The Linguistic Architecture of Mathematical Thought (Part 3): The Symbolic Interface, Historical Necessity, and Bottlenecks in Abstract Thought

Synthesizing the frameworks of "Verb Metabolism" and "Demonstrative Injectivity," this paper examines the historical emergence and cognitive role of mathematical symbolization. Focusing on the non-face-to-face communication network of the "Republic of Letters," the study argues that mathematical notation became indispensable when natural language demonstratives could no longer rely on shared physical contexts. While acknowledging historical algebraic achievements such as 13th-century Chinese Tian Yuan Shu, the paper posits that the prohibitive cognitive cost of maintaining referential injectivity across long, one-dimensional textual strings drove the transition to language-independent symbols—alongside phonological working-memory factors in numerical processing. Finally, while distinguishing thermodynamic entropy from information-theoretic entropy, the paper proposes a testable hypothesis for science education: even when formally introduced at the university level in Japan, entropy remains notoriously difficult to grasp because natural language demonstratives fail to preserve injectivity in highly abstract textual spaces, obscuring what abstract concepts actually refer to for learners before symbolic formalization takes over.