  Consider a finite-sized, multidimensional system in parameter state a. The
system is either at statistical equilibrium or general non-equilibrium, and may
obey either classical or quantum physics. L. Hardy's mathematical axioms
provide a basis for the physics obeyed by any such system. One axiom is that
the number N of distinguishable states a in the system obeys N=max. This
assumes that N is known as deterministic prior knowledge. However, most
observed systems suffer statistical fluctuations, for which N is therefore only
known approximately. Then what happens if the scope of the axiom N=max is
extended to include such observed systems? It is found that the state a of the
system must obey a principle of maximum Fisher information, I=I_{max}. This is
important because many physical laws have been derived, assuming as a working
hypothesis that I=I_{max}. These derivations include uses of the principle of
Extreme physical information (EPI). Examples of such derivations were of the De
Broglie wave hypothesis, quantum wave equations, Maxwell's equations, new laws
of biology (e.g. of Coulomb force-directed cell development, and of in situ
cancer growth), and new laws of economic fluctuation and investment. That the
principle I=I_{max} itself derives, from suitably extended Hardy axioms,
thereby eliminates its need to be assumed in these derivations. Thus, uses of
I=I_{max} and EPI express physics at its most fundamental level -- its
axiomatic basis in math.
