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_{161}and 10

_{162}). The mistake was found almost a century later, in 1974, by Ken Perko, a NY lawyer (!)

For almost a century, when everyone thought they were different knots, people tried their best to find knot invariants to distinguish them, but of course they failed. But the effort was a major motivation to research covering linkage etc., and was surely tremendously fruitful for knot theory.

^{(source)}

**Update (2013):**

This morning I received a letter from Ken Perko himself, revealing the true history of the Perko pair, which is so much more interesting! Perko writes:

The duplicate knot in tables compiled by Tait-Little [3], Conway [1], and Rolfsen-Bailey-Roth [4], is not just a bookkeeping error. It is a counterexample to an 1899 “Theorem” of CN Little (Yale PhD, 1885), accepted as true by PG Tait [3], and incorporated by Dehn and Heegaard in their important survey article on “Analysis situs” in the German Encyclopedia of Mathematics [2].

Little’s “ Theorem ‘was that any two reduced diagrams of the same knot possess the same writhe (number of overcrossings minus number of undercrossings). The Perko pair have different writhes, and so Little’s “Theorem”, if true, would prove them to be distinct!

Perko continues:

Yet still, after 40 years, learned scholars do not speak of Little’s false theorem, describing instead its decapitated remnants as aTait Conjecture– and indeed, one subsequently proved correct by Kauffman, Murasugi, and Thistlethwaite.

I had no idea! Perko concludes (boldface is my own):

I think they are missing a valuable point. History instructs by reminding the reader not merely of past triumphs, but of terrible mistakes as well.

And the final nail in the coffin is that**the image above isn’t of the Perko pair**!!! It’s the “ Weisstein pair ‘$ 10 _ { 161} $ and mirror $ 10 _ {163} $, described by Perko as “those magenta colo red, almost matching non-twins that add beauty and confusion to the Perko Pair page of Wolfram Web’s Math World website. In a way, it’s an honor to have my name attached to such a well-crafted likeness of a couple of Bhuddist prayer wheels, but it certainly must be treated with the caution that its color suggests by anyone seriously interested in mathematics. “

The real Perko pair is this:

You can read more about this fascinating story atRichard Elwes’s blog.

Well, I’ll be jiggered! The most interesting mathematics mistake that I know turns out to be more interesting than I had ever imagined!

1. J.H. Conway,*An enumeration of knots and links, and some of their algebraic properties*, Proc. Conf. Oxford, 1967, p. 329 – 358 (Pergamon Press, 1970). 2. M. Dehn and P. Heegaard, Enzyk. der Math. Wiss. III AB 3 1907), p. 212: “Die algebraische Zahl der Ueberkreuzungen ist fuer die reduzierte Form jedes Knotens bestimmt.” 3. C.N. Little,*Non-alternating /- knots*, Trans. Roy. Soc. Edinburgh**39**(1900), page 774 and plate III. This paper describes itself at p. 771 as “Communicated by Prof. Tait.” 4. D. Rolfsen,*Knots and links*(Publish or Perish, 1976).

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An error of Lebesgue. 1905 or so. Take a Borel set in the plane, project it onto a line, the result is a Borel set. Obvious: the projection of an open set is open, and the Borel sets in the plane are the least family containing the open sets, closed under countable unions and countable decreasing intersections.

But wrong. Projection doesn’t commute with countable decreasing intersection.

Studying this error lead Suslin to begin the line of study now called “descriptive set theory”, 1917 or so.

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All of the (in retrospect) misguided attempts to prove Euclid’s Parallel Postulate, which eventually lead Gauss to develop hyperbolic geometry.

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Kempe’s “proof” of the four-color theorem, which didn’t prove the four-color theorem, but did:

- Prove the
*Five*– color theorem - Somehow manage to go unnoticed for a dozen years
- Lay the foundations for major tools in structural graph theory, and despite being fundamentally flawed, serve as the starting point for the eventual successful proof (s) of 4CT.

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A story I heard in grad school:

Once upon a time, a set theorist was writing a paper on inner models, and in it he wrote, “… and we will call such models*nice*. ” When he got his manuscript back from the typist (this was back in the pre-LaTeX days of technical typists), he saw that his handwriting had been misread, and the line came out as: “… and we will call such models*mice*. ” The name stuck, and to this day if you browse almost any recent volume of the*Journal of Symbolic Logic,*you will find set theory articles on “mice.”

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Maybe it’s not true, but there’s the story of the “Grothendieck prime”:

One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replies, yes, an actual prime number. Grothendieck suggested, “All right, take 57. “

But Grothendieck must have known that 57 is not prime, right? Absolutely not, said David Mumford of Brown University. “He doesn’t think concretely.”

from here:http://www.ams.org/notices/ 200410 / fea-grothendieck-part2.pdf

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It was “proved” in 1961 that the first right derived functor, $ lim ^ 1 _ { leftarrow} $ of the inverse limit functor is zero on Mittag-Leffler systems.

