# What are some famous rejections of correct mathematics?

Dick Lipton has a blog post that motivated this question. He recalled the Stark-Heegner

Theorem: There are only a finite number of imaginary quadratic fields that have unique factorization. They are $$\sqrt{d}$$ for $$d \in \{-1,-2,-3,-7,-11,-19,-43,-67,-163 \}$$.

From Wikipedia (link in the theorem statement above):

It was essentially proven by Kurt Heegner in 1952, but Heegner's proof had some minor gaps and the theorem was not accepted until Harold Stark gave a complete proof in 1967, which Stark showed was actually equivalent to Heegner's. Heegner "died before anyone really understood what he had done".

I am also reminded of Grassmann's inability to get his work recognized.

What are some other examples of important correct work being rejected by the community?

NB. There was a complementary question here before.

• There is, of course, the Cantor-Kronecker debacle, but I think this is too well-known to merit an answer. Feb 3, 2010 at 1:26
• @Qiaochu--I think the Hilbert existence proof ( en.wikipedia.org/wiki/David_Hilbert#The_finiteness_theorem ) is in the same class. Feb 3, 2010 at 1:43
• I don't think the proofs of the four-color theorem and Kepler conjecture were really rejected, but they merit a footnote. Feb 3, 2010 at 2:01
• See also Emch, A. "Rejected Papers of Three Famous Mathematicians". National Mathematics Magazine 11, 186 (1937), in which papers of Schläfli, Riemann, and De Jonquières are discussed. Available at jstor.org/pss/3028220 Feb 3, 2010 at 15:37
• mathoverflow.net/questions/27284/mathematical-controversies/… Jun 7, 2010 at 13:34

Tarski ran into some trouble when he tried to publish his result that the Axiom of Choice is equivalent to the statement that an infinite set $X$ has the same cardinality as $X \times X$.

From Mycielski:
"He tried to publish his theorem in the Comptes Rendus but Frechet and Lebesgue refused to present it. Frechet wrote that an implication between two well known propositions is not a new result. Lebesgue wrote that an implication between two false propositions is of no interest. And Tarski said that after this misadventure he never tried to publish in the Comptes Rendus."

• On p. 215 of his book on the AC, Moore tells the same story, but with Lebesgue and Hadamard. He cites a personal communication from Tarski as source. Jun 16, 2010 at 6:01

From Wikipedia: Higher homotopy groups were first defined by Eduard Čech in 1932 (Čech 1932, p. 203). (His first paper was withdrawn on the advice of Pavel Sergeyevich Alexandrov and Heinz Hopf, on the grounds that the groups were commutative so could not be the right generalizations of the fundamental group.)

• Perhaps Alexandrov and Hopf were right. The higher homotopy groups are not the right generalisation of the fundamental group. The latter classifies covering spaces, but the higher homotopy groups have no corresponding property. Feb 3, 2010 at 15:23
• There are the corresponding Whitehead towers where you "kill" the lowest-dimensional non-trivial homotopy group of a space by a fibration whose fibre is an Eilenberg-Maclane space. In the 1-dimensional case this is a covering space. An example of killing $\pi_2$ of the base would be the Hopf fibration $S^3 \to S^2$. It's maybe not as complete an analogy as you'd like? Feb 4, 2010 at 15:26

The Mordell-Weil theorem, when submitted by Mordell to the London mathematical society's journal, was rejected.

This theorem was the start of the whole set of investigations on elliptic curves, and indeed on arithmetic geometry. Andre Weil in his Ph. D. thesis created the subject of arithmetic of algebraic varieties and Galois cohomology, to prove his strengthened version of this theorem and to understand Mordell's calculations. I also believe that for him the motivation to re-write the foundations of algebraic geometry was also motivated by the desire to give the Mordell-Weil theorem a cleaner form, thoughs the officially stated motivation is for putting his proof of Riemann hypothesis for function fields over finite fields on a firm ground. And, the subject grew, flowered, through greats like Grothendieck, and one must remark the work of Faltings on Mordell conjecture on the same direction proposed in the same paper, which could be proved only so many years later, after Weil failed in his Ph. D. time. Indeed, Fermat's last theorem proof also belongs to the same subject. Looking back, rejection of Mordell's groundbreaking paper is so unbelievable.

