The half-life of a theorem, or Arnold's principle at work
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
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).
ho.history-overview big-list
Igor Rivin
$\begingroup$ +1 for the question, but I think this should be a community wiki. $\endgroup$ – algori May 26 '11 at 17:20
$\begingroup$ I agree this should be CW. $\endgroup$ – Todd Trimble♦ May 26 '11 at 17:36
$\begingroup$ No -- some history questions will have one definitive (and verifiably correct) answer. Whereas here you are explicitly asking for a list, so by definition no one answer will be definitive. $\endgroup$ – Todd Trimble♦ May 26 '11 at 17:56
$\begingroup$ Disturbing as it might often be, rediscovery is one of the surest signs of the importance of a concept.... $\endgroup$ – Suvrit May 26 '11 at 19:34
$\begingroup$ 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.. $\endgroup$ – Franz Lemmermeyer May 27 '11 at 10:41
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.
$\begingroup$ I cannot resist the temptation to cite Etienne Ghys concerning Poincar\'e: "Il faut faire tr`es attention avant de dire que quelque chose n'est pas dans Poincar\'e" (One has to be very careful before saying that something is not in Poincar\'e). $\endgroup$ – Roland Bacher May 27 '11 at 6:48
$\begingroup$ I saw a long talk by A. Borel on Einstein, Poincare, relativity, and the like (at Ann Arbor, in 1999 or 1998). He gave the impression that Poincare really disliked Einstein and his ideas, and offered as evidence a letter of recommendation written by Poincare saying, in effect, this guy is a competent physicist (analogous to: he was in my class, and got a B). $\endgroup$ – Jeff Strom May 27 '11 at 14:15
$\begingroup$ Arnold was not a great fan of Einstein either -- another thing he recounted to me (and I am sure it is in some book of his) was that Einstein's formulation of General Relativity is essentially due to Hilbert. Maybe Borel mentioned this too... $\endgroup$ – Igor Rivin May 27 '11 at 14:51
$\begingroup$ I once heard a talk by a patent lawyer from Silicon Valley. Many times in his career, there were little flurries of activity when people would come to him with the same idea, all within a month or so. He was sure that in most cases, there was no hint of theft or plagiarism - only the connexion that these people were steeped in the same culture and drawing on the same body of ideas, which had become so discussed that everyone had forgotten their importance. Sometimes we forget that we belong to a bigger organism of fellow researchers, hence the phenomenon that "some ideas just have their time". $\endgroup$ – WetSavannaAnimal Jun 10 '11 at 0:21
$\begingroup$ Well, the Wikipedia page on "Relativity priority dispute" mentions Poincaré's paper "La théorie de Lorentz et le principe de réaction", Archives néerlandaises des sciences exactes et naturelles 5: 252–278, in which he said that radiation could be considered as a fictitious fluid with an equivalent mass of $m_r = E/c^2$. English translation here $\endgroup$ – Nik Weaver May 3 '14 at 17:33
This is maybe an extreme example. I don't remember if it was a joke or not, but I recall receiving an e-mail announcement about someone recently inventing the trapezoidal method for approximating Riemann integrals.
