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Interpretation and equivalence; or, equivalence and interpretation

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Abstract

This paper argues that much of the literature on interpreting scientific theories presupposes a certain picture of what interpretation involves: a picture according to which interpreting a theory is like translating from one language to another. In place of this “external” approach to interpretation, this paper proposes an “internal” approach, according to which interpretation is more concerned with delineating a theory’s internal semantic architecture.

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Notes

  1. cf. Jones (1991).

  2. cf. Stanford (2014).

  3. I take this observation from Ruetsche (2011, Chapter 1). Perhaps other virtues, such as simplicity, do not require an interpreted theory; it suffices for the point here that many virtues (plausibly) do.

  4. This certainly does not make it meaningless to inquire about what kinds of formal equivalence might hold between uninterpreted theories: whether they are intertranslatable, mutually interpretable, etc. (for example, Button and Walsh (2018, chap. 5) discuss various such relationships for logico-mathematical theories). But the question of whether two such uninterpreted theories “say the same thing” in virtue of those relationships does not seem a well-posed question: they do in some ways, and not in others. We will return to questions about theoretical equivalence in Sect. 4.

  5. Russell (1993, pp. 77–78).

  6. Russell (1993, pp. 88–89).

  7. Carnap (1929, p. 12); quoted and translated in Coffa (1991, p. 227).

  8. Hence the significance of Craig’s theorem, insofar as it was taken to show that one could find a recursively axiomatisable theory capturing just the “observational content” of any other theory (see Craig (1953), or Putnam (1965) for critical discussion).

  9. See e.g. Reichenbach (1938), or Putnam (1983) for a critique.

  10. I thank Erik Curiel for pressing me on this.

  11. For more on indirect interpretation (and how it compares to direct interpretation of the kind discussed above) see Andreas (2021, Sect. 4).

  12. Carnap (1963, p. 59).

  13. Dialectically, this is because such explications are often associated with Copenhagen-style interpretations of quantum theory, of the kind which primitive ontology seeks to oppose.

  14. Maudlin (2007, pp. 3158–3159).

  15. Dürr (1992, pp. 848–849).

  16. Ghirardi (2016).

  17. Dürr (2008, p. 117); the context makes it reasonably clear that the claim generalises to other forms of primitive ontology.

  18. Dürr and Teufel (2009, p. 38).

  19. In this connection, note that the Aufbau is more pluralist about the choice of basis than one might expect. In particular, Carnap explicitly allows that one could use a physical basis (such as that consisting of elementary material particles or spacetime points), rather than a psychological one, and notes that such a system “would have the advantage that it uses as its basic domain the only domain (namely, the physical) which is characterized by a clear regularity of its processes.” (Carnap, 1967, p. 95).

  20. Allori et al. (2008, p. 365)—although as discussed in n. 36 below, they also seem open to applying the converse direction.

  21. See e.g. Miller (2014), Callender (2015), Esfeld (2014), or Bhogal and Perry (2017).

  22. Chakravartty, (2007, p. 26).

  23. French, (2013, p. 85).

  24. Coffey (2014, pp. 834–835).

  25. I thank an anonymous referee for pressing me on this point.

  26. e.g. Egg and Esfeld (2014) and Esfeld et al. (2014).

  27. It’s in this sense that the internal approach to interpretation makes contact with better-known doctrines under the “internal” label, such as Putnam’s internal realism.

  28. In this, I follow Halvorson (2012), Halvorson (2013), and Lutz (2015).

  29. As regards the syntactic view, see for example Carnap (1939, Part II) or Suppe (1977, Sect. II.E); on the side of the semantic view, see Glymour (2013) and van Fraassen (2014). I’m grateful to an anonymous referee for directing me to the Carnap and Suppe discussions.

