In this essay, I propose a new key to the interpretation of Cavaillès's "philosophical testament" Sur la Logique et la théorie de la science (1942/47), or at least to one of its main philosophical motives: to find a principled answer to the problem of coordination between pure mathematics and physical theory (and thus also to clarify, first of all, the status of mathematical physics). Cavaillès's manifesto, culminating in the idea of a "philosophy of the concept", not only articulates the core ideas of a new philosophy of mathematics around the dynamics of "paradigm" and "thematization"; it also contains a project of a new "doctrine of science" in general. The latter should explain the possibility of conceiving a worldly knowledge that incorporates the intrinsic dialectic of mathematical concepts, but at the same time supplements this internal conceptual development with something radically different: a reason-led action of experimenting and wagering on events in the world. I develop this view of Cavaillès against the background of the ideas of some of the leading thinkers on the coordination problem: Mach, Poincaré, Schlick, Reichenbach, Carnap, Brunschvicg, Gonseth, Suzanne Bachelard, Bas van Fraassen...