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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.

Ievgeni
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Denis Serre
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    Abbas Bahri wrote a famous paper criticizing the rigour of various papers. I don't give his paper as an answer, since it is not really an endless controversy, but his concerns seem to have been very serious. http://sites.math.rutgers.edu/~abahri/papers/five%20gaps.pdf – Ben McKay Oct 05 '17 at 11:18
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    Freedman's work on 4-manifolds has also been strongly criticized, and might be what you are looking for: https://mathoverflow.net/questions/87674/independent-evidence-for-the-classification-of-topological-4-manifolds – Ben McKay Oct 05 '17 at 11:21
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    If the contributions remain respectful and considered, this is an important and useful question. The following is meant as a constructive suggestion of improvement: it would be better if you thought of some way to impose conditions which rule out the 'usual' examples like Almgren's 900+pages regularity paper, or Hales' proof of Kepler's conjecture. (Please note: I am not disputing these papers; I am mentioning them since on the superficial verbal level it is a fact that the latter two are controversial.) Sorry not to have a good suggestion of what additional conditions you could impose. – Peter Heinig Oct 05 '17 at 11:30
  • It was not an endless controversy but Wiles' proof of Fermat's last theorem contained a hole which was raised and he was allowed the privilege of filling it himself. – it's a hire car baby Oct 05 '17 at 11:34
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    There is also the objectivist vs the subjectivist views of probability and as such no paper which permits the choice of something "at random" is rigorously correct, since there is no such thing as "at random" as demonstrated by Bertrand's paradox https://en.wikipedia.org/wiki/Bertrand_paradox_(probability) which mathematicans routinely ignore; e.g. when saying that the probability of choosing a $1$ at random from the natural numbers is zero. – it's a hire car baby Oct 05 '17 at 11:36
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    [Technical subcomment to Robert Frost's very interesting comment: roughly speaking, 'objectivist' is to 'subjectivist' as 'frequentist' is to 'Bayesian'. That is, roughly, 'objectivist'='frequentist' and 'subjectivist'='Bayesian'.] – Peter Heinig Oct 05 '17 at 11:46
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    Though my understanding is based largely on just this article for the layperson, I believe there was apparently controversy around Kuranishi structures and a proof of (a version of) the Arnold conjecture for a time. Not endless but apparently non-trivial. – Oliver Nash Oct 05 '17 at 12:09
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    Would Shinichi Mochizuki's purported proof of the $abc$-conjecture qualify? – Stanley Yao Xiao Oct 05 '17 at 13:13
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    Relevant: https://mathoverflow.net/questions/13896/ – Steve Huntsman Oct 05 '17 at 13:23
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    If I judge from the votes in favour of closing it, the question itself is controversial :) – Denis Serre Oct 05 '17 at 14:52
  • I take it controversy over notation and representation are also not considered here? Otherwise, how to write (in this forum format particularly) composition of functional relations and whether the algebra with empty sub universe should be included in a class have been ongoing items of contention. Gerhard "Maybe More Endless Than Endless" Paseman, 2017.10.05. – Gerhard Paseman Oct 05 '17 at 15:18
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    Depending on interpretation of the question, foundational mathematics may be rife with examples (sets, axiom of choice, consistency, ...). – Kimball Oct 05 '17 at 17:15
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    I see three votes to close, and no comments explaining why one should want to close. Please, speak up! – Gerry Myerson Oct 05 '17 at 21:34
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    @Gerry Myerson: I voted to close because on my opinion it is poorly stated (what does it mean "ensless"? There were long controversies. How long is "endless"? ) and not within the scope of this site. – Alexandre Eremenko Oct 05 '17 at 23:07
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    @GerryMyerson: re "Please, speak up!": I for one did not vote to close, yet favorably see its having been closed. This isn't any criticism of the opening poster. The intention was well-meant, but I think that in its present form the question much too broad to be mathematically informative (it is so broad that it invites all sorts of 'opinion-pieces'). Such questions can be informative if technically formulated (with proof-theoretic/epistemological constraints imposed, and without inflammatory English words like 'controversy', 'famous', 'guilty', and another word too strong to cite. – Peter Heinig Oct 06 '17 at 06:02
  • Would the question be more acceptable were "endless" replaced by "unresolved"? It seems that doubts of this sort (with respect to his own work) were in part what motivated Voevodsky's program related to computer assisted proof verification. Additionally, there are some such controversies and those who do not work in an area do not always know what their status is, so the question is potentially useful, if properly channeled. – Dan Fox Oct 06 '17 at 13:31
  • How would you rate the computer based proof of the four colour theorem? Is it accepted as a proof? Does one still look for a proof that does not rely on the use of a computer? – Joel Adler Oct 06 '17 at 14:09
  • @DanFox. Could be. It turns out that English is not my mother tongue. Sometimes, I employ a word as it was a french one. Feel free to edit the question. Your concern was raised also by Peter Heinig. – Denis Serre Oct 06 '17 at 14:12
