Question

Many famous results were discovered through non-rigorous proofs, with correct proofs being found only later and with greater difficulty. One that is well known is Euler's 1737 proof that

$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots =\frac{\pi^2}{6}$

in which he pretends that the power series for $\frac{\sin\sqrt{x}}{\sqrt{x}}$ is an infinite polynomial and factorizes it from knowledge of its roots.

Another example, of a different type, is the Jordan curve theorem. In this case, the theorem seems obvious, and Jordan gets credit for realizing that it requires proof. However, the proof was harder than he thought, and the first rigorous proof was found some decades later than Jordan's attempt. Many of the basic theorems of topology are like this.

Then of course there is Ramanujan, who is in a class of his own when it comes to discovering theorems without proving them.

I'd be interested to see other examples, and in your thoughts on what the examples reveal about the connection between discovery and proof.

Clarification. When I posed the question I was hoping for some explanations for the gap between discovery and proof to emerge, without any hinting from me. Since this hasn't happened much yet, let me suggest some possible explanations that I had in mind:

Physical intuition. This lies behind results such as the Jordan curve theorem, Riemann mapping theorem, Fourier analysis.

Lack of foundations. This accounts for the late arrival of rigor in calculus, topology, and (?) algebraic geometry.

Complexity. Hard results cannot proved correctly the first time, only via a series of partially correct, or incomplete, proofs. Example: Fermat's last theorem.

I hope this gives a better idea of what I was looking for. Feel free to edit your answers if you have anything to add.

Answer 1

The Alternating Sign Matrix Conjecture in combinatorics was discovered (by researchers in the National Security Agency, so we don't know the motivation) in the late 1970s, but not proved for nearly 20 years. There is a wonderful book about it: Proofs and Confirmations, by David Bressoud.

Answer 2

Collatz Conjecture

The 3x+1 problem and its generalizations

The 3x+1 problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem, concerns the behavior of the iterates of the function which takes odd integers n to 3n+1 and even integers n to n/2. The 3x+1 Conjecture asserts that, starting from any positive integer n, repeated iteration of this function eventually produces the value 1.

The 3x+1 Conjecture is simple to state and apparently intractably hard to solve. It shares these properties with other iteration problems, for example that of aliquot sequences and with celebrated Diophantine equations such as Fermat's last theorem. Paul Erdos commented concerning the intractability of the 3x+1 problem: "Mathematics is not yet ready for such problems." Despite this doleful pronouncement, study of the 3x+1 problem has not been without reward. It has interesting connections with the Diophantine approximation of the binary logarithm of 3 and the distribution mod 1 of the sequence {(3/2)^k : k = 1, 2, ...}, with questions of ergodic theory on the 2-adic integers, and with computability theory - a generalization of the 3x+1 problem has been shown to be a computationally unsolvable problem. In this paper I describe the history of the 3x+1 problem and survey all the literature I am aware of about this problem and its generalizations.

Answer 3

In 1983 or 84, Frey announced that he could prove that Taniyama-Weil conjecture implies Fermat's last theorem. The proof was flawed but this announcement had spectacular consequences:

$\bullet$ Serre pulled out an unpublished conjecture of his and strengthened it so that Taniyama-Weil + $\varepsilon$ would imply FLT,

$\bullet$ Ribet proved enough of $\varepsilon$ so ensure that TW would imply FLT,

$\bullet$ Wiles realized that FLT would be proved as TW could not be ignored and so decided that it had to be by him (in doing so, he completely changed the way people thought about the field and this has led to impressive results including the proof of TW or of Sato-Tate conjecture),

$\bullet$ Shimura decided that he wanted his name attached to the conjecture and Lang made a campaign to remove Weil's...

Answer 4

According to M. Meo, Cauchy's proof of Cauchy's theorem (existence of elements of order a given prime p in every finite group of order divisible by a p) is wrong.

