Semiotics

Reading with the hands and the ears: Braille and the Nemeth code

Ordinary mathematical writing is two-dimensional. In $\frac{x+1}{2}$, what is above the bar is divided by what is below it; in $x^2$, the $2$ is up high and therefore an exponent; in $\sqrt{x+1}$, the stroke of the root covers exactly what is inside. Position is part of the sign, like the place of a digit in a numeral. The eye reads all this at once. But what about someone who does not read with the eye?

October 25, 2026 Read

Writing to think: the notebooks of Gauss and Ramanujan

All mathematical writing has a reader. Sometimes the reader is another person; sometimes it is the author, days or years later. The two call for very different kinds of writing, and two famous notebooks show what happens when a text written for oneself falls into other people’s hands. Gauss’s diary On 30 March 1796, an 18-year-old student in Göttingen opened a small notebook and noted, in Latin, that he had discovered how to divide the circle into 17 equal parts with straightedge and compass, a problem open since the Greeks. It was Carl Friedrich Gauss, and it was this result that made him choose mathematics over philology. Over the next eighteen years he wrote 146 short entries in the notebook, almost always a single sentence, sometimes a formula.

October 25, 2026 Read

Errors that make sense: false rules

Ask a class what $(a+b)^2$ is and some will answer $a^2 + b^2$. Ask what $\sqrt{9 + 16}$ is and someone will say $3 + 4 = 7$. These errors are so common, and so similar from one school to another, that they cannot be chance. They follow a rule. The rule that stretches too far In 1982 the American researcher Marilyn Matz studied the algebra errors of high-school students and proposed that many of them come from false rules (malrules) built out of true ones. The most productive of all is the distributive law:

October 25, 2026 Read

The conventions nobody writes down

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.

October 25, 2026 Read

The captain's age: from word problem to equation

At the end of the 1970s, researchers at the Institute for Research on Mathematics Teaching (IREM) in Grenoble, France, gave primary-school children a strange problem: There are 26 sheep and 10 goats on a ship. How old is the captain? The problem has no answer: nothing in it mentions anyone’s age. Even so, according to the account that the educator Stella Baruk made famous in her book L’âge du capitaine (1985), 76 out of 97 children answered with a number, almost always $36$, the sum of the two that appeared in the text.

October 25, 2026 Read

Two languages in one sentence: the mathematics register

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.

October 25, 2026 Read

Read before you calculate: Bogdanov-Belsky's painting

In 1895 the Russian painter Nikolai Bogdanov-Belsky exhibited a painting called Mental Arithmetic. In the Public School of S. A. Rachinsky. In a village classroom, barefoot peasant boys stare at the blackboard, some with a hand on their chin, one whispering in the teacher’s ear. On the blackboard is written: $$\frac{10^2 + 11^2 + 12^2 + 13^2 + 14^2}{365}.$$ Nikolai Bogdanov-Belsky, Mental Arithmetic. In the Public School of S. A. Rachinsky (1895). Tretyakov Gallery, Moscow.Nikolai Bogdanov-Belsky · Wikimedia Commons · public domain The teacher really existed. Sergei Rachinsky was a professor of botany at Moscow University who gave up his career to teach peasant children in the village of Tatevo, where he had been born. He liked to set sums that looked enormous and had a shortcut. Bogdanov-Belsky himself had been one of his pupils.

October 25, 2026 Read

Counting, saying and writing numbers

The oldest surviving piece of written mathematics is a scratched bone. The Ishango bone, found in 1950 on the shore of Lake Edward, on the border between the Democratic Republic of the Congo and Uganda, is about 20,000 years old and carries groups of parallel notches carved in three columns. Nobody knows for sure what they counted. But the idea is clear: one mark for each thing. The Ishango bone, with the groups of notches carved along the handle. Museum of Natural Sciences, Brussels.Joeykentin · Wikimedia Commons · CC BY-SA 4.0 In the language of chapter 2, the notch is an icon: five marks look like five sheep, at least in number. That is why this writing does not need to be taught. You only have to look. The price is size: to write a thousand, a thousand marks.

