# Widely accepted mathematical results that were later shown wrong?

I wonder if there are 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.)

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I have a déjà vu :) Also, maybe community wiki (~big list)? –  efq 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 :) –  romkyns 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. –  romkyns 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

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 $L$: $$Vol_{n-1}(K\cap \xi^\perp)\le Vol_{n-1}(L\cap \xi^\perp)\qquad\text{for all } \xi\in S^{n-1},$$ does it follow that the volume of $K$ is less than the volume of $L$: $Vol_n(K)\le Vol_n(L)?$

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 $\neg P$!

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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

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 http://www.math.usma.edu/people/Rickey/hm/CalcNotes/CauchyWrgPr.pdf. [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: http://books.google.com/books?id=oV1aLqag6WwC&pg=PA43&lpg=PA43&dq=Ampere+1806&source=bl&ots=lV-HfqHza5&sig=Q_3U85rv2o8YEok--GjYWeJSZPs&hl=en&ei=-BZmTLm8CIL-8Aatpon8Dg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBYQ6AEwAA#v=onepage&q=Ampere%201806&f=false. 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 http://www.scribd.com/doc/29256978/Continuous-Nowhere-Differentiable-Functions-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, 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 uncertainy 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.

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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

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..."

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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
Nate: Yes! (((( –  KConrad Aug 18 '10 at 20:45
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

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

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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
@Sean: your comment needs to be interpreted according to one and not the other sense in which "true in general" is used in mathematics. But I guess the context makes it clear. –  Pete L. Clark Mar 27 '11 at 2:49
The sense of "tomorrow" used by Sean is actually less clear ;). –  Lennart Meier Oct 22 '13 at 20:01

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 $\mathbb P^n$

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Did Vahlen just make the claim, or did he have some kind of (false) argument to back it up? –  KConrad Aug 14 '10 at 2:47
Eisenbud/Evans: springerlink.com/index/M8780037409K6214.pdf  K. T. Vahlen, Bemerkung zur vollställndigen Darstellung algebraischer Raumkurven. J. Reine Angew. Math. 108, 346-347 (1891). gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002161885  The latter is a 1 page paper. The only paper it quotes is from M. Noether, resolver.sub.uni-goettingen.de/purl?GDZPPN002158523 –  Junkie Aug 14 '10 at 5:55
Perron's paper: resolver.sub.uni-goettingen.de/purl?GDZPPN002379600 –  Junkie Aug 14 '10 at 6:18
Just a fun comment: I was thinking of reading Eisenbud's paper for the Commutative Algebra class taught by him, and he said it is his favorite paper. :) –  Ngoc Mai Tran Mar 1 '11 at 0:57
There are some subtelties here, see for example: math.unipd.it/~frossi/schappacher-abs.html and www.math-berlin.de/images/stories/polspacecurvesberlin.pdf –  Laurent Berger Jul 7 '11 at 9:21

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).

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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.

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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

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.

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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 reknown 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.

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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
The word is "renown", not "reknown", BTW. (Sorry for this trivial comment.) –  shreevatsa Aug 14 '10 at 5:09

In the 1960s, John Horton Conway verified the 1920s efforts of Alexander and Seifert in Knot Theory to tabulate all the knots of at most 10 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:

)

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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

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 $Q$.)

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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

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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

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.

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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.

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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.

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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

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.

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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 $x=\omega\approx 0$.

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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

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.

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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

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.

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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.

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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).

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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.

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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

William Shanks's calculation of pi to 707 digits in 1873 seems to have been accepted for 72 years before Ferguson discovered an error in the 528th place.

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It probably wasn't a real error, just a drool stain :-D –  Adam Gal Aug 16 '10 at 11:29
I believe the error was from the 528th decimal and onwards to the 707th, allowing of course for possible `accidentally' correct digits, so it's doubtful that a drool stain could have caused this. –  Granger Aug 23 '12 at 11:52

According to Branko Grunbaum, An enduring error, Elemente der Mathematik 64 (2009) 89-101, reprinted in Mircea Pitici, ed., The Best Writing On Mathematics 2010, Daublebsky in 1895 found that there are precisely 228 types of collections of 12 lines and 12 points, each incident with three of the others. In fact, as found by Gropp in 1990, the correct number is 229.

The Gropp reference is H Gropp, On the existence and nonexistence of configurations $n_k$, J Combin Inform System Sci 15 (1990) 34-48.

Grunbaum has some other examples in this paper, which I may write up for this question.

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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.

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Another one from Grunbaum's paper:

According to Branko Grunbaum, An enduring error, Elemente der Mathematik 64 (2009) 89-101, reprinted in Mircea Pitici, ed., The Best Writing On Mathematics 2010, Bruckner enumerated 4-dimensional simple polytopes with eight facets, in 1909. But one of Bruckner's polytopes does not exist, according to Grunbaum and Sreedharan, An enumeration of simplicial 4-polytopes with 8 vertices, J Combin Theory 2 (1967) 437-465.

