Abstract
In the recent philosophical literature, several counterexamples to the interpretative principle that symmetry-related models are physically equivalent have been suggested (Belot, in: Batterman (ed) The Oxford handbook of philosophy of physics, Oxford University Press, Oxford, 2013, Noûs 52(4):946–981, 2018; Fletcher in Found Phys 50:228–249, 2020). Arguments based on these counterexamples can be understood as arguments from scientific practice of roughly the following form: because in scientific practice such-and-such symmetry-related models are treated as representing distinct physical situations, these models indeed represent distinct physical situations. In this paper, a strategy for analysing arguments of this type is presented and applied to the examples that can be found in the literature. I argue that if we are exclusively interested in models understood as representing entire possible worlds (not their subsystems), arguments from scientific practice should involve some additional assumptions to guarantee they are relevant for models understood in this way. However, none of the examples presented in the literature satisfy all these additional assumptions, which leads to the conclusion that arguments from scientific practice based on these examples do not undermine the interpretative principle that different symmetry-related models represent the same possible world.
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Notes
To avoid any confusion, below I present the formulation of our principles that is more explicit formally:
SYM-MANY: For any physical theory T and for any two models of T, \(M_1\) and \(M_2\): \(M_1\) and \(M_2\) are symmetry-related iff there is a class of possible physical situations P such that both \(M_1\) and \(M_2\) can represent all and only possible physical situations belonging to P.
SYM-ONE: For any physical theory T and for any two models of T, \(M_1\) and \(M_2\): \(M_1\) and \(M_2\) are symmetry-related iff there is a possible physical situation such that both \(M_1\) and \(M_2\) can represent it and they cannot represent any other possible physical situation.
Transformations here are understood passively, so this is not in conflict with what is claimed on p. 8.
Does this mean that if a model can represent exactly one physical situation/possible world, then it always actually represents it? Not necessarily so, as we might choose to not represent anything physical by this model and consider only its mathematical features, or change our standard representative conventions (to take an extreme example, one can use all models of classical mechanics to represent the Eiffel Tower). In all our considerations we assume that the standard representative conventions are used (i.e., \(\mathbf {x}\) in classical mechanics represent positions, \(\mathbf {E}\) in electrodynamics represents the electric field, an so on).
Cf. Fletcher (2020: p. 236), who claims that breaking this equivalence would be ad hoc.
By an “observer” I mean just any reference object, not necessarily human, which is a common way of speaking in physics.
S-W-Equivalence would be more plausible if we were restricted to the internal features of the represented subsystems and abstracted away from their relations to any external, implicitly represented physical objects. This way of thinking can be further supported by observing that the whole idea of implicit representation may sound rather suspicious—one could claim that either something is represented explicitly in the model or is not represented at all. However, the concept of implicit representation seems to be needed to account for how models under the subsystem-interpretation are actually used. When scientists test the predictions of a physical theory, they make observations and measurements in a particular reference frame (often called “laboratory reference frame”), which is associated with a real physical object (namely, the concrete laboratory, where these scientists are working) and is often not represented explicitly in the model. This way of thinking is also visible in textbook presentations of classical mechanics and special relativity, where reference frames are usually associated with “observers” (see, e.g., Gregory 2006: pp. 40, 259; Morin 2008: pp. 509–512; Rindler 1982: p. 7). Therefore, I assume that the idea of an implicitly represented reference object and its (also implicitly represented) relations to the explicitly represented system is tenable.
Perfect qualitative identity is rather rare, but sufficient qualitative similarity in the relevant respects is quite common—without it, scientific experiments would not be repeatable.
Both types of haecceistic differences mentioned in the main text are taken into account here.
Therefore, the more explicit formulation of these principles is as follows:
SYM-ONE-Q-W: For any physical theory T and for any two models of T, \(M_1\) and \(M_2\): if \(M_1\) and \(M_2\) are considered under the world-interpretation, then \(M_1\) and \(M_2\) are symmetry-related iff there is a class of possible worlds P such that the elements of P do not differ qualitatively, and both \(M_1\) and \(M_2\) can represent all and only possible worlds belonging to P.
SYM-ONE-H-W: For any physical theory T and for any two models of T, \(M_1\) and \(M_2\): if \(M_1\) and \(M_2\) are considered under the world-interpretation, then \(M_1\) and \(M_2\) are symmetry-related iff there is a possible world such that both \(M_1\) and \(M_2\) can represent it and they cannot represent any other possible world.