  • @JoelAdler. I don't have problems with a computer-aided proof if the community (in this case, that of graph theory) is convinced. My concern is more that in some cases, the community might not reach a consensus of whether a proof is reliable or not. And such a proof could be computer-aided, or not. – Denis Serre Oct 06 '17 at 14:15
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    The controversy about the correctness of the proof of the classification theorem for the finite simple groups lasted from its announcement 1983 until 2004, where the last known gap has been filled. See e.g. https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups. I don't know if this qualifies, since that classification comprises hundreds of articles. – Martin Seysen Oct 06 '17 at 18:53
  • @AlexandreEremenko "endless" by exact definition means "without end." Surely you know that. The proposed rephrasal to "unresolved" sounds reasonable. – Wildcard Oct 07 '17 at 07:30
  • @DenisSerre I suppose that there are requirements for a computer program to be used in a proof. Its correctness must be shown somehow. – Joel Adler Oct 08 '17 at 18:00
  • @GerryMyerson This is a list type question, with no objectively right or wrong answer. As such it could be a list of significant length, and the voting becomes a popularity contest. If the list becomes longer than a page, then you often find better and/or later answers languishing at the bottom while early and/or popular answers sit near the top. See https://meta.stackexchange.com/questions/124450 and https://meta.stackexchange.com/questions/238219 and, perhaps most salient, https://mathoverflow.net/help/dont-ask where this ticks many boxes for questions that shouldn't be asked. – Adam Davis Oct 09 '17 at 14:06
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    @Adam, some of the most informative pages here are the result of list type questions. – Gerry Myerson Oct 09 '17 at 21:55
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    @GerryMyerson You asked why people might be closing. Whether those reasons are valid is a topic for meta. Note that I do not have the ability to close, so I'm only providing a possible explanation. Consider contributing to questions such as https://meta.mathoverflow.net/questions/1325/why-this-question-is-not-welcomed-but-others-are-welcomed-in-mo so the community may further refine its requirements. – Adam Davis Oct 10 '17 at 01:43
  • While this is most certainly a math site, since you mentioned fluid dynamics, it's worth considering that when dealing with physics, experimental results trump mathematical purity to a degree. If the dispute you mention is ever settled, it will probably be settled by whether the theories that arise from it make accurate predictions about our world. If they do, then I expect someone will eventually resolve the mathematical issues. – jpmc26 Oct 10 '17 at 06:27
  • @RobertFrost, the en.wikipedia.org/wiki/Bertrand_paradox_(probability) is interesting in that it has similarities to the issues in Quantum Mechanics of Bell's Theorem and the way folk get confused about relative polarisation probabilities (which are around a circle). As stated, the mechanism drives the probability, and the misunderstandings. One can be too un-informed about a prior! – Philip Oakley Oct 10 '17 at 14:57
  • @PhilipOakley I'd not seen that before; it's very interesting. In the unlikely event you're interested my opinion on Q Theory, I think to amplify some miniscule phenomenon to the point we can make a boolean statement about it, we must be triggering our "boolean" decision on some minute switching component. This is the principle of how amplification works. So I suspect there's a continuum going on down there but we can only "see" the switches which we can build out of the waves. Based on the quick video I watched on Bell's theorem this can explain it. https://www.youtube.com/watch?v=5JCm5FY-dEY – it's a hire car baby Oct 10 '17 at 20:35
  • Fermat famously wrote in his copy of Arithmetica that he had a proof too long to fit in the margin... – none Jan 01 '19 at 09:58
  • I know of plenty of examples of such controversies in topology -- at least one has been mentioned already. The controversies that stick around the longest often are the ones where it is difficult to state what the problem is. In Freedman's work you see it in complicated iterated induction arguments where one applies subtle topological operations. In other papers you might see a thought-process develop through the paper and if you're not fully into it, you get lost at the latter stages as the argument gets more complicated. I'm fairly agnostic about such controversies -- the truth is stable. – Ryan Budney Jan 01 '19 at 12:40
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    @DenisSerre A side question for my own interest: could you please tell us about the example you have about wave stability in PDE models for fluid dynamics? Not that I want to use the example to answer your question. It's more that I am interested in those topics and would like to read about it. – Gateau au fromage Jun 26 '19 at 13:03
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    Gromov's famous and great book "Hyperbolic groups" contains a lot of statements with somewhat sketchy proofs (or no proof at all, such as "by an easy argument"), and sometimes statements that are technically false (missing assumptions). This applies to other famous works of Gromov. "What should we do in order to salvage mathematical topics that suffer such tensions?" Simply, subsequent authors prove or correct these in detail. I'm not sure there was any tension in this case, but it's largely thanks of the work of subsequent authors. – YCor Jun 01 '20 at 18:32
  • Since Abbas Bahri's death, his web site is no longer available, but the paper I mentioned seems to have been published in Advanced Nonlinear Studies, Volume 15: Issue 2. – Ben McKay Jun 01 '20 at 19:07
  • Adjunct question: Examples of instances in which authoritarian shortsightness/turf wars impeded progress in a field? E.g., Heaviside's mathematics on transmission of signals in telegraph cables, and the history of the Belousov-Zhabotinsky reaction. – Tom Copeland Jan 28 '21 at 14:33

