Toward a Unified Mathematics of Concepts

Concepts are commonly defined as abstract, compact representations of knowledge and treated as basic units of intelligent behavior. Yet, cognition, psychology, and AI lack a shared mathematical language for them. Modern systems represent concepts as vectors, distributions, symbols, graphs, and other structures, but these formalisms are typically treated as competing rather than as solutions to a common problem. We propose an operation-based view that evaluates mathematical frameworks by the conceptual operations they support, identifying thirteen operations (including similarity, composition, generalization, and grounding) that recur across cognition, psychology, and AI. We show that ten frameworks embody distinct commitments to concepts as self-contained content, relational structure, or evolving process, and that these commitments determine which operations each supports naturally. For example, vector-based models facilitate graded similarity and generalization but struggle with explicit composition, whereas symbolic models support composition but offer but generalize poorly. No single framework we examined naturally supports all operations without extension. We test this account empirically using categorization as a case study, operationalizing nine theories on the same items against human judgments. Despite addressing the same conceptual question, the theories produce different procedures and results, demonstrating that mathematical commitment shapes what a theory can explain. We call for hybrid formalisms that treat content, relation, and process as jointly primary.