# Widely accepted mathematical results that were later shown to be wrong?

Are there any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time later, possibly even after being used in proofs of other results?

(I realise it's a bit vague, but if there is significant doubt in the mathematical community then the alleged proof probably doesn't qualify. What I'm interested in is whether the human race as a whole is known to have ever made serious mathematical blunders.)

• I have a déjà vu :) Also, maybe community wiki (~big list)? – M.G. Aug 13 '10 at 10:41
• @ex Given that I have no other way to earn reputation than by asking questions (my math is a mere long-forgotten-university-level), I'd like at least one extra upvote before marking this CW so I can at least upvote some answers :) – Roman Starkov Aug 13 '10 at 10:44
• mathoverflow.net/questions/27749/… is a similar question - only the subject of the proof was later confirmed to be true. – Roman Starkov Aug 13 '10 at 12:16
• The story around the Grunwald-Wang theorem takes the cake on this one, especially Tate's commentary on his reaction to it as a graduate student (but one also has to keep in mind that in those days and earlier, the number of active research mathematicians was a tiny fraction of the number today). See section 5.3 of rzuser.uni-heidelberg.de/~ci3/brhano.pdf – BCnrd Aug 13 '10 at 14:35
• en.wikipedia.org/wiki/Dehn%27s_lemma – Richard Borcherds Aug 15 '10 at 20:35

The Busemann-Petty problem (posed in 1956) has an interesting history. It asks the following question: if $K$ and $L$ are two origin-symmetric convex bodies in $\mathbb{R}^n$ such that the volume of each central hyperplane section of $K$ is less than the volume of the corresponding section of $L$: $$\operatorname{Vol}_{n-1}(K\cap \xi^\perp)\le \operatorname{Vol}_{n-1}(L\cap \xi^\perp)\qquad\text{for all } \xi\in S^{n-1},$$ does it follow that the volume of $K$ is less than the volume of $L$: $\operatorname{Vol}_n(K)\le \operatorname{Vol}_n(L)?$

Many mathematician's gut reaction to the question is that the answer must be yes and Minkowski's uniqueness theorem provides some mathematical justification for such a belief---Minkwoski's uniqueness theorem implies that an origin-symmetric star body in $\mathbb{R}^n$ is completely determined by the volumes of its central hyperplane sections, so these volumes of central hyperplane sections do contain a vast amount of information about the bodies. It was widely believed that the answer to the Busemann-Problem must be true, even though it was still a largely unopened conjecture.

Nevertheless, in 1975 everyone was caught off-guard when Larman and Rogers produced a counter-example showing that the assertion is false in $n \ge 12$ dimensions. Their counter-example was quite complicated, but in 1986, Keith Ball proved that the maximum hyperplane section of the unit cube is $\sqrt{2}$ regardless of the dimension, and a consequence of this is that the centered unit cube and a centered ball of suitable radius provide a counter-example when $n \ge 10$. Some time later Giannopoulos and Bourgain (independently) gave counter-examples for $n\ge 7$, and then Papadimitrakis and Gardner (independently) gave counter-examples for $n=5,6$.

By 1992 only the three and four dimensional cases of the Busemann-Petty problem remained unsolved, since the problem is trivially true in two dimensions and by that point counter-examples had been found for all $n\ge 5$. Around this time theory had been developed connecting the problem with the notion of an "intersection body". Lutwak proved that if the body with smaller sections is an intersection body then the conclusion of the Busemann-Petty problem follows. Later work by Grinberg, Rivin, Gardner, and Zhang strengthened the connection and established that the Busemann-Petty problem has an affirmative answer in $\mathbb{R}^n$ iff every origin-symmetric convex body in $\mathbb{R}^n$ is an intersection body. But the question of whether a body is an intersection body is closely related to the positivity of the inverse spherical Radon transform. In 1994, Richard Gardner used geometric methods to invert the spherical Radon transform in three dimensions in such a way to prove that the problem has an affirmative answer in three dimensions (which was surprising since all of the results up to that point had been negative). Then in 1994, Gaoyong Zhang published a paper (in the Annals of Mathematics) which claimed to prove that the unit cube in $\mathbb{R}^4$ is not an intersection body and as a consequence that the problem has a negative answer in $n=4$.

For three years everyone believed the problem had been solved, but in 1997 Alexander Koldobsky (who was working on completely different problems) provided a new Fourier analytic approach to convex bodies and in particular established a very convenient Fourier analytic characterization of intersection bodies. Using his new characterization he showed that the unit cube in $\mathbb{R}^4$ is an intersection body, contradicting Zhang's earlier claim. It turned out that Zhang's paper was incorrect and this re-opened the Busemann-Petty problem again.

After learning that Koldobsky's results contradicted his claims, Zhang quickly proved that in fact every origin-symmetric convex body in $\mathbb{R}^4$ is an intersection body and hence that the Busemann-Petty problem has an affirmative answer in $\mathbb{R}^4$---the opposite of what he had previously claimed. This later paper was also published in the Annals, and so Zhang may be perhaps the only person to have published in such a prestigious journal both that $P$ and that $\neg P$!

