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I am interested in the situation where you have a very interesting result. For instance, you have solved a very important open problem. However, you are not known in the field and do not have any remarkable publications. Your supervisor thinks the work is good and you submit the work to a high profile journal, but you get rejected.

The thing is that the contribution is very strong. It breaks what most people believe or what they have already proven: e.g., you solve the P vs. NP problem or any other well known open problem.

The reviewers strongly reject your work with no justification and they do not state why the result is wrong. Examples of reviewer comments include:

  • "The proof must be wrong."
  • "You cannot achieve such a result."
  • "You do not understand well the notion of ..."

My question is what to do in this situation? Where to go? If your advisor accepts the work, but the reviewers from the top journal reject the work without even explaining the mistakes, what should you do?

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In that case, the supervisee is probably wrong in their perception of their own work; detecting that is what peer review is for. But then again, maybe not; maybe the presentation is just poor, or the claim too outrageous for some hearts/minds. Upload to arXiv for the time stamp and keep improving form and submitting. Your name does not (read: should not) matter when submitting an article so being unknown is not (read: should not be) an issue. Being known for half-baked crank stuff, on the other hand, is: avoid creating that impression at all cost! –  Raphael Mar 24 at 23:56
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See also here and here for more advice. And remember that Nobel prizes and centuries of fame went to people no one took serious in their (life) time. –  Raphael Mar 24 at 23:58
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You should read this page: research.microsoft.com/en-us/um/people/cohn/Thoughts/… –  Neil Strickland Mar 25 at 0:12
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@Raphael: I'll remind you that the Nobel Prize has only ever been awarded to living persons. I'll also point out that just because they all laughed at Einstein doesn't mean that if they're laughing at you, you're a new Einstein. Good links though -- thanks for those. –  Eric Lippert Mar 25 at 5:59
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"They laughed at Columbus, they laughed at Fulton, they laughed at the Wright brothers. But they also laughed at Bozo the Clown." -Carl Sagan (c2.com/cgi/wiki?TheyLaughedAtEinstein) –  Joel Reyes Noche Mar 25 at 7:14

10 Answers 10

up vote 151 down vote accepted

Your question has some issues. Given some of the questions you have asked on other SE sites in the last few days, I have some reservations about whether your question is being asked in good faith, but taken on its own merits it is a reasonable question so I will try to answer it.

The main issue is that, even in asking this relatively simple question, your writing is far from clear. If you cannot write clearly in this situation, your chances of writing up a difficult piece of mathematics or theoretical computer science are less than good. For instance:

His/Her supervisor(s) accept the work and they published it in a highly known journal and they get rejected.

Laying aside issues of subject/verb agreement and consistency of tense, the entire sentence doesn't make sense: you can't publish a paper and get rejected.

It breaks what most people believe

I don't know what it means to "break what most people believe".

or what they have already proven,

What? Are you saying that your proof contradicts other proven results? Taken literally, that would mean that you have shown mathematics to be inconsistent. In practice this could only mean that if your result is correct then some previously published work is incorrect. If that's the case then you need to be very clear about that and explain the flaws in the earlier work. It distresses me that you don't really seem to believe this but are just throwing it off as loose language.

i.e., He/She solves the P vs. NP problem or any other well known open problem.

Solving an open problem would not "break what people have already proven"....that's what it means for the problem to be open. Also saying "P vs. NP problem or any other well known open problem" is a strange bit of coyness: there is no other problem in theoretical computer science (and very few to none in mathematics as a whole) which is "like" P vs. NP. So it doesn't make sense to give that as an example. It's like saying "i.e., he found the Holy Grail or some other famous cup".

In other questions you have spoken specifically about having a proof of P vs. NP and then upon questioning have retreated from this. This sort of vacillation about what you have done is a red flag of "crankiness" that will make professionals wary.

The reviewers strongly reject his/her work with no justification and they said that the result must be wrong.

Saying that the result must be wrong is not just a justification for rejection, it's the best justification. No professional reviewer will say something is wrong lightly. Almost any reviewer who says this will point to at least one specific error. If they do not, then in practice it almost certainly means that the entire document did not make enough sense to them to be more specific.

If your advisor accepts the work, the reviewers reject the work without even explain the mistakes (it is the "best" journal in his/her domain) then what he/she must do?

