Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-18T17:54:55.612Z Has data issue: false hasContentIssue false
  • English
  • Français

ON ONTOLOGY AND REALISM IN MATHEMATICS

Published online by Cambridge University Press:  21 February 2012

HAIM GAIFMAN
HAIM GAIFMAN*
Affiliation:
Philosophy Department, Columbia University
*
*PHILOSOPHY DEPARTMENT COLUMBIA UNIVERSITY NEW YORK, NY 10027 E-mail:hg17@columbia.edu
Get access Rights & Permissions [Opens in a new window]

Extract

The paper is concerned with the way in which “ontology” and “realism” are to be interpreted and applied so as to give us a deeper philosophical understanding of mathematical theories and practice. Rather than argue for or against some particular realistic position, I shall be concerned with possible coherent positions, their strengths and weaknesses. I shall also discuss related but different aspects of these problems. The terms in the title are the common thread that connects the various sections.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 5 , Issue 3 , September 2012 , pp. 480 - 512
Copyright
Copyright © Association for Symbolic Logic 2012

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

BIBLIOGRAPHY

Appel, K., & Haken, W. (1977). Every planar map is four colorable. Part I. Discharging. The Illinois Journal of Mathematics, 21, 429–490.Google Scholar
Appel, K., & Haken, W. (1989). Every Planar Map is Four Colorable. Providence, RI: American Mathematical Society.Google Scholar
Carnap, R. (1934) Logische Syntax der Sprache, Viena: Springer. Translated into English as The Logical Syntax of Language, 1937.Google Scholar
Dummett, M. (1975) Wang’s paradox. Synthese, 30, 301–324.CrossRefGoogle Scholar
Dummett, M. (1978) Truth and Other Enigmas, Harvard University Press.Google Scholar
Esenin-Volpin, A. (1961). A. Le programme ultra-intuitionniste des fondements des mathématiques. In Infinitistic Methods, Proceedings of the Symposium on the Foundations of Mathematics. Warsaw, Poland, pp. 201–223.Google Scholar
Esenin-Volpin, A. (1970). The ultra-intuitionistic criticism and the antitraditional program for the foundations of mathematics. In Kino, A., Myhill, J., and Vesley, R., editors. Intuitionism and Proof Theory, Proceedings of the Summer Conference at Buffalo, N.Y. 1968. North Holland, pp. 3–45.Google Scholar
Field, H. (1998). Which undecidable mathematical sentences have determinate truth values. In Dales, H. G., and Oliveri, G., editors. Truth in Mathematics. Oxford, UK: Clarendon Press, pp. 291–310.CrossRefGoogle Scholar
Fischer, M., & Rabin, M. (1974). Super-exponential complexity of Presburger arithmetic. In Symposium on Applied Mathematics, SIAM-AMS Proceedings. Vol. 7. pp. 27–41.Google Scholar
Franzén, T. (2006). Gödel’s theorem: An incomplete guide to its use and abuse. Text available on the internet.Google Scholar
Frege, G. (1884) Die Grundlagen der Arithmetik, English translation byAustin, J. L., Foundations of ArithmeticBlackwell & Mott, 1950, revised edition 1953.Google Scholar
Gaifman, H. (1975). Ontology and conceptual framework part I. Erkenntnis, 9, 329–353.Google Scholar
Gaifman, H. (1976). Ontology and conceptual framework part II. Erkenntnis, 10, 21–85.CrossRefGoogle Scholar
Gaifman, H. (2000). What Gödel’s incompleteness result does and does not show. The Journal of Philosophy, 462–470.Google Scholar
Gaifman, H. (2006). Naming and diagonalization, from Cantor to Gödel to Kleene. Logic Journal of the IGPL, 709–728.Google Scholar
Gödel, K. (1931) “Über formal unentscheidebarre Sätze der Principia Mathamtica und verwandler System I” Monathshefte für Mathematik und Physik, 38 pp. 173–198. English translation in Kurt Gödel’s Collected Works, vol. I.CrossRefGoogle Scholar
Gödel, K. (1934). On undecidable propositions of formal mathematical Systems. In Kurt Gödel Collected Works. Vol. 3. Unpublished essays and lectures, 1995. Oxford University Press, pp. 346–371.Google Scholar
Gödel, K. (1951). Some basic theorems on the foundations of mathematics and their implications. In Kurt Gödel Collected Works. Vol. 3. Unpublished essays and lectures, 1995. Oxford University Press, pp. 310–325.Google Scholar
Goodman, N., & Quine, W. V. (1947). Steps toward a constructive nominalism. The Journal of Symbolic Logic, 12, 105–122.CrossRefGoogle Scholar
Gontier, G. (2008). Formal proof - the four-color theorem. Notices of the AMS, 55, 1382–1393.Google Scholar
Hales, T., Harrison, J., McLaughlin, T., et al. . (2010). A revision of the proof of the Kepler conjecture. Discrete Computational Geometry, 44, 1–34.Google Scholar
Hamkins, J. (2011). The set theoretic multiverse. In this volume.Google Scholar
Hilbert, (1904) “On the foundations of logic and arithmetic”, address delivered at the third International Congress of Mathematicians, in van-Heijenoort’s anthologyFrom Frege to Gödel, 1967, Harvard University Press, pp. 129–138.Google Scholar
Hilbert, (1925) “On the infinite”, address delivered at meeting held in Münster, in memory of Weierstrass, in van-Heijenoort’s anthologyFrom Frege to Gödel, 1967, Harvard University Press, pp. 367–392.Google Scholar
Iwan, S. T. (2000). On the untenability of Nelson’s Predicativism. Synthese, 53, 147–154.Google Scholar
Matiyasevich, Y. (1970). Enumerable sets are Diophantine. Soviet Mathematics. Vol. 11, no.2, 354–357. (English translation of the original Russian, errata in Vol. 11, no. 6, 1971).Google Scholar
Nelson, E. (1986). Predicative Arithmetic, Mathematical Notes 12. Princeton University Press, pp. 329–353.CrossRefGoogle Scholar
Parikh, R. (1971). Existence and feasibility in arithmetic. The Journal of Symbolic Logic, 36, 494–508.Google Scholar
Quine, W. V. (1948). On what there is. Review of Metaphysics, 5, 21–37.Google Scholar
Quine, (1969) Ontological Relativity and Other Essays, Columbia University Press.Google Scholar
Robertson, N., Sanders, D., Seymour, P., & Tomas, R. (1996). A new proof of the four-colour theorem. Electronic Research Announcements of the American Mathematical Society, 2, 17–15.Google Scholar
Robinson, A. (1965). Formalism 64. In Bar-Hillel, Y., editor. Proceedings of the 1964 International Congress for Logic, Methodology and Philosophy of Science. North-Holland, pp. 228–246.Google Scholar
Shapiro, S. (2000) Thinking about mathematics, Oxford University Press, pp. 12–15and in part IV “The Contemporary Scene”.Google Scholar
Shelah, S. (2003). Logical dreams. Bulletin of the American Mathematical Society, 40, 203–209.CrossRefGoogle Scholar
Tait, W. (1981). Finitism. The Journal of Philosophy, 78, 524–546. The Journal of Symbolic Logic, 12, 105–122.Google Scholar
Woodin, H. (2001). The continuum hypothesis, Part I. Notices of the American Mathematical Society, 48, 567–576.Google Scholar