Here's the Wikipedia page about the paper / controversy: http://en.wikipedia.org/wiki/Tai%27s_method
Ryan Budney
$\begingroup$ "Other formulas widely applied by researchers under- or overestimated total area under a metabolic curve by a great margin". I find this sentence a bit unsetteling.. $\endgroup$ – J.C. Ottem May 26 '11 at 17:58
$\begingroup$ The real question is whether any of the referees took calculus in college... $\endgroup$ – Qiaochu Yuan May 26 '11 at 18:01
$\begingroup$ I remember once reading a review article in a mechanics journal is which the completeness of eigenfunctions for a self-adjoint fourth order ODE problem was referred to as "Collatz's theorem." Collatz wrote an ODE textbook which is widely used to teach engineers. $\endgroup$ – Michael Renardy May 26 '11 at 18:17
$\begingroup$ I read the paper at one point. The way she explains it, you should pick some nice-looking collection of points on the graph of the function, and interpolate between them with straight lines. Then the area below the straight lines is a union of rectangles and triangles whose area you can compute. She gives data indicating that this gives a more accurate value than simply approximating the area with a union of rectangles, although it is not clear how she found the "true" value of the area. She does make a point of the fact that her method will work no matter what points on the graph you choose. $\endgroup$ – Dan Petersen May 26 '11 at 18:41
$\begingroup$ What boggles me most is: what did people studying metabolic curves do before Tai rediscovered the Trapezium rule? $\endgroup$ – known google May 26 '11 at 23:12
Cantor proved that for any two countable dense subsets of the real line there is a homemorphism from the reals to the reals mapping one countable set to the other. Can the homeomorphism be analytic? This question has been answered periodically (about every twenty years) since Cantor's result was published. However, looking at Cantor's original paper reveals that the very next article in the journal is by a student who extends Cantor's theorem to analytic functions.
Juris Steprans
I am unsure it fits the OP's requirements but, in connection with Ryan Budney's most-upvoted answer regarding the rediscovery of trapezoidal method, let me recall that Grothendieck spent about three years working in isolation in French provinces developing the Lebesgue theory of integration. It was not before he went to Paris that he was told someone had already done that.
Jonathan Chiche
$\begingroup$ Perhaps I should have added that this story is told at the beginning of "Récoltes et Semailles". $\endgroup$ – Jonathan Chiche May 27 '11 at 7:17
If $p$ is a prime, then every minor of the Fourier matrix $(e^{2\pi i jk/p})_{1 \leq j,k \leq p}$ is non-singular.
This fact was proven by Chebotarev in 1927 (answering a question of Ostrowski), Danilevskii in 1937, Resetnyak in 1955, Dieudonne(^) in 1970, Newman in 1975, Evans and Isaacs(^) in 1977, Goldstein-Guralnick-Isaacs in 2005, and by myself(^) in 2005 also, with the authors marked (^) initially unaware of Chebotarev's original work (see this survey of Stevenhagen and Lenstra for some related discussion). A short proof was also given by Frenkel, who solved it in a 1998 mathematics competition as a question posed by Andras Biro. There may well be other proofs in the literature also. (But according to Wikipedia at least, the theorem continues to be attributed primarily (and correctly) to Chebotarev, though it is safe to say that it is not his most well known theorem.)
I ran across the following (to me startling) example in Robert Cromie 1895 techno-thriller The Crack of Doom (reprinted in The End of the World: Classic Tales of Apocalyptic Science Fiction, Michael Kelehan, ed.)
Page 102: "If you consult a common text-book on the physics of the aether, you will find that one grain of matter, contains sufficient energy, if etherised, to raise a hundred thousand tons nearly two miles."
Here "grain" is a standard unit of jewelers (one gram = 15.4 grains). Then it is easy to verify, that within ±2% error, Cromie's "etherised" mass-energy relation is $E = m c^2/2$.
Einstein was 16 years old when Cromie's book appeared (published by a European publishing house) ... a very impressionable age, needless to say. Yet despite the clue that Cromie so generously provided to science fiction fans in Europe, ten years passed before Einstein got the factor of two right.
