The Effective as the Actual and as the Calculable in Jean Cavaillès

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.