Have any abstractionists discussed the possibility of multiple abstraction principles?

Question

An abstractionist is someone who claims that some version of Frege's abstraction principle accounts for or justifies talk of abstract objects. The abstraction principle says that an equation of the form

f(x)=f(y) <=> x~y

justifies talk of objects like f(x) based on an equivalence relation x~y. For example, let d(x) be the direction of line x. What is a direction? Well it is defined using the parallel relation as follows:

d(x)=d(y) <=> x||y

So the fundamental objects are line and the relation of being parallel, and directions are derived from that base, along with all of the properties of directions.

All of the abstractionists I've read (Frege, Wright and Linnebo are the only ones that come to mind, but I've read others) seem to take that specific equation form as fundamental and consider no others, but what about Whitehead's extensive abstraction based on the limit of a sequence? Whitehead abstraction can be used to justify abstractions that are hard to do with Frege abstraction; for example, the abstraction of a geometric point or an ideal gas or a completed infinity. And then there is what we might call a Hilbert abstraction, namely just the existence of an abstract object that answers to any formal theory.

My impression is that the abstractionist program could be fruitfully extended by including these and other sorts of object-forming formalisms in addition to Frege's abstraction principle, but I've never seen a discussion of it. The closest I've seen is a paper by Wright where he contrasts Frege abstraction and Hilbert abstraction as if they were competitors and mutually exclusive.

Answer 1

Ebels-Duggan[16] does not quite seem committed to one abstraction principle per theory:

Further complicating the neo-logicist’s hopes is the fact that not all abstraction principles—even those based on permutation invariant equivalence relations—should count as logical: so not only does said neo-logicist need an account of how an abstraction principle could be logical, but that account must also sort the good principles from the bad. So more needs to be said for what kinds of logical equivalence relations there are on concepts, and which are apt to yield suitably logical abstraction principles. It is thus of interest to classify logical equivalence relations on concepts.

The paper goes on:

This paper proves a classification theorem for such equivalence relations. ... our version of [Fine's theorem] sorts equivalence relations, and their abstraction principles, more usefully. In other words: there are abstraction principles well-discussed in the literature on neologicism, but as Fine states the theorem in [7], one needs to squint to see how their equivalence relations are classified. The version here given allows for more clear-eyed recognition of this sorting...

But the phrase "the finest abstraction principle" (emphasis added) is also used, there, so I'm not sure whether the plurality of such principles occurs on a meta-theoretic level or not (per that author's ambient sense of the word "theory"). Offhand, I think that a theoretical unit can sustain multiple types of equivalence relations (e.g. systems with both absolute and relative identity?), so would be able to sustain distinctive abstraction principles per the types, yet insofar as abstraction and generalization are similar ("isomorphic"?) processes, and generalization tending towards "theoretical unity," what we might end up with would be a hierarchy of abstraction principles, so still with one we might distinguish as "the" abstraction principle par excellence (the principle abstracted from the other principles).

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A late addition: the other day, I found this remarkable text in a posting about a metaphysics workshop that Hamkins attended:

One may, for instance, understand abstraction generatively, holding that the values of abstraction operations are generated from the inputs. Different abstraction principles characterize different abstraction operations - Hume’s Principle characterizes the cardinal abstraction operation, other abstraction principles characterize structural abstraction.

I don't know how to get more information from this, though, unless you emailed Hamkins about how the workshop went. I also don't know if the technical matters he's alluding to are already in your (the OP's) purview, so this might not be a useful update, but I'll hope it could be. (Hamkins is good about responding to emails from people he doesn't know, I might add, and he's always really nice, or that's been my experience, so don't be shy about messaging him if you think it could help!)