The language and grammar of mathematics

It is a remarkable phenomenon that children can learn to speak without ever being consciously aware of the sophisticated grammar they are using. Indeed, adults too can live a perfectly satisfactory life without ever thinking about ideas such as parts of speech, subjects, predicates or subordinate clauses. Both children and adults can easily recognise ungrammatical sentences, at least if the mistake is not too subtle, and to do this it is not necessary to be able to explain the rules that have been violated. Nevertheless, there is no doubt that one’s understanding of language is hugely enhanced by a knowledge of basic grammar it is almost tautologous to say so and this understanding is essential for anybody who wants to do more with language than use it unreflectingly as a means to a non-linguistic end. The same is true of mathematical language. Up to a point, one can do and speak mathematics without knowing how to classify the different sorts of words one is using, but many of the sentences of advanced mathematics have a complicated structure that is much easier to understand if one knows a few basic terms of mathematical grammar. The object of this section is to explain the most important mathematical “parts of speech”, some of which are similar to those of natural languages and others quite different. These are normally taught right at the beginning of a university course in mathematics. Much of the Companion can be understood without a precise knowledge of mathematical grammar, but a careful reading of this section will help the reader who wishes to follow some of the more advanced parts of the book. The main reason for the importance of mathematical grammar is that the statements of mathematics are supposed to be precise, and it is not possible to achieve a high level of precision unless the language one uses is free of many of the vaguenesses and ambiguities of ordinary speech. Mathematical sentences can also be highly complex: if the parts that made them up were not clear and simple, then the unclarities would rapidly propagate and multiply and render the sentences unintelligible.