# Mathematicians whose works were criticized by contemporaries but became widely accepted later

Gauss famously discarded Abel's proof that an algebraic equation of degree five or more cannot have a general solution (Abel himself had rejected divergent series as the work of the devil). Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive—even shocking—that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Ramanujan's work on divergent series was rejected by three leading English mathematicians of the time before he was discovered by Hardy.

The above stories have become mathematical folklore. I would like to know the examples of other mathematicians whose works were initially criticized or rejected by contemporaries but later became widely accepted famous. I am particularly interested in modern mathematicians or lesser known mathematicians of the classical era who stories may not be as popular as those of other mathematical giants.

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Heegner proved that the well-known list of im quad fields with class number 1 was complete, but his proof was rejected by many in the mathematical community, I believe because the paper was hard to read and Heegner was not a professional mathematician. Birch checked it was OK though, many years later. But this comment is really just to say that within mathematics this sort of thing is far rarer than in other fields. I can think of more artists/musicians who died obscure and/or paupers but whose work was celebrated later. –  user30035 Feb 12 '13 at 7:32
The question says "Ramanujan's work on divergent series was rejected by three leading English mathematicians". Is that true? Or did they just ignore the letters he sent them? –  Michael Zieve Feb 12 '13 at 7:56
I strongly feel this question should limit itself to mathematicians who died before, say, 1960. Otherwise it borders on gossip- I can name contemporary mathematicians whose papers were ridiculed but later accepted, but to do so would be entirely gossip. –  Daniel Moskovich Feb 12 '13 at 8:57
Grothendieck is as far from a plausible example of this phenomenon as can be imagined. He won the Fields Medal in 1966. I'm sure there were people who didn't like Grothendieck's influence and style of doing mathematics, but the only reason they would care is that he was so influential. –  arsmath Feb 12 '13 at 14:25
We should distinguish between work that was rejected as wrong and work that was rejected as uninteresting or useless. With Grothendieck, for instance, people may have found his work too abstract to be of interest or to be not real mathematics, but I doubt (although I could be wrong) that anybody believed his results to be incorrect. –  Toby Bartels Feb 13 '13 at 22:17

Higher homotopy groups were defined by Eduard Čech in 1932 in a paper for the International Congress of Mathematicians in Zurich, but Alexandroff and Hopf thought that since they were abelian, they were obviously a rediscovery of the known case of homology and not the true generalization of the fundamental group. So they let him know his work was bunk, he withdrew his paper and, as I've heard, was so discouraged that he didn't do further work in the field. It was not until Hurewicz's work that it was realized that these higher homotopy groups, though abelian, provided essentially different information than homology. (Does anyone know the earliest space which was shown to have identical homology and fundamental group, yet different higher homotopy groups? An example is $S^2 \vee S^4$ vs $\mathbb{CP}^2$; I don't know if that is the first.)

There is some discussion on Ronnie Brown's website:

On this ground, and because it was felt that the groups must be the same as the already known homology groups, Alexandroff and Hopf persuaded Cech to withdraw his paper and only a small paragraph appeared in the Proceedings of the Congress. Three years later, however, a Dutch mathematician, W. Hurewicz, published four Notes explaining the main properties of these higher homotopy groups, but without referring to Cech's paper, so they have come to be known as the Hurewicz homotopy groups. These higher homotopy groups became very important concepts, with many people working on them, despite or even because of the difficulty of calculating them for some standard spaces. Both Alexandroff and Hopf later admitted their mistake over Cech's paper. In the 1960s, when higher homotopy groups, despite their being commutative, had become a fundamental tool in topology and geometry, Hopf told E. Dyer that it showed the error of people regarding themselves as so great they are able to know what shall be the future.

It is also mentioned on the nLab page for Homotopy Group and here on Wikipedia.

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A good example that I had not heard of before! –  Timothy Chow Feb 13 '13 at 16:36
W.Hurewicz's first name--Witold--was a typical Dutch name NOT :-) –  Włodzimierz Holsztyński Mar 3 '13 at 10:30
Yes, Hurewicz was not Dutch. On the other hand, I know some Chinese people with Irish names. –  S. Carnahan Apr 24 '13 at 4:04

This is not an answer but a longish comment, which moreover is certainly "subjective and argumentative". Reading all the stories given in the 12 answers, I find I can classify them in three categories:

(1) the stories that have no factual basis and are pure myths (e.g. the one about Hilbert rejected by Gordan, or Grothendieck rejected by you-know-who, etc.). I'd like to add Fourier to this category but here I don't know the history well enough to be sure. What is certain is that Cauchy, faced with a contradiction between Fourier's result and a theorem he "proved" (limit of continuous function s is continuous) did not dismiss Fourier, and that others quickly dismissed (rightly) Cauchy's result.

