  We argue that, ideally, the ways to measure magnitudes in non-quantum
theories of physics (spacetime, field theory), limit drastically their possible
mathematical models. In particular, gauge invariance in the Yang-Mills
framework, is a necessity of our way of measuring rather than an a priori
imposition on symmetry.
  A general postulational basis for the geometric aspects of classical field
theories is introduced, and the permitted models are studied. Some of them (for
example, compatible with signature-changing metrics or variations of the speed
of interactions) are new, and require a generalization of the concept of
principal fiber bundle, which may be of interest both, physically and
mathematically.
