Symmetries in Foundations

Our mathematical relation to space originated by the symmetries at the core of Greek geometry as well as Riemann’s manifolds. Symmetries continued to provide the conceptual tools for further constructions of mathematical structures, from Poincare’s Geometry of Dynamical Systems to Category Theory, but were disregarded in logical foundations as Arithmetic has been considered, since Frege, the (only or paradigmatic) locus for foundational analyses. They are back now also in Logic, but a direct link to their role in Physics is still missing. As a matter of fact, geodetic principles, in Physics, originate in symmetries and provide an effective foundational frame for the main theoretical approaches, since the work by E. Noether and H. Weyl. The common “construction principles”, largely grounded on symmetries, may renew the foundational links between these two disciplines. Can computability fit into this renewed frame? What can it tell us about physical dynamics?