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Beyond Hilbert’s Reach?

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Logicism, Intuitionism, and Formalism

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

Abstract

Work in the foundations of mathematics should provide systematic frameworks for important parts of the practice of mathematics, and the frameworks should be grounded in conceptual analyses that reflect central aspects of mathematical experience. The Hilbert School of the 1920s used suitable frameworks to formalize (parts of) mathematics and provided conceptual analyses. However, its analyses were mostly restricted to finitist mathematics, the programmatic basis for proving the consistency of frameworks and, thus, their instrumental usefulness. Is the broader foundational quest beyond Hilbert’s reach? The answer to this question seems simple: “Yes & No”. It is “Yes”, if we focus exclusively on Hilbert’s finitism; it is “No”, if we take into account the more sweeping scope of Hilbert and Bernays’s foundational thinking. The evident limitations of Hilbert’s “formalism” have been pointed out all too frequently; in contrast, I will trace connections of Hilbert’s work, beginning in the late 19th century, to contemporary work in mathematical logic. Bernays’s reflective philosophical investigations play a significant role in reinforcing these connections. My paper pursues two complementary goals, namely, to describe a global, integrating perspective for foundational work and to formulate some more local, focused problems for mathematical work.

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References

  1. Ackermann, W., 1924 Begründung des “tertium non datur” mittels der Hilbertschen Theorie der Widerspruchsfreiheit; Mathematische Annalen 93, 1–36.

    Google Scholar 

  2. Aczel, P., 1977 An introduction to inductive definitions; in: Handbook of Mathematical Logic, J. Barwise (ed.), Amsterdam, 739–782.

    Google Scholar 

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

    Google Scholar 

  4. Bernays, P., 1922 Über Hilberts Gedanken zur Grundlegung der Mathematik; Jahresberichte DMV 31, 10–19.

    Google Scholar 

  5. Bernays, P., 1928 Über Nelsons Stellungnahme in der Philosophie der Mathematik; Die Naturwissenschaften, 16 (9), 142–145.

    Article  Google Scholar 

  6. Bernays, P., 1928a Die Grundbegriffe der reinen Geometrie in ihrem Verhältnis zur Anschauung; Die Naturwissenschaften, 16 (12), 197–203.

    Google Scholar 

  7. Bernays, P., 1930 Die Philosophie der Mathematik und die Hilbertsche Beweistheorie; in: (Bernays 1976), 17–61.

    Google Scholar 

  8. Bernays, P., 1930a Die Grundgedanken der Fries’schen Philosophie in ihrem Verhältnis zum heutigen Stand der Wissenschaft; Abhandlungen der Fries’schen Schule, Neue Folge, vol. 5 (2), 97–113. (Based on a talk presented on August 10, 1928.)

    Google Scholar 

  9. Bernays, P., 1934 Über den Platonismus in der Mathematik; in: (Bernays 1976), 62–78.

    Google Scholar 

  10. Bernays, P., 1935 Hilberts Untersuchungen über die Grundlagen der Arithmetik; in: (Hilbert 1935), 196–216.

    Google Scholar 

  11. Bernays, P., 1937 Grundsätzliche Betrachtungen zur Erkenntnistheorie; Abhandlungen der Fries’schen Schule, Neue Folge, vol. 6 (3–4), 278–290.

    Google Scholar 

  12. Bernays, P., 1950 Mathematische Existenz und Widerspruchsfreiheit; in: (Bernays 1976), 92–106.

    Google Scholar 

  13. Bernays, P., 1954 Zur Beurteilung der Situation in der beweistheoretischen Forschung; Revue internationale de philosophie 8, 9–13; Dicussion, 15–21.

    MathSciNet  Google Scholar 

  14. Bernays, P., 1970 Die schematische Korrespondenz und die idealisierten Stukturen; in: (Bernays 1976), 176–188.

    Google Scholar 

  15. Bernays, P., 1976 Abhandlungen zur Philosophie der Mathematik; Wissenschaftliche Buchgesellschaft, Darmstadt.

    MATH  Google Scholar 

  16. Brouwer, L.E.J., 1927 Über Definitionsbereiche von Funktionen; Mathematische Annalen 97, 60–75; translated in: van Heijenoort, 446–463.

    Google Scholar 

  17. Brouwer, L.E.J., 1927a Intuitionistische Betrachtungen über den Formalismus; Koninklijke Akademie van wetenschappen te Amsterdam, Proceedings of the section of sciences, 31, 374–379; translated in: (van Heijenoort), 490–492.

