de re beliefs about numbers?

I just read some Azzuni which seemed to attribute the following argument to Burge:

"The difference between de re and de dicto thought, is that de dicto thought can have content that `goes beyond' your concepts and picks up info from the environment. So if I think de dicto "the nearest vase is green" the proposition which this expresses is purely determined by my concepts. Whereas, if I think de re "*that* vase is green", this picks up content from the context (in particular it claims something which would be false if that vase got painted white, but some other vase wound up getting put in front of me instead).

Now, (one might go on to think) , this suggests mathematical thought involves a de re component. Why? A de re component would explain how mathematical talk can pick up content from facts about matheamatical objects outside the head. Hence, it could explain why the truth conditions for mathematical facts go beyond my (probably recursively axiomatizable) inference dispositions."

The problem with this is that, as Burge himself is famous for pointing out, what someone means by concept-words like arthritis can also require one to `go to the context', (in a slightly broader sense) of how experts near the speaker use the word arthritis, to determine what proposition/truth conditions a sentence about ``arthritis'' has.

So, if the evidence is just that mathematical truths can depend on stuff that `goes beyond'*[Yuck, if there were some typographic convention stronger than academic shudder quotes I'd be using it here :)] our presumably recursively axiomatizable inference dispositions, then I see no evidence for the claim that our number talk is de re. Dependence on broader context could be achieved either by the the object-word "3" functioning as some kind of hidden de re ostension, or the concept word "is 3rd in a number-sequence" (or whatever other pseudo-definite description you would want to associate with three) having a meaning that isn't entirely determined by stuff in the head.

Many other things seem shady too, but I should really read more of Burge's own words before getting too dismissive :).
[edit: ok Burge himself does not seem to be making this argument in the relevant article which is here]