On Ambiguity: The case of fraction, its meanings and roles

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.

Mathswitch is an open-source project that imports mathematical concept records from sources such as Wikidata, Wikipedia, MathWorld, Encyclopedia of Mathematics, nLab, ProofWiki, and Agda-Unimath, and links records that refer to the same concept. It does not reorganize or redefine the imported content; each source retains its own structure. The current focus is on importing concept data from Wikidata and the resources it links to, with plans to expand to further sources and better concept linking. Because the concept set is approximated through queries over Wikidata's collaboratively edited graph, the imported data is noisy: some items are non-mathematical, while others are ambiguous. In this paper, we test whether a voting ensemble of LLM judges can filter this noise. We evaluate it on Wikidata items with known MathWorld identifiers as a positive control, and examine how classification changes when database identifiers are removed from context. We then inspect the cases where the judges disagree with MathWorld and group these disagreements into three categories (degenerate descriptions, narrow scope bias, and editorial-scope mismatches) that suggest different remediation strategies.

Large language models (LLMs) achieve strong results on mathematical reasoning benchmarks yet remain unreliable on elementary numerical tasks, including magnitude comparison, large-integer arithmetic, fractions, and scientific notation. This survey examines basic numerical understanding as a capability distinct from high-level mathematical reasoning. We propose the Numerical Grounding Framework (NGF), which decomposes numeracy into Representational Grounding (RG), mapping numeral forms to value, magnitude, and equivalent representations, and Procedural Grounding (PG), executing arithmetic operations in accordance with their mathematical definitions. Using NGF, we organize recent diagnostic benchmarks, failure modes, structural explanations, and mitigation strategies. We review evidence concerning tokenization, positional encoding, embedding geometry, and pretraining-data distribution. We also apply NGF in a coordinated evaluation of three frontier model families across Number Cookbook, NumericBench, and GSM-Symbolic, comparing atomic, contextual, and reasoning-assisted numeracy. Architectural interventions such as digit-aware tokenization and Abacus Embeddings can improve models trained from scratch but are generally unavailable to users of pretrained systems, for whom supervised fine-tuning, reasoning scaffolds, and external tools are more practical. We conclude with deployment recommendations and research directions for more reliable numerical behavior in foundation models.