  30. Rather than, say, homotopy type theory (see The Univalent Foundations Program (2013)).

  31. Kaplan (1975) and Pooley (2006).

  32. This is how I understand Weatherall (2018)’s critique of the usual dialectic surrounding the “Hole Argument”.

  33. Exactly why we should do so is a matter of some dispute: see e.g. Saunders (2003), Roberts (2008), Baker (2010), Dasgupta (2016), Caulton (2015), and references therein.

  34. See Dewar (2015), Møller-Nielsen (2017), Dewar (2019) for discussion.

  35. This is, in a sense, the converse of Coffey’s approach (albeit applied to models rather than theories).

  36. It should be noted that Allori et al. (2008) are sympathetic to such an idea. The quotation given in Sect. 2 above continues, “Conversely, one could define the notion of PO [primitive ontology] in terms of physical equivalence: The PO is described by those variables that remain invariant under all physical equivalences.” (Allori et al., 2008, p. 365).

  37. See e.g. Rosenstock et al. (2015), Weatherall (2016), Rosenstock and Weatherall (2016), Dewar (2019), Hudetz (2019), Barrett (2019), Halvorson (2019), Dewar (2022).

  38. See e.g. Teitel (2021) and references therein.

  39. My thanks to an anonymous referee for suggesting this way of putting things.

  40. cf. the proposal in Carnap (1956) to explicate necessity as logical validity. However, whereas Carnap’s proposal is intended to give a general semantics for modal logic, the account here is intended merely as a characterisation of which modal claims are true or false (not what modal inferences are valid or invalid). So the fact that Carnap’s semantics are widely regarded as defective [see Williamson (2013)] does not impugn the present proposal.

  41. Despite its naturalness (especially, the way it meshes with the way working scientists tend to talk of possibility), this view of possible worlds has not been very popular amongst metaphysicians. Indeed, I am not sure that it has been explicitly defended. Its closest relative, so far as I am aware, is the view Lewis describes as “pictorial ersatzism” (Lewis, 1986, Sect. 3.3), although even that is only a partial match. (Which may be for the best, given that pictorial ersatzism seems to generally be reckoned implausible: e.g. “[Pictorial ersatzism is] an odd, hybrid view that, I suspect, no one has or ever will hold” (Bricker, 2006, p. 42); “pictorial ersatzism is a puzzling view, and may have no actual adherents” (Nolan, 2015, p. 64).)

  42. cf. Bohnert (1967), and the role of “constraints” in the Munich structuralist program (Andreas, 2010; Stegmüller, 1976).

  43. See Glymour (1970), Quine (1975), Barrett and Halvorson (2015).

  44. See Andréka et al. (2008), Barrett and Halvorson (2016), Washington (2018).

  45. Indeed, much of the research on categorical equivalence of theories (see note 37) can be thought of as exploring what the most general and abstract constraints on a notion of translation might be.

  46. See e.g. Maudlin (2007)’s claim that one can have two versions of electromagnetism: one in which charge density is primitive, and correlated by the laws with the divergence of the electric field; and one in which charge density is defined as the divergence of \(\underline{E}\). Hicks and Schaffer (2017) also discuss the relationship between definability and non-fundamentality.

  47. See (Halvorson 2019, Sect. 5.4).

  48. Of course, in practice we often don’t have distinct variables: one typically uses \(\omega \), for instance, to denote the angular frequency of whatever SHO system is under investigation, whether it be a mass on a spring, an inductive circuit, or whatever. (This also illustrates that the same goes for theoretical but natural-language terms such as “angular frequency”.) But this is no more a cause for puzzlement than the existence of many bearers of the name “John” (or even “John Smith”).

  49. See Quine (1960, chap. 2).

  50. cf. Suppes’ discussion of “the theory of the experiment” (Suppes, 1969) or Nagel’s notion of an “experimental law” (Nagel, 1979, chap. 5).

  51. So, a theory being falsified is better described as our larger theory (the conjunction of the particular theory with the empirical theory, together with appropriate bridging claims) turning out to be inconsistent. This conception of truth in terms of consistency was defended by the early Reichenbach: see Reichen (1965, chap. IV).

