Abstract
Merely approximate symmetry is mundane enough in physics that one rarely finds any explication of it. Among philosophers it has also received scant attention compared to exact symmetries. Herein I invite further consideration of this concept that is so essential to the practice of physics and interpretation of physical theory. After motivating why it deserves such scrutiny, I propose a minimal definition of approximate symmetry—that is, one that presupposes as little structure on a physical theory to which it is applied as seems needed. Then I apply this definition to three topics: first, accounting for or explaining the symmetries of a theory emeritus in intertheoretic reduction; second, explicating and evaluating the Curie-Post principle; and third, a new account of accidental symmetry.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Sundermeyer (2014, pp. 12–13) is an ambiguous case: he calls being approximate and being broken different “qualities” of non-exact symmetries, but does not explain how (if at all) they are to be distinguished. His views as presented there seem to be slightly better interpreted as distinguishing broken from approximate symmetries, but this will not make a difference in what follows.
See also Castellani (2003a, §4).
The connection between broken and approximate symmetry is so attractive that their conceptual differences are easy to overlook. As Weinberg (1997, p. 38) points out, in the early 1960s many physicists had in fact hoped to explain the approximate symmetries of particle phenomena as spontaneously broken exact symmetries, when in fact these are in general separate phenomena.
See also Brading et al. (2017, §4).
What about symmetries that preserve relations between possibilities, not just properties thereof? For example, the inner product of a Hilbert space fixes the transition amplitude between a pair of quantum states (and much else besides); the unitary transformation of the space are the symmetries that preserve these relations. Rosen (1995, Ch. 4.7, 2008, Ch. 10.8) implicitly takes the same approach for relations as with properties, using equivalence relations on the Cartesian product \(\mathcal {S}^n\) to represent elements of \(\mathcal {S}^n\) for which the same n-place relation holds. For simplicity of presentation I will confine attention to the unary case, but see also footnote 10 for more on symmetries that approximately preserve relations.
What of the compatibility of this proposal with the claim above that approximate symmetry is a form of broken symmetry? Rosen (1995, 2008) takes an “exact-symmetry-first” approach that requires approximate symmetry to be defined in relation to an exact symmetry that it approximates. As I describe in the sequel, there are good reasons instead to take an “approximate-symmetry-first” approach that defines exact symmetry as a special (but not always present) case of approximate symmetry. (See also especially footnote 19.)
Redhead (1975, p. 101) has made a similar suggestion to the effect that d should be a metric, which would strengthen null self-distance to the identity of indiscernibles: for all \(u,v \in \mathcal {S}\), \(d(u,v) = 0\) if and only if \(u=v\).
One can extend the application of pseudometrics on \(\mathcal {S}\) to \(\mathcal {S}^n\) using the Hausdorff distance—cf. footnote 17—or some other relevant generalization. This allows one to define approximate symmetries that preserve not just properties of possibilities, but also relations between them, as discussed in footnote 7.
See also Tversky (1977) for various arguments against these properties when interpreted as pertaining to a similarity measure.
It also avoids technical problems that arise when trying to fix an exactly symmetric possibility as a reference to which one must compare an approximately symmetric possibility, particularly when there are infinitely many such choices available (Petitjean 2003, pp. 294–295).
These assumptions can well be challenged. Tversky (1977) argues against them on psychological and linguistic grounds, as they arise for a distance function, even thought he opts for a different sort of numerical representation of similarity nonetheless. Elsewhere I have provided further concise arguments for dropping symmetry and weakening reflexivity to quasi-reflexivity (Fletcher forthcoming b). I believe the framework for approximate symmetry developed in this essay could be extended to these cases, but I shall set that development aside for now.
Gorham (1996) has suggested understanding similarity specifically through a pseudometric, although he applies it not just to models of a theory but between models and concrete worlds to define verisimilitude. Such applications are much beyond the present scope of investigation.
Just as a similarity structure on \(\mathcal {S}\) represents relevant ways for individual possibilities to be similar, a similarity structure on \(\mathcal {S}^n\) represents ways for n-tuples of possibilities to be similar. A pseudometric on \(\mathcal {S}^n\) can induce such a structure, allowing for an analogous definition of an \(\sim \)-approximate symmetry of \(\mathcal {S}^n\). (Cf. footnotes 7 and 10.)
Perhaps this is compatible with the statement from Castellani (2003a, p. 321) cited above, that an approximate symmetry is one “valid under certain conditions”, if one takes “certain conditions” to refer to the set of similarity relations for which a transformation is \(\sim \)-preserving.
In more detail, given two compact sets A and B in a Euclidean space, the Hausdorff distance between them is \(d_H(A,B) = \max \{ \sup _{a \in A} \inf _{b \in B} d(a,b), \sup _{b \in B} \inf _{a \in A} d(a,b) \}\), where d is the Euclidean distance function between points (Steen and Seebach (1995 [1978]), pp. 154–155).
