Abstract
In a 1900 paper entitled “On the Number Concept”, the formalist mathematician David Hilbert proposed a set of axioms from which he hoped arithmetic might be derived. The last of these axioms was an “Axiom of Completeness” stipulating that: “It is not possible to adjoin to the system of numbers any collection of things so that in the combined collection the preceding axioms are satisfied; that is, briefly put, the numbers form a system of objects which cannot be enlarged with the preceding axioms continuing to hold.”1
First of all I must thank Mr Dagfinn Føllesdal for his thorough reading of my text and his insightful suggestions as to how to improve it.
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Notes
Hilbert’s “über den Zahlbegriff” was first published in the Jahresbericht der Deutschen Mathematiker-Vereinigung,vol. 8, 1900, pp. 180–184, and subsequently as an appendix to post-1903 editions of his Grundlagen der Geometrie. I have cited the translation of Hilbert’s axioms for arithmetic appearing in Morris Kline, Mathematical Thought from Ancient to Modern Times, New York, Oxford University Press, 1972, vol. 3, pp. 990–991.
See Edmund Husserl’s Ideas, General Introduction to Pure Phenomenology,New York: Colliers, 1962, §72 and note; Formal and Transcendental Logic, The Hague: M. Nijhoff, 1969, §§28–36; Crisis, The Hague: M. Nijhoff, 1954, §9f and note. In these texts Husserl refers back to his discussions in Logical Investigations, New York: Humanities Press,1970, vol. 1 §§69 and 70, and to then unpublished material from his Göttingen period now published in appendices to his Philosophie der Arithmetik, The Hague: M. Nijhoff, 1970, Husserliana vol. XII. As usual there are some terminological obstacles that make it hard to see the connection Husserl’s ideas have with the logical tradition most familiar to readers of English. First of all, for complete and completeness Husserl uses the German words “definit” and “definitheit” in the place of Hilbert’s “vollstandig” and “vollstandigkeit”. Since in the passages cited above Husserl maintains that his concept of definitheit is exactly the same as Hilbert’s vollstandigkeit,I have tried to avoid the terminological confusion by translating Husserl’s terms by the more familiar “complete” and “completeness” although Husserl translators have understandably chosen “definite” and “definiteness”. Second, in the above texts Husserl refers to his theory of complete Mannigfaltigkeiten, a term which has been translated by “multiplicity” or “manifold”. For Husserl complete Mannigfaltigkeiten are the objective correlates of complete axiom systems.
Suzanne Bachelard, A Study of Husserl’s Formal and Transcendental Logic, Evanston: Northwestern Press, 1968, pp. 59–61. Jean Cavaillès, Sur la logique et la théorie de science, Paris: Vrin, pp. 70–73. Roger Schmit, Husserls Philosophie der Mathematik, Bonn: Bouvier, 1981, pp. 67–86. Hans Lohmar, Husserls Phänomenologie als Philosophie der Mathematik, Dissertation, Cologne, 1987, pp. 151–162. Guillermo Rosado-Haddock, Edmund Husserls Philosophie der Logik und Mathematik im Lichte der Gegenwärtigen Logik und Grundlagenforschung, Dissertation, Bonn: Rheinischen Friedrich-Wilhelms Universität, 1973. Bernold Picker, Die Bedeutung der Mathematik für die Philosophie Edmund Husserls, Dissertation, Münster, 1955.
See for example the note to Husserl’s Ideas § 72.
Karl Schuhmann, Husserl Chronik, The Hague: Martinus Nijhoff, 1977, pp. 6–11. I also discuss Husserl’s background throughout my Word and Object in Husserl, Frege, and Russell: Roots of Twentieth Century Philosophy, Athens: Ohio University Press, 1991 and in an article entitled “Husserl and Frege on Substitutivity”, Mind, Meaning and Mathematics, L. Haaparanta ed., Dordrecht: Kluwer, 1994, pp. 113–140.
Linda Mc Alister, The Philosophy of Franz Brentano, London: Duckworth, pp. 45, 49, 53. Andrew Osborn, The Philosophy of E. Husserl in its Development to his First Conception of Phenomenology in the Logical Investigations, New York: International Press, pp. 12, 17, 18, 21. Michael Dummett, The Interpretation of Frege’s Philosophy, Cambridge, MA: Harvard University Press, pp. 72–73, and his Frege, Philosophy of Language, London: Duckworth, 2nd ed., 1981, p. 683. Hill, Word and Object…, pp. 59–67 and Chapter 7.
Adolf Fraenkel, “Georg Cantor”, Jahresbericht der deutschen Mathematiker Vereinigung, 39, pp. 221, 253n., 257. Edmund Husserl, Introduction to the Logical Investigations: A Draft of a Preface to the Logical Investigations (1913),The Hague: M. Nijhoff, p. 37 and notes. Jean Cavaillès, Philosophie Mathématique, Paris: Hermann, 1962, p. 182. Schmit, pp. 40–48, 58–62. Lothar Eley: 1970, “Einleitung des Herausgebers”, in the Husserliana Philosophie der Arithmetik, pp. XXIII—XXV. Georg Cantor Briefe, ed. by Herbert Meschkowski, New York: Springer, pp. 321, 373–374, 379–380, 423–424. Two Cantor letters dating from 1895 are published in Walter Purkert and Hans Ilgauds, Georg Cantor 1845–1918, Basel: Birkhäuser, 1991, pp. 206–207.