• This is an amazing story. – KConrad Oct 25 '10 at 14:59
• Agreed. I looked just now on MathSciNet and (i) the review of Zhang's 1994 Annals paper gives no indication that later work of the author attained the opposite result (although Zhang's 1999 paper is one of two Citations From Reviews) and (ii) as far as I could see, there is no erratum to the 1994 paper other than the 1999 paper. I find this most curious, to put it mildly. – Pete L. Clark Mar 27 '11 at 2:45
• The review of the 1994 paper was modified in August 2016. The new version of the review gives a link to the review of the 1999 paper, "for further information pertaining to this review". 1994: MR1298716. 1999: MR1689339 – Goldstern Dec 3 '16 at 17:47

Mathematicians used to hold plenty of false, but intuitively reasonable, ideas in analysis that were backed up with proofs of one kind or another (understood in the context of those times). Coming to terms with the counterexamples led to important new ideas in analysis.

1. A convergent infinite series of continuous functions is continuous. Cauchy gave a proof of this (1821). See Theorem 1 in Cours D'Analyse Chap. VI Section 1. Five years later Abel pointed out that certain Fourier series are counterexamples. A consequence is that the concept of uniform convergence was isolated and, going back to Cauchy's proof, it was seen that he had really proved a uniformly convergent series of continuous functions is continuous. For a nice discussion of this as an educational tool, see "Cauchy's Famous Wrong Proof" by V. Fred Rickey. [Edit: This may not be historically fair to Cauchy. See Graviton's answer for another assessment of Cauchy's work, which operated with continuity using infinitesimals in such a way that Abel's counterexample was not a counterexample to Cauchy's theorem.]

2. Lagrange, in the late 18th century, believed any function could be expanded into a power series except at some isolated points and wrote an entire book on analysis based on this assumption. (This was a time when there wasn't a modern definition of function; it was just a "formula".) His goal was to develop analysis without using infinitesmals or limits. This approach to analysis was influential for quite a few years. See Section 4.7 of Jahnke's "A History of Analysis". Work in the 19th century, e.g., Dirichlet's better definition of function, blew the whole work of Lagrange apart, although in a reverse historical sense Lagrange was saved since the title of his book is "Theory of Analytic Functions..."

3. Any continuous function (on a real interval, with real values) is differentiable except at some isolated points. Ampere gave a proof (1806) and the claim was repeated in lots of 19th century calculus books. See pp. 43--44, esp. footnote 11 on page 44, of Hawkins's book "Lebesgue's theory of integration: its origins and development". Here is a Google Books link. In 1872 Weierstrass killed the whole idea with his continuous nowhere differentiable function, which was one of the first fractal curves in mathematics. For a survey of different constructions of such functions, see "Continuous Nowhere Differentiable Functions" by Johan Thim.

4. A solution to an elliptic PDE with a given boundary condition could be solved by minimizing an associated "energy" functional which is always nonnegative. It could be shown that if the associated functional achieved a minimum at some function, then that function was a solution to a certain PDE, and the minimizer was believed to exist for the false reason that any set of nonnegative numbers has an infimum. Dirichlet gave an electrostatic argument to justify this method, and Riemann accepted it and made significant use of it in his development of complex analysis (e.g., proof of Riemann mapping theorem). Weierstrass presented a counterexample to the Dirichlet principle in 1870: a certain energy functional could have infimum 0 with there being no function in the function space under study at which the functional is 0. This led to decades of uncertainty about whether results in complex analysis or PDEs obtained from Dirichlet's principle were valid. In 1900 Hilbert finally justified Dirichlet's principle as a valid method in the calculus of variations, and the wider classes of function spaces in which Dirichlet's principle would be valid eventually led to Sobolev spaces. A book on this whole story is A. F. Monna, "Dirichlet's principle: A mathematical comedy of errors and its influence on the development of analysis" (1975), which is not reviewed on MathSciNet.

• In the general category of your point four, there's a whole panoply of descent arguments for the isoperimetric inequality going back thousand of years. I believe Weierstrass was the first to show that the infimum is attained, by a compactness argument in the space of shapes. – Per Vognsen Aug 14 '10 at 4:59
• It is slightly confusing to say that "Weierstrass presented a counterexample to the Dirichlet principle" and then to say that "Hilbert gave a correct proof of Dirichlet's principle" - do you mean that Hilbert also reformulated it correctly? – Qiaochu Yuan Aug 14 '10 at 8:07
• Qiaochu: I reworded that part in (4). – KConrad Aug 14 '10 at 13:48
• Thank you for giving a detailed popular explanation for all four of these. I have heared about the uniform convergence problem, but not about the rest. – Zsbán Ambrus May 31 '13 at 17:59

The (in)famous Jacobian Conjecture was considered a theorem since a 1939 publication by Keller (who claimed to prove it). Then Shafarevich found a new proof and published it in some conference proceedings paper (in early 1950-ies). This conjecture states that any polynomial map from C^2 to C^2 is invertible if its Jacobian is nowhere zero. In 1960-ies, Vitushkin found a counterexample to all the proofs known to date, by constructing a complex analytic map, not invertible and with nowhere vanishing Jacobian. It is still a main source of embarrassment for arxiv.org contributors, who publish about 3-5 false proofs yearly. Here is a funny refutation for one of the proofs: http://arxiv.org/abs/math/0604049

"The problem of Jacobian Conjecture is very hard. Perhaps it will take human being another 100 years to solve it. Your attempt is noble, Maybe the Gods of Olympus will smile on you one day. Do not be too disappointed. B. Sagre has the honor of publishing three wrong proofs and C. Chevalley mistakes a wrong proof for a correct one in the 1950's in his Math Review comments, and I.R. Shafarevich uses Jacobian Conjecture (to him it is a theorem) as a fact..."

