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My apologies for such a general question, but as a 'math enthusiast', I've often wondered what the optimal strategy is to increase my mathematical maturity. Broadly, should I aim to hunker down & study one topic in detail (measure theory/analysis, which is what i'm doing right now), or should I study a broad swathe of different topics?

Also, are there specific fields of math that are better for developing mathematical maturity? I've often wondered whether number theory/graph theory/combinatorics are better for developing mathematical maturity as even basic proofs need one to think imaginatively.

My apologies again for such a broad question that will likely have many different contradicting responses. However, I don't have anyone to discuss this with, so thought I'd throw out my question here.

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    $\begingroup$ Can you give us a better idea of where you are as a mathematician? We could probably give a better next step with this information. $\endgroup$
    – ml0105
    Apr 18, 2014 at 6:09
  • $\begingroup$ Write proofs. Get feedback on them. $\endgroup$
    – Potato
    Apr 18, 2014 at 6:10
  • $\begingroup$ I know a good bit of point-set topology (Munkres), some algebra (Herstein). Really enjoyed Baby Rudin. Currently trying to learn measure theory. I've enjoyed combinatorics/graph theory/number theory but don't really know too much other than the basics there. $\endgroup$
    – Tejus
    Apr 18, 2014 at 7:00

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Hint: Try to teach what you learned (also what you want to learn) to people with different background!

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