Cauchy works with subgroups of $S_n$, and his proof depends on the construction of what we now call a Sylow subgroup of $S_n$. This subgroup is obtained as a semidirect product, which Cauchy seems to say is actually a direct product (which would be abelian). I am not completely sure whether Cauchy was really wrong, or he did know what was going on, and simply lacked the appropriate language. In any case, would be an example of Lack of foundations.

Answer 5

The Kronecker-Weber theorem needed 3 proofs spanned upon 30 years before being completely proved (it states that all abelian extensions of ${\mathbb Q}$ can be found inside cyclotomic fields). It lead to class field theory.

Answer 6

Results in complexity theory such as $P \neq NP$.

Philosophically, it makes sense that there is a difference between verification and search, and no-one has discovered a counterexample.

(Note, that this result is not strictly speaking known to be correct. However, it is believed to be correct and routinely used as if it were simply true. No one, as far as I can tell, would ever begin a proof in complexity theory by assuming $P = NP$. )

Answer 7

Renormalizations in QFT

Renormalizations as discarding perturbative corrections to masses and charges were not easily accepted, even by their inventors, because of being obviously anti-mathematic. It remains to be a prescription, lucky in some rare cases and wrong in the others.

In Physics we use a perturbation theory where the perturbation is supposed to be small but it is "big" in QFT. First we write down a non perturbed Hamiltonian, let's say:

$\hat H_0 = -\frac{\hbar^2}{2m_e}\frac{d^2}{dx^2} + \hat{V}_0 (x)$ (1)

Everything in it is quite physical including the electron mass. Then we "develop" our theory and include, as we think, a small interaction that has also a kinetic and a potential term:

$\hat H_{int} = -\epsilon\frac{d^2}{dx^2} + \hat{V}_1 (x)$ (2)

The kinetic term shifts the particle mass, it is obvious. But our mass is already good in (1) and any its shifting worsens agreement with experiment. Discarding this correction "restores" the right kinetic part of the Hamiltonian, and taking $\hat{V}_1$ into account improves agreement with experiment. So the discarding practice became a part of QFT calculations.

Appearance of a kinetic perturbative term is due to our misunderstanding interactions. Some part of interactions cannot be treated perturbatively but should be present in the zeroth-order approximation. Discarding is a very bad practice. For (2) it may luckily work, but for other our guesses of interactions it can be more complicated and be just "non renormalizable".

Although shown on a simplest example, the renormalizations in QFT have nothing else in their meaning but repairing a wrongly guessed Hamiltonian via repairing the corresponding solutions. Normally it is difficult to see explicitly that some part of guessed interaction, namely a "self-action" term, is of a kinetic nature. That is why presently they "explain" renormalizations differently.

A correct theory development should not include kinetic perturbative terms. Then the perturbative series will be reasonable, in my opinion.

Answer 8

According to Weierstrass, Riemann knew about the existence of continuous nowhere differentiable functions. (Weierstrass' celebrated example was published in 1872, some 6 years after Riemann's death.) In his lectures, Riemann allegedly suggested the example $$f(x)=\sum\limits_{k=1}^{\infty}\frac{\sin k^2x}{k^2}$$ as early as 1861. He gave no proof and just mentioned that it could had been obtained from the theory of elliptic functions (see the historical note "Riemann’s example of a continuous “nondifferentiable” function continued" by S.L. Segal).

Hardy proved in 1916 that $f$ has no finite derivative at any $x=\pi\xi$ where $\xi$ is irrational but left the general case open.

It was only in 1970 that J. Gerver finally proved that the Riemann function is in fact differentiable when $$x=\pi\frac{2m+1}{2n+1},\qquad m,n\in\mathbb Z,$$ and $f'(x)=-1/2$ at these points ("The Differentiability of the Riemann Function at Certain Rational Multiples of π", ).

Answer 9

Looman (1923) proved that existence of partial derivatives of a function defined on an open subset of the complex plane is a sufficient condition for the function to be analytic. His proof had a gap that was fixed by Menchoff (1936) and we now have the Looman-Menchoff theorem.

