Monday, December 17, 2007

Absolute Generality 15: Lavine on the problems

Shaughan Lavine's is one of two fifty-page papers in Absolute Generality (I'm not sure that the editors' relaxed attitude to overlong papers does either the authors or the readers a great service, but there it is). In fact, the paper divides into two parts. The first six sections review four anti-absolutist arguments, and criticizes McGee's response on behalf of the absolutist to (in particular) the Skolemite argument. The last five sections are much more interesting, arguing that we can in fact do without absolute unrestricted generality -- rather, "full schematic generality will suffice", where Lavine is going to explain at some length what such schematic generality comes to.

But first things first. What are the four anti-absolutist arguments that Lavine considers? (1) First, there's the familiar argument from the paradoxes that suggests that certain concepts (set, ordinal) are indefinitely extensible and that it is not possible for a quantifier to have all sets or all ordinals in its domain. Now, unlike McGee, Lavine recognizes that that the "objection from paradox" raises serious issues. However, he evidently thinks that any direct engagement with the objection just leads to a stand-off between the absolutist and anti-absolutist sides, "each finds the other paradoxical", so he initially sets the argument aside.

(2) The second argument is the "framework objection" that Hellman also discusses in his contribution. "Different metaphysical frameworks differ on what there is ... If the answers to [questions like are there any mathematical entities? is space composed of points?] are not matters of facts, but of choice of framework, ... [then there is only] quantification over everything there is according to the framework [and not] absolutely unrestricted quantification." Well, as I noted before, if two frameworks differ just in what they take to be ontologically basic and what they take to be (in some broad sense) constructions out of the basics, then that is beside the present point. We can still quantify over all the same things in the different frameworks -- for quantifying over everything isn't to be thought of as restricted quantification over whatever is putatively basic. So to make trouble here, the idea would have to be that there can be equally good rival frameworks, with only a conventional choice to be made between them, where Xs exist according to one framework, and cannot even be constructed according to the other. If there are such cases, then there may be an argument to be had: but that is a pretty big "if", and Lavine doesn't give us any reason to suppose that the condition can be made good, so let's pass on.

[To be continued.]

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