For readers interested in more fundamental subject matters than gears: the gear example is analogous to one involving particles. All known particle interactions are invariant under rotational symmetries, while only those not involving the weak interaction (cf. gears with a positive helix angle) are also invariant under parity transformations (cf. reflection symmetry) and charge conjugation (cf. color inversion).
By beginning with similarity relations and then asking whether they have any restrictions which are non-trivial equivalence relations, I am following the “approximate-symmetry-first” approach to approximate symmetry. In contrast, Słowiński and Vanderpooten (1997) develop an analogous formalism (though for different purposes) by beginning with equivalence relations and then extending them to similarity relations (although they do not assume that these relations are symmetric). While the two approaches are extensionally equivalent where they are both defined—every similarity relation has the trivial equivalence relation as a restriction—the differing order of construction and dependence of the Słowiński and Vanderpooten formalism exemplifies the “exact-symmetry-first” approach.
Another kind of example is in the designation of a symmetry as being “accidental”, about which I will say more in Sect. 5.
See Senechal (1995) and references therein for more on the mathematics and symmetry properties of quasicrystals.
The empirical equivalence relation on an abstract possibility space is the equivalence relation on that space the members of whose equivalence classes represent the same of empirical phenomena. For the purposes of this section, I am assuming that how the elements of the space represent empirical phenomena has been given.
The domain of application of a theory is the collection of phenomena to which the theory is intended to apply or represent. This demand for explanation is also considered to be an important heuristic in the search for and development of new theories (Redhead 1975, p. 107).
If the successfully applied possibilities of \(\mathcal {O}\) are a proper subset thereof, one would restrict attention to those presently, but this doesn’t make a difference in the formal aspects of the analysis.
In other words, there are elements of \(\mathcal {N}\) arbitrarily similar to elements of \(\mathcal {O}\); this is formally analogous to \(\mathcal {O}\) lying in the topological closure of \(\mathcal {N}\). For more on the connections between similarity structure and topological structure, see Fletcher (forthcoming b).
This also shows that it is not enough in general for T to be a \(\sim \)-approximate symmetry of \(\mathcal {N}\)—really, the subset thereof witnessing the explanation—for \(n \sim T(o)\) and \(n \sim T(n)\) does not imply that \(T(n) \sim T(o)\) for an arbitrary similarity relation \(\sim \).
Technically, Redhead applies this restriction antecedently to the domain restriction and directly to the models to produce a subtheory of the new theory, but for present purposes we may let this restriction apply instead to similarity relations included in the similarity space for the new and old theories’ possibilities.
Redhead (1975, p. 109n24) himself in a note to his main text seems to suggest restricting attention to continuous symmetries, but does not follow through with the suggestion in his official account.
Redhead (1975, p. 104) already gives another version of the Principle (his “P2\('\)”) that does not falter under the sort of counterexample I have described, but only by restricting to cases where the old and new theory correspond exactly (rather than merely approximately) for the domain of phenomena on which they overlap. However, it is difficult to find realistic examples of this (so-called “exact correspondence”) in the development of physics.
Astute readers will notice that in this passage Kosso seems to speak categorically about a symmetry being accidental or fundamental, instead of it being a matter of degree as I had described in Sect. 5.1, but he switches back in the paragraph following: “A symmetry is really fundamental if it is really interdependent with many other things in nature” (Kosso 2000, p. 119). Similar switches occur throughout his article. Thus perhaps the most charitable reading is to interpret the categorical language as elision of the repetitive qualifications of degree.
In a scholarly lapse, Lange (2011) attributes the term “accidental symmetry” to Weinberg (1995, p. 529), but it appears with a similar sense in the physics literature already in Christian (1952, p. 77). Redhead (1975, p. 81) is the first I could find who introduced it into the philosophical literature.
See also the remarks in footnote 29.
Notwithstanding this general pattern, see footnote 6 of Kantorovich (2003, p. 662).
References
Bangu, S. (2008). Reifying mathematics? Prediction and symmetry classification. Studies in History and Philosophy of Modern Physics, 39(2), 239–258.
Bangu, S. (2013). Symmetry. In R. Batterman (Ed.), The Oxford handbook of philosophy of physics (pp. 287–317). Oxford: Oxford University Press.
Brading, K., & Castellani, E. (2007). Symmetries and invariances in classical physics. In J. Butterfield & J. Earman (Eds.), Philosophy of physics (pp. 1331–1368). Amsterdam: Elsevier.
Brading, K., Castellani, E., & Teh, N. (2017). Symmetry and symmetry breaking. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Winter 2017 ed.). Stanford: Metaphysics Research Lab, Stanford University.
Carnap, R. (1967). The logical structure of the world: Pseudoproblems in philosophy (R. A. George, Trans.) (2nd ed.). Berkeley: University of California Press.
Castellani, E. (2003a). On the meaning of symmetry breaking. In K. Brading & E. Castellani (Eds.), Symmetries in physics: Philosophical reflections (pp. 321–334). Cambridge: Cambridge University Press.
Castellani, E. (2003b). Symmetry and equivalence. In K. Brading & E. Castellani (Eds.), Symmetries in physics: Philosophical reflections (pp. 425–436). Cambridge: Cambridge University Press.
