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WE HOLD THESE TRUTHS TO BE SELF-EVIDENT: BUT WHAT DO WE MEAN BY THAT?

Published online by Cambridge University Press:  01 March 2009

STEWART SHAPIRO
STEWART SHAPIRO*
Affiliation:
Department of Philosophy, The Ohio State University and Arché Research Centre, University of St. Andrews
*
*THE OHIO STATE UNIVERSITY COLUMBUS, OH 43210 AND ARCHÉ RESEARCH CENTRE UNIVERSITY OF ST. ANDREWS ST. ANDREWS, FIFE, SCOTLAND, KY16 9AL E-mail:shapiro.4@osu.edu
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Abstract

At the beginning of Die Grundlagen der Arithmetik (§2) [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of self-evidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that both programs are undermined at a crucial point, namely when self-evidence is supported by holistic and even pragmatic considerations.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 2 , Issue 1 , March 2009 , pp. 175 - 207
Copyright
Copyright © Association for Symbolic Logic 2009

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References

BIBLIOGRAPHY

Benacerraf, P., & Putnam, H. (1983). Philosophy of Mathematics (second edition). Cambridge, UK: Cambridge University Press.Google Scholar
Bernays, P. (1967). Hilbert, David. In Edwards, P., editor. The Encyclopedia of Philosophy, Vol. 3. New York: Macmillan Publishing Company and The Free Press, pp. 496–504.Google Scholar
Bolzano, B. (1837). Theory of Science: Wissenschaftslehre. Vol. 4. Sulzbach: Seidel. Translated by George, R., University of California Press, Berkeley, 1972.Google Scholar
Boolos, G. (1989). Iteration again. Philosophical Topics, 17, 5–21. Reprinted in Boolos (1998a), pp. 88–104.CrossRefGoogle Scholar
Boolos, G. (1998a). Logic, Logic, and Logic. Cambridge, MA: Harvard University Press.Google Scholar
Boolos, G. (1998b). Must we believe in set theory? In Boolos (1998a), pp. 120–132.Google Scholar
Burge, T. (1998). Frege on knowing the foundation. Mind, 107, 305–347. Reprinted in Burge (2005), Chapter 9.CrossRefGoogle Scholar
Burge, T. (2005). Truth, Thought, Reason: Essays on Frege. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
Cantor, G. (1932). Gesammelte Abhandlungen mathematischen und philosophischen Inhalts (Zermelo, E., editor). Berlin: Springer.Google Scholar
Coffa, A. (1991). The Semantic Tradition from Kant to Carnap. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Detlefsen, M. (1986). Hilbert's Program. Dordrecht, The Netherlands: D. Reidel Publishing Company.CrossRefGoogle Scholar
Detlefsen, M. (1988). Fregean hierarchies and mathematical explanation. International Studies in the Philosophy of Science, 3, 97–116.CrossRefGoogle Scholar
Detlefsen, M. (1996). Philosophy of mathematics in the twentieth century. In Shanker, S., editor. Philosophy of Science, Logic and Mathematics in the Twentieth Century: Routledge History of Philosophy, Vol. 9. London, UK: Routledge, pp. 50–123.Google Scholar
Frege, G. (1884). Die Grundlagen der Arithmetik, Breslau: Koebner. The Foundations of Arithmetic, translated by Austin, J. (second edition). New York: Harper, 1960.Google Scholar
Frege, G. (1893). Grundgesetze der Arithmetik 1. Olms: Hildescheim.Google Scholar
Frege, G. (1903). Grundgesetze der Arithmetik 2.Olms: Hildescheim.Google Scholar
Frege, G. (1906). On Schoenflies: Die logischen Paradoxien der Mengenlehre. In Frege (1969), pp. 191–199. Translated by Long, P., & White, R. in Frege (1979), pp. 176–183.Google Scholar
Frege, G. (1914). Logic in mathematics. Translated by Long, P., & White, R. in Frege (1979), pp. 203–250.Google Scholar
