This chapter read notation as the product of people, places and tools. The guide below gathers the questions that help to make that reading of any sign. Five questions for a notation What tool was it born of? The rod board, the punched card, the APL keyboard, the blackboard. The shape of the signs often keeps the shape of the tool. What community uses it? Physicists, engineers, programmers, market sellers. The same letter can mean different things in each tribe, like $\log$ or $j$. What did it replace, and why? The metre replaced local measures because trade gained from it; decimal time replaced nothing because nobody gained. Whom does it leave out? Every abbreviation charges a toll to those who do not know the convention. Whoever writes for many readers undoes the shortcut at least once. What does it make thinkable? Red and black rods made negative numbers thinkable; the Iverson bracket turns counts into sums; formal writing makes it possible for a machine to check a proof. A complete reading of a sign: the equals sign in a program Consider the line total = total + price, in a supermarket checkout program.
A notation does not belong to all mathematicians in the same way. Each field, sometimes each school, has its shortcuts, and outsiders trip over them. This is not a flaw: it is what happens with any living language, which splits into accents and jargons. But it has consequences for readers and for learners. Einstein erases the summation sign In 1916, in the paper that presented general relativity, Albert Einstein had to write hundreds of expressions like
In 1793, at the height of the French Revolution, the National Convention decided that time too would be decimal. The day would have 10 hours, each hour 100 minutes and each minute 100 seconds. Clockmakers made new dials, some with both scales side by side so that people would get used to them. A paper dial presented by the clockmaker Hanin to the National Convention, with the new hours in Arabic numerals and the old ones in Roman numerals, for comparison: “relation of the new hours to the old, adopted by the National Convention”.Hanin · Wikimedia Commons · public domain Decimal time became compulsory at the end of 1794 and was suspended in April 1795, seventeen months after it was decreed. Nobody wanted to change their household clock, church clocks went on striking twelve, and the new time had no practical advantage to make up for the bother. The Republican calendar, with months of 30 days divided into three “decades” of 10, lasted a little longer: Napoleon abolished it at the start of 1806.
Every script has rules nobody remembers learning. In English we read from left to right and from top to bottom; in Arabic, from right to left. Mathematical writing has rules of this kind too, and some are so strong that they look like mathematical truths. They are not: they are conventions, agreements that could have been different. The order of operations What is $2 + 3 \cdot 4$? Almost everyone answers $14$: first the multiplication, then the addition. But nothing in mathematics forces this order. If the convention were “from left to right”, the answer would be $20$. The rule we use (powers, then multiplications and divisions, then additions and subtractions) settled slowly in the algebra books of the sixteenth to nineteenth centuries, because it saves brackets in the expressions that appear most often: polynomials such as $3x^2 + 2x + 1$ would be unreadable otherwise.