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.
Similar content being viewed by others
Notes
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.
Cf. Weatherall [56, p. 332].
This sense of abstraction is sometimes also known as Aristotelian idealization [29].
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.
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].
See also Earman and Norton [24, p. 522].
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.
See also Dewar [21].
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.
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.)
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.
Of course, this is compatible with a skeptical attitude towards such properties but in this case no problem about determinism arises.
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.”
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.
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].
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.
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
Arntzenius, F.: Space, Time, and Stuff. Oxford University Press, Oxford (2012)
Awodey, S.: Structure in mathematics and logic: a categorical perspective. Philos. Math. 4, 209–237 (1996)
Awodey, S.: Category Theory, 2nd edn. Oxford University Press, Oxford (2010)
Baker, D.J.: Symmetry and the metaphysics of physics. Philos. Compass 5(12), 1157–1166 (2010)
Baker, D.J.: Broken symmetry and spacetime. Philos. Sci. 78(1), 128–148 (2011)
Barbour, J.: The End of Time: The Next Revolution in Physics. Oxford University Press, Oxford (2000)
Belot, G.: Fifty million Elvis fans can’t be wrong. Noûs (2017). https://doi.org/10.1111/nous.12200
Belot, G.: New work for counterpart theorists: Determinism. Br. J. Philos. Sci. 46(2), 185–195 (1995)
Belot, G.: Symmetry and equivalence. In: Batterman, R. (ed.) The Oxford Handbook of Philosophy of Physics, pp. 318–339. Oxford University Press, Oxford (2013)
Black, M.: The identity of indiscernibles. Mind 61(242), 153–164 (1952)
Boesch, B.: Scientific representation. In: The Internet Encyclopedia of Philosophy. https://www.iep.utm.edu/sci-repr/. Accessed 29 August 2017 (2017)
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)
Brighouse, C.: Determinism and modality. Br. J. Philos. Sci. 48(4), 465–481 (1997)
Brighouse, C.: Understanding indeterminism. In: Dieks, D. (ed.) The Ontology of Spacetime II, pp. 153–173. Elsevier, Oxford (2008)
Butterfield, J.: The hole truth. Br. J. Philos. Sci. 40(1), 1–28 (1989)
Christodoulou, D.: Mathematical Problems of General Relativity, vol. 1. European Mathematical Society, Zurich (2008)
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)
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)
Dewar, N.: Sophistication about symmetries. Br. J. Philos. Sci. (2017). https://doi.org/10.1093/bjps/axx021
Dewar, N.: Symmetries in physics, metaphysics, and logic. Ph.D. thesis, Oxford University (2016)
Dewar, N.: Symmetries and the philosophy of language. Stud. Hist. Philos. Mod. Phys. 52, 317–327 (2015)
Earman, J.: Why space is not a substance (at least not to first degree). Pac. Philos. Q. 67, 225–44 (1986)
Earman, J.: World Enough and Space-Time: Absolute versus Relational Theories of Space and Time. MIT Press, Cambridge (1989)
Earman, J., Norton, J.: What price spacetime substantivalism? The hole story. Br. J. Philos. Sci. 38(4), 515–525 (1987)
Eddon, M.: Quantitative properties. Philos. Compass 8(7), 633–645 (2013)
Feintzeig, B.: On broken symmetries and classical systems. Stud. Hist. Philos. Mod. Phys. 52, 267–273 (2015)
Feintzeig, B.: Unitary inequivalence in classical systems. Synthese 193(9), 2685–2705 (2016)
Freire Jr., O.: Quantum Dissidents: Rebuilding the Foundations of Quantum Mechanics (1950–1990). Springer, Berlin (2015)
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)
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)
Iftime, M., Stachel, J.: The hole argument for covariant theories. Gen. Relat. Gravit. 38, 1241–1252 (2006)
Jammer, M.: The Philosophy of Quantum Mechanics. Wiley, New York (1974)
Lawvere, F.W., Schanuel, S.H.: Conceptual Mathematics: A First Introduction to Categories, 2nd edn. Cambridge University Press, Cambridge (2009)
Leeds, S.: Holes and determinism: another look. Philos. Sci. 62, 425–437 (1995)
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)
Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, New York (1998)
Maddy, P.: Second Philosophy: A Naturalistic Method. Oxford University Press, Oxford (2007)
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)
Maudlin, T.: Substances and spacetimes: what Aristotle would have said to Einstein. Stud. Hist. Philos. Sci. 21(1), 531–61 (1990)
Melia, J.: Holes, haecceitism and two conceptions of determinism. Br. J. Philos. Sci. 50(4), 639–664 (1999)
Muller, F.A.: How to defeat Wüthrich’s abysmal embarrassment argument against space-time structuralism. Philos. Sci. 78(5), 1046–1057 (2011)
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)
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)
O’Neill, B.: Semi-Riemannian Geometry, with Applications to Relativity. Academic Press, San Diego (1983)
Pooley, O.: Substantivalist and relationalist approaches to spacetime. In: Batterman, R. (ed.) The Oxford Handbook of Philosophy of Physics. Oxford University Press, Oxford (2013)
Rickles, D.: Symmetry, Structure, and Spacetime. Elsevier, Amsterdam (2008)
Rosenstock, S., Barrett, T., Weatherall, J.O.: On Einstein algebras and relativistic spacetimes. Stud. Hist. Philos. Mod. Phys. 52, 309–16 (2015)
Rovelli, C.: Why gauge? Found. Phys. 44(1), 91–104 (2014)
Ruetsche, L.: Interpreting Quantum Theories. Oxford University Press, Oxford (2011)
Rynasiewicz, R.: Rings, holes and substantivalism: on the program of Leibniz algebras. Philos. Sci. 59, 572–89 (1992)
Rynasiewicz, R.: Is there a syntactic solution to the hole problem? Philos. Sci. 63, S55–S62 (1996)
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)
Stachel, J.: The hole argument and some physical and philosophical implications. Living Rev. Relat. 17(1), 1 (2014)
Suárez, M.: Scientific representation: against similarity and isomorphism. Int. Stud. Philos. Sci. 17, 225–244 (2003)
Suárez, M.: An inferential conception of scientific representation. Philos. Sci. 71, 767–779 (2004)
Weatherall, J.O.: Regarding the ‘hole argument’. Br. J. Philos. Sci. 69(2), 329–350 (2018)
Weatherall, J.O.: Fiber bundles, Yang-Mills theory, and general relativity. Synthese 193(8), 2389–2425 (2016)
Weatherall, J.O.: Understanding gauge. Philos. Sci. 85(5), 1039–1049 (2016)
Wilson, M.: There’s a hole and a bucket, dear Leibniz. Midwest Stud. Philos. 18(1), 202–241 (1993)
Wüthrich, C.: Challenging the spacetime structuralist. Philos. Sci. 76(5), 1039–1051 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
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).
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-018-0208-6