Dear Angel,
I totally agree with you that the idea of symmetry, invariance, and the fundamental constants - are keys to understanding the deep ontological connection mathematics and physics. I believe that Tradition tells us a principle that substantiates the symmetry and asymmetry in nature and directs the search to the single base of fundamental knowledge.
Today in fundamental science there is a deep crisis of understanding, crisis of representation and interpretation. This profound existential crisis of basic science - universal human problems. Mathematics and physics require a deep ontological justification (basification).
I invite you to see my analysis of the philosophical foundations of mathematics and physics, the method of ontological constructing of the primordial generating structure, "La Structure mère" as the ontological framework, carcass and foundation of knowledge, the core of which - the ontological (structural, cosmic) memory and information - polyvalent phenomenon of the ontological (structural) memory of Universum as a whole. I believe that the scientific picture of the world should be the same rich senses of the "LifeWorld"(E.Husserl), as a picture of the world poets and philosophers.
Kind regards,
Vladimir