Mathematical frameworks vary in representing and combining concepts
Toward a Unified Mathematics of Concepts
Computation and Language
Summary
People think about concepts in many different ways, like pictures, words, or groups of things, but there isn’t one shared math language to describe them all. The authors looked at thirteen common ways people use concepts, like seeing how similar they are or putting them together. They found that different math methods are better at some of these tasks but not others, and no single method covers everything well. They tested this by seeing how nine theories handle categories and found different results depending on the math used. They suggest combining math methods that treat ideas, relationships, and changes as equally important.
What this means in practice
- •For ai developers: Build concept models combining vectors, symbols, and relations to cover more cognitive tasks like similarity and composition together.
- •For cognitive modelers: Choose conceptual frameworks based on which operations like generalization or composition their models need to support accurately.
Authors
Chen Shani
Abstract
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.