The only way to rectify our reasonings is to make them as tangible as those of the mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: let us calculate!, without further ado, to see who is right. — Leibniz, The Art of Discovery (1685) Leibniz was a lawyer, diplomat and philosopher. He spent his life watching arguments that never ended, between religions, between countries, between schools of philosophy. A mathematician, on the other hand, does not argue about whether $2+2$ is 4: he does the sum. What if arguments could be calculated like numbers?
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.