(2) The ones that don't really concern mathematics: Boltzmann, Bolzano (whose work in mathematic become admired as soon as it was known, and was controversial for something else), Giordano Bruno, and even Brouwer, who as a mathematician was respected and even admired by about everybody else, and was only controversial as a philosopher of mathematics - and certainly no more than any other philosopher is controversial.

(3) The few ones that have a factual basis and do concern mathematics. I will restrict my attention to these cases as they are the only ones that really answer the question. Now, I am afraid that in each of these stories, where a romantic genius makes a discovery that is ignored or rejected by the conservative establishment of mathematics, my heart is with that so-called establishment, whom I can accuse of no wrongdoing, even with the benefit of the hindsight. Indeed, in none of these cases has been the "romantic genius" persecuted or even bullied (as was for example Giordanno Bruno, or to a lesser extent Gallileo). We, the mathematician community, have no autodafe (not to speak of bonfire) in our history to apologize for. What happens in all those cases is that there was a genial mathematician whose works suffered of serious shortcomings, and it was those shortcomings, and not the ones of the mathematical community, that made the process of assimilation of these works by the community longer that it could have been.

Let me explain my point by discussing some cases:

Lévy, are we said, was a great probabilist, but not very rigorous. I agree with this description. Now is its non being rigorous a plus, or a minus? To me the answer is obvious, and I hope everyone here agrees with that. At roughly the same time, Kolmogorov was founding rigorously abstract probabilities on Lebesgue's theory, and this gave his theorems a convincing power that Lévy's have not. The mathematical community has done actually a pretty good and relatively quick work in making Lévy's results rigorous and putting them in the mainstream theory.

Cantor is an interesting case. An absolute genius, for sure, with sometimes almost idiotic remarks -- like when he writes to Dedekind that his bijection between the line and plane refutes the basic idea of dimension. Dedekind kindly answers to him that people working in geometry only consider continuous functions. Now it is perfectly normal and healthy that his works in set theory was exposed to such an harsh criticism in his time. There were serious foundations problem in what he was doing. From the important point of view of rigor, he was putting mathematics back to the time of the early calculus, forgetting all the progresses in rigor made in the nineteenth century, and indeed, there were as is now well-known some serious paradoxes hidden in his theory. The harsh criticism against Cantor's work (such as Poincaré's) was the anti-thesis in a dialectical process, where the role of the synthesis was played by lovers of the Cantor's paradise that wouldn't want to lose rigor and admit paradoxes, such as Hilbert, who were forced by those very criticism to developed a far-reaching program of mathematics to active its goal. Now the partial failure of Hilbert's program (Gödel's incompleteness and inconsistency theorems) shows that there really was something rotten in Cantor's paradise, and the indecidability of Cantor's favorite problem (the continuum's hypothesis) gives retrospectively weight to Poincaré's criticism: arguably, Poincaré never asked a question which was letter shown to be undecidable, as the continuum hypothesis and the questions of the gender of angels.

Galois? Well, he has the best possible excuses for having written his genial discoveries in such an unreadable way: he wrote them partly in jail, partly the night before his death, and all before he was 22. Now for the very same reasons the mathematical establishment (the "Académie des Sciences", including people with a very different mind, like Fourier) has good excuses not to understand what he had done immediately. And again, very soon after his death (about 10 years after), his work was exhumed, intensely admired and integrated in the living mathematics (especially by the German school).

PS: please feel free to vote down this unromantic post. My earlier self would probably have done so.

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+1 for the interesting commentary. –  quid Feb 13 '13 at 1:38
+1 for drawing distinctions and preferring analysis to wallowing in myth :) –  Yemon Choi Feb 13 '13 at 4:00
Another category would be young upstart mathematicians who were criticized and as a result went into some other career. These we have never probably heard about. –  Gerald Edgar Feb 13 '13 at 14:47
There is a larger point I wanted to make. It is self-contradictory to consider the work of someone "revolutionary", "highly original", "in advance on his time", and to at the same time complain that this work was not immediately understood and accepted b the contemporaries. Or to state the contraposite, if the contemporaries, when explained one of those "revolutionary" work, answer "ah yes, of course. Good work. Now let's have lunch, shall we?", then it means that the work was not as revolutionary as claimed. Taking this into account, the works of revolutionary mathematicians has been... –  Joël Feb 14 '13 at 16:39
in general accepted fast, when one compares with other fields (e.g. physics, musics). And there is still another point I want to make, more related to my post. While it is in the nature of things that profoundly original works are not understood right away, I would say that for this kind of work, a relatively quick recognition is a sign of their great qualities. Originality is a good thing, obscurity is not. I can't think of a more original creation than Nietzche's books. Yet the interest of his work was widely recognized by the time of his death, and he became soon after universally famous. –  Joël Feb 14 '13 at 16:46

