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It seems to me that almost all conjectures (hypotheses) that were widely believed by mathematicians to be true were proved true later, if they ever got proved. Are there any notable exceptions?

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This is a misconception, IMO. –  quid May 3 '12 at 15:00
Just like the question @KCnonrad alludes to, this one should be CW. –  Igor Rivin May 3 '12 at 15:38
I do not get the 'cw-drama'; as soon as there are answers one anyways needs a moderator to do it properly...just flag and wait. (Which btw I just did, in case reaally noone did yet.) In any case some including me consider it as slighty against etiquette to close just due to cw-issues; that being said I can see plenty of other reasons why one might want to close (yet did not yet vote accordingly). –  quid May 4 '12 at 0:38
@Joël: personally I'd assume that if something was widely believed decades or centruies ago yet was proved wrong, then the fact/knowledge that something else was widely believe before gets lost over time. (Personally, I am simply unable to judge how widely believed the Hauptvermutung was.) In addition there is the phenomenon that if something turns out to be just slightly wrong then this is somewhat swept under the carpet. Over time the orginal conjectures and believes get morphed into something that then is actually true, while the original in fact was false. To name something specific: –  quid Jul 2 '12 at 13:47
For the morphing process maybe Hodge's conjecture can serve as an example. But, for something famous but not very old were the original belives apparently were wrong one could consider Carelson's therorem. People including him did believe in counterexamples to the claim he then proved true. See his interview in the Feb 2007 Notices AMS. Or for old things, I think (but I am not a math historian) at some point in time (though perhaps not up to the moement were refuted) people were quite convinced one would be able to solve the quintic with radicals, or 'prove' the parallel postulate. –  quid Jul 2 '12 at 14:10

18 Answers 18

In 1908 Steinitz and Tietze formulated the Hauptvermutung ("principal conjecture"), according to which, given two triangulations of a simplicial complex, there exists a triangulation which is a common refinement of both.

This was important because it would imply that the homology groups of a complex could be defined intrinsically, independently of the triangulations which were used to calculate them.

Homology is indeed intrinsic but this was proved in 1915 by Alexander, without using the Hauptvermutung, by simplicial methods.

Finally, 53 years later, in 1961 John Milnor (some topology guy, apparently) proved that the Hauptvermutung is false for simplicial complexes of dimension $\geq 6$.

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I like the story with Milnor arriving late to class and copying down the ten homework questions, then could solve only half of them. Milnor says it's true. –  Will Jagy May 3 '12 at 21:09
Oh, and the presumed homework questions were actually open problems in knot theory. –  Will Jagy May 3 '12 at 21:11
No, this way works well. You know who it is but deliberately play it down. Very fashionable. I got a brief bit of grief for not knowing who Fields Medalist Voevodsky was, in a Meta thread about a video in which he espoused a radical intuitionism. But I actually did not know of him. I don't follow sports. –  Will Jagy May 3 '12 at 21:21
It seems that this story was conclusively disproved on MO last year. See mathoverflow.net/questions/54513/… –  Nik Weaver May 3 '12 at 22:53
You seem very sure ... sounds to me like you didn't take the proverb to heart. –  Nik Weaver May 4 '12 at 2:46

This can perhaps be considered more of a meta-conjecture than a conjecture: Hilbert's program, http://en.wikipedia.org/wiki/Hilbert's_program. The conjecture would be: that set theory (or some set of axioms suitable for doing math) can be proven consistent. Gödel's Incompleteness Theorem disproved this conjecture.

I don't have a reference, but I have the impression that, at the time, Hilbert's program seemed attainable, and Gödel's result came as a surprise.

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Your impression is perfectly correct. It seemed attainable at least by Hilbert and his school. –  Joël Jun 29 '12 at 21:42

Luzin's conjecture was widely believed to be false, until it was proven by Carleson in 1966.

I'm citing from Lennart Carleson's biography: "In 1913 Luzin conjectured that if a function $f$ is square integrable then the Fourier series of $f$ converges pointwise to $f$ Lebesgue almost everywhere. Kolmogorov proved results in 1928 which seemed to suggest that Luzin's conjecture must be false but Carleson amazed the world of mathematics when he proved Luzin's long-standing conjecture in 1966. He explained how he was led to prove the theorem:-

... the problem of course presents itself already when you are a student and I was thinking about the problem on and off, but the situation was more interesting than that. The great authority in those days was Zygmund and he was completely convinced that what one should produce was not a proof but a counter-example. When I was a young student in the United States, I met Zygmund and I had an idea how to produce some very complicated functions for a counter-example and Zygmund encouraged me very much to do so. I was thinking about it for about 15 years on and off, on how to make these counter-examples work and the interesting thing that happened was that I realised why there should be a counter-example and how you should produce it. I thought I really understood what was the background and then to my amazement I could prove that this "correct" counter-example couldn't exist and I suddenly realised that what you should try to do was the opposite, you should try to prove what was not fashionable, namely to prove convergence. The most important aspect in solving a mathematical problem is the conviction of what is the true result. Then it took 2 or 3 years using the techniques that had been developed during the past 20 years or so"

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Thanks for the nice answer. Another source for this is this interview with Carleson (Notices AMS, Feb 2007) ams.org/notices/200702/comm-carleson.pdf –  quid Jul 3 '12 at 10:41

Littlewood disproof of the conjecture (maybe of Gauss) that $\text{li}(x) > \pi(x)$.

