In Tony Martin’s paper, “Evidence in Mathematics” (in Truth in Mathematics, edited by Dales and Oliveri), he gives arguments that one should adopt axioms well up the large cardinal hierarchy (countably many Woodins, I believe) because they provide a good explanation for various facts that we can already observe from ZFC. This is because they are (approximately) equivalent to projective determinacy, which states that every “projective” set of real numbers has various nice properties. The investigation of these properties led to new unifying results in recursion theory, stating that various sets of Turing degrees contain cones, and that the Wadge degrees have a particular nice structure up to a very high level. Results related to these properties are now important in recursion theory and wouldn’t have been discovered without the axiom, and every particular consequence of these results that has been considered has in fact been verified directly from ZFC (though often in more difficult ways). Since projective determinacy is known to be independent of ZFC (if consistent), it seems that we need to postulate it in order to properly explain the phenomena we can already observe, just knowing ZFC.
Based on arguments like this, it seems that Quine misstated the naturalist position, when he said that it should tell us to adopt ZFC+V=L. His reasoning was that ZFC should be accepted because it is an indispensable part of our scientific explanations of the world. However, the only particular sets that are indispensable in these explanations are all constructible, so those are the only ones whose existence we should countenance (all the other sets seem to be in some sense idle, like the angels that make sure gravity keeps doing its job, and the elves that make quarks obey the strong nuclear force). Thus, we should believe that the constructible sets are all that exist, so V=L.
But V=L is incompatible even with fairly weak large cardinal axioms (the existence of a measurable cardinal), and therefore with the stronger axioms advocated by Martin as part of an explanation of what’s going on with sets of Turing degrees and such. So I think Martin’s argument suggests that we do in fact have evidence for sets beyond L. This evidence may not be based directly in the physical world, but if Quine is serious about his holistic picture of science, then ZFC is just as much a part of science as relativity, and just as ZFC is justified because we need (large parts of) it for relativity (and just about every other scientific theory we’ve ever considered), and relativity is justified because it gives the best explanations of our observations, it seems that projective determinacy is justified because it gives the best explanations of phenomena in ZFC.
So if we believe the indispensability theorist, then we really should believe most of the large cardinal axioms, and not just ZFC. I mentioned this point in passing in a recent post.
However, I think most of this will be able to go through for the fictionalist just as well as for the indispensability-argument realist. If ZFC isn’t actually indispensable for our science, but is still quite useful, then someone like Hartry Field is willing to accept it at least as a good story, even if not literally the truth. But once we’re considering the story, I think we should adopt projective determinacy within the story as well. It seems to me that what is true in a fiction is not just literally what the author has asserted, but furhter facts may be as well, if they provide good explanations for what the author has in fact asserted. For instance, in a detective novel with a stupid detective, there may be enough clues presented for the reader to find out who did it, even if the detective never does and the author never explicitly says who did. And in a movie, it may become clear that a certain scene was actually a dream and not reality, because that’s the best way to reconcile it with the rest of the characters’ actions and desires. I think the audience discovers these facts in just the same way that we use inference to the best explanation in science (and our ordinary lives). Such inferences are always defeasible (we may find a better explanation, the author may explicitly deny the truth of the inference, further evidence may count against the inference, etc.) but it seems plausible that they are always active, whether in fiction or reality.
Therefore, I think that the fictionalist is just as justified in ascending the large cardinal hierarchy as the indispensability theorist, and both of them are in fact justified. Penelope Maddy is worried that they might not be, because of Quine’s argument I’ve paraphrased above (I’ve paraphrased that argument from my memory of Maddy’s paraphrase of it, so I may have misrepresented one or both of them through an inaccurate memory). This worry is a large part of what drives her to her position in Naturalism in Mathematics, but I think it is unjustified. Both the fictionalist and the Quinean naturalist should accept large cardinal axioms, just as Maddy believes set theorists should.