SYM-ONE-H-W is not a haecceistic principle in the sense of presupposing the truth of haecceitism (i.e., the view that there can be merely haecceistic differences). It might be true even if there is no such thing as merely haecceistic differences at all. “H” in SYM-ONE-H-W indicates that in contrast to SYM-ONE-Q-W, it excludes that two symmetry-related models under the world-interpretation can represent possible worlds differing haecceistically. According to SYM-ONE-H-W, such models can represent possible worlds that differ neither qualitatively nor haecceistically, which is equivalent to saying that there is exactly one possible world they can represent (for possible worlds to be different, they need to differ either qualitatively or haecceistically).
This is easy to show, but let me do this once. Assume SYM-ONE-H-W, that is, for any physical theory T and for any two models of T, \(M_1\) and \(M_2\), under the world-interpretation: \(M_1\) and \(M_2\) are symmetry-related iff there is exactly one possible world that \(M_1\) and \(M_2\) can represent. Now, take any theory T and any pair of its models, \(M_1\) and \(M_2\), considered under the world-interpretation. Assume that \(M_1\) and \(M_2\) are symmetry-related. By SYM-ONE-H-W, there is exactly one possible world that \(M_1\) and \(M_2\) can represent (call it w). Therefore, the class of possible worlds that can be represented by \(M_1\) is the same as the class of possible worlds that can be represented by \(M_2\)—namely, this is the singleton that has w as its only element (and of course the elements of this class differ at most haecceistically—in fact, they do not differ at all, as there is only one element). As we have considered an arbitrary T and an arbitrary pair of its models, this establishes SYM-ONE-Q-W.
For example, in the literature about the Hole Argument, it is usually assumed that there are no qualitative differences between possible worlds represented by two models related by the hole diffeomorphism, and the discussion concerns the question whether they represent two possible worlds differing only haecceistically or just one possible world (and what the consequences of each option for the ontology of space-time are). See, for example, Pooley (2013, section 7) and references therein.
Therefore, the more explicit formulation of these principles is as follows:
SYM-MANY-Q-W: For any physical theory T and for any two models of T, \(M_1\) and \(M_2\): if \(M_1\) and \(M_2\) are considered under the world-interpretation, then \(M_1\) and \(M_2\) are symmetry-related iff there is a class of possible worlds P such that both \(M_1\) and \(M_2\) can represent all and only possible worlds belonging to P.
SYM-MANY-H-W: For any physical theory T and for any two models of T, \(M_1\) and \(M_2\): if \(M_1\) and \(M_2\) are considered under the world-interpretation, then \(M_1\) and \(M_2\) are symmetry-related iff there is a class of possible worlds P such that the elements of P do not differ qualitatively, and both \(M_1\) and \(M_2\) can represent all and only possible worlds belonging to P.
A theory here is understood in a rather fine-grained way. In particular, if \(T_1\) and \(T_2\) have different dynamical equations, then they count as different theories. For example, classical mechanics would not count as a theory under this understanding but rather as a family of theories.
Is this SYM-MANY-H-W (which, to recall, is identical to SYM-ONE-Q-W) or SYM-MANY-Q-W? I think that Belot (2018) has in mind the former, although he does not consider this distinction explicitly. In any case, I am interested in saving the strongest of our interpretative principles, SYM-ONE-H-W, so it does not matter what the exact target of the counterexamples as conceived by the authors was. Therefore, to avoid overinterpretation, in this section I will discuss the counterexamples in terms of less precise principles SYM-ONE and SYM-MANY (understood as holding for models under the world-interpretation). .
Isometries are understood here as diffeomorphisms extended to the metric via the push-forward operation. This should be distinguished from a narrower sense of isometries, according to which they are a subset of diffeomorphisms that do not change the metric.
Here, Belot is interested in the configuration space of the system and not in the phase space.
It seems that it is SYM-MANY-H-W (equivalent to SYM-ONE-Q-W) that is questioned here—that is, according to Belot, we have to do with qualitative differences here.
For technical details, see Williamson (2013).
