-6

Question:

How can an independent researcher publish a well-written research paper / proof in a respected mathematical publication?

Related Questions:

How do I ensure I'll be able to claim authorship over a math proof, as an amateur math student? How can I "time-stamp" my data without publishing it?

I've reviewed these both, and while I find them somewhat insightful, don't believe they answer my question directly. Generally speaking, how should an independent researcher go about submitting to an academic publication?

Suggested Duplicate / Existing Answer:

I believe I have solved a famous open problem. How do I convince people in the field that I am not a crank?

Whether a "crank" could possibly write a valid proof of this nature is outside the scope of this question. Simply how and where an independent researcher should submit their work when they believe it is of sufficient quality to do so.

TCooper
  • 189
  • 1
  • 8
  • 6
    You are in a tough spot: Given what you wrote, you have hard time even writing coherent math. Most likely (with > 0.99 probability), whatever you will write, no professional mathematician will read. My suggestion would be to forget the Collatz Conjecture and try to learn some rigorous math maybe by reading a book or online notes with an introduction to math proofs (you will find many just by googling). Right now, you are in a situation akin to somebody who would like to play, say, basketball, at a professional level, but cannot dribble a ball. – Moishe Kohan Sep 08 '23 at 05:04
  • 2
    I understand you took a Discrete Math course before. Besides that, what math courses have you taken ? How much math proof have you done besides writing homework problems for your Calculus class ? (I assume you took Calc I & II, right ?) Did you take multivariable Calculus ? – Nobody Sep 08 '23 at 05:29
  • 5
    My suggestion: learn multivariable Calculus and complex analysis online first. Do a lot exercises for them. Then go back to your proof to see it makes sense or not. Right now, you don't even have the foundational knowledge to tell if your proof is correct. If you go to a professional mathematician for a co-author, it's very likely you will be laughed at. So, have enough knowledge to review your own proof before you go to anybody else. – Nobody Sep 08 '23 at 06:08
  • 9
    No, I am not suggesting looking for a coauthor (at best, you will find someone incompetent). I am suggesting learning how to do rigorous math. – Moishe Kohan Sep 08 '23 at 12:43
  • 3
    Moishe Kohan's comment was surely not intended to be harsh. It is in the nature of mathematics that without substantial training you can have no idea whether your work is correct or not. (Sorting this out is something that even the professionals spend lots of time and effort on.) What many amateurs don't realize is that it is all but impossible for someone's first mathematical work to resolve a famous problem. – Pete L. Clark Sep 09 '23 at 04:56
  • 8
    Moishe was translating from one event to another event of similar probability but for which the vanishingly smaller probability of the latter event is easier to believe. I did this myself upon reading your question. What I came up with is: it is about as likely that you have solved the Collatz Conjecture as that in the first full tennis match of your life, you beat Novak Djokovic. Yes, it is possible, but it is overwhelmingly more likely that some mistake has been made. If you are serious about tennis, take the process of improvement seriously... – Pete L. Clark Sep 09 '23 at 05:01
  • 12
    @TCooper: Bigoted? That is a very heavy word to apply to someone who is giving academic advice to an anonymous person on the internet. I think I had better not engage further. – Pete L. Clark Sep 09 '23 at 15:48
  • 1
    "I must learn the specific language of professional mathematicians to convey ideas to professional mathematicians effectively is 100% valid." That's part of it, but more importantly you need to learn how serious mathematics is done so that you yourself can have justified confidence in your claims. If you are semi-jokingly saying that you will "keep working on this here and there, probably for at least the next 9 years :)", then it would surely be a wise investment to first learn how serious mathematics is done and only then spend however many years on the problem. – Adam Přenosil Sep 10 '23 at 19:01
  • @AdamPřenosil I apologize for any confusion my previous communication may have caused about my skills. While I'm proficient in using latex, and comfortable writing functions, matrices, sets, and cases (the 'real' math), I struggle to articulate mathematical analysis into professional proofs. My unfamiliarity with mathematicians' language may have suggested a lack of ability in multivariable calculus or complex analysis, which is not the case. I realize this misunderstanding stemmed from my ineffective communication and defensive responses, rather than productive dialogue. – TCooper Sep 11 '23 at 00:56
  • 1
    Does this answer your question? I believe I have solved a famous open problem. How do I convince people in the field that I am not a crank? – Daniel R. Collins Sep 11 '23 at 01:22
  • @DanielR.Collins I appreciate the link, but I believe that's where the confusion stemmed from. I genuinely was looking for how to publish as an unaffiliated researcher, and that's it. The topic should've been irrelevant and I regret ever mentioning it due to the assumptions it led to, and my defensive responses to those assumptions. Both generally detracting from the value of the question/answer. I think the accepted answer is spot on (why I accepted and upvoted). It provides where and how to proceed, acknowledging you can regardless of affiliation (or lack of). – TCooper Sep 11 '23 at 01:28
  • 1
    @PeteL.Clark Thank you for the comment. I used that word because of one mistaken, but key point in my mind. I believed Moishe had no evidence to make the statement that I "have [a] hard time writing coherent math" - I let my ego respond instead of realizing that by everything I had communicated at that point, he had plenty of evidence to make that statement. Further, I had not offered any evidence to the contrary. Whether or not I felt insulted by it is irrelevant and I obviously shouldn't have let it cloud my judgement in responding to anyone. – TCooper Sep 11 '23 at 02:53

