Here are three large-scale issues. First, in what good sense or senses, if any, does ZFC provide a foundation for mathematics. Second, can category theory provide an alternative(?), or better(?) foundation for some or all mathematics? Third, should set theory should be understood as the study of a single universe, or as the study of a multiverse of set-theoretic universes? Each of these questions is many-faceted, and there are evident connections.
Penelope Maddy takes them all on in a substantial paper ‘Set-theoretic Foundations‘ (forthcoming in a festschrift for Woodin). She writes: “I won’t pretend to sort out all these complex and contentious matters, but I do hope to compile a few relevant observations that might help bring illumination somewhat closer to hand.” Written with her customary lucidity, Maddy does indeed do some useful sorting out, it seems to me. Worth a read if you haven’t seen the paper yet.
I think the paper’s great, and its identification of different foundational roles is very useful (and an advance on anything I’ve seen elsewhere). Nonetheless, I can’t quite agree about the two roles she says set theory can’t play: metaphysical insight and epistemic source. In both cases, I think there are alternative versions that, though weaker, aren’t spurious.
Epistemic source
This role is introduced via a view of Frege’s project that says “if mathematics can be reduced to logic, then knowing a mathematical fact is reduced to knowing a logical fact”. But when this is applied to set theory, it becomes “we know the theorems of mathematics because we know the axioms of set theory and prove those theorems from them.” Perhaps that version is obviously false, as Maddy says, but it’s also obviously different. So let’s look at a version without that change: if mathematics can be reduced to set theory, then knowing a mathematical fact is reduced to knowing a set-theoretic fact. Is that obviously false too? I don’t think so.
What makes Maddy’s set theoretic version “obviously false” is “our greatest mathematicians know (and knew!) many theorems without deriving them from the axioms.” But there are (roughly speaking) two ways a reduction of A to B can be understood. In one, an A-fact just is a B-fact. There’s no “gap”. So when you’ve learned an A-fact, you’ve already learned a B-fact. The whole question of which you know first doesn’t arise. In the other, there is a “gap”, and you have to somehow get from the A-fact to the B-fact, where the “somehow” is a translation or some other way of reducing it. But that doesn’t mean you have to know the B-fact first, or have to derive the A-fact from the B. It also doesn’t mean you have to be more certain of the B-fact. You may even learn the A-fact, and have good reason to believe it’s true, without even knowing what the corresponding B-fact is.
Think, for instance, of chemistry as reduced to physics. (I’m not sure that’s actually happened, but let’s suppose it has.) Chemists learned some chemical facts before anyone knew how to reduce them to physics. They were nonetheless physical facts too, all along. The same will be true if phenomenal facts about consciousness are eventually reduced to physical or functional facts.
Metaphysical insight
Maddy’s position appears to be that if we have a faithful representation of a mathematical object, rather than an identity, that gives us no metaphysical insight whatsoever. However, it may not be quite that strong. On page 7, she says set-theoretic reductions don’t give us “any sort of deep metaphysical information”. Perhaps that’s so, for some suitable notion of depth; but I don’t think the only alternative to deep information is information so shallow it’s not worth having.
Perhaps the issue here is, at least in part, about how heavy the emphasis on “metaphysical” should be, as opposed to the emphasis on “insight”. In any case, I think that reductions, embeddings, similations, faithful representations, and so on can give significant insight into the nature of an object even if they don’t take us all the way to revealing the “true metaphysical identity that object enjoyed all along”.
An important paper, I agree, which you had rightly advertised last year already, and of which you do well to remind us. As I observed then, Maddy highlights the work of Michael Ernst, whose dissertation, if you’re interested, can be found here: http://gradworks.proquest.com/3642922.pdf