  52. Unfortunately, I don’t have a general story about how it is that one identifies an empirical theory as such. That is something I hope to address in future work.

  53. cf. Weatherall’s “puzzleball” view of the foundations of physical theories (Weatherall, 2017)—although note that Weatherall’s proposal concerns explanation within such theories, rather than meaning.

  54. I thank two anonymous referees for raising this concern.

  55. cf. Stein’s discussion of “the fallacy of something more” (Stein, 1989).

  56. cf. van Fraassen (1987), Callender and Cohen (2006).

References

  • Allori, V., Goldstein, S., Tumulka, R., & Zanghì, N. (2008). On the common structure of Bohmian mechanics and the Ghirardi–Rimini–Weber theory. The British Journal for the Philosophy of Science, 59(3), 353–389.

    Article  Google Scholar 

  • Andreas, H. (2010). New account of empirical claims in structuralism. Synthese, 176(3), 311–332. https://www.jstor.org/stable/40801382.

    Article  Google Scholar 

  • Andreas, H. (2021). Theoretical terms in science. In Zalta, E. N., (Eds.), The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, fall 2021 edition. https://plato.stanford.edu/archives/fall2021/entriesheoretical-terms-science/.

  • Andréka, H., Madarász, J. X., & Németi, I. (2008). Defining new universes in many-sorted logic. Mathematical Institute of the Hungarian Academy of Sciences, Budapest, 93. https://pdfs.semanticscholar.org/d4da/6b9ff4c12d5d4c4d82a23376129c84ceecd4.pdf

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

    Article  Google Scholar 

  • Barrett, T. W. (2019). Equivalent and inequivalent formulations of classical mechanics. The British Journal for the Philosophy of Science, 70(4), 1167–1199. https://academic.oup.com/bjps/article/70/4/1167/4867963.

    Article  Google Scholar 

  • Barrett, T. W., & Halvorson, H. (2015). Glymour and Quine on theoretical equivalence. Journal of Philosophical Logic, 45(5), 467–483.

    Article  Google Scholar 

  • Barrett, T. W., & Halvorson, H. (2016). Morita equivalence. The Review of Symbolic Logic, 9(3), 556–582.

    Article  Google Scholar 

  • Bhogal, H., & Perry, Z. (2017). What the Humean should say about entanglement. Noûs, 51(1), 74–94.

    Article  Google Scholar 

  • Bohnert, H. G. (1967). Communication by Ramsey–Sentence clause. Philosophy of Science, 34(4), 341–347. https://www.jstor.org/stable/186123.

    Article  Google Scholar 

  • Bricker, P. (2006). Absolute actuality and the plurality of worlds. Philosophical perspectives, 20(1), 41–76.

    Article  Google Scholar 

  • Button, T., & Walsh, S. (2018). Philosophy and model theory. Oxford University Press.

    Book  Google Scholar 

  • Callender, C. (2015). One world, one beable. Synthese, 192(10), 3153–3177.

    Article  Google Scholar 

  • Callender, C., & Cohen, J. (2006). There is no special problem about scientific representation. Theoria. Revista de Teoría, Historia y Fundamentos de la Ciencia, 21(1), 67–85.

    Google Scholar 

  • Carnap, R. (1929). Von Gott und Seele. Unpublished lecture. Item number RC 089-63-02, Archives of Scientific Philosophy in the Twentieth Century, Department of Special Collections, University of Pittsburgh.

  • Carnap, R. (1939). Foundations of Logic and Mathematics. Number 3 in Foundations of the Unity of Science, Vol 1. University of Chicago Press, Chicago.

  • Carnap, R. (1956). Meaning and necessity: A study in semantics and modal logic. University of Chicago Press.

    Google Scholar 

  • Carnap, R. (1963). Intellectual autobiography. In P. A. Schilpp (Ed.), The philosophy of Rudolf Carnap, number 11 in The Library of Living philosophers (pp. 3–84). Open Court.

