Skip to main content
Log in

Hilbert's Program Revisited

  • Published:
Synthese Aims and scope Submit manuscript

Abstract

After sketching the main lines of Hilbert's program, certain well-known andinfluential interpretations of the program are critically evaluated, and analternative interpretation is presented. Finally, some recent developments inlogic related to Hilbert's program are reviewed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  • Bernays, P.: 1930, ‘Die Philosophie der Mathematik und die Hilbertische Beweistheories’, Blätter für deutsche Philosophie 4, 326–367. English translation in P. Mancosu (ed.) 1998.

    Google Scholar 

  • Bernays, P.: 1967, ‘Hilbert, David’, in P. Edwards (ed.), The Encyclopedia of Philosophy, Vol. 3, MacMillan, New York, pp. 496–504.

    Google Scholar 

  • Browder, F. E. (ed.): 1976, Mathematical Developments Arising from Hilbert's Problems, Proceedings of the Symposia in Pure Mathematics 28, American Mathematical Society, Providence.

  • Detlefsen, M.: 1986, Hilbert's Program. An Essay on Mathematical Instrumentalism, D. Reidel, Dordrecht.

    Google Scholar 

  • Detlefsen, M.: 1990, ‘On an Alleged Refutation of Hilbert's Program using Gödel's First Incompleteness Theorem’, Journal of Philosophical Logic 19, 343–377.

    Google Scholar 

  • Detlefsen, M.: 1993a, ‘Hilbert's Formalism’, Revue Internationale de Philosophie 47, 285–304.

    Google Scholar 

  • Detlefsen, M.: 1993b, ‘The Kantian Character of Hilbert's Formalism’, in J. Czermak (ed.), Philosophie der Mathematik. Akten des 15. Internationalen Wittgenstein Symposiums, Verlag Hölder-Pichler-Tempsky, Vienna, pp. 195–205.

    Google Scholar 

  • Ewald, W. (ed.): 1996, From Kant to Hilbert. Readings in the Foundations of Arithmetic, Oxford University Press, Oxford.

    Google Scholar 

  • Feferman, S.: 1977, ‘Theories of Finite Types Related to Mathematical Practice’, in J. Barwise (ed.), Handbook in Mathematical Logic, North-Holland, Amsterdam, pp. 913–971.

    Google Scholar 

  • Feferman, S.: 1988, ‘Hilbert's Program Relativized: Proof-Theoretical and Foundational Reductions’, The Journal of Symblic Logic 53, 364–383.

    Google Scholar 

  • Feferman, S. and W. Sieg: 1981, ‘Proof Theoretic Equivalences between Classical and Constructive Theories of Analysis’, in Buchholz, Feferman, Pohler and Sieg, Iterated Inductive Definitions and Subsystems of Analysis, Lecture Notes in Mathematics, Vol. 897, Springer-Verlag, Berlin, pp. 78–142.

    Google Scholar 

  • Friedman, H.: 1976, ‘Systems of Second Order Arithmetic with Restricted Induction I, II’ (abstracts), The Journal of Symbolic Logic 41, 557–559.

    Google Scholar 

  • Friedman, H.: 1978, ‘Classically and Intuitionistically Provably Recursive Functions’, in Müller and Scott (eds.), Higher Set Theory, Lecture Notes in Mathematics, Vol. 669, Springer-Verlag, Berlin, pp. 21–27.

    Google Scholar 

  • Friedman, H., S. G. Simpson, and R. L. Smith: 1983, ‘Countable Algebra and Set Existence Axioms’, Annals of Pure and Applied Logic 25, 141–181.

    Google Scholar 

  • Gödel, K.: 1933, ‘Einen Interpretation des intuitionistischen Aussagenkalküls’, Ergebnisse eines mathematischen Kolloquiums, Heft 4, pp. 39–40.

  • Giaquinto, M.: 1983, ‘Hilbert's Philosophy of Mathematics’, British Journal of the Philosophy of Science 34, 119–132.

    Google Scholar 

  • Hilbert, D.: 1900, ‘Mathematische Probleme’, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, pp. 253–297. English translation in F. E.Browder (ed.) 1976.

  • Hilbert, D.: 1905, ‘Ñber die Grundlagen der Logik und der Arithmetik’, Verhandlungen des Dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904, Teubner, Leipzig, pp. 174–185. English translation in J. van Heijenoort (ed.) 1967.

