. [17] Quine, Willard Van O.1969.— Epistemology naturalized, in Ontological Relativity and other essays, Columbia Univ. Press, 1969. Zbl0193.30402MR274272
  18. [18] Riemann, Bernhard1854.— 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. [19] Sieg, Wilfried1999.— Hilbert’s programs: 1917–1922, The Bulletin of Symbolic Logic, 5, 1–44. Zbl0924.03002MR1681894

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