Mathematics and Physics as Emergent Aspects of a Universal Rewrite System

Mathematics and physics are shown to have a symbiotic relationship as emergent aspects of a universal rewrite system. In addition to explaining the 'unreasonable effectiveness' of mathematics in physics and the 'unreasonable effectiveness' of physics in mathematics, this emergent nature of both subjects makes sense of the distinction between syntactic and semantic approaches to logical reasoning. The system is also shown to generate constraints on the kinds of mathematics and physics that are possible, explaining, in particular, why symmetry is so significant in the subjects' foundations, and specifying which symmetries are most significant, as well as indicating their points of origin. Quantum mechanics emerges from this structure in a very specific form which enables us to understand the origin of symmetry breaking in physics and many other aspects of fundamental physical theory. Gravity also has special characteristics which explain its uniqueness among the four physical forces.