With all of the above in hand, we turn now to the main event: a critical evaluation of Dewar’s (external) sophistication about symmetries. In the following subsections, we present what we take to be some problems with this approach to articulating the ontology of symmetry-related models of a given theory. In Sect. 4.1, we question the scope of the examples which Dewar presents in Dewar (2019) in favour of sophistication—and argue that these do not provide compelling motivation for universal external sophistication (and a fortiori not for the interpretational approach to symmetries). In Sect. 4.2, we question the extent to which sophistication can preserve explanatory power—we argue that this is not invariably the case. The discussions in these first two subsections allow us to develop in more detail some general concerns regarding external sophistication—concerns which we express in Sect. 4.3.
On universality
In Dewar (2019, §4), several examples of sophistication are presented, before it is proposed that sophistication—specifically external sophistication—can be applied universally to articulate the ontology of symmetry-related models. If these examples were representative, this would go some way towards rendering plausible the generalisation of the applicability of external sophistication to all cases of symmetries, as well as illustrating in more detail what is meant by this form of sophistication. It is, however, unclear whether the examples are truly successful in this regard.
Dewar’s simplest example is instantaneous electrostatics in terms of potentials (2019, pp. 489, 494). In this case (for reasons which will become clear), it will serve to be more explicit than hitherto about the structure of the KPMs of this theory. These (in order to follow Dewar’s understanding of the theory) we take to be sextuples \(\langle M \times {\mathbb {R}}, \delta _{ij} , \epsilon _{\textsf {ab}} , \gamma , \phi ^{\textsf {a}}, \rho \rangle \), where \(\delta _{ij}\) is a Euclidean metric on the three-dimensional differentiable manifold M, and \(\phi \) and \(\rho \) represent, respectively, the electrostatic potential and charge density. In addition, we have included the ‘internal’ manifold \({\mathbb {R}}\) in which \(\phi \) takes its values, the Euclidean metric \(\epsilon _{\textsf {ab}}\) on this \({\mathbb {R}}\), an ‘internal’ scalar field \(\gamma \) which picks out the origin of this space, and ‘internal’ indices on \(\phi \) (used to specify the value of \(\phi \) in \({\mathbb {R}}\) at a particular \(p \in M\)). DPMs of this theory are picked out via the dynamical equation
$$\begin{aligned} \delta _{ij} \nabla ^i \nabla ^j \phi = 4\pi \rho . \end{aligned}$$
(5)
The semantics of this theory are such that \(\phi \) takes value in \({\mathbb {R}}\). Then, adding any constant \(\kappa \) to any \(\phi \) that solves this equation generates a new solution. If we assume that all solutions differing merely by such a \(\kappa \)-shift are empirically equivalent, then this \(\kappa \)-shift constitutes a symmetry transformation (of the kind relevant to this paper—cf. Sect. 2.2). Do these symmetry-related models represent the same physical state of affairs? This question is analogous to the shift arguments in NGT set on Newtonian spacetime—albeit with an internal, rather than external (i.e., spacetime) transformation.Footnote 27
Consider a map \(\psi : \phi ^\textsf {a} \mapsto \phi ^\textsf {a} + \kappa ^\textsf {a}\) on the internal space which implements a constant shift of the value of \(\phi ^\textsf {a}\). Such a map can be used to generate a new model of this theory, \({\mathcal {M}}_{\text {shift}} = \langle M \times {\mathbb {R}} , \delta _{ij}, \epsilon _{\textsf {ab}}, \gamma , \psi _* \phi ^{\textsf {a}} , \rho \rangle \). Since the diffeomorphism acts only on the internal space, we have \( \delta _{ij} = \psi _* \delta _{ij}\) and \(\psi _* \rho = \rho \); moreover, since these transformations are a symmetry of the Euclidean metric \(\epsilon _{\textsf {ab}}\) on \({\mathbb {R}}\), we have \(\epsilon _{\textsf {ab}} = \psi _* \epsilon _{\textsf {ab}}\). However, since these transformations shift the origin on \({\mathbb {R}}\), as given by \(\gamma \), we have \(\gamma \ne \psi _* \gamma \)—meaning that such shifts of the electrostatic potential \(\phi \) do not generate isomorphic models. In this case, as in the kinematic shift, a mere ‘traditional sophisticationist’ metaphysical move (in this case anti-quidditism—see below) is not sufficient to regard these models as representing the same physical state of affairs; rather, one must also mathematically reformulate, to excise the origin \(\gamma \) of \({\mathbb {R}}\), and move to a formulation in which \(\phi ^{\textsf {a}}\) is valued in a one-dimensional metric affine space. Only then does traditional sophistication suffice as a means of regarding the models as representing the same physical state of affairs.