However, recently a counter-example was found by Neeman and Deligne:http://www.springerlink.com/content/aeem2yx 884 nnufxn /

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An insignificant mistake, but amusing nonetheless: in Cayley’s famous 1854 paper where he defines the concept of an abstract group, as an illustration he proves that there are (three) groups of order 6 (up to isomorphism). This is because he does not realize that the groups $ Z_2 times Z_3 $ and $ Z_6 $ are isomorphic.

This is found on page 51 of A. Cayley, Desiderata and suggestions: No. 1. The theory of groups, American J. Math. 1 (1878), 50 – 52. An interesting related paper is G. A. Miller, Contradictions in the literature of group theory, American Math. Monthly 29 (1922), 319 – 328.

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From wikipedia (http://en.wikipedia.org/wiki/Uniform_convergence), about uniform convergence:

“Augustin Louis Cauchy in 1821 published a faulty proof of the false statement that the pointwise limit of a sequence of continuous functions is always continuous. Joseph Fourier and Niels Henrik Abel found counter examples in the context of Fourier series. Dirichlet then analyzed Cauchy’s proof and found the mistake: the notion of pointwise convergence had to be replaced by uniform convergence. “

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Frege’s proposed axioms in*Die Grundgesetze der Arithmetik.*

Frege was trying to derive the concept of “number” from more basic concepts, and he tried to axiomatize higher-order logic (essentially, a kind of set theory), but his intuitive -seeming axioms were logically inconsistent. Russell first found the inconsistency, which we now call Russell’s Paradox.

In fairness to Frege, he was suspicious of his flawed axiom, before Russell wrote to him about his paradox. In the introduction he writes:

“If we find everything in order, then we have accurate knowledge of the grounds upon which each individual theorem is based. A dispute can arise, so far as I can see, only with regard to my Basic Law concerning courses-of-values (V), which logicians perhaps have not yet expressly enunciated, and yet is what people have in mind, for example, where they speak of the extensions of concepts. “

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Poincare defined the fundamental group and the homology groups and proved that $ H_1 $ was $ pi _1 $ abelianized. So the question came up whether there were other groups $ pi_n $ whose abelianization would give the $ H_n $. Cech defined the higher $ pi_n $ as a proposed answer and submitted a paper on this. But Alexandroff and Hopf got the paper, proved that the higher $ pi_n $ were abelian and thus not the solution, and they persuaded Cech to withdraw the paper. Nevertheless a short note appeared and the $ pi_n $ started to be studied anyway …

Taken fromhttp://www.intlpress.com/hha/v1/n1/a1/, page 17

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I believe Kummer’s failed attempt at a proof of Fermat’s last theorem led to the discovery of ideals.

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Pontryagin made a famous mistake while computing the stable homotopy groups of spheres (specifically, π_{2}) which led to the discovery of the Kervaire invariant. I won’t spoil what the mistake was: watch thisvideoof Mike Hopkins’ talk (second video on the page), starting about 7 minutes in.

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Supposedly Stefan Bergman attended a course on orthogonal functions while an undergraduate, and misunderstood what he was hearing, believing that the functions were supposed to be analytic. This led him to the Bergman kernel and Hilbert spaces of analytic functions, which has developed into a whole field of study at the junction of complex analysis and operator theory, and also with important ramifications in the more geometric parts of SCV. If the story is true, this was certainly an extremely fruitful mistake!

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Not just a great mistake, but also a great*documentation*of a mistake: Stallings’sHow not to prove the Poincare Conjecture. (I think this paper is my answer to every community-wiki question.)

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Poincaré’s discovery of homoclinic points grew out of a extremely serious mistake he made in his original submission for a prize essay contest sponsored by Acta Mathematica in 1888. His original 200 page manuscript, on the restricted three-body problem, was evaluated by Weierstrass, Mittag-Leffler, and Phragmén, who had great difficulty following his arguments. Poincaré responded with a dozen further explanations, totaling 100 pages. After many further exchanges, the editors finally decided to accept the manuscript (this was, after all, Poincaré, and he must know what he’s doing) and awarded him the prize.

But around the time of publication, Phragmén was still puzzled by some points and Mittag-Leffler wrote to Poincaré. They received back a telegram from Poincaré asking that publication be stopped immediately! Poincaré realized that his belief that the stable and unstable manifolds could not intersect transversally was wrong, and that such intersection points, which he later called homoclinic points, immediately forced very complicated dynamically behavior, invalidating much of his work. He wrote to Mittag-Leffler:

“I have written this morning to Mr. Phragmén to tell him about an error which I have committed and he has undoubtedly informed you of my letter. But the consequences of this error are more serious than I first tho ught. It is not true that the asymptotic surfaces are closed, at least not in the sense that I meant before. What is true, is that if one considers the two parts of that surface (which I yesterday still believed coincided with each other) they intersect along infinitely many asymptotic trajectories and furthermore their distance is an infinitesimal of higher order than $ mu ^ p $ however big p is.

