5. Sets, Categories and Topoi: Approaches to Ontology in Badiou’s Later Work

Alain Badiou declared in his 1988 book Being and Event that “mathematics is ontology”. His 2006 follow-up work, Logics of Worlds, remains committed to that declaration but deploys a shift in the type of mathematics at stake. Specifically, while Being and Event is primarily concerned with drawing out the ontological lessons of set theory, Logics of Worlds focuses on the theory of categories and topoi, a more recent mathematical innovation that many believe offers an alternative and superior “foundation” to mathematics than set theory. Category theory arose in the latter half of the 20th century and has rapidly expanded to become a lingua franca of modern mathematics, uniting disparate disciplines and revealing deep underlying connections between hitherto unrelated areas of mathematics. In sharp contrast to set theory’s emphasis on articulating the internal structure of mathematical entities, category theory strips out all interiority and reduces mathematical entities to point-like “objects” with no internal structure. The mathematics arises instead from the network of relations between these objects: “arrows” that go from one