# In the history of mathematics, has there ever been a mistake?

I was just wondering whether or not there have been mistakes in mathematics. Not a conjecture that ended up being false, but a theorem which had a proof that was accepted for a nontrivial amount of time and then someone found a hole in the argument. Does this happen anymore now that we have computers? I imagine not. But it seems totally possible that this could have happened back in the Enlightenment heyday.

Feel free to interpret this how you wish!

-
Lots, and yes it still happens nowadays (most mathematicians don't computer-verify their proofs). People aren't perfect (not even mathematicians!). One famous historical example is an incorrect proof of the four-color theorem (en.wikipedia.org/wiki/Four_color_theorem) which stood for 11 years. See also mathoverflow.net/questions/35468/… . –  Qiaochu Yuan May 1 '12 at 17:41
It happened with the Yamabe problem, wich consist basically of the following: given a compact riemannian manifold, find another metric, conformal to the given, with constant scalar curvature. It leads to a partial differential equation and, firstly, Yamabe itself, in 1960, claimed he has a solution. Years after, Trudinger found a critical error on the proof. Until 1984, the error could not be fixed. So, Trudinger, Aubin and Schoen found a correct proof and the fact coud be restablished. en.wikipedia.org/wiki/Yamabe_problem –  matgaio May 1 '12 at 17:44
I have a half-remembered story in my head of an old "proof" of the continuum hypothesis. The paper itself was perfectly sound, but one of the results it cited turned out to be flawed and brought the whole thing down. Perhaps someone else remembers more details. –  Austin Mohr May 1 '12 at 18:45
I recall from "Surely You're Joking, Mr. Feynman" that when Richard Feynman first solved the problem that won him the nobel prize (and started the field of Quantum Electrodynamics), he realized it contradicted some other widely-believed theorem in Physics. It turns out the original paper which "proved" this theorem had a glaring flaw, but no one had ever bothered to double-check it (he later set to work, with a couple of grad students, to re-verify all of the theorems in quantum physics). I'm afraid I don't know enough about quantum physics to know what that theorem was, though. –  BlueRaja - Danny Pflughoeft May 1 '12 at 21:36
This is may be off-topic, but Anders Haugen received a bronze 50 years after the 1924 Winter olympics because of an arithmetic error. As far as I can tell, the raw numbers were publicly known. –  Andrew Grimm May 1 '12 at 23:51
show 8 more comments

[I posted this recently in another thread, but it works much better here, so I've deleted it from there. I spent some time a couple of years ago trying to track down unequivocally incorrect claims of false results, and this was the most remarkable one I found. ]

In 1933, Kurt Gödel showed that the class called $\lbrack\exists^*\forall^2\exists^*, {\mathrm{all}}, (0)\rbrack$ was decidable. These are the formulas that begin with $\exists a\exists b\ldots \exists m\forall n\forall p\exists q\ldots\exists z$, with exactly two $\forall$ quantifiers, with no intervening $\exists$s. These formulas may contain arbitrary relations amongst the variables, but no functions or constants, and no equality symbol. Gödel showed that there is a method which takes any formula in this form and decides whether it is satisfiable. (If there are three $\forall$s in a row, or an $\exists$ between the $\forall$s, there is no such method.)

In the final sentence of the same paper, Gödel added:

In conclusion, I would still like to remark that Theorem I can also be proved, by the same method, for formulas that contain the identity sign.

Mathematicians took Gödel's word for it, and proved results derived from this one, until the mid-1960s, when Stål Aanderaa realized that Gödel had been mistaken, and the argument Gödel used would not work. In 1983, Warren Goldfarb showed that not only was Gödel's argument invalid, but his claimed result was actually false, and the larger class was not decidable.

Gödel's original 1933 paper is Zum Entscheidungsproblem des logischen Funktionenkalküls (On the decision problem for the functional calculus of logic) which can be found on pages 306–327 of volume I of his Collected Works. (Oxford University Press, 1986.) There is an introductory note by Goldfarb on pages 226–231, of which pages 229–231 address Gödel's error specifically.

