The philosophy of the history of mathematics expounded by Cavaillès bears a close and contrasting relationship to that set out by Brunschvicg in La Modalité du Jugement. (1897). The activity of scientific judgement is said to be mixed, between (ideal) judgements of interiority and (realistic) judgements of exteriority. The mixed form of the historical activity of knowledge is the modality of the possible. Hence a historical epistemology that claims Kantian idealist filiation and rejects speculative idealism. Cavaillès, a thinker of the creative necessity of mathematical development, reduces the role of intellectual adventure and possibility in its history, and in so doing distances himself from a master to whom Canguilhem would have remained closer. The recent history of mathematics, while paving the way for Kronecker's revenge on the abstract set-theoretical conceptions that inspired Cavaillès's necessitarianism, leads us to reconsider the radicalism that opposed him to Brunschvicg's philosophy of the modality of judgement.