Cognitive effects at the interfaces: An exploratory comparative essay on linguistic architecture and the accessibility of mathematics

Mathematical relationships do not change across languages, but the cost of reaching them may. This essay asks where, and through which mechanisms, linguistic architecture changes the cognitive cost of representing, interpreting and manipulating a mathematical relationship that itself stays fixed. Its central distinction comes from Larkin and Simon (1987): two representations can be informationally equivalent yet computationally different. The essay does not rank languages. It treats linguistic features as task-relevant variables at five levels (numeral lexicon, grammar, register, notational interface, modality), acting through six candidate mechanisms on five outcomes (interpretation, formalisation, execution, explanation, transfer). The method is a desk-based comparative synthesis of published evidence in arithmetic, logic and geometry. The feature-driven sample covers English, Mandarin Chinese, Welsh, Russian, Arabic, Basque, Turkish, Guarani, Inuit languages, Mundurukú, Pirahã and American Sign Language. Twenty-three claims are graded under an explicit protocol, each with its principal residual confound. Three observations organise the results. First, most well-evidenced effects sit at interfaces, the points where one representation must be converted into another: number word to digit string, grammatical number to exact quantity, habitual reading to formal reading. Away from those interfaces, in approximate magnitude and core geometric intuition, effects are weak or absent. Second, cost is often redistributed rather than removed: a language can make one step cheaper and another dearer. It is not always conserved, and some irregularity has no identifiable compensating gain. Third, the best-supported claims are the narrowest. The three that grade Strong concern transcoding errors under number-word inversion, the task-specificity of number-word transparency, and the effect of digit-name length on forward digit span. None concerns mathematical ability. The claim is deliberately modest. For the relationships considered here, languages appear broadly intertranslatable at the level of propositional content. They differ in the availability, conventionality, ambiguity, length and alignment of the forms that express those relationships. Those differences move cognitive and educational cost between steps. They do not establish differences in mathematical capacity. The essay closes with open questions and with predictions that could fail. Files in this record: - Cognitive_effects_at_the_interfaces_v1_0_0.pdf: the working paper (51 pages), including the evidence ledger and a glossary - lmccm_evidence_ledger_v1_0_0.csv: the 23 graded claims (Section 11) - lmccm_language_matrix_v1_0_0.csv: the language sample and its features (Sections 4 and 5) - lmccm_references_v1_0_0.csv: the 102 references - lmccm_fig1_chain_v1_0_0.svg: Figure 1, the LMCCM chain All CSV files are UTF-8, comma-separated, with CRLF line endings.