-
Should someone care to know more about this decision problem of predicate logic and the precise meaning of $\lbrack\exists^*\forall^2\exists^*, {\mathrm{all}}, (0)\rbrack$, please consult The Classical Decision Problem by Börger, Grädel, and Gurevich. This book also mentions Gödel's error, although I think the discussion is abstracted from Goldfarb's. –  MJD May 1 '12 at 17:48
What do I need to know/read to understand what is being said here? I guess it's a naive question and i'm also declaring myself as a complete dummy, but I would love to understand what is there. –  Igäria Mnagarka May 1 '12 at 19:18
@GustavoBandeira I don't think it's a naïve question or that you appear to be a dummy. The theorem is obscure and quite technical. You need to know what is a formula of first-order logic. You need to know what it means for a problem to be decidable. In particular, you need to understand that there is no way in general to tell if a predicate formula represents a true statement. I suggest you ask your question in a post and see what you get. –  MJD May 1 '12 at 20:54
@GustavoBandeira I know how you feel! Just keep learning and you will get there. –  MJD May 3 '12 at 16:43
@GustavoBandeira Also, there is a lot of very interesting mathematics you can do without trigonometry or calculus, and this is some of it. You can start learning about mathematical logic right now if you want to. –  MJD May 3 '12 at 16:53
show 8 more comments

When trying to enumerate mathematical objects, it's notoriously easy to inadvertently assume that some condition must be true and conclude that all the examples have been found, without recognizing the implicit assumption. A classic example of this is in tilings of the plane by pentagons: for the longest time everyone 'knew' that there were five kinds of pentagons that could tile the planes. Then Richard Kershner found three more, and everyone knew that there were eight; Martin Gardner wrote about the 'complete list' in a 1975 Scientific American column, only to be corrected by a reader who had found a ninth - and then after reporting on that discovery, by Marjorie Rice, a housewife who devoted her free time to finding tessellations and found several more in the process. These days, she has a web page devoted to the subject, including a short history, at http://home.comcast.net/~tessellations/tessellations.htm

-
I love this example. –  Grumpy Parsnip May 2 '12 at 16:46
A rare and pleasant example of early crowdsourcing. Maths has a long history of public discussion leading to this sort of thing happening; one of my big regrets is that this has become steadily less common since the 19th century. –  Tynam May 3 '12 at 9:48
Is there even a finite number of tiling pentagons? –  D. Thomine May 10 '12 at 20:55
@D.Thomine It would be more accurate to say families of tiling pentagons, rather than individual; you can see the list of 14 families at mathpuzzle.com/tilepent.html . I don't recall seeing compelling proofs that this is an exhaustive list, and I doubt anyone would be shocked now if more were to be found (though it's been more than 25 years since the last now). –  Steven Stadnicki May 10 '12 at 21:30

Several examples come to my mind:

1. Hilbert's "proof" of the continuum hypothesis, in which an error was discovered by Olga Taussky when she was editing his collected works. This was shown to be undecidable by Paul Cohen later.

2. Cauchy's proof (published as lecture notes in his collected papers) of the fact that the pointwise limit of continuous functions is continuous. At the time, there was a poor understanding of the concept of continuity, until Weierstrass came along.

3. Lamé's proof of Fermat's last theorem, erroneous in that it was supposing unique factorization in rings of algebraic integers, which spurred the invention of ideals by Kummer.

-
The Cauchy example is widely cited, but when I looked into it a few years ago I found that the closer I looked the more complex it became. As you pointed out, the notions of continuity and convergence were still evolving. Under the ideas of continuity and convergence current at the time, Cauchy's result may have been correct, but the concepts evolved out from under him to include finer distinctions that did not exist at the time he wrote his proof. This sort of thing happens all the time in mathematics, and is not usually counted as an error. –  MJD May 2 '12 at 13:30
The gap in Lame's proof was immediately pointed out by Liouville, even before publication. –  franz lemmermeyer May 2 '12 at 17:28
I would like to add to my earlier comment that if the Cauchy thing were as simple as you say it would be impossible to believe. If the issue were really whether “the pointwise limit of continuous functions is continuous”, as you say, it can be easily falsified by the example of $x\mapsto x^n$, and Cauchy was certainly aware of this. But as Lakatos points out, nobody even complained of any error until 26 years later. In fact Cauchy was talking about the convergence of a sum of continuous functions. –  MJD Jan 4 at 12:46

One of the classic examples surely is the Perko pair of knots. For 75 years people thought that these two knots were distinct, even though they had found no invariants to distinguish between them. Then in 1974 Kenneth Perko (a lawyer!) discovered that they were actually the same knot. Even Conway, apparently, in compiling his table, had missed this.

It is not by any means a significant error, but it is an intriguing one nonetheless.

