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

Q: Who proved the Cayley-Hamilton Theorem?

A: Frobenius!

We now have the interesting question: Is this a maximal example of Stigler's law? That is, can we find distinct persons A, B, and C who are given credit for a result proven by D? Or A and B who are given credit for a result proven by C and D?

Answer 2

According to Wikipedia, Markov's inequality is due to Chebyshev, and Chebyshev's inequality is due to Bienaymé.

On top of that, Hölder's inequality was first proved by Rogers, and Jensen's inequality by Hölder. What a mess!

Answer 3

The Frobenius automorphism associated to a prime ideal in a Galois extension of number fields was actually developed by Dedekind, who wrote about it (and the associated ramification groups, later found by Hilbert) in a letter to Frobenius on June 8, 1882. Frobenius published this construction in a paper in 1896. Some citations:

1. Frobenius, Collected Works, Vol. 2, pp. 719--733.

2. van der Waerden, Modern Algebra, Vol. 1 (Spring 1966), p. 203.

3. Zassenhaus, Canadian Math. Bulletin 18 (1975), p. 448.

Answer 4

Our linear algebra professor had a joke he told every year at the same spot in the lectures, for some 30 or 40 years. He'd say in an absolutely dry voice and facing the blackboard: "And this is the Cauchy–Bunyakovsky–Schwarz inequality, named like this because it was first proved by Lebesgue". Apparently, Cauchy just did it just as an inequality for sums (ie findim spaces), and Bunyakovsky and Schwarz independently as an inequality for integrals (ie for L2).

Answer 5

Stokes's theorem was stated by William Thomson (Lord Kelvin) in a letter to Stokes. The letter is reproduced on the cover of Spivak's Calculus on Manifolds. I believe the theorem was named after Stokes because he frequently put it on the Tripos exam in Cambridge.

Answer 6

The Pell equation was named so because Euler thought that John Pell was responsible for some key results involving this equation. While Pell was a notable mathematician, he had essentially no connection to the equation. The common belief is that Euler mistook Pell for Lord Brouncker who indeed had a number of results related to the "Pell" equation.

Answer 7

The Cantor-Schroeder-Bernstein theorem was proved by Dedekind; this link is to Dedekind's collected works and there is an informative note at the end.

Answer 8

The most amazing example I know is the Cayley formula which was discovered by Carl Borchardt nearly 30 years earlier. Not only Cayley knew about this, in his paper he specifically wrote that this formula is due to Borchardt, and all he wants to do is give a new simple proof (without determinants as in the matrix tree theorem).

Answer 9

Bézout's theorem

This result was discovered first by Newton in 1665. Even though MacLaurin (1720) and Leonhard Euler gave proofs, the theorem is usually attributted to Etienne Bézout who much later (1776) gave an incorrect proof of the result.

Answer 10

Chow varieties were invented by Van der Waerden (Chow was his student). And Hilbert schemes were invented by Grothendieck (who called them Hilbert schemes himself, however).

Answer 11

If you search for almost any eponymous topic in Wikipedia, you'll find that it was first studied by someone else. For example, the Gaussian distribution (according to Wikipedia) was first studied by de Moivre. It seems that in many cases, naming the body of work was given to the person who first applied its study to some other field (using the earlier example, Gauss used the distribution in astronomy).

The common story goes that L'Hôpital bought "the rights" to L'Hôpital's rule, as he was a nobleman and not a mathematician by trade, although I am not sure about the veracity of that story.

Although I am no expert on the history of Mathematics, it seems as though ideas or formulae assumed their names from certain mathematicians due either to a.) the more notable application or publication of the theory or b.) attribution by mathematicians of a later generation to pay tribute to (or garner attention from) the work of their predecessors.

Answer 12

I was once told that Riemann's integral is due to Darboux, while Lebesgue integral is due to Borel. Riemann invented the Cauchy integral instead.

Answer 13

In my first algebra book the Eisenstein criterion for irreducibilty of a polynomial is named Schönemann criterion and is left as an exercise. This is confusing when all others are talking about the Eisenstein criterion ;-). In fact, here is a quote from Wikipedia:

The criterion is named after Ferdinand Eisenstein. It was published by T. Schönemann in Crelle's Journal 32 (1846), p. 100, and was popularized by Eisenstein in Crelle's Journal 39 (1850), pp. 166-169. Eisenstein's application of this theorem was to polynomials with coefficients in Z[i], not Z.