Frege, G. (1969). Nachgelassene Schriften. (Hermes, H., Kambartel, F., & Kaulbach, F., editors). Hamburg: Felix Meiner Verlag.Google Scholar
Frege, G. (1976). Wissenschaftlicher Briefwechsel. Gabriel, G., Hermes, H., and Thiel, C., editors. Hamburg: Felix Meiner.CrossRefGoogle Scholar
Frege, G. (1979). Posthumous Writings. (Hermes, H., Kambartel, F., & Kaulbach, G., editors). Chicago, IL: The University of Chicago Press.Google Scholar
Frege, G. (1980). Philosophical and Mathematical Correspondence. Oxford, UK: Basil Blackwell.Google Scholar
Gödel, K. (1964). What is Cantor's continuum problem. In Benacerraf and Putnam (1983), pp. 470–485.Google Scholar
Hilbert, D. (1899). Grundlagen der Geometrie. Leipzig, Berlin: Teubner. Foundations of Geometry, translated by Townsend, E., La Salle, IL, Open Court, 1959.Google Scholar
Hilbert, D. (1925). Über das Unendliche. Mathematische Annalen, 95, 161–190. Translated as “On the infinite”, in van Heijenoort (1967), pp. 369–392 and Benacerraf and Putnam (1983), pp. 183–201.CrossRefGoogle Scholar
Hilbert, D. (1935). Gesammelte Abhandlungen, Dritter Band. Berlin: Julius Springer.Google Scholar
Jeshion, R. (2000). On the obvious. Philosophy and Phenomenological Research, 60, 333–355.CrossRefGoogle Scholar
Jeshion, R. (2001). Frege's notions of self-evidence. Mind, 110, 937–976.CrossRefGoogle Scholar
Jeshion, R. (2004). Frege: evidence for self-evidence. Mind, 113, 131–138.CrossRefGoogle Scholar
Kim, J. (1994). Explanatory knowledge and metaphysical dependence. Philosophical Issues, 5, 51–69.Google Scholar
Kitcher, P. (1989). Explanatory unification and the causal structure of the world. In Kitcher, P., & Salmon, W., editors. Scientific Explanation. Minneapolis: University of Minnesota Press, pp. 410–505.Google Scholar
Lavine, S. (1994). Understanding the Infinite. Cambridge, MA: Harvard University Press.Google Scholar
Maddy, P. (1990). Realism in Mathematics. Oxford, UK: Oxford University Press.Google Scholar
Martin-Löf, P. (2009). 100 years of Zermelo's axiom of choice: What was the problem with it? In Lindström, S., Palmgren, M., Segerberg, K., and Stoltenberg-Hansen, , editors. Logicism, intuitionism, and formalism: What has become of them? Dordrecht: Springer, pp. 209–219.Google Scholar
Moore, G. H. (1982). Zermelo's Axiom of Choice: Its Origins, Development, and Influence. New York: Springer-Verlag.CrossRefGoogle Scholar
Peano, G. (1895). Formulaire de mathématiques. Bocca, Turin, Vol. 1 (Vol. 2, 1897).Google Scholar
Russell, B. (1993). Introduction to Mathematical Philosophy. New York: Dover (first published in 1919).Google Scholar
Shapiro, S. (1997). Philosophy of Mathematics: Structure and Ontology. New York: Oxford University Press.Google Scholar
Shapiro, S. (2005). Categories, structures, and the Frege-Hilbert controversy: the status of meta-metamathematics. Philosophia Mathematica, 13(3), 61–77.CrossRefGoogle Scholar
Taylor, R. (1993). Zermelo, reductionism, and the philosophy of mathematics. Notre Dame Journal of Formal Logic, 34, 539–563.CrossRefGoogle Scholar
Van Heijenoort, J. (1967). From Frege to Gödel. Cambridge, MA: Harvard University Press.Google Scholar
Waismann, F. (1982). Lectures on the Philosophy of Mathematics (an introduction by Wolfgang, G., editor). Amsterdam, The Netherlands: Rodopi.Google Scholar
Zermelo, E. (1904). Beweis, dass jede Menge wohlgeordnet werden kann. Mathematische Annalen, 59, 514–516. Translated as “Proof that every set can be well-ordered”, in van Heijenoort (1967), pp. 139–141.CrossRefGoogle Scholar
Zermelo, E. (1908). Neuer Beweis für die Möglichkeit einer Wohlordnung. Mathematische Annalen, 65, 107–128. Translated in van Heijenoort (1967), pp. 183–198.Google Scholar