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There are many, many errors in the published mathematical literature, and the older you get, the more of them you see. What I find even more amusing is when peo

by wbhart 9y ago

There are many, many errors in the published mathematical literature, and the older you get, the more of them you see. What I find even more amusing is when people don't withdraw their papers after you convince them their paper is fundamentally wrong, and they agree the argument can't be salvaged. I used to think of mathematical arguments that I couldn't follow as being "advanced" and written by "elite" people. I now consider them "heuristic" or worse still, "the spam of confused minds". At the top of various fields, you basically have a bunch of experts writing for each other, using heuristic arguments, and writing only to a level that will convince one another that the result is "probably true". Important results of great public interest are picked to pieces. But this is not true of less prominent work (of which the majority of the literature consists), probably each paper is read an average of 0.6 times, including by the authors and referees.

dsacco 9y ago
> There are many, many errors in the published mathematical literature, and the older you get, the more of them you see. Can you provide any examples? I know this is true, but I'm interested in examples you have off the top of your head.
- wbhart 9y ago
  For the most part, I'd prefer not to give explicit examples, since it amounts to public shaming of (often) still living authors. In broad generality, I work on algorithms for number theory and fast arithmetic. If you look at published, peer reviewed, conference proceedings in this area, there are many, many well known cases of incorrect algorithms. Sometimes in the mathematical literature there are multiple definitions given for similar concepts, and later authors assume without checking that they are the same thing. I'm aware of papers that are entirely wrong because of this. Another area where errors are common is in stating identities, which the author didn't try to check with a computer. But, I am also aware of examples in less computational, pure mathematics too. The phenomenon also extends to famous textbooks and undergraduate notes. Simply look at the errata section of any sufficiently well-read textbook. Sometimes incorrect or inadequate assumptions are carried right through the book without explicitly mentioning them. A famous cryptographer used to have online a rant about half-gcd algorithms. As the story goes, the original publication was correct. One famous improvement is only correct for polynomials. A later attempt by another author is not correct for polynomials or integers. And a famous textbook in the field took until their third edition to get the proof/algorithm correct. I've even seen an incorrect definition given for something as fundamental as a field, even by the most famous of mathematicians, now two or three times I think (admittedly once by someone who wasn't an algebraist). You can find many examples easily by Googling, "proof was unfortunately incorrect", although it is usually stated more generously than that, e.g. "we have been unable to reconstruct their argument", "we supply some missing details", "it is not clear exactly what the authors intended", "where we correct an obvious typographical error", etc. And this is to say nothing of the almost infinite supply of papers that are missing critical elements of the proof, or where folklore theorems are relied upon without proof or citation. Simply search for the ubiquitous phrases, "the well-known result" and "it is well-known that".

by wbhart 9y ago

dsacco 9y ago

> There are many, many errors in the published mathematical literature, and the older you get, the more of them you see. Can you provide any examples? I know this is true, but I'm interested in examples you have off the top of your head.

wbhart 9y ago
For the most part, I'd prefer not to give explicit examples, since it amounts to public shaming of (often) still living authors. In broad generality, I work on algorithms for number theory and fast arithmetic. If you look at published, peer reviewed, conference proceedings in this area, there are many, many well known cases of incorrect algorithms. Sometimes in the mathematical literature there are multiple definitions given for similar concepts, and later authors assume without checking that they are the same thing. I'm aware of papers that are entirely wrong because of this. Another area where errors are common is in stating identities, which the author didn't try to check with a computer. But, I am also aware of examples in less computational, pure mathematics too. The phenomenon also extends to famous textbooks and undergraduate notes. Simply look at the errata section of any sufficiently well-read textbook. Sometimes incorrect or inadequate assumptions are carried right through the book without explicitly mentioning them. A famous cryptographer used to have online a rant about half-gcd algorithms. As the story goes, the original publication was correct. One famous improvement is only correct for polynomials. A later attempt by another author is not correct for polynomials or integers. And a famous textbook in the field took until their third edition to get the proof/algorithm correct. I've even seen an incorrect definition given for something as fundamental as a field, even by the most famous of mathematicians, now two or three times I think (admittedly once by someone who wasn't an algebraist). You can find many examples easily by Googling, "proof was unfortunately incorrect", although it is usually stated more generously than that, e.g. "we have been unable to reconstruct their argument", "we supply some missing details", "it is not clear exactly what the authors intended", "where we correct an obvious typographical error", etc. And this is to say nothing of the almost infinite supply of papers that are missing critical elements of the proof, or where folklore theorems are relied upon without proof or citation. Simply search for the ubiquitous phrases, "the well-known result" and "it is well-known that".