Bowman essentially does a Bourdieuian number on Badiou (how does Badiou function in terms of questions of taste and distinction, whose cultural status and identity is shored up by that function, etc.).
Probably the best line of counter-attack would be Rancierian: most of what I've been saying about the will to understand, the ability to participate in the practico-theoretic construction of a truth etc. presupposes a generic intellectual and political (and artistic, and amorous) capacity that Ranciere/Badiou thinks everybody just has, and Bourdieu thinks is the by-product of an uneven social process of cultivation. The idea here would not be to argue, against the evidence of one's own eyes, that the cultivation of people's capacities is not uneven, but to uphold Ranciere's point that such cultivation presupposes a capacity that does not depend upon it. Anyway, I'll work out the details later on.