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On Representational Capacities, with an Application to General Relativity

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Abstract

Recent work on the hole argument in general relativity by Weatherall (Br J Philos Sci 69(2):329–350, 2018) has drawn attention to the neglected concept of (mathematical) models’ representational capacities. I argue for several theses about the structure of these capacities, including that they should be understood not as many-to-one relations from models to the world, but in general as many-to-many relations constrained by the models’ isomorphisms. I then compare these ideas with a recent argument by Belot (Noûs, 2017. https://doi.org/10.1111/nous.12200) for the claim that some isometries “generate new possibilities” in general relativity. Philosophical orthodoxy, by contrast, denies this. Properly understanding the role of representational capacities, I argue, reveals how Belot’s rejection of orthodoxy does not go far enough, and makes better sense of our practices in theorizing about spacetime.

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Notes

  1. See also Earman [23]. For reviews of the vast literature on the subject, from a range of philosophical and physical perspectives, including its bearing on broader debates about the metaphysics of spacetime, see Pooley [45], Stachel [53], Norton [43], and references therein.

  2. Exceptions include Mundy [42] and Leeds [34], to whose syntactic or formal responses Rynasiewicz [51] has critically replied.

  3. See Rynasiewicz [50] and Rosenstock et al. [47] for critical discussion of Earman’s proposal for Leibniz/Einstein algebras as one such formal replacement.

  4. While my arguments essentially use examples only from spacetime theories, I optimistically expect the same theses to hold for physical theories generally and even any scientific theory sufficiently formalized.

  5. Cf. Weatherall [56, p. 332].

  6. This sense of abstraction is sometimes also known as Aristotelian idealization [29].

  7. For more on these debates, see, e.g., Frigg and Nguyen [30] and Boesch [11].

  8. This is not the occasion for an analysis of the “representation-as” relation [30, §7], since the details thereof should not matter for the use to which I shall put it.

  9. A notable exception is the relevant notion of sameness for categories themselves, which is typically the weaker concept of categorical equivalence rather than categorical isomorphism. For more on category theory and the notions of isomorphism and equivalence therein, see, in order of increasing sophistication, Lawvere and Schanuel [33], Awodey [3], and Mac Lane [36].

  10. Cf. Weatherall [56, pp. 331–332]. The intended contrast with these intentions and purposes is with the (possibly careless or misleading) statements of individual actors. This distinction plays an important role in Sect. 6.

  11. See also Earman and Norton [24, p. 522].

  12. Dewar is restricting attention in this statement to theories with a first-order logical formulation, but this conditional assertion of RUME and denial of RDMI is enough for my illustrative purposes.

  13. See also Dewar [21].

  14. In contrast with Belot [7], this is not to say that general relativity needs two sectors of models with different identity conditions, one of which is used for cosmology and another for localized astrophysical modeling, independently of the intentions of the users of those models. I will return to this point in Sect. 7.

  15. The example is an amalgam of Wilson’s buckling beam [59] and Belot’s Epicurean-fashioned swerve theory [7], set in Minkowski spacetime.

  16. Belot [9, p. 331] suggests that homothetic spacetimes in general are “physically equivalent” for those motivated by relationism to “deny that there are possible worlds that agree about distance ratios but disagree about matters of absolute distance.” However, the arguments of Sect. 3.2 show that no such metaphysical assumption is needed for this conclusion (if one reads “physically equivalent” as “having the same representational capacities”). (A further caution: Belot describes homotheties as “scaling symmetries,” but this description may be misleading when matter fields introducing their own length and time scales are present.)

  17. Within the hole argument literature, Butterfield [15] and Maudlin [38, 39] developed positions which are incompatible with REME, in the sense that once one has set a particular Lorentzian manifold to represent a spacetime, those related by to it by a non-identity isomorphism do not. (As I discuss in Sect. 5.2, whether they do in fact depends on a choice of map by which to compare the two.) For discussion of these positions, including their demerits, see also illuminating discussion in Rickles [46, ch. 5] and Pooley [45]. Perhaps others have on other topics, but I have not canvassed the literature.

