This was a question I posed to a friend of mine. We decided to define an “amateur” as someone without a PhD in physics, mathematics or something close.
We came to the conclusion that it is very unlikely that some one with out a PhD could in fact make a real contribution. This is despite the fact that things are far more open today than they ever have been. I mean, we have the arXiv and open access journals online. Almost everyone has the internet at home these days, and if not the local libraries do.
From time to time undergraduate students can contribute, but these are with close supervision. The supervisor will guide the student and nurture the natural ability. This was really the closest to an “amateur” that could contribute we agreed on.
So, why can’t “amateurs” contribute? Here are my thoughts…
1) Without spending some time in academia, “amateurs” are not aware of the culture and what is expected of anyone wishing to contribute to mathematical science. They do not know how to do research.
2) “Amateurs”, although interested and very keen at times do not often realise just how much of a prerequisite can be required to conduct research. They can often lack the mathematical skills to contribute. Claims like “I can solve the Riemann hypothesis using high school mathematics” only suggests that they don’t understand the hypothesis correctly in the first place. Trying to rewrite particle theory using high school maths is also redundant. We have a great construct for doing particle physics, it is called the standard model.
3) Theoretical physics, mathematical physics and mathematics as a whole is split up into smaller sections. One can only hope to get acquainted properly with a small subset of what is out there. Without specialising to a large extent, it is unlikely that one can discover something new and interesting. Trying to find smaller, specialised problems to work on is usually the way forward: unless you are a genius and can discover a whole new branch of mathematics! “Amateurs” seem to be focused on very well-known and published open questions. In number theory the Riemann hypothesis is a great example of this. In physics, a theory of quantum gravity is an example.
4) Because the individual does not understand it, it must be wrong. “Amateurs” fall into this mind set quite often. Finding a simpler more elegant approach to things is a large part of the mathematical sciences. However, trying to show that special relativity or quantum mechanics are mathematically inconsistent or do not agree with nature is futile. This also includes the desire to use nothing but high school maths to explain all of physics.
Not that I want to discourage anyone from thinking about mathematics and physics. I encourage it, but with a caveat: reading Wikepedia and popular accounts of science are not enough for one to start to do research.
UPDATE (15th May 2014)
Please do not post about your pet theories in the comments here. If you have something to say related to this post about the role of amateurs in science then please by all means share it here. Thank you.