  Analysing Quantum Measurement requires analysing the physics of amplification
since amplification of phenomena from one scale to another scale is essential
to measurement. There still remains the task of working this into an axiomatic
logical structure, what should be the foundational status of the concepts of
measurement and probability. We argue that the concept of physical probability
is a multi-scale phenomenon and as such, can be explicitly defined in terms of
more fundamental physical concepts. Thus Quantum Mechanics can be given a
logically unexceptionable axiomatisation. We introduce a new definition of
macroscopic observable which implements Bohr's insight that the observables of
a measurement apparatus are classical in nature. In particular, we obtain the
usual non-abelian observables as limits of abelian, classical, observables.
This is the essential step in Hilbert's Sixth Problem.