I don’t conceal from you the trouble this discovery gives me. “

Mittag-Leffler immediately halted the presses and recalled all copies of this issue he could get, destroying them all (except for a few, one of which remains in the library of the Mittag- Leffler Institute). They asked Poincare to pay for the suppression of this issue, which he did.

Poincare then wrote a new essay, incorporating many of the added notes from the original, and this was the version that Acta Mathematica published (with no mention of the earlier one). Eventually Poincaré used this as the basis of his three volume classic*Les méthodes nouvelles de la mécanique céleste*.

A riveting account of this story is contained in*Poincaré’s discovery of homoclinic points*by KG Anderson, Archive History of Exact Sciences, 48 (2) (1994), 133 – 147.

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Steiner’s count 7776 of the number of the number of plane conics tangent to 5 general plane conics certainly deserves a mention here. He gave this answer in 1848, and it wasn’t fixed until 1864, when Chasles pointed out the error and came up with the correct value of 3264. You can regard this as the first recognition of needing appropriate compactifications in order to do valid calculations in enumerative geometry.

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Hilbert’s program, whose development was induced by on assumptions shattered by Gödel.

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Goodrick’s “story from Grad school” is incorrect. According to Ronald Jensen, the set theorist in question, he felt that the concept was important enough that it deserved a name which had not already been used elsewhere in mathematics. And ‘mice’ was it. (Also, note that ‘mice’ is a noun, and ‘nice’ is an adjective — it would not make sense.)

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Perhaps not under this heading but I enjoy reading in Marshall Hall Group Theory book:

“Let p be any old prime.”

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Bringing in sort of tragic flavor to this question, – the following came to my mind:

He was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction. I tried to imitate him. But I’ve realized that it’s very difficult to make good mistakes.

) Shimura onTaniyama, seen it in the “BBC Horizon Season 1996 Episode 2 – Fermat’s Last Theorem “available on Youtube)

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In chapter 3 of**What Is Mathematics, Really?**(pages 43 – 45), Prof. Hersh writes:

How is it possible that mistakes occur in mathematics?

René Descartes’s Method was so clear, he said, a mistake could only happen by inadvertence. Yet, … his Géométrie contains conceptual mistakes about three-dimensional space.

Henri Poincaré said it was strange that mistakes happen in mathematics, since mathematics is just sound reasoning, such as anyone in his right mind follows. His explanation was memory lapse — there are only so many things we can keep in mind at once.

Wittgenstein said that mathematics could be characterized as the subject where it’s possible to make mistakes. (Actually, it’s not just possible, it’s inevitable.) The very notion of a mistake presupposes that there is right and wrong independent of what we think, which is what makes mathematics mathematics. We mathematicians make mistakes, even important ones, even in famous papers that have been around for years.

Philip J. Davis displays an imposing collection of errors, with some famous names. His article shows that mistakes aren’t uncommon. It shows that mathematical knowledge is fallible, like other knowledge.

…

Some mistakes come from keeping old assumptions in a new context.

Infinite dimensionl space is just like finite dimensional space — except for one or two properties, which are entirely different.

…

Riemann stated and used what he called “Dirichlet’s principle” incorrectly [when trying to prove

hismapping theorem].

Julius König and David Hilbert each thought he had proven the continuum hypothesis. (Decades later, it was proved undecidable by Kurt Gödel and Paul Cohen.)

Sometimes mathematicians try to give a (complete classification) of an object of interest. It’s a mistake to claim a complete classification while leaving out several cases. That’s what happened, first to Descartes, then to Newton, in their attempts to classify cubic curves (Boyer). [

cf.thisannotation by Peter Shor.]

Is a gap in a proof a mistake? Newton found the speed of a falling stone by dividing 0/0. Berkeley called him to account for bad algebra, but admitted Newton had the right answer … Mistake or not?

…

“The mistakes of a great mathematician are worth more than the correctness of a mediocrity.” I’ve heard those words more than once. Explicating this thought would tell something about the nature of mathematics. For most academic philosopher of mathematics, this remark has nothing to do with mathematics or the philosophy of mathematics. Mathematics for them is indubitable — rigorous deductions from premises. If you made a mistake, your deduction wasn’t rigorous, By definition, then, it wasn’t mathematics!

So the brilliant, fruitful mistakes of Newton, Euler, and Riemann, weren’t mathematics, and needn’t be considered by the philosopher of mathematics.

