On automorphism criteria for comparing amounts of mathematical structure

Abstract

AbstractWilhelm (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.