2 Removing excess structure

The background situation is that we have a theory that we believe has excess structure, in the following sense: there are models of the theory that are distinct, or more precisely, non-isomorphic, that nonetheless are related by a symmetry. ^{Footnote 2} The symmetry between these distinct models is such that we think these models ought to be taken to represent a single physical situation. And yet, in being distinct models, the natural interpretation of the theory is that these models represent distinct physical situations. Therefore, the theory attributes structure to the world that we do not believe corresponds to anything physical.

There are several interpretive assumptions in the background of this argument for the presence of excess structure. ^{Footnote 3} For one, there is an assumption that when models are non-isomorphic, they cannot be interpreted as representing the same physical situation. This relies on taking seriously the claims of the theory in the sense that differences between the models of the theory are taken to correspond to differences in the physical situations represented by such models. Inasmuch as we take the aim of physical theories to be that they capture the structure that the world and its parts have, this seems to be a natural assumption. Notice, however, that the argument does not seem to require that we take our theory to be a full description of all the structure of the world; only that the structure it does describe corresponds to structure in the world. We will return to this point later on.

Given that we find ourselves in such a situation, the aim is to find a way to adapt the theory such that it no longer has excess structure. In a recent paper, Dewar (Reference Dewar2019) presents two ways to do this.

The first is reduction. Reduction alters the syntax of a theory by defining the theory in terms of quantities that are invariant under the relevant symmetries. This effectively equivocates between the non-isomorphic models related by the relevant symmetries in the original theory such that they correspond to identical models in the reduced theory. This is arguably the standard method for removing excess structure from a theory.

The second is sophistication. Sophistication keeps the syntax of the theory the same but instead alters the semantics of the theory with respect to which the theory is interpreted so that the relevant symmetries act as isomorphisms under the new semantics. ^{Footnote 4} Inasmuch as isomorphisms preserve the structure of the models of a theory, the natural interpretation of isomorphic models is that they correspond to a single physical situation. Therefore, one has removed excess structure not by redefining the models of the theory, but by redefining the symmetries of the theory. ^{Footnote 5}

To see this distinction in action, we will consider an example from Dewar (Reference Dewar2019) of a theory that describes physical situations in which every object is “handed.” Call this theory ${T_{\rm{H}}}$ in the signature ${\rm{\Sigma }} = \left\{ {L,R} \right\}$ and take it to be described by the following axioms, which say that every object is (excusively) either “left” or “right” handed:

$$\matrix{ {\forall x\left( {Lx \vee Rx} \right),} \hfill \cr {\forall x\neg \left( {Lx \wedge Rx} \right).} \hfill \cr } $$

This theory has excess structure in the following sense. Consider, for example, two models of the theory with the same domain where in one model all objects have the property $L$ while in the second model all objects have the property $R$ . These models are non-isomorphic within ${T_{\rm{H}}}$ ; the map taking $L$ to $R$ and vice versa doesn’t preserve the extension of these properties. And yet, these models are symmetry-related: intuitively, if we imagine the physical situation represented by one of these models, it is indistinguishable from the physical situation represented by the other model; both models correspond to a situation where everything is handed in the same way. More generally, any two models that are related by “flipping” the handedness for each object are symmetric in this way. Therefore, there are distinct models in the theory that are symmetry related, and this indicates that ${T_{\rm{H}}}$ has excess structure.

To describe a reduced version of the theory, ${T_{\rm{R}}}$ , we need to define the theory in terms of quantities that are invariant under the relevant symmetry. An obvious candidate is a congruence relation, ${Cxy}$ , that specifies whether two objects have the same handedness or not. It can be defined via

$$\forall x\forall y\left( {{Cxy} \Longleftrightarrow \left( {\left( {{Lx} \wedge {Ly}} \right) \vee \left( {{Rx} \wedge {Ry}} \right)} \right)} \right).$$

We can then specify ${T_{\rm{R}}}$ in terms of axioms for ${Cxy}$ that say that this relation is an equivalence relation with two equivalence classes.

To define a sophisticated version of the theory, ${T_{\rm{S}}}$ , we need to alter the semantics of ${T_{\rm{H}}}$ such that the symmetry-related but non-isomorphic models of ${T_{\rm{H}}}$ are treated as isomorphic. We can do this by defining an invertible homomorphism $h$ from a model $m$ to a model $n$ that consists of a map ${h_1}:\left| m \right| \to \left| n \right|$ and a bijection ${h_2}:\left\{ {L,R} \right\} \to \left\{ {L,R} \right\}$ such that

$$\matrix{ {{h_1}\left[ {{L^m}} \right] = {{({h_2}\left( L \right))}^n},} \hfill \cr {{h_1}\left[ {{R^m}} \right] = {{({h_2}\left( R \right))}^n}.} \hfill \cr } $$

This homomorphism is such that it can map “left” hands to “right” hands across models (and vice versa). ^{Footnote 6} In this way, the homomorphism need not preserve the extension of the properties given by $L$ and $R$ and so there is no longer a well-defined notion of trans-model identity for the terms “left” and “right.” Thus, while models related by a change in handedness for all objects were non-isomorphic in ${T_{\rm{H}}}$ , they are isomorphic in ${T_{\rm{S}}}$ .

