List of misnamed theorems
Encyclopedia
This is a list of misnamed theorems in mathematics
. It includes theorems (and lemmas, corollaries, conjectures, laws, and perhaps even the odd object) that are well known in mathematics, but which are not named for the originator. That is, these items on this list illustrate Stigler's law of eponymy
(which is not, of course, due to Stigler, who credits Merton).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. It includes theorems (and lemmas, corollaries, conjectures, laws, and perhaps even the odd object) that are well known in mathematics, but which are not named for the originator. That is, these items on this list illustrate Stigler's law of eponymy
Stigler's law of eponymy
Stigler's law of eponymy is a process proposed by University of Chicago statistics professor Stephen Stigler in his 1980 publication "Stigler’s law of eponymy". In its simplest and strongest form it says: "No scientific discovery is named after its original discoverer." Stigler named the...
(which is not, of course, due to Stigler, who credits Merton).
- Benford's lawBenford's lawBenford's law, also called the first-digit law, states that in lists of numbers from many real-life sources of data, the leading digit is distributed in a specific, non-uniform way...
. This was first stated in 1881 by Simon NewcombSimon NewcombSimon Newcomb was a Canadian-American astronomer and mathematician. Though he had little conventional schooling, he made important contributions to timekeeping as well as writing on economics and statistics and authoring a science fiction novel.-Early life:Simon Newcomb was born in the town of...
, and rediscovered in 1938 by Frank BenfordFrank BenfordFrank Albert Benford, Jr., was an American electrical engineer and physicist best known for rediscovering and generalizing Benford's Law, a statistical statement about the occurrence of digits in lists of data.Benford is also known for having devised, in 1937, an instrument for measuring the...
. The first rigorous formulation and proof seems to be due to Ted Hill in 1988.
- Bézout's theoremBézout's theoremBézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees...
. The statement may have been made first by Isaac NewtonIsaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
in 1665. The matter of a proof was taken up by Colin MacLaurinColin MaclaurinColin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series, are named after him....
(c. 1720) and Leonhard EulerLeonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
as well as Étienne BézoutÉtienne Bézout-External links:...
(c. 1750). However, Bézout's "proof" was incorrect. The first correct proof seems to be due mostly to Georges-Henri Halphen in the 1870s.
- Burnside's lemmaBurnside's lemmaBurnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. Its various eponyms include William Burnside, George...
. This was stated and proved without attribution in Burnside's 1897 textbook, but it had previously been discussed by Augustin CauchyAugustin Louis CauchyBaron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...
, in 1845, and by Georg FrobeniusFerdinand Georg FrobeniusFerdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
in 1887.
- Cayley–Hamilton theoremCayley–Hamilton theoremIn linear algebra, the Cayley–Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation....
. The theorem was first proved in the easy special case of 2×2 matrices by CayleyArthur CayleyArthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
, and later for the case of 4×4 matrices by HamiltonWilliam Rowan HamiltonSir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
. But it only proved in general by FrobeniusFerdinand Georg FrobeniusFerdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
in 1898.
- Cramer's paradoxCramer's paradoxIn mathematics, Cramer's paradox is the statement that the number of points of intersection of two higher-order curves can be greater than the number of arbitrary points needed to define one such curve.Cramer's paradox is the result of two theorems: Bézout's theorem In mathematics, Cramer's...
. This was first noted by Colin MaclaurinColin MaclaurinColin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series, are named after him....
in 1720, and then rediscovered by Leonhard EulerLeonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
in 1748 (whose paper was not published for another two years, as Euler wrote his papers faster than his printers could print them). It was also discussed by Gabriel CramerGabriel CramerGabriel Cramer was a Swiss mathematician, born in Geneva. He showed promise in mathematics from an early age. At 18 he received his doctorate and at 20 he was co-chair of mathematics.In 1728 he proposed a solution to the St...
in 1750, who independently suggested the essential idea needed for the resolution, although providing a rigorous proof remained an outstanding open problem for much of the 19th century. Even though Cramer had cited Maclaurin, the paradox became known after Cramer rather than Maclaurin. Halphen, Arthur CayleyArthur CayleyArthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
, and several other luminaries contributed to the earliest more or less correct proof. See for an excellent review.
- Cramer's ruleCramer's ruleIn linear algebra, Cramer's rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution...
