107
$\begingroup$

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

  1. arXiv:math/0303036 [pdf, ps, other] A property of alternating groups Henry Cejtin, Igor Rivin Subjects: Group Theory (math.GR)

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).

$\endgroup$
6
  • 51
    $\begingroup$ Disturbing as it might often be, rediscovery is one of the surest signs of the importance of a concept.... $\endgroup$
    – Suvrit
    May 26, 2011 at 19:34
  • 48
    $\begingroup$ As someone who has read lots of original papers in the 19th and 20th century I'd like to remark that the list of answers would probably be shorter if you had asked for results that have been proved just once.. $\endgroup$ May 27, 2011 at 10:41
  • 17
    $\begingroup$ @Franz: "As someone who has read lots of original papers in the 19th and 20th century" -- so, how old are you? $\endgroup$ Sep 29, 2013 at 4:19
  • 10
    $\begingroup$ After E. de Giorgi: "chi cerca trova ; chi ricerca retrova." $\endgroup$ Oct 2, 2015 at 8:55
  • 19
    $\begingroup$ Once mathematicians start writing papers understandable by a non-expert in their particular subfield and without days-long divination, we might get to a point where people start reading prior work... $\endgroup$ Oct 2, 2015 at 11:56

32 Answers 32

1
2
2
$\begingroup$

A family $\cal F$ of subsets of a finite set is $r$-cover-free if no member of $\cal F$ is contained in the union of $r$ other members of $\cal F$. Let $T(n,r)$ denote the maximum cardinality of an $r$-cover-free family of subsets of an $n$-element set. This concept has arisen independently in several different contexts—information theory, combinatorics, and group testing—under various names (superimposed codes, $ZFD_r$ codes), and bounds on $T(n,r)$ have been rederived several different times.

I almost added to the confusion myself because I rediscovered these objects and was calling them $k$-Sperner sets. Fortunately, before my paper was published, I discovered that my results were already known. See the paper by Miklós Ruszinkó, "On the upper bound of the size of the $r$-cover-free families," J. Combin. Theory Ser. A 66 (1994), 302–310, for a list of the disparate previous papers on the subject, and a proof of the result that for sufficiently large $n$, $\log_2 T(n,r) \le 8n (\log_2 r)/r^2$.

$\endgroup$
-2
$\begingroup$

The discovery of the Mandelbrot set by Udo of Aachen circa 1250 should perhaps count for something, no ? (see http://en.wikipedia.org/wiki/Udo_of_Aachen).

$\endgroup$
3
  • 7
    $\begingroup$ Since it is a hoax, I don't know that it should count for something, though the Mandelbrot (aka the Brooks-Macielski) set is certainly a poster boy (literally) for the Arnold principle. $\endgroup$
    – Igor Rivin
    Nov 15, 2012 at 13:31
  • 1
    $\begingroup$ Closer to our times, Salvatore Pincherle had some ideas related to what is now known as the Mandelbrot set (in the first quarter of the XX century). More details can be found in the book Alexander, Daniel S.; Iavernaro, Felice; Rosa, Alessandro Early days in complex dynamics. A history of complex dynamics in one variable during 1906–1942. History of Mathematics, 38. American Mathematical Society, Providence, RI; London Mathematical Society, London, 2012. xviii+454 pp. ISBN: 978-0-8218-4464-9 $\endgroup$ Nov 15, 2012 at 15:34
  • 4
    $\begingroup$ Another completely fictional creator of fractals is a precocious teenage girl (living in Lord Byron's times) in Tom Stoppard's play "Arcadia". $\endgroup$ Nov 15, 2012 at 15:42
1
2

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.