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A metasemantic challenge for mathematical determinacy

Abstract

This paper investigates the determinacy of mathematics. We begin by clarifying how we are understanding the notion of determinacy (Sect. 1) before turning to the questions of whether and how famous independence results bear on issues of determinacy in mathematics (Sect. 2). From there, we pose a metasemantic challenge for those who believe that mathematical language is determinate (Sect. 3), motivate two important constraints on attempts to meet our challenge (Sect. 4), and then use these constraints to develop an argument against determinacy (Sect. 5) and discuss a particularly popular approach to resolving indeterminacy (Sect. 6), before offering some brief closing reflections (Sect. 7). We believe our discussion poses a serious challenge for most philosophical theories of mathematics, since it puts considerable pressure on all views that accept a non-trivial amount of determinacy for even basic arithmetic.

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Notes

  1. 1.

    Throughout we assume classical logic—an untendentious assumption in mainstream mathematics—for full generality we would need to distinguish between, for different logics L, different notions of L-independence.

  2. 2.

    From a modern point of view, the first clear independence proof is found in Beltrami (1868a, b).

  3. 3.

    See Gödel (1931), Rosser (1936), Goodstein (1944), Kirby and Paris (1982), Gödel (1939), Cohen (1963, 1964), Suslin (1920), Jech (1967), Tennenbaum (1968), and Solovay and Tennenbaum (1971).

  4. 4.

    See Sect. 4 of Warren (2015) for a relevant general characterization of conceptual pluralism.

  5. 5.

    Williamson (1994) sees the indeterminacy induced by vagueness as being merely epistemic in this sense.

  6. 6.

    This is roughly equivalent to notions of “super-truth” as used in supervaluationist treatments of vagueness.

  7. 7.

    Of course, if ZFC is inconsistent then no sentence in the language of set theory, including both CH and its negation, is independent of the theory.

  8. 8.

    Something like this argument is endorsed in Putnam (1967a) but then mocked in Putnam (1967b).

  9. 9.

    The Rosser sentence R intuitively “says” that if R is provable, then there is a shorter proof of \(\lnot R\). Unlike the standard Gödel sentence, the negation of the Rosser sentence for a theory extending arithmetic can be shown to be independent of the theory without assuming that the theory is \(\omega \)-consistent. See Rosser (1936).

  10. 10.

    Just to make sure that there isn’t any confusion here: recall that Indeterminacy applies to subject matters by way of a notion of determinate truth that applies to sentences concerning the subject matter.

  11. 11.

    There are explicit arguments for this, but in the background to such arguments is typically the thought that syntax is determinate, and so, since the canonical examples of undecidable arithmetical sentences (Gödel sentences, Rosser sentences, consistency sentences) all concern, in some sense, syntactic facts about provability, they must be fully determinate. However, it is worth noting that not all examples of arithmetical incompleteness are of this nature; see Paris and Harrington (1977) for an example with no prima facie connection to the syntax of arithmetic.

  12. 12.

    Other possible disanalogies include (i) the extent of disagreement over proposed alternative axioms and related claims (see Clarke-Doane 2013) and (ii) the potentially different role that considerations concerning nonstandard models may legitimately play in indeterminacy arguments in set theory as opposed to arithmetic (see Gaifman 2004). The first type of disanalogy noted here may also be relevant to the plausibility of mathematical pluralism.

  13. 13.

    Our challenge has been influenced by Putnam (1980) and the surrounding literature, especially Hartry Field’s series of papers—(1994, 1998a, b)—on arithmetical and set-theoretic determinacy.

  14. 14.

    Here and throughout we assume that words and sentences are individuated syntactically. Of course the very same metasemantic questions will arise, in a slightly altered form, if linguistic items are individuated semantically.

  15. 15.

    Of course, saying that indeterminacy is often easy to explain, metasemantically, is not the same thing as saying that the concept of indeterminacy, or the semantics of an indeterminacy-operator, is easy to explain.

  16. 16.

    See Benacerraf (1973) and the introduction to Field (1989) for the most important contributions to the literature on epistemological arguments against realism. And see Warren (2016) for a recent generalization and defense of such arguments.

  17. 17.

    Cf. Schechter (2010). In Warren and Waxman (Unpublished) we develop and endorse a generalized epistemological argument against realism along the lines suggested by Schechter.

  18. 18.