Kline, vol. 3, pp. 950–956, 960–966. Mc Alister, p. 49; Osborne, p. 18; Husserl, Introduction…, p. 37.
Hao Wang, From Mathematics to Philosophy, London: Routledge and Kegan Paul, 1974, pp. 145–152 in reference to Bernard Bolzano’s 1837 Wissenschaftlehre §§148 and 155. Bolzano’s book has been partially translated as Theory of Science by R. George, Oxford: Blackwell, 1972, and B. Terrell, Dordrecht: Reidel, 1973.
See Husserl texts cited in note 1.
Alfred North Whitehead, An Introduction to Mathematics, Oxford, Oxford University Press, 1958 (1911), pp. 62–64.
Appendix VI to the Husserliana Philosophie der Arithmetik, p. 433 and FTL § 31. ‘3 Husserl, Introduction…, pp. 33–36.
Hill, Word and Object…,pp. 80–95.
Good accounts of Husserl’s work during the 1890s are given in the editors’ introductions to the Husserliana editions of Husserl’s Philosophie der Arithmetik,Logische Untersuchungen, and Studien zur Arithmetik und Geometrie.
Edmund Husserl, Philosophie der Arithmetik, Halle: Pfeffer, 1891 p. viii, (note this is not the Husserliana edition cited above for the posthumously published material, but Husserl’s 1891 book). Hill, Word and Object…, pp. 84–86.
Husserl, Philosophie der Arithmetik, pp. 340–429 of the Husserliana edition.
Cited in Hill, Word and Object…, p. 85. See also Dallas Willard, Logic and the Objectivity of Knowledge, Athens: Ohio University Press, 1984, pp. 115–116.
Husserl, Philosophie der Arithmetik (1891), pp. 104–105, 132–134. I discuss his arguments in depth in “Husserl and Frege on Substitutivity”.
Husserl, Logical Investigations, note p. 179. Husserl actually retracted pp. 129–132, not pp. 124–132 as a typographical error in the English edition indicates.
Husserl, Philosophie der Arithmetik (1891), pp. 130–131.
Gottlob Frege, Philosophical and Mathematical Correspondence, Oxford: Blackwell, 1979, p. 65 in reference to Frege’s article “On Formal Theories of Arithmetic” now published in Frege, Collected Papers on Mathematics,Logic and Philosophy, Oxford: Blackwell, 1984, pp. 112–112.
Frege, Collected…, pp. 118–119. Husserl’s own copy of Frege’s article is now in the Husserl library in Leuven, Belgium.
Gottlob Frege, Posthumous Writings, Oxford: Blackwell, 1979, p. 122. See also Gottlob Frege, Translations from the Philosophical Writings of Gottlob Frege, Oxford: Blackwell, 1980, pp. 22–23,32–33,162–213.
Frege, Translations..., pp. 69–70.
Frege, Posthumous…, P. 123.
David Hilbert, “On the Infinite” in From Frege to Gödel, ed. by Jean van Heijenoort, Cambridge, MA: Harvard University Press, 1967, p. 379.
Ibid., p. 383.
See the introduction to the Husserliana edition of Husserl’s Logical Investigations. 30 See the introduction to Husserliana vol. 21, Studien zur Arithmetik und Geometrie, The Hague: M. Nijhoff, 1984, p. XII where a 1901 letter from Husserl’s wife is cited.
Constance Reid, Hilbert, New York: Springer Verlag, 1970, pp. 67–68. Reid’s Hilbert and Courant in Göttingen and New York, New York: Springer Verlag, 1976, provide anecdotal material about Husserl’s time in Göttingen. Schuhmann, Husserl Chronik, p. 10, Husserl’s thesis entitled Beiträge zur Theorie der Variationsrechnung.
As Hilbert makes evident in his, “On the Infinite”, anthologized in Van Heijenoort, pp. 369–392.
Frege, Philosophical and Mathematical Correspondence, pp. 34–51, and Gottlob Freges Briefwechsel mit D. Hilbert,E. Husserl, B. Russell,Hamburg: Meiner, 1980, pp. 3, 47. Also Claire Hill, “Frege’s Letters”, pp. 97–118.
The notes for Husserl’s lecture are published as an appendix to the Husserliana edition of Philosophie der Arithmetik,pp. 430–506. I cite them in the text as GL. Concerning the invitation see Husserl’s wife’s letter cited in note 30.
Edmund Husserl, “Rezension von Palagyi”, Zeitschrift fir Psychologie und Physiologie der Sinnesorgane 31 (1903), p. 290. Translation by Dallas Willard in The Personalist 53 (Winter 1972), pp. 5–13. I cite the passage in question in Hill, Word and Object..., p. 20.
Husserl, Introduction…, pp. 36–38, 48. FTL pp. 184–185, 225.
Hilbert in Van Heijenoort anthology, p. 392.
Ibid., pp. 464–465, and 376.
Frege, Translations…, pp. 22–23, 120–121, 141n., 146n., 159–161, for example.
I discuss this at length in my Word and Object…,especially chapter 4, and in “Husserl and Frege on Substitutivity”.
Van Heijenoort, p. 437.
Husserl, Logical Investigations, pp. 293–294.
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Hill, C.O. (1995). Husserl and Hilbert on Completeness. In: Hintikka, J. (eds) From Dedekind to Gödel. Synthese Library, vol 251. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8478-4_7
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