John Sidles
$\begingroup$ Interesting. We're getting a bit off-topic, but is there any evidence that Einstein would have read the book? Was it translated into German - would there have been whiffs of it in an 1895 German "Boy's own"? $\endgroup$ – WetSavannaAnimal Jun 10 '11 at 0:28
$\begingroup$ I can't say I'm surprised. The "physics" of the aether was motivated by electromagnetism, which is inherently a relativistic theory in which the number $c$ naturally appears. The concept of mass-energy (or -aether) equivalence seems plausible given the qualities of both that aether was supposed to possess. Finally, $E = mc^2$ is dimensionally correct (of course) and so getting something that looks like it from the above considerations seems to be not that hard. If so, this is not the only "classical" factor of 2 that Einstein fixed with relativity. $\endgroup$ – Ryan Reich May 4 '14 at 13:33
In his monograph (D. V. Anosov: Geodesic flows on closed Riemann manifolds with negative curvature, Proceedings of the Steklov Institute 90 (1967) AMS 1969), Anosov writes: "Every five years or so, if not more often, someone 'discovers' the theorem of Hadamard and Perron proving it either by Hadamard's method or Perron's. I myself have been guilty of this."
Michael Renardy
$\begingroup$ Related: mathoverflow.net/a/152835/22971 $\endgroup$ – Benjamin Dickman May 4 '14 at 8:03
The Cooley--Tukey FFT algorithm (1965) was already known to Gauss (ca. 1805).
Perhaps a slight stretch, but then the temporal gap is quite impressive.
Details e.g. http://en.wikipedia.org/wiki/Fast_Fourier_transform (under Algorithms)
$\begingroup$ I heard Gauss also solved the Poincare Conjecture before anyone knew about it. $\endgroup$ – Samuel Reid Apr 1 '12 at 1:33
$\begingroup$ @Samuel Reid: is the date you posted this significant? If not, I never heard of this. That he had a solution by todays standars, seems however essentially impossible. $\endgroup$ – user9072 Apr 4 '12 at 12:53
$\begingroup$ Frank Yates working in Design of Experiments also had (some special case of) FFT around 1939-40. $\endgroup$ – kjetil b halvorsen Nov 15 '12 at 2:35
$\begingroup$ The temporal gap is not so big as it looks like. There were a lot of rediscoverings, see "Gauss and the history of the fast Fourier transform" ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1162257 $\endgroup$ – Alexey Ustinov Dec 21 '13 at 13:50
I think the number of examples is roughly half the number of published theorems, so this could be a very long list, indeed. But that won't stop me from making a contribution or two.
The Cauchy-Davenport Theorem. Harold Davenport published it in 1935. Then in 1947 he wrote a paper in which he noted that Cauchy had published the same result in 1813.
Cauchy, A. Recherches sur les nombres, J. Ecole Polytech, Volume 9, 1813, pgs. 99-116.
Davenport, H. On the addition of residue classes, Journal of the London Mathematical Society, Volume 10, 1935, pgs. 30-32.
Davenport, H., A historical note, Journal of the London Mathematical Society, 22, (1947) 100-101.
(References taken from Paul Balister and Jeffrey Paul Wheeler, The Cauchy-Davenport Theorem for Finite Groups, which I found on the web).
Gerry Myerson
$\begingroup$ This is an example I like a lot. Note however that the pages of the Cauchy paper 123 and not 116 is correct. (I did not want to simply change it as then I would also have to change the reference for the references and this seemed a bit invasive.) Since I know that the 116 is somewhat widespread, I guess I should back up my 123. Here gallica.bnf.fr/ark:/12148/cb34378280v/date are the digitized volumes of JEP, one has to scroll down a bit to arrive at 1813. $\endgroup$ – user9072 May 27 '11 at 17:26
This may not strictly count because the time between the independently derived results may not have been long, but the significance of the result makes this example interesting and the two workers concerned certainly were unaware of each other's result.
The Russell Paradox was known to Cantor independently of Russell's announcement of the result and highly likely some years before Russell hit on it. See Jean van Heijenoort, "From Frege to Goedel: a source book in mathematical logic, 1879-1931" (1967) on page 114, where a letter from Cantor to Dedekind is reproduced. Cantor writes to Dedekind in 1899, two years before Russell announced his paradox:
"...If we start from the notion of a definite multiplicity (a system, a totality) of things, it is necessary, as I have discovered, to distinguish two kinds of multiplicities (by this I mean definite multiplicities).