  18. For more arguments that could be mustered in favor of REME, see Dewar [20, ch. 2]. Although Dewar’s thesis is in fact RUME, much of his argumentation could be adapted in support for REME in light of the considerations of Sect. 3.

  19. Of course, this is compatible with a skeptical attitude towards such properties but in this case no problem about determinism arises.

  20. As Weatherall [56, p. 334] emphasizes, “All assertions of relation between mathematical objects—including isomorphism, identity, inclusion, and so on—are made relative to some choice of map.”

  21. There is an analogy here with Muller’s [41] defense of spacetime structuralism against the charge by Wüthrich [60] that each event of a homogeneous spacetime has the same profile of properties, so if events are discerned from one another by such properties, then there would only be one such event. Muller simply points out events may be discerned not only by their (absolute) properties but also their relations with other points.

  22. See also Dees [18, ch. 5] for metaphysical arguments in parallel to mine about the representation of physical units in Sect. 3.2.

  23. Cf. the position of Weatherall [56] vis-à-vis those of Butterfield [15], Brighouse [12], and Pooley [45, §7].

  24. Cf. the demand that “we need to be sure that we are using the formalism correctly, consistently, and according to our best understanding of the mathematics” [56, p. 330].

  25. The qualification, “well-understood,” is important here, for physicists’ attitude towards less understood formalism is more liberal, as the historical use of infinitesimals, Dirac delta functions, etc., attest.

  26. I think my conclusions drawn here also extend to Belot’s claims about non-relativistic spacetimes, but I shall not argue my case here.

References

  1. Arntzenius, F.: Space, Time, and Stuff. Oxford University Press, Oxford (2012)

    MATH  Google Scholar 

  2. Awodey, S.: Structure in mathematics and logic: a categorical perspective. Philos. Math. 4, 209–237 (1996)

    MathSciNet  MATH  Google Scholar 

  3. Awodey, S.: Category Theory, 2nd edn. Oxford University Press, Oxford (2010)

    MATH  Google Scholar 

  4. Baker, D.J.: Symmetry and the metaphysics of physics. Philos. Compass 5(12), 1157–1166 (2010)

    Google Scholar 

  5. Baker, D.J.: Broken symmetry and spacetime. Philos. Sci. 78(1), 128–148 (2011)

    MathSciNet  Google Scholar 

  6. Barbour, J.: The End of Time: The Next Revolution in Physics. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  7. Belot, G.: Fifty million Elvis fans can’t be wrong. Noûs (2017). https://doi.org/10.1111/nous.12200

    Article  Google Scholar 

  8. Belot, G.: New work for counterpart theorists: Determinism. Br. J. Philos. Sci. 46(2), 185–195 (1995)

    MathSciNet  Google Scholar 

  9. Belot, G.: Symmetry and equivalence. In: Batterman, R. (ed.) The Oxford Handbook of Philosophy of Physics, pp. 318–339. Oxford University Press, Oxford (2013)

    Google Scholar 

  10. Black, M.: The identity of indiscernibles. Mind 61(242), 153–164 (1952)

    Google Scholar 

  11. Boesch, B.: Scientific representation. In: The Internet Encyclopedia of Philosophy. https://www.iep.utm.edu/sci-repr/. Accessed 29 August 2017 (2017)

  12. Brighouse, C.: Spacetime and holes. In: PSA 1994: Proceedings of the Biennial Meeting of the Philosophy of Science Association, vol. 1, pp. 117–125 (1994)

  13. Brighouse, C.: Determinism and modality. Br. J. Philos. Sci. 48(4), 465–481 (1997)

    MathSciNet  MATH  Google Scholar 

  14. Brighouse, C.: Understanding indeterminism. In: Dieks, D. (ed.) The Ontology of Spacetime II, pp. 153–173. Elsevier, Oxford (2008)

    Google Scholar 

  15. Butterfield, J.: The hole truth. Br. J. Philos. Sci. 40(1), 1–28 (1989)

    MathSciNet  MATH  Google Scholar 

  16. Christodoulou, D.: Mathematical Problems of General Relativity, vol. 1. European Mathematical Society, Zurich (2008)

    MATH  Google Scholar 

  17. Dasgupta, S.: Absolutism vs comparativism about quantity. In: Bennett, K., Zimmerman, D.W. (eds.) Oxford Studies in Metaphysics, vol. 8, pp. 105–148. Oxford University Press, Oxford (2013)

