It seems to me that the three topics of my qualifying exam relate to three different questions about the objectivity of mathematics. The first is Hartry Field’s question about whether or not mathematical objects exist. The second is the one involved in higher set theory about whether or not there are methods of verifying mathematical statements that go beyond proof from already established axioms. And the third is Dummett’s question about whether or not there is a fact of the matter about mathematical statements that go beyond whatever verification procedures we have. Of course, in the latter two questions, the term “verification” should be taken with a grain of salt, because unless the answer to the first question is affirmative, mathematical statements won’t be literally true. But I think that in whatever sense one approves of standard mathematical theorems as opposed to their negations, one can raise the corresponding two questions.
Now, it seems plausible to me that these three questions are in fact independent, and I’ll try to sketch views that would be characterized as each of these combinations.
Y,Y,Y – I would think that something like this is the view of most set theorists working on large cardinals, and in particular was probably Gödel’s view. He took a naively platonistic view of sets, thought that there were means to discover new axioms beyond ZFC, and that every question about sets had a resolution in this platonic universe. Contemporary set theorists like Woodin and Martin retreat a little bit and call themselves “conditional platonists” or something of the sort, and also have a more sophisticated view of what counts as confirmation for an unprovable axiom. For instance, for them the axiom of projective determinacy (and thus the existence of infinitely many Woodin cardinals) is confirmed not just by the fact that each of its stateable instances is provable, but also that it predicted claims about sets of Turing degrees and continuous functions that not only have themselves been individually confirmed, but have also since become important topics in the study of their respective structures. Thus, the axiom makes many important and divergent explanatory predictions, and thus should be accepted as true.
Y,Y,N – Any position that states that mathematical objects exist, but that the facts of the matter don’t extend beyond our means of verification is going to be at least somewhat awkward. But I may be able to say that a position like this is what Steel suggests when he says that the continuum hypothesis might be ambiguous. That is, there might be two equally good set-theoretic universes, each isomorphic to a generic extension of the other, one of which satisfies CH and one of which doesn’t. Every statement a set theorist working in one structure makes can be translated into a statement about a generic extension of the universe by a set theorist working in the other. There is no fact of the matter as to which one of the two is talking about the “real” universe of sets, because each can be seen as being contained in the other. Most interesting mathematical statements will be decidable as above, but some (like CH) will just be considered ambiguous because they vary between the two models.
Y,N,Y – This is perhaps the naive view one takes after reading about Gödel’s incompleteness results. One believes that the universe of sets exists, and there are many facts of the matter about it, but that we have no access to any of these facts beyond ZFC. Never mind how we have knowledge about the axioms of ZFC. At any rate, there are strict limits to how much we can know.
Y,N,N – This is perhaps the slightly more sophisticated version of the above view. There are many universes of sets, and each consistent extension of ZFC is true in one of them. However, there is no fact of the matter as to which consistent extension is the “right” one, and we have no principled way to adjudicate between them.
N,Y,Y – This position also seems slightly awkward, but I can imagine someone like Hartry Field holding it. Mathematical entities don’t exist, but to the extent that we pretend they do, we might as well pretend they have a complete theory. Presumably the methods of set theorists will alow us to determine what statements should be satisfied beyond the axioms of ZFC. Or maybe we’re just allowed to make it up freely?
N,Y,N – I think this is the position that I am closest to. I agree with Field that mathematical objects don’t exist, and I agree with Dummett that there’s no point in saying there’s a fact of the matter about things we can’t assert (perhaps even more so when the “fact” of the matter is about a fiction). However, I am also convinced by Martin and others about the fact that certain extensions of ZFC are far more natural than others. If an author leaves enough hints in a novel, it seems that there can be a fact of the matter as to who the murderer was in the story, even though it was never explicitly stated. Similarly, if there is enough evidence in the theory of ZFC, it seems that projective determinacy must be true in the fiction as well, even though it was never explicitly stipulated.
N,N,Y – I’m not sure what would persuade one to adopt this position. However, there may be something like this in an instrumentalist view of science like van Fraassen’s. An agnosticism about the facts of the theory might allow one to believe that they could be secretly true, even though we have no way of knowing them to be true, and there aren’t objects to make them true. But enough agnosticism may allow one to consider this position while considering the next one as well.
N,N,N – This seems to be closest to the one that Dummett describes. If mathematical truth is identified with provability from the axioms, then negative answers to the second and third question follow immediately from Gödel’s incompleteness results. And a negative answer to the first question seems to be presupposed in identifying mathematical truth with provability rather than facts about the mathematical objects.
Now of course, all these views can probably be further multiplied by considering their restrictions to particular domains of mathematical discourse, like Peano arithmetic, ZFC, small large cardinals, large large cardinals, and statements like CH and beyond. So someone like Sol Feferman might take the views “Y,Y,Y” about number theory, but “N,N,N” about statements beyond ZFC. But this is just a general outline to show how these different questions might interact, and that they are at least somewhat independent.