If you submit a paper to the top journal in your field claiming a solution to the top problem in your field, and your paper does not make sense or does not evince even a correct understanding of the problem, then the editors are likely not to want to spend much time in response. On the other hand, if you are sincerely interested in getting their expertise, it seems reasonable to write back very politely and ask for more specifics about the error. If your response is in any way argumentative then you risk the editorial staff thinking that you will keep hounding them ad infinitum, and at some point they have to stop replying. So you should write back saying that you are not considering resubmitting the paper to that journal but for your own progress it would be extremely helpful to know what is wrong with it. You could also mention that your supervisor found the paper to be correct.

In fact you could be getting more help on this from your supervisor. If you have really "solved P vs. NP problem or any other well known open problem" and your supervisor believes your solution to be correct, why isn't your supervisor moving heaven and earth to be sure your work is getting the attention it deserves? That doesn't add up. The two possible explanations seem to be (i) your supervisor is being too polite with you: s/he does not actually believe that you have solved P vs. NP; and (ii) your advisor's imprimatur does not carry any weight in the community whatsoever. The latter unfortunately means his/her opinion on the correctness of your work is not worth very much.

A good way to find out whether it's (i), (ii) or -- I do admit that anything is possible! perhaps the top journal in your field is unfairly ignoring your revolutionary work -- is to seek your advisor's help in getting another faculty member to evaluate the work, preferably someone in the department that you can speak to recently.

Finally, you seem to have some real worries that if an unknown person solves a famous problem then it somehow doesn't count. This is really not the way academia works, provided the unknown person is capable of presenting the work in a way which makes sense to the experts (and if not, what a shame, but what else could one possibly expect?). Have you heard of the recent example of Yitang Zhang? Zhang was a non-tenure-track lecturer at the University of New Hampshire when he stunned the mathematical world by proving the existence of bounded prime gaps. He submitted his work to the top mathematical journal...and by all accounts they accepted it with unusual speed. In other words, they received a paper from someone they had probably never heard of, looked at it quickly and saw that it was a plausible attack on a huge open problem, and as a result they sprung into action much more rapidly and thoroughly than for most submissions they get. This is an amazing story, but a true one, and it shows how the community responds to a real situation like this.

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Gievn that I myself am a non-native English speaker, I tend to have more tolerence to the writing from a non-English speaker. Y. Zhang may not be a good example in this case. He went to the US in 1985 and received PhD in math in the US. He has lived in the US since then. From what I know, Zhang has no problem in English while the OP may have serious English language problem. I agree with many parts in your answer, though. –  scaaahu Mar 25 at 2:49
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@scaaahu: Not being able to speak English well is not a character flaw. It is however a problem if English is the language that you're writing your papers in. There is also a distinction to be made between speaking a language imperfectly and expressing yourself poorly. In my answer I tried not to harp much on issues of grammar and usage. –  Pete L. Clark Mar 25 at 3:02
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@scaaahu: Sure, the OP has used entirely the wrong word here. Again, this is not a crime, but doing this in one's academic work could certainly lead to its lack of understanding and thus rejection. Look at it this way: wouldn't it be nice if language issues were most of the source of the OP's difficulties? If so, they can be overcome provided they are acknowledged and addressed. Not to tell someone when their writing is unclear is not doing them a favor, in my opinion. –  Pete L. Clark Mar 25 at 3:19
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"He found the Holy Grail or some other famous cup" - fantastic phrase. –  Tobias Kildetoft Mar 25 at 9:04
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@Selfishness: "[I]f a person is not known and has done very good thing, they will ignore his work (at first at least)." Given that my answer contained a specific, recent, clear counterexample to this, I'm not sure what further to say in response. –  Pete L. Clark Mar 25 at 16:51

Regardless of whether the work is correct or not, the following statement applies:

The burden of proof is on the author to convince the reader of the result.

The community (e.g., editors, reviewers) has no responsibility to evaluate your work to your satisfaction. If the reviewers made a good faith effort to read your paper and were not convinced, then you must make your argument more convincing.

(This does not mean, make a few trivial edits and resubmit. This means, prove your results so thoroughly and in such excruciating detail, and with such demonstrably excellent understanding of the problem context, that they become inarguable. Then figure out a way to express the results in a convincing way.)

If in the process of doing so you find an error, well, you'd be in good company.