    Our distinction between the first and second grades of platonic involvement is similar to Field’s (1984) distinction between “moderate” and “heavy-duty” platonism, though we don’t focus just on relations between abstract and physical objects but rather on the explanatory role played by the former.

  19. 19.

    See Woods (2016) for some related thoughts.

  20. 20.

    This is important to note because the physical significance of some scientific theories—such as quantum mechanics—is hotly contested. For some relevant discussion of the quantum mechanical case, in the context of whether and how nominalist programs such as Field’s (1980) could apply to quantum mechanics, see Malament (1982) and Balaguer (1996) for discussion.

  21. 21.

    See Leeds (1978) for relevant discussion.

  22. 22.

    For Neo-Fregeanism, see Wright (1983), Hale (1987), and the essays in Hale and Wright (2001).

  23. 23.

    For this reason, we also accept a general version of the metaphysical constraint, applying to the metasemantics of all other domains. However, since our concern here is with mathematics alone, that is the focus of our attention.

  24. 24.

    (McGee, 1991, p. 117).

  25. 25.

    See any recursion theory textbook for details, e.g., Odifreddi (1989).

  26. 26.

    It also has non-trivial repercussions; see McGee (1991).

  27. 27.

    The primary sources here are Lucas (1961), Penrose (1989, 1994); helpful responses include Benacerraf (1967), Boolos (1990) and Shapiro (1998, 2003).

  28. 28.

    One approach we will not discuss attempts to use Tennenbaum’s theorem to argue that arithmetic must be categorical and hence determinate—see Dean (2002) and Button and Smith (2011) offer a critique.

  29. 29.

    Quoted from (Carnap, 1934, p. 173).

  30. 30.

    See Chalmers (2012) and Weir (2010) for use of the \(\omega \)-rule as an idealization and Horwich (1998) for unexplained appeal to it.

  31. 31.

    See Shapiro (1991) for details of standard and other semantics for second-order logic.

  32. 32.

    This type of criticism was pioneered by Weston (1976).

  33. 33.

    See Lavine (Unpublished), McGee (1997) and Parsons (2001).

  34. 34.

    See the discussion in Field (2001).

  35. 35.

    Field (1994, 1998a, b).

  36. 36.

    (Field, 1998b, p. 342); a similar quote from page 418 of his earlier (1994) is still ambivalent, but slightly less skeptical of his own approach and slightly more skeptical of alternatives: “I am sure that some will feel that making the determinateness of the notion of finite depend upon cosmology is unsatisfactory; perhaps, but I do not see how anything other than cosmology has a chance of making it determinate”.

  37. 37.

    (Parsons, 2001, p. 22).

  38. 38.

    See Zermelo (1930) and Isaacson (2011). McGee (1997) proves a full categoricity result for open-ended set theory, but only by both (i) adding an urelemente set axiom and (ii) making the crucial assumption that the quantifiers in each theory range over the very same domain.

  39. 39.

    A referee for this journal has noted to us that it might be possible to develop certain conditional limits on Field’s approach in moving from first to second-order arithmetic. The idea being that even if Field’s approach secures determinacy for first-order arithmetic, given the cognitive constraint, this determinacy couldn’t plausibly be extended to (for example) all claims concerning \(\Pi _{2}^{1}\)-sets. One interesting point here is that this criticism can be made while allowing that the interaction of the world and our practice extends determinacy beyond what is accounted for by either factor, all on its own.

  40. 40.

    See Gödel (1964) for some relevant further discussion.

  41. 41.

    See the final section of Quine (1951).

  42. 42.

    See Koellner (2010) and Maddy (1997, 2007). We are unsure whether either of these philosophers thinks that full and complete determinacy can be attained in this fashion.

  43. 43.

    The talk of “singling out” here concerns our practice singling out a determinate theory, rather than our theory singling out (determinately) certain mathematical objects. That is: we are still talking about determinate truth rather than determinate reference to mathematical objects.

  44. 44.

    We’re grateful to Jack Woods and two referees for comments. Thanks also to audiences at NYU and Vienna, where an early version of this paper was presented by DW.

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Warren, J., Waxman, D. A metasemantic challenge for mathematical determinacy. Synthese (2016). https://doi.org/10.1007/s11229-016-1266-y

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Keywords

  • Determinacy
  • Indeterminacy
  • Metasemantics
  • Philosophy of mathematics
  • Incompleteness