[Math] Endless controversy about the correctness of significant papers

Question

In principle, a mathematical paper should be complete and correct. New statements should be supported by appropriate proofs. But this is only theory. Because we often cannot enter into the smallest details, we "prove" wrong statements here and then. I plead guilty, having published myself one or two false (fortunately minor) papers.

So far, this is not harmful. The research community is able to point out incorrect statements, at least among those which have some importance in the development of mathematics. In time, the errors are fixed; this is the role of monographs to present a universally accepted state of the art of a topic.

But sometimes, hopefully rarely, the technicalities are such that a consensus does not emerge and a controversy raises, between the author and their critics. I have an example in the realm of wave stability in PDE models for fluid dynamics. The controversy has lasted for a decade or two and I don't see how it can be resolved some day; it could just kill the topic.

Are there famous endless controversies about the correctness of a significant paper? Are there some significant mathematical questions, that remain unsettled because people disagree on the status of released proofs? What should we do in order to salvage mathematical topics that suffer such tensions?

In this question, I am not concerned with other kinds of controversy, about priority or citations.

Answer 1

(Also mentioned in Oliver Nash's comment)

From a February 2017 article in Quanta Magazine called "A fight to fix geometry's foundations" (the original has relevant links in the text):

...in 2012, a pair of researchers — Dusa McDuff, a prominent symplectic geometer at Barnard College and author of a pair of canonical textbooks in the field, and Katrin Wehrheim, a mathematician now at the University of California, Berkeley — began publishing papers that called attention to the problems, including some in McDuff’s own previous work. Most notably, they raised pointed questions about the accuracy of a difficult, important paper by Kenji Fukaya, a mathematician now at Stony Brook University, and his co-author, Kaoru Ono of Kyoto University, that was first posted in 1996.

This critique of Fukaya’s work — and the attention McDuff and Wehrheim have drawn to symplectic geometry’s shaky foundations in general — has created significant controversy in the field. Tensions arose between McDuff and Wehrheim on one side and Fukaya on the other about the seriousness of the errors in his work, and who should get credit for fixing them.

More broadly, the controversy highlights the uncomfortable nature of pointing out problems that many mathematicians preferred to ignore. “A lot of people sort of knew things weren’t right,” McDuff said, referring to errors in a number of important papers. “They can say, ‘It doesn’t really matter, things will work out, enough [of the foundation] is right, surely something is right.’ But when you got down to it, we couldn’t find anything that was absolutely right.”