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(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.”

Steve Huntsman
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    It is my understanding that some players in the story had approached Fukaya prior to 2012 with concerns, but these concerns were taken less seriously by Fukaya back then. – Student Oct 05 '17 at 14:42
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    I don't know anything about symplectic geometry, but according to the Quanta article, the mathematical part of this controversy has now been resolved, and so it might not quite count as an "endless controversy." – Timothy Chow Dec 31 '18 at 19:23
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The Jordan curve theorem asserts that a simple continuous closed curve separates the plane into two distinct connected open sets. Whether Camille Jordan's original proof is correct or not seems to be a subject of controversy even today.

Here is a 2007 article by Thomas Hales that "defends Jordan’s original proof of the Jordan curve theorem" (from the abstract) and contains opinions from mathematicians who have pondered on the matter.

Here is an excerpt of the paper.

In view of the heavy criticism of Jordan’s proof, I was surprised when I sat down to read his proof to find nothing objectionable about it. Since then, I have contacted a number of the authors who have criticized Jordan, and each case the author has admitted to having no direct knowledge of an error in Jordan’s proof. It seems that there is no one still alive with a direct knowledge of the error.

You can find the proof of Jordan in his cours de l'ecole polytechnique p92ff if you want to form your own opinion.

Don't forget to contribute your own short proof of the Jordan curve theorem, and receive my applause for adding to the long list of proofs of the Jordan theorem, which are probably not short enough, and that contain an argument that is completely trivial (Hales), unsatisfactory to many mathematicians (Veblen), essentially correct (Reeken), invalid (Courant and Robbins), incorrect (M. Kline), not sufficient (another Kline) or simply in need of additional details (Mathoverflow).

coudy
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    I don't see a proof of the Jordan Curve theorem at page 92, are you sure? – Jack M Oct 05 '17 at 21:35
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    @Jack The proof starts at page 92 and covers paragraphs 98-104. You may want to start from paragraph 96 at page 90 for the context. The theorem is actually stated as the last sentence of paragraph 102. Paragraphs 98-102 show that a simple closed curve is between two polygonal simple closed curves. – coudy Oct 06 '17 at 20:26
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Edit: a (methodo-)logical proposal to make this thread more transparent

It can be argued that, broadly, there are three quite distinct 'types' of such controversies (and I propose that each answer in here gets tagged, by the respective contributor, if so inclined, by one of the following three tags):

(non-sequitur) This is the nightmare of anyone who has had to referee a long submitted paper and felt the responsibility to make a judgemental statement about the 'Is it true?'-part of a referee's three Littlewoodian responsitbilities: the proof contains many true things, but the goal seems not to be reached, but it is so difficult to justify why one is not convinced, all one can say is 'I am not convinced'.

