Teitel (2021) argues that formal approaches to equivalence cannot be illuminating, since they run afoul of trivial semantic conventionality: the idea that “any representational vehicle can in principle be used to represent the world as being just about any way whatsoever.” In this paper, I consider the relationship between theoretical equivalence and convention. First, I review the notion that conventions may give rise to equivalences: in particular, that we should regard theories differing “merely by a choice of convention” as equivalent to one another. I consider the importance of this idea for both the debates over geometrical conventionalism, and for Carnap’s Principle of Tolerance (Carnap 1934). However, this idea encounters difficulties when we confront it with the following question: can there be different conventions about equivalence? That is, consider two communities which use the same representational vehicles; but whereas one community regards those vehicles as equivalent when they stand in a certain relation, the other community does not. On the face of it, affirmation of the Principle of Tolerance would appear to force the Carnapian to side with the former community, suggesting that tolerance has its limits—in other words, that it cannot tolerate the intolerant. I suggest that this conclusion is too hasty. A thoroughgoing Carnapianism is, in fact, possible. In order to do so, we should apply the Principle of Tolerance at two different ‘levels’. On the one hand, at the meta-level, it instructs us to regard these two communities as working in different frameworks, which differ from one another over which inferences are permissible but not over substantive matters of fact. On the other hand, at the object-level, it constitutes an advertisement for the more liberal framework: here, it amounts to the observation that admitting more equivalences, and hence permitting more inferences, has pragmatic advantages. I return to the question of trivial semantic conventionality. If we subscribe to the Principle of Tolerance, do we risk collapsing to the position where all representational vehicles whatsoever have the same content? I argue that this is not the case, and that the above analysis indicates why. Although the Principle of Tolerance is presented as a general injunction to regard frameworks as equivalent, there is always the possibility of semantically ascending; when we do so, we are considering the merits of a (meta-)framework that regards those frameworks as equivalent, relative to one that does not do so. And although the (object-level) Principle of Tolerance constitutes a pragmatic advantage for the more tolerant framework, it may be outweighed by other pragmatic considerations. I conclude by suggesting that this gives us the resources to understand the significance of formal work on theoretical equivalence. The existence of an appropriate formal relationship between two theories indicates that there are not significant pragmatic advantages to distinguishing them—and in particular, suggests that regarding the two theories as equivalent will not obstruct our empirical theorising. Hence, a formal equivalence is what licences application of the (object-level) Principle of Tolerance in cases such as these.

I formulate an account of theoretical equivalence for effective quantum field theories. To start, I propose the `Easy Ontology' approach to interpreting what effective theories say about the physical world. Then I show how the Easy Ontology approach can be used to articulate an account of theoretical equivalence. Finally, I discuss the relationship between this account of equivalence and accounts based on formal, mathematical structures. According to the Easy Ontology approach, for each energy level Λ and each effective theory ????Λ at that energy level, the physical propositions described by ????Λ are propositions about quantum fields, particles composed of quanta, physical interactions among fields and particles, transition amplitudes, correlation functions, structures, and so on, all of which interact at Λ. Some effective theories are formulated using terms that denote physical fields rather than physical particles; the contents of those theories include propositions that mention fields only. Other effective theories are formulated in terms that denote physical fields and physical particles both; the contents of those theories include propositions that mention fields and particles. There are further questions, of course, about which of these theories are true, which are more fundamental, which theories' mathematical expressions provide the most perspicuous representation of reality, and so on; and the Easy Ontology approach does not answer questions like these. But with regards to the question of what these theories actually say about the world, there is—according to the Easy Ontology approach—a very straightforward answer: what exists, according to these theories, is whatever these theories say exists. The Easy Ontology approach can be used to formulate an attractive account—call it the `Same Content' account—of theoretical equivalence among effective theories. According the Same Content account, effective theories ????1,Λ and ????2,Λ are equivalent just in case they describe exactly the same class of physical propositions. So ????1,Λ is equivalent to ????2,Λ just in case the physical propositions they describe—as given by the Easy Ontology approach—are the same. The Same Content account preserves certain basic insights of the literature on formal structures, formal symmetries, and theoretical equivalence. According to some effective theories, various formal, mathematical structures correspond to structures in the physical world. And the Same Content Account implies that for two such theories to be equivalent, those physical structures must be identical. But according to the Same Content account, there is more to theoretical equivalence than any one formal criterion captures. (In this way, Same Content respects both a view of equivalence discussed by North (2021) and a pluralist approach to equivalence discussed by Dewar (2019).) In fact, even isomorphism is not sufficient for equivalence: two effective theories may be mathematically isomorphic, and yet be non-equivalent, according to the Same Content account. (In this way, Same Content respects the view—a version of which is discussed by Bradley (this symposium)—that isomorphic theories may be physically distinct.) Theoretical equivalence is, ultimately, a matter of sameness of theoretical meaning; not (solely) a matter of formal correspondence.

