Anti-realist Classical Logic and Realist Mathematics

  • Greg Restall
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 23)


I sketch an application of a semantically anti-realist understanding of the classical sequent calculus to the topic of mathematics. The result is a semantically anti-realist defence of a kind of mathematical realism. In the paper, I begin the development of the view and compare it to orthodox positions in the philosophy of mathematics.


Logical Consequence Classical Logic Mathematical Object Intuitionist Logic Sequent Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This paper has a webpage: Check there to post comments and to read comments left by others. Thanks to Dan Isaacson for hospitality during my visit to Wolfson College and the Philosophy Faculty at Oxford University, and to Allen Hazen, Dan Isaacson, Øystein Linnebo and Alexander Paseau, the audience at the (Anti)Realism conference in Nancy, and two anonymous referees, for helpful comments and discussion. This research is supported by the Australian Research Council, through grant DP0343388.


  1. 1.
    Aczel, P. 1988. Non-Well-Founded Sets. Number 14 in CSLI Lecture Notes. Stanford, CA: CSLI Publications.Google Scholar
  2. 2.
    Balaguer, M. 1998. Platonism and Anti-Platonism in Mathematics. Oxford: Oxford University Press.Google Scholar
  3. 3.
    Brandom, R. B. 1994. Making It Explicit. Cambridge, MA: Harvard University Press.Google Scholar
  4. 4.
    Brandom, R. B. 2000. Articulating Reasons: An Introduction to Inferentialism. Cambridge, MA: Harvard University Press.Google Scholar
  5. 5.
    Coffa, J. A. 1993. The Semantic Tradition from Kant to Carnap. Cambridge, MA: Cambridge University Press. Edited by Linda Wessels.Google Scholar
  6. 6.
    Dummett, M. 1991. The Logical Basis of Metaphysics. Cambridge, MA: Harvard University Press.Google Scholar
  7. 7.
    Feferman, S. 1995. “Definedness.” Erkenntnis 43(3):295–320, 11.CrossRefGoogle Scholar
  8. 8.
    Field, H. 1980. Science Without Numbers: A Defence of Nominalism. Oxford : Blackwell.Google Scholar
  9. 9.
    Field, H. 1991. Realism, Mathematics and Modality. Oxford: Blackwell.Google Scholar
  10. 10.
    Friedman, M. 1999. Reconsidering Logcal Positivism. Cambridge, MA: Cambridge University Press.Google Scholar
  11. 11.
    Friedman, M. 2001. Dynamics of Reason: The 1999 Kant Lectures at Stanford University. Stanford, CA: CSLI Publications.Google Scholar
  12. 12.
    Hacking, I. 1979. “What Is Logic?” The Journal of Philosophy 76:285–319.CrossRefGoogle Scholar
  13. 13.
    Isaacson, D. 1992. “Some Considerations on Arithmetical Truth and the ω-Rule.” In Proof, Logic and Formalization, edited by M. Detlefsen, 94–138. London: Routledge.Google Scholar
  14. 14.
    Kremer, M. 1988. “Logic and Meaning: The Philosophical Significance of the Sequent Calculus.” Mind 97:50–72.CrossRefGoogle Scholar
  15. 15.
    Lance, M. 1996. “Quantification, Substitution, and Conceptual Content.” Noûs 30(4):481–507.CrossRefGoogle Scholar
  16. 16.
    Maddy, P. 1988a. “Believing the Axioms 1.” Journal of Symbolic Logic 53:481–511.CrossRefGoogle Scholar
  17. 17.
    Maddy, P. 1988b. “Believing the Axioms 2.” Journal of Symbolic Logic 53:736–64.CrossRefGoogle Scholar
  18. 18.
    Maddy, P. 2005. “Mathematical Existence.” The Bulletin of Symbolic Logic 11(3):351–76.CrossRefGoogle Scholar
  19. 19.
    Price, H. 1990. “Why ‘Not’?” Mind 99(394):222–38.Google Scholar
  20. 20.
    Read, S. 2004. “Identity and Harmony.” Analysis 64(2):113–15.CrossRefGoogle Scholar
  21. 21.
    Restall, G. 2003. “Just What Is Full-Blooded Platonism?” Philosophia Mathematica 11:82–91.Google Scholar
  22. 22.
    Restall, G. 2005. “Multiple Conclusions.” In Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress, edited by Petr Hájek, Luis Valdés-Villanueva, and Dag Westerståhl, 189–205. London: KCL Publications.Google Scholar
  23. 23.
    Restall, G. 2007. “Proofnets for S5: Sequents and Circuits for Modal Logic.” In Logic Colloquium 2005, Number 28 in Lecture Notes in Logic, edited by Costas Dimitracopoulos, Ludomir Newelski, and Dag Normann. Cambridge, MA: Cambridge University Press.Google Scholar
  24. 24.
    Rumfitt, I. 2000. “ ‘Yes’ and ‘No’.” Mind 109(436):781–823.CrossRefGoogle Scholar
  25. 25.
    Schütte, K. 1977. Proof Theory. Berlin: Springer. Translated from the German by J. N. Crossley.Google Scholar
  26. 26.
    Shapiro, S. 1997. Philosophy of Mathematics: Structure and Ontology. Oxford: Oxford University Press.Google Scholar
  27. 27.
    Shapiro, S. 2000. Thinking About Mathematics: An Introduction to the Philosophy of Mathematics. Oxford: Oxford University Press.Google Scholar
  28. 28.
    Streicher, T., and B. Reus. 1998. “Classical Logic, Continuation Semantics and Abstract Machines.” Journal of Functional Programming 8(6):543–72.CrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.School of Historical and Philosophical StudiesMelbourneAustralia

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