This paper introduces an axiomatic framework for a mathematics of dialectics, where contradiction and synthesis are treated as fundamental operations. It establishes a developmental order that ensures dialectical progression always produces more advanced states. From these axioms, several structural results are proven on progression, cycles, and fixed points, and a concrete integer model is presented. The framework is compared with existing areas such as logic, dynamical systems, order theory, and category theory, and open problems and research directions are outlined.