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Unformatted text preview: The language and grammar of mathematics 1 Introduction It is a remarkable phenomenon that children can learn to speak without ever being consciously aware of the sophisticated grammar they are us- ing. Indeed, adults too can live a perfectly satis- factory life without ever thinking about ideas such as parts of speech, subjects, predicates or subor- dinate clauses. Both children and adults can eas- ily 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 ones understanding of language is hugely en- hanced by a knowledge of basic grammar - it is almost tautologous to say so - and this understand- ing 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 struc- ture 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 im- portant 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 un- derstood without a precise knowledge of mathe- matical grammar, but a careful reading of this sec- tion will help the reader who wishes to follow some of the more advanced parts of the book. The main reason for the importance of mathe- matical grammar is that the statements of math- ematics 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 vague- nesses and ambiguities of ordinary speech. Math- ematical sentences can also be highly complex: if the parts that made them up were not clear and simple, then the unclarities would rapidly propa- gate and multiply and render the sentences unin- telligible. To illustrate the sort of clarity and simplicity that is needed in mathematical discourse, let us consider the famous mathematical sentence Two plus two equals four as a sentence of English rather than of mathematics, and try to analyse it grammatically. On the face of it, it contains three nouns (two, two and four), a verb (equals) and a conjunction (plus). However, looking more carefully we may begin to notice some oddities. For example, although the word plus resembles the word and, the paradigm ex- ample of a conjunction, it doesnt behave in quite the same way, as is shown by the sentence, Mary and Peter love Paris. The verb in this sentence, love, is plural, whereas the verb in the previ- ous sentence, equals was singular. So the wordous sentence, equals was singular....
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