Is it a problem being unable to understand a mathematical definition without examples?

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
@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! – Voyska 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