Semiotics

When the letter misleads: students, professors and apples

In 1981 three researchers at the University of Massachusetts, John Clement, Jack Lochhead and George Monk, set first-year engineering students a simple problem: Write an equation using the variables $S$ and $P$ to represent the following statement: “There are six times as many students as professors at this university.” Use $S$ for the number of students and $P$ for the number of professors. About 37% got it wrong. And almost all of those who got it wrong wrote the same thing:

October 15, 2026 Read

Dummy letters: the k of the sum and the t of the integral

These two expressions have the same value: $$\sum_{k=1}^{4}k^2=1+4+9+16=30,\qquad\sum_{j=1}^{4}j^2=1+4+9+16=30.$$Swapping $k$ for $j$ changed nothing. The letter is there only to mark where the calculation passes; once the sum is done, it disappears. Letters like these are called dummy, or bound. They are the opposite of unknowns: an unknown keeps a value that matters; a dummy letter keeps no value at all once the operation is over. Free and bound Now compare:

October 15, 2026 Read

f, f(x) and x ↦ x²: three ways to name a function

Chapter 1 told how Euler, in 1734, invented $f(x)$ and how the function became a sentence with a hole in it (Euler and functions). This text looks at the ways of naming a function, and at the confusions each one brings. The function and its value Compare two sentences: “the function $f$ is increasing”; “$f(x)$ is positive for $x>0$”. In the first, $f$ is the whole function, the rule, the object that can be differentiated, composed, drawn. In the second, $f(x)$ is a number: the value the function gives when the input is $x$. The difference is the same as between “the calculator” and “the number on its display”.

October 15, 2026 Read

Parameters: letters that stay still while others move

In the equation $$y=ax^2+bx+c$$there are five letters, and they are not on the same level. The $x$ and the $y$ vary: they run through the points of a parabola. The $a$, the $b$ and the $c$ stay still while this happens, but they can be chosen at will beforehand. Each choice gives a different parabola. Letters like these are called parameters, from the Greek pará, beside, and métron, measure: what stands beside the measure, regulating it.

October 15, 2026 Read

The variable: the letter that moves

Variable quantities are those that continually increase or decrease; and, on the contrary, constant quantities are those that remain the same while the others change. — Marquis de l'Hôpital, Analysis of the Infinitely Small (1696), definition I The unknown is a letter that stands still: it hides a value, and the value does not change. The variable is a letter that moves. In $y=x^2-4x+3$, nobody asks “what is $x$?”. The question is a different one: “when $x$ changes, what happens to $y$?”.

October 15, 2026 Read

Letters that generalise: from 3 + 5 = 5 + 3 to a + b = b + a

Add $3+5$ and $5+3$: you get $8$ both times. Try $12+7$ and $7+12$, $100+1$ and $1+100$. It always works. But how many examples do you need to be sure? A thousand? A million? No number of examples is enough, because there are infinitely many pairs of numbers. The sentence $$a+b=b+a$$solves the problem in another way. It speaks of no particular pair: it speaks of all of them at once. The letters $a$ and $b$ here do not hide a value, as the unknown does; they stand for any value. This is what Usiskin called generalised arithmetic.

October 15, 2026 Read

The unknown: a place for what is being sought

An unknown is, literally, something in-cognita, not known. But it is not just anything: it is a definite number that we do not yet know. The equation $x+3=7$ does not speak of all numbers; it speaks of one, and the task is to find out which. Names for what is not known For a long time the unknown had the name of a word, not of a letter. Around the year 250, Diophantus of Alexandria, in the Arithmetica, called the unknown arithmós, “the number”, and abbreviated it with a sign like a final sigma, $\varsigma$. He also had abbreviations for the square of the unknown, $\Delta^{\Upsilon}$ (dýnamis, power), and for its cube, $K^{\Upsilon}$ (kýbos). In Baghdad, around 820, al-Khwarizmi called it shay’, thing, and its square māl, wealth. He wrote everything out in words, without any symbol. In the fifteenth and sixteenth centuries the Italians translated shay’ as cosa, and the German algebraists who copied the Italians were nicknamed cossists. In 1544 Michael Stifel still used different signs for the thing, its square and its cube: The “cossic” signs of Michael Stifel’s Complete Arithmetic (1544): each power of the unknown had its own symbol, with no visible relation between them.Michael Stifel · Wikimedia Commons · public domain In these signs, the square and the cube of the thing do not look like the thing. It is a sterile notation, in the sense of this book: to know that the cube times the square gives the fifth power, one had to memorise a table. When Descartes wrote $x$, $x^2$, $x^3$, the rule $x^3\cdot x^2=x^5$ became visible in the exponents. Chapter 1 tells this passage in more detail, in the text from rhetoric to symbol.