(propositional-contradiction) By this I mean that the result contradicts, on the coarse boolean level of propositional logic, another published result, and both proofs are long, so ferreting out the error is literally a dilemma, a διλήμματος, with two horns (which most of the time, sadly, are not so easy as to be formalizable in Horn logic). This the dream of anyone who has to refereee a long paper, since then there is an undisputable and documentable reason why one cannot give the go-ahead, if the traditional standards of truth are to be upheld at all (which they should), namely that propositional logic is a conditio-sine-qua-non, something like 'checking an arithmetical calculation modulo two'.

(many-small-gaps) By this I mean that neither (non-sequitur) nor (propositional-contradition) are applicable; the overall line of argumentation is convincing, and, by itself, the claimed conclusion seems credible, too, especially as there is no other proposition proved elsewhere which would propositionally contradict it, but there are lots of small mistakes. This is something between the dream and the nightmare: one can then with good conscious recommend publication, or at least, a second round, but the task of patching up all the small errors still is nightmarishly work-intensive.

None of the above three seems to imply any of the others. On a rough intuitive level, these seem mutually distinct 'types' of controversies around a manuscript (in my experience).

I'll 'tag' my proposed contribution to this thread with the second-named 'type'.

A proposed contribution to this thread.

(propositional-contradiction) With trepidation (since I am only beginning to understand what the real issues are), and due respect, let me mention one of the most famous examples these days. To repeat myself: I know that there are many many others round here whom it would behoove more to mention this.

Endlessly fascinatingly- and fertilely-controversial is:

M. M. Kapranov, V. A. Voevodsky: ∞-groupoids and homotopy types. Cahiers de Topologie et Géométrie Différentielle Catégoriques (1991) Volume: 32, Issue: 1, page 29-46

Now the question is of course whether this qualifies as 'endless controversy' since even one of the authors readily acknowledged that there was an error, but a fruitful error, indicative of the traditional methods (both the formal-methods and the social-methods) being inadequate to give 'durable wings' with which to do the 'flights of fancy' (in a positive sense) of higher category theory.

But, while still learning some of the relevant subject matter (and, myself, being mostly working to understand the comparably humble example of the unambiguous interpretability of pasting schemes in good-old-bicategories), I think I can recognize that the above example satisfies each of the requirements

  • famous (why? look around...)

  • endless (why? since this dedicated MO thread seems so unconclusive (to me); after as yet 2624 views on a professional focused site, said thread contains only a "guess" and detailed confirmation *that there is an incorrectness in the sense of propositional logic but it still seems not clear (to me) how to pin down the reason for why the authors 'went wrong'.

  • controversial (why? since one of the authors himself in public lectures said that at first he did not take Simpson's statement that something was wrong serious, rather thought that it was wrong to state that something was wrong; what is endlessly fascinating about this example is the expressiveness of the mathematics which gave rise to this 'controversy')

  • significant (why? because, similar to e.g. Poincaré fertile errors in 'Analysis Situs' and the 5 subsequent 'patches', Kapranov-Voevodsky's error turned out to be a fertile error, for example by motivating one of the authors to find an alternative formal system for mathematics)

A micro-summary is given on a page hosted by the Institute of Advanced Study in Princeton:

During these lectures, Voevodsky identified a mistake in the proof of a key lemma in his paper. Around the same time, another mathematician claimed that the main result of Kapranov and Voevodsky’s “∞-groupoids” paper could not be true, a flaw that Voevodsky confirmed fifteen years later. Examples of mathematical errors in his work and the work of other mathematicians became a growing concern for Voevodsky, especially as he began working in a new area of research that he called 2-theories, which involved discovering new higher-dimensional structures that were not direct extensions of those in lower dimensions. “Who would ensure that I did not forget something and did not make a mistake, if even the mistakes in much more simple arguments take years to uncover?” asked Voevodsky in a public lecture he gave at the Institute on the origins and motivations of his work on univalent foundations.