Answer 14

The number of plane trees with no vertex of degree one and with $n$ endpoints is known as a Schröder number, from a 1870 paper by Ernst Schröder. In 1994 David Hough discovered that these numbers were known to Hipparchus (c. 190 - after 127 B.C.)! For a popular account, see http://math.mit.edu/~rstan/papers/hip.pdf. For a more scholarly treatment, see http://stl.recherche.univ-lille3.fr/sitespersonnels/acerbi/acerbipub5.pdf.

As an irrelevant aside, how do you make accent marks in MathOverflow? Schroder is supposed to have an umlaut over the o.

Answer 15

The Vandermonde Determinant/Matrix. Apparently Vandermonde never explicitly discussed his eponymous determinant. According to Lebesgue in his survey of Vandermonde work, it was probably due to somebody misinterpreting Vandermonde's notation.

Answer 16

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

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 17

An article in the current issue of American Mathematical MONTHLY (G. Folland, "A tale of topology," Am. Math. Monthly 117 (8) pp.663-672, Oct. 2010) quotes Walter Rudin as follows:

Thus it appears that Čech proved the Tychonoff theorem, whereas Tychonoff found the Čech compactification -- a good illustration of the historical reliability of mathematical nomenclature.

Folland's article suggests the truth is more complicated, as it usually is.

Answer 18

Burnside's Lemma, which asserts that the number of orbits of a group action is the average number of fixed points, was known to Cauchy. Burnside himself even attributed it to Frobenius in his book.

Answer 19

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

Answer 20

1. Cauchy–Riemann equations were known to d'Alembert and Euler.

2. Two-dimensional Voronoi diagrams were used by Descartes, three-dimensional - by Dirichlet. Also should be noted, that this construction has several other names in physics: Wigner–Seitz cells, Thiessen polygons.

Answer 21

Currying should, I believe, be referred to as Schönfinkeling.

Answer 22

In honour of the recently departed Benoit Mandelbrot, perhaps it is appropriate to offer up the example of the Mandelbrot set, the first pictures of which were drawn in 1978 by Robert Brooks and Peter Matelski (according to Wikipedia).

Answer 23

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

Answer 24

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

Answer 25

There was a paper published in 2006 entitled "Simpson's Paradox in the Farey Sequence". The paradox is not Simpson's nor is the sequence Farey's. Bonus points.

Answer 26

In logic:

• Tarski's undefinability theorem was obtained by Gödel before Tarski, who obtained it independently. Gödel did not publish the theorem. See Roman Murawskia (1998), "Undefinability of truth. The problem of priority: Tarski vs Gödel", History and Philosophy of Logic, v. 19 n. 3. pp. 153-160

• The result sometimes known as Gödel's diagonal lemma was first stated by Carnap. Gödel (1934) explicitly attributed the result to Carnap (see Kurt Gödel, Collected Works, v. 1, p. 363).

Answer 27

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 28

Expanding on the example given in the original post, here's an excerpt from Borel's "Essays in the History of Lie Groups and Algebraic Groups" (p. 5):

It has been remarked that, as far as terminology is concerned, posterity has not been kind to [Killing]: Cartan subalgebras, Weyl groups, fundamental reflections, roots, and the Coxeter transformation first appeared in his papers in some form. On the other hand, what now goes by his name, the "Killing form" seems to be a misnomer, and it may well be that I am the culprit. Cartan, Chevalley and Weyl never used this terminology. Once, J.J. Duistermaat and J.A.C. Kolk pointed out to me that, to their knowledge, its first occurence is in a paper of mine (Sém. Bourbaki, Exp. 33, May 1951). I do not remember why I chose it, though I probably felt I was innovating, since it is between quotation marks. It is rather likely that discussions with some members of Bourbaki had influenced me, but I cannot blame it directly on Bourbaki, since "Killing form" appears in Bourbaki drafts only from 1952 on. It is true that Killing was the first to remark that the coefficients of the characteristic equation (of a regular semisimple element), i.e. the elementary symmetric functions of the roots, are invariant under the adjoint group, but he did not make much use of the remark and did not single out the sum of the squares of the roots, of which Élie Cartan made such fundamental use in his thesis (1894). It would be more correct to speak of the Cartan form.

Answer 29

The Cayley numbers (also known as the Octonions) were discovered earlier by John T. Graves. The story is nicely explained in John C. Baez's paper, The Octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205.

Answer 30

Quoting from Alan Sokal's 2005 paper on the multivariate Tutte polynomial:

"The Potts model was invented in the early 1950s by Potts’ thesis advisor Domb. The $q = 2$ case, known as the Ising model, was invented in 1920 by Ising’s thesis advisor Lenz. (I hasten to add that these are the only two cases I know of where the thesis advisor’s invention was named after the graduate student, rather than the other way around.)"