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+1. "Extraordinary claims require extraordinary proof." If it isn't convincing, make it so -- and remember that all it takes is one counterexample to show that you haven't solved the problem, so you really do need to consider every possible edge case before you can make that claim. If there is an exception, you haven't solved it but you may have solved a subset... which may or may not be new information. –  keshlam Mar 25 at 14:52
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@user3439590 If the reviewers understand the contribution but think the English should be improved, they will write something like "This paper makes a useful contribution, but has English writing issues." If the reviewers didn't write that, then you didn't convince them that you made a useful contribution. –  ff524 Mar 25 at 15:39
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@Selfishness_has_equilibrium keep in mind though that It is also possible that the English barrier might lead you to not understanding some subtle things with some definition and notions. Which could further easily lead to solving a slightly different version of the problem... And especially in graph theory, sometimes a minor detail is the difference between an open problem and an easy exercise in an introductory course.... –  Nick S Mar 25 at 20:46
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@Selfishness_has_equilibrium, if poor English skills get in the way of reviewers being able to understand what you're trying to say, of course they will reject it. That goes double for making extraordinary claims (and triple if you claim persecution or compare yourself to Einstein). I would suggest that you collaborate with someone who has excellent English skills to review/sanity check your work and edit it for readability. The cost may be that (depending on how much they contribute) you will have to list them as co-author. –  Phil Perry Mar 26 at 13:26
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"If in the process of doing so you find an error, well, you'd be in good company." Reminded me of this news report (Youtube) –  Basic Mar 28 at 10:35

First, make sure you are not really a crank before trying to convince others. Read these common characteristics of cranks. If they apply to you then get professional help.

For the rest of the answer I will assume that you have really solved a famous open problem. In the following "he" refers to a typical non-expert claiming to have a solution for a famous open problem and "she" refers to an expert in the topic.

  1. There is no easy shortcut for you!
    If you are looking for a simple easy shortcut to get your solution verified by an expert then this answer is not for you and I can assure you what you want is not going to happen.

  2. Understand the magnitude of your claim!
    E.g. If you are claiming to have a proof of P is not equal to NP then you are the guy who is claiming to have a design for a rocket that can be built with the currently technology and resources to take a human to Andromeda and back safely while experts are having hard time sending a human to mars. If you are claiming to have a proof of P is equal to NP then you are the guy who is claiming to have a time travel machine.

  3. Understand why experts are reluctant to directly engage non-experts.
    Many experts would be interested to know about any major progress in their field. E.g. there are complexity theorists who do read every P vs. NP related paper posted on arXiv (arXiv has a very lenient acceptance policy regarding P vs. NP claims). They will definitely let other experts know if they notice something interesting. But

    • You are not the only one with such claims.
      There are thousands of people who regularly make such claims.

    • All previous ones suffered from trivial issues no expert would have made.
      It is your job to show you are not one of them.

    • Her time is valuable.
      For most it is not really monetary. But I think giving some numbers would be helpful. In my university a graduate student is paid over $40/hour to mark simple undergraduate assignments. This is nothing compared to what an expert might charge for consulting in the industry.

    • Non-experts often lack basic skills and knowledge to understand her replies.
      E.g. he lacks mathematical maturity, he does not know basic definitions and terminology, etc. It is not uncommon that an expert tells a non-expert what he has is not a proof. She does not mean the proof is incorrect, she means it is not even a proof in the sense that an apple is not a proof. He does not understand when he is told it is "not even wrong!". To make him understand her reply she would have to teach him those required skills and knowledge, too much work just to convince him he does not have a solution. Often he is not patient nor interested in learning (e.g. reading a textbook), he is only interested in a confirmation of what he believes to be a solution. Way too much work in that case.

    • It is often impossible to satisfy him.
      Because of the points mentioned above, he often insists on the validity of his claim even after she tells him it is not. At other times where he understands the reply he considers it a simple easy-to-fix error, not a fundamental one. He tries to fix it and get her verify it. This leads to back and forth.

    • He underestimates the required time and effort on her part to answer his claim.
      He thinks it is a simple easy job for her to answer his claim. E.g. he expects her to give him a counterexample where his algorithm fails. Finding a counterexample for an algorithm is a very difficult task (as anyone who has marked undergraduate algorithms or complexity theory assignments would know). Finding an explanation why an idea is fundamentally flawed and cannot work is even more difficult.

  4. He does not understand it is not a puzzle.
    She is not interested in the question just for its own sake. She expects the solution to the question will be accompanied with major advances in her field. E.g. complexity theorists do not care about P vs. NP just for its own sake. They expect the solution for P vs. NP will come with major progress in our understanding about the nature of efficient computation and its limitations. Often he does not understand this. He thinks of the question as a game or puzzle that he thinks he has won and that is it. This attitude is frustrating for experts.


