# Is there a way to discuss the correctness of the proof of the RH by Atiyah in MO?

I just made a question in MO to discuss the correctness of the proof provided by Prof. Atiyah for the Riemann hypothesis (link here: Is there an error in the pre print published by Atiyah with his proof of the Riemann hypothesis? ). I understand that's usually outside the scope of MO, but this time is the work of an Fields medalist claiming to solve one of the most important math problems currently. Is there a way to reword my question so it could be discussed here? In case the answer is negative, could someone provide a forum with similar quality where such proof could be discussed? Thanks and sorry for the problems caused.

Héctor

• Héctor, I think this one is for the blogs, if anyone wants to take it up. Atiyah has done some of the greatest mathematics in the past 50+ years, but it looks very likely that here he's "not even wrong", and it is for this very reason that he should be accorded some dignity in view of his epoch-making past work -- not a public raking at MO. That's just my opinion.
– Todd Trimble Mod
Sep 23 '18 at 21:01
• The New Scientist article said it best: "New Scientist contacted a number of mathematicians to comment on the claimed proof, but all of them declined." I don't think the professional mathematics community wants to go near this, for good reason. It's a sad situation. If you really want to discuss that preprint, try a more general forum like Reddit r/math.
– none
Sep 23 '18 at 21:40
• Speaking of Reddit: reddit.com/r/math/comments/9icamx/… No need to wait until the lecture, there are preprints (yes, two) already. I think we should cut the guy some slack and not drag this through the mud. The preprint is dedicated to his late wife, who died earlier this year... :-( Sep 23 '18 at 23:08
• The only thing I'm willing to say is that the organizers of the event aren't my heroes (while Atiyah still is). Sep 24 '18 at 2:13
• @ToddTrimble: Can you please post an answer to this meta question, announcing your policy, so that it is easy to see, and can be voted on? Sep 24 '18 at 5:14
• I completely understand the reaction of mathematicians here, but Atiyah's claim got a lot of press. We live in a different era now, where everything is public. It doesn't necessarily have to be Math Overflow, but someone somewhere has to go on record discussing the claims, before it ends up on Alex Jones as part of some bizarre conspiracy theory. Sep 24 '18 at 7:10
• It would help if someone close (professionally) to Atiyah, who knows the situation for what it is, were to address this issue publicly. I work in an engineering school and today already several faculty members have asked me about Atiyah's proof of the Riemann hypothesis, having read about it online. It's important that the underlying situation (whatever it is - although I can guess, I do not know) be made clear, so that the reputation of a great mathematician is not damaged, and the credibility of the mathematical community is not adversely effected. Sep 24 '18 at 10:48
• "but someone somewhere has to go on record discussing the claims" There isn't much to discuss, sadly. If you want somebody on record for MO, I can say it openly that the properties of $T$ are self-contradictory. As to the general public opinion and "conspiracy theories" I couldn't care less: we may be living in a "public era" but our craft has none of these two words in its vocabulary. As to "damage to the reputation", IMHO, all of it goes to the HLF organizers and if somebody asks me, I wouldn't hesitate even a second to attribute it there. Sep 24 '18 at 11:38
• I think, that MO community must have some mathematical point of view on what happened. Atiyah is doing this not for the first time! He claimed two years ago that 6-sphere has no complex structure: arxiv.org/abs/1610.09366 . I never saw a professional statement from some prominent geometer that this paper is wrong. I think that trying to keep this situation under the carpet will be damaging for the math community on many levels. Some people all over the globe will spend time reading the text. A duty of math community will be to repeat $10^{10}$ times that the paper is wrong (I guess) Sep 24 '18 at 12:02
• @Dmitri, MO community $\ne$ math community. Sep 24 '18 at 13:13
• @Dmitri I think you're wrong. Robert Bryant has cast severe doubt on the correctness of Atiyah's work on the question of whether $S^6$ has a complex structure: mathoverflow.net/questions/304071/…
– Todd Trimble Mod
Sep 24 '18 at 14:20
• @ZubinMukerjee "Not even wrong" is a well-known expression, attributed to Wolfgang Pauli. en.wikipedia.org/wiki/Not_even_wrong It's a surmise on my part that it's not just that something was overlooked in his proof, but it seems hard to make any sense of it at all. But I understand your frustration. People are seemingly being cryptic because the situation is embarrassing. (And that's all I'll say about this for now.)
– Todd Trimble Mod
Sep 24 '18 at 19:29
• @JoséHdz.Stgo. I know you posted in good faith. In fact I think just about everyone participating in this discussion is acting in good faith. As mentioned above, I cast a 5th vote to delete, in the name in consistency since other questions which asked about the truth of Atiyah's proof were also deleted. The situation here is very raw and delicate. Please stay tuned for an answer from the moderators that will hopefully clarify our position. I do not think the post about Rademacher is quite in the same category, as he has been dead for almost 50 years.
– Todd Trimble Mod
Sep 24 '18 at 22:23
• @JackBauer no not the same way, for reasons that should be apparent from the thread.
– none
Sep 26 '18 at 16:18
• @user40276 You're making assumptions about the situation. I will now ask that the public speculations be brought to a close, and that everyone keep his or her own counsel. (I may in fact begin deleting a bunch of comments, or moving them to chat.)
– Todd Trimble Mod
Sep 27 '18 at 0:47