For a multiplicity can be such that the assumption that all of its elements 'are together' leads to a contradition, so that it is impossible to conceive of the multiplicity as a unity, as 'one finished thing'. Such multiplicities I call absolutely infinite or inconsistent multiplicities.
As we can readily see, the 'totality of everything thinkable', for example, is such a multiplicity ..."
Cantor was aware that if you applied the Cantor slash argument to the set of all sets, a contradiction would follow. In this letter he shows he was aware of the contradiction inherent in the conception of this 'multiplicity as a unity, as one finished thing', and the Russell paradox in the words that Russell used was likewise found by Russell when he was seeking a flaw in Cantor's slash argument and applied it to set of all sets or, the 'totality of everything thinkable'. See the wikipedia article for the history of Russell's take on things.
WetSavannaAnimal
$\begingroup$ Very interesting. A trivial observation: in the letter you quote, Cantor states only the result that there can be no set of all sets, he doesn't explicit the proof. Yet I agree that he surely knew it. More to the point: this discovery should have been devastating for Cantor. After all, most main stream mathematicians considered all his set theory has pure non-sense, and by his discovery that his usual meted of reasoning could lead to plain contradictions, he would have given a solid point to his adversaries. Maybe that's the reason he didn't publish it. $\endgroup$ – Joël Sep 14 '11 at 12:28
$\begingroup$ @Joel: "this discovery should have been devastating for Cantor". But MAYBE, just maybe it wasn't. The world was only barely ready for Cantor's thinking at the time, and he drew the ire of people like Brouwer and Poincar'e: he was likely a bit careful about what he published. The language Cantor uses here begets my SPECULATION that he had taken the contradiction pretty much in his stride: he realised there were two kinds of "multiplicities" and that it only made sense to talk about one kind as a finished whole, or a set as we now call it. $\endgroup$ – WetSavannaAnimal Sep 15 '11 at 0:23
$\begingroup$ @Joel: Or, to put it another way, he seems to know there is a pitfal, and sidesteps it neatly by definition: he seems almost to be defining a set as something which doesn't suffer the paradox we now call the Russel paradox. It would be really interesting to hear the comments of a real mathematical historian on this one. $\endgroup$ – WetSavannaAnimal Sep 15 '11 at 0:27
Often, this happens when a mathematician does not recognize the problem (s)he faces as a problem already studied in another branch. This happened to me as follows. While studying the Riemann problem for the Euler equations of a compressible gas obeying a Chaplygin equation of state, I encountered the following equation $${\rm div}\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}+\frac{2}{u\sqrt{1+|\nabla u|^2}}=0.$$ I had to solve the Dirichlet boundary-value problem ($u=0$ on $\partial\Omega$) in strictly convex domains $\Omega\subset{\mathbb R}^2$. This I did in Multi-dimensional shock interaction for a Chaplygin gas. Arch. Rational Mech. Anal., 191 (2009), pp 539--577.
Two years later, Lihe Wang pointed out to me that such solutions describe complete minimal surfaces in the 3D-hyperbolic space. The result is therefore due originally to M. Anderson (Inventiones Math. 1982). The boundary regularity, which I left open, was actually proved by Fang Hua Lin (Inventiones Math. 1989).
Denis Serre
$\begingroup$ I guess this is particularly tricky for equations, since they tend to have different name in different fields, and the actual formula is a little hard to look up... $\endgroup$ – Igor Rivin Nov 14 '12 at 13:10
In 1983, at the request of a referee, I added to a paper of mine a proof that (under certain hypotheses which need not detain us here) a certain norm could be given as a certain resultant. Since then I have found that the same theorem had already been published by (at least) half-a-dozen authors going back to Cebotarev in 1936, none of them citing any of their predecessors.
Oh what the heck. It's a nice result, and not all that hard to state. Let $A$ be a commutative ring with unity. Let $f$ and $g$ be in $A[x]$, with $f$ monic. Let $B=A[x]/(f)$. Then the resultant of $f$ and $g$ equals the norm from $B$ to $A$ of the class of $g$ in $B$.
