Oooh! Another good one! Math is a language, and language is a productive process. Axiomatic systems are the products of math as a language, but math resists a complete formalization because language itself resists a complete formalization - implying that language itself is complex - entangled and embedded from within the context it was born.

From what I've been thinking about - axiomatization of an environment must be done by a system with goals to accomplish them, but by definition it is a simplification and kills complexity. The problem with this is that you can't go from simple to complex because (hyper?) complexity is the state of the natural world.