Matt Hare

Jean Cavaillès’ definition of mathematics as _ effective _ _ thought _ or _ effective work _ is central to his analysis of the “objective becoming” of mathematics. This concept crosses two referents: the effective as the _ actual _ and the history of _ effective calculability _. I examine Cavaillès’ treatment of two phases of the mathematical history of the concept: first, debates between the French analysts around effective definability, and, second, Gödel, Church and Kleene’s work on effective computability. Cavaillès’ temporalisation of the concept of the effective is seen to inform his rejection of any _ a priori _ theory of science and correlative relativisation of the transcendental, and thus his theory of discontinuous logical time.