What are examples of mathematical concepts named after the wrong people? (Stigler's law)

Question

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.

Answer 1

And the Bianchi identities are due to Ricci (according to Levi-Civita).

Answer 2

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

Answer 3

Pell's equation

By a mistake of Euler, the Diophantine equation y^2 - Ax^2 = 1 has been erroneously known as "Pell's equation"; but, in fact, the English mathematician John Pell (1611-1685) did no more than copy it down in his papers, from Fermat's letters of 1657 and 1658.

For an extensive historical account on "Pell's equation", see Sir Thomas L. Heath, Diophantus of Alexandria : A Study in the History of Greek Algebra (Dover Pub., New York, 1931-1963, 552 pages), Supplement, Section II: "Equation y^2 - Ax^2 = 1, pp. 277-292. Particularly in page 285, after a presentation of the history of the equation up to Fermat's time (including citations to Pythagoreans, Archimedes, Diaphanous, and the Indian solution), one can read that:

" ... Fermat rediscovered the problem and was the first to assert that the equation x^2 - Ay^2 = 1, where A is any integer not a square, always has an unlimited number of solutions in integers. His statement was made in a letter to Frénicle of February, 1657 (cf. Oeuvres de Fermat, II, pp.333-4). Fermat asks Frénicle for a general rule for finding, when any number not a square is given, squares which, when they are respectively multiplied by the given number and unity is added to the product, give squares. If, says Fermat, Frénicle cannot give a general rule, will he give the smallest value of y which will satisfy the equations 61y^2 + 1 = x^2 and 109y^2 + 1 = x^2 ? (Footnote 3: Fermat evidently chose these cases for their difficulty; the smallest values satisfying the first equation are y=226153980, x=1766319049, and the smallest values satisfying the second are y=15140424455100, x=158070671986249)." And, after a extensive quotation of Fermat's letter, in page 286, one can read that: "The challenge was taken up in England by William, Viscount Brouncker, first President of the Royal Society, and Wallis (Footnote 1: An excellent summary of the whole story is given in Wertheim's paper "Pierre Fermat's Streit mit John Wallis" in Abhandlungen zur Gesch. der Math., IX. Heft (Cantor-Festschrit), 1899, pp.557-576). See also H. Konen, Geschichte der Gleichung t^2-Du^2=1, Leipzig (S. Hirzel), 1901). At first, owing apparently to some misunderstanding, they thought that only rational, and not necessarily integral solutions were wanted, and found of course no difficulty in solving this easy problem. Fermat was, naturally, not satisfied with this solution, and Brouncker, attacking the problem again, finally succeeded in solving it. The method is set out in letters of Wallis (Footnote 2: Oeuvres de Fermat, III, pp.457-480, 490-503) of 17th December, 1657, and 30th January, 1658, and in chapter XCVIII of Wallis' Algebra; Euler also explains it fully in his Algebra (Footnote 3: Part II, chap. VII), wrongly attributing it to Pell (Footnote 4: This was the origin of the erroneous description of our equation as the "Pellian" equation. Hankel (in Zur Geschichte der Math. im Alterthum und Mittlelalter, p.203) supposed that the equation was so called because the solution was reproduced by Pell in an English translation (1668) by Thomas Brancker of Rahn's Algebra; but this is a misapprehension, as the so-called "Pellian" equation is not so much as mentioned in Pell's additions (Wertheim in Bibliotheca Mathematica, III, 1902, pp.124-6); Konen, pp.33-4 note). The attribution of the solution to Pell as a pure mistake of Euler's, probably due to a cursory reading by him of the second volume of Wallis' Opera where the solution of the equation ax^2 + 1 = y^2 is given as well as information as to Pell's work in indeterminate analysis. But Pell is not mentioned in connexion with the equation at all (Eneström in Bibliotheca Mathematica, III, 1902, p.206)."

