If mathematical notation were a transparent code, a dictionary would be enough: $+$ means “plus”, $=$ means “equals”, and that’s that. This chapter has shown that it is not so. The same stroke is an operation or the name of a number; equals is a balance and not an arrow; blank space multiplies in $ab$ and does not in $23$; the root gives a single value, although the equation has two. Reading mathematics means rebuilding, for each formula, the relations between the script, the object and the rules.

This text gathers the chapter’s tools into a practical guide.

Seven questions for an unfamiliar formula

  1. What is written? Look at the signs, the letters, the position of each (above, below, attached, apart) and the groupings. This is syntax.
  2. Which conventions do I need? Order of operations, the meaning of particular letters ($\pi$, $e$, $i$), notations of the subject. If a convention is not the usual one, the text should say so.
  3. What object does the expression refer to? A number, a function, a set, a statement? In what domain? This is semantics.
  4. What is it asking of me? To calculate, solve, prove, compare, draw? “Find $x$” and “let $x$ be any” ask for opposite things. This is pragmatics.
  5. Which transformations are allowed? Every step in a calculation needs a rule that authorises it, and sometimes a hypothesis: dividing by $a$ requires $a\ne0$; $\sqrt a\sqrt b=\sqrt{ab}$ requires $a,b\ge0$.
  6. Is there another register for the same content? A figure, a table, a sentence, a graph? What does each show best?
  7. What does the representation hide? A figure is not to scale; a table shows only some values; a formula can hide hypotheses; a graph can have a stretched scale.

A test for active notation

Since the book’s first text we have used the idea of active and sterile notation. With Peirce’s tools it can be turned into a test with three questions:

  • Does the shape reproduce relations of the object? $\frac{dy}{dx}$ is shaped like a ratio because the derivative is a limit of ratios. $\dot y$ reproduces nothing.
  • By looking at it, do I discover something I did not deliberately put there? This is Peirce’s criterion for the icon. Place value lets you see the pattern in the nine times table; Roman numerals do not.
  • Does it keep working on new problems? Cayley’s matrices, created for linear systems, later served for rotations, quantum mechanics and computers.

And a warning, made clear by Euler’s square root: an active notation also suggests wrong steps. The more the shape invites, the more important it is to know how far the rule holds.

A full reading of dy/dx

To finish, a reading of $\dfrac{dy}{dx}$ with the seven questions:

  1. Writing: two expressions, $dy$ and $dx$, stacked with a fraction bar.
  2. Conventions: $d$ is not a variable multiplying $y$; it is Leibniz’s operator, from differentia.
  3. Object: the derivative of $y$ with respect to $x$, nowadays defined as a limit: $\displaystyle\lim_{h\to0}\frac{y(x+h)-y(x)}{h}$.
  4. What it asks: depends on the context; it may be computing a rate, finding a tangent, setting up a differential equation.
  5. Transformations: the fraction shape licenses the chain rule and substitution in integrals, but does not license everything that would hold for ordinary fractions.
  6. Other registers: the slope of the tangent line to the graph; Lagrange’s notation, $y'$; Newton’s, $\dot y$, if the variable is time.
  7. What it hides: that the limit exists (not every function has a derivative) and that $dy$ and $dx$ on their own only have a rigorous meaning in more advanced theories.

A notation of five characters holds three hundred years of history, a definition, a handful of rules and at least three ways of drawing it.

Glossary

Sign: something that, for someone, stands for something else.

Signifier and signified: in Saussure, the perceived form of a sign and the concept it evokes.

Arbitrariness: the absence of any natural reason linking signifier and signified; $+$ does not look like a sum.

Syntax, semantics and pragmatics: in Morris, the relations of signs to one another, to their objects and to those who use them.

Interpretant: in Peirce, the effect of meaning a sign produces, which is itself a new sign.

Semiosis: the ongoing process by which signs produce interpretations, which produce other signs.

Icon, index and symbol: in Peirce, signs that represent by resemblance, by factual connection and by convention.

Register of representation: in Duval, a system of signs with its own rules: natural language, algebra, table, graph, figure.

Treatment: a transformation within one register, such as solving an equation.

Conversion: a passage from one register to another, such as drawing the graph of a formula.

Active and sterile notation: this book’s terms (and those of the site’s old texts, which spoke of mirabilis abbreviatio) for a script that helps discover new relations and one that merely records what is already known.

Exercise: apply the guide to Σ

Read $\displaystyle\sum_{k=1}^{4}k^2$.

  1. A large $\Sigma$, with $k=1$ below, $4$ above and $k^2$ to the right.
  2. Euler’s convention: $\Sigma$ denotes a sum; $k$ is an index running through the integers from $1$ to $4$.
  3. The object is a number: $1+4+9+16=30$.
  4. It asks for a calculation.
  5. You may, for example, take a constant outside: $\sum 3k^2=3\sum k^2$.
  6. In another register: the sum of the areas of four squares of sides $1$, $2$, $3$ and $4$.
  7. It hides that $k$ is a “dummy”: $\sum_{j=1}^4 j^2$ is exactly the same thing.
Exercise: active or sterile?

Compare two ways of writing “a plus b, all squared”: the sentence in English and $(a+b)^2$. In the formula, replacing $b$ by $-b$ gives $(a-b)^2$ at once, and the expansion $a^2+2ab+b^2$ becomes $a^2-2ab+b^2$ effortlessly. In the sentence, the same discovery requires rewriting everything. The formula passes the second and third tests; the sentence, as a script for calculation, is sterile. But the sentence is still the best script for saying the formula aloud to someone who does not yet know it.

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