• Great example! Just to emphasize, a few more highly respectable mathematicians at various times advanced what they thought was a proof of the Jacobian conjecture. – Victor Protsak Aug 18 '10 at 20:04
• One more thing: while "B. Sagre" is the name written in the arxiv paper, that name doesn't sound right (it's certainly not on the footing of Chevalley or Shafarevich). But change it to B. Segre and then it makes more sense. I confirmed it is Segre from Section 3 of ams.org/journals/bull/1982-07-02/S0273-0979-1982-15032-7/… (Bass, Connell, Wright, "The Jacobian Conjecture: Reduction of Degree and Formal Expansion of the Inverse", Bull. AMS 7 (1982), 287--330). – KConrad Aug 18 '10 at 21:03
• The end paragraph of that paper is just lovely. What a nice way to say that someone was wrong, but that they should keep trying. – Gunnar Þór Magnússon Oct 24 '10 at 20:46
• It should be remarked that Yitang Zhang, the mathematician who first proved that there exist infinitely many bounded gaps between primes, produced an incorrect proof of the Jacobian conjecture as his PHD thesis. – Stanley Yao Xiao Jan 5 '14 at 8:36
• @StanleyYaoXiao I dont think this is correct. Zhangs's thesis can be found here. As far as I can see he doesnt claim to prove the Jacobian conjecture. – Kasper Andersen Jan 30 '15 at 18:30

The Euler Characteristic V-E+F has an interesting history. It was initially stated that, for all polyhedra,

$$V(ertices)-E(dges)+F(aces)=2$$

and its proof was widely accepted, until people found counter-examples.

Imre Lakatos' book Proofs and Refutations has an imagined dialogue between teacher and student giving arguments and counter-examples leading to the correct formulation, which, he explains in his footnotes, traces the actual historical development of the statement and proof of the theorem.

• I never suspected how subtle the issue could be until I picked up Lakatos's book. Definitely a must-read. – Thierry Zell Aug 15 '10 at 14:49
• @Kjetil: (also unknown): This is the central example in the book. It starts right there in the beginning with "A problem and a conjecture" and continues for pages and pages. I suspect you are thinking of a different book if you cannot see it. – ex0du5 May 31 '13 at 20:01
• I cannot believe this, just take a point. – Fernando Muro Oct 5 '13 at 20:16
• @FM: it was meant for polyhedra, apparently missing the convexity assumption. (i.e. that the polyhedron is homeomorphic to the sphere) – ThiKu Nov 2 '14 at 17:38
• @FernandoMuro: The issue (as extensively explored in Lakatos’s book) is that this formula was known “for all polyhedra” before a precise (by modern standards) definition of polyhedron had been established. So the “obvious counterexamples” were not seen as counterexamples, because they obviously weren’t polyhedra. However, when people did start exploring definitions for polyhedron, then (for some of those) this expected result became false. – Peter LeFanu Lumsdaine Feb 7 '16 at 9:42

This question reminded me of the following article of A. Neeman with an appendix by P. Deligne:

Neeman, Amnon, A counterexample to a 1961 “theorem” in homological algebra, Invent. Math. 148, No. 2, 397-420 (2002). ZBL1025.18007.

• 1961-2002, impressive! Excerpt from the abstract:  In 1961, Jan-Erik Roos published a “theorem”, which says that in an abelian category, $\operatorname{lim}^1$ vanishes on Mittag–Leffler sequences. [...] This is a “theorem” that many people since have known and used. In this article, we outline a counterexample. [...] The idea is to make the counterexample easy to read for all the people who have used the result in their work. – Victor Protsak Aug 18 '10 at 20:10
• Bravo for Amnon (who was a classmate in graduate school). – Deane Yang Oct 24 '10 at 23:35
• I think it should be noted that the result is true in general, that is that most abelian categories satisfy enough extra structure that $lim^1$ vanishes on mittag-leffler sequences. I will double check some things and update this comment tomorrow (with the appropriate institutional access). – Sean Tilson Oct 25 '10 at 5:21
• The sense of "tomorrow" used by Sean is actually less clear ;). – Lennart Meier Oct 22 '13 at 20:01
• Roos has rectified things in jlms.oxfordjournals.org/content/73/1/65. A sufficient extra structure as hinted to by Sean Tilson is: AB3, AB4* and having a set of generators. – Torsten Schoeneberg Dec 7 '13 at 15:19

In 1882 Kronecker proved that every algebraic subset in $\mathbb P^n$ can be cut out by $n+1$ polynomial equations.