I think this was widely believed before.

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Euler's sum of powers conjecture, if a sum of $k$th powers is a $k$th power, then the sum has at least $k$ terms.

Proposed by Euler in 1769. Counterexample for $k=5$ found in 1966, for $k=4$ in 1986.

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By our own @Noam Elkies in the latter case. –  Igor Rivin May 3 '12 at 15:36

Euler's conjecture about the nonexistence of $n\times n$ Graeco-Latin squares for $n=4k+2$. Disproved for all $k>1$ by the so called Euler's Spoilers Bose, Shrikhande, and Parker.

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In complex analysis of one variable, Liouville's theorem says that a bounded entire function is constant. Bernstein (1915-17) proved an analogous result in differential geometry, namely, if the graph of a function $f:\mathbf R^2\to\mathbf R$ of class $C^2$ is a minimal surface in $\mathbf R^3$, then the graph a plane. He then posed the classical Bernstein problem, namely, whether the same result also holds for real functions of $n>2$ variables. In terms of differential equations:

(Classical) Bernstein problem: Let the function $f:\mathbf R^n\to\mathbf R$ of class $C^2$ be a solution of $$\sum_{i=1}^nD_i\left(\frac{D_i f}{\sqrt{1+|D f|^2}}\right)=0.$$ Must $f$ be a linear function?

Recall that a hypersurface in $\mathbf R^{n+1}$ is defined to be minimal if its mean curvature vanishes, where its mean curvature is simply the sum of the principal curvatures (sometimes divided by $n$). Equivalently, the hypersurface is a critical point for the $n$-volume with respect to compactly supported variations. The equation above is the condition that the mean curvature of the graph of $f$ vanishes everywhere.

Part of the importance of the Bernstein problem is that it has a direct bearing on the existence of minimal cones and singularities of minimal hypersurfaces in $\mathbf R^{n+1}$. The answer to the problem was proved to be affirmative in the cases $n=3$ by de Giorgi (1965), $n=4$ by Almgren (1966), and $n\leq7$ by Simons (1968), and apparently there was some hope to extend the result to all dimensions.

However, in 1969 Bombieri, de Giorgi and Giusti constructed a counter-example for $n=8$, which yields a counter-example in each dimension $n>8$ by a standard construction, closing the problem. The complete solution of the Bernstein problem turned out to involve a good deal of geometric measure theory and non-linear analysis.

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I believe that Fefferman's disproof in 1971 of the L^p boundedness of disc multiplier for any $p \neq 2$ was considered a great surprise at the time; it showed that the classical result of Bochner and Riesz establishing norm convergence of Fourier series of L^p functions in one dimension failed in two and higher dimensions, if one summed the series in the order of the magnitude of the frequencies (i.e. spherically summed Fourier series). The construction was one of the first applications of Kakeya sets (also known as Besicovitch sets) to harmonic analysis (though there was an earlier paper of Stein and Weiss that also used a related idea). Nowadays, the connection is taken for granted, but it was certainly not obvious at the time of Fefferman's result. (Fefferman himself writes in his paper "... It therefore comes as a surprise, at least to me, that the disc conjecture is false.")

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Mersenne's conjecture on primes is a famous example (although I am not sure how widely it was believed to be true).

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The solution in negative of the isomorphism problem for integral group rings. A counterexample was found by Martin Hertweck:


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There have been multiple conjectures of this type - seemingly motivated, commonly believed, yet false - about the structure of the partial order of Turing degrees of c.e. sets. Two in particular were due to Shoenfield:

  • In 1963, he conjectured that, given any finite poset $P$ which embeds (via $f$) into the c.e. degrees preserving $0, 1$, and $\vee$, and $P'\supseteq P$ has the same maximal and minimal elements, and l.u.b.s, as $P$, then $P'$ embeds into the c.e. degrees via an embedding extending $f$.

This was refuted by the construction of a minimal pair of c.e. degrees, that is, a pair of noncomputable c.e. degrees $\underline{a}$, $\underline{b}$ such that no noncomputable set is computable in both $\underline{a}$ and $\underline{b}$.