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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

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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

EDIT: The episode I had in mind turns out to be the work of Robert Coleman repairing a gap in a paper by Manin about the Mordell conjecture over function fields. See comments by KConrad below, giving specific references. Note that this is not about a false result, it is about an accepted proof with a gap that was found 20 to 25 years later and repaired.

Original:Requesting assistance with a memory.

This being community wiki, I will give my vague memory. I think someone who was actually there could tell a good story. I have been searching with combinations of words on google with no success.

Anyhoo, when I was in graduate school at Berkeley in the 1980's, a professor, whom I think was likely Robert Coleman, told us a story about a celebrated result on "function fields" or the function field version of something... The accepted proof was by someone really big, on google I kept running across the name Manin but I am not at all sure about the name. Prof. Coleman decided to present the proof to a class/seminar. Partway through it became clear that the accepted proof just did not work. I have a sense that the class and professor were able to clean up the proof but I have no idea what publication may have come of this. There is also the chance that the seminar did not occur at Berkeley, rather at an earlier job of the professor concerned. Sigh.

So, there are a few ways this story could be filled in. Many MO people are students or postdocs at Berkeley, somebody could walk down the hall and ask Prof. Coleman if that was really him, and if so what actually happened, or ask Ken Ribet, etc.. Again, someone on MO with encyclopedic knowledge of every possible use of the phrase "function field" might be able to say. Or someone very old, yea, verily stricken in years, like unto me.

Finally, note that the title and text of the OP's question disagree a little, and people have posted both "results" that remained false and correct results with incorrect proofs. Also, my memory is really quite good, but I heard this story once and did my dissertation on differential geometry and minimal submanifolds.

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Coleman, Manin's proof of the Mordell conjecture over function fields, L'Enseign. Math. 36(1990),393-427. From introduction: "In the process of translating Manin's proof of Mordell's conjecture (1963) into modern language we found a gap. The arguments given by Manin do not suffice to prove Manin's theorem of the kernel. We were able to fill this gap by using those arguments to prove a weaker theorem and combining this with the function field analogue of Siegel's theorem and Manin's ideas to complete the proof of Mordell's conjecture for function fields." This isn't a proof of a false theorem. –  KConrad Aug 16 '10 at 1:02
Thanks, Keith...I was not sure of much of anything at first except the phrase "function fields" and a location. When I saw the name Coleman I thought, yes, that seems right. Anyway, right, it turns out the OP really wanted proofs of false theorems, so I edited the question title to reflect that, and this episode does not therefore qualify but seems interesting to me...I see, Manin's work was 1963, Coleman's article did not appear until 1990 and Manin's letter to the editor in Izvestiya dates the preprint as 1988. The name "Manin" did not ring any bells even after I saw it on google... –  Will Jagy Aug 16 '10 at 1:49
Will, I was involved in a similar story concerning Wiles' proof. At my request, in the fall/winter of 1993 Beilinson agreed to give a few talks (in Moscow) on the 1993 paper. Eventually, he came to the point which he couldn't explain. Unfortunately, in spite of catalyzing the whole process, I didn't attend, so to this day I don't know whether he discovered the notorious gap! –  Victor Protsak Aug 16 '10 at 5:01
Victor, of course, Wiles. There was a very nice televivion program about that, just called "The Proof," in a series on public TV called NOVA. Evidently the gap appeared when one of the referees ( a chapter each!) I think Nicholas Katz, was going through line by line and demanding that Wiles explain any problems. Then there came a day when the explanations did not work either, but we know the rest. –  Will Jagy Aug 16 '10 at 22:49
Apparently the story about Manin's proof of the Mordell conjecture over function fields goes a long way : - 1963 Manin's original paper, - 1990 Coleman's correction, - 1991 Chai publishes 'A Note on Manin's Theorem of the kernel' (Am. J. of Math. 1991) saying "In this note, it will be shown that Manin was right after all", - 2008 Bertrand writes 'Manin’s theorem of the kernel : a remark on a paper of C-L. Chai' (see his homepage) saying "Chai gave two proofs. (...) The first proof concerns a more general situation, but contains a gap". –  Matthieu Romagny Aug 28 '10 at 12:12

According to Branko Grunbaum, An enduring error, Elemente der Mathematik 64 (2009) 89-101, reprinted in Mircea Pitici, ed., The Best Writing On Mathematics 2010, Andreini in 1905 claimed that there are precisely 25 types of uniform tilings of three-dimensional space. But in 1994 Grunbaum showed that the correct number is 28. The reference is B Grunbaum, Uniform tilings of 3-space, Geombinatorics 4 (1994) 49-56.

Grunbaum has yet more examples in this paper, which I may write up for this question.

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it appears to be a duplicate of itself. –  Sean Tilson Mar 1 '11 at 1:01
Say, what? I suppose that, technically, every answer to every question is a duplicate of itself, but that doesn't get us very far. This answer starts out the same as one of my other answers, but it cites a completely different result than that cited in the other answer. So I repeat: of what is this a duplicate? –  Gerry Myerson Mar 1 '11 at 3:43

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.

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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

## protected by François G. Dorais♦Oct 5 '13 at 17:43

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