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Supertasks and arithmetical truth

  • Jared WarrenEmail author
  • Daniel Waxman
Article
  • 48 Downloads

Abstract

This paper discusses the relevance of supertask computation for the determinacy of arithmetic. Recent work in the philosophy of physics has made plausible the possibility of supertask computers, capable of running through infinitely many individual computations in a finite time. A natural thought is that, if true, this implies that arithmetical truth is determinate (at least for e.g. sentences saying that every number has a certain decidable property). In this paper we argue, via a careful analysis of putative arguments from supertask computations to determinacy, that this natural thought is mistaken: supertasks are of no help in explaining arithmetical determinacy.

Keywords

Arithmetical truth Determinacy Supertasks 

Notes

Acknowledgements

Thanks to Sharon Berry, Hartry Field, Tomi Francis, Casper Storm Hansen, Beau Madison Mount, James Studd, Jack Woods, and two (possibly identical) referees for this journal.

References

  1. Benacerraf, P., & Putnam, H. (1983). Introduction. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics: Selected readings (2nd ed.). Cambridge: Cambridge University Press.Google Scholar
  2. Berry, S. (2014). Malament–Hogarth machines and Tait’s axiomatic conception of mathematics. Erkenntnis, 79(4), 893–907.CrossRefGoogle Scholar
  3. Bishop, E. (1967). Foundations of constructive analysis. New York: McGraw-Hill.Google Scholar
  4. Button, T. (2009). SAD computers and two versions of the Church–Turing thesis. British Journal for the Philosophy of Science, 60(4), 765–792.CrossRefGoogle Scholar
  5. Davies, E. B. (2001). Building infinite machines. British Journal for the Philosophy of Science, 52(4), 671–682.CrossRefGoogle Scholar
  6. Dummett, M. (1977). Elements of intuitionism. Oxford: Clarendon Press.Google Scholar
  7. Earman, J., & Norton, J. D. (1993). Forever is a day: Supertasks in Pitowsky and Malament–Hogarth spacetimes. Philosophy of Science, 60(1), 22–42.CrossRefGoogle Scholar
  8. Earman, J., & Norton, J. D. (1996). Infinite pains: The trouble with supertasks. In A. Morton & S. P. Stich (Eds.), Benacerraf and his critics (pp. 231–259). Oxford: Blackwell.Google Scholar
  9. Field, H. (1994). Are our mathematical and logical concepts highly indeterminate? Midwest Studies in Philosophy, 19(1), 391–429. P. French, T. Uehling, & H. Wettstein (Eds.).CrossRefGoogle Scholar
  10. Hogarth, M. (2004). Deciding arithmetic using SAD computers. British Journal for the Philosophy of Science, 55(4), 681–691.CrossRefGoogle Scholar
  11. Kreisel, G. (1967). Informal rigour and completeness proofs. In I. Lakatos (Ed.), Problems in the philosophy of mathematics (pp. 138–157). Amsterdam: North-Holland.CrossRefGoogle Scholar
  12. Lavine, S. (n.d.). Skolem was wrong.Google Scholar
  13. McGee, V. (1997). How We Learn Mathematical Language. The Philosophical Review, 106(1), 35.CrossRefGoogle Scholar
  14. Parsons, C. (2008). Mathematical Thought and its Objects. Cambridge: Cambridge University Press.Google Scholar
  15. Putnam, H. (1980). Models and reality. Journal of Symbolic Logic, 45(3), 464–482.CrossRefGoogle Scholar
  16. Salmon, W. C. (Ed.). (1970). Zeno’s paradoxes. Indianapolis and New York: The Bobbs–Merrill Company Inc.Google Scholar
  17. Shapiro, S. (1991). Foundations without foundationalism: A case for second-order logic. Oxford: Oxford University Press.Google Scholar
  18. Tennant, N. (1978). Natural logic. Edinburgh: Edinburgh University Press.Google Scholar
  19. Warren, J., & Waxman, D. (forthcoming). A metasemantic challenge for mathematical determinacy. Synthese.  https://doi.org/10.1007/s11229-016-1266-y.
  20. Warren, J., & Waxman, D. (2016). A metasemantic challenge for mathematical determinacy. Synthese.  https://doi.org/10.1007/s11229-016-1266-y Google Scholar
  21. Weyl, H. (1949). Philosophy of mathematics and natural science. Princeton: Princeton University Press.CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Stanford UniversityPalo AltoUSA
  2. 2.Department of PhilosophyLingnan UniversityTuen MunHong Kong SAR

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