October 25, 2026 Read

References and credits

Further reading Euler, Leonhard. Lettres à une princesse d’Allemagne sur divers sujets de physique et de philosophie. St Petersburg, 1768–1772. Letters 102 to 108 deal with syllogisms using circles. Venn, John. On the diagrammatic and mechanical representation of propositions and reasonings. The London, Edinburgh and Dublin Philosophical Magazine, 10, 1880, pp. 1–18. Peirce, Charles S. Collected Papers, vol. 4. Cambridge, MA: Harvard University Press, 1933. On diagrams and diagrammatic reasoning. Clagett, Marshall (ed.). Nicole Oresme and the Medieval Geometry of Qualities and Motions. Madison: University of Wisconsin Press, 1968. Playfair, William. The Commercial and Political Atlas. London, 1786. Tufte, Edward R. The Visual Display of Quantitative Information. Cheshire: Graphics Press, 1983. Huff, Darrell. How to Lie with Statistics. New York: W. W. Norton, 1954. Cleveland, William S. The Elements of Graphing Data. Monterey: Wadsworth, 1985. Argand, Jean-Robert. Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques. Paris, 1806. Nahin, Paul J. An Imaginary Tale: The Story of √−1. Princeton: Princeton University Press, 1998. Nelsen, Roger B. Proofs Without Words: Exercises in Visual Thinking. Washington: Mathematical Association of America, 1993. Abbott, Edwin A. Flatland: A Romance of Many Dimensions. London: Seeley, 1884. Garland, Ken. Mr Beck’s Underground Map. Harrow Weald: Capital Transport, 1994. Image sources Image Source Licence Exports and imports with Africa, by Playfair (1786) Wikimedia Commons public domain John Snow’s cholera map (1854) Wikimedia Commons public domain Florence Nightingale’s diagram of the causes of mortality (1858) Wikimedia Commons public domain The Russian campaign, by Charles Minard (1869) Wikimedia Commons public domain Title page of Argand’s essay (1806) Wikimedia Commons public domain Königsberg in 1652, engraving by Merian’s heirs Wikimedia Commons public domain Title page of Flatland, American edition of 1885 Wikimedia Commons public domain Plan of a house in Flatland, drawing by Abbott Wikimedia Commons public domain The diagrams of Euler’s circles, Oresme’s latitudes and the staircase sum were produced for this book and contain no text, so they serve all four languages. The interactive figures (the scale of a chart, Argand’s plane and the hypercube) work in all four languages of the site. The record of the downloaded images, with the hash of each file, is in data/semiotica_imagens.json.

October 19, 2026 Read

How to read a diagram: a guide and a glossary

This chapter read drawings. The guide below gathers the questions that help read them with the same care with which one reads a formula. Five questions for a diagram What does the drawing copy? A diagram is an icon of some relations, never of all. In a graph, only the connections; in a Venn diagram, only the inclusions. Find out what matters and ignore the rest. What is convention? The scale, where the axes start, the colours, the shape of the box. These choices do not appear as statements, and they change what the eye concludes. Length, area or angle? The eye compares lengths well, areas badly and angles worse. Ask which of them carries the information and whether it was drawn in the right proportion. Does the drawing prove or illustrate? If every relation the eye uses can be guaranteed by an argument, the drawing proves. If not, it illustrates, and it may deceive. What was left out? An extra dimension, a variable that did not fit, a value the axis cut off. Every drawing is a choice, and the choice says something. A complete reading of a newspaper chart A newspaper shows, in a line chart, the price of a product that went from 10.00 to 10.50 in a year, with the vertical axis running from 9.90 to 10.60. The line crosses the whole chart from bottom to top.

October 19, 2026 Read

The fourth dimension: drawing what cannot be seen

All the diagrams in this chapter have one advantage: the object represented fits, one way or another, on paper. But mathematics speaks quite naturally of spaces of four, ten or a million dimensions. Algebra has no difficulty at all: a point in four dimensions is just a list of four numbers, $(x,y,z,w)$. The eye, however, lives in three. How do you draw what cannot be seen? Flatland The title page of Flatland: a romance of many dimensions, “by A Square”, in the American edition of 1885. The real author was the English schoolmaster Edwin Abbott.Edwin A. Abbott · Wikimedia Commons · public domain In 1884 Edwin Abbott, headmaster of a school in London, published under the pseudonym A Square a short novel called Flatland. The narrator is a square living in a two-dimensional world inhabited by triangles, squares, pentagons and circles. The inhabitants see only segments: viewed edge-on, every polygon is a line.

October 19, 2026 Read

Proofs without words

In 1993 the Mathematical Association of America published a curious book, edited by Roger Nelsen: Proofs Without Words. Each page carries a drawing and, sometimes, a formula underneath. Not one sentence. The idea is that the reader looks, understands and is convinced without anyone explaining anything. The book was so successful that it gained two more volumes, and the journal Mathematics Magazine still publishes such proofs today. The staircase that becomes a rectangle A staircase with 1, 2, 3 and 4 small squares, and two identical staircases fitted together to form a 4 by 5 rectangle.Teaching diagram produced for this book. What is $1+2+3+4$? Draw a staircase with those rows of small squares. Now make a copy, turn it round and fit it alongside: the two staircases form a $4$ by $5$ rectangle, with $20$ small squares. Each staircase is half:

October 19, 2026 Read