3 The representational benefit of sophistication

The alternative reason to prefer sophisticated theories is that they can have representational benefits: Isomorphic models in a sophisticated theory that correspond to identical models in a reduced theory can be used to represent distinct physical situations in a way that the corresponding reduced models cannot.

Varieties of this point have been made previously. For one, there is a strand of literature on the empirical significance of symmetries that aligns closely with the idea that for representing subsystems, symmetry-related models can characterize empirically distinct situations. ^{Footnote 9} Meanwhile, Belot (Reference Belot2018) and Fletcher (Reference Fletcher2020) argue that there is a sense in which isomorphic models can be said to generate distinct possibilities through the maps that relate them. Finally, the partial observables approach to gauge variables pioneered by Rovelli (Reference Rovelli2004) presents a picture under which symmetry-variant features of a theory can be representationally useful. However, I take the novelty of my approach to be the focus on solving the following puzzle:

My resolution to this puzzle draws on a distinction between two kinds of structure that one can define in the context of a theory, which I call “theoretical structure” and “auxiliary structure” respectively.

Theoretical structure is the structure that a theory attributes to the world in virtue of its “invariant” content: the content that is invariant under isomorphisms of the models of the theory. In other words, when models are isomorphic, they are equivalent in terms of theoretical structure. Moreover, one can exemplify the theoretical structure through the equivalence classes of the theory. This is indeed the standard way of explicating mathematical structure, and it is the structure that notions such as “categorical equivalence” aim to capture.

Auxiliary structure is the structure that one can define in the models of a theory in virtue of the way that one characterizes the theoretical structure. The auxiliary structure goes past the theoretical structure in that the mathematical tools one uses to characterize the theoretical structure may have further resources, such that one can differentiate more structure than simply the theoretical structure. In particular, it is only through auxiliary structure that one can talk about differences between isomorphic models. Of interest here is precisely the auxiliary structure that goes past the theoretical structure in this way.

To see this distinction in the handedness theory example, consider a model $M$ of ${T_{\rm{H}}}$ consisting of two objects labelled $a$ and $b$ where ${R^M} = a$ and ${L^M} = b$ . Now consider a second model $N$ of the theory that consists of permuting the domain of the first model and pushing forward the properties such that ${R^N} = b$ and ${L^N} = a$ . These two models agree on theoretical structure since they agree on how many objects have each handedness property. However, there is a difference between the models that one can describe using auxiliary structure: in model $M$ object $a$ is right-handed, while in model $N$ , $a$ is left-handed (and vice versa for object $b$ ).

While reference to the domain of the models is one place where auxiliary and theoretical structure can diverge, the importance of these two kinds of structure in the handedness theory example arises when thinking about how they differ between the original, reduced, and sophisticated theories. In ${T_{\rm{H}}}$ , theoretical structure and auxiliary structure both include handedness structure, i.e. structure through which one can define “left” and “right” hands as distinct. In ${T_{\rm{R}}}$ , neither theoretical structure nor auxiliary structure include handedness structure; while one can say whether hands are congruent or not, one cannot say which are “left” or “right.” ^{Footnote 10} In ${T_{\rm{S}}}$ , theoretical structure and auxiliary structure diverge in a special way: the theoretical structure does not include handedness structure but the auxiliary structure does, since, unlike in the reduced theory, one has access to the properties $L$ and $R$ in describing the models of the theory.

Usually, theoretical structure is understood to give the content of a theory and auxiliary structure is regarded as mere descriptional redundancy; it has to do with one’s pragmatic choice of representation. On this understanding, isomorphic models have equivalent content and any differences between them described by auxiliary structure do not have any bearing on the representational capabilities of these models. However, I will argue against this view: auxiliary structure is not just descriptional redundancy but can have a representational role, and this is true even if one understands the theoretical structure to give the content of the theory. This will provide support for the claim that the sophisticated version of a theory can have representational benefits over the reduced version. I argue for this position through three claims.

This claim captures one made elsewhere that isomorphic models have the same “representational capacities” (Weatherall Reference Weatherall2018; Fletcher Reference Fletcher2020). However, the emphasis here is on the role of theoretical structure—it is the theoretical structure that determines the extent to which models are able to represent some physical situation. The reason is that it is the theoretical structure that captures the physical content of the models of the theory, in the sense of being the structure attributed by the theory to the world.

The second claim is that there is an importantly different sense in which isomorphic models in a sophisticated theory can be physically distinguished.

In order to unpack this claim, let us return to the models of the sophisticated theory that consist of only “left-handed” objects and only “right-handed” objects respectively. One might argue that there is a natural sense in which they can represent distinct situations: they can represent two physical situations where the objects are in different congruence classes. But how is this compatible with the fact that these models have the same invariant structure that captures the physical content of the theory?

