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In science, mistakes happen: bad experiment design, noisy measurements, incomplete data, expensive experimentations, and other factors. Sometimes theories are wrong because we don't know enough.

In mathematics, mistakes can be found right in the paper or traced back to the original, if that paper employs some erroneous results. We have complete information, and the number of mistakes depend mostly on our desire to find them.

The question is: What approximate percentage of published papers contains mistakes? Maybe some data available. And if not, what does your experience say in this respect?

P.S.: By mistakes, I mean mistakes that invalidate proofs by (1) being incomplete, e.g., not all cases considered, (2) proving a correct conjecture with incorrect reasoning, (3) proving an incorrect conjecture.

By "published," I mean papers published in peer-reviewed math journals.

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    $\begingroup$ Where on the spectrum of "typo" to "provably false statament being claimed to be true" would you define a mistake to lie? $\endgroup$
    – Dan Rust
    Oct 1, 2013 at 15:38
  • $\begingroup$ @DanielRust Clearly, not a typo, but mistakes that invalidate proofs by (1) being incomplete, e.g., not all cases considered, (2) proving a correct conjecture with incorrect reasoning, (3) proving an incorrect conjecture. $\endgroup$
    – Anton Tarasenko
    Oct 1, 2013 at 15:42
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    $\begingroup$ If you tried to answer this question rigorously you'd need an analysis of all mathematical papers. But such an analysis A itself would consist of a mathematical work itself NOT taken into account by A in which case you'll need to factor in that omission of A for the error term of A. You'd also want to precisely define "published paper" to have a meaningful analysis for this problem. On top of that, not all (new) mathematics appears only in published papers. $\endgroup$
    – Doug Spoonwood
    Oct 1, 2013 at 15:53
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    $\begingroup$ You might get an idea by searching MathSciNet for titles containing "Errata" or "Correction" etc. $\endgroup$
    – lhf
    Oct 1, 2013 at 15:55
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    $\begingroup$ @lhf That will just give you a lower bound, and probably not a very good one. From my own experience, the vast majority of errors in the mathematical literature do not result in an erratum. Anyhow, the “incompleteness” criterion of the question is still hopelessly fuzzy. What seems incomplete to an outsider may be totally clear to an expert in the field, after all. That even goes for “incorrect reasoning” sometimes, in that a correct argument in the author's draft may have been “improved” in the writing of the paper – but the mistake being easily reversible by someone who knows the field. $\endgroup$
    – Harald Hanche-Olsen
    Oct 1, 2013 at 16:03

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As in @HaraldHanche-Olsen's comment, the question itself is more ambiguous than the questioner may realize. But this meta-point itself deserves to be considered. That is, "correctness/incorrectness" is not objective, _in_practice_, because (as @Harald H-O notes) arguments perceived as incomplete by outsiders may be seen as completely adequate by experts. To elaborate slightly: even the issue of what's omitted may itself be tacit, thus adding a layer of impenetrability for novices. Similarly, statements that are false-as-stated, but irrelevantly so since they are only applied in a sub-case or variant that is correct, and easily verifiably by an expert, will not receive erratum treatment. At best, but also "worst", the virtue of a paper can be in its over-all narrative, which, if sufficiently intelligible to an expert, can be considered "correct" by experts, even if the thing is riddled with literal "errors", as many expert-perceptions have substantial "error-correcting" tendencies built in. And, of course, line-by-line "correctness" does not necessarily prevent other sorts of "mistakes", especially those that are not "localized" in writing.

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