Unlike most books with similar titles, this one is about linguistics and aimed at mathematicians and not the other way around. The preface tells us right away who it is for: ‘The book is accessible to anyone with sufficient general mathematical maturity (graduate or advanced undergraduate). No prior knowledge of linguistics or languages is assumed on the part of the reader’ (viii).
What is covered? Four traditional linguistic topics—phonology, morphology, syntax, and semantics—are well delineated, with a chapter devoted to each. A fifth, phonetics, is also well represented but scattered through various other sections and chapters. Interleaved with these are four mathematical topics: formal systems for discrete objects, finite-state machines, applications of randomness, and models for continuous phenomena.
In the last chapter, Kornai outlines the themes and challenges of the field (if it is one) and asks: ‘Is there, then, a single thread that binds mathematical linguistics together? This book makes the extended argument that there is, and it is the attempt to find fast algorithms’ (249). Actually, this does not really explain the book’s design. A clue to it can be found in K’s dissertation (1995). That work is an extended example of explication, in the sense of Carnap (1950), combined with the attempt to bridge the gap between phonological theory and speech engineering, to the potential improvement of both. In Mathematical linguistics, K attempts to do the same for theoretical linguistics as a whole. It seems to us, however, that K’s presentation reflects the idea of classifying systems by their architectural features, rather than by the running times of algorithms. He also goes well beyond the topics that would be covered in introductory linguistics courses.
K is a phonologist (as well as a mathematician), so a lot of detail and original thinking is evident in the sections on phonology and phonetics. He starts with Bloomfield’s postulates (1926) and Pāṇini, and presents the problems and perspectives of phonology from the ground up. Readers with no background in linguistics—supposedly the targeted readers—might do well to examine an introductory linguistics text, lest their eyes glaze over with the free use of articulatory terms such as postalveolar, coronal, and the like. Later parts of the chapter deal with information-theoretic aspects of phonology. Some aspects of phonology are also dealt with in the next chapter (morphology): the prosodic hierarchy, syllables, and stress systems. This chapter deals in depth with the logic of distinctive features, natural classes, and so on.
In the chapter on morphology (Ch. 4), there are no big surprises in K’s discussion of word structure and derivational and inflectional morphology. But there are a few mistakes: K misuses the term ‘incorporating’ for ‘polysynthetic’ or the like (62), and in discussing the nature of inflections (63), he cites person (in Gunwinggu) as having four values: singular, dual, trial, and plural. [End Page 970] But these are of course values for number, not person. Finally, ‘unmarked (i.e. expressed by a zero morpheme)’ (63) is surely not right.
Turning to the next chapter (Ch. 5), what syntax is included? As might be expected, K favors frameworks with formally respectable characterizations: dependency grammar, categorial grammar of various persuasions, and the like, but in general the array of theories discussed is very broad, much broader than would be found in typical linguistics textbooks, which tend to make one choice for development in depth. Again, what makes K’s exposition unique is the development from the ground up from first principles.
More precisely, K treats three subtopics: combinatorical theories, that is, categorial grammar, phrase structure (§5.1); grammatical theories (making central use of grammatical case), that is, dependency grammars, linking theories, valency (not so much different theories but rather aspects or dimensions of theories) (§5.2); and semantics-driven theories (§5.3). Further sections deal with topics that are usually not treated in linguistic syntax courses and textbooks at all: weighted theories (§5.4) and the ‘regular domain’ (§5.5), both treated below in some depth.
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