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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!

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    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 (http://en.wikipedia.org/wiki/Four_color_theorem) which stood for 11 years. See also http://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-wrong .2012-05-01
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    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. http://en.wikipedia.org/wiki/Yamabe_problem2012-05-01
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    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.2012-05-01
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    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.2012-05-01
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    Also, the wikipedia article on [Wang Tiles](http://en.wikipedia.org/wiki/Wang_tile) used to say that Wang presented a false proof that whether a set of tiles can tile the plane or not is decidable, but that it was shown to actually be undecidable 5 years later. However, that part of the article has since been removed, so I don't know if it was true or not.2012-05-01
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    @BlueRaja-DannyPflughoeft That's always been something that's been on my mind. What if it's all wrong, do I have to go through it all and reverify everything for myself.2012-05-01
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    This is may be off-topic, but [Anders Haugen](http://en.wikipedia.org/wiki/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.2012-05-01
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    @Austin: Perhaps you're talking about the Jones lemma? It was used by Jones to "prove" the normal Moore space conjecture, and to do so he needed the auxilary fact now know as the continuum hypothesis. He had no idea it didn't follow from ZFC.2012-05-02
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    @BlueRaja-DannyPflughoeft I know the particular story you told, and a bit about the context. Unfortunately, the "theorem" was actually an experimental result if I remember correctly. Stripping out all the physics, some experimentalists took data and fit it to a model, concluding that a certain parameter was linear in a certain variable. What Feynman noticed was that this fit was almost entirely based on a single extreme data point, and by discounting that point the experiment was consistent with either a linear or constant model, the latter of which was consistent with his calculations.2012-05-02
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    If you had a dollar for each time this has happened in mathematics since the time of the pre-Socratics,you could balance the national deficit all by yourself!2012-05-02
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    You may want to read the following from David Mumford on the Italian school of Algebraic Geometry: http://ftp.mcs.anl.gov/pub/qed/archive/209.2013-06-18
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    This is an interesting page: "Lost Causes in Theoretical Physics" by math-physicist R. F. Streater. http://www.mth.kcl.ac.uk/~streater/lostcauses.html2014-01-04
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    @BlueRaja-DannyPflughoeft: Your story is probably about [Case's Theorem](http://blogs.msdn.com/b/msr_er/archive/2011/02/04/celebrating-richard-feynman-at-tedxcaltech.aspx). After this story Case sent his paper to Feynman who had to learn second-quantization from a student to understand the paper and find the flaw. This is from Schweber's [book](http://books.google.com/books?id=61n5dE7FJQgC&pg=PA455).2014-01-10
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    Not quite what is asked for, as it is about an opinion rather than a theorem and proof, but I had to think of Hilbert's "In mathematics there is no *ignorabimus*" - enter Gödel.2014-10-23
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    You can find [errata](http://www.math.lsa.umich.edu/~pscott/errata8geoms.pdf) to various books and articles. Maybe this isn’t what you’re looking for, since the mistakes were eventually caught—but they weren’t caught before the words went to print, which implies the mistakes made it through more than one check-point.2017-03-27

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