Some issues concerning eliminative structuralism for second-order arithmetic

Eliminative structuralist philosophers of mathematics insist that what mathematicians study is structures rather than specific realizations of these structures, like a privileged natural number system would be. One example of such an approach would be to take the axioms of second-order Peano Arithmetic PA2, and say an arithmetical sentence ϕ is true if and only if it is true in every standard model of PA2. Since all such models are well-known to be isomorphic, it follows that for every arithmetical sentence ϕ, either ϕ or  ∼ ϕ is true, which is delightful.

The hitch here is the insistence on standard (rather than Henkin) models, since the concept of a standard depends on something very much like a background set theory—a standard model is a second-order model where every subset of Dⁿ is available as a possible value for the second-order n-ary variables, where D is the first-order domain. Thus, such an eliminative structuralism in order to guarantee that every arithmetical sentence has a truth value seems to have to suppose a privileged selection of subsets, and that’s just not structural.

One way out of this hitch is to make use of a lovely internal categoricity result which implies that if we have any second-order model, standard or not, that contains two structures satisfying PA2, then we can prove that any arithmetical sentence true in one of the two structures is true in the other.

But that still doesn’t get us entirely off the hook. One issue is modal. The point of eliminative structuralism is to escape from dependence on “mathematical objects”. The systems realizing the mathematical structures on eliminative structuralism don’t need to be systems of abstract objects: they can just as well be systems of concrete things like pebbles or points in space or times. But then what systems there are is a contingent matter, while arithmetic is (very plausibly) necessary. If we knew that all possible systems satisfying PA2 would yield the same truth values for arithmetical sentences, life would be great for the PA2-based eliminative structuralist. But the internal categoricity results don’t establish that, unless we have some way of uniting PA2-satisfying systems in different possible worlds in a single model. But such uniting would require there to be relations between objects in different worlds, and that seems quite problematic.

Another issue is the well-known issue that assuming full second-order logic is “too close” to just assuming a background set-theory (and one that spans worlds, if we are to take into account the modal issue). If we could make-do with just monadic second-order logic (i.e., the second-order quantifiers range only over unary entities) in our theory, things would be more satisfying, because monadic second-order logic has the same expressiveness as plural quantification, and we might even be able to make-do with just first-order quantification over fusions of simples. But then we don’t get the internal categoricity result (I am pretty sure it is provable that we don’t get it), and we are stuck with assuming a privileged selection of subsets.