Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-17T23:10:22.594Z Has data issue: false hasContentIssue false
  • English
  • Français

Gödel's Program Revisited Part I: The Turn to Phenomenology

Published online by Cambridge University Press:  15 January 2014

Kai Hauser
Kai Hauser*
Affiliation:
Icrea (Institució Catalana de Recerca I Estudis Avançats) and, Universitat de Barcelona, Spain Departament de Lògica, Història I Filosofia de La Ciència, Facultat de Filosofia, C. Montalegre, 6, 08001 Barcelona, Spain Institut Für Mathematik, Technische Universität, MA 8-1, 10623 Berlin, GermanyE-mail: hauser@math.tu-berlin.de
Get access Rights & Permissions [Opens in a new window]

Abstract

Convinced that the classically undecidable problems of mathematics possess determinate truth values, Gödel issued a programmatic call to search for new axioms for their solution. The platonism underlying his belief in the determinateness of those questions in combination with his conception of intuition as a kind of perception have struck many of his readers as highly problematic. Following Gödel's own suggestion, this article explores ideas from phenomenology to specify a meaning for his mathematical realism that allows for a defensible epistemology.

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 12 , Issue 4 , December 2006 , pp. 529 - 590
Copyright
Copyright © Association for Symbolic Logic 2006

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

[1] Bagaria, J., Natural axioms of set theory and the Continuum Problem, Logic, Methodology and Philosophy of Science. Proceedings of the Twelfth International Congress (Hájek, P. et al., editors), King's College Publications, London, 2005, pp. 43–64.Google Scholar
[2] Benacerraf, P., Mathematical truth, Journal of Philosophy, vol. 70 (1973), pp. 661–679.Google Scholar
[3] Cairns, D., Guide for translating Husserl, Nijhoff, The Hague, 1973.Google Scholar
[4] Cantor, G., Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Über unendliche lineare Punktmannigfaltigkeiten, Nr. 5), Mathematische Annalen, vol. 21 (1883), pp. 545–586, Page numbers cited in the text refer to [6], pp. 165–208.Google Scholar
[5] Cantor, G. Beiträge zur Begründung der transfiniten Mengenlehre, Mathematische Annalen, vol. 46 (1895), pp. 481–512, and vol. 49 (1897), pp. 207–246. Page numbers cited in the text refer to [6], pp. 282–351.Google Scholar
[6] Cantor, G. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts (Zermelo, E., editor), Julius Springer, Berlin, 1932, reprographic reproduction, Georg Olms Verlagsbuchhandlung, Hildesheim, 1962.Google Scholar
[7] Church, A., A formulation of the logic of sense and denotation (abstract), The Journal of Symbolic Logic, vol. 12 (1946), p. 31.Google Scholar
[8] Church, A. A formulation of the logic of sense and denotation, Structure, method and meaning (Henle, Paul et al., editors), The Liberal Arts Press, New York, 1951, pp. 3–24.Google Scholar
[9] Church, A. Introduction to mathematical logic, Princeton University Press, Princeton, N.J., 1956.Google Scholar
[10] Church, A. A revised formulation of the logic of sense and denotation, Alternative (1), Noûs, vol. 27 (1993), pp. 141–157.Google Scholar
[11] Davis, M., What did Gödel believe and when did he believe it?, this Bulletin, vol. 11 (2005), pp. 194–206.Google Scholar
[12] Feferman, S., Does mathematics need new axioms?, American Mathematical Monthly, vol. 106 (1999), pp. 99–111.Google Scholar
[13] Feferman, S. et al.(Editors), Kurt Gödel collected works, vol. I, Oxford University Press, New York, 1986.Google Scholar