After some internal discussion, the moderators have agreed to post the following text.

There are calls from the community to enunciate a firm and consistent policy for what questions are permissible or not regarding Atiyah's attempted proof of RH.

If for a moment we blind ourselves to the name on the paper, the natural and appropriate reaction would be to dismiss it out of hand, without fuss and without worry that we were missing out on anything. In the name of consistency, I think that would be the right reaction here as well, and that should really be the policy. So any question that asks for responses on whether this is right or wrong, or if there is something there, should be treated in the way it's always been treated: we at MO do not vet papers that purport to solve famous open problems. And in the case of 5-page articles that announce a solution of RH: we don't discuss them at all. The question will be closed.

My personal opinion is that the situation and the appropriate reaction is made only slightly more complicated by knowing it was Atiyah. There are indeed allusions to sophisticated concepts such as the Todd genus (there is a "Todd function" which is somehow related to the Todd genus). Careful and precise questions about such concepts, at a professional level, are always in principle allowable, and they may mention that some such concept was spotted in Atiyah's article, but these should not turn into cover for discussing the current topic.

We don't think it's possible to spell out policy in advance that will cover every such circumstance; some matters have to be treated on a case-by-case basis, with input from the community. (What's the expression? "Hard cases make bad law"?) Please: just be prudent, and avoid drama.

We have no doubt there will be those who cry "censorship!", "suppression of free thought!", etc. Or, less dramatically: how do you know there's nothing there? Well, none of us know for a fact, and obviously Atiyah's track record from when he was in his prime will give some people pause: for many years he has led mathematicians across difficult country. There may well be a tragic aspect to what is unfolding before us. Let's please preserve some dignity for the man.

Don't worry: if there is truly mathematical takeaway, it will be picked up by the experts. RH will never be solved in a day: any serious solution will take months or years to be pored over and analysed by thousands at home, in offices, in seminar rooms around the world. MathOverflow, especially where it is vulnerable to regrettable social media contaminants, can not play all the necessary roles here. We do not have to be the public bulwark against the conspiracy theorists. Let's keep doing what we do best: provide a clearinghouse to deal with well-circumscribed, sober mathematical questions that can be authoritatively addressed in less than three pages.

Any general public statements we want to make for now can go on this Meta page. If the mathematics begins to look serious, then well-delineated parts can be discussed at Main.