Voevodsky determined that he needed to use computers to verify his abstract, logical, and mathematical constructions. The primary challenge, according to Voevodsky, was that the received foundations of mathematics (based on set theory) were far removed from the actual practice of mathematicians, so that proof verifications based on them would be useless.

The

fifteen years later

seems to approximate "endless" rather closely.

Again, my apologies if this is off-topic for some reason that I do not see, and I know it is debatable whether this counts as endless controversy, maybe indefinite fertility would be a more fitting heading for this example.

Community
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Peter Heinig
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  • On the subject of formal verification, Neil Strickland gave a talk on the subject. I don't think it's a great talk, but he does mention some of the same qualms as Voevodsky, if that's the way to put it. Personally I don't think formal verification has all the answers, and not only because there is obvious dispute about foundations. I think it is partly, perhaps mostly a problem of culture. – James Smith Oct 06 '17 at 18:33
  • @JamesSmith: thanks for pointing out. Personally, I like this talk of Neil Strickland very much and watched it at least three times over the last months. I don't think it is an accurate summary to say that he 'mentions some of the same qualms as Voevodsky'. In my understanding of the English language, 'qualm' is always used ironically and criticizingly to mean 'unjustified concern'. I think the concern are justified. While I recognize you didn't say the proponents of formalization say they have "all the answers", this reminds me of (please do not take offense) a 'straw man' argument. [...] – Peter Heinig Oct 06 '17 at 18:51
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    [...] Surely not an intentional 'straw man', yet this is some of those patterns of arguing where an extreme view, which none from the 'other side' ever voiced, is negated. Again, I am not criticizing you, but one should say that the new 'school' of formalization is informed, humble, and 'embraces' incompleteness, and diversity, and what not. They know what they are doing. One of the deepest things VV has said in the relevant lectures is that this is all about reasoning reliably with inconsistent formal systems. This statement contains much in one sentence. We shouldn't continue here. – Peter Heinig Oct 06 '17 at 18:53
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    Sorry for the off-topic comment, but: as a native English speaker, I don't think the word "qualm" has a connotation of irony or unjustified concern. For me, it does sometimes have a connotation of a vague or imprecise concern. But I would have no qualms about saying that I have qualms about a proof of type (non-sequitur). – Alex Kruckman Oct 07 '17 at 03:52
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    My experience, and the definition in Wiktionary (sense 3), agrees with @AlexKruckman. – LSpice Jan 01 '19 at 01:14
  • I don’t know what to make of the nonsequitur type. If the referee is unable to formulate a specific objection or question, isn’t that more their own problem? – Monroe Eskew Jan 01 '19 at 11:14
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    @MonroeEskew: well, there are degrees of clarity and thoroughness of a proposed proof. Needless to say, most proofs still are not being written in a formal proof system, and hence no step in usual paper can be verified mechanically, but needs human judgement to assess. By non-sequitur I mean that (1) an author makes a claim C, (2) an author proposes a proof P, (3) P does not contain any wrong statement, (4) the referee disagrees with the implicit claim that P proves C. Often, this is not easy to justify. Sometimes, the author then writes to the editor denouncing the referee as incompetent. – Peter Heinig Jan 05 '19 at 15:14
  • @MonroeEskew: to a large degree, the non-sequitur type of controversy is nothing else than the familiar platitude that in sentential logic (aka propositional logic, aka zeroth-order logic), ANY true statement follows from ANY true statement. Of course, this kind of coarse implication is not accepted as a mathematical proof. As there is no generally recognized standard of the 'resolution' with which the implication ought to be explained, controversies still abound in practice. – Peter Heinig Jan 05 '19 at 15:21
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    @PeterHeinig I was taught early on that we can resolve such controversies by inching a bit closer to the application formal deduction rules to precisely stated assumptions. I was convinced that in principle, there is a rock bottom to hit. – Monroe Eskew Jan 05 '19 at 15:26
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Stanley Yao Xiao's comment has been upvoted so highly that it seems worth posting as an answer.

