What theoretical equivalence could not be

Abstract

Formal criteria of theoretical equivalence are mathematical mappings between specific sorts of mathematical objects, notably including those objects used in mathematical physics. Proponents of formal criteria claim that results involving these criteria have implications that extend beyond pure mathematics. For instance, they claim that formal criteria bear on the project of using our best mathematical physics as a guide to what the world is like, and also have deflationary implications for various debates in the metaphysics of physics. In this paper, I investigate whether there is a defensible view according to which formal criteria have significant non-mathematical implications, of these sorts or any other, reaching a chiefly negative verdict. Along the way, I discuss various foundational issues concerning how we use mathematical objects to describe the world when doing physics, and how this practice should inform metaphysics. I diagnose the prominence of formal criteria as stemming from contentious views on these foundational issues, and endeavor to motivate some alternative views in their stead.

How to Be a Spacetime Substantivalist

“…In all of this I am very much in agreement with how Dasgupta (2011) sets up the issues, who forcefully drives home how questions about mathematical solutions are largely irrelevant to the debate. 24 For further discussion and defense of the claims in this paragraph, see Teitel (2021). Compare also Earman (1986b, 236-37), where he describes the process of declaring some equivalence class of solutions merely gauge different as "easy words," and disparagingly contrasts proposing an actual metaphysic with the less impressive feat of "drawing circles around groups of space-time models and labeling them equivalence classes.…”

“…Mathematical doctrines like no-mathematical-shifts, despite their prominence in extant discussions, simply fail to make contact with the questions about the metaphysics of spacetime and modality at issue. They thus do not offer a way around engaging with the ques- 23 The point in the preceding three paragraphs was a central theme throughout Teitel (2019a) and Teitel (2021).…”

Symmetries and Representation

“…In particular, what we say here suggests that an interpretationalist cannot simply declare that symmetry‐related models are equivalent (e.g., see Dewar [2019] or Bradley [2023]) because even if they try to stipulate that, the stipulation might simply fail to achieve anything because of background norms and stipulations elsewhere. For more on the debate about whether principles in the vicinity of Sym1 and Sym2 could be used to infer interesting conclusions about shifted worlds see, in particular, Teitel (2021) and Jacobs (2023).…”

“…In particular, what we say here suggests that an interpretationalist cannot simply declare that symmetry-related models are equivalent (e.g., seeDewar [2019] orBradley [2023]) because even if they try to stipulate that, the stipulation might simply fail to achieve anything because of background norms and stipulations elsewhere. For more on the debate about whether principles in the vicinity of Sym1 and Sym2 could be used to infer interesting conclusions about shifted worlds see, in particular,Teitel (2021) andJacobs (2023).41 It is worth pointing out that the symmetry literature has focused most of their attention on the question of whether two models are "equally capable" or "equally apt" to represent the same situation, but have not discussed in detail the question of what it means to say that a particular model is capable of representing a particular situation.42 Up to certain approximations that depend on the context, e.g., at low speeds, we do not need to consider relativistic laws.43 In some cases philosophers talk of models of physical theories representing possible worlds. By this, they do not mean that the model explicitly represents each and every fact that obtains in that world.…”

Equivalence and Convention

“…Indeed, one might even think that providing this kind of interpretation isn't even needed in addition to the sort of "internal" interpretational work I outlined earlier; one could argue that once it has been specified what the mathematical structures represent, that will determine when two mathematical structures represent the same thing. Wilhelm's contribution to this symposium makes just such a claim, as do Coffey (2014) and Teitel (2021)-the latter of whom, incidentally, also describes interpretations as "mappings from representational vehicles to contents" (4125). Now, it's surely true in some sense that an interpretation consists of a mapping from representations to contents.…”