V. I. Arnold’s book recommendations
Books for amateurs that reveal the beauty of math:
The return of mathematical teaching at all levels from the scholastic chatter to presenting the important domain of natural science is a specially hot problem for France. I was astonished that all the best and most important in methodical approach mathematical books are almost unknown to students here (and, seems to me, have not been translated into French). Among these are (sidenote: On the same spirit of Arnold’s list, there is this great Stack Exchange thread, Best Maths Books for Non-Mathematicians.) :
- Essays on Numbers and Figures by V. V. Prasolov
- The Enjoyment of Math by Hans Rademacher and Otto Töplitz
- Geometry and the Imagination by David Hilbert and Stefan Cohn-Vossen
- What is Mathematics? by Richard Courant and Herbert Robbins
- How to Solve It by George Pólya
- Mathematics and Plausible Reasoning also by George Pólya
- Development of Mathematics in the 19th Century by Felix Klein
Hermite’s, Picard’s, and Goursat’s calculus textbooks:
I remember well what a strong impression the calculus course by Hermite (sidenote: Cours D’analyse De L’école Polytechnique.) made on me in my school years. Riemann surfaces appeared in it, I think, in one of the first lectures (all the analysis was, of course, complex, as it should be). Asymptotics of integrals were investigated by means of path deformations on Riemann surfaces under the motion of branching points. Nowadays, we would have called this the Picard-Lefschetz theory.
Picard, by the way, was Hermite’s son-in-law — mathematical abilities are often transferred by sons-in-law. The dynasty J. Hadamard-P. Levy-L. Schwarz-U. Frisch is yet another famous example in the Paris Academy of Sciences.
The “obsolete” course by Hermite of one hundred years ago (probably, now thrown away from student libraries of French universities) was much more modern than those most boring calculus textbooks with which students are nowadays tormented.
Beginning with L’Hôpital’s first textbook on calculus (sidenote: Analyse des infiniments petits.) and roughly until Goursat’s textbook (sidenote: A Course in Mathematical Analysis.) , the ability to solve such problems was considered to be (along with the knowledge of the times table) a necessary part of the craft of every mathematician.
Mentally-challenged zealots of “abstract mathematics” threw all the geometry (through which connection with physics and reality most often takes place in mathematics) out of teaching. Calculus textbooks by Goursat, Hermite, Picard (sidenote: Traité d’analyse.) were recently dumped by the student library of the Universities Paris 6 and 7 (Jussieu) as obsolete and, therefore, harmful (they were only rescued by my intervention).
When I was a first-year student at the Faculty of Mechanics and Mathematics of the Moscow State University, the lectures on calculus were read by the set-theoretic topologist L. A. Tumarkin, who conscientiously retold the old classical calculus course of French type in the Goursat version. He told us that integrals of rational functions along an algebraic curve can be taken if the corresponding Riemann surface is a sphere and, generally speaking, cannot be taken if its genus is higher, and that for the sphericity it is enough to have a sufficiently large number of double points on the curve of a given degree (which forces the curve to be unicursal: it is possible to draw its real points on the projective plane with one stroke of a pen).
The classic 10-volume series on Physics by Landau:
A teacher of mathematics, who has not got to grips with at least some of the ten volumes of the course by Landau and Lifshitz (sidenote: The Course of Theoretical Physics.) will then become a relict like the one nowadays who does not know the difference between an open and a closed set.
His own series of lectures about group theory:
In the 1960s I taught group theory to Moscow schoolchildren. Avoiding all the axiomatics and staying as close as possible to physics, in half a year I got to the Abel theorem on the unsolvability of a general equation of degree five in radicals (having on the way taught the pupils complex numbers, Riemann surfaces, fundamental groups and monodromy groups of algebraic functions). This course was later published by one of the audience, Valery Borisovich Alekseev, as the book Abel’s Theorem in Problems and Solutions.