Riemann’s incorrect statement of Dirichlet’s principle was corrected, implemented, and flowered into the calculus of variations. On the other hand, thousands of

correcttheorems are published every week. Most lead nowhere.

A famous oversight of Euclid and his students (don’t call it a mistake) was neglecting the relation of “between-ness” of points on a line. This relation was used

implicitlyby Euclid in 300 BC It was recognizedexplicitlyby Moritz Pasch over 2 , 000 years later, in 1882.For two millennia, mathematicians and philosophers accepted reasoning that they later rejected.

Can we be sure that we, unlike our predecessors, are not overlooking big gaps? We can’t. Our mathematics can’t be certain.

The reference to the said article by Philip J. Davis is:

Fidelity in mathematical discourse: Is one and one really two?*Amer. Math. Monthly***79**(1972), 252 – 263.

From that article, I find particularly amusing the following couple of paragraphs from page 262:

There is a book entitled

Erreurs de Mathématiciens, published by Maurice Lecat in 1935 in Brussels. This book contains more than 130 pages of errors committed by mathematicians of the first and second rank from antiquity to about 1900 .There are parallel columns listing the mathematician, the place where his error occurs, the man who discovers the error, and the place where the error is discussed. For example, J. J. Sylvester committed an error in “On the Relation between the Minor Determinant of Linearly Equivalent Quadratic Factors”, Philos. Mag., (1851) pp. 295 – 305. This error was corrected by HE Baker in theCollected Papers of Sylvester, Vol. I, pp. 647 – 650.

…

A mathematical error of international significance may occur every twenty years or so. By this I mean the conjunction a mathematician of great reputation and a problem of great notoriety. Such a conjunction occurred around 1945 when H. Rademacher thought he had solved the Riemann Hypothesis. There was a report in

Timemagazine.

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I don’t know if this is really a mistake: Fermat’s “missing proof” for Fermat’s last theorem.

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Petrovskiĭ-Landis solution to the second part of Hilbert 16 th problem. They “proved” the existence of a bound for the number of limit cycles of planar polynomial vector fields of fixed degree. Ilyashenko pointed out the mistake.

The problem remains wide open but the basic idea of Petrovskiĭ-Landis (complexification of real differential equations) lead to the study of holomorphic foliations.

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Samuel I. Krieger made many attempts at significant contributions to the field of mathematics, unfortunately some of his efforts did not pan out.

In 1934, he claimed that the 72 – digit composite number 231, 584, 178, 474, 632, 390, 847, 141, 970, 017, 375, 815, 706, 593, 969, 331 , 281, 128, 078, 915, 826, 259, 279, 871 was the largest known prime number.

He also attempted to show that the number 2 ^ 256 (2 ^ 257 – 1) was perfect, implying that 2 ^ 257 -1 is a prime number. 2 ^ 257 – 1 is actually a composite number: its smallest prime factor is 535, 006 , 138, 814, 359.

Finally, he claimed to have a counter example to Fermat’s Last Theorem x ^ n y ^ n=z ^ n using the numbers x=1324, y=731 and z=1961 with an undisclosed n. A reporter supposedly called Krieger to ask how the left and the right hand side could be equal, when the left hand side could only end in a 4 or a 6 plus 1, and the right hand side could only end in 1.

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For surfaces of constant mean curvature, it is alleged that Hopf thought that all compact CMC surfaces in $ mathbb {R} ^ 3 $ were round spheres. CMC surfaces are what you get if you have a soap film bounding a fixed volume, so after a childhood full of blowing bubbles this is a pretty reasonable thing to think. And it even happens to be mostly true: Hopf proved that immersed CMC spheres are round, and Alexandrov proved with a nice reflection argument that (embedded) CMC surfaces of any genus must actually be round spheres.

But a bit later, Wente discovered a collection of CMC tori. Ivan Sterling has some nice pictures of theseon his website, as doesMSRI. There are many very pretty connections between these surfaces and algebraic geometry, so to me they sort of mark the start of the modern “integrable systems” era of CMC research.

I should probably add that nobody actually seems sure if Hopf believed that compact CMC surfaces are spheres, but it makes a good creation story for the subfield!

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Lakatos’ work “Proof and refutation” contains many examples of mistakes concerning the development of Euler’s polyhedron formula, along with an extensive treatment of what mistakes are and how they can crucially contribute to the development of mathematics.

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Euler conjectured that there were no pairs of orthogonal Latin squares for orders $ n equiv 2 ( text {mod} ~ 4) $. Nearly two hundred years later, this was proved false for every $ n equiv 2 ( text {mod} ~ 4) $ except $ 2 $ and $ 6 $.Here‘s the link to Euler’s paper. Regardless, Euler’s work certainly helped spur research into Latin squares.

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