3 Answers3

15

If a professional mathematician had a proof of the Collatz Conjecture, they would submit a paper to the Annals of Mathematics. You can look at their webpage for submission guidelines, but I believe it just involves sending a PDF of your paper to the appropriate email address. There are no costs.

I am about 99.99% certain that there is no elementary proof (one that does not involve significant techniques from real and or complex analysis) for the Collatz Conjecture. If someone submitted such a purported proof to the Annals of Mathematics, particularly if it's an amateur, they won't bother to look at it because it's almost certainly a waste of their time.

Any other reputable journal would do the same thing because it's also a waste of their time (plus if you really believed your proof had a chance, you would've submitted it to the Annals of Mathematics).

You should know that the vast majority of mathematicians will not submit a proof to a journal unless they are 99.95% certain their proof is correct. It is very unusual for a paper to be rejected only for errors in a proof (though it does happen, more than 0.05% of the time, indicating most mathematicians are overconfident about their proofs). You don't seem to have that level of certainty in your work.

Alexander Woo
  • 24,741
  • 3
  • 59
  • 91
  • Thank you Alexander. I greatly appreciate the information. I believe my approach is novel. Although I'm not the right person to judge this, I don't believe it is elementary either. Obviously only time will tell. – TCooper Sep 08 '23 at 05:06
  • 5
    @TCooper - I think you misunderstood my statement - for the purposes of that sentence I am defining the word "elementary" to mean not using significant techniques from real or complex analysis. Presumably, you do know whether you're using real/complex analysis or not, so you should know for certain, not just believe, whether your proof is elementary or not according to the definition stipulated for the purposes of this argument. – Alexander Woo Sep 08 '23 at 05:13
  • I think I'm just not making my ignorance of the fields common language and practices clear enough. I realize you defined it, but I'm not sure what qualifies as significant techniques from real or complex analysis. I've done extensive research and analysis, sporadically over 9 years as a way to hone my software engineering/development skills when I needed a side project to work on. But to your points, and the comment on the question - seems like a co-author is the only possible way to translate the information so its useful for practicing mathematicians. – TCooper Sep 08 '23 at 05:20
  • 7
    If you cannot understand this paper: https://arxiv.org/abs/1909.03562 you did not prove the Collatz Conjecture. – Alexander Woo Sep 08 '23 at 05:31
  • I think I'm far better comprehending mathematical language than producing. Similarly to how I can understand Spanish and Japanese to a much higher degree than I can speak them. In the paper, the authors seem to show that for any function that increases without bound as the number increases, the smallest number you get to by applying the Collatz map is less than or equal to the value of that function for most numbers. So it would be a probabilistic "proof" that doesn't provide a deterministic proof of any kind? Just going off the abstract - will read tomorrow. Thank you either way – TCooper Sep 08 '23 at 05:46
  • 4
    @TCooper - yes your summary of the abstract is correct. That paper does not prove the Collatz Conjecture, though I think the experts in the area believe it is significant progress towards a potential proof. Keep in mind that a professional would need at least a week or two to actually read the whole paper and understand all its details. It's relatively easy as math papers go - I actually would have some chance of understanding it despite my expertise being in a completely different area of mathematics. – Alexander Woo Sep 08 '23 at 05:54
  • 7
    @TCooper: "I'm not sure what qualifies as significant techniques from real or complex analysis. I've done extensive research and analysis..." Sounds like you may be mistaking the branch of pure math known as "Analysis" https://en.wikipedia.org/wiki/Mathematical_analysis#Main_branches with everyday uses of the word. If so -- if you're not familiar with the branch of math called analysis -- it's implausible for you to have independently reinvented the tools (built over centuries of work) needed for problems like this one. Take a course like "Real Analysis" first. – civilstat Sep 08 '23 at 18:51
  • @civilstat You're right, why I tried to make it clear I'm a layman, not a mathematician. Some of my best insights have come from the relationship of the change of length of binary representations for each integer in the generated sequences. I might be giving away my "proof", but fresh eyes this morning let me realize that my excitement and belief in my proof last night was just from spending 9+ hours working on it straight and convincing myself... I think the concepts are strong (derived from insights found in binary representation patterns, not directly related) - no proof though. – TCooper Sep 09 '23 at 03:30
  • 2
    @TCooper - sounds like you're looking at limits under the 2-adic norm - something Tao touches on briefly as a well-studied but not ultimately fruitful approach in his paper. – Alexander Woo Sep 09 '23 at 03:33
2