$\begingroup$ I just noticed this (again), and this result [I am guessing slightly less generally) can be found in Susan Landau's paper on factoring in algebraic number fields (about the same time as your paper). $\endgroup$ – Igor Rivin Oct 2 '15 at 10:55
Benford's Law ... so called not because Benford was the first to publish it, but merely because Benford was the first to publish it in a physics journal.
Gerald Edgar
$\begingroup$ Interesting, but who WAS the first to publish it? $\endgroup$ – Igor Rivin May 26 '11 at 22:07
$\begingroup$ Physics? Philosophy! (Proc. Am. Philosophical Soc. 1938) But Benford was a Physicist. Simon Newcomb published the law in American J. Math. in 1881, and was an Astronomer. $\endgroup$ – user9072 May 27 '11 at 0:15
$\begingroup$ Interesting. Proc. Am. Philos. Soc., though, was not a philosophy journal, but a more general journal. The paper following Benford's is on the scattering of electrons by Bethe et.al. I have heard that the reason Benford's paper was noticed by physicists was that it was adjacent to this physics paper. (and I wrongly assumed it was a physics journal where this happened) $\endgroup$ – Gerald Edgar May 27 '11 at 14:43
$\begingroup$ I admittedly just went by the name of the journal, now I see it is a journal with a general scope ('various disciplines in the humanities and sciences'). Thank you for the clarification. $\endgroup$ – user9072 May 27 '11 at 17:42
"On teaching mathematics" by V.I. Arnold:
Prof. M. Berry once formulated the following two principles:
The Arnold Principle. If a notion bears a personal name, then this name is not the name of the discoverer.
The Berry Principle. The Arnold Principle is applicable to itself.
It is also interesting, how Arnold's Principle can be applied to Arnold's works.
1) Arnold's Problem (problem 1993-11 from "Arnold's Problems" Springer, 2005) on statistical properties of finite continued fraction was essentially solved by Lochs in 1961 (32 years before Arnold's conjecture, see "Statistik der Teilnenner der zu den echten Brüchen gehörigen regelmässigen Kettenbrüche", Monatsh. Math., 1961, 65, 27-52).
2) His question about weak asymptotic for Frobenius numbers (problem 1999-8 from "Arnold's Problems" Springer, 2005) was asked earlier by Davison (only for three arguments, but in fact the question is the same, see "On the linear Diophantine problem of Frobenius", J. Number Theory, 1994, 48, 353-363)
3) In the article "Geometry of continued fractions associated with Frobenius numbers" (Funct. Anal. Other Math., 2009, 2, 129-138) he almost rediscovered Rodseth's formula for Frobenius numbers.
4) His "Napkin conjecture" (see Is the "Napkin conjecture" open? (origami)) was solved (see the answer by Andrey Rekalo) in 1797, in the Japanese origami book "Senbazuru Orikata".
Alexey Ustinov
$\begingroup$ I didn't know of Lochs's work. Thanks for the reference. $\endgroup$ – Andrés E. Caicedo Nov 14 '12 at 7:12
$\begingroup$ Interestingly, in statistics this principle is known as "Stigler's law of epinomy" : en.wikipedia.org/wiki/Stigler's_law_of_eponymy $\endgroup$ – kjetil b halvorsen Nov 15 '12 at 2:43
$\begingroup$ mathoverflow.net/questions/24132/… $\endgroup$ – Andrés E. Caicedo Nov 15 '12 at 3:59
$\begingroup$ See also ncatlab.org/nlab/show/Baez's+law $\endgroup$ – David Roberts Oct 2 '15 at 8:11
There is a paper by Kruskal entitled "The theory of well-quasi-ordering: A frequently discovered concept".
