  We propose a generalized finiteness principle for physical theories, in terms
of the concept of tameness in mathematical logic. A tame function or space can
only have a finite amount of structure, in a precise sense which we explain.
Tameness generalizes the notion of an analytic function to include certain
non-analytic limits, and we show that this includes many limits which are known
to arise in physics. For renormalizable quantum field theories, we give a
general proof that amplitudes at each order in the loop expansion are tame
functions of the external momenta and the couplings. We then consider a variety
of exact non-perturbative results and show that they are tame but only given
constraints on the UV definition of the theory. This provides further evidence
for the recent conjecture of the second author that all effective theories that
can be coupled to quantum gravity are tame. We also discuss whether
renormalization group flow is tame, and comment on the applicability of our
results to effective theories.