-
I heard Curtis McMullen mention this in a talk once! It's a pretty cool story. –  Steven-Owen May 2 '12 at 1:09
This answer was given earlier by Jose Brox on the Math Overflow page mathoverflow.net/questions/35468/… –  KCd May 2 '12 at 1:36
Oh I didn't notice it had been given there - apologies, that one's a better answer.... –  Chuck May 2 '12 at 3:02
@Zarrax I suspected he must've had some serious formal training in math, but never knew that - edited accordingly –  Chuck May 2 '12 at 3:16
See also the answer by Daniel Moskovich at mathoverflow.net/questions/879/… –  KCd May 2 '12 at 12:15
show 2 more comments

In 2003 a startling breakthrough was made (Review text only available to MathSciNet subscribers) in the theory of combinatorial differential manifolds. This theory was started by Gel'fand and MacPherson as a new combinatorial approach to topology, and one of the objects of its study is the matroid bundle. Much effort was spent in clarifying the relationship between real vector bundles and matroid bundles. From various previous results, the relationship is expected to be "complicated".

The Annals of Mathematics published in 2003 an article by Daniel Biss whose main theorem essentially showed that the opposite is true: that morally speaking there is no difference between studying real vector bundles and matroid bundles. This came as quite a shock to the field. (For an expert's account of the importance of this result, one should read the above-linked MathSciNet review.)

Unfortunately the article was retracted in 2009 after a flaw was found by (among others) Mnev. The story was popularised by Szpiro in his book of essays.

From Wikipedia one also finds the following account of the incident by someone familiar with the details and has expertise in the field, which contradicts some of the assertions/descriptions in Szpiro's essay. According to the various accounts, "experts" may have known about the error in the proof as early as 2005. But in the "recorded history" the first public announcement was not until 2007, and the erratum only published in 2009. So depending on your point of view, this may or may not count as a theorem accepted for some "nontrivial" amount of time.

-

A fairly recent example that I know of is a paper by the name of "A counterexample to a 1961 'theorem' in homological algebra" by Amnon Neeman (2002). It was a fairly big deal for some people when they realized the 'theorem' was false. I don't know enough about the specifics to discuss it in depth, since it's not terribly close to what I work on, so here is the abstract of Neeman's paper in lieu of any discussion:

In 1961, Jan-Erik Roos published a “theorem”, which says that in an $[AB4 * ]$ abelian category, $\lim^1$ vanishes on Mittag–Leffler sequences. See Propositions 1 and 5 in [4]. This is a “theorem” that many people since have known and used. In this article, we outline a counterexample. We construct some strange abelian categories, which are perhaps of some independent interest.These abelian categories come up naturally in the study of triangulated categories. A much fuller discussion may be found in [3]. Here we provide a brief, self contained, non–technical account. The idea is to make the counterexample easy to read for all the people who have used the result in their work.In the appendix, Deligne gives another way to look at the counterexample.

-
This already was mentioned by Liran Shaul in an answer to the question mathoverflow.net/questions/879/…. –  KCd May 2 '12 at 12:17

The "telescope conjecture" of chromatic homotopy theory is an interesting example.

In 1984, Ravenel published a seminar paper called "Localization with respect to certain periodic homology theories" where he made a series of 7 or 8 important conjectures about the global structure of the ($p$-local) stable homotopy category of finite spaces. Four years later, Devinatz-Hopkins-Smith published "Nilpotence I" (while Hopkins was still a grad student!!), which along with the follow-up paper "Nilpotence II" proved all but one of Ravenel's conjectures, the telescope conjecture. Then in 1990, Ravenel published a disproof of this conjecture, and went so far as to write a paper entitled "Life after the telescope conjecture" in 1992 that detailed a new way forward. But then it turned out that his disproof had a flaw in it too! The telescope conjecture remains open to this day, although I think most experts believe that it is false.

-
The telescope conjecture says: If you start with a $p$-local finite space (or spectrum) of type $n$ and take the mapping telescope of any $v_n$-self map, then the result has the same Bousfield class as $K(n)$ itself. It's known that the Bousfield class of any such mapping telescope is independent of the original choices (this was proved in Nilpotence II); what's not known is whether this always coincides with $\langle K(n) \rangle$. –  Aaron Mazel-Gee May 2 '12 at 1:32
In the sentence "But then it turned out that his disproof had a flaw in it too!" what does "too" refer to? I.e., what else in this story had a flaw in it? –  Omar Antolín-Camarena May 10 '12 at 21:10
@Omar: Sorry, that was unclear. I just meant to emphasize that at first people thought it was true, and then Ravenel thought he proved it was false, but that development was in fact a misstep. –  Aaron Mazel-Gee May 11 '12 at 17:50

A famous example of this involves Vandiver's 1934 "proof" of one of the two steps in a line of attack on (an important case of) Fermat's Last Theorem. In algebraic number theory, there arise important positive integers called class numbers. In particular, for each prime p, a certain class number $h_p^+$ can be defined that is intimately connected with Fermat's Last Theorem.