In the case of substantivalism, this renouncing of a (primitive) non-qualitative, transworld identity of manifold points (here, points in \({\mathbb {R}}\)) means renouncing haecceitism; in the case of electrostatics the manoeuvre goes under the name of anti-quidditism. The idea is that just as haecceities allow for the primitive identification of objects, such as spacetime points, across possibilities, so too do quiddities allow for the primitive identification of property-holdership across possibilities. It is important to note, though, that while (anti-)quidditism often (implicitly) refers to determinable properties, such as ‘having mass’ or ‘being charged’, we are here concerned with transworld identification of determinate properties, such as ‘having that mass’ or ‘having a mass of 1kg’.Footnote 28 What is at stake in our discussion of electrostatics is not determining whether the counterpart of the electrostatic potential \(\phi \) in a \(\kappa \)-shifted world is still an electrostatic potential or instead, say, a gravitational potential. The issue is whether there is a determinate magnitude of the electrostatic potential \(\phi \) in one world—say the magnitude that is represented by \(\phi \) having the numerical value zero—that can be identified with a determinate magnitude of the electrostatic potential \(\phi \) in another world. In other wor(l)ds, is there a matter of fact about determinate magnitudes of the electrostatic potential, or only about differences in magnitudes? To sum up: anti-quidditism (about determinate properties) denies the existence of quidditistic facts about the (determinate) properties in our theories. Given this thesis, no meaningful sense can be made of \(\kappa \)-shifted \(\phi \) as fields representing distinct possibilities. Terminology aside, however, both the anti-haecceitist and anti-quidditist strategies are clearly forms of traditional sophistication—for they identify distinct but isomorphic models of our physical theories.
So: on Dewar’s understanding of the electrostatics case, \(\kappa \)-shifted models are not isomorphic, but one can (in our reconstruction) implement the internal sophisticationist strategy by (a) modifying the semantics of the theory in order to excise \(\gamma \) from its models, and (b) interpreting the points in \({\mathbb {R}}\) anti-quidditistically. It is worth noting here, however, that there is another understanding of electrostatics—different from Dewar’s—in which the origin \(\gamma \) is not included in the semantics, and transworld comparison of (field values at) points in \({\mathbb {R}}\) is facilitated entirely by a quidditistic understanding of the points in that manifold; in that case (just as in the static shift in NGT), no mathematical reformulation is necessary in order to understand these models as representing the same physical state of affairs; rather, an anti-quidditist metaphysical move suffices. (Indeed, there is a sense in which including the origin \(\gamma \) of \({\mathbb {R}}\) is unnatural, as one must then make two conceptually similar philosophical moves in order to regard models of electrostatics differing at most by \(\kappa \)-shifted \(\phi \) as being physically equivalent; for this reason, a version of the theory without inclusion of \(\gamma \) is our preferred initial formulation of the theory.) In either case, though, it should be clear that what Dewar has illustrated here is an instance of internal sophistication (whether involving (a) both semantic reformulation and traditional sophistication, as on the former version of the theory, or (b) just traditional sophistication, as on the latter); the example provides no positive illustration of how external sophistication is supposed to work. Indeed, this is especially so as this particular example merely excises structure naturally associated with quidditistic differences: it remains unclear, absent further details, how the sophisticationist strategy is supposed to work in more complex cases of symmetries.
