Mathematics as Evolving Language: A Formal Framework for Mathematical Change

The Mathematics as Language (MaL) framework models mathematics as an evolving linguistic practice. Rule systems occupy nodes in a directed theory space; admissible transitions satisfy structural, functional, and genealogical continuity. Invariant cores — statements stable across all admissible descendants — are formally defined and shown to be non-empty and non-decreasing. Case studies of non-Euclidean geometry, intuitionistic logic, and forcing validate the model. The framework operates from Constitutive Geometric Projection, which grounds the three continuity conditions as irreducible projections of constitutive structure and explains why independently evolving mathematical lineages converge on the same cores, unlike natural languages.