Kummer proposed that (an important case of) Fermat's Last Theorem could be proved by

i) Proving that $h_p^+$ is not divisible by p

ii) Proving that $h_p^+$ not being divisible by p implies the "first case" of Fermat's Last Theorem.

In 1934, Vandiver published a proof of ii). In the introduction to "Cyclotomic Fields I and II", Serge Lang stated:

"...many years ago, Feit was unable to understand a step in Vandiver's 'proof' that $p$ not dividing $h_p^+$ implies the first case of Fermat's Last Theorem, and stimulated by this, Iwasawa found a precise gap which is such that there is no proof."

(In fact, Vandiver passed away believing that his proof was correct.)

I would like to know more about this history of this myself, and would gladly edit this post with more reliable information. For instance,

http://gow.epsrc.ac.uk/NGBOViewGrant.aspx?GrantRef=EP/C549074/1

says that Feit's observation occurred "around" 1980, which suggests that it was never published.

-
@Barry: I emailed Walter Feit about this back in 1992, asking him what the nature of the gap was. He told me that he had gotten stuck at a step in Vandiver's proof and asked Iwasawa about it, who couldn't straighten things out. By this time Feit had forgotten the details. I then wrote to another mathematician (still alive, so let me not mention the name in case there is an error in what follows) whose reply was this: "On page 119, he says 'Let $p$ be a prime divisor of one of the $w$'s in (2).' Then on p. 120 in the middle he seems to be using the "fact" that $x+\zeta{y}$ is divisible [contd.] –  KCd May 2 '12 at 12:38
by $p$ (I can't guarantee that this is what he is doing since I didn't quite get (7) to work). However, in (2) there does not seem to be any guarantee that $p$ does not also occur in the denominator somewhere. In fact, it must, if $p$ does not divide $x+\zeta{y}$. He cannot ignore some ideals, since he needs them all on page 122." –  KCd May 2 '12 at 12:39

Some technical results in the disintegration theory of von Neumann algebras (roughly speaking, results expressing an algebraic object as a "direct integral" of "simpler" algebraic objects) stated by Minoru Tomita in the 1950s turned out to not be OK. There was an entire chapter following Tomita's approach in Naimark's book Normed Rings that vanished from later editions when the errors came to light.

I am not clear on the details of exactly how Tomita's stuff was wrong. (This happened before I was born, and I am not that interested in the history of mathematics, so I only know what I have heard about this from people who were there when it happened.) I have heard one person say that Tomita made use of certain technical results that only held under certain "nice" hypotheses that were not met at the level of generality at which he was working. Another person said that Tomita's arguments simply weren't clear enough to admit close analysis of how he went wrong, but that flaws were evident once people produced counterexamples to statements of the results. I don't personally know which of these stories is closer to the truth.

I am not sure to what extent this work was "accepted for a nontrivial amount of time." The person who told me most of what I know about this conveyed to me that at the time, there was a sense in the air that there was something "fishy" about some of the theorems, and that counterexamples were circulated among people working in the area long before it all worked itself out in print.

-

This seems to be related enough to deserve to be in an answer:

The April 2013 issue of the Notices of the AMS features a long article Errors and Corrections in Mathematics Literature written by Joseph F. Grcar.

Not a specific mistake, rather an analysis of how mathematics journals and mathematicians deal with mistakes in general, compared to other sciences.

-

I don't know how long some of his proofs standed, but Legendre is infamous for his repeated attempts at proving the parallel postulate.