"Hermann Graßmann submitted [Die lineale Ausdehnungslehre] as a Ph. D. thesis, but Möbius said he was unable to evaluate it and forwarded it to Ernst Kummer, who rejected it without giving it a careful reading." [Edit: This Wikipedia quote is at least misleading, see the addendum below.]

From a webpage : "His 1844 work Die lineale Ausdehnungslehre: Ein neuer Zweig der Mathematik [The Theory of Linear Extension, a New Branch of Mathematics], effectively single-handedly founded Linear Algebra. This work was submitted as a Ph.D. thesis in Mathematics, however its formulation of linear vector space in opposition to the canonical Euclidean geometry of the time was too radical for his contemporary mathematics establishment and was rejected. Consequently, the significance of contribution to the mathematical sciences were largely unrecognized in his lifetime, but they were eventually re-discovered towards the end of the nineteenth century and early twentieth century. It was because of these rejections and early lack of recognition of his work in the mathematics that he suffered he turned himself to Vedic studies, and made those discoveries in there which we know him best for [!]" -- Hermann Graßmann: Philologist and Mathematician

The claim about "single-handedly founding linear algebra" seems exaggerated. For a closer investigation, one might e.g. look at the articles by D. Fearnley-Sander quoted and referenced in the wikipedia article.

Added: After reading (small) parts of Engel's biography (vol. III.2 in Graßmann's Gesammelte Werke), I feel that the quotations above are kind of unfair towards Moebius and Kummer. Moebius actually put quite an effort into supporting Graßmann, for several years. Regarding Kummer, it should first of all be noted that Graßmann did not submit his work as a Ph.D. thesis in the modern sense, but sent it, along with another work, to the ministry, to apply for a professorship at some university. Kummer's report (reprinted there, pp. 126--129) is ambiguous, in that he harshly criticises the form, but admits that "diese Schrift wirklich neue und interessante Gesichtspunkte gewährt, so daß ich über den wissenschaftlichen Wert des Inhalts mich wirklich lobend und anerkennend äußern kann". Kummer indicates that more profound results may be found in Graßmann's work with more effort. But for a teaching position, he suggests, there are younger excellent mathematicians with much better style of exposition. He also says that he has no reservations against awarding Graßmann the title "Professor", but given Graßmann's deficits in exposition, he has doubts about him as lecturer; however, he says, it could still be enquired whether his oral teaching abilities are better (given that Graßmann was a school teacher). Apparently, the ministry did not take the last suggestion serious, but wrote to some lower school office whether it would be OK to award Graßmann solely the title; the office advised against this, saying it would cause trouble with superior teachers who did not have this title (...). So the ministry wrote back to Graßmann, dismissing his wishes. -- Frustrating as this must have been for Graßmann, but given Peter Michor's comment, I agree that many statements in Joël 's answer do apply very much to Graßmann's case.

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That's a good one. I think one of the first established mathematicians to recognize his work is Clifford who used Grassmann's anticommuting variables. –  Abdelmalek Abdesselam Feb 12 '13 at 14:15
The 1844 edition was written in the style of Kant, carefully pondering philosophical aspects of notions. The intended audience seems to be philosophers more than mathematicians. This edition seems to be the first instance where "space" was considered with possibly more than 3 dimensions. The revised edition of 1862 reads already like a modern text of linear algebra. These opinions are based on my own reading of (small) parts of Grassmann's books. –  Peter Michor Feb 12 '13 at 17:18
@Abdelmalek: Actually, Hamilton had read Graßmann's works and was quite enthusiastic about it, suggesting that Graßmann had come close to discovering the quaternions (the highest praise Hamilton knew). See pp. 204-208 of the cited biography, cf. Hamilton's foreword to his Lectures. "[M]y own researches, or speculations, would have a better chance of being appreciated in these countries, if readers had first been put through a sufficient course (or dose) of Graßmann" he wrote to De Morgan. True: Linear algebra is quite helpful for thinking about quaternions! –  Torsten Schoeneberg Mar 17 '13 at 17:13