Answer 10

Euler "proved" that $\sum \mu(n)/n = 0$ by observing that $\sum \mu(n) n^{-s} = 1/\zeta(s)$ and setting $s = 1$. Actually, the result $\sum \mu(n)/n = 0$ was later proved by von Mangoldt, and shown to be equivalent to the prime number theorem by Landau.

Answer 11

This identity is still not proven:

$$\sum_{n=0}^\infty \left(\frac{1}{(7n+1)^2}+\frac{1}{(7n+2)^2}-\frac{1}{(7n+3)^2}+\frac{1}{(7n+4)^2}-\frac{1}{(7n+5)^2}-\frac{1}{(7n+6)^2}\right)=\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2} \log \left| \frac{\tan t + \sqrt{7} }{\tan t - \sqrt{7} } \right| dt$$

It arose from physical applications.

Answer 12

Another classic example is the Littlewood-Richardson rule for decomposing products of Schur polynomials. It was discovered and proved in some special cases in 1934 by Littlewood and Richardson. In 1938 Richardson published a purported proof which had some gaps; however, apparently he managed to write so obscurely that the result was accept at least until the '50's. The first complete proofs were found in the '70's by Schützenberger and Thomas.

This is definitely an example in which the trouble arose from the difficulty of the result, which involves from pretty thorny combinatorics. In his paper "The representation theory of the symmetric groups", Gordon James said the following : "Unfortunately the Littlewood–Richardson rule is much harder to prove than was at first suspected. The author was once told that the Littlewood–Richardson rule helped to get men on the moon but was not proved until after they got there."

Remark : The above chronology is taken from wikipedia. I learned the Littlewood-Richardson rule from modern accounts, but I have to admit that I've never tried to go back and read the early papers on the subject.

Answer 13

Grunwald's Theorem on the existence of extensions satisfying local data was well known and widely used; Whaples even gave a second proof of this result before Wang found a counterexample and closed the gap. A similar mistake occurred when Shafarevich proved that solvable groups are Galois groups over the rationals - the case of 2-groups was "problematic".

On a more fundamental level, Kummer's proofs of unique factorization into prime ideal numbers had gaps because he did not know about the concept of integral closure. This gap was noticed and closed only by Dedekind.

Answer 14

When Stephen Smale was a graduate student, he thought he had a proof of the Poincaré Conjecture as follows: Take a compact simply-connected 3-manifold M and remove the interiors of two disjoint 3-balls to get, say, M₁ having as boundary two copies of S². It is easy to show that M₁ has a nonsingular vector field entering along one S² and exiting along the other. Clearly by the simply-connectedness of M, each orbit entering on one boundary component must exit on the other one. Thus M₁ must be S² x [0,1] and hence by replacing the removed 3-balls, M must have been S³. QED.

I'm not sure who first pointed out the error, but undoubtedly understanding examples like this helped him appreciate the subtlety of the problem and ultimately prove the Generalized Poincaré Conjecture for dimensions ≥ 5.

Answer 15

The "Yamabe problem": Every compact Riemannian manifold admits a conformally-related metric with constant scalar curvature. Yamabe thought he had proved this in 1960, but his proof had--I'm not making this up--a sign error. The error was discovered by Neil Trudinger in 1968, after Yamabe's death. As I understand it, Trudinger was working on a similar nonlinear elliptic PDE problem (with a critical Sobolev exponent) and got stuck, so he looked at Yamabe's paper to see how Yamabe had dealt with the same issue. Turned out he hadn't. Trudinger was able to give a partial solution to the problem; later Aubin expanded it to cover more cases, and finally in 1984 Rick Schoen was able to prove it the cases that Aubin had left open (with a small gap in the higher-dimensional case that was repaired by Schoen and Yau in 1988). The proof surprisingly used the positive mass theorem from general relativity.

Yamabe's original paper never attracted much attention until the error was found. But because of the subtlety of the methods required to fill in the gap, it has become a model for applications of nonlinear elliptic PDE to geometry, especially to conformally invariant problems and other problems with critical regularity.

