Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

• Here is one which I learnt from this answer here

Answer 4

Polya conjecture was proved to be false in 1958.

Answer 5

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.")

Answer 6

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"

Answer 7

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

I think this was widely believed before.

Answer 8

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.

Answer 9

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

Answer 10

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.

Answer 11

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.

Answer 12

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

http://www.jstor.org/discover/10.2307/3062112?uid=3738296&uid=2129&uid=2&uid=70&uid=4&sid=56137438593

Answer 13

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.

Answer 14

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 $\geq 6$.

Answer 15

How about the Pythagorean tenet that all numbers are rational?

Answer 16

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

Answer 17

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.