这是 Tribute to Vladimir Arnold 一文的续篇，可在此下载。
Dmitry Fuchs
One day we met in a long line at the student canteen. “Listen,” he said, “can you explain to me what a spectral sequence is?” I began uttering the usual words: a complex, a filtration, differentials, adjoint groups, etc. He frowned and then said, “Thus, there is something invariant [‘invariant’ in his language meant ‘deserving of consideration’] in all this stuff, and this is the spectral sequence, right?” I thought for amoment and said, yes.At this moment we got our meals, and our conversation changed its direction.
Whatever he did—mathematics, skiing, biking—he preferred not to learn how to do it but just to do it in the most natural way, and he did everything superlatively well.
He always considered algebra and topology as something auxiliary. Once I heard him saying respectfully, “Siegel’s case, this is a true analysis,” and this sounded like “true mathematics”. Whatever he did, his unbelievably deep understanding of analysis was always his main instrument.
More than that, I knew something that he did not know: the Stokes theorem as it is stated in modern books, (int_C d varphi = int_{partial C} varphi), was first proved and published by the French mathematician E. Goursat (1917).
Yakov Eliashberg
In 1972 Vladimir Igorevich was one of my Ph.D. dissertation referees or, as it was called, an “official opponent”. I remember that on the day of my defense, I met him at 5 a.m. at the Moscow Train Station in Leningrad. He immediately told me that one of the lemmas in my thesis was wrong. It was a local lemma about the normal form of singularities, and I thought (and, frankly, still do) that the claim is obvious. I spent the next two hours trying to convince Vladimir Igorevich, and he finally conceded that probably the claim is correct, but still insisted that I did not really have the proof. A year later he wrote a paper devoted to the proof of that lemma and sent me a preprint with a note that now my dissertation is on firm ground.
During his lectures he liked to make small mistakes, expecting students to notice and correct him. Apparently, this method worked quite well at the Moscow University. Following the same routine during his Syktyvkar lecture, he made an obvious computational error—something like forgetting the minus sign in the formula ((cos x)' = −sin x)—and expected somebody in the audience to correct him. No one did, and he had to continue with the computation, which, of course, went astray: the terms which were supposed to cancel did not. Very irritated, Arnold erased the blackboard and started the computation all over again, this time without any mistakes. After the lecture, he told me that the undergraduate students at Syktyvkar University are very bad. The next day, after my regular class, a few students came to me and asked how is it possible that such a famous mathematician is making mistakes in differentiating (cos x)?
Each Stanford lecture he would usually start with a sentence like “What I am going to talk about now is known to most kindergarten children in Moscow, but for Stanford professors I do need to explain this.”What followed was always fascinating and very interesting.
Yulij Ilyashenko
His presentation was coordinate-free: all the constructions were invariant with respect to coordinate changes. “When you present material in coordinates,” he said, “you study your coordinate system, not the effect that you want to describe.”...The language of pictures was even more important in his course than that of formulas. He always required a student to present the answer in both ways, a formula and a figure, and to explain the relation between them.
Boris Khesin
Arnold finished this story by quoting someone's definition of youth in mathematics which he liked best: “A mathematician is young as long as he reads works other than his own!”
Speaking of writing, once I asked Arnold how he managed to make his books so easy to read. He replied: “To make sure that your books are read fast, you have to write them fast.” His own writing speed was legendary. His book on invariants of plane curves in the AMS University Lecture series was reportedly written in less than two days. Once he pretended to complain: “I tried, but failed, to write more than 30 pages a day.…I mean to write in English; of course, in Russian, I can write much more!”