Now here are some tips:

  1. Be humble.
    It is much easier to get her to have a look at your solution if you are genuinely humble and eager to learn and accepting if you are told that you are wrong.

  2. Make sure you understand what is required to solve the question.
    E.g. understand that a program that seems to efficiently solve an NP-complete problem is not a proof, understand that an idea does not make a proof, make sure you understand the definitions and terminology, etc.

  3. Know the basics.
    I keep repeating this: read a good textbook on the topic and solve its exercises. It is beneficial for you as you will know more and will be more convincing. It is beneficial for her because you will not waste her time with simple mistakes that you would have noticed yourself if you had read a good textbook. It is annoying to deal with people who claim to have solved P vs. NP but repeatedly make basic mistakes that a good student who has taken an undergraduate course on the topic will not make.

  4. Use your real name.
    Not using your real name indicates that you are trying to avoid suffering any potential negative consequence of your claim being incorrect. Using your real name indicates that you are sure enough to be ready to suffer potential negative professional consequences if you are mistaken, so you can be taken more seriously. If you are not completely sure about your claim do not waste her time.

  5. Don't shirk. Do your share before expecting help from others.
    If you want her to look at your solution you should spend 10 times more time and effort than she will spend helping you. For claims about P vs. NP you have to do way more.

  6. You will not get more than one chance.
    Make it count. If on the first page of your paper she finds a silly mistake or a basic error (e.g. you do not even know the definitions of P and NP) then she will be done with your claims forever.

  7. Understand the known obstacles for solving the question and why they do not apply to your solution.
    E.g. if you are claiming P is not equal to NP then you should have a good idea why relaltivization and natural proofs barriers do not apply to your solution. Similarly if you are claiming P is equal to NP.

  8. Try to prove simpler more acceptable claims.
    E.g. if you have a proof of P is equal to NP then you should also have a proof of simpler weaker major results like Factoring is in P. If you can extract a clean proof for such claims then you can first try publishing them. Such results can be much easier to get verified as they are considered more likely.

  9. Make sure your solution is not too strong.
    In other words, make sure it does not contradict other known results. E.g. if your argument for P is equal to NP would also show that P is equal to ExpTime (which we know is false) then you are in trouble (Scott Aaronson mentions a few more cases of too strong results in his blog post Eight Signs A Claimed P≠NP Proof Is Wrong).

  10. Check your solution.
    Make sure there are no mistakes. All steps should easily seen to follow from the previous ones. Make sure you do not make extra assumptions at any point.

  11. Recheck your solution.
    Put your proof aside completely for two weeks or more. Do not think about it. Then go back and recheck it with a fresh mind as if you were checking someone else's solution.

  12. Build evidence for your claims.
    E.g. if you have a really efficient algorithm (i.e. its running time is a polynomial with small constants) which you have proven to solve an NP-complete problem then it should not be a difficult task to beat the state-of-art SAT-solvers or to break various cryptographic protocols based on hardness conjectures (those conjectures will be false if P is equal to NP).

  13. Write easy-to-read concise clean abstract and introduction.
    Do not put any unnecessary background/history/philosophical consequences/discussion of importance/general commentary. It is a famous open problem; every expert knows its significance. Save them for your final version. Right now you should focus on convincing her that your claim is correct. She first wants an easy-to-read short error-free convincing high-level explanation of your solution. It should also explain why any known obstacles do not apply to your solution. It should also contain any other evidence that can support the correctness of your claim. If you fail the reader is not likely to continue reading.

  14. Make sure the rest of your paper matches your abstract and introduction.
    If you fail the reader is not likely to continue reading.

  15. Make sure every detail in your paper is correct.
    Follow the standard structure of papers in the topic. Check a few famous well-written papers in the area that have solved major open problems. All definitions should be clear, easy to understand, and rigorous. Every theorem (lemma, etc.) should be clearly and rigorously stated, and the proof of each of them should follow their statement. She should be able to see why each claim in the proof is correct based on the previous steps, definitions, and lemmas without too much trouble. If you fail the reader is not likely to continue reading.

