We discuss the Hilbert program for the axiomatization of physics in the contextof what Hilbert and von Neumann came to call the analytical apparatus and itsconditions of reality. We suggest that the idea of a physical logic is the basisfor a physical mathematics and we use quantum mechanics as a paradigm case foraxiomatics in the sense of Hilbert. Finite probability theory requires finitederivations in the measurement theory of QM and we give a polynomial formulationof local complementation for the metric induced on the topology of the Hilbertspace. The conclusion hints at a constructivist physics.
"Hilbert's idea of a physical axiomatics: the analytical apparatus of quantum mechanics." J. Phys. Math. 2 1 - 14, December 2010. https://doi.org/10.4303/jpm/P100601