zbMATH — the first resource for mathematics

Is there a “Hilbert thesis”? (English) Zbl 1428.03021
The paper is devoted to the so called Hilbert thesis stating that – as J. Barwise put it – “The informal notion of provable used in mathematics is made precise by the formal notion provable in first-order logic”. Different variations of this thesis are discussed. The author also considers whether the very name “Hilbert thesis” is justified. The thesis is compared with Church’s thesis concerning computability. The problem of the possibility of providing arguments towards the considered thesis is discussed.

03A05 Philosophical and critical aspects of logic and foundations
03-03 History of mathematical logic and foundations
01A60 History of mathematics in the 20th century
Full Text: DOI
[1] Ackermann, W., Begründung des “tertium non datur” mittels der Hilbertschen Theorie der Widerspruchsfreiheit, Mathematische Annalen, 93, 1-36, (1925) · JFM 50.0023.02
[2] Azzouni, J., The derivation-indicator view of mathematical practice, Philosophia Mathematica 12(3):81-105, 2004. · Zbl 1136.00301
[3] Barwise, J., An introduction to first-order logic, in J. Barwise, (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, pp. 5-46.
[4] Beklemishev, L., and A. Visser, Problems in the logic of provability, in D. Gabbay, S. Goncharov, and M. Zakharyaschev, (eds.), Mathematical Problems from Applied Logic I: Logics for the XXIst Century, vol. 4 of International Mathematical Series, Springer, Berlin, 2005, pp. 77-136. · Zbl 1100.03051
[5] Berk, L.A., Hilbert’s Thesis: Some Considerations about Formalizations of Mathematics, Ph.D. thesis, MIT, 1982. http://hdl.handle.net/1721.1/15650.
[6] Bernays, P., Abhandlungen zur Philosophie der Mathematik, Wissenschaftliche Buchgesellschaft, Darmstadt, 1976. · Zbl 0335.02002
[7] Boolos, G.S., J.P. Burgess, and R.C. Jeffrey, Computability and Logic, 4th edn., Cambridge University Press, Cambridge, 2003.
[8] Boolos, G.S., and R.C. Jeffrey, Computability and Logic, 3rd edn., Cambridge University Press, Cambridge, 1989.
[9] Davis, M., and W. Sieg, Conceptual confluence in 1936: Post and Turing, in G. Sommaruga, and T. Strahm, (eds.), Turing’s Revolution: The Impact of His Ideas about Computability, Springer, Berlin, 2015, pp. 3-27. · Zbl 1400.01008
[10] Ebbinghaus, H.-D., Ernst Zermelo, Springer, Berlin, 2007.
[11] Feferman, S., H.M. Friedman, P. Maddy, and J.R. Steel, Does mathematics need new axioms?, Bulletin of Symbolic Logic 6(4):401-446, 2000. · Zbl 0977.03002
[12] Gentzen, G., Untersuchungen über das logische Schließen I, II, Mathematische Zeitschrift 39:176-210, 405-431, 1935. · Zbl 0010.14501
[13] Gentzen, G., The consistency of elementary number theory, in M.E. Szabo, (ed.), The Collected Papers of Gerhard Gentzen, North-Holland, Amsterdam, 1969, pp. 132-201. English translation of Die Widerspruchsfreiheit der reinen Zahlentheorie originially published in 1939.
[14] Gödel, K., On undecidable propositions of formal mathematical systems, in M. Davis, (ed.), The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Raven Press, 1965, pp. 41-73.
[15] Gödel, K., Vortrag über Vollständigkeit des Funktionenkalküls, in S. Feferman, et al., (eds.), Collected Works, vol. III of Unpublished Essays and Lectures, Oxford University Press, 1995, pp. 16-29. Lecture delivered on 6 September 1930 at the Conference on Epistemology of the Exact Sciences in Königsberg; German original and English translation.
[16] Hales, T.C., Developments in formal proof, Séminaire Bourbaki, (2014), 1086, 66ème année, 2013-2014.
[17] Hilbert, D., Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Kongreß zu Paris 1900, Nachrichten von der königl. Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-physikalische Klasse aus dem Jahre 1900, 1900, pp. 253-297.
[18] Hilbert, D., Über den Zahlbegriff, Jahresbericht der Deutschen Mathematiker-Vereinigung 8:180-184, 1900. · JFM 31.0165.02
[19] Hilbert, D., Axiomatisches Denken, Mathematische Annalen 78(3/4):405-415, 1918. (Lecture delivered on 11 September 1917 at the Swiss Mathematical Society in Zurich). · JFM 46.0062.03
[20] Hilbert, D., Über das Unendliche, Mathematische Annalen 95:161-190, 1926. · JFM 51.0044.02
[21] Hilbert, D., Die Grundlagen der Mathematik, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 6(1/2):65-85, 1928.
[22] Hilbert, D., Die Grundlegung der elementaren Zahlenlehre, Mathematische Annalen 104:485-494, 1931. (Lecture delivered in December 1930 in Hamburg). · JFM 57.0054.04
[23] Hilbert, D., The foundations of mathematics, in J. van Heijenoort, (ed.), From Frege to Gödel, Harvard University Press, 1967, pp. 464-479. English translation of [21].