• Briefly addressing the actual question here directly: no, there is not a rewording of the question that will make it appropriate for MathOverflow in the near future. As for other forums, you can always start a blog. :-) But my personal advice would be to resist the temptation to pay this too much attention right now. Sep 24 '18 at 22:49
• Further, I am going to clean up this page slightly, deleting answers which are address the manuscript, talk, or proof, rather than the MathOverflow policy on this matter. Sep 24 '18 at 22:50
• If we have a special event in honor of Dr. Trimble, will that too be a Todd Function?
– Asaf Karagila Mod
Sep 25 '18 at 6:44
• @Asaf, I trimble at the thought. Sep 25 '18 at 7:25
• Just as a practical matter, what should we do if another question on this pops up on MO? Point a link to this and vote to close? Leave it to the moderators? Sep 25 '18 at 7:41
• @arsmath I put a link to this and voted to close. Sep 25 '18 at 8:21
• @arsmath You could also flag for attention.
– Todd Trimble Mod
Sep 25 '18 at 10:20
• the reference to "conspiracy theorists" seems a bit ridiculous. the conspiracy theorist probably couldnt care less about the affair or math in general. on other hand there might be a bigger issue of tabloids writing some Perelman-like articles on Atiyah in the not too distant future... at least you guys tried to address the issue more directly with this public statement & do appreciate the effort on that.
– vzn
Sep 28 '18 at 18:23
• With many of the most interesting questions on MO, the truth is, we don't know at the moment of raising them whether they can be authoritatively addressed and on how many pages. Still, many of them turn out to can, while many of the rest have generated interesting discussions and observations. I agree, of course, with the main point of the statement. (And none of this takes away anything from Atiyah's accomplishments.) Sep 30 '18 at 9:41
• Just wanted to say WELL DONE to the moderators for the way they've handled this. I don't know anything at all about the situation here, but it's clearly a tricky issue... Reading the discussion here, and the statement above, I really appreciate the considered and thoughtful way that things have been dealt with. Oct 1 '18 at 12:18
• re mod TT "I think this one is for the blogs, if anyone wants to take it up". then allow a few refs! oops, some mathematical bloggers such as Woit/ Tao et al have opted for total silence/ near blackout. think it would have been so much better if there had been some prior alert by insiders. filling the vacuum, very thorough bkg/ links on the affair vzn1.wordpress.com/2018/09/26/…
– vzn
Oct 3 '18 at 20:26

I will take a different viewpoint than some and say that the correctness of the proof can be discussed, and maybe it should. But as to how, that's a question of tactfulness and precision.

