Invariance, intrinsicality and perspicuity

It is now standard to interpret symmetry-related models of physical theories as representing the same state of affairs. Recently, a debate has sprung up around the question when this interpretational move is warranted. In particular, Møller-Nielsen (Philos Sci 84(5):1253–1264, 2017) has argued that one is only allowed to interpret symmetry-related models as physically equivalent when one has a characterisation of their common content. I disambiguate two versions of this claim. On the first, a perspicuous interpretation is required: an account of the models’ common ontology. On the second, stricter, version of this claim, a perspicuous formalism is required in addition: one whose mathematical structures ‘intrinsically’ represent the physical world, in the sense of Field (Science without numbers: a defence of nominalism, Princeton University Press, Princeton, 1980). Using Dewar’s (Br J Philos Sci 70(2):485–521, 2019) distinction between internal and external sophistication as a case study, I argue that the second requirement is decisive. This clarifies the conditions under which it is warranted to interpret symmetry-related models as physically equivalent.