The language and grammar of gate and multiply and render the sentences unin- mathematics telligible. To illustrate the sort of clarity and simplicity that is needed in mathematical discourse, let us consider the famous mathematical sentence “Two 1 Introduction plus two equals four” as a sentence of English rather than of mathematics, and try to analyse It is a remarkable phenomenon that children can it grammatically. On the face of it, it contains learn to speak without ever being consciously three nouns (“two”, “two” and “four”), a verb aware of the sophisticated grammar they are us- (“equals”) and a conjunction (“plus”). However, ing. Indeed, adults too can live a perfectly satis- looking more carefully we may begin to notice factory life without ever thinking about ideas such some oddities. For example, although the word as parts of speech, subjects, predicates or subor- “plus” resembles the word “and”, the paradigm ex- dinate clauses. Both children and adults can eas- ample of a conjunction, it doesn’t behave in quite ily recognise ungrammatical sentences, at least if the same way, as is shown by the sentence, “Mary the mistake is not too subtle, and to do this it is and Peter love Paris”. The verb in this sentence, not necessary to be able to explain the rules that “love”, is plural, whereas the verb in the previ- have been violated. Nevertheless, there is no doubt ous sentence, “equals” was singular. So the word that one’s understanding of language is hugely en- “plus” seems to take two objects (which happen to hanced by a knowledge of basic grammar - it is be numbers) and produce out of them a new, single almost tautologous to say so - and this understand- object, while “and” conjoins “Mary” and “Peter” ing is essential for anybody who wants to do more in a looser way, leaving them as distinct people. with language than use it unreflectingly as a means to a non-linguistic end. Reflecting on the word “and” a bit more, one The same is true of mathematical language. Up finds that it has two very different uses. One, as to a point, one can do and speak mathematics above, is to link two nouns whereas the other is without knowing how to classify the different sorts to join two whole sentences together, as in “Mary of words one is using, but many of the sentences of likes Paris and Peter likes New York”. If we want advanced mathematics have a complicated struc- the basics of our language to be absolutely clear ture that is much easier to understand if one knows then it will be important to be aware of this dis- a few basic terms of mathematical grammar. The tinction. (When mathematicians are at their most object of this section is to explain the most im- formal, they simply outlaw the noun-linking use of portant mathematical “parts of speech”, some of “and” - a sentence such as “3 and 5 are prime num- which are similar to those of natural languages and bers” is then paraphrased as “3 is a prime number others quite different. These are normally taught and 5 is a prime number”.) right at the beginning of a university course in This is but one of many similar questions: any- mathematics. Much of the Companion can be un- body who has tried to classify all words into the derstood without a precise knowledge of mathe- standard eight parts of speech will know that the matical grammar, but a careful reading of this sec- classification is hopelessly inadequate. What, for tion will help the reader who wishes to follow some example, is the role of the word “six” in the sen- of the more advanced parts of the book. tence, “This section has six subsections”? Unlike, The main reason for the importance of mathe- “two” and “four” earlier, it is certainly not a noun. matical grammar is that the statements of math- Since it modifies the noun “subsection” it is some- ematics are supposed to be precise, and it is not what adjectival, but it does not behave like an ordi- possible to achieve a high level of precision unless nary adjective: the sentences “My car is red” and the language one uses is free of many of the vague- “Look at that tall building” are perfectly gram- nesses and ambiguities of ordinary speech. Math- matical, whereas the sentences “My car is six” and ematical sentences can also be highly complex: if “Look at that six building” are not just nonsense the parts that made them up were not clear and but ungrammatical nonsense. So do we invent a simple, then the unclarities would rapidly propa- new part of speech called a “numeral”? Perhaps 1 2 we do, but then our troubles will only just be be- (3) symbolically, another way to do it is to define ginning: the more one tries to refine the classifica- P to be the collection, or set, of all prime numbers. tion of English words, the more one realizes just Then (3) can be rewritten, “5 belongs to the set how great is the variety of different ways we use P ”. This notion of belonging to a set is sufficiently them. basic to deserve its own symbol, and the symbol used is ∈. So a fully symbolic way of writing the 2 Four basic concepts sentence is 5 ∈ P . The members of a set are usually called its el- Another word, which famously has three quite dis- ements, and the symbol ∈ is usually read “is an tinct meanings, is “is”. The three meanings are element of”. So the “is” of sentence (3) is more illustrated in the following three sentences. like ∈ than =. Although one cannot directly sub- (1) 5 is the square root of 25. stitute the phrase “is an element of” for “is”, one (2) 5 is less than 10. can do so if one is prepared to modify the rest of (3) 5 is a prime number. the sentence a little. In the first of these sentences, “is” could be re- There are three common ways to denote a spe- placed by “equals”: it says that two objects, 5 and cific set. One is to list its elements inside curly the square root of 25, are in fact one and the same brackets: {2, 3, 5, 7, 11, 13, 17, 19}, for example, is object, just as it does in the English sentence “Lon- the set whose elements are the eight numbers 2, 3, don is the capital of the United Kingdom.” In the 5, 7, 11, 13, 17 and 19. The majority of sets consid- second sentence, “is” plays a completely different ered by mathematicians are too large for this to be role. The words “less than 10” form an adjectival feasible - indeed, they are often infinite - so a sec- phrase, specifying a property that numbers may or ond way to denote sets is to use dots to imply a list may not have, and “is” in this sentence is like “is” that is too long to write down: for example, the ex- in the English sentence “grass is green.” As for pressions {1, 2, 3,..., 100} and {2, 4, 6, 8,... } rep- the third sentence, the word “is” there means “is resent the set of all positive integers up to 100 and an example of”, as it does in the English sentence the set of all positive even numbers respectively. “Mercury is a planet.” A third way, and the way that is most impor- These differences are reflected in the fact that tant, is to define a set via a property: an exam- the sentences cease to resemble each other when ple that shows how this is done is the expression they are written in a more symbolic way. An ob- √ {x : x is prime and x < 20}. To read an expres- vious way to write (1) is 5 = 25. As for (2), it sion such as this, one first says, “The set of”, be- would usually be written 5 < 10, where the sym- cause of the curly brackets. Next, one reads the bol < means “is less than”. The third sentence symbol that occurs before the colon. The colon would normally not be written symbolically be- itself one reads as “such that”. Finally, one reads cause the concept of a prime number is not quite what comes after the colon, which is the property basic enough to have universally recognised sym- that determines the elements of the set. In this in- bols associated with it. However, it is sometimes stance, we end up saying, “The set of x such that useful to do so, and then one must invent a suit- x is prime and x is less than 20,” which is in fact able symbol. One way to do it would be to adopt equal to the set {2, 3, 5, 7, 11, 13, 17, 19} considered the convention that if n is a positive integer, then earlier. P (n) stands for the sentence “n is prime”. An- other way, which doesn’t hide the word “is”, is to Many sentences of mathematics can be rewrit- use the language of sets. ten in set-theoretic terms. For example, sentence (2) earlier could be written as 5 ∈ {n : n < 10}. 2.1 Sets Often there is no point in doing this - as here where it is much easier to write 5 < 10 - but there are Broadly speaking, a set is a collection of objects, circumstances where it becomes extremely conve- and in mathematical discourse these objects are nient.