About this book
Nobody is born knowing how to read $\frac{dy}{dx}$. Or $3+4=7$. Every sign in mathematics was invented by someone, somewhere, to solve some problem, and every one had to be learned by generations of readers before it looked natural. Some opened paths their inventors never imagined; others recorded well what was already known and led nowhere.
This book looks at mathematical notation as a system of signs. Semiotics, the study of signs, helps us ask not only what a symbol stands for, but also how it works, which rules guide the way we read it and, above all, what it lets us do. The old texts on this site called it mirabilis abbreviatio, the marvellous abbreviation: writing that shortens the text without shortening the thought. Here we call it active notation, and its opposite sterile notation.
The plan of the book
- From word to symbol: the history of mathematical notation
- The mathematical sign: convention, reference and interpretation
- Variables, unknowns and functions: letters that organise relations
- Equality, equivalence and proof
- Writing calculus: change, accumulation and infinity
- Diagrams, graphs and the visual side of mathematics
- Learning to read and write mathematics
- Notation, culture and the making of knowledge
The first four chapters are already published, in 46 short texts. The first is historical: it runs from the Indian zero and al-Khwarizmi’s algebra in words, through Aryabhata’s sine, Napier’s logarithms, Leibniz’s $d$, the integral, Cauchy’s limit, Galois’s table and Gibbs’s vectors, to the quantifiers of logic. The second is conceptual: it uses Saussure, Peirce, Morris and Duval to read the minus sign, equality, the square root and graphs. The third looks at letters: the unknown that is sought, the arbitrary number that generalises, the variable that moves, the parameter that stays still, the function that gets a name and the dummy letter that can be swapped for another. The fourth follows the equals sign: the question, the law and the definition it hides, Frege’s sense and reference, Gauss’s $\equiv$, the arrows $\Rightarrow$ and $\Leftrightarrow$ that link the lines of a solution, and the chains of equalities and small words that turn a sequence of calculations into a proof. Each text can be read on its own and ends with two worked exercises. The remaining chapters make up the planned course of the book.