One can understand Dewar’s second example—of full-blown electromagnetism, in its vector potential formulation—as being intended to address this concern. KPMs of this theory (eschewing an explicit presentation of the ‘internal’ machinery, which we included in the previous example) are quadruples \(\langle M, \eta _{ab}, A^a, J^a \rangle \), where \(\eta _{ab}\) is a fixed Minkowski metric field on the four-dimensional differentiable manifold M, \(A^a\) is a four-vector encoding Maxwell fields, and \(J^a\) is a source term. DPMs of this theory are the Maxwell equations,
$$\begin{aligned} \nabla _a \left( \nabla ^a A^b - \nabla ^b A^a \right) = J^b , \end{aligned}$$
(6)
where \(\nabla \) is the derivative operator compatible with \(\eta _{ab}\). It is very well-known that (6) is invariant under electromagnetic ‘gauge transformations’ of the form
$$\begin{aligned} A^a \mapsto A^a - \nabla ^a \Lambda , \end{aligned}$$
(7)
for some scalar function \(\Lambda \); moreover, models of electromagnetism related by such gauge transformations are typically taken to be empirically equivalent. In this case (as in Dewar’s understanding of electrostatics), models related by (7) are not isomorphic: the transformation under consideration is the analogue of the Leibnizian kinematic shift, rather than the static shift. So: how does Dewar’s sophisticationist strategy proceed in the context of this theory?
As before, in this case, Dewar does more than merely declare that these symmetry-related models may be treated as being isomorphic, as per the external sophisticationist agenda. In addition, Dewar mathematically reformulates the theory, such that the correlates of the symmetry-related models in the original version of electromagnetism are indeed isomorphic in the reformulated theory. In this case, Dewar proposes that recourse to the machinery of fibre bundles is the appropriate strategy:
Finally, consider the electromagnetic theory. This time, models of the theory are to be connections on a principal \(U\left( 1\right) \)-bundle over \({\mathbb {R}}^4\). [Footnote suppressed.] Once more, we retain [(6)], but now interpreted in a way that makes use only of the more minimalist structure available in the models: [\(A^a\)] is now interpreted as the vector potential of the target connection relative to some arbitrarily chosen flat connection on the principal bundle. (Dewar 2019, p. 501)
Note that here, Dewar is explicitly presenting a different (more impoverished) mathematical formalism for electromagnetism—that is, a different space of kinematic (a fortiori dynamic) possibilities. In this new formalism, solutions of the theory related by (7) are isomorphic—so that one may (finally—as in the previous example) apply traditional sophistication in order to regard all such models as representing the same physical state of affairs.Footnote 29
We have absolutely no qualms about this overall (internal sophisticationist) strategy (beyond issues raised in Sect. 4.2) as affording, at least prima facie, a novel tool for articulating the ontology of symmetry-related models, compared to reduction. However, it is important to note that the mathematical reformulation here—just as in the case of reduction when faced with non-isomorphic models—was essential to the project: external sophistication of the theory under consideration was insufficient to do this, absent carrying through the project of also mathematically reformulating the theory under consideration—that is, absent the project (just as in the case of reduction) of finding a new formalism via which models of the original theory may ultimately be interpreted. Given that, in general, it may be very difficult to identify what the appropriate mathematical reformulation is supposed to consist in (a fact which Dewar also acknowledges: “it is often very opaque what kind of internal construction will correspond to an external construction.” Dewar (2015, p. 503)), in our view, the external sophisticationist strategy is insufficient as a means of articulating the ontology of symmetry-related models. Moreover, this difficulty highlights that in this case there was no guarantee that such a formulation was even possible, and a fortiori there is no universal generalisation of such a guarantee for all symmetries in all theories. Sometimes one first needs to explicitly perform non-trivial mathematical feats, such as moving to the fibre bundle formalism of electromagnetism. Without such a universal guarantee, this otherwise-promising strategy of internal sophistication is most naturally paired with motivationalism (i.e. options (e) or (f) in Sect. 3.2), instead of providing a justification of interpretationalism (option (c).Footnote 30
