Suppose you prove a theorem, and then sleep well at night knowing that future generations will remember your name in conjunction with the great advance in human wisdom. In fact, sadly, it seems that someone will publish the same (or almost the same) thing $n \ll \infty$ years later. I was wondering about what examples of this people might have. Here are two:
Bill Thurston had remarked in the late seventies that Andre'ev's theorem implies the Circle Packing Theorem. The same result was proved half a century earlier by Koebe (so the theorem is now known as the Koebe-Andre'ev-Thurston Circle Packing Theorem). However, in the book
Croft, Hallard T.(4-CAMBP); Falconer, Kenneth J.(4-BRST); Guy, Richard K.(3-CALG)
Unsolved problems in geometry.
Problem Books in Mathematics. Unsolved Problems in Intuitive Mathematics, II. Springer-Verlag, New York, 1991. xvi+198 pp. ISBN: 0-387-97506-3
the question of existence of mid-scribed polyhedron (which is obviously equivalent to the existence of circle packing) with the prescribed combinatorics is listed as an open problem.
Another example: In the early 2000s, I noticed that every element in ${\frak A}_n$ is actually a commutator, and Henry Cejtin and I proved this in
However, this result was already published by O. Ore a few years earlier:
Ore, Oystein
Some remarks on commutators.
Proc. Amer. Math. Soc. 2, (1951). 307–314.
But that's not all: in D. Husemoller's thesis, published as:
Husemoller, Dale H.
Ramified coverings of Riemann surfaces.
Duke Math. J. 29 1962 167–174.
Only a few years after Ore's paper, this result is reproved (by Andy Gleason) — this is actually the key result of the paper.
Another example (which actually inspired me to ask the question):
If you look at the comments to
(un)decidability in matrix groups
, you will find a result proved by S. Humphries in the 1980s reproved by other people in the 2000s (and I believe there are other proofs in between).
It would be interesting to have a list of such occurrences (hopefully made less frequent by the existence of MO).
Best Answer
According to Arnol'd himself (we are not worthy, we are not worthy...) Poincare had published a paper in the proceedings of the French Philosophical Society in the late nineteenth century sometime, on electromagnetism, but since the paper was intended for philosophers, there were no equations in it, except one, on the last page, which was $E = m c^2$ -- apparently this was too cool to resist. Apparently, Poincare was, in fact, the referee of Einstein's special relativity paper, around a decade later, and recommended it for acceptance, and when people asked him why, given that he had known the stuff for a while, he responded that this seems to be a bright young guy, and such should be encouraged. I am not sure where this is written down (I heard it from Arnold in person). Perhaps someone here knows the actual facts.