In 1891 Vahlen asserted that the result was best possible by exhibiting a curve in $\mathbb P^3$ which he claimed was not the zero locus of 3 equations. It is only 50 years later, in 1941, that Perron gave 3 equations defining Vahlen's curve, thus refuting Vahlen's claim which had been accepted for half a century.

Finally, in 1973 Eisenbud and Evans proved that $n$ equations always suffice to describe (set-theoretically) any algebraic subset of $\mathbb P^n$

Hilbert's 21st problem, on the existence of linear DEs with prescribed monodromy group, was for a long time thought to have been solved by Plemelj in 1908. In fact, Plemelj died in 1967 still believing he had solved the problem.

However, in 1989, Bolibruch discovered a counterexample. Details are in the book The Riemann-Hilbert Problem by Anosov and Bolibruch (Vieweg-Teubner 1994), and a nice popular recounting of the story is in Ben Yandell's The Honors Class (A K Peters 2002).

• – Victor Protsak Aug 14 '10 at 8:10
• See also Beauville, Monodromie des systèmes différentiels linéaires à pôles simples sur la sphère de Riemann, Séminaire Bourbaki, 35 (1992-1993), Exposé No. 765, available at  numdam.org/item?id=SB_1992-1993__35__103_0  – Chandan Singh Dalawat Mar 27 '11 at 6:23
• Plemelj (and every careful reader of Plemelj) knew that he solved the problem in "general position" only. Who claimed before Bolibruch that the problem is solved completely? – Alexandre Eremenko Dec 22 '17 at 23:29

In 1959 Kravetz published a proof that the Teichmuller metric on Teichmuller space is negatively curved in the sense of Buseman. This was widely quoted and used until Linch found a gap in 1971.

In 1974, Howard Masur showed that the Teichmulller metric is not negatively curved, by exhibiting two distinct geodesic rays which have a common starting point but stay a bounded distance apart. There is now a whole subfield studying Teichmuller geometry, which grew out of the failure of Kravetz's theorem.

In 1803, Gian Francesco Malfatti proposed a solution to the problem of how to cut out three circular columns of marble of maximal area from a triangular piece of stone. Malfatti's solution was three circles that are tangent to each other and to the sides of the triangle (known as Malfatti circles). His solution was believed to be correct until 1930, when it was shown that Malfatti circles are not always the best solution. Then, in 1967, Goldberg showed that Malfatti circles are never the optimal solution. Finally, in 1992, Zalgaller and Los' gave a complete solution to the problem.

http://en.wikipedia.org/wiki/Malfatti_circles

• In fact, a common way to see that Malfatti's solution isn't always right is to consider the limiting case, an isosceles triangle with a fixed base and side angles approaching the right angle, so that the triangle becomes a strip of a fixed width. So the real question is, why did it take so long? – Victor Protsak Aug 21 '10 at 2:29
• Note the article, Marco Andreatta, Andras Bezdek, Jan P Boronski, The problem of Malfatti: two centuries of debate, to appear in the Mathematical Intelligencer (published online 13 July 2010). – Gerry Myerson Sep 13 '10 at 7:20

One part of Hilbert's 16th problem is to determine whether a polynomial vector field in $\mathbb R^2$, $$V(x,y) = (P(x,y),Q(x,y)),$$ has at most a finite number of limit cycles.

In 1923, Dulac published a paper supposedly proving this.

Around 1980–81, Ecalle and Ilyashenko independently recognized that the proof had serious gaps.

In 1991–92, Ilyashenko and Ecalle independently published (quite different) proofs that a polynomial vector field in the plane does indeed have at most a finite number of limit cycles.

See Ilyashenko's paper, "A centennial history of Hilbert's 16th problem".

(Many related questions remain unsolved, such as finding sharp or even good upper bounds for the maximum number of limit cycles in terms of the degrees of the polynomials $P$ and $Q$.)

• I want to add, that the proofs of Ilyashenko and Ecalle are long and technical and also haven't been read by anybody but their authors... So that makes us think about this temporary value of mathematical theories... – Olga Dec 5 '14 at 9:20
• Here is a recent progress on this problem. For some related question see this post. – Ali Taghavi Jun 8 '15 at 10:19

Kempe's "proof" of the four-color theorem springs to mind. Wikipedia says that Kempe published it in 1879 and it was proven to be incorrect by Heawood in 1890. As I recall, the flaw in the original argument was as follows: Kempe defined a structure on a planar graph called a Kempe chain, and argued that certain of these chains could not intersect. There was a subtle flaw in this argument (which I can't seem to find a decent explanation of) and it failed for certain large graphs - the chains can in fact intersect. Heawood provided a 25-node example of intersecting chains; the smallest counterexamples are the Fritsch and Soifer graphs on 9 nodes.

Edit: I didn't address the renown of Kempe's proof. Wikipedia says that it was "widely acclaimed" (interesting coincidence of wording) while Thomas 1998 provides an excellent history but says little on this matter. I don't know if this could be truly considered "widely acclaimed" based on an uncited Wikipedia entry.

