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It's a common observation in Lie theory that Cartan matrices and the Killing form are named after the wrong people; they were discovered by Killing and Cartan, respectively. I remember learning about many other examples of this phenomenon, but can't think of too many at the moment. Wikipedia has some examples here and here, but I'm curious about more obscure examples.

Bonus points for an interesting story behind why the concept was incorrectly named. Concepts that were deliberately named in honor of another mathematician don't count.

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closed as no longer relevant by Kevin H. Lin, quid, Qiaochu Yuan, Zev Chonoles, Gjergji Zaimi Jul 9 '11 at 3:12


By the way, in the mathematical community "Stiegler's law" is often referred to as "Arnol'd's law", inclusive of the corollary "Arnol'd's law applies to Arnol'd's law as well". –  pasquale zito May 10 '10 at 20:24
To further complicate things, there is also Whitehead's law: "Everything of importance has been said before by someone who did not discover it." –  bhwang May 10 '10 at 21:38
Oh gosh, I could not imagine that there are SO many wrong names. Perhaps some day there will be a big important Brandenburg theorem, of course just because another one has proven it. ;-) –  Martin Brandenburg May 10 '10 at 23:09
Not that I have a problem with the question per se, but "the wrong people" is pretty ambiguous. The first person to study something might not be the most deserving -- often a crucial application or popularizations trumps the actual innovation. Nor is it necessarily the case that the intent of the naming was to honor the inventor -- frequently the naming is done for reasons of analogy ("Euler systems" come to mind). –  Cam McLeman May 11 '10 at 1:10
Stigler's law is called Boyer's Law by H.C. Kennedy in "Who Discovered Boyer's Law?" (Amer. Math. Monthly vol. 79 1972, 66--67). It says that "Mathematical formulas and theorems are usually not named after their original discoverers." The label Boyer's law was chosen because Boyer gave many examples of this phenomenon in his book A History of Mathematics. –  KConrad Sep 8 '10 at 17:12

56 Answers 56

Banach algebras should probably be called Gelfand Algebras, or something similar. I'm not sure of the history here, but presumably the "Banach" is attached because this is the study of "complete" normed algebras. I don't believe that Banach actually did much work on algebras (as opposed to Banach spaces).

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Nakayama's lemma was first proved by Krull in a special case, and by Goro Azumaya in the general case.

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I think that's why many people prefer to call it NAK lemma. –  VA. May 12 '10 at 2:51
It's not so clear. In the historical notes to his book, Nagata says that in the case N=0 and M an ideal, the lemma was given by Krull, and that the general case is in a paper of Azumaya. But Nagata learned it from Nakayama and Azumaya when he was an undergraduate. Since some mathematicians were calling it Nakayama's lemma, he asked Nakayama who had this formulation first, to which Nakayama responded that he did not remember whether Nakayama or Azumaya was the first. Probably it should be considered folklore, and the name "Nakayama's lemma" seems appropriate. –  JS Milne May 12 '10 at 11:34
"History is a myth men agree to believe."-Napoleon Bonaparte –  Andrew L May 13 '10 at 14:32

Pell's equation








Raul Nunes ( raul_nunes@uol.com.br ) NEST Nunes' Exposition of Scientific Truths ( http://www.geocities.com/raulnunes )

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Jones' Conjecture. Jones does not even exist; it's a Western pseudonym of Chuan-Min Lee.

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Perhaps counterexamples to Stigler's/Arnol'ds law are actually the rare items. The most significant one that I know is the Cartesian coordinate system which, strangely, seems to have actually been invented by Descartes!

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Farey series, attributed to Farey, were actually first studied by Haros.

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Pythagoras' Theorem apparently predates Pythagoras.

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And similarly much of Euclid predates Euclid. –  Tom Smith May 10 '10 at 20:51
But so far as we know, Pythagoras (or one of his followers) was the first to prove it, or even to attempt to do so. –  Charles Staats May 11 '10 at 2:39

Euler's nine point circle was never discussed by Euler. This is an error of the "argument by authority" type: Catalan propagated that incorrect attribution made by another scholar, the "learned Terquem", without checking it himself.

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The "Lichnerowicz formula" relating the square of the Dirac operator to the Laplacian has been proved more than 30 years earlier by Schrödinger.

See: E. Schrödinger, Dirac'sches Elektron im Schwerefeld, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1932, 105-128 (1932).

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Farey fractions were introduced by C. Haros. John Farey asked a question about them that reached Cauchy, and Caucy then attributed the question and result to Farey, and the rest is history.

