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I was reading a book on coding theory, there was a definition fot the Hamming's Distance and also one example. Understanding purely from the definition was hard but the example helped to give meaning to the definition. I felt the same when reading some other books so as I'm still self-learning mathematics, I just got this curiosity:

Is it a problem to understand mathematical definitions only with examples? Should I aim at a level of understanding with definitions only? Will this level eventually come?

It may be a naive question, but I feel insecure of bulding wrong study practice. I'm a also a piano student, when reading about piano study I've discovered that it's study isn't really intuitive then I've expanded and started to get worried also with studying habits for other things I study.

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I think it's quite impossible to really understand a definition without examples. As for "increased level" - you can try to cook up your own examples. –  user8268 Sep 4 '13 at 7:53
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@DanBrumleve I've discovered where Grothendieck is! 57 is the Grothendieck prime, look here and there is a movie staring Wesley Snipes! Obviously Wesley Snipes is Grothendieck in disguise! Gotcha! –  Vÿska Sep 4 '13 at 8:24
    
@Gustavo I guess my reason for mentioning it is just that the uncomfortable possibility of discovering a Grothendieck prime is a pitfall of learning by definition or rule or pattern rather than example. $57$ looks like a prime. My answer I hope has a more balanced implication, there is so much to learn about numbers without thinking about any of them. On the other hand, $57$ is a great example of what I mean. :) –  Dan Brumleve Sep 4 '13 at 8:27

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Almost every mathematician learns from specific examples first, and then goes on to the generalized abstraction. Just keep practising with the specific cases until the general idea seems easy and natural. You don't need to worry about this.

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There are at least two sides of mathematics, on one you have an intricate structure whose tricks and aphorisms yield obviously true conclusions, and on the other there are computations and contests to find counterexamples to the conventional wisdom. The beauty is always somewhere in between.

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