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(mathematical subject is intended)

It's obviously not a simple question, but I thought it could lead to some interesting discussion. I was reading Lewis Campbell's biography of James Clerk Maxwell and he mentioned somewhere that when they started learning Euclid at school, Maxwell's peers (Campbell included) could feel that Maxwell was already at the heart of the subject, while they were all on the edge. I have felt this myself many times (being on the edge; almost not being able to bridge the gap between an artificial understanding and a "creative understanding"), and I am really interested in how it is that someone reaches the heart of a subject. I don't know how to define this, but it's usually when they can give simple proofs that you couldn't even dream of giving. The proofs have this magic property about them, such as auxiliary lines that seem to drop out of nowhere. Such minds are able to pursue successful research in the field. They have a full command of the subject even very early on, as in Maxwell's case for instance.

This might seem like a vague question, but I have tried my best to give a feel of what I really mean, so I hope we can get a other people's views on this.

NOTE: If you understand what I mean and have a better way of phrasing the question please leave a comment, I really don't want this to get closed because of my bad wording.

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    $\begingroup$ Two comments: it doesn't seem likely to be the case in all examples that this deep understanding is acquired early on; and I'm skeptical that there's a clear algorithm to achieve this level of understanding, given that many more people would like to work at the level of Maxwell than actually do. $\endgroup$ – Kevin Arlin Nov 6 '14 at 0:03
  • $\begingroup$ @KevinCarlson: In many cases it is acquired early (if we acccept that teenage years is early), e.g. Gauss, Abel, Galois, Euler, Lagrange, etc. But overall I'm hoping we can at least get some views on this, even if - and you're right - there isn't a clear-cut path. $\endgroup$ – user45220 Nov 6 '14 at 0:08
  • $\begingroup$ Yes, of course there are many great young mathematicians, and of course it would be interesting to learn how to join them. $\endgroup$ – Kevin Arlin Nov 6 '14 at 0:09
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Well, the real answer is "I don't know". But I too think about this question, and here are my thoughts:

1) Understand the subject as thoroughly as possible. This seems obvious, but there's some creativity here. Do you understand the history? And what current thinking is? Have you read all the papers that you can on it?

2) A combination of quiet solo study and invigorating conversations with others - both have tremendous benefits to understanding.

3) Repetition, experimentation, and practice - restating things, writing for others, trying new ways of looking at things... these all contribute to a deeper understanding.

4) Drugs - substances alter mental states and can provide insights to those who are already well-versed in a topic. Note: I do not recommend taking drugs nor am I responsible for you breaking the law or for any adverse effects from you taking drugs. See Engineers on LSD and Sagan on Marijuana

5) Personality/brainpower - obviously some people are better able to dive deep into a topic than others. Better concentration ability, more mental horsepower, other gifts.

6) Lateral relationships - the more you know about diverse subjects, the more likely you are to tie insights from others into the subject you're interested in.

7) Finally, learn the Trivium, which gives you a framework for understanding, and becoming an expert at, any topic.

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