Saturday, November 21, 2009

Plenetudinous Platonism, Boolos and Completeness

Plenetuindous Platonism tries to resolve worries about access to mathematical objects by saying that there are mathematical objects corresponding to every "coherent theory".

The standard objection to this, based on a point by Boolos, is that if 'coherent' means first-order consistent, then this has to be false because there are first order consistent theories which are jointly inconsistent- but if 'coherent' doesn't mean first-order consistent, the notion is obscure.

I used to think this objection was pretty decisive, but I don't any more.

For, contrast the following two claims:
- TRUE: all consistent first-order theories have models in the universe of sets (completeness theorem)
- FALSE:all consistent first-order theories are true (Boolos point)

Which of these is relevant to the plenetudinous platonist?

What the plenetudinous platonist needs to say is that whichever kind of first-order consistent things we said about math, we would have expressed truths. But remember that quantifier restriction is totally ubiquitous in math and life (if someone says all the beers are in the fridge they don't mean all the beers in the universe, and if some past mathematician says there's no square root of -2, they may be best understood as not quantifying over a domain that includes the complex numbers).

So, what the plenetudinous platonist requires is that that every first order consistent theory comes out true for some suitable restriction of the domain of quantification, and interpretation of the non-logical primitives. And this is something the reductive platonist must agree with, because of the completeness theorem! The only difference is that the reductive platonist thinks there are models of these theories built out of sets, whereas the plenetudinous platonist thinks there's a structure of fundamental mathematical objects corresponding to each such theory.

Thus, plentudinous platonism's ontological commitments can be stated pretty crisply, as in the bold section above. And there's nothing inconsistent or about these commitments, unless normal set theory is inconsistent as well!

1. You have a typo in the second to last paragraph.

As far as the philosophy goes I think you again overlook the fact that the whole point of platonism is to be able to refer to mathematical objects without any implicit quantifier restriction. If you take away the platonists ability to say, 'I mean set not some other object that might behave similarly' then you've gutted platonism. The whole point is that every statement in the language of set theory has a definite truth value even if we can't know what it is and that means you have to be sure that set always refers to the real sets not elements in some countable model satisfying all the conditions you could name.

So now you really do have a problem. Does the plentitudinous platonist also get the super power of refering to sets with the certainty he is talking about the real sets not some quantified restriction? If no then in what sense is he really a platonist? (again platonism can't be defined as the willingness to endorse some finite list of claims) If yes then does this super power only work for sets or does it work for all his other mathematical objects too? If so does it do so in a way that gives Boolos's objection weight?

Ultimately, however, I don't think this is actually a meaningful question since I think the whole idea of this magical reference magnetism is fundamentally confused.

2. Also on a more general point there seems to be something deeply wrong about your approach to platonism and these other philosophies of math.

I mean if you think that on a totally unconfused view people would never bring up platonism because it's irrelevant to understanding mathematical truth and applications then there seems to be something deeply confused about straightforwardly talking about whether it is true or not.

Put differently it seems to me that platonism, fictionalism, etc.. can only really be understood as answers to a certain percieved philosophical problem. Indeed, I would take it to be a core component of what it means to be a platonist/fictionalist/etc is to take your theory as the right answer to the problem of knowledge/access/etc..

3. Saying that there are lots of mathematical objects out there doesn't mean that you can't talk about more restricted classes of them.

If you have a categorical conception of what structure "the numbers" have (e.g. the smallest collection containing a 0 elt and closed under a sucessor oportion) your statements will have definite (and probably unknowable in some cases) truthvalues.

Similarly for the sets. If you can mean that the heirarchy of sets is supposed to contain "really all possible subcollections" of the stuff at each stage, then you will only count as talking about the objects with that structure (hence not about countable models).