Here is a “mile high” interpretation of Williamson’s argument against conceptual-turn-methodology as it applies to McDowellian conceptualism:
1) Conceptualism, as spelled out by Dummett’s first two tenants, is too restrictive unless ‘thought’ is taken in a broad sense.
2) If a methodology (i.e. conceptualism) is too restrictive, then (13) that methodology must me rejected.
3) If ‘thought’ is taken in the broad sense McDowell endorses, then everything is thinkable.
4) It is conceivable that ‘the nature of [sub-sub-atomic particles] may preclude the kind of separable causal interaction with complex beings that isolating them in thought would require’.
5) (4) -> (6) it is possible that there are necessarily objects that cannot be thought of.
7) If (6), then (8) it is false that it is necessarily possible that everything can be thought of.
9) If everything is thinkable, then the unthinkable things allegedly described in (4) do not exist.
10) If (3)&(9), then (11) if we take ‘thought’ in the broad sense, then conceptualism is ontologically restrictive (in particular it denies the existence of the unthinkable things described in (4)).
12) If (11)&(8), then taking ‘thought’ in the broad McDowellian sense is too restrictive.
I think that (12) is this argument’s weak link. The basic intuition that Williamson seems to be invoking here is that methodology shouldn’t restrict ontology. If that’s what’s going on, I think he is wrong. Consider the following case: certainly, if anything should be built into our methodology it is that we should ceteris paribus prefer simple theories to complicated ones. But this implies that theories should be ontologically parsimonious when ever possible. In other words, the simplicity requirement puts a restriction on ontology and since the simplicity requirement is a legitimate methodological constraint, methodology can legitimately restrict ontology. One might still worry, rightly I think, that conceptualism places illegitimate restrictions on philosophical methodology. However, the case in question does not appear to show that. As Dan noted, losing elusive objects isn’t a theory breaker. If it turns out that elusive objects do play important roles in our best theories, then of course this might be a serious concern. Similarly, if we can prove elusive objects exist, McDowellian conceptualism is in trouble. But, if neither of the preceding antecedents are true, it seems we are cutting off excess weight from our theories in advance, which is good.
I apologize for both the sky-highness and the heavy-handedness of my argument reconstruction. It is not to be taken as a complete reconstruction of Williamson’s argument against McDowell, as again his other points seem sound. However, I think it brings to the forefront an important issue concerning philosophical methodology and reveals a questionable assumption that Williamson may be straying dangerously close to making.