> For example, not only students but also modern algebro-geometers on the whole do not know about the Jacobi fact mentioned here: an elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system.
This one resonated with me. At one point I had a problem that had to do with Kahler manifolds, but I knew nothing about them and only had some worked examples using basic ideas from Hamiltonian mechanics that I learned from Arnol'd's book. In hopes of resolving my issues, I spoke to some symplectic geometers and used Hamilton-Jacobi language. They vaguely knew what I was talking about but couldn't carry out any calculations.
I once heard an anecdote about Arnol'd that he came to France and lambasted the French mathematicians on a similar basis -- for all their writing, they couldn't carry out "simple" calculations with a clear vision (as once can when one has a physics motivation). It might have been the anecdote was referring to the Serre-Arnold debate -- thanks for that link.
Of course, Arnol'd has... high standards, to put it mildly. In his book, "Problems for children from 5 to 15", he writes in the preface,
"My long experience has shown that, very frequently, dimwits falling at
school behind solve them better than A-grade pupils, since – for their survival
at the back of the classroom – they must permanently think more than required
“for governing the whole Seville and Granada”, as Figaro used to say about
himself, while A-graders cannot catch “what should be multiplied by what”
in these problems. I have also noticed that five year old kids solve similar
problems better than pupils spoiled by coaching, which in their turn cope with
the questions better than university students used to swotting who anyway
beat their professors (the worst in solving these simple problems are Nobel
and Fields prize winners)."
One of the problems is to sum 1/n^2 from 1 to infty (this is for children not older than 15, remember). Not only would I be unable to do this without modern technology (like Fourier analysis), I also find it amazing that Arnold writes:
"prove that the sum is pi^2/6, that is, approximately 3/2",
as though the approximation were harder (or perhaps more important) than finding the exact value. To me, that small comment really underlines just how hands-on he was.
Edit: I remembered a joke.
Why did Bourbaki stop writing textbooks?
They found out Serge Lang was one person.