Christian, R. S. (1952). The analysis of high energy neutron-proton and proton-proton scattering data. Reports on Progress in Physics, 15(1), 68–100.
Costa, G., & Fogli, G. (2012). Symmetries and group theory in particle physics: An introduction to space-time and internal symmetries. Berlin: Springer.
Debs, T. A., & Redhead, M. L. G. (2007). Objectivity, invariance, and convention: Symmetry in physical science. Cambridge, MA: Harvard University Press.
Ehlers, J. (1986). On limit relations between, and approximative explanations of, physical theories. In R. B. Marcus, G. J. Dorn, & P. Weingartner (Eds.), Logic, methodology and philosophy of science VII, proceedings of the international congress of logic methodology and philosophy of science (Vol. 8, pp. 387–403). Amsterdam: Elsevier.
Fletcher, S. C. (2016). Similarity, topology, and physical significance in relativity theory. British Journal for the Philosophy of Science, 67, 365–389.
Fletcher, S. C. (2019). On the reduction of general relativity to Newtonian gravitation. Studies in History and Philosophy of Modern Physics. https://doi.org/10.1016/j.shpsb.2019.04.005.
Fletcher, S. C. (forthcoming a) Approximate local Poincaré spacetime symmetry in general relativity. In C. Beisbart, T. Sauer, & C. Wüthrich (Eds.), Thinking about space and time. Birkhäuser.
Fletcher, S. C. (forthcoming b) Similarity structure on scientific theories. In B. Skowron (Ed.), Topological Philosophy. de Gruyter.
Gorham, G. (1996). Similarity as an intertheory relation. Philosophy of Science, 63, S220–S229.
Kantorovich, A. (2003). The priority of internal symmetries in particle physics. Studies in History and Philosophy of Modern Physics, 34, 651–675.
Kosso, P. (2000). Fundamental and accidental symmetries. International Studies in the Philosophy of Science, 14(2), 109–121.
Lange, M. (2011). Conservation laws in scientific explanations: Constraints or coincidences? Philosophy of Science, 78(3), 333–352.
Lichtenberg, D. B. (1970). Unitary symmetry and elementary particles. New York: Academic Press.
Nickles, T. (1973). Two concepts of intertheoretic reduction. The Journal of Philosophy, 70(7), 181–201.
Nozick, R. (2001). Invariances: The structure of the objective world. Cambridge, MA: Harvard University Press.
Petitjean, M. (2003). Chirality and symmetry measures: A transdisciplinary review. Entropy, 5, 271–312.
Polchinski, J. (2017). Dualities of fields and strings. Studies in History and Philosophy of Modern Physics, 59, 6–20.
Post, H. (1971). Correspondence, invariance and heuristics: In praise of conservative induction. Studies in History and Philosophy of Science, 2(3), 213–255.
Redhead, M. L. G. (1975). Symmetry in intertheory relations. Synthese, 32(1–2), 77–112.
Rickles, D. (2008). Symmetry, structure, and spacetime. Amsterdam: Elsevier.
Rosen, J. (1975). Symmetries discovered: Concepts and applications in nature and science. Cambridge: Cambridge University Press.
Rosen, J. (1995). Symmetry in science: An introduction to the general theory. New York: Springer.
Rosen, J. (2008). Symmetry rules: How science and nature are founded on symmetry. Berlin: Springer.
Senechal, M. (1995). Quasicrystals and geometry. Cambridge: Cambridge University Press.
Słowiński, R., & Vanderpooten, D. (1997). Similarity relation as a basis for rough approximations. In P. P. Wang (Ed.), Advances in machine intelligence and soft computing (Vol. 4, pp. 17–33). Durham: Duke University Press.
Steen, L. A., Seebach, J. A., Jr. (1995 [1978]). Counterexamples in topology (2nd ed.). Mineola, NY: Dover.
Stewart, I., & Golubitsky, M. (1992). Fearful symmetry: Is God a geometer?. Oxford: Blackwell.
Sundermeyer, K. (2014). Symmetries in fundamental physics (2nd ed.). Cham: Springer.
Tversky, A. (1977). Features of similarity. Psychological Review, 84(4), 327–352.
van Fraassen, B. C. (1989). Laws and symmetry. Oxford: Clarendon Press.
Weinberg, S. (1995). The quantum theory of fields (Vol. 1). Cambridge: Cambridge University Press.
Weinberg, S. (1997). Changing attitudes and the standard model. In L. Hoddeson, L. Brown, M. Riordan, & M. Dresden (Eds.), The rise of the standard model: Particle physics in the 1960s and 1970s (pp. 36–44). Cambridge: Cambridge University Press.
Weyl, H. (1952). Symmetry. Princeton: Princeton University Press.
Wigner, E. P. (1967). Symmetries and reflections. Bloomington, IN: Indiana University Press.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Fletcher, S.C. An invitation to approximate symmetry, with three applications to intertheoretic relations. Synthese 198, 4811–4831 (2021). https://doi.org/10.1007/s11229-019-02371-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-019-02371-x