    Google Scholar 

  • Carnap, R. (1967). The logical structure of the World; Pseudoproblems in philosophy. University of California Press.

    Google Scholar 

  • Caulton, A. (2015). The role of symmetry in the interpretation of physical theories. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 52, 153–162.

    Article  Google Scholar 

  • Chakravartty, A. (2007). A metaphysics for scientific realism: Knowing the unobservable. Cambridge University Press.

    Book  Google Scholar 

  • Coffa, J. A. (1991). The semantic tradition from Kant to Carnap: To the Vienna station. Cambridge University Press.

    Book  Google Scholar 

  • Coffey, K. (2014). Theoretical equivalence as interpretative equivalence. The British Journal for the Philosophy of Science, 65(4), 821–844.

    Article  Google Scholar 

  • Craig, W. (1953). On axiomatizability within a system. The Journal of Symbolic logic, 18(01), 30–32.

    Article  Google Scholar 

  • Dasgupta, S. (2016). Symmetry as an epistemic notion (twice over). The British Journal for the Philosophy of Science, 67(3), 837–878.

    Article  Google Scholar 

  • Dewar, N. (2015). Symmetries and the philosophy of language. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 52, Part B:317-327.

    Article  Google Scholar 

  • Dewar, N. (2019). Sophistication about symmetries. The British Journal for the Philosophy of Science, 70(2), 485–521. https://academic.oup.com/bjps/article/70/2/485/4111183.

    Article  Google Scholar 

  • Dewar, N. (2022). Structure and equivalence. Cambridge Elements in Philosophy of Physics. Cambridge University Press.

  • Dürr, D. (2008). Bohmian mechanics. In J. Bricmont, D. Dürr, M. C. Galavotti, G. Ghirardi, F. Petruccione, & N. Zanghi (Eds.), Chance in physics: Foundations and perspectives (pp. 115–132). Springer.

    Google Scholar 

  • Dürr, D., Goldstein, S., & Zanghí, N. (1992). Quantum equilibrium and the origin of absolute uncertainty. Journal of Statistical Physics, 67(5–6), 843–907.

    Article  Google Scholar 

  • Dürr, D., & Teufel, S. (2009). Bohmian mechanics: The physics and mathematics of quantum theory. Springer.

    Google Scholar 

  • Egg, M., & Esfeld, M. (2014). Primitive ontology and quantum state in the GRW matter density theory. Synthese, 192(10), 3229–3245.

    Article  Google Scholar 

  • Esfeld, M. (2014). Quantum Humeanism, or: Physicalism without properties. The Philosophical Quarterly, 64(256), 453–470.

    Article  Google Scholar 

  • Esfeld, M., Hubert, M., Lazarovici, D., & Dürr, D. (2014). The ontology of Bohmian mechanics. The British Journal for the Philosophy of Science, 65(4), 773–796.

    Article  Google Scholar 

  • Everett, H. (1957). “Relative State’’ formulation of quantum mechanics. Reviews of Modern Physics, 29(3), 454–462.

    Article  Google Scholar 

  • French, S. (2013). Handling humility: Towards a metaphysically informed naturalism. In J. I. Galparsoro & A. Cordero (Eds.), Reflections on naturalism (pp. 85–104). Springer.

    Chapter  Google Scholar 

  • Ghirardi, G. (2016). Collapse theories. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Metaphysics Research Lab, Stanford University.

    Google Scholar 

  • Glymour, C. (1970). Theoretical realism and theoretical equivalence. In PSA: Proceedings of the Biennial meeting of the philosophy of science association, (pp. 275–288)

  • Glymour, C. (2013). Theoretical equivalence and the semantic view of theories. Philosophy of Science, 80(2), 286–297.

    Article  Google Scholar 

  • Halvorson, H. (2012). What scientific theories could not be. Philosophy of Science, 79(2), 183–206.