Paul Lévy

Paul Lévy was an extraordinarily productive mathematician: in parallel with and independently from the Soviet mathematicians Kolmogorov and Khinchin, he discovered the major part of what is known today as the theory of stochastic processes. Among his contributions where the study of various properties of Brownian motion and the discovery of necessary and sufficient conditions in limit theorems for sums of independent random variables. He proved the Central Limit Theorem using characteristic functions, independently from Lindeberg who proved the same theorem using convolution techniques. He discovered the class of probability distributions known as "stable distributions" and proved the generalized version of the Central Limit Theorem for independent variables with infinite variance. He also introduced the notion of Brownian local time in the context of study of the properties of Brownian motion: today this concept plays a key role in the study of fine properties of diffusion processes. Michel Loeve gives a vivid description of Lévy's contributions: Paul Lévy was a painter in the probabilistic world. Like the very great painting geniuses, his palette was his own and his paintings transmuted forever our vision of reality... His three main, somewhat overlapping, periods were: the limit laws period, the great period of additive processes and of martingales painted in pathtime colours, and the Brownian pathfinder period."

Although he was a contemporary of Kolmogorov, Lévy did not adopt the axiomatic approach to probability. Joseph Doob writes of Lévy: "[Paul Lévy] is not a formalist. It is typical of his approach to mathematics that he defines the random variables of a stochastic process successively rather than postulating a measure space and a family of functions on it with stated properties, that he is not sympathetic with the delicate formalism that discriminates between the Markov and strong Markov properties, and that he rejects the idea that the axiom of choice is a separate axiom which need not be accepted. He has always travelled an independent path, partly because he found it painful to follow the ideas of others."

This attitude was in strong contrast to the mathematicians of his time, especially in France where the Bourbaki movement dominated the academic scene. Adding this to the fact that probability theory was not regarded as a branch of mathematics by many of his contemporary mathematicians, one can see why his ideas did not receive in France the attention they deserved at the time of their publication. P.A. Meyer writes: "Malgré son titre de professeur, malgré son élection à l'Institut ... Paul Lévy a été méconnu en France. Son oeuvre y était considérée avec condéscendance, et on entendait fréquemment dire que ce n'était pas un mathématicien." Translation: Although he was a professor and a member of the Institut [i.e., the Academy of Sciences], Paul Lévy was not well recognized in France. His work was not highly considered and one frequently heard that "he was not a mathematician".

However, Paul Lévy's work was progressively recognized at an international level. The first issue of Annals of Probability, an international journal of probability theory, was dedicated to his memory in 1973, two years after his death.

See also what Laurent Schwartz writes in his book "Un Mathématicien aux prises avec le siècle" about the relations between Paul Lévy and the Bourbaki group, http://books.google.fr/books?id=Eqc0cyFR0AEC&lpg=PA173&ots=qx9f0eMmcd&dq=paul%20levy%20bourbaki&hl=fr&pg=PA173#v=onepage&q=paul%20levy%20bourbaki&f=false

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Louis Bachelier (see en.wikipedia.org/wiki/Bachelier, fr.wikipedia.org/wiki/Louis_Bachelier) –  Uwe Franz Feb 12 '13 at 17:35

The axiom of choice was formulated by Zermelo in 1904 in order to prove his well-ordering theorem. The axiom of choice was strongly criticized by many famous mathematicians, including Baire, Borel, and Lebesgue. Nowadays, of course, although a minority of mathematicians still reject it, the axiom of choice is accepted as part of mainstream mathematics.

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Of course, Gödel's result that consistency of ZF implies consistency of ZFC (1940) was helpful to accept its use. –  YCor Mar 17 '13 at 19:17

Not in pure mathematics but in applied mathematics we have the case Ludwig Boltzman, the Austrian physicist (also the founder of the Austrian Mathematical Society) whose greatest achievement was in the development of statistical mechanics. He spent a life time trying to defend the now famous equation

$$S = k\log W$$

Boltzmann's mentor and colleague Josef Loschmidt criticized Boltzmann's demonstration of entropy increase on the grounds that dynamical laws are reversible. If all the particles could be turned around exactly (or if time could be reversed), Boltzmann's work indicated the entropy should decrease, violating the second law. Eventually he committed suicide out of depression.

Today the above equation is one of the most important and fundamental equations of science.