Excerpt from source:

Mordell submitted his subsequent work on indeterminate equations of the third and fourth degree when he became a candidate for a Fellowship at St John's College, but he was not successful. His paper on this topic was rejected for publication by the London Mathematical Society but accepted by the Quarterly Journal. Mordell was bitterly disappointed at the way his paper had been received. He wrote at the time on an offprint of the paper:-

This paper was originally sent for publication to the L.M.S. in 1913. It was rejected ... Indeterminate equations have never been very popular in England (except perhaps in the 17th and 18th centuries); though they have been the subject of many papers by most of the greatest mathematicians in the world: and hosts of lesser ones ...

Such results as [those in the paper] ... marks the greatest advance in the theory of indeterminate equations of the 3rd and 4th degrees since the time of Fermat; and it is all the more remarkable that it can be proved by quite elementary methods. ... We trust that the author may be pardoned for speaking thus of his results. But the history of this paper has shown him that in his estimation, it has not been properly appreciated by English mathematicians.

The details of Weil's work can be found in his autobiography, "Apprenticeship of a mathematician"..

• Good story, but you've mixed two papers: the one that was rejected by LMS in 1913 is Mordell, L. J. Indeterminate equations of the third and fourth degrees. Quart. J., 170-186 (1914). The "Mordell-Weil theorem" paper is JFM 48.0140.03 Mordell, L. J. On the rational solutions of the indeterminate equations of the third and fourth degrees. Cambr. Phil. Soc. Proc. 21, 179-192 (1922). Jun 7, 2010 at 4:40

Wigner's (motivated by quantum mechanics) classification of the irreducible unitary representations of the Poincaré group (the group of automorphisms of R4 with the Lorentz metric) was not accepted by the "American Journal of mathematics" on the ground of not being mathematically interesting. Later it was published by "Annals of mathematics" on von Neumann's suggestion. In this work Wigner introduced the method of induced representations and “normal subgroup analysis”. Both became very important building blocks in the representation theory of Lie groups.

• I wish I could remember where I first heard another version, where rejection was by Physical Review (different slant!). Articles by Irving Segal (1996, p. 459) and Wigner himself (1979; pdf) say the paper was rejected by “a Springer physics journal”, resp. “one of our mathematical journals”. Oct 17, 2017 at 0:22
• @Francois Ziegler, Thank you very much for the corrections. I have heard this story a long time ago from Prof. Louis Michel. I don't remember which version it was. Nov 8, 2017 at 20:02

Galois theory, maybe?

Quoting Wikipedia:

Galois returned to mathematics after his expulsion from Normale, although he was constantly distracted in this by his political activities. After his expulsion from Normale was official in January 1831, he attempted to start a private class in advanced algebra which did manage to attract a fair bit of interest, but this waned as it seemed that his political activism had priority. Simeon Poisson asked him to submit his work on the theory of equations, which he submitted on January 17. Around July 4, Poisson declared Galois' work "incomprehensible", declaring that "[Galois'] argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor"; however, the rejection report ends on an encouraging note: "We would then suggest that the author should publish the whole of his work in order to form a definitive opinion." While Poisson's rejection report was made before Galois' Bastille Day arrest, it took some time for it to reach Galois, which it finally did in October that year, while he was imprisoned. It is unsurprising, in the light of his character and situation at the time, that Galois reacted violently to the rejection letter, and he decided to forget about having the Academy publish his work, and instead publish his papers privately through his friend Auguste Chevalier. Apparently, however, Galois did not ignore Poisson's advice and began collecting all his mathematical manuscripts while he was still in prison, and continued polishing his ideas until he was finally released on April 29, 1832.

