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I am not saying I don’t believe some people are naturally more gifted at certain things or develop stronger skills or have stronger talents than others.
For instance, if after failing to solve a problem, you receive the solution and study it carefully, you may discover an insight or problem-solving technique that eluded you before, and will now be able to solve similar problems that were previously out of reach.
One should also bear in mind that being able to partially solve a problem (e.g.
to expand out the definitions, solve some special cases, and isolate key difficulties) is also a very important measure of progress (see this previous post of mine on this topic), as is the practice of constantly asking yourself “dumb” questions in the subject (as discussed in this post).
One should also not focus on the most difficult questions, but rather on those just outside your current range.
And I greatly appreciated the article on your page where you say that someone does NOT have to be a genius to be a mathematician.
Does that mean you don’t consider yourself a genius or you don’t really see a distinction between yourself and others who apply themselves and are ambitious?
That’s why I disagree with this post by astronomer Julianne Dalcanton which i found linked from your page where she doesn’t believe most people can reach the level of Feynman-Einstein-Hawking smart. And isn’t part of their fame due to circumstance and perhaps even chance-not their intellectual ability but I mean their status and the fact that their discoveries happened to be earth-shattering or were given more attention by the public at-large beyond the scientific community?
Surely some of the scientists working today will make equally groundbreaking or insightful discoveries or develop innovative theories and thus can fairly be labeled “genius” or as having the same level of smarts? I think I have developed a stronger aptitude for language than for math and due to suffering from depression in high school and middle school I didn’t push myself nearly as much as I could and lost much of my motivation.
I do have a book on how to solve mathematical problems at this level; in particular, the first chapter discusses general problem-solving strategies.
There are of course several other problem-solving books, such as Polya’s classic “How to solve it“, which I myself learnt from while competing at the Mathematics Olympiads.