Finally, consider the third example presented by Dewar, which is posed in the language of first-order predicate logic. ‘Kinematically’, the objects of this theory are two predicates, L and R; ‘dynamically’, the following two sentences are satisfied:
$$\begin{aligned} \forall x&(Lx \vee Rx), \end{aligned}$$
(8)
$$\begin{aligned} \forall x \lnot&(Lx \wedge Rx). \end{aligned}$$
(9)
It is true that if we have a solution to this theory, swapping all L and R predicates—that is, predicating (only) L of all objects originally instantiating R and vice versa—takes a solution to another solution, i.e. preserves the ‘dynamics’. What is not the case is that this suffices to call such a swap a ‘symmetry’ in the relevant sense (cf. Sect. 2.2), despite Dewar’s claim to the contrary (Dewar 2019, p. 488). Without any further context, the predicates L and R could be anything—and so it need not be the case that the models related by the L/R-switch are empirically equivalent, which, as we have seen, is the sense of ‘symmetry transformation’ relevant to these considerations. (To take an example, let R stand for ‘being a rhinoceros’ and L stand for ‘being a leopard’. A world with one rhino and six leopards is not empirically equivalent to a world with one leopard and six rhinos!) In any case, let us restrict to the empirically equivalent L/R-swaps, as seems to be what Dewar has in mind when he makes the suggestion—which he presents as a mere heuristic whilst it thus actually does a lot of work—that we think of L and R as referring to the ‘left-handedness’ and ‘right-handedness’ of gloves (Dewar 2019, p. 487). (Here we should presumably further assume parity conservation, as would be true of gloves governed by, say, NGT.) As in the electrostatics case, Dewar proposes that sophistication in this case can proceed via the rejection of quiddities (Dewar 2019, pp. 498–499).Footnote 31 According to such anti-quiddittism about determinate handedness, there is no way of identifying the handedness of gloves across worlds; they are after all (at least intrinsically) qualitatively identical (paceVan Cleve 1987Footnote 32), since the shape and size of all the components and the angles between them are identical between gloves. In other words, there is no absolute, primitive sense in which a glove is ‘left-handed’ beyond it being enantiomorphically related to what we conventionally call, in that same world, a ‘right-handed’ glove (i.e., there is no primitive, non-qualitative, non-conventional determinate property of ‘being left-handed’ to be instantiated beyond such a pair of incongruent gloves each instantiating the determinable, qualitative property ‘being handed’ as well as the determinate, qualitative property ‘standing in the opposite configuration as the other glove’.) If this is what Dewar proposes, though, then this analysis is clearly once again (at bestFootnote 33) an instance of traditional sophistication—non-qualitative distinctions between isomorphic worlds are simply ‘forgotten’.
If Dewar’s examples were illustrative, representative instances of external sophistication, then this would go some way towards supporting the proposal that symmetry-related models may invariably be externally sophisticated, thereby in turn supporting the interpretational approach. Instead, however, we have seen that Dewar’s examples provide no motivation at all for external sophistication: either they are merely cases of traditional sophistication, or they explicitly appeal to mathematical reformulation (and so internal sophistication), which is a strategy more naturally paired with motivationalism than with interpretationalism. Of course, none of this proves that there could be no illustrative examples of external sophistication, nor does it provide any argument against external sophistication. Discussion of these more directly critical matters is deferred to Sect. 4.3.
