Abstract

As well as playing a foundational role in mathematics, set theory formalises the possibility of working with collections of objects. Intuitive, vague and general before Cantor, the set-theoretic approach would assume a central role in twentieth-century mathematics, notably through the axiomatisation of Zermelo-Fraenkel and the subsequent extension of von Neumann-Bernays-Gödel set theory. To solve some problems of set theory Samuel Eilenberg and Saunders Mac Lane, in 1945, introduced a new speculative tool for studying mathematical structures: Category Theory. Conceived as a generic environment for studying the transfer of information between mathematical structures, category theory represents one of the research areas of contemporary mathematics that can be effectively interwoven with the current frontiers of philosophical thought. Through a dialogue between category theory and the philosophy of Alfred North Whitehead and Alain Badiou, this essay attempts to show how this new approach suggests a transitory and processual ontology in which the theoretical core is not objects but morphisms between different categories and the properties of these relational transits: a relational, synthetic, contextual way of thinking that reflects well much of the structuralist and post-structuralist critique of the solidity and claimed absoluteness of foundations.