Abstract
This paper presents a defense of Epistemic Arithmetic as used for a formalization of intuitionistic arithmetic and of certain informal mathematical principles. First, objections by Allen Hazen and Craig Smorynski against Epistemic Arithmetic are discussed and found wanting. Second, positive support is given for the research program by showing that Epistemic Arithmetic can give interesting formulations of Church's Thesis.
Similar content being viewed by others
REFERENCES
Anderson, C. A.: 1993, 'Analyzing Analysis', Philosophical Studies 72, 199–222.
Boolos, G.: 1975, 'On Second-Order Logic', Journal of Philosophy 82, 509–27.
Boolos, G.: 1993, The Logic of Provability, Cambridge University Press, Cambridge.
Dummett, M.: 1982, 'The Philosophical Basis of Intuitionistic Logic', reprinted in P. Benacerraf, and H. Putnam (eds.), Philosophy of Mathematics. Selected Readings, 2nd edition, Cambridge University Press, Cambridge, pp. 97–129.
Flagg, R.: 1985, 'Church's Thesis is Consistent with Epistemic Arithmetic', in S. Shapiro (ed.), Intensional Mathematics, pp. 121–72.
Flagg, R.: 1986a, 'Integrating Classical and Intuitionistic Type Theory', Annals of Pure and Applied Logic 32, 27–51.
Flagg, R. and H. Friedman: 1986b, 'Epistemic and Intuitionistic Formal Systems', Annals of Pure and Applied Logic 32, 53–60.
Goodman, N.: 1984, 'Epistemic Arithmetic is a Conservative Extension of Intuitionistic Arithmetic', Journal of Symbolic Logic 49, 192–203.
Goodman, N.: 1986, 'Flagg Realizability in Epistemic Arithmetic', Journal of Symbolic Logic 51, 387–92.
Hazen, A.: 1990, 'The Myth of the Intuitionistic “Or”', in M. Dunn and A. Gupta (eds.), Truth or Consequences?, Kluwer, Dordrecht, pp. 177–95.
Hellman, G.: 1989, 'Never Say “Never”! On the Communication Problem between Intuitionism and Classicism', Philosophical Topics 17, 47–67.
Heyting, A.: 1930, 'Die formalen Regeln der intuitionistischen Logik', Sitzungsberichte der Preussischen Akademie von Wissenschaften. Physikalisch-Mathematische Klasse, pp. 42–56.
Horsten, L.: 1993, 'Note on an Objection of Lifschitz against Shapiro's “Epistemic Arithmetic”', in J. Czermak (ed.), Proceedings of the 15th International Wittgenstein-Symposium. Part I: Philosophy of Mathematics, Hölder-Pichler-Tempsky, Vienna, pp. 289–96.
Horsten, L.: 1994, 'Modal-Epistemic Variants of Shapiro's System of Epistemic Arithmetic', Notre Dame Journal of Formal Logic 35, 284–91.
Horsten, L.: 1997, 'Provability in Principle and Controversial Constructivistic Principles', Journal of Philosophical Logic 26, 635–60.
Kreisel, G.: 1987, 'Church's Thesis and the Ideal of Informal Rigour', Notre Dame Journal of Formal Logic 28, 499–519.
Lifschitz, V.: 1985, 'Calculable Natural Numbers', in S. Shapiro (ed.), Intensional Mathematics, pp. 173–90.
Mendelson, E.: 1990, 'Second Thoughts about Church's Thesis and Mathematical Proofs', Journal of Philosophy 87, 225–33.
Montague, R.: 1963, 'Syntactical Treatments of Modality, with Corollaries on Reflection Principles and Finite Axiomatizability', Acta Philosophica Fennica 16, 153–67.
Myhill, J.: 1952, 'Philosophical Implications of Mathematical Logic', The Review of Metaphysics 6, 165–98.
Myhill, J.: 1985, 'Intensional Set Theory', in S. Shapiro (ed.), Intensional Mathematics, pp. 47–62.
Shapiro, S.: 1980, 'On the Notion of Effectiveness', History and Philosophy of Logic 1, 209–30.
Shapiro, S.: 1981, 'Understanding Church's Thesis', Journal of Philosophical Logic 10, 353–65.
Shapiro, S. (ed.): 1985a, Intensional Mathematics, Amsterdam, North-Holland.
Shapiro, S.: 1985b, 'Epistemic and Intuitionistic Arithmetic', in S. Shapiro (ed.), Intensional Mathematics, pp. 11–46.
Smorynski, C.: 1991, 'Review of S. Shapiro, “Intensional Mathematics”', Journal of Symbolic Logic 56, 1496–9.
Troelstra, A. S. and D. van Dalen: 1988a, Constructivism in Mathematics. An Introduction, Vol. I, North-Holland, Amsterdam.
Troelstra, A. S. and D. van Dalen: 1988b, Constructivism in Mathematics. An Introduction, Vol. II, North-Holland, Amsterdam.
Rights and permissions
About this article
Cite this article
Horsten, L. In Defense of Epistemic Arithmetic. Synthese 116, 1–25 (1998). https://doi.org/10.1023/A:1005016405987
Issue Date:
DOI: https://doi.org/10.1023/A:1005016405987