• Neumann's Ford Prize-winning book review of a book by Edwards on Galois theory, mathdl.maa.org/images/upload_library/22/Ford/Neumann407-411.pdf, implies that Poisson's complaints are justified: Galois' original paper is very hard to read, and Edwards' book-length treatment is mainly about rewriting its material in a more understandable way. Feb 3, 2010 at 1:30
• @DE: Thanks for the link -- that was one of the best mathematical book reviews I've ever seen. Feb 4, 2010 at 3:43
• I think it's also important to note that Galois died about a month after being released. It was then Liouville who filled in the details and presented Galois' discoveries to the community. Jan 6, 2015 at 7:25
• An updated link to the review is maa.org/sites/default/files/pdf/upload_library/22/Ford/…. Oct 9, 2017 at 14:26

Smale's eversion of the 2-sphere was first thought to be an "obvious counterexample" to a result he proved in his 1958 thesis. See the Wikipedia article "Smale's paradox" for further information.

• I can't find anything in the Wikipedia article about Smale's eversion looking like an "obvious counterexample" to anything. Can you give us some more sources or information? Feb 3, 2010 at 1:27
• Sorry, I didn't make myself clear. Smale proved a general result about immersions of spheres of arbitrary dimension (I don't know the details). His advisor Raoul Bott pointed out that the general result implied that the 2-sphere could be smoothly turned inside-out in $\mathbb{R}^3$, which he thought was obviously wrong. But Smale's general result was correct, and this led to the discovery of the explicit eversions of the 2-sphere mentioned in the Wikipedia article. Feb 3, 2010 at 1:51
• @JohnStillwell I think it's worthwhile to incorporate this comment into the answer for completeness (remember: comments are not meant to be of lasting value
– Danu
Nov 6, 2015 at 22:11

Ross Street's groundbreaking paper on higher categorical descent and higher nerves

R. Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987) 283-335

was first submitted to Topology but Greame Segal rejected it without review. Street had expected that Segal would appreciate a paper on higher categorical nerve, in view of the fundamental work of Segal on nerves and classifying spaces published in Topology and Publ. IHES. But curiously, Segal considered Street's work on higher nerves of little relevance to topology. The contemporary merger of homotopy theory and higher category theory in the works of Joyal, Simpson, Lurie, Rezk, Cisinski, Jardine and others has of course proven Segal to be wrong in that statement.

Ludwig Schläfli discovered the regular polytopes in $$\mathbb{R}^4$$, including the 24-cell, 120-cell, and 600-cell, among many results of n-dimensional geometry, between 1850 and 1852. He wrote up his results in a big manuscript, Theorie der vielfachen Kontinuität, which was rejected by the Vienna Akademie, and also in Berlin. It was finally published after his death, in 1901. In the meantime, the regular polytopes had been rediscovered by Stringham in 1880. See Coxeter's Regular Polytopes and the Wikipedia article on Schläfli.

• You’ve got this backwards. It was Schläfli who rejected correct mathematics through his refusal to accept the faceted 600-cell, great faceted 600-cell, small stellated 120-cell, great grand 120-cell, great 120-cell, and grand stellated 120-cell. Dec 20, 2021 at 16:45

René Schoof once told me that when he submitted his PhD Thesis, the chapter containing his algorithm to compute the number of points of elliptic curves over finite fields did not appeal at all to the referee, who wondered whether such questions had some interest at all... History decided otherwise!

Qiaochu Yuan already mentioned in a comment Kronecker's negative impact on Cantor's first paper on set theory which was ahead of its time about 20 years, when published in 1874, and which had been delayed several months, unusually long (for that time) such that Cantor considered to withdraw it. Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Crelles Journal f. Mathematik Bd. 77, S. 258 - 262 (1874).

But it is less well known that Cantor's paper PRINCIPIEN EINER THEORIE DER ORDNUNGSTYPEN, ERSTE MITTHEILUNG had to be withdrawn from Mittag-Leffler's Acta in 1884 and had to wait for publication until Ivor Grattan-Guinness published it in Acta Mathematica 124 (1970) 65 - 107. This paper contains a tremendous richness of proposed applications of set theory. Its enforced withdrawal seems to have given Cantor the first really hard stroke.