How should one remove “excess structure” from a physical theory? Dewar (2019) presents two ways to undertake such a task: first, one could move to a reduced version of the theory, where one defines the models of the theory only in terms of structure that is invariant under the symmetries of the theory. Second, one could move to a sophisticated version of the theory, where symmetry-related models of a theory are isomorphic. Dewar argues that despite these alternatives attributing the same structure to the world, the sophisticated version can have explanatory benefits over the reduced version. Here, I consider a different argument that might be considered in favour of the sophisticated view: the differences between isomorphic models in a sophisticated theory can have representational significance. Thus, a reduced version of a theory may be unable to distinguish physical situations that can be distinguished in the sophisticated version of the theory. That isomorphic models can be used to represent distinct situations has been argued before (for instance, Belot (2018); Fletcher (2018); Roberts (2020)). However, these arguments do not directly show that the reduced version of a theory is unable to represent such situations. Indeed, if the reduced version of a theory posits the same structure as the sophisticated alternative, how can the sophisticated version represent a greater number of physical possibilities? I will argue that this tension can be resolved by considering more carefully the ways that isomorphic models can be used to represent distinct situations. This will provide support for the claim that the sophisticated version of a theory can be representationally advantageous to its reduced version, without rejecting the claim that the sophisticated and reduced version of a theory ascribe the same structure. In particular, I will argue that the sense in which isomorphic models can represent distinct situations is that one has the freedom to define additional structure within the models of the theory that can be used to represent a physical standard of comparison. It is the ability to define this additional structure that may be lacking in the reduced version of the theory, since one may not have the same representational freedom within the models of the reduced theory. I will use this argument to make a more general claim: the language that one uses to define the models of a theory is not always merely a conventional choice. One may choose one way of representing the models of a theory because it provides one with the resources to define additional structure that plays a physical role, even though this structure is not part of the structure of the theory. I suggest that more attention should therefore be paid to the role that model theory plays in determining which version of a theory should be adopted.

There has been a widening divide between two broad approaches to theoretical equivalence in physics: to what we mean when we say that two physical theories are fully equivalent, saying all the same things about the world but perhaps in different ways. On the one side is the formal approach to equivalence. Formal accounts say that physical theories are equivalent when they are formally or structurally or mathematically equivalent (in addition to being empirically equivalent). Proponents then work on figuring out which formal notion is the right one. On the other side is a growing resistance to formal approaches. Opponents note that physical theories consist of more than their formal apparatus, so that questions concerning the equivalence of theories must involve more than their formal features. They point to cases of theories that are equivalent in various formal and empirical respects, but nonetheless differ in what they say about the world. Some have gone so far as to conclude that the formal results being generated have no significance beyond pure mathematics. I advocate a middle ground. A formal equivalence of the right kind is important to questions of equivalence in physics: this is necessary (if not sufficient) for wholesale theoretical equivalence, as we can see in some familiar examples. More, it is not immediately clear, and is worth investigating, what type of formal equivalence is relevant to reasonable judgments of equivalence in physics. At the same time, since any formalism can be made to represent any kind of physical reality simply by brute stipulation, it’s also not right to claim that a formal equivalence must be physically significant in all cases, without further ado. A more nuanced position is in order. In actual scientific practice and theorizing, the choice of formalism is not completely up for stipulational grabs, in that there are better and worse choices of representational vehicle, given standard theoretical criteria: there are good scientific reasons for choosing one formalism over another. Some interpretive and physical stipulations will be made, that is, but certain choices of formalism will be more natural or well-suited than others, given those assumptions. (We should distinguish between the equivalence of descriptions, in the sense of their saying or representing the same things, and the relative naturalness or well-suitedness of descriptions, in the sense of their saying or representing those things in better or worse ways.) As a result, we can learn things of physical significance by examining a theory’s (best) formulation, and its formal relationships to other mathematical formulations. We just have to be careful to mind the stipulations we make, and to be explicit about the different respects, formal and not, in which theories can be equivalent to one another. I will give examples of familiar, reasonable judgements of equivalence and naturalness in physics to illustrate all this; suggest that, properly understood, a structural equivalence of some kind is necessary for wholesale equivalence in physics; and draw connections to the structured view of scientific theories recently advocated by Hans Halvorson.