October 15, 2026 Read

One letter, five roles

Look at these five lines: $$x+3=7,\qquad A=b\cdot h,\qquad a+b=b+a,\qquad y=kx,\qquad \sin^2x+\cos^2x=1.$$All of them use letters, and all of them look like algebra. But in none of them do the letters do the same thing. In $x+3=7$, the $x$ is a fixed, unknown number. Only $4$ will do. It is an unknown. In $A=b\cdot h$, the letters hide nothing: they say how three quantities are related. It is a formula, and the letters are quantities that vary together. In $a+b=b+a$, the letters are arbitrary numbers. The sentence holds for every pair. The letter generalises. In $y=kx$, $x$ and $y$ move, and $k$ stays still. The $k$ is a parameter: fixed in each problem, but arbitrary. In $\sin^2x+\cos^2x=1$, the $x$ is also arbitrary, but the sentence is not an equation to solve: it is an identity, true for every $x$. Usiskin’s conceptions In 1988 the American educator Zalman Usiskin, in an article that became a classic, proposed that school algebra mixes at least four conceptions, each giving the letter a different role:

October 15, 2026 Read

Chapter 2: references and credits

Further reading Saussure, Ferdinand de. Course in General Linguistics. Original: Cours de linguistique générale. Paris: Payot, 1916. Peirce, Charles Sanders. Collected Papers of Charles Sanders Peirce. 8 vols. Cambridge, MA: Harvard University Press, 1931–1958. Volume 2 contains the texts on icon, index and symbol. Peirce, Charles Sanders. The Essential Peirce: Selected Philosophical Writings. 2 vols. Bloomington: Indiana University Press, 1992–1998. Morris, Charles W. Foundations of the Theory of Signs. Chicago: University of Chicago Press, 1938. Duval, Raymond. Sémiosis et pensée humaine: registres sémiotiques et apprentissages intellectuels. Bern: Peter Lang, 1995. Duval, Raymond. A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 2006, p. 103–131. Falkner, Karen P.; Levi, Linda; Carpenter, Thomas P. Children’s understanding of equality: a foundation for algebra. Teaching Children Mathematics, 6(4), 1999, p. 232–236. Mazur, Joseph. Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers. Princeton: Princeton University Press, 2014. Swetz, Frank J. Capitalism and Arithmetic: The New Math of the 15th Century. La Salle: Open Court, 1987. Includes a translation of the Treviso Arithmetic (1478), with the rule of three and gelosia multiplication. Cajori, Florian. A History of Mathematical Notations. 2 vols. Chicago: Open Court, 1928–1929. On the fraction bar, the names of the rule of three and the signs of proportion. The old texts on this site about fractions, multiplication and the rule of three supplied the etymology of numerator and denominator, Egyptian multiplication, Pacioli’s gelosia and the table of names for the rule of three.

October 13, 2026 Read

How to read a notation: a guide and a glossary

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.

October 13, 2026 Read

Formula, table and graph: treatment and conversion

When Descartes published La Géométrie in 1637, he invented neither curves nor equations. Both had existed for centuries. What he invented was a bridge between them: a way of turning every curve into an equation and every equation into a curve, by measuring distances from a reference line (the two perpendicular axes we now call Cartesian became established only after him). The Cartesian plane is, above all, a machine for translating from one script into another.

October 13, 2026 Read

Square root: the side of the square and the sign that forgot

Two questions that look the same: Question Answer What is $\sqrt9$? $3$ Which real numbers satisfy $x^2=9$? $3$ and $-3$ Many people find the first line strange. If $(-3)^2$ also gives $9$, why isn’t $\sqrt9$ “$\pm3$”? The answer is in chapter 1, in the old name of the symbol. Radix quadrati was the side of the square. A square of area $9$ has side $3$, and no side measures $-3$.

October 13, 2026 Read