[14] Feferman, S. et al.(Editors), Kurt Gödel collected works, vol. II, Oxford University Press, New York, 1990.Google Scholar
[15] Feferman, S. et al.(Editors), Kurt Gödel collected works, vol. III, Oxford University Press, New York, 1995.Google Scholar
[16] Feferman, S. et al.(Editors), Kurt Gödel collected works, vol. IV, Oxford University Press, New York, 2003.Google Scholar
[17] Feferman, S. et al.(Editors), Kurt Gödel collected works, vol. V, Oxford University Press, New York, 2003.Google Scholar
[18] Feferman, S., Friedman, H., Maddy, P., and Steel, J. R., Does mathematics need new axioms?, this Bulletin, vol. 6 (2000), pp. 401–446, edited and expanded contributions to a panel discussion held at the ASL annual meeting in June 2000.Google Scholar
[19] Fodor, J. A., The elm and the expert: Mentalese and its semantics, MIT Press, Cambridge, MA, 1994.Google Scholar
[20] Føllesdal, D., Husserl's notion of the noema, Journal of Philosophy, vol. 66 (1969), pp. 680–687.Google Scholar
[21] Føllesdal, D. Brentano and Husserl on intentional objects and perception, Grazer philosophische Studien, vol. 5 (1978), pp. 83–94.Google Scholar
[22] Føllesdal, D. Husserl on evidence and justification, Edmund Husserl and the phenomenological tradition (Sokolowski, R., editor), 1988, pp. 107–129.Google Scholar
[23] Frege, G., Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Nebert, Halle, 1879, English translation by Bauer-Mengelberg, St. in [100, 1–182].Google Scholar
[24] Friedman, H., Finite functions and the necessary use of large cardinals, Annals of Mathematics, vol. 148 (1998), no. 3, pp. 803–893.Google Scholar
[25] Gödel, K., The present situation in the foundations of mathematics, 1933, lecture at a meeting of the Mathematical Association of America, reprinted in [15], pp. 45–53.Google Scholar
[26] Gödel, K. The consistency of the axiom of choice and the generalized continuum hypothesis, Proceedings of the National Academy of Sciences, U.S.A., vol. 24 (1938), pp. 556–557, reprinted in [14], pp. 26-27.CrossRefGoogle ScholarPubMed
[27] Gödel, K. The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory, Annals of Mathematics Studies, vol. 3, Princeton University Press, Princeton, 1940, reprinted in [14], pp. 33–101.Google Scholar
[28] Gödel, K. Russell's mathematical logic, The philosophy of Bertrand Russell (Schilpp, P. A., editor), Northwestern University, Evanston, IL, 1944, pp. 123–153, reprintedin[14], pp. 119–141.Google Scholar
[29] Gödel, K. Some observations about the relationship between the theory of relativity and Kantian philosophy, 1946, in [15], pp. 230–246.Google Scholar
[30] Gödel, K. What is Cantor's Continuum Problem?, American Mathematical Monthly, vol. 54 (1947), pp. 515–525, reprinted in [14], pp. 176-187.Google Scholar
[31] Gödel, K. Some basic theorems on the foundations of mathematics and their implications, 1951, in [15], pp. 304–323.Google Scholar
[32] Gödel, K. Is mathematics syntax of language?, 1953, in [15], pp. 334–362.Google Scholar
[33] Gödel, K. The modern development of the foundations of mathematics in the light of philosophy, 1961, pp. 374–386, transcription of shorthand draft, with translation by Köhler, Eckehart and Wang, Hao, revised by J. Dawson, C. Parsons, and William Craig, in [15].Google Scholar
[34] Gödel, K. What is Cantor's Continuum Problem?, Philosophy of mathematics: Selected readings (Benacerraf, P. and Putnam, H., editors), Cambridge University Press, Cambridge, 1964, revised and expanded version of [30], reprinted in [14], pp. 254–270.Google Scholar
[35] Gödel, K. Some remarks on the undecidability results, 1972, in [14], pp. 305–306.Google Scholar
