Generative Theories: A Categorical Model of Incremental Concept Formation in Mathematics

Mathematical theories are often presented as if they were static collections of axioms from which theorems are simply deduced. Actual theory-building is usually more dynamic: new concepts are introduced, old structures are reorganized, fragments of prior theories are merged, and the resulting framework gains both greater expressive power and stronger explanatory compression. This paper proposes a formal model of that process. Working in the general setting of an institution, we define a generative theory system as a category of theory presentations equipped with four primitive operations: conservative extension, definitional refinement, amalgamating composition, and abstraction. A development is modeled as a path or diagram in this category, and a stable theory is defined as the semantic colimit of such a path. The framework captures three essential features of mathematical innovation: (i) compatibility with old results, (ii) gradual increase in expressive power, and (iii) the emergence of new theorems as consequences of universal properties. We prove several structural results, including closure of conservative extension under composition, conservativity of definitional expansion, stability of filtered generative limits, and an amalgamation theorem for pushouts of compatible theory fragments. We also explain how the categorical viewpoint makes the “internal necessity” of new results precise: once a structure is chosen to satisfy a universal property, many later results are forced by its functorial semantics. The resulting model is not intended as a replacement for ordinary proof theory or model theory, but as a meta-theoretical account of how mathematical theories evolve by controlled generation.