Elementary Equivalence and Structuralism

22 05 2005

I was just grading an essay question on my students’ logic final asking them to contrast two of the views of philosophy of mathematics that we’ve discussed. In discussing structuralism, a few of them made statements that were ambiguous between talking about isomorphism and talking about elementary equivalence. But now that I think about it, maybe there is a reasonable version of structuralism where all that one wants to preserve is elementary equivalence rather than isomorphism.

Standardly, a structuralist (of one particular sort) says that there is no fact of the matter about what objects we’re talking about when doing mathematics beyond the isomorphism-type of the structure as a whole. That is, maybe the natural numbers really are the von Neumann ordinals (n={0,…,n-1}, so 0 is empty, 1 is {0}, 2 is {0,{0}}, etc.), and maybe they really are the Zermelo ordinals (n={n-1}, so 0 is empty, 1 is {0}, 2 is {{0}}, etc). But they most definitely are not one of the non-standard models of Peano arithmetic that contain “numbers” larger than any actual natural number.

But now what if there is no fact of the matter there either? The standard structuralist runs into a problem when trying to explain why she thinks we’re talking about one isomorphism class rather than the other, unless she brings in second order logic and maybe some other heavy (and questionable) machinery. But as long as we just limit ourselves to questions that can be stated in the language of number theory, these issues just don’t arise. We could try to say that in the standard model, no natural number has infinitely many predecessors – but “infinitely many” isn’t something we can say in the language of (first-order) number theory. If it’s set theory that we’re talking about, there really is an internal statement that says something like “there are uncountably many real numbers” – but this statement is true even in countable models, because it’s just talking about what pairing functions exist within the model. Unless we’ve decided once and for all what model of set theory we’re using, there’s no way to say what cardinality means in any given model – cardinality is only understood by talking about what actual functions exist, not what can be stated in some first-order language.*

This view isn’t exactly structuralism, because it says that we don’t even know what structure the natural numbers instantiate. But it’s more than empiricism, because it accepts all sentences we can have evidence for as equal, rather than prioritizing the observational ones. Once we’ve fixed the complete theory, then we’ve fixed everything there is to know. The other questions are somehow metaphysical only in a vague and problematic way, rather than the questions that can be stated in the internal language. I think this view might be related to the one Carnap proposed in “Empiricism, Semantics, and Ontology”, but of course that paper came out long before structuralism, so it might have been thought of as just a type of formalism.

*I think this issue points out why all those problems in mereology asking how many objects there are are troubling. I don’t know what it might mean to say that there are (or aren’t) inaccessibly many objects in the world, unless you’ve got an antecedent notion of set. And even more so for how many sets there are. Sure ZFC proves that the universe has certain closure properties just like those of V_k for k inaccessible. But this doesn’t mean that there “really are” inaccessibly many ordinals or whatever. That statement seems to me to mean nothing. After all, Skolem’s paradox shows us that there really could be just countably many in some sense, and we wouldn’t know the difference.

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