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–  jjcale Feb 13 '13 at 19:44

Brouwer's intuitionistic mathematics was heavily criticized by his contemporaries, most notably Hilbert. For almost a century it was casually ridiculed by mathematicians who had no clue whatsoever about it. However, in the late 20th and early 21st century the importance of intuitionistic logic was recognized by mathematicians who worked in areas close to computer science. By the late 21st century the tables were turned and most mathematicians were educated in the tradition of Martin-Löf type theory (a starting development for many mathematicians born in the 20th century, who dismissed Martin-Löf's work on the grounds of it being useful exclusively for philosophically commited constructivists) . In their ignorace they now considered ridicule of Zermelo and Fraenkel an appropriate activity. Will they ever learn?

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@Bauer: Do you really think that at some moment "most mathematicians were educated in the tradition of constructive Martin-Löf type theory"? –  boumol Feb 12 '13 at 11:01
@boumol: The late 21st century is still some way off, but we can hope... –  Zhen Lin Feb 12 '13 at 11:07
My mistake, I was reading late 20th century. @Bauer:I haven't downvote yet, I will wait for the late 21st century. –  boumol Feb 12 '13 at 11:13
This will never happen. There's no trade-off between ZF and Martin-Lof type theory, unless you are explicitly philosophically committed to constructivism. As a proportion of mathematicians, classical complex function theory is a much smaller proportion of mathematics than it was in the late 19th century, but nobody makes fun of complex analysts. –  arsmath Feb 12 '13 at 12:06
Yes, yes, yes, and 1/2 yes. (How do you formalize classical mathematics in Martin-Lof type theory? Doesn't strong normalization prevent it?) My point is that unless you are a constructivist, Martin Lof type is just another thing you can study, like complex analysis, or homological algebra, or whatever. People who study the fundamental groups of hyperbolic 3-manifolds don't make fun of 19th century complex analysts. There's no reason why type theorists of the 22nd century need to make fun of today's set theorists. –  arsmath Feb 12 '13 at 13:38

Perhaps the canonical example is Nikolai Ivanovich Lobachevsky? His work on hyperbolic geometry was subject to severe ridicule, stemming from a negative review of Ostrogradsky, the leading Russian mathematician of the time. Perhaps the severity of the ridicule piled on Lobachevsky was part of the inspiration for Tom Lehrer's song Lobachevsky?

Needless to say, Lobachevsky's work became widely accepted later.

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Note that Lobachevsky's work was paralleled by János Bolyai (en.wikipedia.org/wiki/J%C3%A1nos_Bolyai) whose work was (and still is) even less recognized. –  GH from MO Feb 12 '13 at 9:00
If my memory serves me right, Lehrer made this choice for prosodic reasons only. Or euphonic ones. I mean, it sounds better than other names and fits well. I recall Lehrer talking about that in an interview, to which one could probably find a link on a Youtube channel devoted to him. –  Jonathan Chiche Feb 12 '13 at 9:04
Found: home.broadpark.no/~emeyn/tl/radio2.html. "I took the name of Lobachevsky, only for prosodic reasons, just because it fit". (With an emphasis on "only".) –  Jonathan Chiche Feb 12 '13 at 9:11
@Jonathan Chiche: Hence the word "perhaps" :-). It would be quite a coincidence if "prosodic reasons" were indeed literally the only reasons, given that Lobachevsky was both famously ridiculed, and later famously falsely accused of plagiarism (he was accused of stealing his ideas from Gauss). –  Daniel Moskovich Feb 12 '13 at 11:11
@GHfromMO, "Note that Lobachevsky's work was paralleled by János Bolyai" --- pun intended? –  Gerry Myerson May 4 '14 at 0:39

Surprised no one has mentioned Fourier. His idea that you could express arbitrary functions as infinite sums of sines was initially rejected.

(Of course "arbitrary" isn't quite right, but far more functions than his contemporaries would have imagined. It took many years to carefully define the boundaries of Fourier's theory, and to extend those borders through generalizations.)

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Who: Lucjan Emil Boettcher (1872-1937), working in the theory of iteration, dynamics of rational maps and functional equations.

Criticism/rejection: Boettcher studied mathematics in Warsaw and engineering in Lvov, and completed a doctorate in mathematics in Leipzig with Sophus Lie. Afterwards he became a lecturer ("docent") at Lvov Polytechnics, teaching also some courses at Lvov University. In the years 1901-1919 he made four attempts at getting habilitation at the University (a process similar to tenure review). All were unsuccessful. Here are samples of the evaluations:

...the results of Dr. B\"ottcher seem to be too formalistic developments, and therefore can be only one-sided contributions to the theory of solutions of functional equations." (1911)

....The method used by the Candidate in his works cannot be considered scientific. The author works with undefined, or ill-defined, notions (e.g., the notion of an iteration with an arbitrary exponent), and the majority of the results he achieves are transformations of one problem into another, no less difficult. In the proofs there are moreover illegitimate conclusions, or even fundamental mistakes." (1918)

...Dr. B\"ottcher's works do not yield any positive scientific results. There are many formal manipulations and computations in them; essential difficulties are usually dismissed with a few words without deeper treatment. The content and character diverges significantly from modern research." (1918)

What he is nowadays famous for: Boettcher theorem, Boettcher equation and Boettcher coordinate. All these related notions describe behavior of an analytic function $f(z)=a_pz^p+..., \ p \geq 2$ in a neighborhood of the fixed point $z=0$. They are important in holomorphic dynamics.

