Hilbert and the Problem of Clarifying the Infinite

  • Sören Stenlund
Part of the Synthese Library book series (SYLI, volume 341)


Classical Mathematics Proof Theory Elementary Number Theory Mathematical Part Deductive Science 
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  1. 1.
    Becker, O., 1927, Mathematische Existenz, Jahrbuch für Philosophie und phänomenologische Forschung, VIII, pp. 439–809.Google Scholar
  2. 2.
    Benacerraf, P. and Putnam, H., (eds.), 1983, Philosophy of Mathematics, Selected Readings, Second Edition, Cambridge University Press, Cambridge.Google Scholar
  3. 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. 4.
    Brouwer, L.E.J., Mathematics and Logic, in Heyting 1975, pp. 72–97 [3].Google Scholar
  5. 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. 6.
    Detlefsen, M., 1990, Brouwerian Intuitionism, Mind 99, pp. 501–534.Google Scholar
  7. 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. 8.
    Detlefsen, M., 1998, Mathematics, foundations of, in Routledge Encyclopedia of Philosophy, Version 1.0, London and New York.Google Scholar
  9. 9.
    Edwards, H., 1988, Kronecker’s Place in History, in Kitcher and Aspray 1988, pp. 139–144.Google Scholar
  10. 10.
    Ewald, W., 1996, From Kant to Hilbert. Readings in the Foundations of Mathematics, Oxford University Press, Oxford.Google Scholar
  11. 11.
    Feist, R., 2002, Weyl’s Appropriation of Husserl’s and Poincaré’s Thought, Synthese 132, pp. 273–301.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 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. 13.
    Heyting, A. (ed.), 1975, Brouwer Collected Works I, North-Holland, Amsterdam.Google Scholar
  14. 14.
    Hilbert, D., 1918, Axiomatisches Denken, Mathematische Annalen 78, pp. 405–415. English translation as “Axiomatic Thought” in Ewald 1996 [10].CrossRefMathSciNetGoogle Scholar
  15. 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.CrossRefGoogle Scholar
  16. 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].CrossRefMathSciNetGoogle Scholar
  17. 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.zbMATHCrossRefGoogle Scholar
  18. 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. 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.CrossRefMathSciNetGoogle Scholar
  20. 20.
    Kant, I., Critique of Pure Reason, translated by N.K. Smith, MacMillan, London, 1973.Google Scholar
  21. 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. 22.
    Kitcher, P. 1976, Hilbert’s Epistemology, Philosophy of Science 43, pp. 99–115.CrossRefMathSciNetGoogle Scholar
  23. 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. 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. 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. 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. 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. 28.
    Poincaré, A., 1908, Science et Méthode, Paris, Flammarion. English translation as “Science and Method”, Dover, New York, 1952.Google Scholar
  29. 29.
    Raatikainen, P., 2003, Hilbert’s Program Revisited, Synthese 137, 157–177.CrossRefMathSciNetGoogle Scholar
  30. 30.
    Reid, C., 1970, Hilbert, Springer, New York.zbMATHGoogle Scholar
  31. 31.
    Sieg, W., 1999, Hilbert’s Programs: 1917–1922, The Bulletin of Symbolic Logic, Vol. 5.MathSciNetGoogle Scholar
  32. 32.
    Sieg, W., 2002, Beyond Hilbert’s Reach?, in D.B. Malament, Reading Natural Philosophy, Open Court, Chicago, pp. 363–405.Google Scholar
  33. 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. 34.
    Van Heijenoort, J. 1967, (ed.), From Frege to Gödel: A Source-Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge MA.zbMATHGoogle Scholar
  35. 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. 36.
    Wittgenstein, L., 1980, Wittgenstein’s lectures, Cambridge 1930–1932, from the notes of J. King and D. Lee, Blackwell, Oxford.Google Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Sören Stenlund
    • 1
  1. 1.Department of PhilosophyUppsala UniversitySweden

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