Question

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincare's 3d sphere charaterization or the search to prove that Euclid's parallel axiom is really necessary unnecessary.

But I also think there are less famous mistakes worth hearing about. So, here's a question:

What's the most interesting mathematics mistake that you know of?

This question is community wiki, meaning neither the question nor the answers receive points (which are reserved for "hard" questions). So please post as much as you like (indeed please post one answer per post so that others can upvote the ones easier), vote a lot and vote freely.

(should there be a tag 'not-math-related' or similar?)

EDIT: There is a similar question which has been closed as a duplicate to this one, but which also garnered some new answers. It can be found here:

http://mathoverflow.net/questions/77425/failures-that-lead-eventually-to-new-mathematics

Answer 1

C.N. Little listing the Perko pair as different knots in 1885 (10₁₆₁ and 10₁₆₂). 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.

Answer 2

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

Answer 3

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 intersections.

But wrong. Projection doesn't commute with countable intersection.

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

Answer 4

Kempe's "proof" of the four-color theorem, which didn't prove the four-color theorem, but did:

1. Prove the five-color theorem
2. Somehow manage to go unnoticed for a dozen years
3. 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.

Answer 5

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 worte, "... 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."

Answer 6

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

Answer 7

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. (See my comment for the correct Cayley reference.)

Answer 8

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

Answer 9

Pontryagin made a famous mistake while computing the stable homotopy groups of spheres (specifically, π₂) which led to the discovery of the Kervaire invariant. I won't spoil what the mistake was: watch this video of Mike Hopkins' talk (second video on the page), starting about 7 minutes in.

Answer 10

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 from http://www.intlpress.com/hha/v1/n1/a1/ ,page 17

Answer 11

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.

Answer 12

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/aeem2yx884nnufxn/

Answer 13

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

Answer 14

Hilbert's program, whose development was induced by on assumptions shattered by Goedel.

Answer 15

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!

Answer 16

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."

Answer 17

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.

Answer 18

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.)

Answer 19

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

"Let p be any old prime."

Answer 20

I find this one (it is not in the same vein as the ones that have been posted here so far, this is not a pure math mistake) to be interesting and instructive to students: patriot missile failure due to poor understanding of binary decimals

Answer 21

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. 852,133,201 is a factor of 2^257-1.

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.

Answer 22

I don't know if this is really a mistake: Fermat's "missing proof" for Fermat's last theorem.

Answer 23

Then there's always the Martian Climate Orbiter Newtons vs Pounds of thrust embarrassment.

Answer 24

If Hilbert's program was a "mistake", then surely so was Russell-Whitehead's Principia Mathematica.

Answer 25

Petrovisky-Landis solution to the second part of Hilbert 16th 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 Petrovisky-Landis ( complexify ) lead to the study of holomorphic foliations.

Answer 26

Karl Pearson's contributions in the development of statistics are so ubiquitous that most users take his assumptions for granted. One key contribution and mistake of his was to claim that all distributions are parametric. Such models are still predominantly used in social and behavioral sciences, but his insistence led to a lot of interesting and very useful developments in mathematical statistics and its applications by people who published refutations of his work (like R.A. Fisher).

As a non-math mistake, Karl Pearson avidly advocated eugenics towards racial purity. Big mistake.

Answer 27

William Shanks (1812-1882), who calculated pi to the 707th place, by hand, but it was only correct for the first 527 places.

Answer 28

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 these on his website, as does MSRI. 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!

Answer 29

Euler conjectured that there were no pairs of orthogonal Latin squares for orders $n \equiv 2 \pmod 4$. Nearly two hundred years later, this was proved false for every $n \equiv 2 \pmod 4$ except $2$ and $6$. Here's the link to Euler's paper. Regardless, Euler's work certainly helped spur research into Latin squares.

Answer 30

The mother of all examples: Euclid's Elements contains errors from start to finish.