Axiomatization is uncommon outside mathematics, partly for being often viewed as embalming, partly because the best-known axiomatizations have serious shortcomings, and partly because it has had only one eminent champion, namely David Hilbert. The aims of this paper are to describe what will be called dual axiomatics, for it concerns not just the formalism, but also the meaning of the key concepts; and to suggest that every instance of dual axiomatics presupposes some philosophical view or other. To illustrate these points, (...) a theory of solidarity will be crafted and axiomatized, and certain controversies in both classical and quantum physics, as well as in the philosophy of mind, will be briefly discussed. The upshot of this paper is that dual axiomatics, unlike the purely formal axiomatics favored by the structuralists school, is not a luxury but a tool helping resolve some scientific controversies. (shrink)