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The Hole Argument, take n

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Abstract

I apply homotopy type theory (HoTT) to the hole argument as formulated by Earman and Norton. I argue that HoTT gives a precise sense in which diffeomorphism-related Lorentzian manifolds represent the same spacetime, undermining Earman and Norton’s verificationist dilemma and common formulations of the hole argument. However, adopting this account does not alleviate worries about determinism: general relativity formulated on Lorentzian manifolds is indeterministic using this standard of sameness and the natural formalization of determinism in HoTT. Fixing this indeterminism results in a more faithful mathematical representation of general relativity as used by physicists. It also gives a substantive notion of general covariance.

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Notes

  1. I will use “HoTT” ecumenically, to refer to any research program interested in a theory the ballpark of the theory presented in the HoTT Book [24]—which see for an introduction to HoTT. At minimum I mean to adopt a perspective in the spirit of abstract homotopy theory interpreted as a theory of higher equalities [23]. The theory of the HoTT Book is one way to make this precise, but my argument doesn’t turn on all of its details.

  2. I use the words “identical”, “equal”, and “the same” interchangeably.

  3. A comprehensive introduction to HoTT is beyond the scope of this paper. See [22] for a treatment of type-theoretical formalization in general and [24] for a full introduction to HoTT in particular. What follows is a compressed exposition of some aspects of the propositions-as-types paradigm [24, Sect. 1.11], dependent types [24, Sects. 1.6 and 2.7], identity types [24, Sects. 1.12 and 2.11], and univalence [24, Sect. 2.10]. The role of identity in HoTT, its most interesting feature for our purposes, is discussed from a philosophical perspective by [1, 8, 14, 15, 23, 26].

  4. See the HoTT Book [24, Sect. 1.11] for an introduction to this paradigm and [22] for a book-length treatment of natural language in this paradigm.

  5. This point has also been made by Shulman [23, p. 40].

  6. See also [21, p. 99] and [27, p. 335], for example, for similar reconstructions of Earman and Norton’s indeterminism dilemma.

  7. In particular, I set aside failures of determinism related to the extendability of spacetimes [10, 16]. I also set aside versions of the indeterminism worry that merely reclothe the verificationist argument in dynamical terms. I take the important feature of the indeterminism argument to be that the equation of motion for GR is diffeomorphism-invariant.

  8. Alternatively, we could follow Butterfield [7] by investigating a ramified notion of S-determinism, where S is “a kind of region that occurs in manifolds of the kind occurring in the models [of the theory]” [7, p. 7]. Each generalization of “moment of time” would then give a different S, giving rise to a different notion of determinism. The Minkowski space example will then be relevant as long as agreement outside of the unit ball induces agreement on some region of kind S; this should be the case for any reasonable S.

  9. Why the same equality, rather than an arbitrary equality? After all, as I said above, any term of an equality type gives a unique translation between statements involving the terms flanking the equality. But this translation is only unique relative to a choice of some equality—i.e., we need some particular equality on which to rely, and different choices give different translations. The antecedent of the conditional defining determinism supposes that the two models agree on some region with respect to some standard of sameness, while the consequent of the conditional states that these models also agree outside of the given regions. The consequent is only well-formed if it relies on a particular equality, which grammatically can only be supplied by the equality already referred to by the antecedent. Switching to some heretofore unmentioned equality halfway through the conditional would be like switching the referent of a pronoun to some heretofore unmentioned person halfway through a conditional. So this analysis of determinism is similar to a conditional donkey sentence: the “agree” of the consequent is tacitly anaphoric on the equality tacitly referred to by the “agree” of the antecedent [22, Sect. 3.7].

  10. The distinction in HoTT between inducing a unique equality and inducing some equality or other and the distinction in category theory between equivalence of categories and bijection of isomorphism classes can both be cashed out in terms of the distinction between contractibility and connectedness that can be found in both theories. Corfield [8] gives a more detailed treatment of definite descriptions in HoTT that makes the role of this distinction explicit.

  11. This is essentially the claim that the initial value problem of GR is well-posed [25, Theorem 10.2.2].

  12. This discussion of perturbation theory in GR in HoTT is admittedly far too brief to be convincing. I gesture at it only to indicate that general covariance in my sense has substantive consequences and they are the correct ones. A detailed treatment of perturbation theory, as well as initial value problems and boundary conditions, will be given elsewhere.

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Acknowledgements

An early version of this paper was presented at IPP 2015 and received helpful feedback from the audience. Thanks, too, to Craig Callender, Kathleen Connelly, Nat Jacobs, Chip Sebens, Sebastian Speitel, Anncy Thresher, and Christian Wüthrich for extensive feedback and discussion since then.

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  1. Munich Center for Mathematical Philosophy, LMU Munich, Geschwister-Scholl-Platz 1, 80539, Munich, Germany

    John Dougherty

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Dougherty, J. The Hole Argument, take n. Found Phys 50, 330–347 (2020). https://doi.org/10.1007/s10701-019-00291-x

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