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?

11$\begingroup$ This is a misconception, IMO. $\endgroup$ – user9072 May 3 '12 at 15:00

2$\begingroup$ Obviously the relative weakness of the examples below (after two months) show that the PO's impression was not a misconception. Except perhaps the example of Hilbert's program, non of the example given above strikes me as a real "widely believed conjecture", as say the Riemann Hypothesis, the Birch and SwinnertonDyer, the Serre's conjecture or FontaineMazur in the theory of modular forms, the Poincare's conjecture, etc. $\endgroup$ – Joël Jun 29 '12 at 21:48

2$\begingroup$ @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: $\endgroup$ – user9072 Jul 2 '12 at 13:47

3$\begingroup$ 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. $\endgroup$ – user9072 Jul 2 '12 at 14:10

4$\begingroup$ Let me reformulate my critic to most answers of this question. An answer should not only point to a conjecture now proved false, but also provide evidence that it used to be "widely believed". $\endgroup$ – Joël Jul 3 '12 at 23:44
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 $\geq 6$.

3$\begingroup$ Some TOPOLOGY GUY?! Oh, mine! How could you not know Milnor. The Milnor! $\endgroup$ – Behnam Esmayli Apr 4 '18 at 15:15

23$\begingroup$ @BehnamEsmayli: Not your kind of humour? I liked that joke. $\endgroup$ – j.p. Apr 4 '18 at 17:29
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 longstanding 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 counterexample. 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 counterexample 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 counterexamples work and the interesting thing that happened was that I realised why there should be a counterexample 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" counterexample 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"

2$\begingroup$ Thanks for the nice answer. Another source for this is this interview with Carleson (Notices AMS, Feb 2007) ams.org/notices/200702/commcarleson.pdf $\endgroup$ – user9072 Jul 3 '12 at 10:41
This can perhaps be considered more of a metaconjecture 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.

$\begingroup$ Your impression is perfectly correct. It seemed attainable at least by Hilbert and his school. $\endgroup$ – Joël Jun 29 '12 at 21:42
Littlewood's disproof of the conjecture (maybe of Gauss) that $\text{li}(x) > \pi(x)$.
I think this was widely believed before.
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.

4$\begingroup$ By our own @Noam Elkies in the latter case. $\endgroup$ – Igor Rivin May 3 '12 at 15:36
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.")
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/19932901/S027309791993003987/S027309791993003987.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 FranklWilson 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 43dimensional polytope of 86 facets with a diameter of more than 43: http://annals.math.princeton.edu/2012/1761/p07.
Also see this survey by Ziegler: http://www.math.uiuc.edu/documenta/volismp/22_zieglerguenter.pdf
In complex analysis of one variable, Liouville's theorem says that a bounded entire function is constant. Bernstein (191517) 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 counterexample for $n=8$, which yields a counterexample 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 nonlinear analysis.
Two widely believed conjectures:
 The proportion of hyperbolic knots amongst prime knots of $n$ or fewer crossings approaches $1$ as $n$ approaches infinity.
 The crossing number (the minimal number of crossings of a knot diagram of that knot) of a composite knot is not less than that of each of its factors.
The first conjecture is widely believed because of massive numerical evidence, and because many topological objects that are related to knots are generically hyperbolic e.g. compact surfaces, various classes of 3manifolds, closures of random braids...
The second conjecture is also widely believed to be true because of lots of numerical evidence it is moreover believed that crossing number should in fact be additive with respect to connect sum (so the crossing number of a composite knot should actually be the sum of the crossing number of its components). It was proved for various classes of knots alternating knots, adequate knots, torus knots, etc.
... and yet ...
Malyutin shows that either Conjecture 1 or Conjecture 2 must be false!!
Update: Malyutin has proven that Conjecture 1 is false. Thus this answer becomes honest, and Conjecture 1 is a conjecture that was widely believed to be true but (much) later shown to be false. arxiv.org/abs/1907.04458

2$\begingroup$ It may be worth pointing out that there are reasons not to believe in Conjecture 1, see e.g. comments on a blog post of yours ldtopology.wordpress.com/2018/04/04/… $\endgroup$ – j.c. Jul 27 '18 at 13:56

2$\begingroup$ Recently Malyutin proved Conjecture 1 is false in arxiv.org/abs/1907.04458 $\endgroup$ – this_is_a_banana Aug 15 '19 at 1:54

1

$\begingroup$ Great! I knew about the two conjectures and also toyed around with those problems when I was introduced to knot theory (and to mathematical proof in general) by Colin Adams' book. But I learned only now about the disproof. $\endgroup$ – Hermann Gruber Aug 16 '19 at 19:21
Euler's conjecture about the nonexistence of $n\times n$ GraecoLatin squares for $n=4k+2$. Disproved for all $k>1$ by the so called Euler's Spoilers Bose, Shrikhande, and Parker.
The solution in negative of the isomorphism problem for integral group rings. A counterexample was found by Martin Hertweck:
Mersenne's conjecture on primes is a famous example (although I am not sure how widely it was believed to be true).
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 counterconjectured 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.

