## 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.

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## Notes

- 1.
For a sampling see Barrett (2015, 2019), Barrett and Halvorson (2016a, 2016b, 2017), Butterfield (2018), Curiel (2014), Coffey (2014), Glymour (2013), Halvorson (2012), Halvorson (2013), Hudetz (2019), North (2009), Rosenstock et al. (2015), Teh and Tsementzis (2017), Tsementzis (2017), and Weatherall (2015). For some older work on the topic see Glymour (1970, 1977), Quine (1975), Sklar (1982), and Putnam (1983). For a helpful overview of the literature see Weatherall (2019). There has also been a burgeoning interest in the related topic of dualities. Much of what I will say also bears on this topic. However, discussing dualities explicitly would require another paper, so I shall confine my attention here to theoretical equivalence.

- 2.
No label here is perfect, but I have found ‘semantic equivalence’ to be the least misleading. Another option would be ‘worldly equivalence’. Other labels one finds in the literature for the target phenomenon include ‘metaphysical equivalence’, ‘full equivalence’, ‘interpretational equivalence’, and ‘representational equivalence’. Readers should feel free to substitute whichever label they prefer throughout.

- 3.
I should flag that I think there are problems with lumping sentences together with mathematical objects in this way. In particular, I am skeptical of the common practice of treating mathematical objects as things that, like sentences, might be or fail to be semantically equivalent to one another. When we use a sentence to describe the non-mathematical world, we do so by using it to express some proposition or content. By contrast, when we use a mathematical object like a set of mathematical models to describe the non-mathematical world, we do say by saying something

*about*that object and the non-mathematical world, usually highlighting some salient respect in which the two are similar. There is little sense in asking, even on some particular occasion of use, what a mathematical model “says about the non-mathematical world”; rather, it is similar in certain respects and different in others. Some of my skepticism about this contrast will crop up below, but I will try to set it aside as much as possible, and acquiesce in the standard practice of treating mathematical objects as things that may be semantically equivalent to one another. Doing so allows me to focus on my concerns about formal criteria of equivalence in particular. - 4.
Choice of font might affect the

*truth*of certain token sentences given standard representational conventions (consider ‘this sentence is written in Times New Roman’). However, the issue in the main text concerns the bearing of font choice on the proposition expressed. - 5.
See Soames (2003, ch. 12–13) for an overview of some reasons for the fall of positivism. In the main text I described the standard characterization of the positivist program, and my comments are directed at the program only understood in this way (according to which it is committed to a flat-footed empiricist criterion of semantic content). An anonymous referee points out that some commentators argue that the positivists in fact held more sophisticated and defensible views than the standard characterization would suggest. For discussion, see Friedman (1999) and Creath (2020).

- 6.
See Barrett and Halvorson (2016a) for a rigorous presentation.

- 7.
Notably, as Weatherall appreciates, for Glymour’s purposes the need for first-order formulations does not arise. We shall see that he regarded definitional equivalence only as a

*necessary*condition for semantic equivalence, and, in the cases he was interested in, the “uniqueness” clause sufficed for his results (which concerned verdicts about only*in*equivalence). - 8.
Compare also van Fraassen (2014, 279): “If the same diffusion equation is presented to describe gas diffusion and, elsewhere, temperature distribution over time, would anyone think that one and only one theory was being presented? [...] A representation has content. A representation of gas diffusion is not the same thing as a representation of temperature distribution, even if the math is the same.” Though because this example involves empirically inequivalent contents, it will likely not worry proponents of formal criteria, for reasons I outline below in the main text.

- 9.
- 10.
- 11.
And the same would be true even of candidates for what physicists sometimes describe as a “total,” “complete,” or “final” theory, such as string theory or some other candidate theory of quantum gravity.

- 12.
Putnam (1983, 39) is aware of this challenge. He offers a list of some candidate phenomena in the context of different Lorentz frames in Special Relativity. Still, one wants some precise characterization of what counts as the phenomena in general, otherwise we still would lack a general proposal for when a formal equivalence proof plus empirical equivalence licenses a substantial non-mathematical conclusion like a claim of semantic equivalence. Moreover, the dialectic I rehearse in this paragraph applies to Putnam’s specific examples (in particular, we still lack justification for the non-mathematical premise that a formal equivalence result plus empirical equivalence plus explaining exactly these specific candidate phenomena suffices for semantic equivalence).