• Of course, Cantor himself originally rejected his set-theoretic arguments: "Je le vois, mais je ne le crois pas!" en.m.wikipedia.org/w/…
– user44143
Oct 17, 2017 at 2:30

You all know about the journal of rejected math papers: http://www.rejecta.org

• This is awesome! Jun 7, 2010 at 4:19
• I'm unsure if it's a practical joke, even. There's only the one issue! Jun 10, 2010 at 1:45
• Sadly, it appears not to exist anymore. Jun 28, 2013 at 21:48
• @CharlesRezk It's on web.archive.org; I've added the link. May 21, 2014 at 11:53
• I thought this was a joke! I think it would be nicer if the full referee reports were published together with the paper and an open letter from the author. May 21, 2014 at 13:57

Ostrogradsky rejected Lobachevsky's paper "A concise outline of the foundations of geometry".

• Gauss also invented hyperbolic geometry, but himself rejected it. See Milnor's article on "150 years of hyperbolic geometry". I think there are also many other examples where Gauss's "few but ripe" policy made him reject amazing results he invented. Link (might need institutional access): ams.org/bull/1982-06-01/S0273-0979-1982-14958-8/home.html Feb 4, 2010 at 6:30
• Also, Beltrami's 1868 "pseudosphere" model of hyperbolic geometry (more like the universal cover of the pseudosphere), which should have settled the issue, was delayed for a year. Apparently Beltrami was at first deterred by criticisms from Cremona. Feb 4, 2010 at 23:37

I just remembered Apery's theorem:

Due to the wholly unexpected nature of the result and Apéry's blasé and very sketchy approach to the subject many of the mathematicians in the audience dismissed the proof as flawed.

According to the nLab category theory entry, Eilenberg and Mac Lane's paper introducing category theory, General theory of natural equivalences, was originally rejected before being published.

What about deBranges' proof of the Bieberbach conjecture:

https://en.wikipedia.org/wiki/Louis_de_Branges_de_Bourcia

• I guess de Branges would probably say "My proof of the Riemann hypothesis". Has anyone been known to take a serious look at it? Feb 3, 2010 at 1:45
• The Bieberbach conjecture is a good example. Regarding the Riemann Hypothesis, I recall that I have seen at least one paper that claimed to show that de Brange's approach (at the time) could not prove it. Doing a quick search, it was probably the Conrey and Li paper from 1998 mentioned on the Wikipedia page. Whether his approach has changed since then I do not know. Feb 3, 2010 at 11:28
• He wished it upon himself: the first two rejected versions were wrong! Jun 7, 2010 at 4:43
• de Branges proof of the Bieberbach conjecture was never rejected. His paper of Riemann hypothesis was never submitted to a journal, thus it also could not be rejected. Jan 29, 2017 at 20:07
• @DeaneYang: The accounts I have been able to find, such as this one and this one, indicate that de Branges circulated a 350-page manuscript privately among a number of mathematicians, but they do not mention any formal submission to a journal or other publisher. As for whether this means the proof was never "rejected," the OP uses the phrase "rejected by the community." I think this is an accurate description in this case, even if there was no official rejection by a journal. Oct 24, 2019 at 21:20

Abel's work on elliptic functions was unappreciated by the referee Fourier. I think this is the Abel part of the famous Abel-Jacobi theorem.

All in all, Abel could not find appreciation in his lifetime, and by the time he got a decent job, he was ill with tuberculosis and died in obscurity, without even money for a treatment, and without being able to marry his sweetheart Christina.

I remember all this from E. T. Bell. I am however unable to dig up an online reference.