  16. Have a general expert who personally knows you check your solution.
    I am assuming that you do not know personally any expert in the area of the question. The closer the general expert is to the area of the question the better it will be. E.g. for P vs. NP, you can ask a mathematician, preferably a theoretical computer scientist. Opinion of people who are not experts in the topic may not have much weight but it will make sure you are not making some simple mistake.
    Understand that at this point someone who does not know you personally has no reason to check your solution.

  17. Have another general expert who knows you personally check your solution.
    Rome was not built in a day. You have to build confidence in your solution little by little. Those you convince can become your bridges to reach the experts.

  18. If they are convinced ask them to show your solution to an expert they know.
    E.g. for P vs. NP, ask them to show it to a complexity theorist they know. At this point you are less likely to be making a basic mistake and you have good evidence to support your claim. Your solution now requires the expertise of an expert in the topic.

  19. If she is convinced she will definitely show it to other experts.
    News about any major progress in an area will spread very fast among the experts in that area. Other experts (complexity theorists in the case P vs. NP) will recheck your solution independently. If they are convinced you will probably get an invitation to submit your paper to a famous journal (something like JACM in the case of P vs. NP).

  20. Do not claim to solve a famous open problem more than once.
    As I wrote above, you will not get more than one chance! You do not have a right to ask her to see what is wrong with your fixed solution if you made a mistake. (The exception is when she explicitly asks you to try to fix your solution and send the fixed version back to her.)

  21. Do not expect an explanation for why your idea cannot work.
    It is unlikely that someone would be able to show formally that an informal idea cannot work. If the idea is formal enough then the reason that it cannot work can be a new interesting result in itself; however, proving such results can be even more difficult than solving the original question. In the case of P vs. NP, if you are claiming to have an efficient algorithm for an NP-hard problem you should not expect her to find an input where your algorithm fails.


In summary,

Understand that she is not required to help you. If she is helping you she is doing so out of generosity. She has a right to stop it whenever she pleases without any explanation. Be mindful of her time, do not waste it for what you could/should have done yourself, try to make her job in helping you as easy as possible, and do not do anything that will make her regret trying to help you.

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@Jack, here is what I meant: some people treat P vs. NP like a one level computer game with a Yes or No answer that they have to win. We (complexity theorist) care about P vs. NP because we believe that settling the question will be accompanied with significant progress in our understanding of the nature of efficient computation and its limitations, we don't care about it just for its own sake. As Scott wrote once: "[We] like to take P vs. NP as our “flagship example” of a huge class of questions about what is and isn’t feasible for computers, none of which we know how to answer." –  Kaveh Mar 26 at 23:36
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"if you a have an algorithm which you have proven to solve an NP-complete problem then it should not be a difficult task to beat..." -- this may be bad advice. There is no reason to think that such an algorithm should be "real" efficient: it may have abysmal performance on all inputs that we can store. Otherwise, good answer. It illustrates perfectly that it takes extraordinary effort to solve an extraordinary problem. Of course, a crank would a) be unable to diagnose themselves and b) refute many of your points because the have these conspiracy theories. (You do assume everbody's friendly.) –  Raphael Mar 27 at 8:27
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+1: What an exceptionally detailed and helpful answer. I hope the OP appreciates that the generosity of time and spirit that went into this. –  Pete L. Clark Mar 27 at 15:02
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@Raphael, you are right in principle (though it happens rarely in practice), I tried to make the statement more precise. –  Kaveh Mar 28 at 7:17
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+1 for making the expert a she –  user1938107 Mar 30 at 5:55

If your interpretation of events is: "I have a heartbreaking work of staggering genius and the only obstacle to acceptance is that I am not well known and the elites are blocking my work", then you're unlikely to get good advice on what to do here or elsewhere.

The problem, as Raphael indicates, is that while it's possible that this interpretation is correct, it's far more likely that in fact your result does NOT solve the major open problem that you think it does.

Once you admit that this possibility exists, then many steps present themselves, all listed in the very good links provided. Reaching out to people who might comment on your work, looking at the literature to see if approaches like yours have been tried before and have failed, seeing if your solution also solves related (simpler) problems, and so on.

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Thanks for your help. –  Learning Mar 25 at 15:33
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yes. few will mention this & maybe it is considered taboo to do so, but as hinted in your answer it seems in some that these grandiose claims could roughly correlate with psychological symptoms/issues eg delusions of grandeur, narcissism, narcissistic personality disorder etc –  vzn Mar 26 at 5:12

What does your advisor say about all this?

