# Mathematical results that were generally accepted but later proven wrong?

I am giving a presentation on mathematical results that were widely accepted for a period of time and then later proven wrong, or vice versa. This talk is geared towards undergraduates who are likely to have not taken calculus. I have a few examples already, but I was wondering if anyone knew of others.

So far, I have the initial proofs of the 4-color problem, and "a continuous function is differentiable almost everywhere." These are both problems that can be explained simply, and their counter examples can be explained relatively simply. The problem I keep running into is that either the question is too advanced, or the counter example is too advanced. I was wondering if any of you kind people knew of examples or resources I could look into.

Any input would be appreciated. Thank you!

• One cannot expect anything very elementary, "widely accepted" means the gap was subtle. Vice-versa may be more interesting, perhaps Ruffini on the quintic, or initial neglect of the work of Galois. Oct 23, 2014 at 16:34
• What do you mean by '4-color problem'? Presuming you mean the usual question about planar graphs being 4-colorable, then there was a flaw in an initial proof attempt and the flaw lasted for about a decade, but I'm not sure how widely-accepted the initial proof is, and the theorem itself is true... Oct 23, 2014 at 16:55
• "Almost everywhere" was not precisely defined until some time after it ceased to be widely believed that differentiable functions are continuous almost everywhere. Oct 23, 2014 at 17:08
• Many cranks believe that they refuted things like Cantor's theorem, or the axiom of choice. Does that count as an answer? :-P Oct 23, 2014 at 18:59
• IMHO, any introductory level presentation on this topic should begin with the discovery of irrational numbers. The Pythagoreans originally believed that all numbers could be written as a ratio of integers. Supposedly a 5th century Pythagorean named Hippasus was the first to prove the irrationality $\sqrt{2}$, and was subsequently drowned at sea as punishment. Oct 25, 2014 at 16:32

Copying the crucial paragraph from my answer here, concerning a result claimed by Kurt Gödel in a 1933 paper:

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.

This isn't elementary enough to show to your calc students, but this example is really outstanding since it was accepted for so long, is so unequivocally wrong, and was committed by an undisputably first-rank mathematician.

Examples like this one are very unusual. Mathematics as a whole has a superb track record of not making this kind of mistake. Most of the examples one finds of “at first we thought $x$, but now we know that $x$ is wrong” are, when you look at them more closely, actually examples of “at first we thought $x$, but then we found a better way to understand the whole subject that $x$ is part of, and after we changed everything around we saw that the status of $x$ had changed.” One can look at an example of this type and say “This shows that mathematicians are sometimes wrong” and completely miss that there is something much more subtle and interesting happening. Imre Lakatos has a whole book, Proofs and Refutations, about this process, which some of your students might enjoy.

While not along the lines of a traditional "theorem/proof," the calculation of $\pi$ by William Shanks in 1873 to 707 decimal places was actually only correct to 527 decimal places--and this error was demonstrated in 1944.

• He was wrong for 180 positions?:O
– MonK
Jun 8, 2015 at 14:13
• @MonK The way that Shanks calculated the digits of $\pi$ was such that successive digits require the correct calculation of previous digits. Therefore, a calculation error at any step will propagate to all digits after that step. That is not to say that there might not have been coincidences (in which his calculation happened to obtain a correct digit in certain places). But once his calculated value deviated from the correct value, all subsequent calculated values are regarded as invalid for the purposes of determining "the number of correct digits." Jun 8, 2015 at 14:21

A problem simple enough to be explained to undergraduats is the Malfatti circles: the solution provided by Malfatti is never the optimal one.

This result didn't stand for long, but in 1847 a reputable mathematician claimed to have proven Fermat's Last Theorem but his proof relied (wrongly) on unique prime factorization in a particular ring. See Fermat's Last Theorem and Kummer's Objection