In this expository article one of the contributions of Jean Cavailles to the philosophy of mathematics is presented: the analysis of ‘mathematical experience’. The place of Cavailles on the logico-philosophical scene of the 30s and 40s is sketched. I propose a partial interpretation of Cavailles's epistemological program of so-called ‘conceptual dialectics’: mathematical holism, duality principles, the notion of formal contents, and the specific temporal structure of conceptual dynamics. The structure of mathematical abstraction is analysed in terms of its complementary dimensions: (...) paradigmatic generalization (domain extension, descriptive definitions, creative role of the symbolism...) and thematic reflexivity of concepts (promotion of operations to objects of a higher type). (shrink)