# Errors in math research papers [duplicate]

Have there been cases of errors in math papers, that were undetected for so long, that they caused subsequent errors in research, citing those papers. ie: errors getting propagated along. My impression is that this type of thing is extremely rare.

What was the worst case of such a scenario? Thanks.

• This question and its answers contain many interesting examples of mathematical mistakes. – J.-E. Pin Dec 19 '15 at 13:13
• The concept of infinity, as some contrarians would have us believe... – arctic tern Mar 29 '16 at 20:31
• Also – MJD Mar 30 '16 at 13:08

One egregious case recently analyzed in detail by Adrian Mathias is Bourbaki's text Theory of sets and a couple of sequels published by Godement and others. Mathias' paper is:

Mathias, A. R. D. Hilbert, Bourbaki and the scorning of logic. Infinity and truth, 47–156, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 25, World Sci. Publ., Hackensack, NJ, 2014.

Mathias analyzes several ubiquitous errors in the book, such as choosing an inappropriate foundation in Hilbert's pre-Goedel epsilon (or tau) operator, confusion of language and metalanguage, missing hypotheses that make certain statements incorrect, and even more serious "editorial comments" suggesting to the reader that certain issues in logic are too complicated to be clarified completely.

The result was not merely perpetuation of errors in other papers, but the stagnation of logic in France for several generations that only recently has begun to be corrected.

• there goes my afternoon.... – mercio Mar 30 '16 at 14:33
• :-) It's well worth a read. @mercio – Mikhail Katz Mar 30 '16 at 14:39
• Could you clarify where in the linked article it says that there are actual technical errors in Bourbaki's Théorie des ensembles? I understand that it is widely argued Bourbaki was misguided in its choice of logical foundations for mathematics, and that its members sometimes displayed ignorance of the results of 20th century research in logic. But I am surprised by the assertion that the actual execution of this system in Bourbaki's treatise is incorrect. If it is, then does this still apply to the latest edition of the book? – user49640 Jun 22 '17 at 19:50
• It's been a while since I read the article but as I recall there were both misconceptions and errors both in Bourbaki and in books and textbooks it inspired. Confusion of language and metalanguage is an error. I agree with @mercio that Mathias's article is a fascinating read. – Mikhail Katz Jun 23 '17 at 8:46
• Looking at the article, and restricting attention to the latest edition of Bourbaki's treatise itself, the actual technical criticisms seem very modest and to my eye don't amount to "errors." The criticism that is emphasized the most seems to be that in cases where there is no object $x$ satisfying a relation $R$, then the term $\tau_x(R)$, which normally represents some object with property $R$, takes on an unspecified value instead of being undefined, and that this leads to counterintuitive results. I would describe this as perhaps a poor choice, but not an error. – user49640 Jun 23 '17 at 16:42