Strictly speaking, Fletcher’s examples do not use the term “symmetry” but rather the term “isomorphism.” Two models are isomorphic if they are structurally the same (the precise meaning of this depends on what we count as structure in a given context). Being isomorphic is a different relation than being symmetry-related, as it does not appeal to the theory’s dynamics. For our current purposes, this distinction will not matter much because all examples considered here involve symmetry-related models (even if the author’s main focus is on their being isomorphic), so they may serve as counterexamples to SYM-ONE, even if the original purpose of invoking them was slightly different. However, in general the distinction between symmetry and isomorphism is worth bearing in mind.
Here SYM-ONE-Q-W seems to be relevant, as the difference in the value of the Schwarzschild radius is a qualitative one.
Is the difference here meant to be qualitative or quantitative? Perhaps each answer is defensible. On the one hand, the shape of the trajectory of the particle is the same in all models, which may incline one to think about physical situations represented by these models as qualitatively the same, differing only in how they are embedded in space-time. On the other hand, the difference in direction is something that can be observed, which suggests that it is qualitative. Our analyses in Sect. 6 will apply to both cases, so the answer does not matter here (cf. footnote 16), but this ambiguity is interesting on its own.
One could suggest that philosophers focusing on the way in which scientists use certain terms are interested in capturing scientific practice as such, irrespective of whether it is empirically successful or not. This might be a worthwhile pursuit, but as here we are ultimately interested in reaching some metaphysical conclusions, the question whether a given scientific practice is empirically successful is clearly relevant.
Here, I understand scientific realism only as an affirmation of the existence of objects denoted by theoretical terms, but in general it comprises several theses about scientific theories (see, e.g., Psillos 1999).
This condition might not seem to be worth mentioning, but it will turn out that it is not satisfied in example 7.
The issue of the observability of differences between symmetry-related models is analysed in the literature about the so-called direct empirical significance of symmetries (see, e.g., Kosso 2000; Brading and Brown 2004; Greaves and Wallace 2014; Teh 2016). Significantly for our discussion, only models under the subsystem-interpretation are considered in this context. As Brading and Brown (2004: p. 646) write (emphases mine):
We maintain that the direct empirical significance of physical symmetries rests on the possibility of effectively isolated subsystems that may be actively transformed with respect to the rest of the universe. (...) The example of Galileo’s ship also illustrates that observing a symmetry involves two observations (...) we first observe the transformation, which involves transforming a subsystem with respect to some reference that is itself observable, and we then observe that the symmetry holds for the subsystem.
According to this quote, we can observe the difference between two symmetry-related states because they are related to some reference objects, which is possible only if these states are not the states of the universe as a whole. This suggests that the primary or even the only way in which a difference between two symmetry-related models can contribute to the predictive success of science is not available if the models are used to represent the universe as a whole, which provides a reason to doubt that condition (g) can ever be satisfied.
Answering this question decisively would require some empirical studies of the physicists’ community (e.g., based on questionnaires). The issue is complicated by the fact that the users of these models might not even have any definite views concerning the preferred understanding of such models under the world-interpretation.
The word “model” in this expression is used not in the sense that is usual in this paper (i.e., a model of a theory), but as a name for a kind of a theory.
See, for example, Krasiński (1997), Bojowald (2015), Joyce et al. (2015), Bull et al. (2016), Kragh and Longnair Kragh and Longair (2019). They can be classified in relation to the standard cosmological model in the following way: (1) GR solutions different than FLRW (inhomogeneous and/or anisotropic), (2) modifications of GR, (3) alternatives to GR, (4) theories that are supposed to be more fundamental than GR and expected to reduce to GR in appropriate limits, (5) semi-classical theories that are between GR and theories of category (5). The boundaries between these categories are not expected to be sharp.
The same is true for another of Belot’s (2013) examples that I did not mention in Sect. 4, namely, a generalised symmetry of the Kepler problem associated with the conservation of the Lenz-Runge vector. Solutions related by this symmetry may have different eccentricities and orientations in space. However, if we consider a further object that can be used to measure the eccentricity of an ellipse (i.e., it can serve as a ruler) and with respect to which the orientation in space can be determined, then the extended system no longer has the mentioned symmetry. (Cf. Wallace 2019 on the role of the extendibility of symmetries.)
Recall that theories here are understood in a fine-grained way; cf. footnote 15.
The subtlety here is that the metric in GR is sensitive to the presence of material objects, so after the introduction of some such objects the metric would no longer be exactly Schwarzschild. However, it could still be approximately Schwarzschild; otherwise, such models would not be applicable to the actual world. The fact that a model represents a physical situation only approximately does not on its own undermine the viability of questions about the way in which it represents.