One response is to say that one could simply stipulate the interpretation of the models to be different, or, following Wallace (Reference Wallace, Read and Nicholas2022),

So in the above example, one might say that one could stipulate a standard of “left” across the models such that they represent different physical situations relative to this standard. But where does this representational convention come from, physically? In particular, since the theoretical structure is not sensitive to such choice, in what sense can one impose it on the models in order to distinguish the interpretation of these models?

Here is where auxiliary structure comes into play. The sense in which one can impose a physical representational convention across isomorphic models of a sophisticated theory, I argue, is that one can use the auxiliary structure of the theory, through which one can distinguish these models descriptionally, to represent further details about the physical situations represented by these models. When one can use auxiliary structure to represent an additional system that acts as a reference frame, one can distinguish the situations represented by isomorphic models that differ relative to the fixed auxiliary structure. For example, in the sophisticated models discussed above, one can use the auxiliary structure corresponding to the ability to define left- and right-handed objects to define a new object that is stipulated to be “left handed” in both models, such that, relative to this hand, the models are distinct; one corresponds to the world where everything is congruent to this new hand, and the other corresponds to the world where everything is not congruent to the new hand. This use of auxiliary structure as defining a reference frame allows one to impose a representational convention that has a physical interpretation.

However, this would be of no interest if auxiliary structure could play the same role in the reduced theory. This leads to the final claim.

The reasoning is as follows. The reduction process equivocates between isomorphic models in the sophisticated theory that are non-isomorphic in the original theory; they correspond to identical models in the reduced theory. Therefore, while one could say that certain isomorphic models in the sophisticated theory correspond to distinct physical situations by using auxiliary structure to define a reference system, one cannot say the same of the corresponding identical models in the reduced theory.

To see this clearly, consider the same models as before and consider adding a new object that is handed in some way. We do not have the resources to stipulate that it represents a “left” or “right” hand in the reduced theory; we can only stipulate that it represents, for example, some hand congruent to all the hands in a model. But now, inasmuch as the models are identified in the reduced theory, the additional structure will not be able to distinguish these models by fixing its interpretation across them; if it is congruent to the hands in the first model, it is congruent in an identical model. Therefore, while physically relevant distinctions could be drawn between isomorphic models in the sophisticated theory using auxiliary structure, one cannot draw such distinctions in the corresponding reduced models.

To say that the additional hand is congruent to the objects in one model but not in an identical model, one would need to stipulate some additional fact about the difference between the two models, namely that all the hands in one model are in a different equivalence class to those of the other. But this fact can only be specified when talking about the models as subsets of a “larger” model that includes the objects of both models, where the difference between the congruence of the objects can be specified in terms of theoretical structure. But this move is something that can also be made in the sophisticated theory; what is lacking in the reduced theory is the ability to distinguish these models without stipulating additional inter-model relations that rely on theoretical structure.

The combination of these three claims highlights that isomorphic models in a sophisticated theory play a multi-faceted role: they can be used to represent a single physical situation (Claim 1), and they can be used to represent distinct situations through a physical interpretation of the auxiliary structure as, for example, a fiducial system (Claim 2). ^{Footnote 11} Inasmuch as this multi-faceted role is beneficial, Claim 3 demonstrates that the sophisticated version of a theory can have a representational advantage over the reduced version. Moreover, these three claims are compatible with the sophisticated and reduced theories being theoretically equivalent in the sense of attributing the same theoretical structure. We therefore have solved the puzzle stated earlier: there is no tension between a sophisticated theory having representational benefits while also being theoretically equivalent to a reduced theory.

This representational argument for sophistication is arguably more robust than the explanatory argument given by Dewar (Reference Dewar2019), for the following reasons. First, one can point to precisely why a theory is superior if its models have the resources to represent a greater number of physical distinctions than the models of another theory: inasmuch as these physical distinctions are ones that one thinks a theory ought to capture, a theory whose tools prevent one from representing them is lacking. Second, unlike the fact that one can point to explanatory benefits of the reduced theory, there is not a corresponding representational benefit that can attributed to the reduced theory, inasmuch as reduction always constrains auxiliary structure. Therefore, the two worries raised previously for the explanatory account are bypassed.

However, the representational argument is aligned with the explanatory argument in the sense that part of the explanatory benefit of sophistication might be seen to come from its representational benefits. For example, consider the fact that two subsystems that each consist only of congruent hands can be such that their combined system does not consist only of congruent hands. This fact can be explained in the sophisticated theory through the difference in auxiliary structure between the representation of the subsystems: using auxiliary structure as a physical representational convention distinguishes the subsystems even though the models are equivalent in terms of theoretical structure. In the reduced theory, there are no relevant differences in auxiliary structure between the models representing these subsystems, and so the difference can only be given by asserting further statements about the relation between the subsystems. Therefore, one might argue, this fact falls out naturally from the sophisticated theory but must be stipulated in the reduced theory. And so, the representational power of sophisticated theories lends itself to explanatory benefits.