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I am an amateur mathematician. I have trained myself on math-related websites and other resources. However, the lack of technical knowledge, methods, and sociological culture usually shared by professional mathematicians has prevented me from going further and publishing rigorous and interesting results.

Many amateurs, often outside areas where universities are easy to reach, face the same hurdles. I know a lot of them are cranks but not all of them are: some have extremely good computational skills, an extremely original vision of maths and sometimes a lot of time on their hands.

What can be done to bridge this gap between amateurs and professionals? How can we give amateurs the codes and techniques needed to enrich the corpus of mathematical results with new, fresh and interesting contributions?

For example, could retired professions create a free resource that links universities and labs to amateurs? In this way, the retired professionals could act as "pre-referees," teaching the amateurs how to write a paper, and collaborating on the research itself? There could even be a magazine in which publications resulting from this are published.

closed as too broad by Buffy, Bryan Krause, user68958, Jon Custer, Azor Ahai Apr 24 at 17:41

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    I think it would be too hasty to close this question here. Where can such a question be asked, if not here? – paul garrett Apr 23 at 21:46
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    What exactly is your question? It looks like a meaningless rant. Please rephrase your question and write a real question with "?" at the end. – Alone Programmer Apr 23 at 22:00
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    I think the main question asked is how to increase the likelihood that genuinely good ideas found by amateurs can be actually used in mainstream research. I think it would be good to make this explicit. Additionally, what can be done by who? Do you want to know what an amateur can do, or an academic, or someone else? – Discrete lizard Apr 24 at 5:51
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    My impression is that while there might be exceptions, what most amateurs really want is to to able to make a contribution while still working with math in exactly the same way they always have, which usually means reading textbooks (though often advanced ones) and playing around with things on their own. And they want to be able to focus on the same problems they always have, since those are the ones they are truly interested in. – Tobias Kildetoft Apr 24 at 7:59
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    what i was trying to express is, that amateurs always aim for the unproven conjectures and many professors receive dozens of emails from amateurs who claimed to have proven one of the famous conjectures on half a page. – Zest Apr 25 at 6:07
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First, I can't help but notice your complaints are extremely vague and the language is evasive. You don't like "sociological culture usually shared by professional mathematicians", but it's hard to guess what you mean by that, and what are specific "aforementioned issues" you have problems with. Put it into plain words, please.

However, I can give a general comment. Amateurs might indeed have brilliant ideas, and this is true for all branches of science from history and linguistics to maths and chemistry. Amateurs can write papers and submit it to journals/conferences as we at universities do.

The problem is usually in, as you say, "lack of technical knowledge and methods", which in practice means that these papers are often poorly written (I had to review some) and aren't ready to be published. Your idea of "bridging the gap" by effort of retired professionals is rather popular among amateurs: we might have brilliant ideas, and all we lack is just knowledge of some technicalities, so we just need someone to take care of them, so we can focus on the big things.

Unfortunately, working in science is a full-time job, and dealing with "technicalities" takes most time. I would go as far as saying that understanding the methods and being able to write a decent paper is as important as having bright mind full of cool ideas.

Potential "gap-bridgers" should 1) be good in several wide areas (while academics usually specialize in narrow fields); 2) devote a lot of free time to find grains of brilliance among 99% waste generated by amateurs; 3) do the hardest and least rewarding work by turning mere ideas into something publishable; 4) neglect own interests in favor of pursuing someone else's agenda.

Summing up, an amateur doesn't necessarily have to undergo a formal training and get a PhD. However, having good understanding of methodology and paper writing is vital. There are books devoted to this subject -- take your time and acquire the necessary knowledge rather than waiting for someone to do the job. If you still need help, try to collaborate with people having more experience and write a joint paper.

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    I don't think what you point out are complaints. Mathematicians share a lot of shorthand that is not readily available in the written literature. There are also many folklore results, things that are known and passed on orally but not written down in readily available or easy to locate sources. – Andrés E. Caicedo Apr 24 at 15:11
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There are many ways to contribute to a scientific body of knowledge without professional training. Nowadays it's actually much easier via platforms such as SE and ArXiv. If you have an amazing new idea - write it down formally, put it on ArXiv/MO and you'll get plenty of reviews (on MO) and a platform to showcase your work (ArXiv). If your idea is good and is related to something that some mathematician is currently studying, I am certain that sending the paper to them will at the very least elicit a reply and possibly a discussion.

In my opinion, the reason you don't see a lot of these bridges being built is that professional academics don't see a need to build them. They already have access to rigorously trained students and fellow professors: why spend time trying to figure out which amateur mathematician is worth talking to? Many amateur mathematicians are simply not rigorous enough and do not produce high-level results. The cases where it does happen are just too rare and not worth the time and effort required to find them.

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    I think the "write it down formally" etc. is precisely the part that an amateur may need most help with. Also, you won't be able to get a paper posted on the arXiv unless you know how to write something that looks like a paper, and that is not necessarily a simple thing. I don't think it is just a matter or having a platform. The question also suggests that the amateur may not have access to the latest scientific literature, which makes it harder for them to know they are actually providing a new contribution. – Andrés E. Caicedo Apr 24 at 14:19
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    @AndrésE.Caicedo There are enough freely available papers around to get an idea on how to write one. And at least the professionals I know will not disregard your idea just because you don't follow a certain style, as long as you put effort into writing it down properly (e.g. checked for typos, wrong calculations, etc.). The accessibility of literature is a major discussion point in many fields already. Pros are paying a lot for it (either themselves or through their institution), so it might be hard to get it without proper funds (or taking non-legal approaches). – Dirk Apr 24 at 14:48
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    @Dirk Maybe. But I doubt all that is needed is to read enough examples. Of course, this may be enough for some people, but others may need a more hands-on approach. – Andrés E. Caicedo Apr 24 at 14:56
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    Usually people spend years perfecting writing skills, usually with the help of a mentor. That’s exactly what a PhD is – Spark Apr 25 at 9:06

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