Which is in fact what he wanted.
This does not mean that Norton’s dome cannot be used in the argumentation for which it was devised—namely, that Newtonian mechanics is not fully deterministic.
A note on the recently made distinction between the “interpretational” approach and the “motivational” approach to symmetries (Møller-Nielsen 2017; Read and Møller-Nielsen 2020) is in place here. According to the interpretational approach, we can regard two symmetry-related non-isomorphic models (call them \(M_1\) and \(M_2\)) as representing the same possible physical situation solely on the basis that they are symmetry-related, whereas according to the motivational approach, the fact that two non-isomorphic models are symmetry-related should only motivate us to find a modified theory in which the counterparts of these two models (call them \(M'_1\) and \(M'_2\)) will be isomorphic. Once such a new theory has been found, we are entitled to regard \(M_1\) and \(M_2\) (as well as \(M'_1\) and \(M'_2\)) as representing the same physical situations; but prior to this, we should rather treat \(M_1\) and \(M_2\) tentatively as representing different physical situations. According to the authors, this is because without the modified theory we do not have “a metaphysically perspicuous characterization of the (putative) reality underlying symmetry-related models” (Møller-Nielsen 2017: p. 1258). Therefore, this is an example of an argument not referring directly to scientific practice but rather based on a general view on what one should expect scientific theories to provide. My claims in this paper are tangential to this “interpretational” versus “motivational” debate. The question I am interested in here is whether various potential counterexamples to SYM-ONE-Q-W or SYM-ONE-H-W succeed in undermining them. If we found any counterexamples to SYM-ONE-Q-W, then both interpretational and motivational approaches would turn out to be wrong, as then there would be positive cases of non-isomorphic symmetry-related models that represent distinct possible worlds, so we should neither interpret them as representing the same possible world, nor should we be motivated to find a modified theory (counterexamples to SYM-ONE-H-W that are not counterexamples to SYM-ONE-Q-W would not have such an effect, as they would concern haecceistic differences only, so the models involved would be isomorphic). If, as I claim, these counterexamples are unsuccessful in undermining SYM-ONE-Q-W and SYM-ONE-H-W, then it is still left open whether one should take an interpretational or motivational attitude towards symmetry-related non-isomorphic models.
References
Baker, D. J. (2010). Symmetry and the metaphysics of physics. Philosophy Compass, 5(12), 1157–1166.
Belot, G. (2013). Symmetry and equivalence. In R. Batterman (Ed.), The Oxford handbook of philosophy of physics (pp. 318–339). Oxford University Press.
Belot, G. (2018). Fifty million Elvis fans can’t be wrong. Noûs, 52(4), 946–981.
Bojowald, M. (2015). Quantum cosmology: A review. Reports on Progress in Physics, 78(2), 023901.
Brading, K. A., & Brown, H. R. (2004). Are gauge symmetry transformations observable? British Journal for the Philosophy of Science, 55(4), 645–665.
Bull, P., et al. (2016). Beyond \(\Lambda \)CDM: Problems, solutions, and the road ahead. Physics of the Dark Universe, 12, 56–99.
Callan, C., Dicke, R. H., & Peebles, P. J. E. (1965). Cosmology and Newtonian mechanics. American Journal of Physics, 33(2), 105–108.
Castellani, E. (2003). Symmetry and equivalence. In K. Brading & E. Castellani (Eds.), Symmetries in physics: Philosophical reflections (pp. 425–436). Cambridge University Press.
Castelnovo, C., Moessner, R., & Sondhi, S. L. (2008). Magnetic monopoles in spin ice. Nature, 451, 42–45.
Caulton, A. (2015). The role of symmetry in the interpretation of physical theories. Studies in History and Philosophy of Modern Physics, 52, 153–162.
Dewar, N. (2022). Structure and equivalence. Cambridge University Press.
Fitzpatrick, R. (2012). An introduction to celestial mechanics. Cambridge University Press.
Fletcher, S. C. (2020). On representational capacities, with an application to general relativity. Foundations of Physics, 50, 228–249.
Greaves, H., & Wallace, D. (2014). Empirical consequences of symmetries. British Journal for the Philosophy of Science, 65(1), 59–89.
Gregory, R. D. (2006). Classical mechanics. Cambridge University Press.
Guth, A. H. (1981). Inflationary universe: A possible solution to the horizon and flatness problems. Physical Review D, 23(2), 347–356.
