Fascinating article in the new Nessie on the differing conceptions of mathematics at stake in Badiou’s metaphysical differend with Deleuze.

Of particular note:

Nicholas Bourbaki puts the point even more strongly, noting that “the axiomatic method is nothing but the ‘Taylor System’ - the ‘scientific management’ - of mathematics.”

and (a really key issue in my view):

In identifying ontology with axiomatic set theory, Badiou is adopting the position of “major” mathematics with its dual programs of “discretization” and “axiomatization.” This contemporary orthodoxy has often been characterized as an “ontological reductionism.” In this viewpoint, as Penelope Maddy describes it, “mathematical objects and structures are identified with or instantiated by set theorematic surrogates, and the classical theorems about them proved from the axioms of set theory.” Reuben Hersh gives it a more idiomatic and constructivist characterization: “Starting from the empty set, perform a few operations, like forming the set of all subsets. Before long you have a magnificent structure in which you can embed the real numbers, complex numbers, quaterions, Hilbert spaces, infinite-dimensional differentiable manifolds, and anything else you like.” Badiou tells us that he made a similar appeal to Deleuze, insisting that “every figure of the type ‘fold,’ ‘interval,’ ‘enlacement,’ ‘serration,’ ‘fractal,’ or even ‘chaos’ has a corresponding schema in a certain family of sets….”