A mathematics text in English is written in two languages at once. One is the language of signs: $=$, $+$, $x$, $\sqrt{\ }$. The other looks like English, but is not quite: it uses everyday words with meanings agreed among mathematicians, and the two mix in the same sentence.

The English linguist Michael Halliday called this variety of language the mathematics register, in the 1970s. A register is a way of speaking tied to an activity, like that of the kitchen (“sauté”, “soft peaks”) or of football (“offside”, “set piece”). The mathematics register has one extra problem: many of its words already exist in ordinary language with another meaning, and the reader is not warned when the meaning changes.

Words that change meaning

The English mathematics educator David Pimm, in his book Speaking Mathematically (1987), collected many such cases. Some of them:

  • A product is the result of a multiplication, not something you buy.
  • A difference is the result of a subtraction; “what is the difference between 7 and 3?” asks for $4$, not “they are both odd”.
  • Volume is a measure of space, not of sound.
  • A root is the side of the square, as we saw in chapter 1, and also the value that makes a polynomial vanish.
  • Normal, rational and real do not mean ordinary, sensible and true.
  • The average can be arithmetic, geometric, weighted; in the newspaper it is almost always just one of them.

The greatest danger is not in the nouns, which a glossary can handle. It is in the small words.

The small words

“Or.” In everyday life, “coffee or tea?” expects a single choice. In mathematics, “$x > 0$ or $y > 0$” is also true when both are positive. Mathematical “or” is inclusive; the exclusive one has to be spelled out: “either one or the other, but not both”.

“If… then.” “If it rains, I won’t go” is usually heard as “and if it doesn’t rain, I’ll go”. In mathematics, “if $n$ is a multiple of 4, then $n$ is even” says nothing about numbers that are not multiples of 4: $6$ is even and not a multiple of 4, and the sentence remains true. To say both things, mathematics has another phrase, “if and only if”, and another sign, the $\Leftrightarrow$ of chapter 4.

“Some.” “Some students were absent” suggests, in conversation, that not all of them were. In mathematics, “some prime number is even” means only “at least one”, and it would still be true if they all were.

“Times more.” “Peter has three times more stickers than Anna.” If Anna has $10$, does Peter have $30$ (three times what Anna has) or $40$ (what Anna has, plus three times that)? Dictionaries and style guides disagree. The mathematics register avoids the doubt by replacing the phrase with triple.

Learning the register

The mathematics register is not learned by decree, from a list of definitions. It is learned like any language: by hearing it in use, speaking it, being corrected. That is why asking a student to read aloud a sentence like $\forall x \in \mathbb{R},\ x^2 \geq 0$ (“for every real $x$, $x$ squared is greater than or equal to zero”) is not a waste of time: it is the passage from one language to the other, and it shows where the reading fails.

Exercise: true or false?

Decide, in the mathematical sense: (a) “Every prime number is odd or equal to 2.” (b) “If a number is divisible by 6, then it is divisible by 3.” (c) “Some multiple of 4 is even.”

(a) True: the only even prime is $2$, and the inclusive “or” does no harm. (b) True: $6k = 3 \cdot 2k$. It does not claim that every number divisible by 3 is divisible by 6 ($9$ is not). (c) True, and in the mathematical sense it remains true even though all multiples of 4 are even: “some” only requires at least one.

Exercise: rewrite without ambiguity

Rewrite in the mathematics register: “Prices rose two times more than wages”, knowing that wages rose $5\%$ and that the intention is to say that the price increase was double the wage increase.

“The percentage increase in prices was double the percentage increase in wages: wages rose $5\%$ and prices $10\%$.” Replacing “two times more” with “double” and writing the numbers removes the two possible readings ($10\%$ or $15\%$).

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