As suggested at the end of my last post, I’m beginning to think that a more important issue in the philosophy of mathematics than the question of whether or not mathematical entities exist is the question of whether questions independent of ZFC (or PA, or some suitable other theory) can be decided. The former question is that of “realism in ontology”, while the latter is more like the “realism in evidence” that I once mentioned. (And there’s also traditionally the question of “realism in truth-values”, about whether or not these statements have truth-values independent of our abilities to come to know them.) Both platonists and anti-platonists have taken both answers to the question of “realism in evidence”.
Gödel motivated his search for new axioms in platonism, but it was platonism of a very definite sort, in which there was one universe of sets that our theory is aiming to describe. The new axioms were supposed to be further truths about this universe. The natural contrast at the time was with something like the formalist position, on which any consistent set of axioms was as good as any other, so that if our interest was in ZFC, then any extension of it was as good as any other. Later, Hartry Field developed a more sophisticated form of anti-realism, though he too has suggested that there is very little we can say beyond ZFC.
Since then, people like Ed Zalta and Mark Balaguer have suggested that there is a sense of platonism on which all consistent theories describe actual universes of mathematical entities. This view has been called “plentiful platonism”, “full-blooded platonism”, and “plenitudinous platonism”, among others. Traces of this view can be found probably at least as far back as Carnap’s “Empiricism, Semantics, and Ontology”. For these philosophers however, the plenitudinousness is more important than the platonism – both have suggested that an anti-realist interpretation of their view is plausible as well.
I think there are problems with such views, and perhaps the most prominent defender of the non-plenitudinous views is Penelope Maddy, whose ontological views are at this point also neither clearly platonist nor anti-platonist. (More accurately, she seems to be against ontological claims of either sort.) Interestingly, this evening I heard a talk by Daniel Isaacson (from Oxford), advocating a sort of structuralism that he suggested would also support the search for new axioms. Although the model of mathematics may not be unique, he suggests that the structure up to isomorphism should be (ie, the theory should be categorical), and thus there is at least a truth-value to statements beyond those of first-order ZFC, even if we can’t immediately find out what that truth-value is.
I still want to defend anti-platonist views of the ontology of mathematics, but questions about the plenitude or not of mathematical entities (or stories) seem to be more pressing, because they affect the actual mathematical practice of set theorists. Now that set theory has pressed so far beyond the widely accepted axioms of ZFC, this question is taking over some of the importance of the traditional foundational questions. We are all fairly confident that mathematics will not turn out to be unjustifiable (though some people I know have suggested they think ZFC might be inconsistent), so one way or another the foundational issues can probably be either resolved or ignored. But whether set theorists are doing anything mathematically worthwhile is a more controversial question.