    Google Scholar 

  18. Brouwer, L.E.J., 1953 Points and spaces; Canadian Journal of Mathematics 6, 1–17.

    MathSciNet  Google Scholar 

  19. Buchholz, W., 2000 Relating ordinals to proofs in a more perspicuous way; to appear in: (size e.a., 2002), 37–59.

    Google Scholar 

  20. Buchholz, W., Feferman, S., Pohlers, W., and Sieg, W., 1981 Iterated inductive definitions and subsystems of analysis: recent prof-theoretical studies; Lecture Notes in Mathematics, Springer Verlag.

    Google Scholar 

  21. Carnap, R., 1930 Die Mathematik als Zweig der Logik; Blätter für Deutsche Philosophie 4, 298–310.

    MathSciNet  Google Scholar 

  22. Church, A. and Kleene, S., 1936 Formal definitions in the theory of ordinal numbers; Fundamenta Mathematicae 28, 11–21.

    Google Scholar 

  23. Dedekind, R., 1872 Stetigkeit und irrationale Zahlen; in: (Dedekind 1932), pp. 315–324.

    Google Scholar 

  24. Dedekind, R., 1877 Sur la théorie des nombres entiers algébriques; Bulletin des Sciences mathématiques et astronomiques, pp. 1–121; partially reprinted in: (Dedekind 1932), pp. 262–96.

    Google Scholar 

  25. Dedekind, R., 1888 Was sind und was sollen die Zahlen; in: (Dedekind 1932), pp. 335–391.

    Google Scholar 

  26. Dedekind, R., 1890 Letter to Keferstein; in: (van Heijenoort), pp. 98–103.

    Google Scholar 

  27. Dedekind, R., 1932 Gesammelte mathematische Werke, Dritter Band; R. Fricke, E. Noether, and ö. Ore (eds.); Vieweg, Braunschweig.

    Google Scholar 

  28. Ewald, W. (ed.), 1996 From Kant to Hilbert - A source book in the foundations of mathematics; two volumes; Oxford University Press.

    Google Scholar 

  29. Feferman, S., 1981 How we got from there to here; in: (Buchholz e.a.), 1–15.

    Google Scholar 

  30. Feferman, S., 1982 Inductively presented systems and the formalization of meta-mathematics; in: Logic Colloquium ‘80, North Holland Publishing Company, 95–128.

    Google Scholar 

  31. Feferman, S. and Sieg, W., 1981 Iterated inductive definitions and subsystems of analysis; in: (Buchholz e.a.), 16–77.

    Google Scholar 

  32. Fraenkel, A., 1930 Die heutigen Gegensätze in der Grundlegung der Mathematik; Erkenntnis 1, 286–302.

    Article  MATH  Google Scholar 

  33. Gödel, K., 1933 The present situation in the foundations of mathematics; in: Collected Works III, 36–53.

    Google Scholar 

  34. Gödel, K., 1938 Vortrag by Zilsel; in: Collected Works III, 86–113.

    Google Scholar 

  35. Gödel, K., 1986 Collected Works I; Oxford University Press, Oxford, New York.

    MATH  Google Scholar 

  36. Gödel, K., 1990 Collected Works II; Oxford University Press, Oxford, New York.

    MATH  Google Scholar 

  37. Gödel, K., 1995 Collected Works III; Oxford University Press, Oxford, New York.

    MATH  Google Scholar 

  38. Hallett, M., 1994 Hilbert’s axiomatic method and the laws of thought; in: Mathematics and Mind, A. George (ed.), Oxford University Press, 158–200.

    Google Scholar 

  39. Hallett, M.,1995 Hilbert and logic; in: Québec Studies in the Philosophy of Science I, M. Marion and R.S. Cohen (eds.) Kluwer, Dordrecht, 135–187.

    Google Scholar 

  40. Hand, M., 1989 A number in the exponent of an operation; Synthese 81, 243–65.

    Article  MathSciNet  Google Scholar 

  41. Hand, M., 1990 Hilbert’s iterativistic tendencies; History and Philosophy of Logic 11, 185–92.

    Article  MATH  MathSciNet  Google Scholar 

  42. Hilbert, D., 1900 Über den Zahlbegriff; Jahresbericht der DMV 8, 180–94; reprinted in: Grundlagen der Geometrie, 3. Auflage, Leipzig 1909, 256–62.

    Google Scholar 

  43. Hilbert, D., 1901 Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Kongress zu Paris 1900; Archiv der Mathematik und Physik, 3rd series, 1, 44–63, 213–237.

    Google Scholar 

  44. Hilbert, D., 1905 Über die Grundlagen der Logik und Arithmetik; in: (Hilbert 1900a), 243–258; translated in: (van Heijenoort), 129–138.