Answer 16

I guess the historically first example is the Theorem of Pythagoras, already known to the Babylonians but probably not discovered by a "proof" satisfying modern standards.

Answer 17

The classification of finite simple groups was announced 1983 when Geoff Mason was still working on the quasithin case. I've heard somewhere that he lost his motivation then and never finished his 600+ pages manuscript. The gap was closed 20 years later by Michael Aschbacher and Steve Smith.

Answer 18

Just to complement Gerhard Paseman's answer. The story of how Girolamo Saccheri in early 1700's "almost" discovered hyperbolic geometry is quite amusing. Actually he died thinking he had proved the fifth postulate, but his argument was weak: "the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines". The sentence referes to his construction of a quadrilateral with two sides of equal length perpendicular to a given one. The acute angles are the ones opposite to the right ones. But Wikipedia explains this too...

In this example an ideological bias prevented the discovery of beautiful mathematics... I wonder if this still happens now a days, probably yes.

Answer 19

According to Atiyah (Responses to: A. Jaffe and F. Quinn, ``Theoretical mathematics: toward a cultural synthesis of mathematics and theoretical physics'' Bull. Amer. Math. Soc. (N.S.) 29(1993), no. 1, 1--13; MR1202292 (94h:00007)) Hodge's proofs on what is now called Hodge Theory (representation of deRham cohomology classes by harmonic forms) were incorrect, because Hodge was not an analyst, though the theory was correct.

Answer 20

There are at least two Hilbert problems that were considered to be solved, but the proofs turned out to be incomplete, as pointed out by Yulii Ilyashenko.

1. In 1923 Dulac published a 140+ page memoir purporting to show that a polynomial vector field on the plane has only finitely many limit cycles, the second part of the 16th Hilbert problem. The memoir was difficult to read, but the claim was generally accepted until in 1981 Ilyashenko found a serious gap. Full proofs were obtained independently by Écalle and Ilyashenko around 1991. Read the full story.

2. Existence of linear differential equations having a prescribed monodromic group was the subject of the 21st Hilbert problem, also known as the Riemann-Hilbert problem. From Wikipedia article:

Josip Plemelj published a solution in 1908. This work was for a long time accepted as a definitive solution; there was work of G. D. Birkhoff in 1913 also, but the whole area, including work of Ludwig Schlesinger on isomonodromic deformations that would much later be revived in connection with soliton theory, went out of fashion. Plemelj produced a 1964 monograph Problems in the Sense of Riemann and Klein, (Pure and Applied Mathematics, no. 16, Interscience Publishers, New York) summing up his work. A few years later the Soviet mathematician Yuliy S. Il'yashenko and others started raising doubts about Plemelj's work. In fact, Plemelj correctly proves that any monodromy group can be realised by a regular linear system which is Fuchsian at all but one of the singular points. Plemelj's claim that the system can be made Fuchsian at the last point as well is wrong. (Il'yashenko has shown that if one of the monodromy operators is diagonalizable, then Plemelj's claim is true.)

Indeed in 1989 Soviet mathematician Andrey A. Bolibrukh (1950–2003) found a counterexample to Plemelj's statement. This is commonly viewed as providing a counterexample to the precise question Hilbert had in mind; Bolibrukh showed that for a given pole configuration certain monodromy groups can be realised by regular, but not by Fuchsian systems.

Answer 21

In 1905, Lebesgue gave a "proof" of the following theorem:

If $f:\mathbb{R}^2\to\mathbb{R}$ is a Baire function such that for every x, there is a unique y such that f(x,y)=0, then the thus implicitly defined function is Baire.

He made use of the "trivial fact" that the projection of a Borel set is a Borel set. This turns out to be wrong, but the result is still true. Souslin spotted the mistake, and named continuous images of Borel sets analytic sets. So a mistake of Lebesgue led to the rich theory of analytic sets. Lebesgue seemingly enjoyed this fact and mentioned it in the foreword to a book of Souslins's teacher Lusin.

