Skip to main content

Hilbert and the Problem of Clarifying the Infinite

  • Chapter
Logicism, Intuitionism, and Formalism

Part of the book series: Synthese Library ((SYLI,volume 341))

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 229.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 299.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 299.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Becker, O., 1927, Mathematische Existenz, Jahrbuch für Philosophie und phänomenologische Forschung, VIII, pp. 439–809.

    Google Scholar 

  2. Benacerraf, P. and Putnam, H., (eds.), 1983, Philosophy of Mathematics, Selected Readings, Second Edition, Cambridge University Press, Cambridge.

    Google Scholar 

  3. Bernays, P., 1930, Die Philosophie der Mathematik und die Hilbertsche Beweistheorie, Blätter für deutsche Philosophie 4, pp. 326–367. Translated as “The Philosophy of Mathematics and Hilbert’s Proof Theory,” in Mancosu 1998.

    Google Scholar 

  4. Brouwer, L.E.J., Mathematics and Logic, in Heyting 1975, pp. 72–97 [3].

    Google Scholar 

  5. Brouwer, L.E.J., 1928, Intuitionistische Betrachtungen über den Formalismus, KNAW Proceedings 31, pp. 374–379. Translated as “Intuitionist Reflections on Formalism” in Mancosu 1998.

    Google Scholar 

  6. Detlefsen, M., 1990, Brouwerian Intuitionism, Mind 99, pp. 501–534.

    Google Scholar 

  7. Detlefsen, M., 1993, The Kantian Character of Hilbert’s Formalism, in Czermak, J. (ed.), Proceedings of the 15th International Wittgenstein-Symposium, Verlag Hölder-Pichler-Temsky, Vienna, pp. 195–205.

    Google Scholar 

  8. Detlefsen, M., 1998, Mathematics, foundations of, in Routledge Encyclopedia of Philosophy, Version 1.0, London and New York.

    Google Scholar 

  9. Edwards, H., 1988, Kronecker’s Place in History, in Kitcher and Aspray 1988, pp. 139–144.

    Google Scholar 

  10. Ewald, W., 1996, From Kant to Hilbert. Readings in the Foundations of Mathematics, Oxford University Press, Oxford.

    Google Scholar 

  11. Feist, R., 2002, Weyl’s Appropriation of Husserl’s and Poincaré’s Thought, Synthese 132, pp. 273–301.

    Article  MATH  MathSciNet  Google Scholar 

  12. Hallett, M., 1995, Hilbert and Logic, in M. Marion and R.S. Cohen, (eds.), Québec Studies in the Philosophy of Science, Part I, Kluwer, Dordrecht, pp. 135–187.

    Google Scholar 

  13. Heyting, A. (ed.), 1975, Brouwer Collected Works I, North-Holland, Amsterdam.

    Google Scholar 

  14. Hilbert, D., 1918, Axiomatisches Denken, Mathematische Annalen 78, pp. 405–415. English translation as “Axiomatic Thought” in Ewald 1996 [10].

    Article  MathSciNet  Google Scholar 

  15. Hilbert, D., 1922, Neubegründung der Mathematik. Erste Mitteilung, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 1, pp. 157–177. English translation as “The New Grounding of Mathematics, First Report,” in Mancosu 1998 pp. 198–214.

    Article  Google Scholar 

  16. Hilbert, D., 1926, Über das Unendliche, Mathematische Annalen 95, pp. 161–190. English translation as “On the Infinite” in Benacerraf, P. and Putnam, H., 1983 [2].

    Article  MathSciNet  Google Scholar 

  17. Hilbert, D., 1928, Die Grundlagen der Mathematik, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 6, pp. 65–85. Translated as “The Foundations of Mathematics” in Van Heijenoort 1967.

    Article  MATH  Google Scholar 

  18. Hilbert, D., 1930, Naturerkennen und Logik, in Hilbert: Gesammelte Adhandlungen, Dritter Band, (New York: Chelsea Publishing Company, 1965). English translation as “Logic and the Knowledge of Nature” in Ewald 1996.

