The half-life of a theorem, or Arnold's principle at work

Question

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

2. 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).

Answer 1

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.

Answer 2

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.

Answer 3

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.)

Answer 4

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.

Answer 5

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

Answer 6

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.

Answer 7

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."

Answer 8

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)

Answer 9

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).

Answer 10

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. 99–123. (Gallica)

Davenport, H. On the addition of residue classes, Journal of the London Mathematical Society, Volume 10, 1935, pgs. 30–32. doi:10.1112/jlms/s1-10.37.30

Davenport, H., A historical note, Journal of the London Mathematical Society, 22, (1947) 100–101. doi:https://doi.org/10.1112/jlms/s1-22.2.100

(References taken from Paul Balister, and Jeffrey Paul Wheeler, The Cauchy-Davenport Theorem for Finite Groups, arXiv:1202.1816).

Note: The last paper is joint work of Balister and Wheeler according to scribd but the version on arXiv gives Wheeler as sole author.

Answer 11

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 $B$.

Answer 12

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.

Answer 13

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.

Answer 14

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.

Answer 15

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.

Answer 16

"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.

(Situation is similar to Hofstadter's law: It always takes longer than you expect, even when you take into account Hofstadter's Law.)

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”.

Answer 17

There is a paper by Kruskal entitled "The theory of well-quasi-ordering: A frequently discovered concept".

Answer 18

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...

and

Now the brilliant ideas of Brahmagupta, Bhaskara II and Narayana were completely unknown to the European mathematicians in the 17th Century.

Answer 19

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.

Answer 20

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.

Answer 21

The Poincaré-Birkhoff-Witt-theorem. Proven independently by Garrett 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.

Answer 22

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

Answer 23

I gave an example in Two different theorems but only one fact?, where I asked a similar question.

Answer 24

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.

Answer 25

Here is a very nice example. The abstract of the article "On planarity of compact, locally connected, metric spaces", by R. Bruce Richter, Brendan Rooney and Carsten Thomassen, starts as follows.

Independently, Claytor [Ann. Math. 35 (1934), 809–835] and Thomassen [Combinatorica 24 (2004), 699–718] proved that a 2-connected, compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain $K_5$ or $K_{3,3}$.

I thought that was quite a witty and honest way of describing this particular case of the phenomenon in question.

Answer 26

The parallelogram of forces is often attributed to Varignon. It was already discovered by Stevin roughly a century earlier.

Answer 27

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.

Answer 28

(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 $\sum_{x \in A} x = \sum_{x \in B} x$. 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:

1. #15.23 in "Problems and Theorems in Classical Set Theory" (solution on pp. 323-324) by Péter Komjáth and Vilmos Totik (2006).

2. AMM 11002 (2003).

3. Liong-shin Hahn (1992) who already proved it in Chinese as:

4. Liong-shin Hahn (1981).

Also, from the editorial comment in the solutions to AMM problem 11002:

This well-known problem has still other solutions. The discrete version (assuming that all weights are integers) was Problem B-1 on the 1973 Putnam Competition, and, as the proposer reported, recently appeared in the French newspaper Le Monde. The real-valued problem appears in Chapter 44 of Mathematical Miniatures by S. Savchev and T. Andreescu (MAA, 2002). Generalizations to complex numbers and Abelian groups appear in articles in Mathematics Magazine.

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.

Answer 29

An example which shows that even MathOverflow does not always prevent a theorem from being forgotten and rediscovered later is the following assertion which has been proved by G. A. Miller [1] in 1900 and proved again by Joachim König [2] in 2016:

Theorem: Let $n \in \mathbb{N}$. Then, for any $a, b, c \in \mathbb{N}$ satisfying $1 < a, b, c \leq n-2$, the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$ whose product has order $c$.

For details, see here.

References:

[1] Miller, G. A., On the product of two substitutions., American J. 22, 185-190 (1900). ZBL31.0134.03.

[2] König, Joachim, A note on the product of two permutations of prescribed orders, Eur. J. Comb. 57, 50-56 (2016). ZBL1339.05006.

Answer 30

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).