I know of a number of mathematical “proofs” that have been accepted for a good while only to be later shown to be false.
However, in every case I know of, it was the reasoning, not the conclusion, that was shown to be incorrect. I do not know of even one case in which the conclusion was not later proved to be correct.
For example, “Iron-clad proofs” of Euclid contained unrecognized assumptions that made the “proofs” to be not proofs. However, Hilbert later proved Euclid’s conclusions to be correct.
Do you know of a “proof” that was demonstrated to be incorrect AND that its conclusion was shown to be incorrect?
Riemann believed that his analysis of the frequency of prime numbers showed that the integral of the logarithmic function remains larger than the prime number counting function for large n, but this turns out not to be the case.
Plenty of people through the years have “proven” Euclid’s Parallel Postulate. Except they all (at least, the not-completely-crackpot ones) just replaced the axiom with another one equivalent to it, like “the sum of the interior angles of a triangle is 180 degrees”.
Mersenne gave a list of exponents he believed generated Mersenne primes. However his list was incorrect in both including composites and omitting primes. The first error was not found for over 200 years.
OTOH, he gave no proof. He might have not actually created the list himself. He was sort of a letter forwarder- getting info from others and copying/passing them on.
Gottlob Frege wrote the two-volume book Basic Laws of Arithmetic in which he developed the foundations of mathematics from logic and naive set theory. Volume 2 of the book was about to go to press when Bertrand Russell pointed out that the axioms Frege used led to contradictions; an additional “Axiom of Regularity” was needed.
Frege responded graciously:
Google Skewe’s number for more details. But Riemann tried a large number of cases; he never claimed to have proved this. It is rather amazing, considering how many flawed proofs have been published that so few of them made claims that turned out to be false. I do know of at least one case where a student in Denmark discovered a counter-example to a published proof, but that is fairly rare. But mainly what you see are cases where people stated conjectures that turned out to be false.
There was a purported counter-example to the Fermat conjecture that circulated one April first probably around 1990, but that was a conscious joke and, anyway, not published.
He didn’t claim to have a proof, but he did assert it as a fact, so I thought that counted…
Agreed - by and large published proofs are right. Wiles’ proof of Fermat’s theorem did have some gaps, but they were resolved before publication.
I know this isn’t what you’re looking for but I have taught a few sections of discrete mathematics at a large US university. I have seen many, many incorrect proofs of things weren’t true to begin with.
Here’s a couple of webpages with examples of what you’re looking for:
I’m not the OP, but thanks Wendell. This is great.
When I read these, it turned out that most were examples of unproved claims that turned out to be wrong or are still undecidable. There were a few that did answer to OP.
I once discovered a flaw in a proof in a book. I was a student and the proof did not work without an additional hypothesis and a couple of professors found a counter-example. But the theorem in question was old and well known but always stated with that hypothesis. Still it is remarkable how rarely it happens, considering how many proofs do contain errors.
There was a really first class mathematician that I knew well and respected enormously who joked that he started every paper by correcting the errors of his previous paper.
For that matter, not all of Euclid’s propositions are actually true, either. For instance, one of his proofs was that the intersection of two planes is a straight line. But it isn’t always: Two planes can also intersect in a single point. He just didn’t consider the possibility of four or more dimensions.
He also proved that there were only five regular solids, but by his definition of “regular solid”, there are at least seven. You need to add more conditions to his definition to restrict it to the standard Platonic five.
Okay, I give up: Example?
(taking a wild swing at it): Suppose you have a four dimensional space with coordinates x,y,z,a. The planes defined by x=y=0, and a=z=0 intersect at one point (0,0,0,0)
Yup, exactly the example I would have used, except that I’d have said w instead of a.
Funny, I’d have used t.
But Euclid is correct if you stick to 3 dimensions, which is certainly what Euclid had in mind. Just as in three space, you can have skew lines that don’t meet at all, even at infinity (so they are not parallel in ordinary space) you can, in sufficiently high dimension (I imagine 5 is enough) have two planes that are not parallel and don’t meet all.
Four dimensions is enough for “skew planes”. Five might be enough for the planes to not even contain any lines parallel to each other-- I’ll have to think about that a bit more.