AbstractThis is an essay review of Jill North’s book Physics, Structure, and Reality. It focuses on two of the main topics of the book. The first is North’s idea that we can use coordinates as a window into the structure that a theory posits; the second is North’s argument for the inequivalence of Lagrangian and Newtonian mechanics.

It has been suggested that we can tell whether two theories are equivalent by comparing the structure that they ascribe to the world. If two theories posit different structures, then they must be inequivalent. The aim of this article is to evaluate the extent to which this desideratum holds for the different standards of equivalence that are currently on the table.

In this paper, we examine the relationship between general relativity and the theory of Einstein algebras. We show that according to a formal criterion for theoretical equivalence recently proposed by Halvorson (2012, 2015) and Weatherall (2015a), the two are equivalent theories.

Wilhelm (Forthcom Synth 199:6357–6369, 2021) has recently defended a criterion for comparing structure of mathematical objects, which he calls Subgroup. He argues that Subgroup is better than SYM ∗ , another widely adopted criterion. We argue that this is mistaken; Subgroup is strictly worse than SYM ∗ . We then formulate a new criterion that improves on both SYM ∗ and Subgroup, answering Wilhelm’s criticisms of SYM ∗ along the way. We conclude by arguing that no criterion that looks only to the automorphisms of mathematical objects to compare their structure can be fully satisfactory.