• A. B. Kempe, ‘On the geographical problem of the four colours’, American Journal of Mathematics 2 (part 3) (1879), 193–200. jstor.org/stable/2369235  P. G. Tait, ‘Note on a theorem in the geometry of position’, Transactions of the Royal Society of Edinburgh 29 (1880), 657–60.  de la Vallee Poussin (in 1896) also found the error of Kempe. Heawood calls the Kempe proof "now apparently recognized"  P. J. Heawood, Map colour theorems, Quart. J. Math. 24 (1890), 332–338. – Junkie Aug 13 '10 at 11:42
• The subtle flaw was that if one makes a modification to achieve some desirable property one must make sure not to lose what has been achieved earlier. As for the early acceptance, it is my understanding that the American Journal of Mathematics was considered a serious journal. – Wilberd van der Kallen Aug 13 '10 at 18:52
• @Wilberd: The AJM was founded a year earlier, so I don't know how much of a reputation it could have accrued by then. – Thierry Zell Aug 13 '10 at 20:16
• The word is "renown", not "reknown", BTW. (Sorry for this trivial comment.) – shreevatsa Aug 14 '10 at 5:09
• Concerning "widely acclaimed". C.S. Peirce in his 1903 Harvard lectures on pragmatism says:"Mr. Alfred B. Kempe, proposed a proof of it, somewhat, though not exactly, of the kind we are supposing our imaginary inventor to be aiming at. Yet I am informed that many years later a fatal flaw was discovered in Mr. Kempe's proof. I do not remember that I ever knew what the fallacy was" [CP 5.490], so it seems the matter did get aired. On the other hand, graphs and diagrams were Peirce's thing, so he may not be representative. – Conifold Sep 6 '17 at 23:10

In the 1960s, John Horton Conway verified the Nineteenth Century efforts of Tait and Little to tabulate all the knots through alternating 11 crossings (1). He found several omissions and one duplication, but somewhat famously failed to discover another one. This error would propagate when Dale Rolfsen added a knot table (as Appendix C) in his influential book Knots and Links in 1976, based on Conway's work. This addition happened in spite that Kenneth Perko had noticed, in 1974, the other pair of entries in classical knot tables that actually represent the same knot (2). In Rolfsen's knot table, this supposed pair of distinct knots is labeled $10_{161}$ and $10_{162}$. Now this pair is called the Perko pair, for obvious reasons :)

(1) An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon, pp. 329–358

(2) On the classification of knots. Proc. Amer. Math. Soc. 45 (1974), 262--266

(The information of this post is quoted from the following Wikipedia articles:

)

• We have the following suggested edit: (One might add that these two knot diagrams counterexemplify a 1899 "Theorem" of the original knot tabulators, that Dehn and Heegaard blessed in their German math encyclopedia article on Analysis situs. --Ken Perko, October 12, 2013) – S. Carnahan Oct 15 '13 at 13:17
• – Torsten Schoeneberg Nov 26 '17 at 10:25

R. B. Kershner's paper "On Paving the Plane," Amer. Math. Monthly 75 (1968), 839–844, announced the classification of all convex pentagons that tile the plane. Kershner said that "The proof...is extremely laborious and will be given elsewhere." As far as I know the proof was never published, but the claim was apparently accepted at least until 1975 when Martin Gardner wrote about the subject. Then, as explained in detail by Doris Schattschneider ("In Praise of Amateurs," in The Mathematical Gardner, ed. David A. Klarner, Wadsworth International, 1981, pages 140–166), Richard James III and Marjorie Rice found examples that had been missed by Kershner.

• And, of course, we're still finding some! – Akiva Weinberger Sep 1 '15 at 21:21
• we might be done now though -- last year Michaël Rao announced a proof that the current classification of 15 convex pentagons is complete – Harry Richman Jan 27 '18 at 16:41

Euler in his 1759 paper on knight's tours claimed that closed tours were not possible on any board with 4 or fewer ranks, though he gave no explicit proof. The claim was repeated by other influential writers such as E. Lucas and W. Ahrens. It was proved true for 4-rank boards by C. Flye Sainte-Marie in 1877. It was finally disproved by Ernest Bergholt in 1918 by constructing closed tours on 3x10 and 3x12 boards. Algorithms for enumerating tours on 3xn boards have now been devised by D. E. Knuth. This is a case of a famous mathematician's statements being taken as gospel and not really subjected to testing.

There are also numerous sources that state that Euler constructed a magic knight's tour on the 8x8 board. Where this mis-statement originated I'm not sure, but it has proved difficult to eradicate from the literature. In fact the first such tours were found by W. Beverley in 1848 and C. Wenzelides in 1849.

I'm surprised that this one has not already been mentioned. Voevodsky wrote an article explaining that one of the main motivations for his interest in homotopy type theory and univalent foundations was his personal experience with incorrect results being widely accepted for many years. For example, a 1989 paper by Kapranov and Voevodsky on ∞-groupoids contained a false result that was accepted until Simpson published a counterexample in 1998 (and even then, it took many more years before the community fully accepted Simpson's counterexample).

I think that this is a particularly important example from a sociological or historical point of view, since it spurred Voevodsky, a "mainstream" mathematician, to take seriously computerized proof assistants, which had been (and perhaps still is!) regarded by most people as a specialized subject of little interest to most mathematicians.

