This work elucidates the structure and complexity of human language in terms of the mathematics of information and computation. It strengthens Chomsky's early work on the mathematics of language, with the advantages of a better understanding of language and a more precise theory of structural complexity. Ristad argues that language is the process of constructing linguistic representations from the forms produced by other cognitive modules and that this process in NP-complete. This NP-completeness is defended with a phalanx of elegant and revealing proofs that rely only on the empirical facts of linguistic knowledge and on the uncontroverted assumption that these facts generalize in a reasonable manner. For this reason, these complexity results apply to all adequate linguistics theories and are the first to do so.