Answer 22

The Nielsen realization problem. Let $S$ be a compact oriented topological surface and let $\text{Mod}(S)$ be its mapping class group, ie the group of orientation preserving diffeomorphisms of $S$ modulo isotopy. There is a natural surjection $\text{Diff}^+(S) \rightarrow \text{Mod}(S)$. The Nielsen realization problem was the conjecture (due to Jacob Nielsen) that every finite subgroup of $\text{Mod}(S)$ can be lifted to a finite subgroup of $\text{Diff}^+(S)$ (and thus is a subgroup of the group of automorphisms of a Riemann surface).

Nielsen proved this for finite cyclic subgroups (this is very nontrivial!), and a number of other people slowly chipped away at other classes of finite subgroups. In 1959, Kravetz published a paper which purported to prove that Teichmuller space is negatively curved. A "center of mass" argument would then establish that every finite subgroup of $\text{Mod}(S)$ fixes a point in Teichmuller space, and it then follows easily that the finite subgroup can be lifted to $\text{Diff}^+(S)$.

This was an important result, and Kravetz's paper was frequently quoted. However, in 1971 Linch pointed out in his thesis that Kravetz's paper had an error! In fact, in his 1974 thesis Howie Masur proved that Teichmuller space is not negatively curved (in a pretty strong sense).

Finally, in 1980 Steve Kerckhoff proved that Teichmuller space, while not negatively curved, did satisfy a subtle negative-curvature like property which gave the desired result.

Answer 23

Dehn's lemma was given an incorrect proof by Dehn in 1910; only in 1956 was a true proof found

Answer 24

How about mirror symmetry of Calabi-Yaus? This started from the observation by physicists that string theory on certain pairs of Calabi-Yaus gave identical theories. This has lead to a lot of work by physicists and mathematicians to understand what's going on, leading to things like the SYZ conjecture, homological mirror symmetry, etc.

So, more specifically physicists theories treat spacetime $M$ as something that locally looks like $M=\mathbb{R}^4\times X$ in such a way that $X$ is "small" by saying (roughly) operators (which represent observables) when "looking at things" below a certain energy scale can't see directly the dynamics associated with $X$. Associated with $M$ is a special kind of quantum field theory called a superconformal field theory (SCFT), which requires that $X$ be a Calabi-Yau 3-fold.

Various topological invariants of $X$ can tell us about how the SCFT behaves.

But it was discovered that the associated SCFTs don't uniquely determine $X$. It turns out there are pairs of Calabi-Yau 3-folds $(X,\hat{X})$ (called mirrors) that give the same SCFT.

From the SCFT point of view, these two mirror manifolds are related by an automorphism of the SCFT, which does not correspond to an automorphism of the Calabi-Yau manifold, but instead gives a mirror manifold in a way that switches cohomology groups around. It can also be thought of as switching complex structures with symplectic ones somehow.

From the rigorous point of view, though, not much of this is well-defined. It relies on the machinery of QFTs which no one has been able to come close to defining axiomatically, as well as string theory which relies on a lot of machinery that has the same kinds of problems.

Out of this came a number of more mathematically precise conjectures, such as the SYZ conjecture, which explains this in terms of special Lagrangian manifolds and fibrations of the mirror manifolds into it.

This also started ideas of homological mirror symmetry, which tries to describe this in terms of homology and derived categories.

Answer 25

In a sense, the entire field of ergodic theory was born from Boltzmann's incomplete proof of the H-theorem.

Answer 26

The four-color theorem.

Answer 27

Fourier analysis.

Answer 28

Calculus before Weierstrass.

Answer 29

There are two famous examples from enumerative algebraic geometry. The Schubert calculus was used by Schubert to solve many elaborate enumerative problems, but it was only fairly recently that these results were verified according to modern standards of mathematical rigor. Also, string theory predicted some enumerative results that the mathematicians were only later able to verify.

More generally, modern theoretical physics has produced enormous numbers of mathematical results whose derivations are non-rigorous. Some of these have been rigorously verified but some remain open problems.

Answer 30

Renormalization.