    Google Scholar 

  18. Dees, M.K.: The fundamental structure of the world: Physical magnitudes, space and time, and the laws of nature. Ph.D. thesis, Rutgers University (2015)

  19. Dewar, N.: Sophistication about symmetries. Br. J. Philos. Sci. (2017). https://doi.org/10.1093/bjps/axx021

    Article  MathSciNet  MATH  Google Scholar 

  20. Dewar, N.: Symmetries in physics, metaphysics, and logic. Ph.D. thesis, Oxford University (2016)

  21. Dewar, N.: Symmetries and the philosophy of language. Stud. Hist. Philos. Mod. Phys. 52, 317–327 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Earman, J.: Why space is not a substance (at least not to first degree). Pac. Philos. Q. 67, 225–44 (1986)

    Google Scholar 

  23. Earman, J.: World Enough and Space-Time: Absolute versus Relational Theories of Space and Time. MIT Press, Cambridge (1989)

    Google Scholar 

  24. Earman, J., Norton, J.: What price spacetime substantivalism? The hole story. Br. J. Philos. Sci. 38(4), 515–525 (1987)

    MathSciNet  Google Scholar 

  25. Eddon, M.: Quantitative properties. Philos. Compass 8(7), 633–645 (2013)

    Google Scholar 

  26. Feintzeig, B.: On broken symmetries and classical systems. Stud. Hist. Philos. Mod. Phys. 52, 267–273 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Feintzeig, B.: Unitary inequivalence in classical systems. Synthese 193(9), 2685–2705 (2016)

    MathSciNet  MATH  Google Scholar 

  28. Freire Jr., O.: Quantum Dissidents: Rebuilding the Foundations of Quantum Mechanics (1950–1990). Springer, Berlin (2015)

    MATH  Google Scholar 

  29. Frigg, R., Hartmann, S.: Models in science. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Spring 2017 edition (2017)

  30. Frigg, R., Nguyen, J.: Scientific representation. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Winter 2016 edition (2016)

  31. Iftime, M., Stachel, J.: The hole argument for covariant theories. Gen. Relat. Gravit. 38, 1241–1252 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  32. Jammer, M.: The Philosophy of Quantum Mechanics. Wiley, New York (1974)

    Google Scholar 

  33. Lawvere, F.W., Schanuel, S.H.: Conceptual Mathematics: A First Introduction to Categories, 2nd edn. Cambridge University Press, Cambridge (2009)

    MATH  Google Scholar 

  34. Leeds, S.: Holes and determinism: another look. Philos. Sci. 62, 425–437 (1995)

    MathSciNet  Google Scholar 

  35. Lehmkuhl, D.: Literal versus careful interpretations of scientific theories: the vacuum approach to the problem of motion in general relativity. Philos. Sci. 84(5), 1202–1214 (2017)

    MathSciNet  Google Scholar 

  36. Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, New York (1998)

    MATH  Google Scholar 

  37. Maddy, P.: Second Philosophy: A Naturalistic Method. Oxford University Press, Oxford (2007)

    MATH  Google Scholar 

  38. Maudlin, T.: The essence of spacetime. In: Fine, A., Leplin, J. (eds.) PSA 1988: Proceedings of the Biennial Meeting of the Philosophy of Science Association, vol. 2, pp. 82–91 (1989)

  39. Maudlin, T.: Substances and spacetimes: what Aristotle would have said to Einstein. Stud. Hist. Philos. Sci. 21(1), 531–61 (1990)

    MathSciNet  Google Scholar 

  40. Melia, J.: Holes, haecceitism and two conceptions of determinism. Br. J. Philos. Sci. 50(4), 639–664 (1999)

    MathSciNet  MATH  Google Scholar 

  41. Muller, F.A.: How to defeat Wüthrich’s abysmal embarrassment argument against space-time structuralism. Philos. Sci. 78(5), 1046–1057 (2011)

