**The mathematics of language.** By **Marcus Kracht**. (Studies in generative grammar 63.) Berlin: Mouton de Gruyter, 2003. Pp. xvi, 589. ISBN 3110176203. $137.20 (Hb).

Reviewed by **Sean A. Fulop**, *California State University Fresno*

Books in mathematical linguistics are not published very often, and this one is certainly a welcome addition to the literature, as it is a highly independent and novel treatment of many mathematical aspects of linguistics. Kracht begins with a brief introduction that serves as an overview of the book, highlighting a major innovation of his overall approach to the mathematical modeling of language. In traditional mathematical linguistics, deriving from Noam Chomsky’s early work in the late 1950s, a formal language is simply a set of sentences, whether treated as strings simpliciter (most commonly) or as syntactic structures of some kind (occasionally done; cf. Philip Miller, *Strong generative capacity*,* *CSLI Publications, 1999), but K brings meanings into the fold and also treats elements of every syntactic category in a more egalitarian kind of formal model. For K, a formal language is an ‘algebra of signs’, where a sign consists of its exponent (string), its syntactic category, and its meaning. In this way, all the strings of every syntactic category can be admitted into the language and given interpretations as a result of his algebraic view—in algebra, one has a set of elements together with operations that act on some of the elements to yield other elements, so that both parts and wholes are included, as it were.

Ch. 1, ‘Fundamental structures’, presents the needed mathematical preliminaries, but K’s methods are quite novel because his command of mathematics is evidently great. Nowhere does he rely on ‘standard’ treatments, preferring to make everything up his own way from more basic principles. This has the advantage of presenting a highly unified perspective that is quite interesting, but it comes with the drawback that even the experienced mathematical linguist will find it essential to actually read the chapter in order to understand the rest of the book. It is clear from the outset that here we have no ordinary mathematical methods book in the vein of Barbara Partee, Alice ter Meulen, and Robert Wall’s *Mathematical methods in linguistics* (Kluwer, 1991). It also becomes clear very quickly that a decent amount of mathematics experience is a prerequisite. The chapter includes an interesting and original section on fundamentals of linguistics.

Ch. 2, ‘Context free languages’, discusses recognition and parsing in connection with regular, context-free, and semilinear languages, concluding with a useful section addressing the perennially confusing question of whether natural languages are context-free. Ch. 3, ‘Categorial grammar and formal semantics’, introduces the central novel contributions of the work mentioned above, viz. treating languages as algebras of signs that admit compositionally interpreted meanings. Here K also treats basic logic, lambda calculus, combinatory logic, and Montague semantics from his own perspective, including a tutorial run-through of Pentus’s proof that Joachim Lambek’s syntactic calculus generates exactly the context-free languages.

Ch. 4, ‘Semantics’, delves into various aspects of formal semantics, including Boolean semantics, intensionality issues, binding and quantification, and even dynamic semantics of discourse. Ch. 5, ‘PTIME languages’, discusses various approaches to formal grammar, all possessing polynomial-time parsing algorithms and loosely describable as ‘mildly context-sensitive’, namely literal-movement grammars, linear context-free rewrite systems, tree-adjoining grammars, index grammars, and K’s own revisionist ‘de Saussure grammars’.

The sixth and final chapter, ‘The model theory of linguistic structures’, presents a model-theoretical approach to various elements of formal linguistic structure, including categories, strings, allophones and phonemes (no, phonology is not ignored here as usual), and also ordered trees. K then treats numerous well-known grammar formalisms using the preceding groundwork, viz. transformational grammar, generalized and head-driven phrase structure grammar, and finally government and binding theory, after which the book simply stops in true mathematician’s form (mathematicians often do without such niceties as conclusions).

At the end, one finds a valuable bibliography that includes all relevant primary sources, and an excellent index that includes the numerous arcane symbols—handy when the reader has forgotten a definition. All in all, it is a wonderful book, highly worth studying (one can hardly ‘read’ the book in the normal sense of the word). The extremely original approaches that are introduced everywhere get a bit draining because no amount of familiarity with the literature will ease the reader’s burden, but they also have the great merit of providing an uncommonly uniform perspective on a very wide range of topics that one would expect to suffer from a disjointed array of approaches. The book includes useful exercises at the end of most sections (247 in all!), and this makes it easy to use for teaching (advanced) courses.

One slight quibble with K’s book is that it might better be titled ‘The mathematics of linguistics’, since it is really a unified set of tools for developing and evaluating linguistic formalisms, and does not really consider very many mathematical properties of languages themselves. That said, the book derives numerous important facts about the mathematical properties of linguistic theories that all linguists should be aware of, not least of which are: (i) Because so many currently studied syntactic formal frameworks possess identical language-generating power, empirical results showing that a particular framework can account for certain syntactic facts in this or that language are as worthless as they are sophistic; (ii) There is no way to determine a unique set of phonemes of a language, so there can be no single correct phoneme inventory.