Jordan, T. F. (2005). Cosmology calculations almost without general relativity. American Journal of Physics, 73(7), 653–662.
Joyce, A., Jain, B., Khoury, J., & Trodden, M. (2015). Beyond the cosmological standard model. Physics Reports, 568, 1–98.
Kosso, P. (2000). The empirical status of symmetries in physics. British Journal for the Philosophy of Science, 51(1), 81–98.
Kragh, H. and Longair, M. S., Eds. (2019). The Oxford Handbook of the History of Modern Cosmology. Oxford: Oxford University Press.
Krasiński, A. (1997). Inhomogeneous cosmological models. Cambridge University Press.
Lazarides, G. (2006). Basics of inflationary cosmology. Journal of Physics: Conference Series, 53, 528–550. https://doi.org/10.1088/1742-6596/53/1/033.
Luc, J. (2021). Context-dependence of statements about quantities and symmetries (undated manuscript).
McCrea, W. H., & Milne, E. A. (1934). Newtonian universes and the curvature of space. The Quarterly Journal of Mathematics, os–5(1), 73–80.
Milne, E. A. (1934). A Newtonian expanding universe. The Quarterly Journal of Mathematics, 5(1), 64–72.
Møller-Nielsen, T. (2017). Invariance, interpretation, and motivation. Philosophy of Science, 84(5), 1253–1264.
Morin, D. (2008). Introduction to classical mechanics with problems and solutions. Cambridge University Press.
Norton, J. D. (2008). The Dome: An unexpectedly simple failure of determinism. Philosophy of Science, 75(5), 786–798.
Pooley, O. (2013). Substantivalist and relationalist approaches to spacetime. In R. Batterman (Ed.), The Oxford handbook of philosophy of physics. Oxford University Press.
Pooley, O. (2017). Background independence, diffeomorphism invariance and the meaning of coordinates. In D. Lehmkuhl, G. Schiemann, & E. Scholz (Eds.), Towards a theory of spacetime theories (pp. 105–143). Birkhäuser.
Psillos, S. (1999). Scientific realism: How science tracks truth. Routledge.
Rajantie, A. (2012). Magnetic monopoles in field theory and cosmology. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 370(1981), 5705–5717.
Read, J., & Møller-Nielsen, T. (2020). Motivating dualities. Synthese, 197, 236–291.
Rindler, W. (1982). Introduction to special relativity. Clarendon University Press.
Roberts, B. (2020). Regarding ‘Leibniz Equivalence’. Foundations of Physics. https://doi.org/10.1007/s10701-020-00325-9.
Sciama, D. W. (1971). Modern cosmology. Cambridge University Press.
Sundermeyer, K. (2014). Symmetries in fundamental physics. Springer.
Teh, N. J. (2016). Galileo’s gauge-understanding the empirical significance of gauge symmetry. Philosophy of Science, 83(1), 93–118.
Wald, R. M. (1984). General relativity. The University of Chicago Press.
Wallace, D. (2019). Observability, redundancy and modality for dynamical symmetry transformations. http://philsci-archive.pitt.edu/id/eprint/18813
Weatherall, J. O. (2018). Regarding the ‘hole argument’. The British Journal for the Philosophy of Science, 69(2), 329–350.
Weinberg, S. (1972). Gravitation and cosmology: Principles and applications of the general theory of relativity. Wiley.
Williamson, T. (2013). Modal logic as metaphysics. Oxford University Press.
Wulfman, C. E., & Wybourne, B. G. (1976). The Lie group of Newton’s and Lagrange’s equations for the harmonic oscillator. Journal of Physics A: Mathematical and General, 9(4), 507–518.
Acknowledgements
I would like to thank Jeremy Butterfield, Tomasz Placek and anonymous reviewers for comments on previous versions of this paper. I also benefited from discussing these issues at Philosophy of Physics Graduate Conference (Oxford) as well as Philosophy and History of Physics Reading Group (Cambridge and LSE). This work was supported by National Science Centre in Poland, research grant Preludium 2017/25/N/HS1/00705.
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National Science Centre in Poland, research grant Preludium 2017/25/N/HS1/00705.
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Luc, J. Arguments from scientific practice in the debate about the physical equivalence of symmetry-related models. Synthese 200, 72 (2022). https://doi.org/10.1007/s11229-022-03618-w
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DOI: https://doi.org/10.1007/s11229-022-03618-w