If she really believes you have solved this major problem, she should be moving mountains to help you publish and disseminate it. (It sounds like her name is on it too, so she has an even greater incentive.) But you've used the rather lukewarm phrasing that she "accepts" it. Better get her completely on board first, or get her to explain in more detail her reservations (which may indeed turn out to reveal fatal flaws).

The advantage you have over the average crank is that, as a student, you already have ties to the scientific community, through your advisor. Take advantage of this. Once you and your advisor are satisfied that your manuscript is of the best possible quality (see ff254's answer), post it on arXiv, and circulate. Your advisor surely knows experts in the field, and should have enough reputation that she can get them interested in it.

I'm not sure about your field, but in mathematics, this currently tends to be the way that the community handles solutions to major open problems. You don't just submit it to Annals, have the referees approve it, and then wait a few months until everyone gets their issue of Annals in the mail and is astonished. Instead, you get the community to study it first. You convince a few experts that it is plausible enough to be worth their attention, and they look at it. Either they find a critical flaw right away (the most common case), or they find a lesser flaw that you or someone else fixes, and maybe, gradually, a consensus develops that it is probably right. That's when you send it to Annals.

One thing that worries me in what you wrote is:

It breaks what most people believe or what they have already proven...

Which is it? The distinction is crucial. If it contradicts people's intuition, that raises the bar a little, but scientists are used to being surprised. If it contradicts something previously proved, that raises the bar a lot. It puts on you the burden of not only showing that your work is right, but showing specifically why the previously accepted work was actually wrong. (You can't just say "Mine is right, therefore theirs must be wrong.") You didn't say anything about having done that. (And if you can't find a flaw in the previous work, then your claim is in fact along the lines of "Mathematics is inconsistent". The bar on that one is more or less on the moon.)

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I also left a similar answer, even up to the (independently chosen) phrasing "moving mountains" versus "moving heaven and earth". I should say though that getting the community to study your revolutionary work is a good way to go and common, but I don't think it always happens this way, especially among people that have few ties to qualified experts. I brought up the (amazing) example of Yitang Zhang in my answer. So far as I know, he really did work in isolation and submit his paper to the Annals rather than shop it around much. –  Pete L. Clark Mar 25 at 2:32
    
@PeteL.Clark: Heh. Great minds, etc. Thanks for your nice answer, the example of Zhang, and especially the deconstruction of the question. –  Nate Eldredge Mar 25 at 2:36
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Your answer reminds me of a really interesting story. Short version: serious mathematician thinks he's found a proof that math (PA) is inconsistent, explains it clearly enough that experts can understand, a top mathematician reads the outline and spots the error, and the author retracts the proof. So even really extraordinary claims will be looked it if they're explained clearly and reasonably by a sensible person. –  Noah Snyder Mar 26 at 3:57
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@Lohoris See Gödel's Second Incompleteness Theorem. –  Istvan Chung Mar 26 at 14:45
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It might be possible to prove mathematics consistent but only if it is inconsistent –  Philip Gibbs Mar 26 at 22:23

May I add to Nate Eldredge's comprehensive answer that, if your work shakes or shatters the commonly held views in your community, then it is very important that you reconcile those views with yours, by which I mean: show exactly where the community is "wrong" or "not exactly right" and why. Offer counterexamples, predictions, all you can.

Relativity would be nowhere if it didn't reduce to good old Newtonean mechanics where the latter performed perfectly!

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And sometimes it's appropriate to explain why it seemed that what's been done couldn't be done. –  Mars Mar 26 at 18:40

By attacking your own proof even stronger than the others do.

Seriously, there is a reason why people in your discipline haven't been able to find the answer for centuries. The a priori probability that you are wrong is so high that even when you have created a good looking proof, the a posteriori probability that you are right is way too low. This means that, if you know enough of your own discipline, you should not be convinced that you solved it. Given a problem which has resisted solution for a long time, being convinced that you solved it just because you have a proof you believe in is a sure sign of a crackpot.

So non-crackpot behavior in such a case will be to try to take the proof apart, shoot it down, tear it to pieces from all possible angles. This is what your peers will be doing, and this is what they will expect you to be doing. To forget your pride, your subjective biases, and to be merciless to your own result.

They you only believe you after you have found more ways to disprove your result than they themselves can think of, tried them all, and failed in all of them. And your paper has to clearly show that this is what you did. Anything else will earn you the crackpot title.