    Google Scholar 

  45. Hilbert, D., 1922 Neubegründung der Mathematik; Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 1, 157–177.

    Article  Google Scholar 

  46. Hilbert, D., 1923 Die logischen Grundlagen der Mathematik; Mathematische Annalen 88, 151–165.

    Article  MathSciNet  Google Scholar 

  47. Hilbert, D., 1925 Über das Unendliche; Mathematische Annalen 95, 1926, 161–190; translated in: (van Heijenoort), 367–392.

    Google Scholar 

  48. Hilbert, D., 1928 Die Grundlagen der Mathematik; Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 6 (1/2), 65–85.

    Article  MATH  Google Scholar 

  49. Hilbert, D., 1935 Gesammelte Abhandlungen; Vol. 3; Springer, Berlin.

    Google Scholar 

  50. Hilbert, D. and Bernays, P., 1934 Grundlagen der Mathematik; vol. I, Springer, Verlag.

    Google Scholar 

  51. Hilbert, D. and Bernays, P., 1939 Grundlagen der Mathematik; vol. II, Springer, Verlag.

    Google Scholar 

  52. Jäger, G., 1986 Theories for admissible sets – a unifying approach to proof theory; Bibliopolis, Naples.

    Google Scholar 

  53. Joyal, A. and Moerdijk, I., 1995 Algebraic Set Theory; London Mathematical Society Lecture Notes Series 220; Cambridge University Press.

    Google Scholar 

  54. Kino, A., Myhill, J., and Vesley, R.E. (eds.), 1970 Intuitionism and Proof Theory; Proceedings of the summer conference at Buffalo, N.Y., 1968.

    Google Scholar 

  55. Mancosu, P., 1999 Between, Russell, and Hilbert: Behmann on the foundations of mathematics; Bulletin of Symbolic Logic 5 (3), 303–330.

    Article  MATH  MathSciNet  Google Scholar 

  56. Moerdijk, I. and Palmgren, E., 2000 Well–founded trees in categories; Annals of Pure and Applied Logic 104, 189–218.

    Article  MATH  MathSciNet  Google Scholar 

  57. Moerdijk, I. and Palmgren, E., 2002 Type theories, toposes and constructive set theory: predicative aspects of AST; Annals of Pure and Applied Logic 114, 155–201.

    Article  MATH  MathSciNet  Google Scholar 

  58. Paetzold, H., 1995 Ernst Cassirer – Von Marburg nach New York; Wissenschaftliche Buchgesellschaft, Darmstadt.

    Google Scholar 

  59. Parsons, C.D., 1980 Mathematical intuition; Proc. Aristotelian Society N.S. 80 (1979–80), 145–168.

    Google Scholar 

  60. Parsons, C.D., 1982 Objects and logic; The Monist 65 (4), 491–516.

    MathSciNet  Google Scholar 

  61. Parsons, C.D., 1984 Arithmetic and the categories; Topoi 3 (2), 109–121.

    Article  MathSciNet  Google Scholar 

  62. Parsons, C.D., 1987 Developing arithmetic in set theory without infinity: Some historical remarks; History and Philosophy of Logic 8, 201–213.

    Article  MATH  MathSciNet  Google Scholar 

  63. Parsons, C.D., 1990 The structuralist view of mathematical objects; Synthese 84, 303–346.

    Article  MATH  MathSciNet  Google Scholar 

  64. Parsons, C.D., 1994 Intuition and number; in: Mathematics and Mind, A. George (ed.), Oxford University Press, 141–157.

    Google Scholar 

  65. Peckhaus, V., 1990 Hilbertprogramm und Kritische Philosophie; Vandenhoek & Ruprecht, Göttingen.

    MATH  Google Scholar 

  66. Peckhaus, V., 1994 Hilbert’s axiomatic programme and philosophy; in: The History of Modern Mathematics, vol. III (Knobloch, E. and Rowe, D.E., eds.), Academic Press, 91–112.

    Google Scholar 

  67. Peckhaus, V., 1994a Logic in transition: the logical calculi of Hilbert (1905) and Zermelo (1908); in: Logic and Philosophy of Science in Uppsala, Prawitz, D., and Westrestahl, D. (eds.), Kluwer, 311–323.

    Google Scholar 

  68. Peckhaus, V., 1995 Hilberts Logik. Von der Axiomatik zur Beweistheorie; Intern. Zs. f. Gesch. u. Ethik der Naturwiss., Techn. U. Med. 3, 65–86.

    MATH  Google Scholar 

  69. Pohlers, W., 1989 Proof Theory – an introduction; Lecture Notes in Mathematics 1407, Springer Verlag.

    Google Scholar 

  70. Poincaré, H., 1905 Les mathématiques et la logique; Revue de métaphysique et de morale, 13, 815–35; translated in (Ewald, vol. 2),

    Google Scholar 

  71. Poincaré, H., 1906 Les mathématiques et la logique; Revue de métaphysique et de morale, 14, 17–34; translated in (Ewald, vol. 2), 1038–52.