For more information about "Pell's equation", see Harold M. Edwards, The Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Springer-Verlag, New York, 1977, 410 pages), pp. 25-33. Particularly in page 33 one can read that

"This problem of Fermat is now known as "Pell's equation" as a result of a mistake on the part of Euler. In some way, perhaps from a confused recollection of Wallis's Algebra, Euler gained the mistaken impression that Wallis attributed the method of solving the problem not to Brouncker but to Pell, a contemporary of Wallis who is frequently mentioned in Wallis's works but who appears to have had nothing to do with the solution of Fermat's problem. Euler mentions this mistaken impression as early as 1730, when he was only 23 years old, and it is included in his definitive Introduction to Algebra written around 1770. Euler was the most widely read mathematical writer of his time, and the method from that time on has been associated with the name of Pell and the problem that it solved --- that of finding all integer solutions of y^2 - Ax^2 = 1 when A is a given number not a square --- has been known ever since as "Pell's equation", despite the fact that it was Fermat who first indicated the importance of the problem and despite the fact that Pell had nothing whatever to do with it."

See also André Weil, Number Theory : An approach through history - From Hammurapi to Legendre (Birkhäuser, Boston, 1984, xv+375 pages), in many different pages. In particular, at page 174, one can read that:

"Pell's name occurs frequently in Wallis's Algebra, but never in connection with the equation x^2 - Ny^2 = 1 to which his name, because of Euler's mistaken attribution, has remained attached; since its traditional designation as "Pell's equation" is unambiguous and convenient, we will go on using it, even though it is historically wrong."

Raul Nunes ( [email protected] ) NEST Nunes' Exposition of Scientific Truths ( http://www.geocities.com/raulnunes, Wayback Machine )

Answer 4

To expand on Pasquale's comment, here's a quote from Arnold's article On teaching mathematics:

Similarly to the fact that America does not carry Columbus's name, mathematical results are almost never called by the names of their discoverers.

In order to avoid being misquoted, I have to note that my own achievements were for some unknown reason never expropriated in this way, although it always happened to both my teachers (Kolmogorov, Petrovskii, Pontryagin, Rokhlin) and my pupils. 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.

Perhaps somebody knows which results by Kolmogorov et. al. he is thinking of.

Answer 5

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).

Answer 6

Many of the examples mentioned go back to earlier centuries, when insulated national traditions and slow communications promoted mistaken labelling of results and concepts. A much more recent example from the 1950s involves the notion of Bruhat ordering on a general Coxeter group, motivated at first by the example of finite crystallographic reflection groups in Lie theory. The name seems to have been suggested by D.N. Verma in the late 1960s. For some reason the ordering itself fails to appear (even in the exercises) in Bourbaki's influential 1968 Chapters IV-VI dealing with Coxeter groups, root systems, Weyl groups and affine Weyl groups. Deodhar and others propagated the term "Bruhat ordering" in their papers, and as late as 1990 I routinely used this term in my book Reflection Groups and Coxeter Groups. But by then Borel, who had gotten more deeply involved in sorting out the history of Lie theory, objected that the ordering was not at all found in Bruhat's development of the Bruhat decomposition but had occurred for Weyl groups in Chevalley's treatment of the partial ordering of closures of Bruhat cells (Schubert varieties) in the flag variety.

As a result many of us now try in principle to start with something like Chevalley-Bruhat ordering (shortened to Bruhat ordering) or even Chevalley ordering. But this runs counter to a large body of literature including the 1979 Kazhdan-Lusztig paper.

Side remark: While Coxeter was the first to recognize the special presentation of a finite real reflection group that led to the term Coxeter group in Bourbaki, the general definition owes at least as much to people like Iwahori and Tits. Coxeter was interested in traditional (often intricate) combinatorial geometry and not in Lie theory or its generalizations. But short labels are easier to invent and tend to stick.

Answer 7

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!

Answer 8

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.

Answer 9

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

Answer 10

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.

Answer 11

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.

Answer 12

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

Answer 13

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.

Answer 14

Tannakian categories, a terminology chosen by N. Saavedra in his thesis under Grothendieck.

Grothendieck would prefer (Grothendieck–) Galois–Poincaré $\otimes$-categories.

See Récoltes et Semailles.

Answer 15

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.

Answer 16

Apparently the Robin boundary condition was never studied by Robin. (Gustafson, K. and T. Abe, (1998b). The third boundary condition – was it Robin's?, The Mathematical Intelligencer, 20, 63–71.)

