On the epistemological justification of Hilbert's metamathematics
Philosophia Scientiae, Volume 9 (2005) no. 2, pp. 225-238.

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.

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Legris, Javier. On the epistemological justification of Hilbert's metamathematics. Philosophia Scientiae, Volume 9 (2005) no. 2, pp. 225-238. http://archive.numdam.org/item/PHSC_2005__9_2_225_0/

[1] Abrusci, V. Michele 1978.- 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] Bernays, Paul 1930a.- 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 | JFM

[3] Bernays, Paul 1930b.- 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. | JFM

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

[5] Brouwer, Luitzen Egbertus Jan 1907.- 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] Descartes, René 1701.- Regulae ad directionem ingenii. In Oeuvres de Descartes ed. by Charles Adam and Paul Tannery, Paris: Vrin, 1982.

[7] Goldfarb, Warren D. 1979.- Logic in The Twenties: The Nature of The Quantifier. Journal of Symbolic Logic 44, 351-368. | MR | Zbl

[8] Hilbert, David 1922.- Neubegründung der Mathematik. Erste Mitteilung. Abhandlungen aus dem Mathematischen Seminar der Hamburger Universität 1, 157- 177. English translation in [Mancosu 1998]: 198-214.

[9] Hilbert, David 1923.- Die logischen Grundlagen der Mathematik. Mathematische Annalen 88, 151-165. | JFM

[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. | MR

[11] Hilbert, David 1931.- Die Grundlegung der elementaren Zahlenlehre, Mathematische Annalen 104, 485-494. English translation in [Mancosu 1998]: 266-273 | MR | Zbl

[12] Hilbert, David & Paul Bernays 1934.- Grundlagen der Mathematik, Vol. I, Berlín: Springer, 1934.

[13] Leisenring, A.C. 1969.- Mathematical Logic And Hilbert’s ϵ-Symbol, New York: Gordon and Breach Science Publishers, 1969. | MR | Zbl

[14] Mancosu, Paolo 1998.- From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s, New York - Oxford: Oxford University Press, 1998. | MR | Zbl

[15] Nelson, Leonard 1959.- Beiträge zur Philosophie der Logik und Mathematik, Frankfurt: Öffentliches Leben, 1959.

[16] Parsons, Charles 1998.- Finitism and Intuitive Knowledge. In Matthias Schirn (ed.): The Philosophy of Mathematics Today, Oxford: Clarendon Press, 1998: 249-270. | MR | Zbl

[17] Sinaceur, Hourya 1993.- Du Formalisme à la constructivité: le finitisme, Revue internationale de Philosophie 1993/4 n. 186, 251-283.

[18] Weyl, Hermann 1921.- Über die neue Grundlagenkrise der Mathematik, Mathematische Zeitschrift 10, 39-79. | JFM

[19] Weyl, Hermann 1925.- 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. | JFM

[20] Weyl, Hermann 1953.- Über den Symbolismus der Mathematik und mathematischen Physik, Studium Generale, 6, 219-228. | Zbl