• According to Bell, Legendre and Cauchy were asked to referee Abel's manuscript (on Abel's theorem) in 1826. They stalled, with the excuse that manuscript was not legible, and eventually Cauchy mislaid it. It was not published until 1841. Feb 6, 2010 at 22:06
• If true, Stubhaug's "Niels Henrik Abel and his Times: Called Too Soon by Flames Afar" must have it. Jun 7, 2010 at 4:49

Gelfand-Mazur — every real unital Banach algebra where every non-zero element is invertible is isomorphic to either $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ — was first published without proof by Mazur. Mazur had a (rather short) proof, but the editor demanded he shortened it further. He refused to shorten it, and so it was published without proof.

Later Gelfand published a proof of a weaker version (only for complex commutative Banach algebras), probably without knowing about Mazur's result.

• There is a related 1932 work on locally compact topological skewfields by Pontrjagin, with the "same" classification outcome, Über Stetige Algebraische Korper, Annals of Mathematics 33, No. 1 (Jan., 1932) 163-174 jstor.org/stable/1968110 May 7, 2015 at 17:22

Gauss essentially invented the Fast Fourier Transform in 1805, but the importance of his work was not understood for a century.

"A 1965 paper by John Tukey and John Cooley [2] is generally credited as the starting point for modern usage of the FFT. However, a paper by Gauss published posthumously in 1866 [3] (and dated to 1805) contains indisputable use of the splitting technique that forms the basis of modern FFT algorithms.

"Gauss was interested in the problem of computing accurate asteroid orbits from observations of their positions. His paper contains 12 data points on the position of the asteroid Pallas, through which he wished to interpolate a trigonometric polynomial with 12 coefficients. Instead of solving the resulting 12-by-12 system of linear equations by hand, Gauss looked for a shortcut. He discovered how to separate the equations into three subproblems that were much easier to solve, and then how to recombine the solutions to obtain the desired result. The solution is equivalent to estimating the DFT of the data with an FFT algorithm."

http://www.mathworks.com/access/helpdesk/help/techdoc/math/brentm1-1.html

"Recent studies of the history of the fast Fourier transform (FFT) algorithm, going back to Gauss[1], provide an example of exactly the opposite situation. After having been published and used over a period of 150 years without being regarded as having any particular importance, the FFT was re-discovered, developed extensively, and applied on electronic computers in 1965, creating a revolutionary change in the scale and types of problems amenable to digital processes."

• It was not really rejected, it is simply that until cheap computational power was around, it was a solution waiting for a problem. By the way, it was invented yet another time between Gauss and Cooley and Tukey by some mathematicians in the British admiralty, circa WWI. Feb 4, 2010 at 7:25
• This cannot be completely true: Since Gauss used the method in an actual computation, it must have provided an advantage even then. Were these kinds of calculations rare, or was it just that Gauss didn't make his idea more public?
– user22882
May 21, 2014 at 15:43

The Monty hall problem, surely?

Riemann's work with curved spaces, particularly their applications to physics was at least 40 years ahead of it's time. He pushed forward ideas that space is perhaps curved and that forces such as gravity could be thought of as bending in space. He gave a few lectures on these ideas, but fellow mathematicians and physicists didn't really know what good could come of them and didn't pay much attention. Of course, Einstein finally solved the puzzle many years later.

Also, Joseph Fourier was laughed at when he proposed the notion of Fourier series for solving the heat equation, particularly at the lack of rigor and the overall scope of it's applications. Opinions on the matter changed a decade or two later when the theory began to root itself in rigor thanks to Dirichlet.

• Isn't this - particularly the first - more a case of things being "ignored" rather than "rejected"? Another famous example in this vein would be Poincare's discovery of sensitive dependence on initial conditions (or "chaos" in popular lingo) in the n-body problem. It took until the mid-20th century for the fundamental importance of these phenomena to be recognized, but that does not mean that his result was rejected. Feb 4, 2010 at 10:56
• It sounds like he wasn't ignored or rejected. He had a lot of nice ideas, some of which just didn't have a natural place in physics yet. The success of an idea has a lot to do with whether or not the ambient culture is ready to hear it -- in this case it took Lorentz's work, the Michelson-Morley experiment and Maxwell's equations to "set the stage", I suppose in the opposite order to which I wrote them. :) Feb 4, 2010 at 15:40
• Are you sure you aren't conflating several people here? It was Clifford who first made a vague suggestion that gravitation might be the manifestation of some underlying geometry, I don't remember Riemann ever saying anything like this in writing. Apr 22 at 17:48

