We address a recent proposal concerning 'surplus structure' due to Nguyen et al. [ 'Why Surplus Structure is Not Superfluous.' Br. J. Phi. Sci Forthcoming.] We argue that the sense of 'surplus structure' captured by their formal criterion is importantly different from-and in a sense, opposite to-another sense of 'surplus structure' used by philosophers. We argue that minimizing structure in one sense is generally incompatible with minimizing structure in the other sense. We then show how these distinctions bear on Nguyen et al.'s arguments about Yang-Mills theory and on the hole argument.

We argue that several apparently distinct responses to the hole argument, all invoking formal or mathematical considerations, should be viewed as a unified “mathematical response”. We then consider and rebut two prominent critiques of the mathematical response before reflecting on what is ultimately at issue in this literature.

In this paper, I give a counterexample to a claim made in (Norton [2008]) that empirically equivalent theories can often be regarded as theoretically equivalent by treating one as having surplus structure, thereby overcoming the problem of underdetermination of theory choice. The case I present is that of Lorentz's ether theory and Einstein's theory of special relativity. I argue that Norton's suggestion that surplus structure is present in Lorentz's theory in the form of the ether state of rest is based on a misunderstanding of the role that the ether plays in Lorentz's theory, and that in general, consideration of the conceptual framework in which a theory is embedded is vital to understanding the relationship between different theories.