On explanation
As discussed in Sect. 3.1, Dewar acknowledges the importance of retaining explanatory power in the context of symmetry-to-(un)reality inferences. In fact, we are in agreement that it is a problematic feature of some reduced theories that they exhibit explanatory deficits relative to the original theories from which they are derived (Dewar 2019; Martens 2018b). Consider, for instance, the reduced theory corresponding to the above example of electrostatics, where the invariant mathematical structure is the electric field \(E_i := \nabla _i \phi \). In order to capture the full content of the original theory, it is not sufficient to substitute this definition of the electric field into (5), to obtain
$$\begin{aligned} \delta _{ij} \nabla ^i E^j = 4\pi \rho ; \end{aligned}$$
(10)
rather, one also needs to add the condition
$$\begin{aligned} \epsilon _{ijk}\nabla ^j E^k = 0. \end{aligned}$$
(11)
In the original theory, however, the equivalent expression is a mathematical identity:
$$\begin{aligned} \epsilon _{ijk} \nabla ^j \nabla ^k \phi = 0. \end{aligned}$$
(12)
Thus, in our original formulation of electrostatics, equation (12) is an “analytic or definitional necessity rather than a “mere” law” (Dewar 2019, p. 497)—as it is in the case of equation (11) in our ‘reduced’ formulation of electrostatics. Dewar correctly points out that such a definition counts as a proper explanation on many popular accounts of explanation in philosophy of science; an explanation that is lacking in the reduced theory. The question to be considered now, then, is whether (externally) sophisticated theories invariably preserve explanatory virtues as compared with the unsophisticated theories from which they are constructed.
Dewar’s motivation for seemingly promising as much stems from the fact that sophistication does not change the equations of a given theory—it modifies only the semantics, while leaving the syntax untouched. In the case study of electrostatics from Sect. 4.1, for example, sophistication merely replaces \({\mathbb {R}}\) by a one-dimensional, oriented, metric affine space as the range of \(\phi \). This leaves \(\nabla _i \phi \) well-defined and invariant, such that equations (5) and (12) also still hold and remain well-defined, thereby preserving explanatory power. The natural question to ask at this point is, then, the following: can explanatory power arise only from the dynamical equations of a given theory? In this subsection, we give a negative answer to this question. Explanatory power may be preserved in examples such as electrostatics—but (we contend) such examples are again unrepresentative.
Our first problem case for the claim that sophisticated theories invariably preserve explanatory power can be illustrated by considering NGT set on Newtonian spacetime. Within this theory, absolute velocities are not “idly turning wheels” (Møller-Nielsen 2017, p. 1263), but are used to define the observable relative velocities. Following Dewar’s earlier claim about definitions, this should count as a proper explanation: absolute velocities ‘indirectly’ explain observable phenomena by defining/explaining relative velocities which are observable. It is then somewhat surprising that Dewar does not promise or show that the explanatory power will not decrease when such symmetry-related models are (externally) sophisticated, but instead denies that these putative explanations are truly indispensable (Dewar 2015, p. 322).
A second problem case for the claim that sophisticated theories invariably preserve explanatory power is also drawn from Newtonian gravitation. Consider force-based Newtonian gravitation, in terms of the standard set of initial variables and parameters, i.e. distance r, velocity v, and mass m. For simplicity, focus on models with two equally massive particles with zero total angular momentum. One aspect of the behaviour of these particles is encapsulated in the escape velocity formula
$$\begin{aligned} v_e = \sqrt{\frac{2Gm}{r}}. \end{aligned}$$
(13)
If the initial relative velocity \(v_0 := v\left( t=0\right) \) of the particles is larger than \(v_e\), they will end up escaping each other; otherwise, they will end up colliding. Two sets of initial conditions agreeing on their mass ratios can nevertheless be distinct if these ratios hold in virtue of distinct quidditistic absolute masses. Can these absolute masses make a difference? They can. The following transformation will, for an appropriate value of the scalar \(\alpha \), change whether the escape velocity inequality is satisfied:
$$\begin{aligned} \begin{array}{l l l} m &{} \mapsto &{} \alpha m , \\ r &{} \mapsto &{} r , \\ v &{} \mapsto &{} v. \end{array} \end{aligned}$$
(14)
These non-qualitative mappings of initial states lead to empirically distinguishable evolutions—escape versus non-escape. (This transformation is thus not a symmetry (Martens 2017, 2019a, b)—paceDasgupta 2013). Although this does not make absolute masses detectable in the same sense as relative velocities, it does make them detectable in some weaker sense, that still makes them more empirically relevant than, say, absolute velocities (Martens 2019b). Quidditistic absolute masses may thus explain detectable phenomena.
One may retort that since transformation (14) is not a symmetry, it has no relevance to the current project concerning symmetry-to-(un)reality inferences. In response to this concern, we proceed now by considering another theory for which transformation (14) is a symmetry, but in which absolute masses nevertheless play an indispensable explanatory role.