    Article  Google Scholar 

  • Halvorson, H. (2013). The semantic view, if plausible, is syntactic. Philosophy of Science, 80(3), 475–478. Halvorson, H.

  • Halvorson, H. (2019). The logic in philosophy of science. Cambridge University Press.

    Book  Google Scholar 

  • Hicks, M. T., & Schaffer, J. (2017). Derivative properties in fundamental laws. The British Journal for the Philosophy of Science, 68(2), 411–450.

    Article  Google Scholar 

  • Hudetz, L. (2019). Definable categorical equivalence. Philosophy of Science, 86(1), 47–75.

    Article  Google Scholar 

  • Jones, R. (1991). Realism about what? Philosophy of Science, 58(2), 185–202.

    Article  Google Scholar 

  • Kaplan, D. (1975). How to Russell a Frege-Church. The Journal of Philosophy, 72(19), 716–729.

    Article  Google Scholar 

  • Lewis, D. (1986). On the plurality of Worlds. Blackwell Publishers Ltd.

    Google Scholar 

  • Lutz, S. (2015). What Was the syntax-semantics debate in the philosophy of science about? Philosophy and Phenomenological Research, 91(3), 1–8.

    Google Scholar 

  • Maudlin, T. W. E. (2007). Completeness, supervenience and ontology. Journal of Physics A: Mathematical and Theoretical, 40(12), 3151.

    Article  Google Scholar 

  • Miller, E. (2014). Quantum entanglement, Bohmian mechanics, and Humean supervenience. Australasian Journal of Philosophy, 92(3), 567–583.

    Article  Google Scholar 

  • Møller-Nielsen, T. (2017). Invariance, interpretation, and motivation. Philosophy of Science, 84(5), 1253–1264.

    Article  Google Scholar 

  • Nagel, E. (1979). The structure of science: Problems in the logic of scientific explanation. Hackett.

    Google Scholar 

  • Nolan, D. (2015). David Lewis. Routledge.

    Book  Google Scholar 

  • Pooley, O. (2006). Points, particles, and structural realism. In D. Rickles, S. French, & J. Saatsi (Eds.), The structural foundations of quantum gravity (pp. 83–120). Oxford University Press.

    Chapter  Google Scholar 

  • Putnam, H. (1965). Craig’s theorem. The Journal of Philosophy, 62(10), 251.

    Article  Google Scholar 

  • Putnam, H. (1983). Equivalence. Realism and reason, volume 3 of philosophical papers (pp. 26–45). Cambridge University Press.

    Chapter  Google Scholar 

  • Quine, W. V. O. (1960). Word and object. M.I.T. Press.

    Google Scholar 

  • Quine, W. V. (1975). On empirically equivalent systems of the world. Erkenntnis, 9(3), 313–328. http://springer.longhoe.net/article/10.1007/BF00178004.

    Article  Google Scholar 

  • Reichenbach, H. (1938). Experience and prediction: An analysis of the foundations and the structure of knowledge. University of Chicago Press.

    Google Scholar 

  • Reichenbach, H. (1965). The theory of relativity and a priori knowledge. University of California Press. (English translation by M. Reichenbach of “Relativitätstheorie und Erkenntnis Apriori” (1920)).

  • Roberts, J. T. (2008). A puzzle about laws, symmetries and measurability. The British Journal for the Philosophy of Science, 59(2), 143–168.

    Article  Google Scholar 

  • Rosenstock, S., Barrett, T. W., & Weatherall, J. O. (2015). On Einstein algebras and relativistic spacetimes. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 52, Part B:309-316.

    Article  Google Scholar 

  • Rosenstock, S., & Weatherall, J. O. (2016). A categorical equivalence between generalized holonomy maps on a connected manifold and principal connections on bundles over that manifold. Journal of Mathematical Physics, 57(10), 102902.

    Article  Google Scholar 

  • Ruetsche, L. (2011). Interpreting quantum theories: The art of the possible. Oxford University Press.

    Book  Google Scholar 

  • Russell, B. (1993). Our knowledge of the external World. Routledge.

