Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-25T18:09:47.925Z Has data issue: false hasContentIssue false
  • English
  • Français

Hilbert's program sixty years later

Published online by Cambridge University Press:  12 March 2014

Wilfried Sieg
Wilfried Sieg*
Affiliation:
Department of Philosophy, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213
*
Mathematisches Institut der Ludwig-Maximilians-Universitat, 8000 München 2, West Germany.
Get access Rights & Permissions [Opens in a new window]

Extract

On June 4, 1925, Hilbert delivered an address to the Westphalian Mathematical Society in Miinster; that was, as a quick calculation will convince you, almost exactly sixty years ago. The address was published in 1926 under the title Über das Unendliche and is perhaps Hilbert's most comprehensive presentation of his ideas concerning the finitist justification of classical mathematics and the role his proof theory was to play in it. But what has become of the ambitious program for securing all of mathematics, once and for all? What of proof theory, the very subject Hilbert invented to carry out his program? The Hilbertian ambition reached out too far: in its original form, the program was refuted by Gödel's Incompleteness Theorems. And even allowing more than finitist means in metamathematics, the Hilbertian expectations for proof theory have not been realized: a constructive consistency proof for second-order arithmetic is still out of reach. (And since that theory provides a formal framework for analysis, it was considered by Hilbert and Bernays as decisive for proof theory.) Nevertheless, remarkable progress has been made. Two separate, but complementary directions of research have led to surprising insights: classical analysis can be formally developed in conservative extensions of elementary number theory; relative consistency proofs can be given by constructive means for impredicative parts of second order arithmetic. The mathematical and metamathematical developments have been accompanied by sustained philosophical reflections on the foundations of mathematics. This indicates briefly the main themes of the contributions to the symposium; in my introductory remarks I want to give a very schematic perspective, that is partly historical and partly systematic.

Type
Survey/Expository Papers
Information
The Journal of Symbolic Logic , Volume 53 , Issue 2 , June 1988 , pp. 338 - 348
Copyright
Copyright © Association for Symbolic Logic 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bernays, P. [1930] Die Philosophie der Mathematik unddie Hilbertsche Beweistheorie, [Bernays, 1976], pp. 17–61.Google Scholar
Bernays, P. [1935] Hilberts Untersuchungen über die Grundlagen der Arithmetik, [Hilbert, 1935], pp. 196–216.Google Scholar
Bernays, P. [1967] Hilbert, David, Encyclopedia of philosophy (Edwards, P., editor), Vol. 3, Macmillan and Free Press, New York, pp. 496–504.Google Scholar
Bernays, P. [1976] Abhandlungen zur Philosophie der Mathematik, Wissenschaftliche Buchgesellschaft, Darmstadt.Google Scholar
Bishop, E. [1967] Foundations of constructive analysis, McGraw-Hill, New York.Google Scholar
Buchholz, W., Feferman, S., Pohlers, W., and Sieg, W. [1981] Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies, Lecture Notes in Mathematics, vol. 897, Springer-Verlag, Berlin.Google Scholar
Dedekind, R. [1872] Stetigkeit und irrationale Zahlen, [Dedekind, 1932], pp. 315–334.Google Scholar
Dedekind, R. [1888] Was sind und was sollen die Zahlen? [Dedekind, 1932], pp. 335–391.Google Scholar
Dedekind, R. [1932] Gesammelte mathematische Werke. Vol. 3 (Fricke, R. et al., editors), F. Vieweg and Sohn, Braunschweig.Google Scholar
Feferman, S. [1977] Theories of finite type related to mathematical practice, Handbook of mathematical logic (Barwise, J., editor), North-Holland, Amsterdam, pp. 913–971.CrossRefGoogle Scholar
Hilbert, D. [1900] Über den Zahlbegriff, Jahresberichte der Deutschen Mathematiker-Vereinigung, vol. 8, pp. 180–194.Google Scholar
Hilbert, D. [1902] Sur les problèmes futurs des mathèmatiques, Compte Rendu du Deuxième Congrès International des Mathématiciens, Gauthier-Villers, Paris, pp. 59–114.Google Scholar
Hilbert, D. [1904] Über die Grundlagen der Logik und Arithmetik, Grundlagen der Geometric, 5th ed., Teubner, Leipzig, 1922, pp. 243–258.Google Scholar
Hilbert, D. [1926] Über das Unendliche, Mathematische Annalen, vol. 95, pp. 161–190.CrossRefGoogle Scholar
Hilbert, D. [1935] Gesammelte Abhandlungen. Vol. 3, Springer-Verlag, Berlin.Google Scholar
Hilbert, D., and Bernays, P. [1939] Grundlagen der Mathematik. Vol. II, Springer-Verlag, Berlin.Google Scholar
Kreisel, G. [1963] Stanford report on the foundations of analysis (with contributions also by Howard, W. A. and Tait, W. W.), Stanford University, Stanford, California.Google Scholar
Kreisel, G. [1968] A survey of proof theory, this Journal, vol. 33, pp. 321–388.Google Scholar
Kronecker, L. [1887] Über den Zahlbegriff, Werke. Vol. III, Part 1, Teubner, Leipzig, 1899, pp. 251–274.Google Scholar
Kronecker, L. [1901] Vorlesungen zur Zahlentheorie (Hensel, K., editor), Teubner, Leipzig.Google Scholar
Purkert, W., and Ilgauds, H. J. [1987] Georg Cantor, 1845–1918, Birkhäuser, Basel, 1987.Google Scholar
Reid, C. [1970] Hilbert, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Sieg, W. [1984] Foundations for analysis and proof theory, Synthese, vol. 60, pp. 159–200.CrossRefGoogle Scholar
Sieg, W. [1985] Reductions of theories for analysis, Foundations of logic and linguistics (Papers from the seventh international congress of logic, methodology and philosophy of science, Salzburg, 1983; Dorn, G. and Weingartner, P., editors), Plenum Press, New York, pp. 199–230.Google Scholar
Smoryński, C. [1977] The incompleteness theorems, Handbook of mathematical logic (Barwise, J., editor), North-Holland, Amsterdam, pp. 821–866.CrossRefGoogle Scholar
Van Heijennoort, J. (editor) [1967] From Frege to Gödel: a source-book in mathematical logic, 1879–1931, Harvard University Press, Cambridge, Mass. Google Scholar
Weyl, H. [1918] Das Kontinuum, Verlag von Veit, Leipzig.CrossRefGoogle Scholar