Consider the following Newtonian theory (see Martens 2019a for further discussion of this theory). (Although this theory is of course related to standard Newtonian gravity—both theories are in fact empirically equivalent—it is syntactically distinct from standard Newtonian gravity; here we will consider this theory on its own.) The gravitational force between two masses is given by
$$\begin{aligned} F_{\text {grav}} = \gamma \frac{Mm}{r^2 \sum \limits _k m_k} = \gamma \frac{M}{r^2 \sum \limits _k \frac{m_k}{m}}, \end{aligned}$$
(15)
with \(\gamma \) a constant such that in the actual world \(\gamma = G\sum \limits _k m_k\). The corresponding escape velocity depends only on mass ratios—
$$\begin{aligned} v_e = \sqrt{\frac{\gamma }{r\sum \limits _k\frac{m_k}{m}}}, \end{aligned}$$
(16)
—thereby rendering transformation (14) a symmetry (in the relevant sense—cf. Sect. 2.2) of this theory; uniform mass scalings do not lead to an empirical difference. Note, moreover, that this symmetry relates qualitatively indistinguishable models.
There is some ambiguity as to how one should proceed with sophisticating this theory, as there are at least two options available for ‘forgetting structure’ (without modifying the syntax).Footnote 34 The most obvious option is to throw away absolute masses altogether. It was for this exact reason—obtaining an ontology with no absolute masses but only mass ratios \(m_{ij}\) (i.e., primitive and symmetric relations of ‘comparative masshood’ holding between particles i and j)—that this theory was designed in the first place (Martens 2019a). However, if the fundamental ontology in this theory consists only of mass relations, there is nothing to ensure the transitivity of those mass relations,
$$\begin{aligned} m_{ij} = m_{ik} \cdot m_{kj}, \quad \text {for any three particles } i,j,k. \end{aligned}$$
(17)
To make matters worse, without such a transitivity constraint holding, one could not even coherently interpret these mass relations as ratios in the first place (Martens 2019a). Only if one commits to absolute masses in virtue of which the mass ratios hold does one explain the transitivity of those mass relations, as the following becomes a mathematical truth (Armstrong 1988; Martens 2018a, 2019a; Roberts 2016; Russell 1903):
$$\begin{aligned} \frac{m_i}{m_j} = \frac{m_i}{m_k} \cdot \frac{m_k}{m_j} \quad \text {for any three particles } i,j,k. \end{aligned}$$
(18)
Recall that Dewar agrees that this constitutes an explanation. Thus, in this theory, absolute masses may not be invariant under the symmetry transformation (14), but they are nevertheless explanatorily indispensable: they ensure transitivity of mass ratios as a matter of mathematical fact. It is hard to see how any other ontology could dispense with absolute masses whilst retaining this explanation.
This pushes one towards the second option for sophisticating this theory: retain the absolute masses but ‘forget’ just their quiddities. (As with the previous option, the syntax is not modified.) As Jacobs argues, this suffices to explain the transitivity of mass relations (Jacobs 2019). This therefore seems the best option for a weak interpretationalist. It should be noted though, once more, that the original motivation for introducing (15) was to dispose of absolute masses in virtue of which the mass ratios obtain.Footnote 35 It goes against the spirit of interpretationalism to retain more than what is invariant under the symmetry. Moreover, this confirms once again that explanatory considerations—mass ratios obtain in virtue of, i.e. are metaphysically explained by, underlying absolute masses (albeit non-quidditistic ones)—that go beyond syntax and dynamical equations can play a crucial role in determining the metaphysical consequences of symmetries. Furthermore, even if this option is indeed the best of both options, the fact remains that a non-trivial choice between two options had to be made. The whole point of external sophistication and of interpretationalism is that one is supposed to be able to draw immediate metaphysical consequences from the symmetries of a theory. The fact that we needed to do some work (cf. our discussion of gauge transformations within electromagnetism in Sect. 4.1), that we needed to make non-trivial decisions before the appropriate ontology was revealed, resonates more with motivationalism than interpretationalism.