. It is named after Gabriel CramerGabriel CramerGabriel Cramer was a Swiss mathematician, born in Geneva. He showed promise in mathematics from an early age. At 18 he received his doctorate and at 20 he was co-chair of mathematics.In 1728 he proposed a solution to the St...
(1704–1752), who published the rule in his 1750 Introduction à l'analyse des lignes courbes algébriques, although Colin MaclaurinColin MaclaurinColin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series, are named after him....
also published the method in his 1748 Treatise of Algebra (and probably knew of the method as early as 1729).
- Frobenius theoremFrobenius theorem (differential topology)In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations...
. This fundamental theorem was stated and proved in 1840 by Feodor DeahnaFeodor DeahnaHeinrich Wilhelm Feodor Deahna was a German mathematician, best known for providing proof of what is now known as Frobenius theorem in 1840.- Selected works :...
. Even though Frobenius cited Deahna's paper in his own 1875 paper, it became known after Frobenius, not Deahna. See for a historical review.
- Heine-Borel theorem. This theorem was proved in 1872 by Émile BorelÉmile BorelFélix Édouard Justin Émile Borel was a French mathematician and politician.Borel was born in Saint-Affrique, Aveyron. Along with René-Louis Baire and Henri Lebesgue, he was among the pioneers of measure theory and its application to probability theory. The concept of a Borel set is named in his...
, not by Eduard HeineEduard HeineHeinrich Eduard Heine was a German mathematician.Heine became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Legendre functions . He also investigated basic hypergeometric series...
. Borel used techniques similar to those that Heine used to prove that continuous functions on closed intervals are uniformly continuous. Heine's name was attached because SchönfliesArthur Moritz SchönfliesArthur Moritz Schoenflies , sometimes written as Schönflies, was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology....
noticed the similarity in Heine's and Borel's approaches. In fact, the theorem was first proved in 1852 by Peter Dirichlet, but Dirichlet's lecture notes were not published until 1904.
- L'Hôpital's ruleL'Hôpital's ruleIn calculus, l'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms. Application of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit...
. This rule first appeared in l'Hôpital's book L'Analyse des Infiniment Petits pour l'Intelligence des Lignes CourbesL'Analyse des Infiniment Petits pour l'Intelligence des Lignes CourbesAnalyse des Infiniment Petits pour l'Intelligence des Lignes Courbes , is the first textbook published on the infinitesimal calculus of Leibniz. It was written by the French mathematician Guillaume de l'Hôpital, and treated only the subject of differential calculus...
in 1696. The rule is believed to be the work of Johann BernoulliJohann BernoulliJohann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...
since l'Hôpital, a nobleman, paid Bernoulli a retainer of 300 francs per year to keep him updated on developments in calculus and to solve problems he had. See L'Analyse des Infiniment Petits pour l'Intelligence des Lignes CourbesL'Analyse des Infiniment Petits pour l'Intelligence des Lignes CourbesAnalyse des Infiniment Petits pour l'Intelligence des Lignes Courbes , is the first textbook published on the infinitesimal calculus of Leibniz. It was written by the French mathematician Guillaume de l'Hôpital, and treated only the subject of differential calculus...
and reference therein.
- Maclaurin series. The Maclaurin series was named after Colin MaclaurinColin MaclaurinColin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series, are named after him....
, a professor in Edinburgh, who published this special case of the Taylor seriesTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
in 1742, but never claimed to have discovered it.
- Morrie's law. The name is due to physicist Richard FeynmanRichard FeynmanRichard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
, who used to refer to the identity under that name. Feynman picked that name because he had learned the law during his childhood from a boy with the name Morrie Jacobs.
- Pell's equationPell's equationPell's equation is any Diophantine equation of the formx^2-ny^2=1\,where n is a nonsquare integer. The word Diophantine means that integer values of x and y are sought. Trivially, x = 1 and y = 0 always solve this equation...
. The solution of the equation x2 − dy2 = 1, where x and y are unknown positive integers and where d is a known positive integer which is not a perfect square, is ascribed to John PellJohn Pell-Early life:He was born at Southwick in Sussex. He was educated at Steyning Grammar School, and entered Trinity College, Cambridge, at the age of thirteen. During his university career he became an accomplished linguist, and even before he took his B.A. degree corresponded with Henry Briggs and...
. It seems to have been discovered by FermatPierre de FermatPierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...