• When discussing this proof, it's incredibly important to avoid ad-hominem attacks, benevolent and malicious ones alike. Many people say stuff like "Atiyah has output a lot of unaccepted work in the last years, this is going to be no better" or "He is a Fields medalist for crying out loud, his word is holy!". Both is unacceptable here.
• Waiting for some time is entirely fine. Put questions on hold until the emotions have cooled down. Math is eternal.
• While in principle there is no problem discussing details from papers (even from preprints) this particular situation is somehow delicate, as anyone following past developments can confirm. I would suggest that holding our horses for a while is the best action we can do. Sep 24 '18 at 8:21
• Aside from other concerns, I’m uncomfortable with discussing a preprint before it has been made public by the author, or we are sure that it has been made public with permission of the author. Sep 24 '18 at 8:29
• @JeremyRickard As far as I could see (you know there were technical problems), the Heidelberg talk looks the same as the preprint Sep 24 '18 at 8:46
• And apparently (this was said in the talk) it was submitted to the arXiv, but not accepted, so it's clear the author wants this public... Sep 24 '18 at 8:48
• Math may be eternal but these papers are not math. That Arxiv rejected them but HLF accepted them says something unflattering about HLF. I'm with Fedja in assigning them a lot of blame for this debacle.
– none
Sep 24 '18 at 12:38
• @none I think that interpretation is very uncharitable towards both Heidelberg and the arxiv. Why should they apply the same criteria? They're completely different fora. A simple counterproposal: maybe Atiyah's paper was judged to look less like a fully fleshed-out preprint (because it wasn't) and more like a rough outline of / advertisement for a talk (because it was). Thus it was rejected as a preprint and accepted as a talk. Heidelberg may want to revise its criteria for acceptance after this, but we have no reason to doubt that they were applied fairly, consistently, and in good faith. Sep 24 '18 at 17:48
• Billy and Zubin, at least for established researchers, the arxiv is basically a self-publishing outlet where you upload your paper with no editorial review at all, and it becomes available to everyone on the server. So if they reject something it reflects an unusual situation (Atiyah's recent activities in this case) that HLF also had to be aware of. HLF still accepted the talk and promoted it on their twitter, which came across to some of us as cheap sensationalism, using Atiyah's fame and the claimed RH result to draw attention to their conference without regard to the consequences.
– none
Sep 24 '18 at 19:55
• "if they reject something it reflects an unusual situation (Atiyah's recent activities in this case)" This sounds like a conspiracy theory to me. I'm not sure I want to believe that the arxiv moderators are in the business of prematurely retiring mathematicians whose work has slipped. I'm far more inclined to believe that it was rejected from the arxiv because e.g. it was sketchily written and grossly under-referenced, not for reasons of mathematics, and that HLF accepted it on the basis of (misplaced but good-faith) trust in the mathematical claims. No contradiction there. Sep 24 '18 at 21:46
• (Note that I do agree with the things you said in the latter half of your last comment. I think there are good reasons to believe the HLF acted in bad faith at times. I just don't think that your comparison with the arxiv is a fair criticism. They're completely different institutions, with different goals, communicating via different media. There's no reason they should have all the same opinions.) Sep 24 '18 at 21:51
• Billy, when you say "HLF accepted it on the basis of (misplaced but good-faith) trust in the mathematical claims", I think that is basically accusing them of incompetence given the history. I can't say for sure that you're wrong, but if they are incompetent they shouldn't be running such conferences. Then you add "I think there are good reasons to believe the HLF acted in bad faith at times." Exactly. The outcome was foreseeable and is unfortunate. I think they did everyone a disservice.
– none
Sep 24 '18 at 22:34
• I cannot help but notice that some people here discuss matters that are almost tangential to the original question. We shouldn't be trying to blame someone, but find out how we should react. Sep 25 '18 at 7:16
• @ManuelBärenz yeah that is kind of a tangent, about how we got where we are. Regarding the original question: for reasons already discussed, it's a misdirection to treat this as a normal unconfirmed math result and discuss it in purely mathematical terms. The MO mods handled it the best way under the circumstances, imho.
– none
Sep 26 '18 at 16:15

I don't think there's any good solutions here, but I am going to suggest what I think is the least bad solution, and people can vote on whether it is a good idea or not. There are going to be lots of well-meaning people who want to give Atiyah the benefit of the doubt and are going to ask questions about the proof, so I think leaving it unaddressed on the Math Overflow front page will be difficult to sustain.

I suggest we create a new question that is Community Wiki just for this topic (and Atiyah's fine structure constant paper, since it seems to be related), and close any other questions about it as a duplicate of this question. Then the moderators carefully moderate the question. We avoid any discussion of Atiyah as a person, but people are free to post answers that address the correctness of either paper.

