Hilbert's idea of a physical axiomatics: the analytical apparatus of quantum mechanics

We discuss the Hilbert program for the axiomatization of physics in the contextof\nwhat Hilbert and von Neumann came to call the analytical apparatus and\nitsconditions of reality. We suggest that the idea of a physical logic is the\nbasisfor a physical mathematics and we use quantum mechanics as a paradigm case\nforaxiomatics in the sense of Hilbert. Finite probability theory requires\nfinitederivations in the measurement theory of QM and we give a polynomial\nformulationof local complementation for the metric induced on the topology of\nthe Hilbertspace. The conclusion hints at a constructivist physics.