Grunwald's Theorem (1933) says that an element of a number field is an $$n$$-th power if and only if it is locally almost everywhere. As anyone who studied number theory now should guess, there is a problem with even primes, as discovered by Wang in 1948. This resulted the the corrected Grunwald-Wang theorem.

In the Wikipedia link above, Tate is quoted as saying:

Some days later I was with Artin in his office when Wang appeared. He said he had a counterexample to a lemma which had been used in the proof. An hour or two later, he produced a counterexample to the theorem itself... Of course he [Artin] was astonished, as were all of us students, that a famous theorem with two published proofs, one of which we had all heard in the seminar without our noticing anything, could be wrong.

• Yes, a great example! :) – paul garrett Feb 3 '19 at 5:09

On the matter of Cauchy's "mistaken" proof that a convergent infinite series of continuous functions is continuous, Detlev Laugwitz argues in his paper "Definite values of infinite sums: Aspects of the foundations of infinitesimal analysis around 1820" (in particular pages 211-212) that Cauchy was well aware of the issue that $\displaystyle\sum_{k=1}^{\infty} \frac{\sin(kx)}k$ is not continuous at $x=0$, and that it's not a counterexample to his theorem.

Basically, Laugwitz argues that the mistake is not in Cauchy's proof, but in its interpretation by others; in particular, a direct translation of Cauchy's notions of infinitesimal quantities and convergence into epsilons and deltas fails to capture the intended meaning. The point is that Cauchy understood the series to converge for infinitesimal $x$ as well, which is tantamount to requiring uniform convergence in the modern sense. His line of reasoning can be made rigorous by using non-standard analysis.

Edit: To elaborate, here a faithful reproduction of Cauchy's theorem and Cauchy's (1853) discussion of this trigonometric series.

Theorem: Let $S_m(x)$ be the partial sums of a series on the interval $a \leq x \leq b$. If

• $S_m(x)$ is continuous for all finite $m$
• and $S_m(\xi)$ converges to $S(\xi)$ for all numbers $\xi$ in the interval (including non-standard numbers!)

then the sum $S(x)$ is also continuous. (Continuity in the sense of Cauchy, which is defined with infinitesimals and also very sensitive to $x$ being non-standard or not, but that's not relevant here.) $\square$

Now, consider the series $\sum \frac{\sin(kx)}k$. It's not a counterexample to this theorem because it does not converge for infinitesimal $x$. Namely, let $n=\mu$ infinitely large and $x = \omega := \frac1\mu$ infinitesimally small. Then, the residual sum is

$$S(\omega) - S_{\mu-1}(\omega) = \sum_{k=\mu}^{\infty} \frac{\sin(k\omega)}k = \sum_{k=\mu}^{\infty} \frac{\sin(k\omega)}{k\omega}\omega \approx \int_{\omega\mu}^{\infty} \frac{\sin t}{t} \ dt = \int_1^{\infty} \frac{\sin t}{t} \ dt$$

Clearly, the integral is finite and not negligible; hence, the series does not converge for $x=\omega\approx 0$.

• I can't view the article you link to electronically (past the first page). Only a few years after Cauchy's work appeared, Abel (1826) wrote "It seems to me that this theorem has exceptions" and his specific counterexample was the alternating version of the Fourier series written above: sin x - (1/2)sin(2x) + (1/3)sin(3x) - ... This was long before epsilons and deltas and was contemporaneous with Cauchy, so I'm suspicious that the error is entirely one of "modern" definitions. I'd think Abel understood at that time whatever Cauchy meant when writing about convergence of infinite series. – KConrad Aug 15 '10 at 17:22
• No, Laugwitz argues that Abel misunderstood. In particular, Cauchy (1853) himself shows that this very Fourier series is not convergent in his sense. (The emphasis on epsilons and deltas is mine, not Laugwitz'.) I cannot reproduce the whole article here, but I'll try to elaborate on the main argument. My uni has access to the article, I can send you a copy (for educational purposes) if you like. – Greg Graviton Aug 16 '10 at 9:09
• Greg, okay please send me the article. Let me know if you have trouble from a web search figuring out my email address. – KConrad Aug 16 '10 at 23:36
• @GregGraviton, it should be mentioned that Laugwitz acknowledges that Cauchy's formulation of the sum theorem in the 1821 book was incorrect, and moreover Laugwitz says that Cauchy himself acknowledges as much in his 1853 paper. It is probably worth creating a separate thread on this important question regarding Cauchy interpretation. – Mikhail Katz Jan 5 '16 at 17:45
• @KConrad, ditto. – Mikhail Katz Jan 5 '16 at 17:48

In a 1966 paper (Rational surfaces over perfect fields, Publ. Math. IHES), Manin gave examples of cubic surfaces with Brauer group of order 2. In 1996, Urabe proved a conjecture of Tate on The bilinear form of the Brauer group of a surface (this is the title of his Invent. Math. 1996 paper) after noticing that Manin's examples, that were in contradiction with Tate's conjecture, were false (this he noted in Calculation of Manin's invariant for Del Pezzo surfaces, Math. of Computation 1996). Read a bit more on this story in Liu, Lorenzini, Raynaud, On the Brauer group of a surface, Invent. Math. 2005.

