My specific argument against Badiou’s foundational treatment of mathematics goes one step further than general arguments against the validity of scientism and foundationalism that Badiou himself lays out convincingly.
Badiou’s philosophy is profoundly split between a specifically mathematical exploration that does not, and cannot, have the philosophical consequences that he wants and a more general metaphorical extrapolation of concepts he has derived from his reflection on the mathematics. Badiou declares that mathematics is ontology but the burden of proof is on him.
In the specifics of the working out of his system he will derive certain concepts from the mathematics. For example in his reflexion on omega, the first infinite ordinal, he extracts a typology of infinities: the infinite as point, as place, as horizon, and as repetition. He then turns to poetry to “explore the labyrinth of the different forms taken by the couple finite/infinite”, admitting explicitly that this turn is metaphorical.
The mathematics is a very interesting theoretical tool that is well worth working out, but that has only suggestive value, heuristic rather than apodictic force.
Badiou has aptly remarked that the contemporary transcendental is a mixture of “crass positivism” (his expression) and vacuous moralism. I think that the nadir of this mix in theoretical circles is the rise of mathematical reductionism, an ideology which combines the speculative privileging of mathematics and a sort of hard-nosed pathos reminiscent of what in science fiction used to be called the “cold equations”.
People whose philosophical penetration does not go beyond a glib juggling with jargon but who have some mastery of theoretically fashionable sectors of mathematics pop up and descry, or so they think, their own image writ large in Badiou’s philosophy. Formerly this combination of bland moralism and crass positivism was embodied in an obsession with physics as foundational. We can see relics of this in Laruelle’s appeal to a vague “quantum thought” of his own imagination, that would be better described as a quantum Stimmung.
The advantage of mathematics as de-realised foundational discipline is that any tie with actual scientific theories and empirical evidence. Mathematical reductionism irepresents an attempt at an ontology compatible with any scientific discovery while yet maintaining an aura of scientificity. I do not think that Badiou himself fully falls into this trap, but he is not fully free from it either.