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.

Theo(00:12:09) :See, I think that ZFC, and, in fact, arithmetic, and, in fact, The Physical Universe, is inconsistent. But that’s based more on facetiousness than on any particular evidence.

Yasha Eliashberg, contact geometer at Stanford, once defined not only “mathematically worthwhile” but “interesting” as “useful for string theory.” This is not the sense in which you mean it, but it may not be _too_ far off, ne? Yasha called on that definition when trying to dissuade me from studying Nonstandard Analysis. (Thing is, I’m pretty sure that NSA holds the tools / ways to think about correctly atomizing space-time.)

lumpy pea coat(21:54:13) :I don’t get these “realism in truth-value” realists. What do they take to be the truth-bearers, and what are the truth-conditions under which a sentence is said to be true? They must be giving some story not in terms of satisfiability. But why should the truth conditions of mathematical statements differ from any other sentence (e.g. declarative ones)?

Plentiful platonism is an interesting view from a modal realist point of view. There may be counterparts of numbers that are not exactly like numbers in this universe (assuming there are such things) which allow us to talk about possibilities and counterfactual situations regarding actual numbers. Most people think the intrinsic properties of numbers (if there are any) are necessary, but that might not be the case.

Maybe that is sort of off-topic or irrelevant. Probably.

Kenny(15:39:12) :There’s certainly an analogy I see between plentiful platonism and a certain sort of modal realism. They both seem to rely on the idea that anything that’s possible, in fact exists. Penelope Maddy points out in an extended segment in

Naturalism in Mathematics(which I think she’s also published as a separate paper) that the notions of “function” and “set” in mathematics also eventually reached a similar level of generality. For a while, they thought of functions as just polynomials, then anything piecewise rational, then definable, and finally any arbitrary one-to-one or many-to-one association. I discussed this passage a bit here. (http://www.antimeta.org/blog/archives/2005/07/definabilism_an.html)As for realists in truth-value but not ontology, I must admit that it’s a bit puzzling to me too. However, I think a hermeneutic fictionalist strategy makes some sort of sense of this position, as does a purely syntactically-based logicism, or Dummettian intuitionism. On such an account, to be “true” just means to be provable, and no mathematical entities (except the proofs, which might actually be concrete) need exist.

Aidan McGlynn(11:18:03) :Stewart Shapiro’s examples of realists in truth-value but not ontology are Geoff Hellman’s and Charles Chihara. The linking thought between the two seems to be that mathematics sentences are to be treated as making assertions about what is possible, rather than what is actual. Such modal sentences can be held to have objective truth values (hence we get our realism in truth value), but we need not commit to the actuality of any mathematical objects. And it doesn’t seem like we need to worry about the truth conditions of mathematical sentences being funky on this kind of view anymore than those of modal discourse generally (I’ll let others decide how reassuring that is).