Supertasks and arithmetical truth

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.

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Notes

  1. 1.

    See Field (1994) and Putnam (1980).

  2. 2.

    See the essays in Salmon (1970) for a collection of classic articles on supertasks.

  3. 3.

    Quoted from Benacerraf and Putnam (1983), p. 20. Weyl (1949) also hints at use of supertasks in arithmetic, though negatively.

  4. 4.

    See Davies (2001) and Earman and Norton (1993, 1996).

  5. 5.

    See Hogarth (2004); see also Button (2009) for discussion of how such generalizations relate to the Church–Turing thesis.

  6. 6.

    As a referee has pointed out, our current practice arguably does include dispositions to (a) “take certain combinations of physical data as evidence for the existence of a supercomputer” and (b) “form new mathematical beliefs on the basis of interactions with that apparent supercomputer”. We don’t deny that in such a scenario, mathematicians would likely take the results of the supertask computations as evidence for arithmetical claims. But although they might play such an epistemic role, our point here is that deference to a supertask computer is not part of the basic metasemantics of actual arithmetic. That is to say: the content of our actual arithmetical claims is surely not fixed by the outcome of any empirical process, let alone (merely possible) supertask computations.

  7. 7.

    See Weyl (1949), p. 42, Bishop (1967), and Dummett (1977), for example.

  8. 8.

    Earman and Norton (1996), p. 251.

  9. 9.

    This is because \({\mathsf {PA}}\) (and even weaker arithmetical theories such as Robinson Arithmetic) are \(\Sigma _{1}\)-complete.

  10. 10.

    Obviously, for this particular case we could work with a restricted version of the rule requiring that \(\varphi \) contain no unbounded quantifiers.

  11. 11.

    For a proof of this that doesn’t require indexing proofs with infinite ordinals, see Tennant (1978). It is worth noting that the result also holds with weak arithmetical theories, such as Robinson arithmetic, in place of \({\mathsf {PA}}\).

  12. 12.

    See Kreisel (1967) for relevant discussion.

  13. 13.

    Typically by way of a second-order categoricity theorem; see Shapiro (1991) for details. An analogous result is available in the open-ended, first-order setting; for this approach, see Lavine (n.d.), McGee (1997), and Parsons (2008), chapter 8.

  14. 14.

    We use this notion rather than categoricity, which is stronger. True arithmetic has nonstandard models, for example.

  15. 15.

    Our discussion concerns only attempts to exploit the computational power of supertask computers to establish determinacy. But, as a referee for this journal has noted, an alternative strategy would be to argue that the existence of Malament–Hogarth machines of the kind envisioned here directly explains the truth of universally quantified claims over infinite physical structures. In essence, this approach appeals to the structure that must be physically realized in order for supertask computation to be possible, as well as the presumed determinacy of the physical vocabulary used to refer to it. Considered as a strategy for securing determinacy this is closely related to the cosmological approach of Field (1994). Accordingly, supertask computations aren’t playing an essential role in this type of approach.

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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.

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Warren, J., Waxman, D. Supertasks and arithmetical truth. Philos Stud 177, 1275–1282 (2020). https://doi.org/10.1007/s11098-019-01252-w

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Keywords

  • Arithmetical truth
  • Determinacy
  • Supertasks