# A charming little textbook

Like many others, the library of the mathematics department in Bordeaux used, every once in a while, to put in a bin somewhere duplicate copies of journals for anyone to take home; more rarely books also appeared there. This way, I found one some day which I find particularly fascinating.

It is not a research book, but an old French textbook for the program of the last year of high-school (the “Terminale”), published in 1971. It is part of a series probably encompassing the whole high-school curriculum. It is written by M. Debray, M. Queysanne, and D. Revuz (yes, the one of Revuz-Yor).

More properly, this is only Volume 1 of the Terminale textbook, since there exist two additional ones (which I do not have). The main topics of this first volume are “Nombres, probabilités” (numbers, probability theory), and one can see what the two others treated by looking at the official curriculum for the Terminale, which is reproduced at the beginning of the text: one was for analysis (basic calculus) and the other for geometry.

So the first striking impression is that of sheer size: if the other two volumes are of comparable length, this amounts to about 1000 pages for a single year of study, and this must have been combined with classes in French, Philosophy, Physics and Chemistry, History and Geography and Biology, and at least one foreign language, with Latin as a bougie sur le gâteau for quite a few students (probably; I can’t seem to find the general official program for 1971).

So what did French students know when finishing high-school in 1971? Before going into this, let’s look at what they were already supposed to know when entering the Terminale (as indicated by various Rappels de première in the text). The first chapter is entitled Entiers naturels (integers), which is reasonable enough; Section 1.1 is

Éléments réguliers pour une loi de composition interne. (“Regular elements for an internal operation”).

So an internal operation on a set is a concept taken for granted. Later, we learn that binary relations are just as old hat (hence also equivalence relations and quotient sets), and similarly for vector spaces (over the reals; not necessarily finite-dimensional). We learn that the study of the reals was, in Première, based on the Propriété des segments emboîtés (the fact that a countable decreasing intersection of non-empty compact intervals is non-empty), but it is taken anew now with the axiom that a non-empty set of reals which is bounded above as a least upper bound.

Then here are some highlights of the book:

• In agreement with the official curriculum, the most important structures (groups, rings, vector spaces, and homomorphisms) are pointed out as frequently as possible whenever they occur;
• The complex numbers are constructed as a subring of the 2×2 matrices over the reals (the product of such matrices being another remnant of the previous year);
• The rationals and the integers are reconstructed as subrings of the reals, and shown to satisfy all the properties previously proved about them from the preliminary chapters where they were studied independently;
• But the construction of Z from the integers, by a good old Grothendieck group construction, is relegated to an Appendix;
• Is N really taken for granted? No, another Appendix sketches the construction of the (nonnegative) integers from set theory (!), through cardinals; this is for “interested readers”;
• Limits are (of course) presented in clean and robust ε / δ fashion.

One may get from this the impression of Bourbakism running rampant among schoolchildren, but I confess to having selected these topics just to give such an impression. In fact, the text is not at all a distilled version of Nicolas’s great treatise for younger readers, but its influence is probably present in two respects: an emphasis on structures (and homomorphisms), and on rigor. Where things are not proved, this is stated clearly: nous avons admis les principales propriétés de… (“We have assumed valid the main properties of…”).

The style itself is much more readable for a textbook: most new definitions come after a short description of why they might be interesting or natural, often together with a preliminary example, which may then be the subject of a quick follow-up after the definition itself. In addition, there are many excellently chosen exercises, ranging from direct computational ones to more adventurous escapades (the arithmetic-geometric mean, for instance).

But more importantly perhaps, there is not only an emphasis on abstraction and definitions: the discussion of the reals contains a detailed investigation of approximations (decimal expansions, interval arithmetic, relative and absolute errors), and there is a really nice probability chapter. There, the definitions, as before, are rigorous and correspond to the general theory: although the theory is of course built on finite sets, there are tribus (“σ-algebras”), random variables, distribution laws and probability measures, and this chapter concludes with a version of the weak law of large numbers.

In the end, I get a mixed feeling from reading this book. It seems a fair guess that, in its time and for its intended audience, it must have been a complete disaster: the amount of the material to assimilate, and its abstract sophistication, must have made it a subject of hatred (or of jest…) for most students. But, as a “platonic” incarnation of the concept of a mathematical textbook, independently of its time and place, it is really not bad. If one or more of my children, one day, develop a taste for mathematics, and if they start looking eagerly, at a young and impressionable age, for almost any book to assuage their thirst for mathematics (as I did, and as most other mathematicians have done and still do), then at a reasonable time I will give them this textbook with pleasure.

(Note: I do not know at what rate the curriculum changed; I myself remember learning equivalence relations well before high-school, since vectors in the plane were introduced as equivalence classes of “bipoints équipollents”, but by the time I was finishing high-school, around 1986-87, they had been removed, though I remember that my mathematics teacher, in the first post-high-school year, started using them during the very first lesson without so much as a word of explanation… Nowadays, judging from what the students know when entering the first semester of university in Bordeaux, the program has probably been reduced by at least 90 percent, and all abstraction has vanished).

### Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

## One thought on “A charming little textbook”

1. Elencwajg says:

Cher collègue,
comme d’habitude, j’ai trouvé votre article fort intéressant et agréablement inattendu .
En Belgique dans les années ’60 Georges Papy mit en place une réforme de l’enseignement des mathématiques dont l’audace et l’esthétique étaient sans pareilles dans le monde.
Son manuel “Mathématiques Modernes I” destiné aux elèves de sixième des athénées et lycées belges (enfants de 12 ans)contenait une magnifique initiation à la théorie des ensembles, incluant la démonstration en belles couleurs du théorème de Cantor-Schröder-Bernstein.
Le manuel Mathématiques Modernes V (élèves de 17 ans)contenait une attendrissante initiation à la théorie des corps (rebaptisés “Champs”).
Je n’ai plus vu ces livres depuis plus de quarante ans mais vous parviendrez sans doute à les consulter dans vos adorables bibliothèques suisses, qui contiennent étrangement pas mal d’ouvrages utilisés dans le secondaire belge il y a 50 ans et plus.

Avec mes nostalgiques salutations,
Georges Professausaurus Elencwajg