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It seems to me that Lagrange's theorem may well be one of the most prominent examples of the phenomenon under discussion.

According to J. J. Rotman,

the theorem was inspired by work of Lagrange (1770), but it was probably first proved by Galois.

Curiously enough, the Wikipedia article adscribes the first complete proof of the theorem to Pietro Abbati Marescotti.

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The algebraic numbers that are now commonly called "Gauss sums" were studied in more general form than that introduced in Gauss's Disquisitiones by Lagrange [1]. In that same work, Lagrange shows how to generate an abelian extension of degree n by adjoining an nth root after, if necessary, adjoining the nth roots of unity. These generators were later called "Kummer generators". Jacobi sums, which are closely related to Gauss sums, were studied earlier than Jacobi by Gauss and Cauchy.

Finally, a story best recounted by Weil [2]: "For reference, we recall that the Gauss sums appear among the local constant factors in the functional equations of the $L$ functions; these factors are also called "nombres radiciels" ("root-numbers", "Wurzelzahlen"), undoubtedly because of Hilbert, who a had a sort of genius for bad terminology, where it would have been sensible to name "Wurzelzahl" that which before him had been named a "Lagrange resolvent" , and "Lagrangian Wurzelzahl" that which here has been called a Gauss sum".

[1] Lagrange, Reflexions sur la resolution algebrique des equations, Nouveaux Mem. de l'Acad. R. des Sc. et B.-L. de Berlin, 1770-1771 = Oeuvres, vol. III, p. 332;

[2] Weil, La Cyclotomie Jadis et Naguere.

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If the Russians are to believed, Kuratowski's theorem in graph theory was proved earlier by Pontryagin, but he hadn't published his notes.

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In reference to exactly this phenomenon (and in particular to the case of Pell's equation), Andre Weil once observed that "This has happened many times in mathematics. For example, I live on von Neumann Circle. I live there. Yet still it is called von Neumann Circle".

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The Shimura-Taniyama conjecture was originally known as the Weil conjecture see http://www.ams.org/notices/199511/forum.pdf, also see the comment of Weil on page 7 (with other examples) in his response to Lang on the same issue as in the question posed here.

Additionally, the Frey curve was actually first considered by Yves Hellegouarch.

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Morse theory is named after Marston Morse; it was widely used at least 50 years earlier. Wikipedia mentions Cayley and Maxwell, in the context of topography. Maxwell also used it in his work on electromagnetism, as detailed (complete with extensive passages from Maxwell's treatise) in the appendix of Mystery of point charges (A. Gabrielov, D. Novikov and B. Shapiro) available here (subscription)

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Some people have tried to give examples with as high a multiplicity as possible. I want to try to break the record for the smallest non-zero example: Martin's axiom was introduced by Martin and Solovay. (I judge that to have multiplicity 1/2.)

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Sorry, but this is a non-example. Martin's axiom was introduced by Martin. It is true that this happened in a paper by Martin and Solovay, and that it was after a construction of Solovay and Tennenbaum suggested it, but there is no misnomer here. –  Andres Caicedo Oct 13 '10 at 20:58
So much for trying to find examples on Wikipedia. But perhaps that actually improves my example, making its multiplicity even smaller than 1/2. –  gowers Oct 13 '10 at 21:36

I had a companion observation that almost noone attributes the well-known sum-of-roots, product-of-roots etc. polynomial formulas as Vieta's formulas as I posted on Yahoo!Answers.

Because as user absird pointed out, it makes that sort of topic Google-proof; at least a bad name is better than no name for purposes of searching or discussion.

('Yes it's very hard to refer to something when noone knows it by its proper name or uses that name. I tried many Google searches on "sum-of-roots product-of-roots" and it was almost impossible to get a coherent lead.')

MathWorld notes: The theorem was proved by Viète (also known as Vieta, 1579) for positive roots only, and the general theorem was proved by Girard.