- 13.
For example, the dialectic applies to the version of the reply sketched by Hudetz (2019, 48), that formal criteria are sufficient for semantic equivalence when conjoined both with empirical equivalence and equivalence of “theoretical content beyond the empirical (if there is any)” (48). Hudetz is admirably upfront that this sketch must be fleshed out, but already we can see how the dialectic might go. If ‘theoretical content’ is just non-empirical content, then the proposal amounts to declaring semantic equivalence sufficient for semantic equivalence, and the formal criteria are rendered redundant. So ‘theoretical content’ plus ’empirical content’ must amount to the sort of intermediate-level of content described above in the main text. One precisification of such content makes the strategy exactly akin to the ‘physics deference proposal’ that I shall argue against in Sect. 5. Still, however the notion is made precise, the inference to semantic equivalence must be defended, not just assumed.

- 14.
This diagnosis strikes me as a plausible reading of Tsementzis (2017), and is suggested by the prominence of examples from pure mathematics in Halvorson (2012) and Barrett and Halvorson (2016a, 2016b). (Though, as I will note shortly in the main text, in other places these latter authors make claims that presuppose more than purely mathematical ambitions for sentential criteria.) The diagnosis is also suggested by work applying sentential criteria to different logics, such as Wigglesworth (2017), Dewar (2018), and Woods (2018).

- 15.
- 16.
Earman hoped that the Einstein algebra inspired metaphysics would offer a metaphysics of spacetime that addresses the hole argument (Earman & Norton, 1987 for the classic statement of this argument, and Pooley (2013, Section 7) and Norton (2015) for overviews of the many replies the argument has provoked). This motivation is widely taken to have been undermined by Rynasiewicz (1992), who constructed an analogue of the hole argument in terms of Einstein algebras. However, this issue, and the motivations for Earman’s position generally, will not bear on my arguments. Similarly, Earman used ‘relationism’ to encompass more than just the negation of substantivalism. Hence, he took his Einstein algebra inspired metaphysics to offer a novel third view, that vindicates certain aspects of both substantivalism and relationism, rather than a relationist view. But nothing in what follows turns on the terminological question of which views we label ‘relationist’.

- 17.
This latter issue is contentious. Some metaphysicians (for example Fine, 2001; Schaffer, 2009) argue that most existence questions, whether ‘are there spacetime points?’ or ‘are there numbers?’, are trivially answered in the affirmative. They then employ some additional ideology to carve what they see as more interesting questions, such as ‘are there numbers at the

*fundamental*level?’, or ‘are there*really*numbers?’, and so on. Some have pushed this general line about the substantivalism/relationism debate in particular, proposing that the debate cannot concern merely the*existence*of spacetime points, which even relationists can grant (for different versions of this line, see Field, 1984; North, 2018). I shall ignore this wrinkle in the main text, but incorporating it would not challenge my arguments. The viability of the purely existential framing may also undermine some of RBW’s skepticism about the debate; at one point they concede “of course, it remains open to the person who wants to give [the two formalisms] a metaphysical significance to say that one of them is more fundamental than the other” (315), yet find their deflationary conclusion “far more philosophically interesting” (316). Notice though that even skeptics about ‘fundamentality’ talk can pose the question of whether there are spacetime points. Moreover, if my arguments succeed then their deflationary conclusion is either ill-posed (see Sect. 2) or else garners no support from their formal proof. - 18.
Thanks to an anonymous referee for suggesting that I address this kind of proposal. There are various attempts to spell out the very rough idea in a more plausible and precise manner in the vast literature on scientific modeling. For some helpful surveys of the lay of the land here, see Suarez (2010) and Frigg and Nguyen (2016).

- 19.
For instance, even given my tenuous handle on the notion of a picture-theory interpretation, arguably such interpretations can assign only contents that are

*purely**qualitative*(not about any particular objects). If so, such interpretations can at best assign only contents like*there are some spacetime points or other standing in such-and-such pattern of field values*, rather than contents describing which particular spacetime points have which field values. Yet the latter non-qualitative contents are the ones required to even formulate the hole argument, which is perhaps the central argument that animates the contemporary substantivalism/relationism debate. (For references to some overviews of the hole argument, see footnote 16.) In Sect. 6 we’ll see that arguably even purely qualitative yet topic-specific contents of the sort just described (such as purely qualitative contents about spacetime points) require going beyond the representational resources of anything like a picture-theory interpretation. - 20.
Additional evidence for this claim comes from the common practice amongst philosophers of science, especially in the literature on dualities, of using ‘physical equivalence’ as a label for what I am calling ‘semantic equivalence’. If my arguments against the physics deference proposal are successful then this terminology is highly misleading. For relevant citations and discussion, see Butterfield (2018, 34). Compare also Putnam’s remark—when arguing for the semantic equivalence of traditional continuous conceptions of spacetime and gunky conceptions (on which there are no measure-zero points)—that “it can make no difference to

