On automorphism criteria for comparing amounts of mathematical structure

Abstract 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 this mistaken; strictly worse . then formulate new improves on both and Subgroup, answering Wilhelm’s criticisms along the way. conclude by arguing no looks only to automorphisms objects compare their can be fully satisfactory.