Experimenting with ontologies: sets, spaces, and topoi with Badiou and Grothendieck

The paper explores the ontology and logic of the irreducibly multiple in set theory and in topos theory by considering the differences between Badiou's logical and Grothendieck's ontological approach to topos theory. It argues that Grothendieck's ontological program for topos theory leads to a more radical concept of the multiple than does the set-theoretical ontology, which defines Badiou's view of ontology even in his later, more topos theoretically oriented work. Extending Grothendieck's way of thinking to other fields enables one to give ontological multiplicities-no longer bound by the set-theoretical ontology or ultimately by any mathematical ontology, even in mathematics-a great diversity and richness. It follows that the set-theoretical ontology is not sufficiently rich to accomplish what Badiou thinks it could accomplish even in mathematics itself, let alone elsewhere; and Badiou wants it to work elsewhere-indeed, wherever it is possible to speak of ontology. I shall also consider, in closing, some implications of the arguments for the workings of the multiple in ethics, politics, and culture.