Answer 17

Pythagoras' Theorem apparently predates Pythagoras.

Answer 18

Maybe I've missed it, but it seems no one has mentioned Pascal's triangle. According to Wikipedia it had a slew of earlier discoverers, going back to "The Persian mathematician Al-Karaji (953–1029) [who] wrote a now-lost book which contained the first formulation of the binomial coefficients and the first description of Pascal's triangle."

Wikipedia goes on to say, "The triangle was later named for Pascal by Pierre Raymond de Montmort (1708) who called it 'Table de M. Pascal pour les combinaisons' (French: Table of Mr. Pascal for combinations) and Abraham de Moivre (1730) who called it 'Triangulum Arithmeticum PASCALIANUM' (Latin: Pascal's Arithmetic Triangle), which became the basis of the modern Western name."

Answer 19

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.)

Answer 20

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.)

Answer 21

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.

Answer 22

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.

Answer 23

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)

Answer 24

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 (Wayback Machine).

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.

Answer 25

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

EDIT.

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

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[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

Link, doi: 10.1215/S0012-7094-79-04642-8

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:

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

Link, doi: 10.1215/S0012-7094-79-04605-2

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

GDZ, eudml

---

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.

Answer 26

Well Wigners rotation in special relativity which is the observation that two non-parallel boosts will result not in another boost but a composite of a boost and rotation was actually first formulated by Ludvik Silberstein in 1914, by Llewelyn Thomas in 1926 and then by Wigner in 1939, that is over two decades later.

Since Wigner acknowledged Silberstein's priority, I propose it be renamed Silberstein rotation ...

Answer 27

The Gauss-Bonnet theorem can be proved using the Gauss map, but the theorem itself appears to be due to Blaschke and the original proof procedure is due to Rodrigues.

It seems that neither the theorem nor the proof of Rodrigues were known to Gauss or to Bonnet.

Answer 28

Bell numbers, named after Eric Temple Bell who was not the first to study them.

\begin{align} B_0 & = 1 \\ B_1 & = 1 \\ B_2 & = 2 \\ B_3 & = 5 \\ B_4 & = 15 \\ B_5 & = 52 \\ B_6 & = 203 \\ & \,\,\,\vdots \end{align}

The Bell number $B_n$ is the number of partitions of a set of $n$ members.

It is also the $n$th moment of the Poisson distribution with expected value $1.$

In 1964 Gian-Carlo Rota wrote about these and called them exponential numbers.¹ At some time after that they came to be conventionally called Bell numbers.

In that paper Rota never mentions the connection with the Poisson distribution, although he did in later writings. He defines a linear functional $L$ on the space of polynomials in $u$ by saying $L((u)_n) = 1$ for all $n,$ where $(u)_n$ is the $n$th-degree falling factorial $u(u-1)(u-2)\cdots(u-n+1)$ and states a lemma: $$ L(u p(u-1)) = L(p(u)) $$ for every polynomial $p(u).$ Those who have studied empirical Bayes methods in statistics will recognize that as a form of the Robbins lemma: \begin{align} & \text{If } X\sim\operatorname{Poisson}(\lambda) \\[6pt] & \text{then } \operatorname E(Xf(X-1)) = \lambda\operatorname E(f(X)). \end{align} (The function $f$ need not be a polynomial.) This lemma was discovered by Herbert Robbins in the 1950s as a result of analyzing data on car insurance.

(1) Rota, Gian-Carlo (1964). "The number of partitions of a set". American Mathematical Monthly. 71 (5): 498–504.

doi:10.2307/2312585.

JSTOR 2312585.

MR 0161805.

Answer 29

Wilson's Theorem

According to Wikipedia:

This theorem was stated by Ibn al-Haytham (c. 1000 AD) and in the 18th century, by John Wilson. Edward Waring announced the theorem in 1770, although neither he nor his student Wilson could prove it. Lagrange gave the first proof in 1771. There is evidence that Leibniz was also aware of the result a century earlier, but he never published it.

Answer 30

Introduction of Gröbner basis and its algorithmic computation is due to Bruno Buchberger, but it is named after his advisor Wolfgang Gröbner.