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 before 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.
Feel free to interpret this how you wish!
Solution 1:
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.
Solution 2:
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 https://sites.google.com/site/intriguingtessellations/home
EDIT: True to my 'I doubt anyone would be shocked' comment below, apparently another tiling has recently been found by some folks at the University of Washington in Bothell. There's a pretty good article about it at The Guardian.
EDIT 2: The problem has now seemingly been established; there are exactly $15$ pentagonal tesselations. Quanta’s article covers the subject pretty well.
Solution 3:
Several examples come to my mind:
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.
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.
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.