On the epistemological justification of Hilbert’s metamathematics

Javier Legris

Philosophia Scientiae (2005)

  • Volume: 9, Issue: 2, page 225-238
  • ISSN: 1281-2463

Abstract

top
The aim of this paper is to examine the idea of metamathematical deduction in Hilbert’s program showing its dependence of epistemological notions, specially the notion of intuitive knowledge. It will be argued that two levels of foundations of deduction can be found in the last stages (in the 1920s) of Hilbert’s Program. The first level is related to the reduction – in a particular sense – of mathematics to formal systems, which are ‘metamathematically’ justified in terms of symbolic manipulation. The second level of foundation consists in warranting epistemologically the validity of the combinatory processes underlying the symbolic manipulation in metamathematics. In this level the justification was carried out with the aid of notions from modern epistemology, particularly the notion of intuition. Finally, some problems concerning Hilbert’s use of this notion will be shown and it will be compared with Brouwer’s notion and with the idea of symbolic construction due to Herrmann Weyl.

How to cite

top

Legris, Javier. "On the epistemological justification of Hilbert’s metamathematics." Philosophia Scientiae 9.2 (2005): 225-238. <http://eudml.org/doc/103752>.

@article{Legris2005,
abstract = {The aim of this paper is to examine the idea of metamathematical deduction in Hilbert’s program showing its dependence of epistemological notions, specially the notion of intuitive knowledge. It will be argued that two levels of foundations of deduction can be found in the last stages (in the 1920s) of Hilbert’s Program. The first level is related to the reduction – in a particular sense – of mathematics to formal systems, which are ‘metamathematically’ justified in terms of symbolic manipulation. The second level of foundation consists in warranting epistemologically the validity of the combinatory processes underlying the symbolic manipulation in metamathematics. In this level the justification was carried out with the aid of notions from modern epistemology, particularly the notion of intuition. Finally, some problems concerning Hilbert’s use of this notion will be shown and it will be compared with Brouwer’s notion and with the idea of symbolic construction due to Herrmann Weyl.},
author = {Legris, Javier},
journal = {Philosophia Scientiae},
language = {eng},
number = {2},
pages = {225-238},
publisher = {Éditions Kimé},
title = {On the epistemological justification of Hilbert’s metamathematics},
url = {http://eudml.org/doc/103752},
volume = {9},
year = {2005},
}

TY - JOUR
AU - Legris, Javier
TI - On the epistemological justification of Hilbert’s metamathematics
JO - Philosophia Scientiae
PY - 2005
PB - Éditions Kimé
VL - 9
IS - 2
SP - 225
EP - 238
AB - The aim of this paper is to examine the idea of metamathematical deduction in Hilbert’s program showing its dependence of epistemological notions, specially the notion of intuitive knowledge. It will be argued that two levels of foundations of deduction can be found in the last stages (in the 1920s) of Hilbert’s Program. The first level is related to the reduction – in a particular sense – of mathematics to formal systems, which are ‘metamathematically’ justified in terms of symbolic manipulation. The second level of foundation consists in warranting epistemologically the validity of the combinatory processes underlying the symbolic manipulation in metamathematics. In this level the justification was carried out with the aid of notions from modern epistemology, particularly the notion of intuition. Finally, some problems concerning Hilbert’s use of this notion will be shown and it will be compared with Brouwer’s notion and with the idea of symbolic construction due to Herrmann Weyl.
LA - eng
UR - http://eudml.org/doc/103752
ER -