    Google Scholar 

  • Hilbert, D.: 1918, ‘Axiomatisches Denken’, Mathematische Annalen 78, 405–415. English translation in W. Ewald (ed.) 1996.

    Google Scholar 

  • Hilbert, D.: 1922, ‘Neubegründung der Mathematik. Erste Mitteilung’, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 1, pp. 157–177. English translation in W. Ewald (ed.) 1996 and in P. Mancosu (ed.) 1998.

    Google Scholar 

  • Hilbert, D.: 1923, ‘Die logischen Grundlagen der Mathematik’, Mathematische Annalen 88, 151–1165. English translation in W. Ewald (ed.) 1996.

    Google Scholar 

  • Hilbert, D.: 1926, ‘Ñber das Unendliche’, Mathematische Annalen 95, 161–190. English translation in J. van Heijenoort (ed.) 1967.

    Google Scholar 

  • Hilbert, D.: 1928, ‘Die Grundlagen der Mathematik’, Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität 6, 65–85. English translation in J. van Heijenoort (ed.) 1967.

    Google Scholar 

  • Hilbert, D.: 1929, ‘Probleme der Grundlegung der Mathematik’, Mathematische Annalen 102, 1–9. English translation in P. Mancosu (ed.) 1998.

    Google Scholar 

  • Hilbert, D.: 1931, ‘Die Grundlegung der elementaren Zahlentheorie’, Mathematische Annalen 104, 485–494. English translation in W. Ewald (ed.) 1996 and P. Mancosu (ed.) 1998.

    Google Scholar 

  • Hilbert, D. and W. Ackermann: 1928, Grundzüge der theoretischen Logik, Springer-Verlag, Berlin.

    Google Scholar 

  • Kitcher, P.: 1976, ‘Hilbert's Epistemology’, Philosophy of Science 43, 99–115.

    Google Scholar 

  • Kreisel, G.: 1958a, ‘Hilbert's Programme’, Dialectica 12, 346–372.

    Google Scholar 

  • Kreisel, G.: 1958b, ‘Mathematical Significance of Consistency Proofs’, The Journal of Symbolic Logic 23, 155–182.

    Google Scholar 

  • Kreisel, G.: 1968, ‘A Survey of Proof Theory’, The Journal of Symbolic Logic 33, 321–388.

    Google Scholar 

  • Kreisel, G.: 1976, ‘What have we Learned from Hilbert's Second Problem?’, in F. E. Browder (ed.) 1976, pp. 93–130.

  • Mancosu, P.: 1998, ‘Hilbert and Bernays on Metamathematics’, in P. Mancosu (ed.) 1998, pp. 149–188.

  • Mancosu, P. (ed.): 1998, From Brouwer to Hilbert, the Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, New York.

    Google Scholar 

  • Murawski, R.: 1994, ‘Hilbert's Program: Incompleteness Results vs. Partial Realizations’, in J. Wolenski (ed.), Philosophical Logic in Poland, Kluwer Academic Pulishers, Dordrecht, pp. 103–127.

    Google Scholar 

  • Paris, J. and L. Kirby: 1978, ‘_n Collection Schema in Arithmetic’, in A. McIntyre et al. (eds.), Logic Colloquium '77, North-Holland, Amsterdam, pp. 199–209.

    Google Scholar 

  • Parsons, C.: 1970, ‘On Number Choice Schema and its Relation to Induction’, in Kino et al. (eds.), Intuitionism and Proof Theory, North-Holland, Amsterdam, pp. 459–473.

    Google Scholar 

  • Prawitz, D.: 1981, ‘Philosophical Aspects of Proof Theory’, in G. Flöistad (ed.), Contemporary Philosophy, a New Survey, Vol. 1, Martinus Nijhoff, The Hague, pp. 235–277.

    Google Scholar 

  • Reid, C.: 1970, Hilbert, Springer-Verlag, Berlin.

    Google Scholar 

  • Rosser, J. B.: 1936, ‘Extensions of Some Theorems of Gödel and Church’, The Journal of Symbolic Logic 1, 87–91.

    Google Scholar 

  • Sieg, W.: 1984, ‘Foundations for Analysis and Proof Theory’, Synthese 60, 159–200.

