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

  1. 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.
  2. What community uses it? Physicists, engineers, programmers, market sellers. The same letter can mean different things in each tribe, like $\log$ or $j$.
  3. What did it replace, and why? The metre replaced local measures because trade gained from it; decimal time replaced nothing because nobody gained.
  4. 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.
  5. 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.

  1. The tool: the punched card and the Fortran keyboard, which had few signs and inherited the equals sign from formulas.
  2. The community: programmers, for whom equals means “store”. For a mathematician, the line is an equation forcing $\text{price} = 0$.
  3. What it replaced: the “add and note down” a cashier did by hand, product by product.
  4. Whom it leaves out: the student who has just learned, at school, that the equals sign asserts an equality.
  5. What it makes thinkable: the idea of a name as a box that changes over time, the basis of any iterative algorithm.

Glossary

  • Rod numerals. Chinese writing of numbers, copied from the counting board, with rods that alternate upright and lying from one place to the next.
  • Base 20 (vigesimal). A system in which each place is worth twenty times the previous one, used by the Maya, with a sign of its own for zero.
  • Quipu. A set of knotted cords used by the Inca to record numbers in base 10 with place value.
  • Decimal time. Division of the day into 10 hours of 100 minutes of 100 seconds, imposed by the French Revolution in 1794 and abandoned in 1795.
  • Grad. A unit of angle in which the right angle measures 100. It survives as the GRAD mode of calculators.
  • Einstein convention. A rule that implies a sum over every index appearing twice in a term.
  • Floor and ceiling. $\lfloor x \rfloor$, the largest integer less than or equal to $x$, and $\lceil x \rceil$, the smallest integer greater than or equal to $x$ (Iverson).
  • Iverson bracket. $[P]$ is $1$ if the statement $P$ is true and $0$ if it is false.
  • Assignment. A command that stores a value in a name: x = x + 1 in Fortran and C, x := x + 1 in Algol and Pascal, $x \leftarrow x + 1$ in Knuth.
  • Stigler’s law. No discovery bears the name of the person who made it first. Attributed by Stigler to Merton.
  • Eponym. A person’s name given to a theorem, formula or object, as in “Pascal’s triangle”.
  • Ethnomathematics. The study of the ways of counting, measuring and calculating of cultural groups, inside and outside school (D’Ambrosio).
  • Proof assistant. A program that checks, step by step, a proof written in a formal language, such as Coq and Lean.
Exercise: apply the guide

Apply questions 1, 2 and 5 to the percent sign, $\%$.

  1. Tool: the sign comes from the Italian commercial abbreviation of per cento, handwritten in account books and, over time, shrunk to $\%$. 2. Community: merchants and bankers, before mathematicians. 5. What it makes thinkable: comparing proportions with different denominators on a common standard, $100$: $\frac{3}{8}$ and $\frac{7}{20}$ are hard to compare, $37.5\%$ and $35\%$ are not.
Exercise: three notations, one number

Write the number $2026$ in base 60 and in the Maya base 20 (without the calendar rule).

Base 60: $2026 = 33 \times 60 + 46$, that is, $33\,;\,46$, like $33$ minutes and $46$ seconds. Base 20: $2026 = 5 \times 400 + 1 \times 20 + 6$, that is, the places $5$, $1$ and $6$: a bar on top, a dot in the middle, a bar and a dot at the bottom. The interactive figure in the first text checks both.

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