    MathSciNet  Google Scholar 

  42. Mundy, B.: Space-time and isomorphism. In: PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, vol. 1, pp. 515–527 (1992)

  43. Norton, J.D.: The hole argument. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Fall 2015 edition (2015)

  44. O’Neill, B.: Semi-Riemannian Geometry, with Applications to Relativity. Academic Press, San Diego (1983)

    MATH  Google Scholar 

  45. Pooley, O.: Substantivalist and relationalist approaches to spacetime. In: Batterman, R. (ed.) The Oxford Handbook of Philosophy of Physics. Oxford University Press, Oxford (2013)

    Google Scholar 

  46. Rickles, D.: Symmetry, Structure, and Spacetime. Elsevier, Amsterdam (2008)

    Google Scholar 

  47. Rosenstock, S., Barrett, T., Weatherall, J.O.: On Einstein algebras and relativistic spacetimes. Stud. Hist. Philos. Mod. Phys. 52, 309–16 (2015)

    MathSciNet  MATH  Google Scholar 

  48. Rovelli, C.: Why gauge? Found. Phys. 44(1), 91–104 (2014)

    ADS  MATH  Google Scholar 

  49. Ruetsche, L.: Interpreting Quantum Theories. Oxford University Press, Oxford (2011)

    MATH  Google Scholar 

  50. Rynasiewicz, R.: Rings, holes and substantivalism: on the program of Leibniz algebras. Philos. Sci. 59, 572–89 (1992)

    MathSciNet  Google Scholar 

  51. Rynasiewicz, R.: Is there a syntactic solution to the hole problem? Philos. Sci. 63, S55–S62 (1996)

    MathSciNet  Google Scholar 

  52. Stachel, J.: Einstein’s search for general covariance, 1912–1915. In: Howard, D., Stachel, J. (eds.) Einstein and the History of General Relativity, pp. 1–63. Birkhäuser, Boston (1989)

    Google Scholar 

  53. Stachel, J.: The hole argument and some physical and philosophical implications. Living Rev. Relat. 17(1), 1 (2014)

    ADS  MATH  Google Scholar 

  54. Suárez, M.: Scientific representation: against similarity and isomorphism. Int. Stud. Philos. Sci. 17, 225–244 (2003)

    Google Scholar 

  55. Suárez, M.: An inferential conception of scientific representation. Philos. Sci. 71, 767–779 (2004)

    Google Scholar 

  56. Weatherall, J.O.: Regarding the ‘hole argument’. Br. J. Philos. Sci. 69(2), 329–350 (2018)

    MathSciNet  MATH  Google Scholar 

  57. Weatherall, J.O.: Fiber bundles, Yang-Mills theory, and general relativity. Synthese 193(8), 2389–2425 (2016)

    MathSciNet  MATH  Google Scholar 

  58. Weatherall, J.O.: Understanding gauge. Philos. Sci. 85(5), 1039–1049 (2016)

    MathSciNet  Google Scholar 

  59. Wilson, M.: There’s a hole and a bucket, dear Leibniz. Midwest Stud. Philos. 18(1), 202–241 (1993)

    Google Scholar 

  60. Wüthrich, C.: Challenging the spacetime structuralist. Philos. Sci. 76(5), 1039–1051 (2010)

    Google Scholar 

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Authors and Affiliations

  1. Department of Philosophy, University of Minnesota, Twin Cities, Minneapolis, MN, USA

    Samuel C. Fletcher

  2. Munich Center for Mathematical Philosophy, Ludwig Maximilian University of Munich, Munich, Germany

    Samuel C. Fletcher

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Correspondence to Samuel C. Fletcher.

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I would like to thank Gordon Belot, Neil Dewar, Ben Feintzeig, Jim Weatherall, and an anonymous referee for encouraging comments on a previous draft of this essay, which was written in part with the support from a Marie Curie Fellowship (PIIF-GA-2013-628533).

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Fletcher, S.C. On Representational Capacities, with an Application to General Relativity. Found Phys 50, 228–249 (2020). https://doi.org/10.1007/s10701-018-0208-6

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