Michał Kukieła
Most of the guts of the Kalman filter - the recursive algorithm for estimating the probability density function for the hidden states of an unknown Markov process - was known to Gauss and used by the latter to simplify hand calculations needed to find optimal estimates of planetary orbits from astronomical observations.
The point of the Kalman filter is that, as each new measurement comes in, the parameters of the (assumed) Gaussian PDF are updated by simple recurrence relationships that make use of former estimates of these parameters and the new observation alone. You don't have to go back and get all the old observational data and do a maximum likelihood estimate from the newly augmented full dataset from scratch.
Kalman was unaware of Gauss's work, and indeed his method of proof was quite different and more general, so it may not altogether count as a rederiving of an old result, but still the key ideas are rediscovered by an author unaware of his intellectual forerunners.
See this exposition.
A related issue is that of simultaneity. The best example I recall of that is the independent discovery by Friedberg and by Muchnik of solutions to Post's problem ( two sets of integers neither of which were computable from the other, a.k.a a pair of incomparable and not very complicated Turing degrees below 0', the Turing degree of the halting problem). Perhaps someone can confirm/refute the idea that they were both under 20 years of age at the time of discovery.
Two personal examples are a note on the size of a minimal counterexample to Frankl's union closed conjecture which I showed to my advisor, and then found it over a year later published by Giovanni LoFaro. He never told me directly, but I suspect my advisor thought that Ron Graham had proved and not published a similar lower bound (given some restrictions, a minimal example on a universe of n elements must have at least 4n - 1 sets in the family). The following year I came up with a basis of five equations for an equational theory that was shown to be finitely based with a basis of at most X equations (I forget the value of X, but had something like 5 or 6 decimal digits). Libor Polak found the same basis, and in his paper "On Hyperassociativity" kindly acknowledged my independent discovery, which happened within a month of his. There were other examples among my fellow graduate students, at least one of which resulted in a change of dissertation topic.
It is experiences like this that support the notion "If Gauss didn't know it, either Euler did or it wasn't worth knowing."
Gerhard "I Keep At It Anyway" Paseman, 2011.05.26
Gerhard Paseman
$\begingroup$ Muchnik was born in 1934 and solved the problem in 1956, so he was 21 or 22 at the time. Friedberg solved the problem when he was 20, according to this article: time.com/time/magazine/article/0,9171,824058,00.html $\endgroup$ – Timothy Chow May 26 '11 at 22:27
There is the Hilbert-Burch Theorem, which gives structure of Cohen-Macaulay ideals having projective dimension one in a regular or polynomial ring. The named authors published their results about 80 years apart. Eisenbud wrote (page 506 of his "Commutative Algebra..." book) that "many people have discovered it for themselves (and many have published it) in the intervening years".
This theorem is quite useful in many contexts in algebra and geometry. In fact, you can find a nice historical account at the end of Chapter 8 of Hartshorne's "Deformation Theory", which applies it to study deformations of Cohen-Macaulay subschemes of codimension two.
Hailong Dao
In holomorphic dynamics, there are examples of rational maps on the Riemann sphere having only repelling cycles (i.e., whose Fatou set is empty) , and a class of such examples is associated with elliptic functions. This class is called "Latt\'es examples", because in 1918 Samuel Latt`es constructed an $f$ satisfying $\mathcal{P}(2z)=f(\mathcal{P}(z))$, where $\mathcal{P}$ is the Weierstrass elliptic function coming from a certain lattice in $\mathbb{C}^2$. It was thought to be the first such example. However, an example based on Jacobi elliptic function appeared in 1898 in the PhD thesis of Lucjan Emil B\"ottcher, Beitr\"age zu der Theorie der Iterationsrechnung, published by Oswald Schmidt, Leipzig, pp.78, and another one was given in his paper in Polish, Zasady rachunku iteracyjnego (cz\c e\'s\'c pierwsza i cz\c e\'s\'c druga) [Principles of iterational calculus (part one and two)], {\it Prace Matematyczno - Fizyczne}, vol. X (1899 - 1900), pp. 65 - 86, 86-101.
