CMM reminder

Just a reminder that we're looking at Cian Dorr's 'There are no abstract objects' this Friday in the CMM seminar (9.30-11, CETL Building) . I read it last night, and it's well worth a look at. There will be a lot to talk about, but I thought I might just start us off here.

Dorr is interested in nominalist paraphrases of claims that, on the surface, seem to commit us to abstract objects. (The paraphrases can be of the only-possible-in-principle, perhaps-only-with infinitary-resources, variety). He discusses 'modalised' nominalist paraphrases e.g.

(1) The number of the planets is nine

mapping to

(2) If there were (enough) numbers, and the concrete world were just as it is, then the number of the planets would be nine.

He compares (1) and (2) with e.g.

(3) The things that explain the facts about the motion of atoms are subatomic particles of kinds K related to atoms in way W

mapping to

(4) It is consistent with the facts about the motion of atoms that: the things that explain the motion of atoms are subatomic particles of kinds K related to atoms in way W.

(I've modified Dorr's phrasing slightly, but not anything of substance, I hope).

2 and 4 seem to drop the ontological commitments of 1 and 3 respectively. The nominalist obviously likes this in the case of 1 -> 2. But qua scientific realist, he may be less comfortable in being pressured to retreat from 3 to the more cautious, less commital 4. But how can he justify this difference in attitude to 'ontological retreat'?

Obviously there's a lot which could be said here. Dorr points out that whereas both 2 and 4 can be seen as applying complex modal operators to the original sentences 1 and 3, the character of these operators differ. In the case of the nominalist's counterfactual operator, we plausibly have a universal quantifier over worlds:

5) In all worlds (if (enough) numbers exist and the concrete world were just as it is -> the number of the planets is nine)

In the case of the scientific 'cautious man', the consistency claim is plausibly an existential quantifier over worlds:

6) There is a world at which (the things which explain the (actual) motion of atoms are subatomic particles of kind K related to atoms in way W)

Dorr argues that this is an epistemologically important difference, which can help provide a non-ad hoc reason for the nominalist to engage in one kind of ontological retreat, but not another.

What I don't understand is how that helps with the case of e.g.

(4*) If there were subatomic particles of kind K related to atoms in way W, the things that explained the motion of atoms would be subatomic particles of kind K related to atoms in way W.

which is more directly analoguous to (2).

Presumably Dorr must be thinking that there could be subatomic particles of kind K related to atoms in way W without those being the things that explained the motion of atoms. Perhaps there could be e.g. different laws, or there could be some extra things around in some worlds, not of kind K, that also help explain the motion. But everything seems to hang on what gets packed into W. If W includes specification of the relevant laws, and also e.g. that the K-particles + laws completely explain the motion of the atoms, etc, then 4* starts looking true to me. But it shares the logical structure of the nominalist paraphrase, while being non-scientific-realist in spirit.

Am I missing something? Can anybody think of a plausible story on which 4* comes out false, even if a lot is built into W?