# History of mathematics Question **URL:** \ **Category:** Factual Questions\ **Created:** [February 18, 2018, 7:39pm UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068 "2018-02-18T19:39:47Z")\ **Posts on this page:** 17\ **Page:** 1
**Author:** ![Steven\_Estes](https://avatars.discourse-cdn.com/v4/letter/s/94ad74/32.png) [@Steven\_Estes](https://boards.straightdope.com/u/Steven_Estes)\ **Post date:** [February 18, 2018, 7:39pm UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/1 "2018-02-18T19:39:47Z")
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? ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [February 18, 2018, 7:54pm UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/2 "2018-02-18T19:54:47Z")
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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [February 18, 2018, 8:03pm UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/3 "2018-02-18T20:03:57Z")
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”. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [February 18, 2018, 8:33pm UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/4 "2018-02-18T20:33:00Z")
[Mersenne](https://en.wikipedia.org/wiki/Mersenne_prime#History) 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. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [February 18, 2018, 11:01pm UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/5 "2018-02-18T23:01:51Z")
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: > [@Bertand Russell](#): > > As I think about acts of integrity and grace, I realise that there is nothing in my knowledge to compare with Frege’s dedication to truth. His entire life’s work was on the verge of completion, much of his work had been ignored to the benefit of men infinitely less capable, his second volume was about to be published, and upon finding that his fundamental assumption was in error, he responded with intellectual pleasure clearly submerging any feelings of personal disappointment. It was almost superhuman and a telling indication of that of which men are capable if their dedication is to creative work and knowledge instead of cruder efforts to dominate and be known. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [February 19, 2018, 12:30am UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/6 "2018-02-19T00:30:43Z")
> [@Andy\_L](#): > > 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. 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. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [February 19, 2018, 12:35am UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/7 "2018-02-19T00:35:58Z")
> [@Hari\_Seldon](#): > > Google Skewe’s number for more details. But Riemann tried a large number of cases; he never claimed to have proved this. He didn’t claim to have a proof, but he did assert it as a fact, so I thought that counted… > [@Hari\_Seldon](#): > > 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. Agreed - by and large published proofs are right. Wiles’ proof of Fermat’s theorem did have some gaps, but they were resolved before publication. ---
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [February 19, 2018, 12:42am UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/8 "2018-02-19T00:42:32Z")
> [@Steven\_Estes](#): > > Do you know of a “proof” that was demonstrated to be incorrect AND that its conclusion was shown to be incorrect? 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. ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [February 19, 2018, 1:35am UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/9 "2018-02-19T01:35:29Z")
Here’s a couple of webpages with examples of what you’re looking for: > > ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [February 19, 2018, 1:39am UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/10 "2018-02-19T01:39:06Z")
> [@Wendell\_Wagner](#): > > Here’s a couple of webpages with examples of what you’re looking for: > > [ho.history overview - Widely accepted mathematical results that were later shown to be wrong? - MathOverflow](https://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-to-be-wrong) > > [fake proofs - In the history of mathematics, has there ever been a mistake? - Mathematics Stack Exchange](https://math.stackexchange.com/questions/139503/in-the-history-of-mathematics-has-there-ever-been-a-mistake) I’m not the OP, but thanks Wendell. This is great. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [February 19, 2018, 1:03pm UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/11 "2018-02-19T13:03:30Z")
> [@Wendell\_Wagner](#): > > Here’s a couple of webpages with examples of what you’re looking for: > > [ho.history overview - Widely accepted mathematical results that were later shown to be wrong? - MathOverflow](https://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-to-be-wrong) > > [fake proofs - In the history of mathematics, has there ever been a mistake? - Mathematics Stack Exchange](https://math.stackexchange.com/questions/139503/in-the-history-of-mathematics-has-there-ever-been-a-mistake) 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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [February 19, 2018, 3:03pm UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/12 "2018-02-19T15:03:09Z")
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. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [February 19, 2018, 3:36pm UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/13 "2018-02-19T15:36:30Z")
> [@Chronos](#): > > 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. Okay, I give up: Example? ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [February 19, 2018, 3:44pm UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/14 "2018-02-19T15:44:36Z")
> [@Thudlow\_Boink](#): > > 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) ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [February 19, 2018, 4:35pm UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/15 "2018-02-19T16:35:48Z")
Yup, exactly the example I would have used, except that I’d have said w instead of a. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [February 19, 2018, 10:47pm UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/16 "2018-02-19T22:47:57Z")
> [@Chronos](#): > > 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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [February 20, 2018, 4:13am UTC](https://boards.straightdope.com/t/history-of-mathematics-question/809068/17 "2018-02-20T04:13:41Z")
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.