  • Eleven years later, Shoenfield conjectured that all finite lattices were embeddable into the c.e. degrees in a way that preserved 0.

Manuel Lerman counter-conjectured that the lattice $S_8$ was not so embeddable; this was proved by Lachlan and Soare six years later.

The motivation behind both conjectures was the intuition that the c.e. degrees were a nicely behaved structure; in particular, I think it was believed that the poset of degrees c.e. in and above a given $\underline{d}$ should be isomorphic to the poset of c.e. degrees, that the theory of the c.e. degrees is decidable, that the poset is $\aleph_0$-categorical, etc., and all of these turned out to be false.

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Good sources on the history of these conjectures are Manuel Lerman's paper "The embedding problem for the Recursively Enumerable Degrees," and Richard Shore's "Conjectures and Questions from Gerald Sacks's Degrees of Unsolvability." –  Noah S May 4 '12 at 15:50

Polya conjecture was proved to be false in 1958.

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Borsuk's conjecture was believed to be true for 60 years till its counterexample was found in 1993 by Jeff Kahn and Gil Kalai.

They constructed an infinite family of counterexamples by using a result of Frankl and Wilson: http://www.ams.org/journals/bull/1993-29-01/S0273-0979-1993-00398-7/S0273-0979-1993-00398-7.pdf

Here's what Babai and Frankl say about the demise of Borsuk's conjecture in their manuscript "Linear Algebra Methods in Combinatorics":

Dead at the age of 60. Died after no apparent signs of illness, unexpectedly, of grave combinatorial causes. The news of the demise of Borsuk's venerable conjecture (1933) spread like brushfire among combinatorialists in Summer 1992. The disproof, found by Jeff Kahn (Rutgers) and Gil Kalai (Hebrew University), was the hot topic between lectures at conferences (the result came too late to be included on the regular programs). Countless copies of the manuscript traveled over electronic networks, silently crossing oceans and continents at lightening speed. The authors of this book found out about the result in more conventional ways. One of us heard it from Kahn himself while examining Gabi Bollobás's remarkable sculptures at the reception at a meeting in Cambridge, England. By then, in Tokyo, the other author had learned about it in a telephone conversation with a friend in New Jersey.

What Kahn communicated in a few minutes and without the benefit of paper or blackboard was not just the news of the result but also the complete proof. Remarkably, Borsuk's geometric conjecture was disproved in just a few lines, relying on the Frankl-Wilson Theorem (Theorem 7.15), a modular version of the RW theorem.

Also see this very recent survey article by Gil Kalai: http://arxiv.org/pdf/1505.04952v1.pdf

Hirsch Conjecture is another possible example. In 2010, Francisco Santos constructed a 43-dimensional polytope of 86 facets with a diameter of more than 43: http://annals.math.princeton.edu/2012/176-1/p07.

Also see this survey by Ziegler: http://www.math.uiuc.edu/documenta/vol-ismp/22_ziegler-guenter.pdf

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How about the Pythagorean tenet that all numbers are rational?

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Was this a conjecture or a philosophical position? –  Igor Rivin May 3 '12 at 15:37
My recollection from history of maths lectures is that the Greeks were well aware there were quantities/ magnitudes that could not be constructed as ratios of whole quantities/magnitudes. The issue is what they would have regarded as "number" –  Yemon Choi May 3 '12 at 19:31
A very nice article on this, by Errol Morris. opinionator.blogs.nytimes.com/2011/03/08/… –  Andres Caicedo Jun 29 '12 at 18:11
  • Here is one which I learnt from this answer here
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An example from set theory: My understanding is that it was once widely believed that all reals appearing in canonical inner models of large cardinals (at least up to supercompact cardinals) would be $\Delta^1_3$ in a countable ordinal. This is because it was assumed that linear iterations, the only kind known at the time, would suffice to compare such inner models. This assumption turned out to fail at the level of Woodin cardinals, far below supercompact cardinals. The resulting non-linear iterations (iteration trees) are a basic part of inner model theory today, whereas canonical inner models for supercompact cardinals are still far out of reach.

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In introductory functional analysis one learns that every normed linear space with a Schauder basis is separable. The converse of this was a famous question raised by Banach- does every separable Banach space have a Schauder basis? Since almost all known separable Banach spaces had been shown to possess a Schauder basis it was believed that this must be true. But in 1972 Enflo constructed a counterexample to this.

For this achievement of his Enflo was awarded a live goose by Stanislaw Mazur.

See this.

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Two examples from lattice theory: is every lattice with unique complements distributive? [no] is every distributive algebraic lattice isomorphic to the lattice of congruences of a lattice? [no] See http://www.ams.org/notices/200706/tx070600696p.pdf

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