    Google Scholar 

  72. Poincaré, H., 1906a Les mathématiques et la logique; Revue de métaphysique et de morale, 14, 294–317; translated in (Ewald, vol. 2), 1052–71.

    Google Scholar 

  73. Rathjen, M., 1995 Recent advances in ordinal analysis; Bulletin of Symbolic Logic 1 (4), 468–85.

    Article  MATH  MathSciNet  Google Scholar 

  74. Shapiro, S., 1997 Philosophy of mathematics – Structure and ontology; Oxford University Press.

    Google Scholar 

  75. Sieg, W., 1977 Trees in Metamathematics; Ph.D. Thesis; Stanford.

    Google Scholar 

  76. Sieg, W., 1981 Inductive definitions, constructive ordinals, and normal derivations; in: (Buchholz e.a.), 143–187.

    Google Scholar 

  77. Sieg, W., 1984 Foundations for analysis and proof theory; Synthese 60 (2), 159–200.

    Article  MATH  MathSciNet  Google Scholar 

  78. Sieg, W., 1990 Relative consistency and accessible domains; Synthese 84, 259–97.

    MATH  MathSciNet  Google Scholar 

  79. Sieg, W., 1997 Aspects of mathematical experience; in: Philosophy of mathematics today, E. Agazzi and G. Darvas (eds.), Kluwer Academic Publishers, 195–217.

    Google Scholar 

  80. Sieg, W., 1999 Hilbert’s programs: 1917–1922; Bulletin of Symbolic Logic 5 (1), 1–44.

    Article  MATH  MathSciNet  Google Scholar 

  81. Sieg, W., Sommer, R. and Talcott, C. (eds), 2002, Reflections on the Foundations of Mathematics, Association for Symbolic Logic, A.K. Peters.

    Google Scholar 

  82. Stein, H., 1988 Logos, Logic, Logistiké: Some philosophical remarks on the 19th century transformation of mathematics; in: (Aspray and Kitcher), 238–259.

    Google Scholar 

  83. Tait, W.W., 1968 Constructive Reasoning; in: Proc. 3rd Int. Congress of Logic, Methodology, and Philosophy of Science, Amsterdam; 185–199.

    Google Scholar 

  84. Tait, W.W., 1981 Finitism; Journal of Philosophy 78, 524–546.

    Article  Google Scholar 

  85. Tait, W.W., 2000 Remarks on Finitism; in: (Sieg e.a., 2002), 410–419.

    Google Scholar 

  86. van Heijenoort, J. (ed.), 1967 From Frege to Gödel, a source book in mathematical logic, 1879–1931; Harvard University Press, Cambridge.

    Google Scholar 

  87. Weyl, H., 1925 Die heutige Erkenntnislage in der Mathematik; Symposion 1, pp. 1–32. (Reprinted in vol. 2 of Weyl’s “Gesammelte Abhandlungen”, Springer Verlag, 1968, 511–542.)

    Google Scholar 

  88. Weyl, H., 1928, Diskussionsbemerkungen zu dem zweiten Hilbertschen Vortrag über die Grundlagen der Mathematik; Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 6, pp. 86–88. (Translated in (van Heijenoort), 482–4.)

    Google Scholar 

  89. Whitehead, A.N. and Russell, B., 1910 Principia Mathematica, vol. 1; Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  90. Whitehead, A.N. and Russell, B., 1912 Principia Mathematica, vol. 2; Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  91. Whitehead, A.N. and Russell, B., 1913 Principia Mathematica, Vol. 3; Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  92. Zach, R., 1997 Numbers and Functions in Hilbert’s Finitism; Taiwanese Journal for Philosophy and History of Science 10, 33–60.

    MathSciNet  Google Scholar 

  93. Zermelo, E., 1909 Sur les ensembles finis et le principe de l’induction complète; Acta Mathematica 32, 185–193.

    Article  MathSciNet  Google Scholar 

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  1. Carnegie Mellon University, Pittsburgh, PA, USA

    Wilfried Sieg

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  1. Wilfried Sieg
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  1. Dept. Historical, Philosophical and Religious Studies, Umeå University, SE-901 87 Umeå, Sweden

    Sten Lindström

  2. Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden

    Erik Palmgren  & Viggo Stoltenberg-Hansen  & 

  3. Department of Philosophy, Uppsala University, Box 627, 751 26 Uppsala, Sweden

    Krister Segerberg

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Sieg, W. (2009). Beyond Hilbert’s Reach?. 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_19

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