[36] Gödel, K. Notes from Husserl's Ideen on loose sheets found in the book, contained in Gödel's Nachlass, document #050100 transcribed by Cheryl A. Dawson.Google Scholar
[37] Gödel, K. Notes from Husserl's Ideen, contained in Gödel's Nachlass, document #050101 transcribed by Cheryl A. Dawson.Google Scholar
[38] Gödel, K. Shorthand in Husserl's Ideen, contained in Gödel's Nachlass, document ? transcribed by Cheryl A. Dawson.Google Scholar
[39] Gödel, K. Notes on the Logische Untersuchungen, contained in Gödel's Nachlass, document #060456 transcribed by Cheryl A. Dawson.Google Scholar
[40] Hanf, W. P. and Scott, D. S., Classifying inaccessible cardinals (abstract), Notices of the American Mathematical Society, vol. 8 (1961), p. 445.Google Scholar
[41] Hausdorff, F., Grundzüge einer Theorie der geordneten Mengen, Mathematische Annalen, vol. 65 (1908), pp. 435–505.CrossRefGoogle Scholar
[42] Hauser, K., Indescribable cardinals and elementary embeddings, The Journal of Symbolic Logic, vol. 56 (1991), pp. 439–457.Google Scholar
[43] Hauser, K. Objectivity over objects: A case study in theory formation, Synthese, vol. 128 (2001), no. 3, pp. 245–285.Google Scholar
[44] Hauser, K. Is Cantor's Continuum Problem inherently vague?, Philosophia Mathematica, vol. 10 (2002), pp. 257–285.Google Scholar
[45] Hauser, K. Is choice self-evident?, American Philosophical Quarterly, vol. 42 (2005), no. 4, pp. 237–261.Google Scholar
[46] Heidegger, M., Kant und das Problem der Metaphysik, 1929, 4th edition, V. Klostermann, Frankfurt a. M., 1973.Google Scholar
[47] Husserl, E., Über den Begriff der Zahl. Psychologische Studien, Habilitationschschrift, Halle, 1887, reprinted in Husserliana XII, Martinus Nijhoff, The Hague, 1970, pp. 289–338.Google Scholar
[48] Husserl, E. Philosophie der Arithmetik. Logische und psychologische Untersuchungen, C. E. M. Pfeffer (Robert Stricker), Halle a. d. S., 1891, reprinted in Husserliana XII, Martinus Nijhoff, The Hague, 1970.Google Scholar
[49] Husserl, E. Psychologische Studien zur Elementaren Logik, Philosophische Monatshefte, vol. 30 (1894), pp. 159–191, reprinted in Husserliana XXII, Martinus Nijhoff, The Hague, 1979, pp. 92–123.Google Scholar
[50] Husserl, E. Logische Untersuchungen, Erster Teil, Prolegomena zur reinen Logik, 1 ed., Niemeyer, Halle, 1900.Google Scholar
[51] Husserl, E. Logische Untersuchungen, Zweiter Band. I. Teil. Untersuchungen zur Phänomenologie und Theorie der Erkenntnis, 1901, 2nd edition, Niemeyer, Halle, 1913.Google Scholar
[52] Husserl, E. Logische Untersuchungen, Zweiter Band. II. Teil. Elemente einer phänomenologischen Aufklärung der Erkenntnis, 1901, 4th edition, Niemeyer, Tübingen, 1968.Google Scholar
[53] Husserl, E., Die Idee der Phänomenologie: Fünf Vorlesungen, 1907, Felix Meiner Verlag, Hamburg, 1986.Google Scholar
[54] Husserl, E. Philosophie als strenge Wissenschaft, Logos, vol. 1 (1911), pp. 289–341, I. C. B. Mohr.Google Scholar
[55] Husserl, E. Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie, Erstes Buch, Husserliana III, 1913, Nijhoff, The Hague, 1950.Google Scholar
[56] Husserl, E. Phänomenologische Psychologie, 1925, paperback edition, Felix Meiner Verlag, Hamburg, 2003.Google Scholar
[57] Husserl, E. Draft D of phenomenology, 1927, (article subsequently published in Encyclopaedia Britannica, 14th edition, London (1929), pp. 699–702), revised translation by Richard Palmer, available at http://www.hfu.edu.tw/~huangkm/phenom/husserl-britanica.htm.Google Scholar
[58] Husserl, E. Formale und transzendentale Logik. Versuch einer Kritik der logischen Vernunft, Jahrbuch für Philosophie und phanomenologische Forschung, vol. 10 (1929).Google Scholar