    Google Scholar 

  • Saunders, S. (2003). Indiscernibles, general covariance, and other symmetries: The case for non-eliminativist relationalism. In A. Ashtekar, D. Howard, J. Renn, S. Sarkar, & A. Shimony (Eds.), Revisiting the foundations of relativistic physics: Festschrift in Honour of John Stachel. Kluwer.

    Google Scholar 

  • Stanford, K. (2014). Reading nature: The interpretation of scientific theories. In L. Sklar (Ed.), Physical theory: Method and interpretation. Oxford University Press.

    Google Scholar 

  • Stegmüller, W. (1976). The structure and dynamics of theories. Springer.

    Book  Google Scholar 

  • Stein, H. (1989). Yes, but... Some skeptical remarks on realism and anti-realism. Dialectica, 43(12), 47–65.

    Article  Google Scholar 

  • Suppes, P. (1969). Models of data. In Studies in the methodology and foundations of science, (pp. 24–35). Springer.

  • Suppe, F. (1977). The search for philosophic understanding of scientific theories. In F. Suppe (Ed.), The structure of scientific theories (Second, pp. 3–241). University of Illinois Press.

    Google Scholar 

  • Teitel, T. (2021). What theoretical equivalence could not be. Philosophical Studies, 178(12), 4119–4149. https://doi.org/10.1007/s11098-021-01639-8

    Article  Google Scholar 

  • The Univalent Foundations Program. (2013). Homotopy type theory: Univalent foundations of mathematics. http://homotopytypetheory.org/book, Institute for advanced study

  • van Fraassen, B. C. (1987). The semantic approach to scientific theories. In N. J. Nersessian (Ed.), The process of science: Contemporary philosophical approaches to understanding scientific practice, science and philosophy (pp. 105–124). Springer. https://doi.org/10.1007/978-94-009-3519-8_6

    Chapter  Google Scholar 

  • van Fraassen, B. C. (2014). One or two gentle remarks about Hans Halvorson’s critique of the semantic view. Philosophy of Science, 81(2), 276–283. http://www.jstor.org/stable/10.1086/675645.

    Article  Google Scholar 

  • Washington, E. E. (2018). On the equivalence of logical theories. Senior thesis, Princeton University, Princeton

  • Weatherall, J. O. (2016). Are Newtonian gravitation and geometrized Newtonian gravitation theoretically equivalent? Erkenntnis, 81(5), 1073–1091. http://springer.longhoe.net/article/10.1007/s10670-015-9783-5.

    Article  Google Scholar 

  • Weatherall, J. O. (2017). Inertial motion, explanation, and the foundations of classical spacetime theories. In D. Lehmkuhl, G. Schiemann, & E. Scholz (Eds.), Towards a theory of spaceme theories, number 13 in Einstein studies (pp. 13–42). Birkhäuser.

    Chapter  Google Scholar 

  • Weatherall, J. O. (2018). Regarding the ‘hole argument’. The British Journal for the Philosophy of Science, 69(2), 329–350.

    Article  Google Scholar 

  • Williamson, T. (2013). Modal logic as metaphysics. Oxford University Press.

    Book  Google Scholar 

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Acknowledgements

Many thanks to Dave Baker, Thomas Barrett, Jeff Barrett, Erik Curiel, Josh Eisenthal, Alex Meehan, Tushar Menon, Oliver Pooley, James Read, Katie Robertson, Emma Ruttkamp-Bloem, David Schroeren, Kyle Stanford, Jim Weatherall, David Wallace, and three anonymous referees for comments on and/or discussions of earlier drafts of this and related material; I’m also very grateful to audiences at The Semantics of Theories conference and the SoCal Philosophy of Physics Reading Group for their insightful questions.

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Dewar, N. Interpretation and equivalence; or, equivalence and interpretation. Synthese 201, 119 (2023). https://doi.org/10.1007/s11229-023-04102-9

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