Direct answer to the question that's asked: you write the paper up and submit it to a journal. There's no requirement that the author be affiliated to a university to publish.

How do you submit? You go to the target journal's website and check their submission procedure. There's a high chance you'll need to register an account on the journal's editorial management system, and then there's a submission system not dissimilar to the kind of system you might use to apply for university admission or for a visa to some foreign country.

Which journal? Find something where your proof is within the aims and scope. In the case of the Collatz conjecture, I'm not a mathematician, but a proof would apparently be one of the greatest mathematical achievements of all time, which means you should send to the most prestigious mathematics journals out there - so e.g. Annals of Mathematics. Note the website gives exact submission instructions (in this case you don't need an account at all, you can just send them an email).

Finally, you might be interested: Do journals in general have any kind of policy regarding papers submitted by someone without a research affiliation? Top journals get lots of submissions. They won't desk reject a submission solely because you are an independent researcher, but they will desk reject if your paper is poorly written, poorly formatted, poorly structured, etc. When you say your paper is well-written, sure it really is well-written (not just English-wise, but also in the "language of mathematics"), or the reader might not make it past the abstract.

Allure
  • 127,528
  • 50
  • 325
  • 493
-7

Try to follow the rules for writing using available papers from the field as an example and follow the rules from guidelines for authors from the relevant journals. Then submit your work to preprints.com or arxiv.org.

Don't worry after that. You will find a way to real, reviewed journals if the work is correct and a valid extension of knowledge in the field.

One shouldn't undermine themselves with such statements as "I'm not real mathematician". If you reached the proof as you said, then you are. Initially many "real" scientists will not understand your findings and will tell you that it is nothing more than rubbish if you haven't yet provided any proof.

TCooper
  • 189
  • 1
  • 8
Dejan
  • 11
  • 1
  • Hate to see all the flack you're getting for a valid answer. It's odd to me how often hard problems are solved by people outside the field offering a fresh perspective, and yet people within the field will continually assume only they themselves, with the tools that have failed them for decades or centuries to arrive at a solution, are the only possibility. Sure, in most cases building off existing work is the only way forward - but for the hardest problems, it's most often a novel approach from a novice that, at least starts the ball rolling, if not providing a direct solution. – TCooper Sep 10 '23 at 20:34
  • 4
    @TCooper if you know of examples of a hard problem being solved by someone outside the field offering a fresh perspective, you should write an answer to this question: https://academia.stackexchange.com/questions/181336/is-there-a-single-example-of-an-outsider-considered-a-crank-publishing-a-groun – Allure Sep 10 '23 at 23:10
  • @Allure Rather than citing examples that are anecdotal, here's scientific research covering numerous examples -https://journals.sagepub.com/doi/full/10.1177/0956797616634665 - You can also check out InnoCentive - https://en.wikipedia.org/wiki/InnoCentive - who collaborated on this research. If a counter-argument is simply "this doesn't apply to complex mathematical problems because math is special", I have no contradiction for a logical fallacy. – TCooper Sep 10 '23 at 23:30
  • 1
    @TCooper again, if you know of examples, you should write them as answers to that question. – Allure Sep 10 '23 at 23:38
  • @TCooper many of the answers to that question cite examples from >30 years ago. If you know of examples from >30 years ago or from fields that are not mathematics, you can definitely write them as answers. I'm done.. – Allure Sep 10 '23 at 23:45
  • 1
    @Allure I guess I missed the point of your comments in this context originally. It seemed you were attempting to discredit my statement by implying a thread on this site had no specific examples meeting the criteria. I was simply attempting to support my statement. I have no interest in copy and pasting 230 examples from the research I linked to that thread, or talking about the petroleum engineer that created auto-tune, etc. The fact that outsider's fresh perspectives often lead to novel solutions is well established. – TCooper Sep 10 '23 at 23:52