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Simply brilliant. I wouldn't have phrased it better myself... the idea occurred to me when I was on my first "verge on scientific breakthrough"; I gave myself a cold shower, a serious scolding, a strong criticism to the value of the solution - and only then I realized that although my solution is irrelevant to the problem at hand and doesn't solve it, it presents a potential to solve other, related problems. To achieve, one has to know how to fail... –  vaxquis Mar 29 at 5:26

Some advice is to very carefully check that the proofs are correct, ask one's supervisor for advice, and seek third opinions. Perhaps the supervisor has colleagues in the research area who would be willing to read the draft and offer concrete feedback.

If the journal submitted to is good, yet the reviewers did not give any useful feedback at all, then there is almost certainly a problem with the abstract and introduction.

The abstract and introduction should make clear the new idea that allows this "breakthrough". Presumably many have approached this problem in the past and failed; there may be widespread beliefs about why it is difficult to prove or perhaps even known "barriers" to attempted proofs. The abstract and introduction should clearly and briefly mention why such beliefs, objections, or barriers do not apply or how they were overcome.

In short, the abstract and introduction must give the skeptical reader reason to believe the paper could be correct, given the reader's background knowledge. If this is done, I would hope that reviewers would at least mention why they do not believe the result.

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While most of the answers seems to have much confidence in the academic system, I would like to offer another viewpoint.

I think it is in fact much harder for an unknown (to a specific field) to present a solution to the scientic community than normally expected.

Scientists do screw up and sometimes royally.
First example: The infamous Monty Hall problem.
More than 65% of all professional answers to Marilyn (with all sorts of academic grades including statisticians) strongly rejected their answers, sometimes with outright jeers and taunts. This included Paul Erdos and Straight Dope Cecil Adams. So even the majority of experts can fail.

Second example: The also infamous neutrino anomaly. The interesting thing here is not the error itself, but the reaction on Arxiv. Anyone who would have dared to offer superluminal theories before the announcement would have been immediately declared as relativity crank. After the announcement papers came flooding in offering all sorts of superluminal theories explaining what we know now to be simply a bad cable.

What are the problems an unknown may face ?

  1. Arxiv. You need an affiliation from an university or research institute and/or an endorsement from a know author. Arxiv can revoke or limit your access without explanation. This requirement also applies to fully qualified scientists which are working in companies.

  2. Journals. Too many people are trying to get their results published in too few respectable journals. Journals also pretty scale bad, you have to wait a long time to get published. Lesser known journals may have lower barriers, but you have the real danger that the contribution is missed. And even the lower journals may reject the paper.

  3. Scientists. The situation is different in various countries, but normally scientists are overworked and underpaid. They have not the time nor the resources to review contributions with the very slim chance to get a scientific jackpot.

If someone thinks the viewpoint is not valid, try just for fun to supply a normal paper under a pseudonym and the home address.

The only viable option I see is to get contact to the scientists in the field and try to work with them over the contribution which may be harder than it sounds. The list provided by Kaveh is a good resource to start with.

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Your assessment of the Monty Hall issue is incorrect. The underlying issue was that the problem was not well-specified. The subtle ambiguity in the specification was what caused the disagreement between experts. –  EnergyNumbers Mar 28 at 9:20
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@EnergyNumbers Yes, I heard that excuse. But the original question does not offer any ambiguity. The host knows the position of the car (it is not random), it is a gameshow (he cannot open the car door) and he chooses another door with a blank. And, really, what allmost all critics said was not:"It has two different solutions, the answer depends on the following circumstances etc. pp.". What they mostly said was: "THE CHANCE IS ALWAYS 1/2, YOU STUPID FOOL !!" Both Massimo Piattelli-Palmarini and Gero von Randow, a science journalist, explained the solution and the people still disagreed. –  Thorsten S. Mar 28 at 13:18
    
To stop just a discussion in progress: My argument is simply that scientific consensus can screw up royally. If you are not believing that this can happen, look up some of the original quotations prominent scientists did about Wegener's continental drift and Fritz Zwicky's dark matter at the time of the discovery. Believe me, you do not want to read this. –  Thorsten S. Mar 28 at 13:28
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The neutrino result was a very special case. It was announced very, very cautiously, almost as "Er, guys? We have this kinda crazy result that we can't explain. We tried for ages and we found mistakes and fixed them and still got the same thing. Can you guys see what we did wrong?" With that amount of care taken in the experimental work that seemed to maybe show faster-than-light neutrinos, it becomes worth considering. Most FTL results are just some guy saying, "Yeah, but what if Einstein were wrong and..." or, worse, "Einstein says this is impossible but, if we subtly redefine this, then..." –  David Richerby Mar 29 at 1:27
    