, who set it as a challenge problem in 1657. The first European solution is found in a joint work in 1658 by John Wallis and Lord Brouncker; in 1668, a shorter solution was given in an edition of a third mathemathetician's work by Pell; see The first rigorous proof may be due to LagrangeJoseph Louis LagrangeJoseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
. The misnomer apparently came about when EulerLeonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
confused Brouncker and Pell; see for an extensive account of the history of this equation.
- Poincaré lemma. This was mentioned in 1886 by Henri PoincaréHenri PoincaréJules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
, but was first proved in a series of 1889 papers by the distinguished Italian mathematician Vito VolterraVito VolterraVito Volterra was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations....
. Nonetheless it has become known after Poincaré. See for the twisted history of this lemma.
- Pólya enumeration theoremPólya enumeration theoremThe Pólya enumeration theorem , also known as the Redfield–Pólya Theorem, is a theorem in combinatorics that both follows and ultimately generalizes Burnside's lemma on the number of orbits of a group action on a set. The theorem was first published by John Howard Redfield in 1927...
. This was proven in 1927 in a difficult paper by J. H. RedfieldJ. H. RedfieldJohn Howard Redfield was an American mathematician, best known for discovery of what is now called Pólya enumeration theorem in 1927, ten years ahead of similar but independent discovery made by George Pólya...
. Despite the prominence of the venue (the American Journal of MathematicsAmerican Journal of MathematicsThe American Journal of Mathematics is a bimonthly mathematics journal published by the Johns Hopkins University Press.- History :The American Journal of Mathematics is the oldest continuously-published mathematical journal in the United States, established in 1878 at the Johns Hopkins University...
), the paper was overlooked. Eventually, the theorem was independently rediscovered in 1936 by George PólyaGeorge PólyaGeorge Pólya was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory...
. Not until 1960 did Frank HararyFrank HararyFrank Harary was a prolific American mathematician, who specialized in graph theory. He was widely recognized as one of the "fathers" of modern graph theory....
unearth the much earlier paper by Redfield. See for historical and other information. - Stokes' theoremStokes' theoremIn differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
. It is named after Sir George Gabriel StokesGeorge Gabriel StokesSir George Gabriel Stokes, 1st Baronet FRS , was an Irish mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...
(1819–1903), although the first known statement of the theorem is by William ThomsonWilliam Thomson, 1st Baron KelvinWilliam Thomson, 1st Baron Kelvin OM, GCVO, PC, PRS, PRSE, was a mathematical physicist and engineer. At the University of Glasgow he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging...
(Lord Kelvin) and appears in a letter of his to Stokes. The theorem acquired its name from Stokes' habit of including it in the Cambridge prize examinations. In 1854 he asked his students to prove the theorem in an examination; it is not known if anyone was able to do so. - Zorn's lemmaZorn's lemmaZorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...
is named for Max Zorn. Much work on the theorem now known as Zorn's lemma, and on several closely related formulations such as the Hausdorff maximal principleHausdorff maximal principleIn mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914...
, was done between 1907 and 1940 by Zorn, Brouwer, HausdorffFelix HausdorffFelix Hausdorff was a Jewish German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.-Life:Hausdorff studied at the University of Leipzig,...
, KuratowskiKazimierz KuratowskiKazimierz Kuratowski was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.-Biography and studies:...
, R. L. Moore, and others. But the particular theorem now known as "Zorn's lemma" was never proved by Zorn, and in any event Zorn's results were anticipated by Kuratowski. The theorem was discovered by ChevalleyClaude ChevalleyClaude Chevalley was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, and the theory of algebraic groups...
in 1936, and published and attributed to Zorn by him in Bourbaki's Théorie des Ensembles in 1939. A very similar result was anticipated by S. BochnerSalomon BochnerSalomon Bochner was an American mathematician of Austrian-Hungarian origin, known for wide-ranging work in mathematical analysis, probability theory and differential geometry.- Life :...
in 1928.
See also
- Stigler's law of eponymyStigler's law of eponymyStigler's law of eponymy is a process proposed by University of Chicago statistics professor Stephen Stigler in his 1980 publication "Stigler’s law of eponymy". In its simplest and strongest form it says: "No scientific discovery is named after its original discoverer." Stigler named the...
- Matthew effectMatthew effect (sociology)In sociology, the Matthew effect is the phenomenon where "the rich get richer and the poor get poorer". Those who possess power and economic or social capital can leverage those resources to gain more power or capital. The term was first coined by sociologist Robert K...
- List of theorems