I love this reply to Mancosu, by Thomas Forster:

<< Well, here's a position that has at least the beneifit of clarity and brevity:

No explanations without counterfactuals
No counterfactuals without contingency
All of Mathematics is necessary
------------------------------
No explanations in mathematics!


I'm not saying i believe it, but finding the right place to pick holes
might be a useful discipline. >>

I would say that mathematics has "doxastic counterfactuals" (because, thanks to the nonobviousness of math, we can imagine theorems being false), but that they are different for each person (maybe you can imagine the Pythagorean theorem being false while somebody else cannot). Garber has written something on logical counterfactuals. Gaifman has written on assigning subjective probabilities to logical statements.

I don't understand why they don't turn to cognitive science. The only person I know in this area of "mathematical explanation" is Jeremy Avigad, and he's not at all interested in approaching this from cognitive science. I don't get it.