V. I. Arnold and the ultimate calculus textbook?

V. I. Arnold passed away suddenly, on June 3rd, 2010, in a French hospital while being operated on for peritonitis. This was sudden and unexpected and took away a delightful teacher. It is most regretful.

On March 7th, 1997 He gave an address, “On teaching mathematics” at the Palais de Découverte in Paris. The entire transcript can be found online.

This speech was very forthright and direct. Reading the text one soon appreciates the fact that the audience possibly had one of three reactions: standing on their chairs cheering Arnold on; sitting in embarrassed silence; or rioting in the aisles. Since I have never read of any harsh reaction to his views rioting is out of the question. Perhaps a humiliation of the academic pedagogic mathematics community resulted. Professor Arnold made a clear denunciation of the then, and I suppose even now, technique of teaching mathematics divorced from the realities of physics and insight of geometry.

Towards the end he recommends some books he feels are well within the expectations of the general reader who is interested in becoming more aware of the subtleties of the science of mathematics to have read:

“Among these are Numbers and figures by Rademacher and Töplitz, Geometry and the Imagination by Hilbert and Cohn-Vossen, What is Mathematics? by Courant and Robbins, How to solve it and Mathematics and Plausible Reasoning by Polya...”

These congruency of mathematics with geometry is apparent to any who have opened these books.

Then he makes a remarkable statement:

“I remember well what a strong impression the calculus course by Hermite (which does exist in a Russian translation!) 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 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.

“If mathematicians do not come to their senses, then the consumers who preserved a need in a modern, in the best meaning of the word, mathematical theory as well as the immunity (characteristic of any sensible person) to the useless axiomatic chatter will in the end turn down the services of the undereducated scholastics in both the schools and the universities.

“A teacher of mathematics, who has not got to grips with at least some of the volumes of the course by Landau and Lifshitz, will then become a relict like the one nowadays who does not know the difference between an open and a closed set.”

So, here then, in a few brief paragraphs, we have a recommendation of books that should be read, then a book that should be read if it could be found, literally, and finally a series of books that could be read if one wants to consider himself or herself a serious student.

It is easy to obtain the first mentioned books 

and the last books, the famous Landau and Lifshitz series,

but where does one find the much older text by Hermite? Indeed, what exactly is it’s title?

Giving Professor Arnold full credit for his insight on calculus pedagogy, the problem becomes: What text is Arnold referring to and how can it be brought to life in English if it isn’t already?

I have done some investigation and narrowed down the search to Hermite’s Cours d’ Analyse de L’Ecole Polytechnique, from 1873.

So the chase is on: identify, obtain Hermite or the Russian translation of Hermite, or both, ideally, and initiate an attempt to translate one or both into English! In the meantime one should immediately hie themselves to the bookseller and obtain all of the books in the first list and start reading.