@DavidRicherby And because they asked what they did wrong, people should wait and send proposals instead of submitting FTL papers. Violating relativity is a really terrible problem. All particles have "normal" rest mass (m=0 => v=c / m > 0 => v < c), there is no way to get a non conflicting description of FTL events with separate observers because RT is needed to describe time and space properties correctly. Please explain to a crank that FTL of normal matter should be considered impossible after seeing how the "experts" were willing to accept possible FTL and even had proposals to offer. –  Thorsten S. Mar 31 at 0:36

Despite what people will say it is true that journals will reject papers using author profiling without a proper review. It is hard to say how many papers are rejected this way but Elsevier say that they reject 30 to 50 percent of papers without a review for other "technical reasons". See also this paper about how editors can save time by looking at author attributes such as affiliation to reject papers without looking at them.

I have personal experience of this because I recently made significant progress on a well known 100 year old open problem after experts in the field had said that future progress was likely to be very slow. The journal I submitted the paper to rejected it as soon as I confirmed that I had no affiliation. There was no reviewer report and they did not give any specific reason. I had complied with all their technical requirements for submission.

However, I pointed out to them that according to the code of conduct of the committee on publication ethics to which the journal claims to adhere "Editorial decisions should not be affected by the origins of the manuscript" and "Journals should have a declared mechanism for authors to appeal against editorial decisions." To my surprise they responded after a delay to tell me that they would look at it again.

It is true that there are many claimed resolutions of problems such as P vs NP that can be dismissed at a moments glance. This can be done because there are well understood reasons why these problem are hard and a solution would need to address that. Many claimed proofs of open problems by non-academics descend quickly into non-standard language that makes it hard to even address why they are wrong so they are just ignored by the community. It is up to the authors to make sure they communicate their ideas correctly.

If you do have a solution to an open problem my advice is to submit to an open repository such as arXiv. If you can't get an endorser use viXra or figshare (full disclosure: I am viXra admin) Do not pay attention to negative things said about viXra. It's purpose is just to give you an independent time-stamp and an archived copy you can point to. It does not attempt to review or give your work credibility in any way. The last thing you should do is submit to journals or send to experts without having a verified public copy because if it really is a breakthrough there is a real risk of plagiarism that can only be averted by having a prior copy archived.

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Your first paragraph seems to indicate that the link is to an Elsevier editor admitting that they use author profiling to reject papers, but there is no such thing at that link (he lists perfectly valid technical reasons to reject a paper). Also note that in the mathematical community, putting something on viXra will mark you as less serious (whether this should be the case or not), so I would not advice anyone to upload anything there unless they really have no other options. –  Tobias Kildetoft Mar 25 at 11:45
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You need to read the answer again where it says "full disclosure: I am viXra admin" Note also that it is not a commercial site. We have no need to tout for business. I mention it along with an alternative only because it is a relevant part of the answer. –  Philip Gibbs Mar 26 at 0:03
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"if it really is a breakthrough there is a real risk of plagiarism". This is the sort of thing that often gets people identified as cranks - they believe that editors and reviewers are going to steal their work at the same time as rejecting it as being invalid. Do you really believe this is a risk when submitting to reputable journals, and are you aware of any cases where it has happened? –  jwg Mar 26 at 8:37
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@jwg This risk does exist and it does not help ridiculing it. See Jocelyn Bell and Rosalind Franklin. As student our group found out that one assistant used the data of another assistant without permission. Mark Chu-Carroll from "Good Math, Bad Math" had a bad encounter when he talked to another person about a new idea and found later out that this person put out a paper with this idea without acknowledgment. –  Thorsten S. Mar 28 at 0:15
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@ThorstenS., as David Richerby pointed out, listing various plagiarism cases is not the same as citing a single case when reviewers or editors have rejected submitted material only to plagiarize it. I did not 'ridicule' this - I pointed out, I hope helpfully, that this kind of claim is exactly what people look out for to recognize cranks. If you are submitting serious work, you are doing yourself a disservice by appearing cranky. On the other hand, if your submitted work genuinely has been plagiarized, you should make a fuss about it. –  jwg Mar 28 at 0:25

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