[59] Husserl, E. Cartesianische Meditationen, 1929, Felix Meiner Verlag, Hamburg, 1977.Google Scholar
[60] Husserl, E. Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie, Drittes Buch (Biemel, M., editor), Husserliana V, Nijhoff, The Hague, 1952.Google Scholar
[61] Husserl, E. Erfahrung und Urteil. Untersuchungen zur Genealogie der Logik, Ciaassen, Hamburg, 1964.Google Scholar
[62] Husserl, E. Analysen zur passiven Synthesis. Aus Vorlesungs- und Forschungsmanuskripten, 1918–26 (Fleischer, M., editor), Husserliana XI, Nijhoff, The Hague, 1966.Google Scholar
[63] Jensen, R. B., Inner models and large cardinals, this Bulletin, vol. 1 (1995), pp. 393–407.Google Scholar
[64] Kanamori, A., The higher infinite, second ed., Springer–Verlag, Berlin, Heidelberg, New York, 2003.Google Scholar
[65] Kant, I., Kritik der reinen Vernunft, Felix Meiner Verlag, Hamburg, 1998, Numbers preceded by ‘A’ and ‘B’ are references to pages in the first and second editions of 1781 and 1787 respectively.Google Scholar
[66] Kern, Iso, Husserl and Kant. Eine Untersuchung über Husserls Verhältnis zu Kant und zum Neukantianismus, Nijhoff, The Hague, 1964.Google Scholar
[67] Kunen, K., Elementary embeddings and infinitary combinatorics, The Journal of Symbolic Logic, vol. 36 (1971), pp. 407–413.Google Scholar
[68] Leibniz, G. W., Die philosophischen Schriften von G. W. Leibniz (Gerhardt, C. I., editor), Weidmann, Berlin, 1875–1890.Google Scholar
[69] Leibniz, G. W. Leibniz: The monadology and other philosophical writings, Oxford, 1898, translated by Latta, Robert.Google Scholar
[70] Levy, A., Axiom schemata of strong infinity in axiomatic set theory, Pacific Journal of Mathematics, vol. 10 (1960), pp. 223–238.Google Scholar
[71] Levy, A. and Solovay, R. M., Measurable cardinals and the continuum hypothesis, Israel Journal of Mathematics, vol. 5 (1967), pp. 234–248.Google Scholar
[72] Locke, J., An essay concerning human understanding (Nidditch, P. H., editor), Oxford University Press, Oxford, 1975.Google Scholar
[73] Maddy, P., Realism in mathematics, Oxford University Press, Oxford, 1990.Google Scholar
[74] Mahlo, P., Über lineare transfinite Mengen, Berichte über die Verhandlungen der königlichen Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematische-Physische Klasse, vol. 63 (1911), pp. 187–225.Google Scholar
[75] Mahlo, P. Zur Theorie und Anwendung der ρo-Zahlen, Berichte über die Verhandlungen der königlichen Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematische-Physische Klasse, vol. 64–65 (1912/1913), pp. 108–112 and 268–282.Google Scholar
[76] Martin, D. A., Mathematical evidence, Truth in mathematics (Dales, H. G. and Olivieri, G., editors), Clarendon Press, Oxford, 1998, pp. 215–231.Google Scholar
[77] Martin, D. A. Gödel's conceptual realism, this Bulletin, vol. 11 (2005), pp. 207–224.Google Scholar
[78] Martin, D. A. and Steel, J. R., A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–125.Google Scholar
[79] Merleau-Ponty, M., Phénoménologie de la Perception, Gallimard, Paris, 1945.Google Scholar
[80] Moschovakis, Y. N., Descriptive set theory, North Holland, Amsterdam, 1980.Google Scholar
[81] Mycielski, J. and Steinhaus, H., A mathematical axiom contradicting the axiom of choice, Bulletin de l'Academie Polonaise des Sciences, Serie des Sciences Mathematiques, Astronomiques et Physiques, vol. 10 (1962), pp. 1–3.Google Scholar
[82] Parsons, Ch., Platonism and mathematical intuition in Kurt Gödel's thought, this Bulletin, vol. 1 (1995), pp. 44–74.Google Scholar
[83] Parsons, Ch. Hao Wang as philosopher and interpreter of Gödel, Philosophia Mathematica, vol. 6 (1998), pp. 3–24.Google Scholar
