Challenges and Perspectives in the Mathematical Foundations of Computational Linguistics

Abstract

 
Abstract
Mathematics forms the theoretical foundation of computational linguistics; a multidisciplinary field concerned with the computational modelling of human language. This paper reviews the central role of mathematical frameworks in the evolution of computational linguistics, from early formal language theory inspired by Chomsky to modern statistical and machine learning based approaches in natural language processing. Core mathematical areas such as linear algebra, probability theory, and calculus underpin key techniques including vector space models, word embeddings, language modelling, and the training of deep neural networks through gradient-based optimization. Despite substantial progress, significant challenges remain. Accurately modelling semantics and pragmatics particularly context-dependent meaning, ambiguity, and real-world knowledge continues to be difficult. In addition, the interpretability of complex models such as large language models poses theoretical and practical concerns, alongside issues related to data sparsity in low-resource languages and ethical problems such as bias in AI-driven language systems. Looking forward, the field is expected to benefit from deeper integration of advanced mathematical concepts and emerging architectures, including transformers and graph neural networks. Future research is increasingly shifting toward holistic, goal-oriented models of language use. Continued innovation at the intersection of mathematics, linguistics, and computer science will be essential for addressing the remaining complexities of human language.