    Google Scholar 

  19. Hilbert, D., 1931, Die Grundlegung der elementaren Zahlentheorie, Mathematische Annalen 104, pp. 485–494. Translated as “The Grounding of Elementary Number Theory,” in Mancosu 1998, pp. 266–273.

    Article  MathSciNet  Google Scholar 

  20. Kant, I., Critique of Pure Reason, translated by N.K. Smith, MacMillan, London, 1973.

    Google Scholar 

  21. Kitcher, P. and Aspray, W., (eds.), 1988, History and Philosophy of Modern Mathematics, Minnesota Studies in the Philosophy of Science, vol. XI.

    Google Scholar 

  22. Kitcher, P. 1976, Hilbert’s Epistemology, Philosophy of Science 43, pp. 99–115.

    Article  MathSciNet  Google Scholar 

  23. Klein, F., 1911, The Evanston Colloquium lectures on mathematics, Macmillan, New York. Partially reprinted in Ewald 1996, pp. 957–971 [10].

    Google Scholar 

  24. Mancosu, P., 1998, From Brouwer to Hilbert, The Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, Oxford.

    Google Scholar 

  25. Marion, M., 1995, Kronecker’s ‘Safe Haven of Real Mathematics’, Quebec Studies in the Philosophy of Science, Part I, M. Marion and R.S. Cohen, (eds.), Kluwer, Dordrecht, 1995, pp. 189–215.

    Google Scholar 

  26. Marion, M., 1995b, Wittgenstein and Ramsey on Identity, Essays on the Development of the Foundations of Mathematics, J. Hintikka (ed.), Kluwer, Dordrecht, 1995, pp. 343–371.

    Google Scholar 

  27. Poincaré. A., 1900, Du rôle de l’intuition et de la logique en mathématique. In Compte rendu du Deuxiéme congrés international des mathematiciens tenu é Paris du 6 au 12 août 1900, Gauthier-Villar, Paris, pp. 115–130. English translation as “Intuition and Logic in Mathematics” in Ewald 1996 pp. 1012–1020 [10].

    Google Scholar 

  28. Poincaré, A., 1908, Science et Méthode, Paris, Flammarion. English translation as “Science and Method”, Dover, New York, 1952.

    Google Scholar 

  29. Raatikainen, P., 2003, Hilbert’s Program Revisited, Synthese 137, 157–177.

    Article  MathSciNet  Google Scholar 

  30. Reid, C., 1970, Hilbert, Springer, New York.

    MATH  Google Scholar 

  31. Sieg, W., 1999, Hilbert’s Programs: 1917–1922, The Bulletin of Symbolic Logic, Vol. 5.

    MathSciNet  Google Scholar 

  32. Sieg, W., 2002, Beyond Hilbert’s Reach?, in D.B. Malament, Reading Natural Philosophy, Open Court, Chicago, pp. 363–405.

    Google Scholar 

  33. Stenlund, S., 1996, Poincaré and the Limits of Formal Logic, in J.-L. Greffe, G. Heinzmann, K. Lotenz (eds.), Henri Poincaré, Science and Philosophy, International Congress, Nancy, France, 1994, Akademie Verlag, Berlin, Albert Blanchard, Paris, pp. 467–479.

    Google Scholar 

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

    MATH  Google Scholar 

  35. Weyl, H., 1927, Comments on Hilbert’s second lecture on the foundations of mathematics, in Van Heijenoort 1967, pp. 480–484 [34].

    Google Scholar 

  36. Wittgenstein, L., 1980, Wittgenstein’s lectures, Cambridge 1930–1932, from the notes of J. King and D. Lee, Blackwell, Oxford.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Stenlund, S. (2009). Hilbert and the Problem of Clarifying the Infinite. In: Lindström, S., Palmgren, E., Segerberg, K., Stoltenberg-Hansen, V. (eds) Logicism, Intuitionism, and Formalism. Synthese Library, vol 341. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8926-8_20

Download citation

Publish with us

Policies and ethics