On Automorphism Criteria for Comparing Amounts of Mathematical Structure

Wilhelm (2021) has recently defended a criterion for comparing struc- ture 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 Sub- group, answering Wilhelm’s criticisms of SYM∗ along the way. We con- clude by arguing that no criterion that looks only to the automorphisms of mathematical objects to compare their structure can be fully satisfactory.