There is a currently unresolved controversy over Shinichi Mochizuki's claimed proof of the abc conjecture. In a ten-page note, Peter Scholze and Jakob Stix have stated:

We, the authors of this note, came to the conclusion that there is no proof. We are going to explain where, in our opinion, the suggested proof has a problem, a problem so severe that in our opinion small modifications will not rescue the proof strategy.

Mochizuki, however, maintains that there are no problems with his proof and that Scholze and Stix suffer from "fundamental misunderstandings."

UPDATE 1: Mochizuki's work has been accepted for publication in PRIMS, even though most number theorists believe that the proof is incomplete. For more details, see this April 2020 post on Peter Woit's blog.

UPDATE 2: Mochizuki's papers were published in PRIMS in March 2021. However, the publication of the papers has not ended the controversy; see the March 2021 post on Peter Woit's blog.

UPDATE 3: NHK aired a documentary (in Japanese) in April 2022. It tried to present "both sides of the story," though according to the reviews I've seen, it presented Mochizuki's theory as being "too difficult to understand" rather than having a huge gap. Again, there is some additional commentary on Woit's blog. In other news, there is a guest post by Kirti Joshi on David Roberts's blog which, while not resolving any of the big issues, presents some work that Joshi says "provides new evidence regarding Mochizuki’s work."

Timothy Chow
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    Mochizuki being the editor-in-chief of PRIMS does not contribute to the validity of his work, to put it mildly. – Rafał Gruszczyński Feb 03 '23 at 16:21
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    What muddies the water is that either of the two main parties in the dispute may have flaws in their argument, but have the correct conclusion. Both of these opinions have been expressed to me by people who I know are not just leaping to a conclusion! – David Roberts Feb 06 '23 at 11:19
  • Then again, if I were told in Japanese that my proof is "too difficult to understand" or that "lemma X seems difficult", I would immediately interpret this as my interlocutor trying to convey that my proof is faulty without upsetting social norms. – Olivier Jun 14 '23 at 06:37
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As far as I know, Wu-Yi Hsiang still maintains that his proof of the Kepler conjecture is complete and correct. Perhaps this does not quite meet your criteria because it seems that nobody other than Hsiang believes that his proof is complete and correct; is that enough agreement to use the word "consensus"?

In the introduction to his delightful anthology, 18 Unconventional Essays on the Nature of Mathematics, Reuben Hersh states, regarding the Flyspeck project of Thomas Hales, that he does not know anyone who either believes that the project will be completed or that, even if claimed to be complete, it will be universally accepted as definitively verifying the correctness of the proof. Of course, the completion of Flyspeck was announced in 2014. My impression is that most people accept that Flyspeck has settled the Kepler conjecture, but there are probably still some skeptics. Back in 2008, I had an email exchange with someone who pointed out that HOL Light is based on OCAML, and that the formal correctness of OCAML has not been established. It may be that there will always be a nontrivial minority of mathematicians who remain skeptical of results whose only proof is not "surveyable" or "humanly comprehensible," in which case such results may remain permanently controversial. (One could imagine that one day the Kepler conjecture will have a humanly comprehensible proof, but there will surely be other results, e.g., in extremal combinatorics, that are unlikely to have any proof other than an immense computer verification.)

