Is there any point in sharing a scientific discovery as a "nobody"?

As far as I'm aware, Marjorie Rice had no formal mathematical education, but discovered new pentagonal tilings that are still credited to her. She once got a standing ovation at a meeting of the Mathematical Association of America, as well.

So yes, there is a point, if there really is a scientific discovery. Note the highlight on the word "if". Most "discoveries" by amateurs are obviously wrong, to the point that spending anything more than a few minutes on it is a waste of time. If they're not wrong, they can be pointless/unnecessary (example). If you're going to contact an academic with some claimed advance, be pretty damn sure about what you are claiming, because you only get one chance.

The other answers are focusing on the fact that most of the time, when amateurs claim to have made a discovery, it is incorrect or otherwise without much value. That is true.

But to be a bit more helpful, I might reframe your question as "How can I get professional scientists to engage with my work, as an amateur?" The answer is to learn how to present your results the way professional scientists do. Refusal/inability to do this is the main reason amateurs get ignored. It's not an easy thing to learn, but it is easier than making important scientific discoveries, so if you can do the latter, you can learn to do the former.

Here is a dose of reality. All scientific discovery is subject to scrutiny from peers before publication (and also after publication). After several decades of successful scientific publications, some of my ideas, which are fully substantiated still get rejected, or require a year's worth of back and forth to publish in top venues. Someone without that experience or rigor may think this process is someone gatekeeping discovery for some nefarious or unjust reason, it is just the high bar for maintaining the scientific record.

There is no magic cabal of scientists blocking discovery. It is just that there is an expectation of rigor and professionalism that someone without appropriate training is highly unlikely to have. Unfortunately, most research-active scientists do not have time to entertain work that is so far from being ready to be seriously scrutinized. As a result, those on the outside find themselves kept out.

If you have a high-quality result, done appropriately using the scientific method (or whatever is acceptable in said field, of course not everything follows hypothesis-based processes), and reported in the appropriate/conventional way, then it should be able to find some home for dissemination.

The problem is that without appropriate background, you may not know how to do any of those things, or have a good way to figure out if your discovery is even novel, worthwhile, or actually correct. This is why people spend more than a decade between undergraduate, graduate, and postdoc training to reach the level of being capable independent investigators.

Science is very romanticized. The picture that a non-scientist usually has of science is far from what the reality is. In physics you get stuff sent all the time where they do something I came to know as "Lettercalculation", you rearrange letter in formulas, the more fundamental the better, e.g. Maxwell's Equations, without any deeper knowledge of the mathematical objects, the theories that underlie this particular equation, the microscopic details etc. There is a reason why all the sciences require several years of studying, starting with the basics, that are not glamorous (to most, sorry mathematicians:-) ).

When I was younger I answered politely and pointed out the error. Now I just ignore that. Because it ain't worth the time. The person sending it usually does not understand why they are wrong but insists on being right.

It is different from a person asking something about my field. They are not "right", they listen and try to understand. Those usually are fun conversations.

Does it ever happen that a "nobody" makes major breakthroughs? Yes. Of course it does. Allure brought up Marjorie Rice. A more famous example is Ramanujan, who is one of the all-time great mathematicians.

But it doesn't happen often. In fact, it happens very, very, rarely.

And professors get lots of letters from nobodies. It is not their duty to try to understand what you have done. Nor is it their duty to give you a detailed explanation of why you are wrong. They teach, they do research, they supervise doctoral students, and so on.

You might try offering to pay someone in the field to help you with your question. Grad students are often in need of money, maybe one of them would help you out for a relatively reasonable fee.

Or you could write up your idea, submit it to a journal, and hope you get a detailed response and not just a rejection. Or maybe you could submit it as a presentation somewhere.

Because it is quite likely that you are wrong. Sorry.

If you share your discovery with others you will get feedback and can learn something useful from it. E.g.:

1. you may learn that the ideas you discovered are correct but already known

2. you may learn that the ideas you think you discovered are wrong, and the reasons why they are wrong.

3. you may learn that the ideas you discovered are correct and novel, but not interesting.

4. you may get zero engagement from people on your ideas, which looks like "no feedback" but is actually its own sort of feedback, indicating that your ideas are likely not a productive direction to follow.

5. you may learn that the ideas you discovered are correct, novel, and genuinely interesting. And through sharing them, others will learn about them too!

The last of these outcomes is obviously the most exciting one. If you are worried about not getting the credit in the event this happens, you can easily protect yourself by maintaining a record of your discoveries and when/where you published them. Keep careful logs of your ideas and of emails and web postings: screenshots, timestamps of emails you send, etc. Use a gmail account and send copies of all your emails to another account that uses a different email service if you're really paranoid. That's all the protection you need. In this day and age it would be essentially impossible for a malicious researcher to steal someone else's ideas that were carefully documented in this way.

You should keep in mind however that it is overwhelmingly more likely that one of the outcomes 1-4 will be the actual outcome of sharing your ideas. But my point is that even those outcomes are all positive outcomes. Getting feedback from others about your ideas, even negative feedback, is a learning experience: it can help you correct misconceptions in your knowledge and learn about what people are interested in, and will ultimately make you much better positioned to come up with better ideas in the future.

Is there any point in sharing a scientific discovery as a "nobody"?

what incentive is there for amateur "nobody" researchers [...] to share their discoveries?

To summarize: yes, there is a point. The incentives are all aligned in the direction of sharing your ideas. The only disincentive is the fear of hurt feelings from the possibility of finding out that your ideas were not as exciting as you had hoped. But if you let that fear rule you, then one might indeed ask, as you are asking: is there any point? - To thinking about research at all, that is.

Someone who makes an important discovery should be showing it to others with the possibility of it being shown to be wrong, or maybe being inconclusive or incomplete, and in need of improvement.

If someone has zero interest in finding out how their result may be wrong they're almost automatically put in the"crank" category.

As a concrete example, bridging the gap between quantum field theory and general relativity, as a layperson, I'd say is straight up impossible. Before you even study the basics of the fields, you usually obtain a B.Sc degree, i.e. at least 3 years of studying.

There are instances where doing a Gedankenexperiment can lead to major advances or insights, but usually they have to be accompanied by equations, as equations are the language of science. If your discovery lacks equations, chances are nil they are taken seriously. Additionally, you should familiarize yourself with the fields so you at least use the common terms and symbols to describe and formulize your ideas.

You could also share your idea, as a starting point, in one of the relevant stack exchanges here and get some immediate feedback to the merit of it.