Who first recognized his work: Joseph Fels Ritt, in his paper On the iteration of rational functions. Trans. Amer. Math. Soc. 21 (1920), no. 3, 348-356. Ritt seems to have given the first complete proof of Boettcher's theorem.

What else should he be famous for: Boettcher gets credit for constructing the first Lattes-type example of an everywhere chaotic map (see this MO question: The half-life of a theorem, or Arnold's principle at work). But he also should be recognized for pioneering the Fatou-Julia theory (20 years before Julia and Fatou, and without the advantage given by the notion of normal families) in his study of regions of convergence of iterates of rational maps and their boundaries. E.g. he described the Julia set for a monomial and for a Chebyshev polynomial of an arbitrary degree no less than two. More importantly, he also first stated an upper bound for the number of non-repelling cycles of a rational function in terms of the number of its critical points (in 1920s conjectured again by Fatou and proved to be sharp in 1980s by Shishikura).

An interesting twist: In principle, the committee members were right! At best, Boettcher only sketched his ideas. At worst, he really worked with ill-defined objects (he did study iterates with arbitrary exponents...) or made mistakes (e.g., in describing properties of boundary curves" of regions of convergence, better known as Julia sets). He also published some of his results multiple times and often devoted many pages to detailed analysis of other mathematicians' work (Koenigs, Leau etc.), so his articles could come across as derivative.

More to read, for those interested: Lucjan Emil B\"ottcher and his mathematical legacy, by Stanis\law Domoradzki and Ma\lgorzata Stawiska, http://arxiv.org/pdf/1207.2747.pdf

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Józef Maria Hoene-Wroński was not respected in his time, dismissed as a loony. I think I first read of him in ''The Mathematical Experience.''. Not a very famous mathematician but I had heard of the Wronskian...

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For mathematical achievements of Wro\'nski (including, but not limited to Wro\'nskian), see: Pragacz, Piotr Notes on the life and work of Józef Maria Hoene-Wroński. Translated from the Polish original [Wiadomości Matematyczne (Ann. Soc. Math. Pol.) 43 (2007)] by Jan Spaliński. Trends Math., Algebraic cycles, sheaves, shtukas, and moduli, 1–20, Birkhäuser, Basel, 2008 It probably did not help that Wro\'nski sometimes mixed mathematical and philosophical content in his writing. –  Margaret Friedland Feb 13 '13 at 2:19
This is an excellent example, I think. –  Daniel Moskovich Feb 13 '13 at 5:26
He later opened a lodge in Aspen, called Ron's Ski-In. –  Gerry Myerson Feb 13 '13 at 21:53

Bernard Bolzano (1781--1848), although his mathematical writings were not really rejected, but ignored. (It was his political and philosophical writings that really got him into trouble.)

English wikipedia: "Bolzano's posthumously published work Paradoxien des Unendlichen (The Paradoxes of the Infinite) was greatly admired by many of the eminent logicians who came after him, including Charles Sanders Peirce, Georg Cantor, and Richard Dedekind. [...] To the foundations of mathematical analysis he contributed the introduction of a fully rigorous ε-δ definition of a mathematical limit [...] he was one of the earliest mathematicians to begin instilling rigor into mathematical analysis with his three chief mathematical works Beyträge zu einer begründeteren Darstellung der Mathematik (1810), Der binomische Lehrsatz (1816) and Rein analytischer Beweis (1817). These works presented "...a sample of a new way of developing analysis", whose ultimate goal would not be realized until some fifty years later when they came to the attention of Karl Weierstrass [...] Today he is mostly remembered for the Bolzano–Weierstrass theorem, which Karl Weierstrass developed independently and published years after Bolzano's first proof and which was initially called the Weierstrass theorem until Bolzano's earlier work was rediscovered."