Timothy Chow
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    This reminds me of de Branges – Steve Huntsman Oct 05 '17 at 21:14
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    @SteveHuntsman: All correct proofs are alike; every incorrect proof is incorrect in its own way? – Lucia Oct 06 '17 at 02:54
  • I think the whole point of Hale's blueprint proof is that it is humanly comprehensible, certainly so compared to the original submitted to Annals. I've posted a link to Hale's own talk on the subject somewhere on this page, so I think best not to post it again, but it is well worth watching. It is interesting to note your comments about HOL light not itself being fully trustworthy. I didn't know that, I had assumed that it was. – James Smith Oct 15 '17 at 12:46
  • @JamesSmith : Some time ago Harrison provided a verification of the HOL Light kernel without definitions. However, since HOL Light runs (or originally ran) on top of OCAML, you might wonder about the correctness of OCAML. On the bright side, I gather that there has been some progress in this direction since 2008, e.g., https://link.springer.com/chapter/10.1007/978-3-319-08970-6_20 Regarding the "blueprint proof," it is humanly comprehensible, but there is still an enormous computer calculation involved whose results you simply have to trust. – Timothy Chow Oct 15 '17 at 19:18
  • I agree. However, it is worth pointing out in passing that the software making the calculations was also verified as part of the Flyspeck project. – James Smith Oct 15 '17 at 20:11
  • Leaving that aside, I couldn't prove the Kepler conjecture to my own satisfaction because the amount of effort to assemble and run the software to verify it is considerable. I simply gave up. In my humble opinion the blueprint proof will really come to life when the software used to verify it is simple and accessible to non-experts. We are still some way from that. A distance ambition of mine is to see it happen, though. – James Smith Oct 15 '17 at 20:15
  • Sorry one more thing I should mention to no-one in particular. On re-reading your answer, I see shades of my belief that the acceptance or otherwise of a complex and lengthy proof is not necessarily settled by formal verification. I've written elsewhere that I think that this is as much of a cultural issue as a technical one but again I can't find the words. However, I do strongly believe that it is bound, at least in part, to the accessibility of the software to mathematicians who aren't experts in formal verification. – James Smith Oct 15 '17 at 20:21
  • @JamesSmith : Sorry to be pedantic, but I think people could get confused by your statement that "the software making the calculations was also verified." This is true in the sense that the calculations aren't just being executed by some C code or something, but are being subject to verification by the proof assistant. But on the other hand, everything is being programmed and run in OCAML, and it is not the case that the implementation of OCAML has been formally verified to be correct. – Timothy Chow Oct 16 '17 at 01:35
  • Hi, Timothy. I was referring to the software that made the numerical calculations this time, as you rightly point out. But yes, since all of the software was verified with a verifier written in OCAML, of course there is still an element of doubt. There's no need to apologise for being pedantic, it's par for the course. – James Smith Oct 16 '17 at 09:29
  • It's perhaps worth noting that there is a document called "Defects in the revised definition of Standard ML" that, while it doesn't refer to OCaml, certainly ought to highlight that even a language that is "precisely defined" enough to have a book devoted to the definition has some lacunae. – Steve Huntsman Oct 27 '17 at 20:03
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F. Enriques' claimed in 1904 that, given a smooth projective surface $S$ with irregularity $q>0$, what we call nowadays the Picard scheme of $S$ is an abelian variety of dimension $q$.

Enriques' algebraic proof was considered controversal and led to many disputes among the geometers of the Italian school (in fact, they called Enriques' claimed result the fundamental theorem in the theory of irregular algebraic surfaces, or also the theorem of completeness of the characteristic series).

Actually, Enriques' proof contained lacunae, as well as the subsequent proof by Severi. The first correct proof, using transcendental methods, was given by Poincaré in 1910. For a correct algebraic proof in characteristic $0$ it was necessary to wait for the work of Grothendieck, 50 years later.

Today it is known that the result is false in positive characteristic, as shown by Igusa in 1955. In fact, he constructed a smooth projective surface $S$ with $\mathrm{Pic}^0(S)$ non reduced, and hence not an abelian variety.

Faheem Mitha
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Francesco Polizzi
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    There are two lovely articles in the Notices of the American Mathematical Society about the history of this theorem: http://www.ams.org/notices/201102/rtx110200240p.pdf and http://www.ams.org/notices/201102/rtx110200250p.pdf – Jason Starr Oct 05 '17 at 12:59
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    To help those readers who are as uncultured as I am: "lacunae", the plural of "lacuna", means "gaps" (I had to look that up). – John Pardon Oct 05 '17 at 14:32
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One classic example of this, though now resolved, is Euler's polyhedron formula $V - E + F = 2$. The formula initially was asserted without qualification, and gradually people began to point out cautiously (I suppose in deference to Euler) that there were "exceptions" to the theorem. The history of this formula is explored in Imre Lakatos's lovely book Proofs and refutations, written in the form of a drama, but with copious historical references.

This was a fertile error in the sense given above by Peter Heinig.