• And it becomes more interesting: Liu, Lorenzini, Raynaud used a result in their proof which was wrong. However, their result is still correct (there is a corrigendum to their paper). – Thomas Geisser May 5 at 12:03

Lebesgue famously "proved" that the projection of a Borel set in $\mathbb R^2$ is a Borel set in $\mathbb R$. Famously disproved by Souslin a decade later. See this answer by Gerald Edgar.

The wronskian determinant of $n$ functions which are $(n-1)$ times differentiable is $$W(f_1,\dotsc,f_n)=\det\begin{pmatrix} f_1 & f_2 & \dots & f_n\\ f_1' & f_2' & \dots & f_n'\\ \vdots & & & \vdots\\ f_1^{(n-1)} & f_2^{(n-1)} & \dots & f_n^{(n-1)} \end{pmatrix}.$$

It is clear that if $f_1$, ..., $f_n$ are linearly dependent, $W(f_1,\dotsc,f_n)=0$. For some years, the converse was assumed to be true too, until Peano gave the counterexample: $f_1=x^2$ and $f_2=x\cdot |x|$ are linearly independent, though $W(f_1,f_2)=0$ everywhere. Later, Bôcher even gave counter examples with infinitely differentiable functions.

Bôcher also proved that the converse holds as soon as the functions are analytic. Other conditions are also known for the converse to hold.

Engdahl and Parker describe the history of the wronskian . For a nice proof of Bôcher's result, one can have a look at a paper of Bostan and Dumas .

 Susannah M. Engdahl and Adam E. Parker. Peano on Wronskians: A Translation, Loci (April 2011), DOI:10.4169/loci003642.
 Alin Bostan and Philippe Dumas. Wronskians and linear independence, American Mathematical Monthly, vol. 117, no. 8, pp. 722–727, 2010.

Euclid's proofs were accepted for two thousand years. Only in the late 19th century was it noticed by Hilbert and others that Euclid was making a lot of implicit assumptions and that if you don't make those assumptions the results are false. The text by Prenowitz and Jordan is a good source for details.

• I would say that "not rigorous enough by modern standards" is very different from "wrong". – Angelo Aug 13 '10 at 13:12
• @Angelo, it's not clear to me whether OP insists on the results being wrong, or accepts situations where the result was right but the proof was wrong. But Euclid did stuff like assuming, without ever stating it explicitly, that a line through the center of a circle meets the circle. This is false if you take your plane to be ${\bf Q}^2$, a set where all Euclid's explicit assumptions hold. – Gerry Myerson Aug 13 '10 at 13:40
• @Angelo: No, the proof were wrong by Euclid's standards: His axioms and postulates were not sufficient to prove certain result what he claimed to price. Gerry: are you sure it was only in the late 19th century that the gap in Euclid was discovered? I remember reading in some boon on history of mathematics it was in the 18th century... – Joël Jan 5 '14 at 4:52
• @GerryMyerson But Euclid was not writing about $\mathbb Q^2$, he was writing about (what we would call} $\mathbb R^2$, so he was right. As the inventor of the world's first axiomatic system, he was entitled to decide what kinds of things had to be justified by explicit axioms and postulates, and what could be inferred from a figure or from geometric intuition. If the rules were changed later, that didn't make his work wrong. – bof Jan 5 '14 at 5:38
• @GerryMyerson You might as well say that all of mathematics through the 19th century was erroneous, because set theory had not been axiomatized, and pretty much everything would fail in a set theory where the axiom of pairing or union was false. For that matter, even when they worked from axioms, mathematicians before the 20th century were not justified in drawing any consequences from their axioms, because the rules of predicate calculus had not been formalized. – bof Jan 5 '14 at 5:40

Any rational function field over a finite field has genus $0$ and class number $1$, where the class number of a function field over a finite field is the number of degree-zero elements of the divisor class group. In 1975, Leitzel, Madan, and Queen proved there are exactly $7$ nonisomorphic function fields over finite fields with positive genus and class number $1$. Almost 40 years later, in 2014, Stirpe found an $8$th example (see http://arxiv.org/abs/1311.6318)! A precise gap was then found in the original proof, and once fixed the theorem is that there are $8$ examples (see http://arxiv.org/abs/1406.5365 and http://arxiv.org/abs/1412.3505).

In 1993, Pat Gilmer asserted as Theorem 1 of Classical knot and link concordance, that certain Casson-Gordon invariants vanish for all slice knots, which would be true if the kernel of the inclusion $H_1(M_K;\mathbb{Z}[t^{\pm1}])\rightarrow H_1(N_D;\mathbb{Z}[t^{\pm1}])$ were a metabolizer for the Blanchfield pairing. There, $M_K$ is the $3$--manifold obtained from zero-surgery on a knot K and $N_D$ is the complement of a slice disc in $D^4$.
The statement was believed, and many papers based statements on this theorem, which was taken for granted. It looks plausible, and the similar-looking statements of Levine or of Cochran-Orr-Teichner are certainly true. But it was shown a decade later in Stefan Friedl's 2004 thesis, Sections 8.3 and 8.4, that Gilmer's proof assumes that tensoring with $\mathbb{Q}/\mathbb{Z}$ is exact, which is false. Stefan is forced to do something unnatural and ugly to get his results, and to show that for each choice of Seifert surface, the Casson-Gordon invariants in question vanish for all but a finite number of primes (Theorem 8.6).
I believe that Gilmer's theorem is still open, which is very irritating for people studying knot concordance; because surely it MUST be true, and it is quite fundamental.

