Various views are not mutually exclusive. For example, gamificationgamification is definitely an important part of the SE architecture. And even if reputation and badges should not be the main reason why some users come here, if they help to reinforce good use of the site (for example, if they incentivize users to ask good questions and write good answers), then that is fine. Similarly, if somebody comes here mainly because they like to teach mathematics and they strive to give very accessible and clear explanation in each of their answers, that also helps creating a repository of useful mathematical facts. But on some occasions there are clashes between users who have different views about what this site is/should be. Somewhat related point was mentioned also in this answerthis answer the the recent post: http://meta.math.stackexchange.com/questions/20944/what-do-you-think-is-the-largest-problem-facing-math-se-today-july-2015What do you think is the largest problem facing Math.SE today (July 2015)?
I have found this older question which, however, seems to have a bit different focus: http://meta.math.stackexchange.com/questions/1574/what-is-the-purpose-of-math-se-to-learn-to-provide-nice-answers-or-just-to-getWhat is the purpose of math.se; to learn to provide nice answers or just to get them? (The OP asks about this in connection with the problem whether it is ok to post answer which is similar to the previous answer. And the main stress seems to be on difference between posting answer as somebody in order to teach somebody something and posting answer with the intention that I learn something while working on the answer.)
I have also found this post: http://meta.math.stackexchange.com/questions/191/dissonance-of-purpose-what-kind-of-site-should-mu-beDissonance of purpose: What kind of site should MU be? Already from the number (id) of the post, you can see that it is from beginnings of the site. So I thinks that the positions might have changed since then.