*physical explanation*whether we treat space-time points as ‘real’ or as mere logical constructions” (Putnam, 1983, 43, emphasis original). The difficulties writing down physical laws in gunky spacetimes suggests otherwise (see, for instance, Arntzenius & Hawthorne, 2005 and Arntzenius, 2008, 2012, ch. 4). - 21.
Or alternatively instantiate a “proto-consciousness” intrinsic property that grounds facts about which macroscopic objects are phenomenally conscious. I shall ignore this wrinkle in the main text.

- 22.
Field-theoretically, this could be phrased in terms of whether certain excitations in various quantum fields (which license our talk about a particle being present) also bring about phenomenal consciousness.

- 23.
We can also imagine a difference that affects the mathematical models the physicists use to express their views. The argument in the main text does not turn on the precise details of how the difference gets formally expressed: many options will still render the resulting vehicles equivalent relative to all extant formal criteria.

- 24.
In the main text I am bracketing certain fringe views where the truth of panpsychism would percolate up to spoil physical equivalence; for instance, understandings of quantum mechanics where consciousness triggers wave-function collapse. If even the truth of panpsychism were claimed to fall under the subject matter of physics, we could employ numerous other examples instead (such as debates about whether there are moral properties, abstract objects, and so on).

- 25.
If they were instead to bite the bullet about this case and every other analogous case, I would then lean more on the concerns that I raised but set aside above: (i) the lack of a principled and metaphysically-neutral characterization of physical equivalence, and (ii) the lack of a principled and metaphysically-neutral procedure for couching whatever non-sentential representational vehicles are at issue as categories.

- 26.
Precedents for views in the spirit of the one I am sketching here can be found in Sklar (1980)—who emphasizes the importance of semantic connections to ordinary concepts (like that of an object) via analogies when doing science—and also in Maudlin (2018)—who emphasizes the importance of what he calls a “commentary” to supplement any given mathematical formalism in order to arrive at some semantic content. A similar moral has also been drawn in the vast literature on scientific modeling, where it is now widely recognized that features beyond a mathematical model itself—such as the intentions and natural language glosses of the scientist using the model—are integral to effecting an interpretation of the model. For some helpful overviews of this literature see the works cited in footnote 18. Nguyen (2017) applies this moral about scientific modeling to the debate over theoretical equivalence, supporting the extant critiques mentioned in Sect. 2 (in particular, the critiques of Sklar, 1982; Coffey, 2014).

- 27.
Category-theoretic criteria allow some additional freedom, stemming from the choice of arrows when couching some mathematical physics in category-theoretic terms. However, this point also does not challenge the moral in the main text. The reason is that the distinctions we can draw with additional representational resources also extend beyond those we can draw by different choices of arrows when deciding on a category-theoretic representation, granting the standard contentful significance of such arrows as erasing distinctions between possibilities. This fact also diminishes the interest of the claims made on behalf of the non-mathematical significance of category-theoretic criteria by Barrett (2019) and Weatherall (2019), to the effect that we can use different choices of arrows to diagnose ambiguities in how some mathematical physics can be used to represent the world: restricting this claim to any natural class of relevant interpretations (as we must, recall Sect. 2), these ambiguities can involve distinctions that also cut finer than choosing which differences between possibilities wash out when testing for equivalence, which is what a category’s arrows are used to represent.

- 28.
For related remarks about the importance of more general metasemantical issues in the philosophy of language and mind to questions about how we use mathematical objects to represent the world when doing science, see Callender and Cohen (2006).

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## Acknowledgements

Thanks most of all to Cian Dorr and Ben Holguín, for extremely helpful comments and discussion at every stage of the paper’s development. Many thanks also to David Albert, Thomas Barrett, Dave Chalmers, Hartry Field, and Tim Maudlin, for comments and discussion that led to significant improvements. Finally, for very helpful comments and discussion about this material, thanks to Kevin Coffey, Michael Strevens, Brad Weslake, the anonymous referees, as well as the audiences at NYU’s work in progress seminar and Caltech’s philosophy of physics reading group.

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Teitel, T. What theoretical equivalence could not be.
*Philos Stud* **178, **4119–4149 (2021). https://doi.org/10.1007/s11098-021-01639-8

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### Keywords

- Theoretical equivalence
- Representation
- Logical positivism
- Substantivalism
- Metaphysical realism