    Google Scholar 

  • Sieg, W.: 1985, ‘Fragments of Arithmetic’, Annals of Pure and Applied Logic 28, 33–71.

    Google Scholar 

  • Sieg, W.: 1988, ‘Hilbert's Program Sixty Years Later’, The Journal of Symbolic Logic 53, 338–348.

    Google Scholar 

  • Sieg, W.: 1990a, ‘Relative Consistency and Accessible Domains’, Synthese 84, 259–297.

    Google Scholar 

  • Sieg, W.: 1990b, ‘Review of Stephen G. Simpson. Friedman's Research on Subsystems of Second Order Arithmetic’, Journal of Symbolic Logic 55, 870–874.

    Google Scholar 

  • Sieg, W.: 1998, ‘Proof Theory’, in E. Craig (ed.), Routledge Encyclopedia of Philosophy, Routledge, London and New York.

    Google Scholar 

  • Sieg, W.: 1999, ‘Hilbert's Programs: 1917-1922’, The Bulletin of Symbolic Logic 5, 1–44.

    Google Scholar 

  • Simpson, S.: 1984, ‘Which Set Existence Axioms are Needed to Prove Cauchy/Peano Theorem for Ordinary Differential Equations?’, The Journal of Symbolic Logic 49, 783–802.

    Google Scholar 

  • Simpson, S.: 1985a, ‘Reverse Mathematics’, in Nerode and Shore (eds.), Proceedings of the Recursion Theory Summer School, Proc. Symp. Pure Math., AMS 42, pp. 461–471.

  • Simpson, S.: 1985b, ‘Friedman's Research on Subsystems of Second Order Arithmetic’, in Harrington et al. (eds.), Harvey Friedman's Research on the Foundations of Mathematics, North-Holland, Amsterdam, pp. 137–159.

    Google Scholar 

  • Simpson, S.: 1987, ‘Subsystems of Z2 and Reverse Mathematics’, Appendix in G. Takeuti, Proof Theory, 2nd edn., North-Holland, Amsterdam, pp. 432–446.

    Google Scholar 

  • Simpson, S.: 1988, ‘Partial Realizations of Hilbert's Program’, The Journal of Symbolic Logic 53, 349–363.

    Google Scholar 

  • Smorynski, C.: 1977, ‘The Incompleteness Theorems’, in J. Barwise (ed.), Handbook in Mathematical Logic, North-Holland, Amsterdam, pp. 821–866.

    Google Scholar 

  • Smorynski, C.: 1985, Self-Reference and Modal Logic, Springer-Verlag, Berlin.

    Google Scholar 

  • Smorynski, C.: 1988, ‘Hilbert's Programme’, CWI Quarterly 1, 3–59.

    Google Scholar 

  • Tait, W.: 1981, ‘Finitism’, Journal of Philosophy 78, 524–546.

    Google Scholar 

  • Trakhtenbrot, B. A.: 1950, ‘Nevozmoznost'algorifma dla prblemy razresimosti na konecnyh klassah’ (‘Impossibility of an Algorithm for the Decision Problem in Finite Classes’), Doklady Akademii Nauk SSSR 70, 569–572.

    Google Scholar 

  • Troelstra, A. and D. van Dalen: 1988, Constructivism in Mathematics: An Introduction, Vols. I-II, North Holland, Amsterdam.

    Google Scholar 

  • Van Dalen, D.: 1995, ‘Herman Weyl's Intuitionistic Mathematics’, The Bulletin of Symbolic Logic 1, 145–169.

    Google Scholar 

  • Van Heijenoort, J. (ed.): 1967, From Frege to Gödel: A Source-Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA.

    Google Scholar 

  • Von Neumann, J.: 1931: ‘Die formalistische Grundlegung der Mathematik’, Erkenntnis 2, 116–134.

    Google Scholar 

  • Weyl, H.: 1921, ‘Ñber die neue Grundlagenkrise der Mathematik’, Mathematische Zeitschrift 10, 39–79. English translation in P. Mancosu (ed.) 1998.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Raatikainen, P. Hilbert's Program Revisited. Synthese 137, 157–177 (2003). https://doi.org/10.1023/A:1026287119565

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1026287119565

Keywords

Navigation