Margaret Friedland
$\begingroup$ Some fundamental notions in holomorphic dynamics were discovered independently by different people (cf. Feldmann Denis's answer). One could also mention Julia sets, pioneered by B\"ottcher (the Polish paper cited above) and Pincherle. $\endgroup$ – Margaret Friedland Nov 15 '12 at 15:37
I gave an example in Two different theorems but only one fact?, where I asked a similar question.
Johann Cigler
Two quotes from the MacTutor History page on Pell's equation:
...the first contribution by Brahmagupta was made around 1000 years before Pell's time...
Now the brilliant ideas of Brahmagupta, Bhaskara II and Narayana were completely unknown to the European mathematicians in the 17th Century.
Sam Nead
The Poincaré-Birkhoff-Witt-theorem. Proven independently by Garret Birkhoff and Ernst Witt in 1937, and attributed either to one or both or none of them in the following years (and even to Harish-Chandra, who gave another proof). Then, in the 1950s, some authors seem to have pre-inventend the statement of Etienne Ghys as cited by Roland Bacher on this page (thereby giving another example, although on a meta-level), and began to attribute it to Poincaré, too. -- There seem to be different views on whether Poincaré in his article, which dates from 1900, gave a "complete" proof. Many more details can be found in: T. Ton-That, T.-D. Tran: Poincaré's proof of the so-called Birkhoff-Witt theorem, Rev. Histoire Math., 5 (1999), pp. 249–284. Edit: There is already an MO page about Poincaré's supposed proof.
Another example -- the classification of codomains of epimorphisms from $\mathbb{Z}$ -- was the content of my answer here.
Torsten Schoeneberg
$\begingroup$ The Ton-Tran paper is very interesting! $\endgroup$ – Igor Rivin Nov 14 '12 at 16:52
$\begingroup$ All the more interesting indeed since the first proof, as indicated on the MO page mentioned above, is not due to Poincare, nor Birkhoff, nor Witt but to Alfredo Capelli. $\endgroup$ – Abdelmalek Abdesselam Jan 6 '17 at 20:06
$\begingroup$ I am not sure if Ton-That's and Tran's version of Poincaré's proof is correct. See mathoverflow.net/questions/74923/… . $\endgroup$ – darij grinberg May 23 '17 at 18:30
The parallelogram of forces is often attributed to Varignon. It was already discovered by Stevin roughly a century earlier.
I heard it once from a specialist in dynamical systems that the Hurwitz criterion (which helps determine whether all eigenvalues of a matrix have negative real part, see e.g. http://en.wikipedia.org/wiki/Routh%E2%80%93Hurwitz_stability_criterion) used to get rediscovered on a regular basis by engineers who needed it to check whether a zero of a vector field in a Euclidean space is stable (one has to linearize the field near the fixed point and apply Lyapunov's stability theorem, see e.g. http://en.wikipedia.org/wiki/Lyapunov_stability)
I should also say that this person was referring to the time before computers became widespread, so chances are, nowadays no one bothers.
algori
I had a similar experience. In my paper "Operatormethoden für q-Identitäten" (Mh. Math. 88 (1979), 87-105, I observed that for operators $A, B$ satisfying $BA=qAB$ the $q$-binomial theorem holds in the form $$(A+B)^n = \sum_{k=0}^n {n\choose k}_q A^k B^{n-k}.$$ Later I learned that M.-P. Schützenberger had proved this already in C. R. Acad. Sci. Paris 236 (1953), 352-353. It seems that this fact has been rediscovered several times. Someone (I have forgotten who it was) claimed that already Euler had this result.
$\begingroup$ This was essentially known to H. S. A. Potter [Amer. Math. Monthly 57, (1950). 321–322] and H. Wielandt [Math. Nachr. 4, (1951). 308–314]. See the historical paper in Amer. Math. Monthly 111 (2004), no. 8, 655–667. $\endgroup$ – Denis Serre Oct 2 '15 at 9:17
Irreducible representations of $GL_n$.