-

Not sure if the following fits the criterion for constraint, but Hans Rademacher incident comes to mind (page 82, The Riemann Hypothesis: For the aficionado and virtuoso alike):

8.2 Hans Rademacher and False Hopes

In 1945, Time Magazine reported that Hans Rademacher had submitted a flawed proof of the Riemann Hypothesis to the journal Transactions of the American Mathematical Society. The text of the article follows: A sure way for any mathematician to achieve immortal fame would be to prove or disprove the Riemann hypothesis. This baffling theory, which deals with prime numbers, is usually stated in Riemann’s symbolism as follows: “All the nontrivial zeros of the zeta function of s, a complex variable, lie on the line where sigma is 1/2 (sigma being the real part of s).” The theory was propounded in 1859 by Georg Friedrich Bernhard Riemann (who revolutionized geometry and laid the foundations for Einstein’s theory of relativity). No layman has ever been able to understand it and no mathematician has ever proved it.

One day last month electrifying news arrived at the University of Chicago office of Dr. Adrian A. Albert, editor of the Transactions of the American Mathematical Society. A wire from the society’s secretary, University of Pennsylvania Professor John R. Kline, asked Editor Albert to stop the presses: a paper disproving the Riemann hypothesis was on the way. Its author: Professor Hans Adolf Rademacher, a refugee German mathematician now at Penn.

On the heels of the telegram came a letter from Professor Rademacher himself, reporting that his calculations had been checked and confirmed by famed Mathematician Carl Siegel of Princeton’s Institute for Advanced Study. Editor Albert got ready to publish the historic paper in the May issue. U.S. mathematicians, hearing the wildfire rumor, held their breath. Alas for drama, last week the issue went to press without the Rademacher article. At the last moment the professor wired meekly that it was all a mistake; on rechecking. Mathematician Siegel had discovered a flaw (undisclosed) in the Rademacher reasoning. U.S. mathematicians felt much like the morning after a phony armistice celebration. Sighed Editor Albert: “The whole thing certainly raised a lot of false hopes.” [142]

Edit: This link has further (dis)proofs of RH including de Branges saga.

-

A plentiful source of examples of "theorems" that were "proved" is supplied by the Italian school of algebraic geometry.

The Italians, most prominently Guido Castelnuovo, Federigo Enriques and Francesco Severi, derived some remarkable results on classification algebraic surfaces, relying strongly on geometical insight. The problem was, their reliance on intuition ultimately led them astray, to the point where some of things that were intuitively obvious to Severi were plain wrong. For an extreme example, Severi claimed to show a degree 6 surface in a 3 dimensional projective space has at most 52 nodes, while Mumford exhibited such surface that in fact had 65 nodes. Wikipedia provides a short but informative discussion. There is also a great thread on Mathoverflow.

-

Has there ever been a mistake? LOL! Yeah, just a few. ;-)

OK, so that isn't exactly what you asked...

Well, there have been plenty of conjectures which everybody thought were correct, which in fact were not. The one that springs to mind is the Over-estimated Primes Conjecture. I can't seem to find a URL, but essentially there was a formula for estimating the number of primes less than $N$. Thing is, the formula always slightly over-estimates how many primes there really are... or so everybody thought. It turns out that if you make $N$ absurdly large, then the formula starts to under-estimate! Nobody expected that one. (The "absurdly large number" was something like $10^{10^{10^{10}}}$ or something silly like that.)

Fermat claimed to have had a proof for his infamous "last theorem". But given that the eventual proof is a triumph of modern mathematics running to over 200 pages and understood by only a handful of mathematicians world wide, this cannot be the proof that Fermat had 300 years ago. Therefore, either 300 years of mathematicians have overlooked something really obvious, or Fermat was mistaken. (Since he never write down his proof, we can't claim that "other people believed it before it was proven false" though.)

Speaking of which, I'm told that Gauss or Cauchy [I forget which] published a proof for a special case of Fermat's last theorem - and then discovered that, no, he was wrong. (I don't recall how long it took or how many people believed it.)

-
Is this what you were thinking of with the "over-estimated primes"? –  jwodder May 1 '12 at 20:47
Plausibly, yes. –  MathematicalOrchid May 1 '12 at 20:56
Thanks for your answer; I imagined this stuff kind of stuff happening a lot. I was curious as to whether people say to have proved something though and other people see and believe the proof only for someone later to come and punch hole into it. –  Steven-Owen May 1 '12 at 22:03
"Speaking of which, I'm told that Guass or Cauchy [I forget which] published a proof for a special case of Fermat's last theorem - and then discovered that, no, he was wrong." To my taste this is too much hearsay for a site like this. If you were "told" this by a reliable source, please include the source, so that interested readers can investigate it. If you can't remember where you heard it from, is it really good form to put it in an answer? –  Pete L. Clark Apr 28 '13 at 18:45