Carmichael's totient function conjecture (stating that the equation $\phi(x)=n$ never has a unique solution) was a theorem until an error was found in 1922 (apparently after the proof was left as an exercise in a textbook); since then, it is a conjecture. See: http://mathworld.wolfram.com/CarmichaelsTotientFunctionConjecture.html

• It was problem 8 on page 36 of the Dover edition of Carmichael, The Theory of Numbers, copyright 1914. – Gerry Myerson Aug 23 '12 at 4:52

Here is a list of counterexamples to once accepted theorems on Clifford algebras.

Edit: The original link is broken, I now replaced it by a pointer to the wayback machine. Alternatively, here are two of Lounesto's articles:

P. Lounesto: Counterexamples in Clifford algebras with CLICAL, pp. 3-30 in R. Ablamowicz et al. (eds.): Clifford Algebras with Numeric and Symbolic Computations. Birkh\"auser, Boston, 1996.

P. Lounesto: Counterexamples in Clifford algebras. Advances in Applied Clifford Algebras 6 (1996), 69-104.

• I was only vaguely aware of (the late) Pertti Lounesto through his activities on sci.math, but it is clear he was a controversial figure. I'm curious: are all those counterexamples he lists now generally accepted as valid counterexamples to once-accepted theorems? – Todd Trimble Jun 1 '13 at 3:04

I heard that Bott's theorem on the periodicity of the stable homotopy of the unitary group was delayed for some time by an erroneous computation in dimension 10, possibly due to Pontryagin.

• Toda famously calculated the homotopy of the unitary groups and claimed to have found answers contradicting Bott, though he later corrected his calculations and published a proof of Bott's theorem. – Dan Ramras Jul 7 '11 at 4:09

Verma proved that the multiplicities of all simple modules in a verma module are 1 or 0. When BGG tried to repeat his proof for some other case they found an error. This led to the study of multiplicities in category O etc.

• Do you have a reference for that claim? As far as I know, Verma proved that $\text{Hom}(M_\lambda,M_\mu)$ is at most 1-dimensional (which is true). Dixmier raised the problem of describing the submodule structure of $M_\lambda,$ but the existence of higher multiplicities made this a daunting task. – Victor Protsak Aug 18 '10 at 19:53
• I heard it from Bernstein in a lecture. Maybe he didn't mean exactly what I wrote but he definitely said that Verma had an error of this type. Maybe Verma claimed that what you mentioned implies the multiplicity freeness, I'm not sure. – Adam Gal Aug 28 '10 at 13:09

The Auslander Conjecture states: Every crystallographic subgroup $\Gamma$ of $\mathrm{Aff}(\mathbb{R}^n)$ is virtually solvable, i.e. contains a solvable subgroup of ﬁnite index.

He published an incorrect proof in 1964 of this statement.

In 1983 Fried and Goldman proved Auslander’s conjecture for $n = 3$.

Abels, Margulis and Soifer proved the conjecture for $n\leq 6$ in 2012.

Although it is not my area of expertise, I believe it is considered to be an important open conjecture and has led to active research.

• Sean: More precisely, in this case the proof turned out to be wrong, but the "result" became a major open problem in the field (not a wrong result, as far as we know). – Misha Jun 1 '13 at 13:15

No less a mathematician than Kurt Gödel was guilty of claiming to have proved a result that was accepted for decades, and even used by others, before being shown to be wrong. Stål Aanderaa showed that Gödel's argument was incorrect and Warren D. Goldfarb showed that the result itself was false. The claimed result was about the decidability of a class of formulas including equality; see here for more details.

The following is very far from my areas of competence, but I think it would fit the bill if the authors are correct:

https://arxiv.org/abs/1801.06359 L. Rempe-Gillen, D. Sixsmith, On connected preimages of simply-connected domains under entire functions.

Quoting from the abstract:

Let $f$ be a transcendental entire function, and let $U,V\subset\mathbb{C}$ be disjoint simply-connected domains. Must one of $f^{-1}(U)$ and $f^{-1}(V)$ be disconnected?

In 1970, Baker implicitly gave a positive answer to this question... It was recently observed by Julien Duval that there is a flaw in Baker's argument (which has also been used in later generalisations and extensions of Baker's result). We show that the answer to the above question is negative; so this flaw cannot be repaired. Indeed, there is a transcendental entire function $f$ for which there are infinitely many pairwise disjoint simply-connected domains $(U_i)_{i=1}^{\infty}$, such that each $f^{-1}(U_i)$ is connected... On the other hand, if $S(f)$ is finite (or if certain additional hypotheses are imposed), many of the original results do hold.

For the convenience of the research community, we also include a description of the error in the proof of Baker's paper, and a summary of other papers that are affected.