German wikipedia: "In einem Aufsatz von 1817 bewies er den Zwischenwertsatz und führte Cauchy-Folgen ein, vier Jahre vor Augustin Louis Cauchy. Bolzanos Arbeiten zu einer strengeren Grundlegung der Analysis wurden von seinen Zeitgenossen im Gegensatz zu denen von Cauchy kaum beachtet und erst in der zweiten Hälfte des 19. Jahrhunderts gewürdigt."
(rough translation: "In a paper from 1817 he proved the intermediate value theorem and introduced Cauchy sequences, four years prior to Cauchy. Bolzano's works towards a more rigorous foundation of calculus, unlike Cauchy's, were hardly noticed by his contemporaries, and only began to be recognised in the second half of the 19th century.")

PS: Recognition comes late. According to this site, Czechoslovakia (a country young people do not know anymore) issued this postal stamp honouring Bolzano in 1981.

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So this is more of a case of un-recognition, as opposed to ridicule? –  Andrej Bauer Feb 12 '13 at 11:51
As I recall: Bolzano was a Catholic priest, with a position of professor of philosophy in Prague. He taught theology and also mathematics. Since this was part of the Austro-Hungarian Empire, in the theology courses he should have been teaching (but wasn't) that it is the religious duty of every citizen to obey the edicts of the emperor. So he was dismissed from the university. And it became very difficult for him to publish, even mathematics. So I agree: it is not that his mathematics was criticized, but rather that it was mostly unpublished and therefore unknown. –  Gerald Edgar Feb 12 '13 at 13:56
Yes, Bolzano's case does not really fit the spirit of the question. If you want me to delete this answer, no problem. –  Torsten Schoeneberg Feb 12 '13 at 14:11

Oliver Heaviside probably deserves some mention here. I think he deserves a lot of the credit for the development of vector calculus.

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And some contribution to what has now been formalized as generalized functions. –  John D. Cook Feb 12 '13 at 21:10

Louis de Branges and his proof of the Bieberbach conjecture.

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@Margaret Friedland: I do not understand your comment, could you please elaborate. I was under the firm believe that Bieberbach is considered as proved by de Branges, while there were some issues early on (also some actual ones). And there are other (simpler) proofs meanwhile. –  quid Feb 13 '13 at 1:44
@quid: you are completely right, I removed the previous comment. It's getting late, I somehow read "Riemann" instead of "Bieberbach", and thought of de Branges's (later) interest in proving the Riemann hypothesis. For those who want to read more about the sociological aspects of the proof of Bieberbach conjecture, as well as of de Branges's work on other famous problems, I can recommend the story in: Krantz, Steven G. The proof is in the pudding. The changing nature of mathematical proof. Springer, New York, 2011. xviii+264 pp. ISBN: 978-0-387-48908-7 –  Margaret Friedland Feb 13 '13 at 2:04

I would add Appel and Haken's computer assisted proof of the Four Colour Theorem to the list.

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From Saunders Mac Lane's obituary:

"... In seeking to provide a sound conceptual framework for the subject [of homological algebra], they invented the notions of category and functor. These notions were slow to gain acceptance (the first Eilenberg-Mac Lane paper on categories was nearly rejected by the Transactions of the American Mathematical Society) on account of their seeming lack of content: for a decade or so, category theory was derided by other mathematicians as "abstract nonsense". But in time the substantial new advances made possible by the categorical way of thinking about mathematics won it acceptance: it has by now become an indispensable part of the vocabulary of the great majority of pure mathematicians (and, increasingly, of researchers in theoretical physics and computer science)."

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"...for a decade or so, category theory was derided by other mathematicians as "abstract nonsense"" ...you mean this doesn't still happen? –  Sam Gunningham Mar 17 '13 at 18:26
Again, to have a rejected paper doesn't mean that your work is not accepted, otherwise this would apply to almost all living mathematicians. As for having a paper "nearly rejected", ... If I teach a class of mathematics and a student at first don't understand what I am saying and don't get the point of the material taught until, near the end of the semester, working hard for the final exam, he really gets it and appreciates it, should I say that my work was "first criticized, and then eventually accepted", and present myself as a victim of that student ? Obviously not. Yet this is what... –  Joël Mar 17 '13 at 18:40
... happened of Eilenberg-Maclane, and to many of the mathematicians named as an answer to the OP's question... –  Joël Mar 17 '13 at 18:41
Well, I would respond by saying that this is an example precisely in the vein of your item (3). It is the case that Eilenberg and MacLane's ideas were treated with some skepticism at first, but that they simply continued working them out, writing excellent textbooks, working at making them accessible, finding examples, and illustrating the value of the work. Unlike some of the less fortunate visionaries people have posted about here, they got to see the impact of their ideas throughout their remaining mathematical careers. I don't think the OP required only sad or unjust stories... –  David Jordan Mar 17 '13 at 21:31

Galois maybe?