• Unfortunately I don't see how this approach could end up in anything but a public shaming of Sir Michael Atiyah, which I don't think the MO community wants to be associated with. I certainly don't. So, while I understand the problems that motivated this proposal I am not sure I can agree with it. Sep 24 '18 at 12:31
• @Denis It seems that instead he is receiving a public shaming from laymen across the internet. It may be that an authoritative (and carefully written) account from a professional mathematician minimizes the harm from the public interest. But of course, I do not know.
– mme
Sep 24 '18 at 13:13
• There's nothing intrinsically shameful about producing a proof that turns out to be wrong. We can dispassionately evaluate it without speculating about how it came about. Atiyah's announcement received a great deal of press -- multiple non-mathematicians have asked me about it. Junior people who are aware of Atiyah's reputation are going to ask here if the proof is right, and if it's wrong if it can be salvaged. If we refuse to answer questions about it, it will quite strange. Sep 24 '18 at 16:44
• Instead of a dispassionate evaluation, we are giving dark hints about Atiyah's state of mind, which I would imagine is a worse form of public shaming. Sep 24 '18 at 16:45
• I'm not sure we are darkly hinting at his "state of mind" exactly, not in any diagnostic sense. (Yesterday I watched a video of him addressing the last ICM. The audio was of poor cell-phone quality, but I was impressed by his physical vigor, and he told some good jokes and so forth.) But I think it's fair to say, based on the evidence before us, that there might be something tragic happening here. Let's just say that this is not how we want Sir Michael Atiyah remembered, and that we want to preserve some decorum, as well as speak truth. It's hard to do with all the noise.
– Todd Trimble Mod
Sep 24 '18 at 17:11
• What do you think "something tragic happening here" sounds like? It sounds a lot worse than "the proof has a gap here". Sep 24 '18 at 17:44
• Might be something tragic. What I'm saying is that that possibility must be borne in mind, whatever we do. (There can be various explanations, but it's not for us to speculate.) I can think of several cases in the life of scientists where the quality of their science at some point in life had a tragic aspect, without in any way imputing mental illness or anything like that.
– Todd Trimble Mod
Sep 24 '18 at 18:30
• Well, there has been already a lot of media attention to the purported proof, this will become part of the record. Addressing the purported proof mathematically seems much cleaner than the constant patronizing. Sep 24 '18 at 18:35
• @arsmath I'm not sure to what extent you are making a general argument on principle, but while I agree in principle that "There's nothing intrinsically shameful about producing a proof that turns out to be wrong", reading both of the documents produced by Atiyah ought to show you (from what I remember of your own mathblogging) that they are qualitatively different from e.g. Dunwoody's briefly claim in 2002 to have found a proof of the Poincare conjecture - and not in a good way. I feel able to say this factually, without making any insinuation or inference about Atiyah's state of mind. Sep 24 '18 at 19:26
• @MichaelGreinecker Initially I was in the camp of those who preferred not to make a big song and dance about the announcement and the purported proof, but I am coming round to the point of view in your last comment. I hope it is possible to point out that certain statements simply do not make sense while still allowing the possibility that there are some underlying ideas which have been poorly or vaguely presented Sep 24 '18 at 19:29
• @YemonChoi I'm contemplating the same thing. However, I am very very very far from convinced that it's up to MO to do the job. As I see it, if we try to view this in purely mathematical terms, the choices are (1) this proof makes little to no sense to anyone who could plausibly weigh in, or (2) some such person says, "yes, I see what Atiyah is attempting to do", but points out some step or part that is mistaken or irreparable. I see "the proof has a gap here" as feasibly applying only to possibility (2), whereas there's a real question as to what expert will stand up and declare (1).
– Todd Trimble Mod
Sep 24 '18 at 20:09
• I also see a CW mechanism at Main as not well adapted to the task, for reasons somewhat akin to Feynman's parable of the Chinese emperor's nose length. Ideally it would be best if only experts weighed in. Would that happen? Seems pretty unlikely to me. And then to put it up to votes. Seems like a real mess to moderate as well.
– Todd Trimble Mod
Sep 24 '18 at 20:13
• @YemonChoi I agree with your assessment of the manuscripts. I'm just trying to find a way to accurately communicate the status of the proof while simultaneously minimize the discomfort many people feel with the situation. Sep 24 '18 at 20:29
• Amplifying what Todd Trimble says, I think it's basically a category error to treat these papers as math proofs that can be checked for mistakes. And besides the Chinese emperor's nose (I had to look that one up), we're in something like the blue-eyed islanders' puzzle. It's outside of our boundaries as math people to discuss anyone's eye color in public even though we might reach a conclusion about it privately. So we are left with a dilemma.
– none
Sep 24 '18 at 20:58
• @none Your first sentence is exactly what I meant.
– Todd Trimble Mod
Sep 25 '18 at 0:36

I think the questions are ill timed. Andrew Wiles claimed to have a proof of Fermat's Last Theorem in 1993. It turned out he was wrong, and that it took (at least) weeks of work to produce a proof that is now part of the literature. Wiles deserved and got recognition for the amount of effort he put in and results he got out, even in 1993, but the public (and even most mathematicians) would not understand or appreciate what he did up to that year. If the masses need to understand anything, it is that it takes time to turn claims of a proof into something that properly belongs to the literature.