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Really? People in high school competition circles, at least in the US, are generally pretty good about calling them Vieta's formulas. –  Qiaochu Yuan Jan 18 '11 at 20:10
It may be geographically dependent. At high school level, no. I was educated in Ireland and never saw these referred to as Vieta's formulas. Even in college they would be blankly presented as 'properties of polynomials'. If you google (and manage to find them as 'sum-of-roots') that is typically how you will find them presented. At IMO level, perhaps. Even then I don't recall ever seeing the name (not like Cauchy-Schwarz, Hölder, Pappus, Chinese Remainder Theorem et al). Personally I only found out the name in 2008, i.e. 20 years later. And it's not like I didn't read a lot. –  smci Jan 18 '11 at 20:38
I was educated in the US, and was unaware of the name "Vieta's formulas" until about two minutes ago. –  Charles Rezk Jan 18 '11 at 20:48

A favorite of mine is l'Hôpital's rule. l'Hôpital paid Johann Bernoulli a retainer to keep him updated on developments in calculus and to solve problems he had. Correspondence shows that Bernoulli stated and proved the rule, which l'Hôpital then published.

Heine-Borel was first published by Borel, not Heine. In fact, Heine's name was attached because he was using similar methods to solve related problems. Too bad for both of them that it was actually Dirichlet who was the first recorded to have proved it, but his notes were published posthumously and after Borel's proof.

Cramer's Rule was published first by MacLaurin, and some believe MacLaurin knew the proof some 20 years before Cramer's publication.

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I don't know if this is a real example, but it led to a nice gem in a recent abstract on the arXiv: "Glaisher's correspondence goes back to Euler."

(As far as I know Glaisher generalized Euler's bijection, which is why he gets the eponym -- in addition some people say "Euler-Glaisher" instead.)

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Huh? I don't follow. Euler did not have "bijective proof" as a concept. It was Glaisher who invented the bijection, but if I recall correctly never published it - Sylvester did it and attributed it to him some time later. You are probably referring to Euler's "odd vs. distinct" partitions theorem. Using modern "involution principle" technology one can convert analytic proofs into bijective, and in this case the one-line Euler's proof becomes Glaisher's bijection (this is O'Hara's theorem - see my survey on partitions and a recent paper on O'Hara algorithm, joint with Konvalinka). –  Igor Pak May 10 '10 at 21:50

Cartan discovered the Killing form, and Killing discovered the Cartan matrix.

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That's the example I used in the original post :) –  Qiaochu Yuan May 13 '10 at 1:44
Aw, nuts, that's what I get for not reading things carefully. :) –  David Hansen May 13 '10 at 2:08

I think the Kazhdan-Lusztig Conjectures are due to Vogan.


True or false, the claim is mainly based on the very first two paragraphs of

[II] Irreducible characters of semisimple Lie groups II. The Kazhdan-Lusztig conjectures. David A. Vogan, Jr. Duke Math. J. Volume 46, Number 4 (1979), 805-859. --- The link


gives a universal access to the first page, which contains the two paragraphs in question. In case you don't have access to the full paper, here is a scan of the references (to completely understand the two paragraphs):


Here are two more references:

[I] Irreducible characters of semisimple Lie groups I, David A. Vogan, Jr., Duke Math. J. Volume 46, Number 1 (1979), 61-108.


[KL] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Inventiones Mathematicae, Volume 53, Number 2, 165-184.


I would summarize things as follows.

Step 1. In [I] Vogan made a certain conjecture.

Step 2. [II] and [KL] were written simultaneously. Each paper cites the other. In [KL] Kazhdan and Lusztig also made a certain conjecture. When he learned this, Vogan immediately (or at least very fast) proved that the "Step 1 conjecture" implies that of Kazhdan and Lusztig. (He even showed that the "Step 1 conjecture" generalizes that of Kazhdan and Lusztig.)

But, again, the best is to read carefully the first two paragraphs of [II]. Vogan explains this much more clearly than I, and it's always better to hear things from the horse's mouth.

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The answer is incorrect, since the KL 1979 paper (Invent. Math.) was the first to state the KL Conjecture based on Hecke algebras and KL polynomials. Kazhdan and Lusztig were motivated by questions about singularities of Schubert varieties, Springer's representations of Weyl groups, Jantzen's work on Verma modules, etc. Vogan was approaching similar machinery from the direction of real Lie group representations, which led him after the KL paper to complete parts of his own program using "KLV polynomials". See Steven Kleiman's history of intersection homology (arXiv) for the role of GM. –  Jim Humphreys Aug 7 '10 at 16:00

Liouville talked about the Legendre function when he studied the so-called Euler Gamma function. It made me doubt about who defined the Gamma function first.

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According to this link Steiner Systems were mentioned by by W Woolhouse in 1844 before the famous Kirkman Schoolgirl problem (P Kirkman 1847) - Steiner's work was more systematic and did advance the theory, but it came in 1853.

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The notion of Frobenius manifold is due to Dubrovin

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