Browse > Article

Hilbert's Program as Research Program  

Cheong, Kye-Seop (Duksung Women's University)
Publication Information
Journal for History of Mathematics / v.24, no.3, 2011 , pp. 37-58 More about this Journal
The development of recent Mathematical Logic is mostly originated in Hilbert's Proof Theory. The purpose of the plan so called Hilbert's Program lies in the formalization of mathematics by formal axiomatic method, rescuing classical mathematics by means of verifying completeness and consistency of the formal system and solidifying the foundations of mathematics. In 1931, the completeness encounters crisis by the existence of undecidable proposition through the 1st Theorem of G?del, and the establishment of consistency faces a risk of invalidation by the 2nd Theorem. However, relative of partial realization of Hilbert's Program still exists as a fruitful research program. We have tried to bring into relief through Curry-Howard Correspondence the fact that Hilbert's program serves as source of power for the growth of mathematical constructivism today. That proof in natural deduction is in truth equivalent to computer program has allowed the formalization of mathematics to be seen in new light. In other words, Hilbert's program conforms best to the concept of algorithm, the central idea in computer science.
증명;형식적 공리론;완전성;무모순성;커리-하워드 대응;절단제거;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Sieg W., Hilbert's Program: 1887-1922. Bulletin of symbolic Logic, 5(1), pp.1-44, 1999.   DOI   ScienceOn
2 Murawsk: R., On proofs of the consistency of arithmetic, Studies in logic, grammar and rhetoric 5(18), pp.41-50, 2002.
3 Bernays P., On Hilbert's Thoughts Concerning the Grounding of Arithmetics, 1923. English translation in [Mancosu, pp.223-226].
4 Bernays P., Hilbert, David, Encyclopedia of philosophy, vol.III, Macmillan Publishing Co., 1967.
5 Bernays P., The Philosophy of Mathematics and Hilbert's Proof Theory, 1930. in [Mancosu, pp.234-265].
6 Zach R., The practice of finitism. Epsilon calculus and consistency proofs in Hilbert's Program, Synthese, 137, pp.211-259, 2003.   DOI
7 Cassou-Nogues Pierre, Hilbert, Les Belles Lettres, 2001.
8 Detlefsen M., Hilbert's Program, Dordrecht, 1986.
9 Ewald W. B. (ed), From Kant to Hilbert. A Source book in the foundations of Mathematics, vol. 2, Oxford University press, 1996.
10 Zach R., Hilbert's Program, The Stanford Encyclopedia of Philosophy, 2003.
11 Mancosu P., Hilbert and Berneys on metamathemetics, in [Mancosu(Ed.), pp.149-188]
12 Simpson S. G., Partial realizations of Hilbert's Program, Journal of Symbolic Logic 53(2), pp.349-363, 1988.   DOI   ScienceOn
13 Slater B. H., Epsilon Calculi, Logic Journal of IGPL vol. 14, number 4, pp.535-590, 2006.   DOI   ScienceOn
14 Tait W. W., Finitism, Journal of Philosophy. 78, pp.524-546, 1981.   DOI   ScienceOn
15 Van Heijenoort J., (ed.), From Frege to Godel. A source Book in mathematical Logic, 1897-1931, Harvard University press, 1967.
16 Mancosu P., (ed.), From Brouwer to Hilbert. The Debate of the Foundations of Mathematics in the 1920s, Oxford University press, 1998.
17 Kreisel G., Hilbert's Program, 1983, in philosophy of mathematices, In Benacerraf and Putnam, pp.207-238.
18 Hilbert D., The New Grounding of Mathematics, 1922, English translation in [Mancosu, pp.198-214]
19 Hilbert D., On the infinite, 1926, English translation in [Van Heijenoort, pp.367-392].
20 Hilbert D., The foundation of mathematics, 1927, English translation in [Van Heijenoort, pp.464-479].
21 Feferman S., Hilbert's Program relativized: Proof-Theoretical and foundational reductions, Journal of Symbolic Logic, 53(2), pp.284-304, 1988.   DOI
22 Benacerraf P. and Putnam H., (ed.), Philosophy of Mathematics, Cambridge University press, 1983.
23 Gray J.J., Le Defi de Hilbert, Dunod, 2004.
24 Herbrand J., English translation: 'On the consistency of arithmetic', in: Heijenoort, J. van (Ed.) From Frege to Godel. A source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, Mass, pp.620-628, (1931)1967.
25 Avigad J. and Reck E., Clarifing the nature of the infinite: the development of metamathematics and proof theory, (, 2001.