I think Sir Michael Atiyah has some ideas worth hearing. For reasons personal to me, I do not want to hear them from Sir Michael directly. I do not want to even see them on MathOverflow until they have gone through a filter that makes their presentation suitable for this forum. That means someone other than Sir Michael finds a way to present the ideas, or parts of them, in a fashion that is more clear, more self contained, and more about the math than the media. The recent questions are reasonable ones to ask after enough time has been given to apply such a filter. This means days at a minimum.

Gerhard "Proofs Are Much Like Cakes..." Paseman, 2018.09.24.

• Can someone explain why this answer has so many downvotes? Sep 24 '18 at 19:54
• @Vincent, I didn't downvote, but perhaps because it seems that only the first and last sentences (and arguably the first paragraph) address the question that was asked. Sep 24 '18 at 21:57
• I downvoted because I thought the post showed unjustified optimism about the papers' math content, leading to an imho unrealistic prescription for how the situation should play out. That said, Gerhard was at the ICM talk so he may well be better attuned than I am.
– none
Sep 24 '18 at 22:23
• I didn't downvote. I think Gerhard was just trying to be polite and cautious here and so may have come across as hedging. And indeed, having a really open conversation here is pretty uncomfortable.
– Todd Trimble Mod
Sep 24 '18 at 22:27
• I can think of three down voting plausibilities. The Atiyah fans do not like my not wanting to hear from Sir Michael directly, so downvote. The people who are not signature fans do not like my use of signature to suggest things, so downvote. Finally, the people who enjoy rapid fire gossip may dislike my suggestion to wait days before asking such questions, so downvote. Gerhard "Can Call Myself Doddering Fool" Paseman, 2018.09.24. Sep 25 '18 at 4:15
• I downvoted, not out of any ill will, simply because I don't think it really answers the question. I agree with Gerhard on e.g. what "the masses" should understand about mathematics - I just don't think that should have any bearing on what's allowed to be discussed on MO. More importantly, I'm finding it hard to read through the hedging: I agree that content on MO should be suitably presented (obviously!), but the OP's question is: what does a "suitable" presentation look like, and why have previous questions been unsuitable? "Wait at least a few days" is great advice, but not an answer! Sep 25 '18 at 8:30
• I have not watched the video, but I understand that not enough was given at the lecture yesterday to be judged as a complete proof, and that even as a sketch some key components are absent. So where is the proof the question asks us to judge? Even if the proof was given on Monday, how long should it take to understand? These MathOverflow questions are inspired by media announcements, not by a proper presentation. At least the question on Todd functions (operators?) is based on an articulated idea. Gerhard "'Wait For It' Is Answer" Paseman, 2018.09.25. Sep 25 '18 at 14:51
• Is this a first for nested-quote signatures? Loren "@GerhardPaseman's newest signature is '"Wait For It" Is Answer'" Spice Sep 25 '18 at 17:40
• @Lspice, it may be for you. Since I don't have a twitter feed, you will have to check my user pages (yes, pages) for "all actions" under activity for earlier examples. Gerhard "Might Triple Nest This Year" Paseman, 2018.09.25. Sep 25 '18 at 17:56
• The question on this Todd function is precisely the one I had in mind - e.g. because the introduction of the Todd function was not new during the lecture, and it's appeared in the literature at least twice before. Certainly, the answer to that question may end up being "wait for it", but there's absolutely no reason that we have to wait in order to ask the question, which is what your answer suggests. It's a very good question. I think we can be sensitive to this scenario without calling for a moratorium on all related mathematical discussion. Sep 25 '18 at 19:33
• Anyway, I don't intend to argue over it. I don't feel that strongly about it. I just wanted to give some insight as to why I downvoted, since someone asked. Sep 25 '18 at 19:38