References

top
  1. [1] Abrusci, V. Michele1978.— Autofondazione della matematica. Le ricerche di Hilbert sui fondamenti della matematica, Ricerche sui Fondamenti Bernays, della Matematica by David Hilbert, ed. by V. Michele Abrusci, Naples: Bibliopolis, 1978: 13-131. 
  2. [2] Bernays, Paul1930a.— Die Philosophie der Mathematik und die Hilbertsche Beweistheorie. Blätter für Deutsche Philosophie, 4 (1930-1931), 326-367. Reprinted in [Bernays 1976], 17-61. English translation in [Mancosu 1998]: 189-197 JFM56.0044.02
  3. [3] Bernays, Paul1930b.— Die Grundgedanken der Fries’schen Philosophie in ihrem Verhältnis zum heutigen Stand der Wissenschaft. Abhandlungen der Fries´schen Schule. Neue Folge, 5 (1930), 99-113. JFM57.1313.01
  4. [4] Bernays, Paul1976.— Abhandlungen zur Philosophie der Mathematik, Darmstadt: Wissenschaftliche Buchgesellschaft, 1976. Zbl0335.02002MR444417
  5. [5] Brouwer, Luitzen Egbertus Jan1907.— Over de Grondslagen der Wiskunde. Amsterdam - Leipzig: Maas & Van Suchtelen. English translation. in L. E. J. Brouwer, Collected Works I, ed. by Arendt Heyting and Hans Freudenthal, Amsterdam: North Holland, 1975: 11- 101. 
  6. [6] Descartes, René1701.— Regulae ad directionem ingenii. In Oeuvres de Descartes ed. by Charles Adam and Paul Tannery, Paris: Vrin, 1982. 
  7. [7] Goldfarb, Warren D.1979.— Logic in The Twenties: The Nature of The Quantifier. Journal of Symbolic Logic 44, 351-368. Zbl0438.03001MR540666
  8. [8] Hilbert, David1922.— Neubegründung der Mathematik. Erste Mitteilung. Abhandlungen aus dem Mathematischen Seminar der Hamburger Universität 1, 157- 177. English translation in [Mancosu 1998]: 198-214. 
  9. [9] Hilbert, David1923.— Die logischen Grundlagen der Mathematik. Mathematische Annalen 88, 151-165. Zbl48.1120.01JFM48.1120.01
  10. [10] Hilbert, David 192.— Über das Unendliche, Mathematische Annalen 95, 161-190. English translation in Jean Van Heijenoort (ed.): From Frege to Gödel. A Source Book in Mathematical Logic, 1879-1931, Cambridge (Mass.): Harvard University Press, 1967: 367-392. MR1512272
  11. [11] Hilbert, David1931.— Die Grundlegung der elementaren Zahlenlehre, Mathematische Annalen 104, 485-494. English translation in [Mancosu 1998]: 266-273 Zbl0001.26001MR1512682
  12. [12] Hilbert, David & Paul Bernays 1934.— Grundlagen der Mathematik, Vol. I, Berlín: Springer, 1934. Zbl0009.14501
  13. [13] Leisenring, A.C.1969.— Mathematical Logic And Hilbert’s ϵ -Symbol, New York: Gordon and Breach Science Publishers, 1969. Zbl0188.31501MR276059
  14. [14] Mancosu, Paolo1998.— From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s, New York - Oxford: Oxford University Press, 1998. Zbl0941.01003MR1685779
  15. [15] Nelson, Leonard1959.— Beiträge zur Philosophie der Logik und Mathematik, Frankfurt: Öffentliches Leben, 1959. 
  16. [16] Parsons, Charles1998.— Finitism and Intuitive Knowledge. In Matthias Schirn (ed.): The Philosophy of Mathematics Today, Oxford: Clarendon Press, 1998: 249-270. Zbl0918.03003MR1701943
  17. [17] Sinaceur, Hourya1993.— Du Formalisme à la constructivité: le finitisme, Revue internationale de Philosophie 1993/4 n. 186, 251-283. 
  18. [18] Weyl, Hermann1921.— Über die neue Grundlagenkrise der Mathematik, Mathematische Zeitschrift 10, 39-79. JFM48.0220.01
  19. [19] Weyl, Hermann1925.— Die heutige Erkenntnislage in der Mathematik, Symposium 1, 1-32. Repr. In Hermann Weyl, Gesammelte Abhandlungen, Vol. II ed. By K. Chandrasekharan, Berlin-Heildelberg-N.York: Springer, 1968: 511-542. English translation in [Mancosu 1998]: 123-142. JFM51.0043.05
  20. [20] Weyl, Hermann1953.— Über den Symbolismus der Mathematik und mathematischen Physik, Studium Generale, 6, 219-228. Zbl0050.00309

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.