Mohan Ganesalingam.The Language of Mathematics: A Linguistic and Philosophical Investigation.FoLLI Publications on Logic, Language and Information.

Abstract

This ambitious book sets out to provide a linguistic analysis of the language used in written mathematics, both textual and symbolic. It is a revised version of the author’s Cambridge Ph.D. thesis, a worthy recipient of FoLLI’s E.W. Beth dissertation award for 2011. Mohan Ganesalingam is a linguist with a Ph.D. in computer science, and his work combines insights from these disciplines with a substantial grasp of mathematics. However, there is much in the book that should interest philosophers of mathematics. Firstly, Ganesalingam’s project leads him to confront some significant issues in the foundations of mathematics, for which he proposes a response that is, in part, novel. Secondly, and perhaps more importantly, he demonstrates something which is often discussed but seldom attempted: he shows how his account of mathematics can be applied to a significant body of actual mathematical practice. The book is very clearly structured. Chapter 1 begins with a defence of Ganesalingam’s methodological presuppositions. Critically, and in distinction from earlier projects of more modest scope, notably the work of Aarne Ranta [1997], he insists on sufficient breadth to encompass all of pure mathematics and on what he calls ‘full adaptivity’, that any mathematical content be extracted from the text under analysis, and not baked into the analytic system (p. 3). The latter constraint prevents him from, for example, building set theory into his linguistic model. Although his account is intended to provide an analysis of the content of ‘rigorous, careful textbooks’ he confines it to what he calls the ‘formal mode’ of the language found therein: the statements exclusively concerning mathematical objects and mathematical properties (p. 7). Characteristically, such textbooks also contain much that is in an informal mode — remarks about the context, or value, or interest, of the mathematical results, say — but, as Ganesalingam notes, analysis of these comments would require a full analysis of natural language (p. 8). Conversely, one of the attractions mathematics in the formal mode holds for the linguist is its comparative simplicity.