Let us finally and briefly return to another problematic symmetry: that of gauge shifts of the electromagnetic vector potential (Sect. 4.1), but specifically in the context of the Aharonov-Bohm effect (Aharonov and Bohm 1959). This context has been discussed frequently and in detail in the literature—e.g. Dewar (2019), Healey (2007), Martens (2018b)—so we will not dwell on it here. Suffice it to say that, although we agree that the sophisticated theory can recover analogues of (12) and (17), to the extent that this counts as an explanation it is not an explanation that is, in all relevant senses, separable (pace Jacobs 2019), as Dewar acknowledges (Dewar 2019, fn. 56). At the same time, there is a reduced theory available that is local in all relevant senses (Wallace 2014).Footnote 36
To sum up: the promise of universal external sophistication that explanatory powers will invariably be preserved is motivated by the fact that such sophistication leaves the dynamical equations untouched. We agree that this ensures that explanations provided by those dynamical equations are preserved, and we have seen that this constitutes an improvement upon reductionism for some cases. However, this motivation remains silent on other types of explanation (as well as on other super-empirical criteria). Dewar has thus not provided a universal argument for explanatory power being preserved after sophistication. The examples in this subsection indicate that there cannot be such argument, as sophistication fails to invariably preserve explanations that do not derive from the dynamical equations. On both the weak interpretational approach and the strong motivational approach to symmetry-to-(un)reality inferences, loss of such explanatory power may then be considered as a reason for blocking these inferences, even when these approaches are (partially) combined with (traditional or external) sophistication.
On sophistry
If the examples presented by Dewar (2019), discussed in Sect. 4.1, do not support his proposal for the universal application of external sophistication, might there be independent positive reasons for believing in this proposal, as there are for the restricted application of sophistication (i.e., the traditional notion)? Or (weaker still), can any transparent examples of external sophistication of non-isomorphic symmetry-related models be given?
We are sceptical. To stipulate that qualitatively distinct, i.e. non-isomorphic models, such as models of NGT related by kinematic shifts, are nevertheless isomorphic reads prima facie as nothing more than a flat-out contradiction.Footnote 37 The burden falls on the advocate of external sophistication to explicate what (s)he has in mind here—for to simply insist that all symmetry-related models be regarded as being isomorphic simply appears to be begging the interpretative question—of Russellian theft over honest toil (Russell 1919, p. 71).
We submit that Dewar is seeking to have his cake and eat it. He cannot. One natural worry to have about the external sophisticationist programme is that it is not (fully) realist in spirit to begin with.Footnote 38 As is evident from Sect. 3.1, Dewar suggests that one can be a realist (simpliciter) without having to make any commitments as to which parts of a theory one is realist or anti-realist about—that is, without having to commit to realism about anything specific. This is certainly consonant with external sophistication—but in that case, it is often opaque what reality is being subscribed to; moreover, it is often unclear what grounds or explains or justifies the physical equivalence of models that, on a natural interpretation, represent distinct possible worlds (Møller-Nielsen 2017). Dewar concedes as much when he states that it is “often very opaque” (Dewar 2019, p. 503) what kind of semantics (if any!) corresponds to this stipulation that symmetry-related models should be considered ‘as if’ they are isomorphic. We confess, in line with Møller-Nielsen (2017) and Sider (2018, §5), that we find it difficult to make sense of such an opaque realism.
In this context, it is helpful to emphasise the following belief which we do share with Dewar:
if we want to know the answers to specific questions about the nature of a theory’s ontology and ideology, then [a reformulated theory] is invaluable. (Dewar 2015, p. 326)
We take it that a complete and honest form of realism should not only take these questions—such as the question of which parts of a theory one is to be realist about and which parts one is not—on board, but consider them to be crucial. Leaving them out is at best a dishonest form of realism, and at worst a form of anti-realism. Treating qualitatively distinct models related by a symmetry ‘as if’ they are isomorphic does not help one do the hard interpretative work that is required of realism.