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.
@article{PHSC_2005__9_2_225_0, author = {Legris, Javier}, title = {On the epistemological justification of {Hilbert's} metamathematics}, journal = {Philosophia Scientiae}, pages = {225--238}, publisher = {\'Editions Kim\'e}, volume = {9}, number = {2}, year = {2005}, language = {en}, url = {http://archive.numdam.org/item/PHSC_2005__9_2_225_0/} }
Legris, Javier. On the epistemological justification of Hilbert's metamathematics. Philosophia Scientiae, Aperçus philosophiques en logique et en mathématiques, Tome 9 (2005) no. 2, pp. 225-238. http://archive.numdam.org/item/PHSC_2005__9_2_225_0/
[1] 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.
1978.-[2] 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
1930a.-[3] 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
1930b.-[4] Abhandlungen zur Philosophie der Mathematik, Darmstadt: Wissenschaftliche Buchgesellschaft, 1976. | MR | Zbl
1976.-[5] 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.
1907.-[6] Regulae ad directionem ingenii. In Oeuvres de Descartes ed. by Charles Adam and Paul Tannery, Paris: Vrin, 1982.
1701.-[7] Logic in The Twenties: The Nature of The Quantifier. Journal of Symbolic Logic 44, 351-368. | MR | Zbl
1979.-[8] Neubegründung der Mathematik. Erste Mitteilung. Abhandlungen aus dem Mathematischen Seminar der Hamburger Universität 1, 157- 177. English translation in [Mancosu 1998]: 198-214.
1922.-[9] Die logischen Grundlagen der Mathematik. Mathematische Annalen 88, 151-165. | JFM
1923.-[10] 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
192.-[11] Die Grundlegung der elementaren Zahlenlehre, Mathematische Annalen 104, 485-494. English translation in [Mancosu 1998]: 266-273 | MR | Zbl
1931.-[12] Grundlagen der Mathematik, Vol. I, Berlín: Springer, 1934.
& 1934.-[13] Mathematical Logic And Hilbert’s -Symbol, New York: Gordon and Breach Science Publishers, 1969. | MR | Zbl
1969.-[14] From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s, New York - Oxford: Oxford University Press, 1998. | MR | Zbl
1998.-[15] Beiträge zur Philosophie der Logik und Mathematik, Frankfurt: Öffentliches Leben, 1959.
1959.-[16] Finitism and Intuitive Knowledge. In Matthias Schirn (ed.): The Philosophy of Mathematics Today, Oxford: Clarendon Press, 1998: 249-270. | MR | Zbl
1998.-[17] Du Formalisme à la constructivité: le finitisme, Revue internationale de Philosophie 1993/4 n. 186, 251-283.
1993.-[18] Über die neue Grundlagenkrise der Mathematik, Mathematische Zeitschrift 10, 39-79. | JFM
1921.-[19] 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
1925.-[20] Über den Symbolismus der Mathematik und mathematischen Physik, Studium Generale, 6, 219-228. | Zbl
1953.-