3$\begingroup$ 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." $\endgroup$ – Noah Schweber May 4 '12 at 15:50
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.
Lusztig's conjecture in modular representation theory, which would describe the characters of simple $G_1$modules for $p$ larger than the Coxeter number, was generally believed to be true for a long time. Here $G$ is a reductive algebraic group in characteristic $p$ and $G_1$ is its Frobenius kernel. The conjecture in turn leads to character formulae (or at least, algorithms for calculating characters) for simple representations of $G$.
They were proven for large $p$ by AndersenJantzenSoergel, with later contributions by others, including an explicit bound by Fiebig. But recently Geordie Williamson proved that the original condition on $p$ is not enough.
See What to do now that Lusztig's and James' conjectures have been shown to be false? for more details.
The dynamical degree of a dominant rational map $f:\mathbb{P}^N\to\mathbb{P}^N$ is defined by the limit $$ \delta(f) := \lim_{n\to\infty} \deg(f^{\circ n})^{1/n}, $$ where $f^{\circ n}$ is the $n$th iterate of $f$. It was conjectured by Bellon and Vialet [1] that $\delta(f)$ is always an algebraic integer, and over the succeeding two decades, this was proven for many classes of maps. But Bell, Diller, and Jonsson [2] recently gave an example of a map on $\mathbb{P}^2$ whose dynamical degree is a transcendental number.
[1] Algebraic entropy, Bellon, M. P. and Viallet, C.M., Comm. Math. Phys. 204 (1999), 425437.
[2] A transcendental dynamical degree, Jason P. Bell, Jeffrey Diller, Mattias Jonsson, https://arxiv.org/abs/1907.00675


$\begingroup$ @SamHopkins The example is indeed very simple to write down. The magic, aka the knowledge and skill of the authors, is in (1) knowing to look at that example and (2) actually proving $\delta(f)$ is transcendental. $\endgroup$ – Joe Silverman Aug 16 '19 at 20:39
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 nonlinear 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.
Fermat's conjecture that all numbers of the form $ F_{n} : =2^{2^{n}}+1 $ are prime. Euler proved that $ 641\mid F_{5} $ .
The MarkusYamabe Conjecture in differential equations was posed in 1960. It states that if $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ is a $C^1$ map such that $f(0)=0$ and the eigenvalues of the Jacobian matrix $Df(x)$ have negative real part for every $x\in\mathbb{R}^n$, then $x=0$ is globally attractive. In the early 1990s, proofs for the $n=2$ case were given, but in 1996 a complicated counterexample in $n=4$ was constructed, and in 1997 a simple polynomial counterexample for $n\geq 3$ was produced.
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
My impression is that Stanley's partionability conjecture and his depth conjecture (which was shown to imply the partionability conjecture in 2008) were both believed to be true, until Duval, Goeckner, Klivans, and Martin found a counterexample to the partionability conjecture in 2015. See this AMS survey article: https://www.ams.org/journals/notices/201702/rnotip117.pdf. This fits with a few other examples already mentioned (the Hauptvermutung, the Hirsch conjecture, ...) which warn us that although simplicial complexes/polytopes may appear to be intuitively simple objects, they can in fact be extremely complicated.
Let $k \geq 3$ be fixed. Ramsey's theorem says that if $n$ is sufficiently large and we color the edges of the complete graph $K_n$ red or blue, there must be at least one monochromatic $K_k$. As it turns out, it's not just "at least one" but many: An averaging argument shows that as $n \rightarrow \infty$ a positive fraction of all the $\binom{n}{k}$ copies of $K_k$ in our coloring must be monochromatic.
This leads to a natural followup question: How few copies can we get? If we consider all $2$colorings of $K_n$, which one (asymptotically) minimizes the number of monochromatic copies of $K_k$?
This was first studied for the case $k=3$ (monochromatic triangles) by Goodman, who in 1959 gave an explicit answer asymptotic to $\frac{1}{4} \binom{n}{3}$. The fraction $\frac{1}{4}$ has a natural interpretation here  if we color randomly, this is the expected fraction of monochromatic triangles. Three years later, Erdős observed that the random coloring gives an upper bound of $2^{\binom{k}{2}+1}$ on the minimum fraction of monochromatic $K_k$, and said it "seems likely" this was asymptotically optimal.
By 1980 Burr and Rosta conjectured that something even stronger was true: For any fixed graph $H$ the asymptotic way to minimize monochromatic copies of $H$ was just to color randomly. It wasn't until 1989 that Sidorenko gave a counterexample to the BurrRosta conjecture (a triangle with a pendant edge) and Thomason disproved Erdős's original conjecture by giving a coloring with significantly fewer monochromatic $K_4$ then random.
It is still an open question to determine the optimal coloring to minimize monochromatic $K_4$, and also still an open question to determine for which graphs the BurrRosta conjecture is true (such graphs are termed "common" in the literature).
 Here is one which I learnt from this answer here, on the asymptotic behavior of $\log (n)  \frac{n}{\pi(n)}$ (Legendre conjectured it tended to something other than the correct limit $1$).

5$\begingroup$ This answer refers to the expected volume of a tetrahedron with vertices chosen randomly in a unit volume tetrahedron, widely believed to be rational, until it turned out that it was in fact irrational. $\endgroup$ – Daniel Moskovich Apr 4 '18 at 13:51

$\begingroup$ When I clicked on the link, the answer I saw had nothing to do with tetrahedra, but with the prime number theorem. Not sure what is going on. $\endgroup$ – Todd Trimble♦ Aug 17 '19 at 11:55

$\begingroup$ @ToddTrimble Daniel is referring to the 2nd most voted answer to the linked question, while the link goes to the most voted answer. $\endgroup$ – Will Sawin Aug 17 '19 at 12:06
How about the Pythagorean tenet that all numbers are rational?

12$\begingroup$ Was this a conjecture or a philosophical position? $\endgroup$ – Igor Rivin May 3 '12 at 15:37

2$\begingroup$ 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" $\endgroup$ – Yemon Choi May 3 '12 at 19:31

$\begingroup$ A very nice article on this, by Errol Morris. opinionator.blogs.nytimes.com/2011/03/08/… $\endgroup$ – Andrés E. Caicedo Jun 29 '12 at 18:11