Robert Haraway
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Nearly two decades after the publication of Adam Elga, Self-locating belief and the Sleeping Beauty problem, Analysis, 60(2): 143-147, 2000, there is apparently no consensus as to the resolution of the problem. See Peter Winkler, The Sleeping Beauty Controversy, The American Mathematical Monthly, Vol. 124, No. 7 (August-September 2017), pp. 579-587

Gerry Myerson
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  • Sleeping beauty paradox discussed at Cross Validated: https://stats.stackexchange.com/questions/41208/the-sleeping-beauty-paradox – kjetil b halvorsen Oct 05 '17 at 22:21
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    This seems to me to be a philosophical controversy rather than a mathematical one. – Timothy Chow Oct 05 '17 at 23:09
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    @Timothy, maybe so, though it has been the subject of an article in the Math Intelligencer (J S Rosenthal, A mathematical analysis of the Sleeping Beauty problem, 31 (2009) 32-37) and Winkler's essay in the Monthly. – Gerry Myerson Oct 06 '17 at 03:07
  • The Sleeping Beauty problem looks to hav a lot of similarities to the Monty-Hall problem, but done in reverse, with a slightly different twist as to what one needs to (not) know to change a 1/3 probability into a 1/2 probability, and vice versa – Philip Oakley Oct 10 '17 at 14:32
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    @Philip, I think a difference is that once one states Monty Hall carefully there is one answer that is clearly correct, whereas there seems to be genuine controversy about Sleeping Beauty. – Gerry Myerson Oct 10 '17 at 21:56
  • @GerryMyerson, The similarity with Monty Hall (a game show host) is the misdirection of the audience and the contestants as to what the over all scenario is. Both cases have a three (boxes/sleeps) versus two (still-closed-boxes/sides-of-a-coin) confusion of human comprehension. If one states the problem too carefully, then neither are a problem! – Philip Oakley Oct 10 '17 at 22:58
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    @Philip, if you have a conclusive argument to settle Sleeping Beauty, I would encourage you to publish it. – Gerry Myerson Oct 11 '17 at 00:50
  • @GerryMyerson The paradox is solved thus: She knows in advance that she will be double-sampling from the tails. Therefore the strategy to take over multiple samples, in order to maximise accuracy, is one third. But if you were to ask her this question: "when we ask you what the coin turned up, please bear in mind that if it transpires that we have asked you twice about the same coin toss then your two answers will only be given the same weight as the one answer you give in relation to some coin toss about which you are only asked once" then she should answer half. – it's a hire car baby Oct 13 '17 at 09:43
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    So it's a paradox of whether accuracy of probability is measured per coin toss or per "ask". – it's a hire car baby Oct 13 '17 at 09:45
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    The paradox arises because the problem involves two random variables: C, the result of the coin toss, and D, the current day of the week). Since SB will exist (but not necessarily be conscious) for both Monday and Tuesday, the prior distribution of (C,D) is equally distributed among all four possibilities (Heads,Monday), (Heads,Tuesday), (Tails,Monday), (Tails,Tuesday). Thus the prior probability of heads is 1/2. But when SB learns that she is conscious today, she now knows that (Heads,Tuesday) is eliminated, and the posterior probability of heads is now 1/3. – Terry Tao Dec 31 '18 at 22:50
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    Terry (and Winkler) focus on probability. But the decision-theory aspect of the paradox is important. E.g. (Conitzer 2015) shows how evidentiary decisionmaking arguments that are used to justify a belief of 1/3 are vulnerable to Dutch books in some variants of the paradox. I think it is about justifying "locally correct" beliefs that are also globally consistent (or at least not exploitable). [It is easier to tell sleeping beauty how to gamble optimally than to tell her how to form beliefs such that she gambles optimally.] – usul Feb 06 '23 at 14:29
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A seemingly endless controversy that has already lasted 15 years surrounds Yaroslav Sergeyev's proposed theory of infinity, first published in 2003. He has over twenty publications on this topic. Critics maintain that his theory is inconsistent, with whatever valid material actually derived from earlier rigorous sources. He also has some backers including one that posted an answer at MO.

Complete disclosure: I am one of the authors of this 2017 publication in Foundations of Science critical of Sergeyev's production.

Some other possibilities are mentioned in this question.

Mikhail Katz
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