The Pre-History of Mathematical Structuralism (excellent book, strongly recommend) pointed me to Linnebo & Pettigrew’s “Two types of abstraction for structuralism”, which compares Dedekind and Frege’s ideas of abstraction in the context of non-eliminative structuralism. It’s a worthwhile read, but even setting aside my eliminative leanings I have some issues with their presentation and conclusions.
They do a pretty good job laying out the desirable properties of a non-eliminativist system of abstracted structures – “instantiation” (the structure we abstract from a system is isomorphic to that system), “purity” (positions in the structure have no “fundamental” properties except structural ones, with the definition of “fundamental” being a crucial question for structuralists), “uniqueness” (e.g. the Cauchy and Dedekind reals both abstract to the same “pure” complete ordered field). In the Dedekind section they also introduce “self-occupancy”, based on Shapiro’s “places-as-objects” vs “places-as-offices” dichotomy.
However, they focus excessively on isomorphism of structures. this seems partly to be a standard convention among philosophical structuralists, and additionally their paradigmatic structure is the real numbers as a complete ordered field, which is categorical (its models are all isomorphic). but imo any structuralism worth its salt needs to handle non-categorical structures, it’s incomplete without an account of what we’re doing when we say “this is an abelian group” or “this is a group with an n-element generating set”
Instead of isomorphism, we want something that we can abstract out in a consistent way but we don’t need to be able to reverse it. for “group with an n-element generating set” we clearly want something that behaves like the free group on that set — there are many non-isomorphic groups with the same generating set but all of them can be embedded into the free group by “forgetting” that certain products of the generating set are equal. with what we might call the “free abelian group”, otoh, I don’t know how we’d even formulate it set-theoretically. this is a case where category theory can be useful as a language for talking about these things even if we don’t build our whole foundation on it like awodey et al.
Frege-style abstraction seems to be founded on an isomorphism condition, for which they bring up difficulties (first it has to be embedded into set theory to avoid paradoxes, then it runs into a problem with counting isomorphic objects). they allow that it works for “rigid” systems with no nontrivial automorphisms. Dedekind-style abstraction is more like, we set up the Dedekind cuts to define the structure of the real numbers, then we abstract away everything except that structure and posit new entities which have that structure. The problems they pose for it are basically also about automorphisms.
I think they’ve got a bit of conceptual sloppiness where they define certain properties in terms of the existence of an isomorphism between a system and its abstraction, and then notice that the properties don’t necessarily function when they’re set up with different isomorphisms as the witnesses to these properties. But this is just a problem of bad definitions that don’t maintain coherence!
It seems like most of their problems fall out of the idea that there is precisely one correct way to abstract a system. Certainly a non-eliminative structuralist will want to say that, if we abstract some property of a system, we will always get the same structure from it. But the same system can be abstracted into the same structure multiple ways. They treat this as absurd, for some reason.
Take lengths, for an example. We can map lengths to the real numbers, with natural interpretations for ordering and addition but not multiplication, and indeed this is one of the key motivating examples for the real numbers. But the same length can be mapped to 1 (as meters), 100 (as centimeters), 3.28084 (as feet), and 39.3701 (as inches). Addition, ordering, and geometry will remain untouched as long as we maintain a single consistent unit across our measurements and thus a single isomorphism/functor/whatever. (Once we’ve chosen a unit we can define multiplication as well.) Many other structures behave similarly in one way or another, like the interchangeability of i and -i.