Supposedly Vizing had some difficulty publishing what is now known as Vizing's theorem (the result that every degree-d graph can be edge-colored with at most d+1 colors) leading to it eventually being published in a very obscure journal, Akademiya Nauk SSSR. Sibirskoe Otdelenie. Institut Matematiki. Diskretny˘ı Analiz. Sbornik Trudov. Soifer recounts the story in The Mathematical Coloring Book, pages 136–137.

This does not fit the original poster's question but is vaguely related:

I have heard it said that Carleson did not get the Fields Medal in '66 because his proof of Carleson's Theorem was too difficult to read and verify at the time. (Granted, the result was only published in '66.)

And alas, he was too old to get it in '70.

It is only a slight exaggeration to say that the collected works of Hannes Alfven would fit this description.

• Could you expand on that? Feb 19, 2010 at 1:48
• I think the easiest way to fit that into the limited characters allowed here is to point at the wikipedia page for Alfven waves: en.wikipedia.org/wiki/Alfv%C3%A9n_wave This was the rule, not the exception for Alfven. Even after his Nobel he didn't get much respect, (except from guys like Fermi and Chandrasekhar - you could do worse). It's also worth looking at his wikipedia page: en.wikipedia.org/wiki/Hannes_Alfv%C3%A9n Feb 19, 2010 at 20:34
• @AndrewMullhaupt: You can edit the answer instead of writing in comments, if the limit on length is an issue. May 21, 2014 at 12:54

The idea of Van Kampen diagrams in group theory was developed by Van Kampen in the 1930s but no one really used them until the 1960s when they were rediscovered independently by Lyndon and Ol'shanskii. Today they are an important tool in geometric group theory.

• I would say these were "ignored" rather than "rejected". Feb 3, 2010 at 14:02
• Ol'shanskii did not rediscover van Kampen diagrams independently of Lyndon. The book of Lyndon and Schupp appeared first and was widely known. However, there were errors in L-S including the proof of the van Kampen lemma (about the diagrams), so many leading group theorists rejected the method in the 60-80s. Ol'shanskii did make everything completely rigorous in his book and earlier papers.
– user6976
Oct 28, 2010 at 2:49

This is somehow related to, yet it is not about a specific result, but it's about a life.

Namely, according to these sources: https://en.wikipedia.org/wiki/Nikolai_Luzin#The_Luzin_affair_of_1936, and https://mathshistory.st-andrews.ac.uk/Extras/Luzin/, it was claimed in "Pravda", in the summer of '36, that Nikolai Luzin published "would-be scientific papers", within foreign journals (!). Also, he was accused of "praising weak work" (!!). And this accusation was certified by some sort of court, AFAIK. Great shame on these former students of Luzin!

• That was not a mathematical rejection, but a political one, not unlike the campaign led by Teichmüller against Landau and his own adviser Hasse (who was not Nazi enough for him). Some of the actors probably had opportunistic reasons as well. This is a very interesting subject, but I think it is off-topic. Feb 4, 2010 at 12:26
• There is a recent book on the "Lusin affair" that compiled previously unavailable archive materials, "Дело Лузина". Jun 7, 2010 at 4:15

Regarding Carleson's theorem, one of the Fields committee member is supposed to have said that "it would be an insult for Carleson, to give him the Fields medal for that". (I remember the anecdote from Paul Koosis's class, but I might have distorted it in details). To this day, I'm puzzled by what the committee member really meant.

• perhaps... that would be a good explanation! (i'm not a complex analyst, and i'm not familiar with carleson's work).
– maks
Feb 7, 2010 at 7:33