Their discovery is usually attributed to Schur in his 1901 dissertation. However, they can be found in the 1891 "Essai d'une théorie générale des formes algébriques" by Deruyts. In fact, they are even older than that since they can also be found in the 1872 acticle "Ueber eine Fundamentalaufgabe der Invariantentheorie" by Clebsch in the super well known journal Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Gottingen. A reasonably modern account of Clebsch's contribution is in the article "Invariant Theory, Tensors and Group Characters" by Littlewood.
Abdelmalek Abdesselam
I have rediscovered several theorems:
Using a (very) recent theorem, I proved some properties of the root measures one obtain from $$P_n(z)-z=0$$ where $P_{n+1} = P_n(z)^2 + c,P_0(z)=z,$ (Thus almost all roots of $$P_n$$ lie in the julia set). For example, they give an invariant measure, and the root measures obtained from the derivatives of $$P_n$$ all converge to the same measure (this last part follows from a theorem by Hans Rullgård, in his phd thesis).
However, the result about this invariant measures are proved in the 50:s by Hans Brolin, (under Lenart Carlessons supervision).
My first article was a generalization on some recent results (papers) by my advisor, on the Schroedinger equation, but it turns out that the full generalization, proved by a method similar to ours, was done in the 30:s (long before people knew/cared about quantum mechanics).
Per Alexandersson
$\begingroup$ Isn't there a paper by Jenkinson and Pollicott on this? $\endgroup$ – Igor Rivin May 27 '11 at 12:33
$\begingroup$ @Igor : Maybe, what paper in particular do you refer to? $\endgroup$ – Per Alexandersson May 27 '11 at 17:33
$\begingroup$ I think this one, but I am not 100% certain (don't have enough time to look at the paper right now): MR1773814 (2001i:37015) Pollicott, Mark(4-MANC); Jenkinson, Oliver(F-PROV-IM) Computing invariant densities and metric entropy. (English summary) Comm. Math. Phys. 211 (2000), no. 3, 687–703. 37A35 (37E05 37M25) In any case, you might find it interesting... $\endgroup$ – Igor Rivin May 27 '11 at 20:05
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 $n$, $\log_2 T(n,r) \le 8n (\log_2 r)/r^2$.
(Not quite a theorem, but an interesting problem.) In an earlier MO question I wondered:
Given $x_1 \leq \ldots \leq x_{2n + 1}$ with each $x_i \in \mathbb{R}$ (or $\mathbb{C}$), suppose that for any $x_i$ we remove that the remaining numbers can be assigned to disjoint multisets $A$ and $B$ such that $|A| = |B| (= n)$ and $\sum_{x \in A} A = \sum_{x \in B} B.$ Is it true that all the $x_i$ must be equal?
The answer is yes.
Moreover, in subsequent updates, it became clear that the question had already been solved as:
#15.23 in "Problems and Theorems in Classical Set Theory" (solution on pp. 323-324) by Péter Komjáth and Vilmos Totik (2006).
AMM 11002 (2003).
Liong-shin Hahn (1992) who already proved it in Chinese as:
Liong-shin Hahn (1981).
In addition, the question over $\mathbb{Z}$ came up as Putnam B1 (1973) and #3.4.31 in "The Art and Craft of Problem Solving" (2006, 2e, p. 107) by Paul Zeitz.
Benjamin Dickman
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).
Feldmann Denis
$\begingroup$ 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 '12 at 13:31
$\begingroup$ 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 $\endgroup$ – Margaret Friedland Nov 15 '12 at 15:34
$\begingroup$ Another completely fictional creator of fractals is a precocious teenage girl (living in Lord Byron's times) in Tom Stoppard's play "Arcadia". $\endgroup$ – Margaret Friedland Nov 15 '12 at 15:42
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