Also, a famous example is Hilbert's work on invariant theory. I don't know if there is truth in the "theology and not mathematics" story regarding Hilbert's first paper with the basis theorem, but in any case it took a while before this new way of doing algebra became accepted.

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Is this true? Of course not everyone is going to switch mathematical styles instantly, but was there any actual resistance? (The evidence that Paul Gordan said anything like "That is not mathematics, that is theology" is pretty thin.) –  arsmath Feb 12 '13 at 15:18
It is true that it took some decades for the mathematical community to digest and elaborate the ingenious but only sketched ideas of Galois. But who criticized his work? –  Martin Brandenburg Feb 12 '13 at 15:19
For more information about the "theology" urban legend, see people.math.jussieu.fr/~harris/theology.pdf –  Timothy Chow Feb 12 '13 at 15:25

One often hears established mathematicians commenting on the work of younger ones as "pointless generalizations". Sometimes, these pointless generalizations are used by somebody else to prove a result that attracts attention, and then the same established mathematicians will say "OK, this proved to be useful, but it was just an easy exercise that could have been included in the proof of the big result". It might take some more time before the younger generation actually recognizes the value of these pointless generalizations: not everything can be world shaking or even first class.

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Vague. Hearsay. Subjective. –  Yemon Choi Feb 14 '13 at 16:53
Well, I am not going to give any concrete examples because they would concern people who are still (very) active. I was just trying to point out that "Mathematicians whose works were criticized by contemporaries but became widely accepted later" is happening constantly (depending on the definition of the vague and subjective "later", "widely"). –  practical Feb 14 '13 at 17:40
As someone who has just recently rejected an article for being "pointless generalization" (and also for showing shallow and poor taste), I still think your answer inadvertently feeds an unfortunate perception –  Yemon Choi Feb 14 '13 at 19:37
Personally I find the emphasis on "younger" quite strange. Or is this meant purely figuratively. –  quid Feb 15 '13 at 0:19
I agree with the established mathematicians. –  Vivek Shende Mar 3 '13 at 13:18

Perhaps the papers by Black and Scholes, and Merton on the asset pricing of an European option.

The paper was rejected by two journals before it was accepted for publication by a third journal.

1997 these guys (Black died before) became the Nobel Prize in Economics. Their work is regarded as the corner stone and pillar of financial mathematics.

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Like with most other answers, I profoundly disagree with this one. For one thing, this is is not mathematics, but let us forget about this. Black, Scholes, and Merton were three young bright very well integrated student, graduate from the best university in their field (Chicago, MIT, Harvard). The Advisor of Merton was Paul Samuelson then and now considered by many the greatest economists of his generation. Scholes' advisor was Eugene Fama, one of the founding father of modern finance (and by the way a student and nemesis of our own Mandelbrot). To be sure, Black's adviser was not an... –  Joël Mar 2 '13 at 18:44
economist, but it was the great Marvin Minsky. So they were all three the exact opposite of the forgotten scents working in obscurity. Now what happen specifically with Black and Scholes equation was that their proof of it was unnecessarily complicated, and with a lot of very disputable hypotheses (it was based on Fama's modeling on Financial market, which are in many respects difficult to believe). For this reason, they had some difficulties publishing it. At about the same time, Merton found independently a much cleaner, simpler, beautiful proof with less hypothesis (still disputable)... –  Joël Mar 2 '13 at 18:49
and nowadays it is only's Merton's proof which is taught anymore. Merton, which heard of Black and Scholes's work, helped them having it published and waited this was done to publish his version: a great example of honesty. Finally, in less than three years both papers were published, and a few years after, their equation was used in financial markets and taught in all mathematical finance classes. To conclude, if everyone who got a paper rejected one or two time could be on this list, we would all be there. –  Joël Mar 2 '13 at 18:52
@Joel: I still think the Black--Scholes formula has this mathematical flavour with fine and difficult probability theory. Your last sentence: ??? –  Hans Mar 2 '13 at 19:15
Hans: on this I agree. The formula and its proof is beautiful. I remember when I first learn it saying to myself: this can't be true. This formula can not follow from this set of hypothesis. There must be a mistake or a hidden assumption somewhere in the proof. And I read it, several times, and at the end I understood. That's a mathematical proof as its best. one which does everything: explaining why the result is true and at the same time forbidding any doubt about its truth (as a mathematical result I mean -- the question of its applicability in real world is an other matter). –  Joël Mar 2 '13 at 20:14