[24] Hilbert, D., On the infinite, in Jean van Heijenoort, (ed.), From Frege to Gödel, Harvard University Press, 1967, pp. 367-392. English translation of [20].
[25] Hilbert, D., and W. Ackermann, Grundzüge der theoretischen Logik, vol. XXVII of Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Springer, Berlin, 1928.
[26] Hilbert, D., and W. Ackermann, Grundzüge der theoretischen Logik, vol. XXVII of Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, 2nd edn., Springer, Berlin, 1938.
[27] Jäger, G., Inductive definitions and non-wellfounded proofs, Talk given in Tübingen in honor of Peter Schroeder-Heister’s 60th birthday, 2013.
[28] Kahle, R., Von Dedekind zu Zermelo versus Peano zu Gödel, Mathematische Semesterberichte 64(2):159-167, 2017. · Zbl 1395.01020
[29] Kleene, S.C., Introduction to Metamathematics, North Holland, Amsterdam, 1952. · Zbl 0047.00703
[30] Kleene, S.C., Origins of recursive function theory, Annals of the History of Computing 3(1):52-67, 1981. · Zbl 0998.03501
[31] Kleene, S.C., Gödel’s impression on students of logic in the 1930s, in P. Weingartner, and L. Schmetterer, (eds.), Gödel Remembered, vol. IV of History of Logic, Bibliopolis, Berkeley, 1987, pp. 49-64.
[32] Kreisel, G., Informal rigour and completeness proofs, in I. Lakatos, (ed.), Problems in the Philosophy of Mathematics, vol. 47 of Studies in Logic and the Foundations of Mathematics, Elsevier, Amsterdam, 1967, pp. 138-186.
[33] Kripke, S.A., The Church-Turing “Thesis” as a special corollary of Gödel’s completeness theorem, in B.J. Copeland, C.J. Posy, and O. Shagrir, (eds.), Computability, MIT Press, Cambridge, 2013, pp. 77-104. · Zbl 1372.03091
[34] Marfori, M.A., Informal proofs and mathematical rigour, Studia Logica 96:261-272, 2010. · Zbl 1208.03011
[35] Moschovakis, Y., Notes on Set Theory, 2nd edn., Undergraduate Texts in Mathematics, Springer, Berlin, 2006.
[36] Naibo, A., M. Petrolo, and T. Seiller, On the computational meaning of axioms, in J. Redmond, O.P. Martins, and Á.N. Fernández, (eds.), Epistemology, Knowledge and the Impact of Interaction, Springer, Berlin, 2016, pp. 141-184.
[37] Naibo, A., M. Petrolo, and T. Seiller, Verificationism and classical realizability, in C. Başkent, (ed.), Perspectives on Interrogative Models of Inquiry: Developments in Inquiry and Questions, Springer, Berlin, 2016, pp. 163-197.
[38] Odifreddi, P., Classical Recursion Theory, vol. 125 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1989. · Zbl 0661.03029
[39] Parsons, C., Finitism and intuitive knowledge, in M. Schirn, (ed.), The Philosophy of Mathematics Today, Oxford University Press, Oxford, 1998, pp. 249-270. · Zbl 0918.03003
[40] Post, E.L., Finite combinatory processes—formulation 1, Journal of Symbolic Logic 1(3):103-105, 1936. · JFM 62.1060.01
[41] Prawitz, D., Natural Deduction, A Proof-Theoretical Study, Almquist and Wiksell, 1965. · Zbl 0173.00205
[42] Shapiro, S., The open texture of computability, in B.J. Copeland, C.J. Posy, and O. Shagrir, (eds.), Computability, MIT Press, Cambridge, 2013, pp. 153-181.
[43] Shoenfield, J.R., Mathematical Logic, Addison-Wesley, 1967. Reprinted by ASL, AK Peters, 2000.
[44] Sieg, W., In the shadow of incompletenss, in Hilbert’s Programs and Beyond, Oxford University Press, Oxford, 2013, pp. 155-192. First published in another collection in 2011.
[45] Smullyan, R.M., Fixed points and self-reference, International Journal of Mathematics and Mathematical Sciences 7(2):283-289, 1984. · Zbl 0585.03001
[46] Soare, R.I., Interactive computing and relativized computability, in B.J. Copeland, C.J. Posy, and O. Shagrir, (eds.), Computability, MIT Press, Cambridge, 2013, pp. 203-260.
[47] Streett, R.S., and E.A. Emerson, An automata theoretic decision procedure for the propositional mu-calculus, Information and Computation 81:249-264, 1989. · Zbl 0671.03023
[48] Tapp, C., An den Grenzen des Unendlichen, Mathematik im Kontext, Springer, Berlin, 2013. · Zbl 1267.03005
[49] Weir, A., Informal proof, formal proof, formalism, The Review of Symbolic Logic 9(1):23-43, 2016. · Zbl 1381.03014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.