[84] Parsons, Ch. Reason and intuition, Synthese, vol. 125 (2000), pp. 299–315.Google Scholar
[85] Piaget, J., Einführung in die genetische Erkenntnistheorie, Suhrkamp, Frankfurt a. M., 1973.Google Scholar
[86] Reinhardt, W. N., Remarks on reflection principles, large cardinals, and elementary emdeddings, Proceedings in Symposia in Pure Mathematics (Jech, T., editor), vol. 13, Part II, 1974, pp. 189–205.Google Scholar
[87] Reinhardt, W. N. Set existence principles of Shoenfield, Ackermann, and Powell, Fundamenta Mathematiche, vol. 84 (1974), pp. 5–34.Google Scholar
[88] Reinhardt, W. N. Satisfaction definitions and axioms of infinity in a theory of properties with necessity operator, Mathematical logic in Latin America (Arruda, A. I. et al., editors), North Holland, Amsterdam, 1980, pp. 267–303.Google Scholar
[89] Ross, D., Aristotle. De anima, Clarendon, Oxford, 1961.Google Scholar
[90] Rota, G. C., Husserl, Discrete thoughts: Essays on mathematics, science andphilosophy (Kac, M., Rota, G. C., and J. T.Schwartz, editors), Birkhäuser, Boston, 1986, pp. 175–181.Google Scholar
[91] Rota, G. C. Indiscrete thoughts, Birkhäuser, Boston, 1997.Google Scholar
[92] Sierpiński, W. and Tarski, A., Sur une propriété caractéristique des nombres inaccessibles, Fundamenta Mathematicae, vol. 15 (1930), pp. 292–300.Google Scholar
[93] Simon, H. A. and Newell, A., Information processing in computer and man, American Scientist, vol. 52 (1964), pp. 281–300.Google Scholar
[94] Solovay, R. M., can be anything it ought to be, The Theory of Models, Proceedings of the 1963 International Symposium at Berkeley (Addison, J. W. et al., editors), North Holland, Amsterdam, 1965, p. 435.Google Scholar
[95] Stumpf, C., Über den psychologischen Ursprung der Raumvorstellung, Hirzel, Leipzig, 1873.Google Scholar
[96] Tait, W. W., Truth and proof: the Platonism of mathematics, Synthese, vol. 69 (1986), pp. 341–370.Google Scholar
[97] Tarski, A., Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica, vol. 1 (1936), pp. 261–405, German translation by Blaustein, L. of Projecieprawdy w jezykach nauk dedukcyjnych, Prace Towarzystwa Naukowego Waszawskiego, wydział III, no. 34 (1933).Google Scholar
[98] Tieszen, R., Husserl and the philosophy of mathematics, Cambridge companion to Husserl (Smith, B. and Smith, D., editors), Cambridge University Press, Cambridge, 1995, pp. 438–462.Google Scholar
[99] Ulam, S., Zur Masstheorie in der allgemeinen Mengenlehre, Fundamenta Mathematicae, vol. 16 (1930), pp. 140–150.Google Scholar
[100] van Heijenoort, J. (editor), From Frege to Gödel, Harvard University Press, Cambridge, 1971, 2nd printing.Google Scholar
[101] Wang, H., From mathematics to philosophy, Routledge and Kegan Paul, London, 1974.Google Scholar
[102] Wang, H. A logical journey. From Gödel to philosophy, MIT Press, Cambridge, 1996.Google Scholar
[103] Wittgenstein, L., Philosophische Untersuchungen, Suhrkamp, Frankfurt, 1982.Google Scholar
[104] Woodin, W. H., The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter, Berlin, New York, 1999.Google Scholar
[105] Woodin, W. H. The continuum hypothesis, Notices of the American Mathematical Society, (2001), pp. 567–576 and 681–690.Google Scholar
[106] Woodin, W. H., Mathias, A. R. D., and Hauser, K., The axiom of determinacy , de Gruyter Series in Logic and its Applications, to appear.Google Scholar
[107] Wynn, K., Addition and subtraction in human infants, Nature, vol. 358 (1992), pp. 749–750.Google Scholar
[108] Zermelo, E., Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta Mathematicae, vol. 16 (1930), pp. 29–47.Google Scholar