Skip to main content Accessibility help
×
×
Home

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

  • STEWART SHAPIRO (a1)

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.

Copyright

Corresponding author

*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

References

Hide All
Benacerraf, P., & Putnam, H. (1983). Philosophy of Mathematics (second edition). Cambridge, UK: Cambridge University Press.
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. 496504.
Bolzano, B. (1837). Theory of Science: Wissenschaftslehre. Vol. 4. Sulzbach: Seidel. Translated by George, R., University of California Press, Berkeley, 1972.
Boolos, G. (1989). Iteration again. Philosophical Topics, 17, 521. Reprinted in Boolos (1998a), pp. 88–104.
Boolos, G. (1998a). Logic, Logic, and Logic. Cambridge, MA: Harvard University Press.
Boolos, G. (1998b). Must we believe in set theory? In Boolos (1998a), pp. 120132.
Burge, T. (1998). Frege on knowing the foundation. Mind, 107, 305347. Reprinted in Burge (2005), Chapter 9.
Burge, T. (2005). Truth, Thought, Reason: Essays on Frege. Oxford, UK: Oxford University Press.
Cantor, G. (1932). Gesammelte Abhandlungen mathematischen und philosophischen Inhalts (Zermelo, E., editor). Berlin: Springer.
Coffa, A. (1991). The Semantic Tradition from Kant to Carnap. Cambridge, UK: Cambridge University Press.
Detlefsen, M. (1986). Hilbert's Program. Dordrecht, The Netherlands: D. Reidel Publishing Company.
Detlefsen, M. (1988). Fregean hierarchies and mathematical explanation. International Studies in the Philosophy of Science, 3, 97116.
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. 50123.
Frege, G. (1884). Die Grundlagen der Arithmetik, Breslau: Koebner. The Foundations of Arithmetic, translated by Austin, J. (second edition). New York: Harper, 1960.
Frege, G. (1893). Grundgesetze der Arithmetik 1. Olms: Hildescheim.
Frege, G. (1903). Grundgesetze der Arithmetik 2.Olms: Hildescheim.
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.
Frege, G. (1914). Logic in mathematics. Translated by Long, P., & White, R. in Frege (1979), pp. 203–250.
Frege, G. (1969). Nachgelassene Schriften. (Hermes, H., Kambartel, F., & Kaulbach, F., editors). Hamburg: Felix Meiner Verlag.
Frege, G. (1976). Wissenschaftlicher Briefwechsel. Gabriel, G., Hermes, H., and Thiel, C., editors. Hamburg: Felix Meiner.
Frege, G. (1979). Posthumous Writings. (Hermes, H., Kambartel, F., & Kaulbach, G., editors). Chicago, IL: The University of Chicago Press.
Frege, G. (1980). Philosophical and Mathematical Correspondence. Oxford, UK: Basil Blackwell.
Gödel, K. (1964). What is Cantor's continuum problem. In Benacerraf and Putnam (1983), pp. 470–485.
Hilbert, D. (1899). Grundlagen der Geometrie. Leipzig, Berlin: Teubner. Foundations of Geometry, translated by Townsend, E., La Salle, IL, Open Court, 1959.
Hilbert, D. (1925). Über das Unendliche. Mathematische Annalen, 95, 161190. Translated as “On the infinite”, in van Heijenoort (1967), pp. 369–392 and Benacerraf and Putnam (1983), pp. 183–201.
Hilbert, D. (1935). Gesammelte Abhandlungen, Dritter Band. Berlin: Julius Springer.
Jeshion, R. (2000). On the obvious. Philosophy and Phenomenological Research, 60, 333355.
Jeshion, R. (2001). Frege's notions of self-evidence. Mind, 110, 937976.
Jeshion, R. (2004). Frege: evidence for self-evidence. Mind, 113, 131138.
Kim, J. (1994). Explanatory knowledge and metaphysical dependence. Philosophical Issues, 5, 5169.
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. 410505.
Lavine, S. (1994). Understanding the Infinite. Cambridge, MA: Harvard University Press.
Maddy, P. (1990). Realism in Mathematics. Oxford, UK: Oxford University Press.
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. 209219.
Moore, G. H. (1982). Zermelo's Axiom of Choice: Its Origins, Development, and Influence. New York: Springer-Verlag.
Peano, G. (1895). Formulaire de mathématiques. Bocca, Turin, Vol. 1 (Vol. 2, 1897).
Russell, B. (1993). Introduction to Mathematical Philosophy. New York: Dover (first published in 1919).
Shapiro, S. (1997). Philosophy of Mathematics: Structure and Ontology. New York: Oxford University Press.
Shapiro, S. (2005). Categories, structures, and the Frege-Hilbert controversy: the status of meta-metamathematics. Philosophia Mathematica, 13(3), 6177.
Taylor, R. (1993). Zermelo, reductionism, and the philosophy of mathematics. Notre Dame Journal of Formal Logic, 34, 539563.
Van Heijenoort, J. (1967). From Frege to Gödel. Cambridge, MA: Harvard University Press.
Waismann, F. (1982). Lectures on the Philosophy of Mathematics (an introduction by Wolfgang, G., editor). Amsterdam, The Netherlands: Rodopi.
Zermelo, E. (1904). Beweis, dass jede Menge wohlgeordnet werden kann. Mathematische Annalen, 59, 514516. Translated as “Proof that every set can be well-ordered”, in van Heijenoort (1967), pp. 139–141.
Zermelo, E. (1908). Neuer Beweis für die Möglichkeit einer Wohlordnung. Mathematische Annalen, 65, 107128. Translated in van Heijenoort (1967), pp. 183–198.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed