 Consider classical mechanics first. The observables are functions on phase space, functions of the positions and momenta of the particles. The axioms governing the behavior of the basic observables—Newton’s equations for the positions or Hamilton’s for positions and momenta—define the theory. What would be the point of making additional axioms, for other observables? After all, the behavior of the basic observables entirely determines the behavior of any observable. For example, for classical mechanics, the principle of the conservation of energy is a theorem, not an axiom. 