Q1: What is your view on the purpose, aims and benefits of MO?
Q2: What are your specific aim, targets and benefits in participation at MO?
I treat MO to some extent as a substitute for a large mathematics department with experts in all sorts of areas. My university is small compared to many, and with only a few specialised research groups, and I can't always find an expert to wander up to and
talk ask about various things I need that are at arms length from what I know.
For me, MathOverflow is a relaxing way to learn some math. It's like walking into a bookstore and just randomly browsing the selection. Sometimes you come away empty-handed, but but just as often you're pleasantly surprised. And you might come across some stuff you wouldn't have thought to actually look for.
Since I've gotten so much from MathOverflow, I also try to contribute to it myself, out of a sense of balance. But in my mind that's secondary -- morally like a member's fee, or something.
The purpose of MO is to provide a place for people to ask (and answer) research-level mathematical questions. The aim is thus to provide the best possible conditions towards this end, and the benefit is that having such a site improves the infrastructure for mathematical research.
It gives a unique opportunity to ask a question to a very large audience. In the usual conversations in a math department or in a conference, the audience is small. Here the audience is larger and much more diverse than even for a question published in a journal. And the feedback is usually immediate.
Mathematics is my profession and also my hobby. I enjoy discussing mathematics with people. This is a nice outlet for such activity. Sometimes I learn new things useful for my research.
I thought that it would be a good place to make available to a wide audience problems which I've been unable to make progress on despite years of effort. To my great surprise, the effort to alter my questions and make them easier led to my posing a question which I could actually answer partially.
First let me say that these are very good questions.
My answer to the second one would be similar to many others, let me just add that MO has immensely enhanced my feeling of being a member of one of the most worthy communities in the world.
Concerning the first - I think MO is similar (in a minor way) to things like ethnic entities or languages which develop according to their own metapurposes that might be not known or even unreachable to people participating in their development.
I use it to find out what questions people find interesting in order to gear my studies towards them. I search around for questions in things I am interested in and if no one is answering these questions, I take it as a sign that the area has interest but needs answers (and then check the literature of course). This gives me a better sense of "what are the hot topics" than the literature alone. (Plus if I know the answer, I answer it!)
Also, it's a good way to find out who the experts are your field. I keep seeing the same names answering all of the questions in my topic, and so I am going to that person's workshop this summer!
So I don't tend to answer as much on this site (likely because I am a graduate student and some of this is hard!), but still "participate" and get a lot out of it.
I recently started with MO, but I see it as a brainstorming session in a way. I was specifically trying to see how others would connect a Rubik's cube and calculus, as I was having a bit of trouble. I received several helpful answers, and although many people posted an answer only to tell me that it could not be done or question who planted the notion that they were related, I have recently completed the essay, which I am very proud of. Thank you to everyone who posted something they thought might be useful (even if they weren't sure it could be done).
I really see MO as a way for people involved in math to encourage and help each other, which is why I was slightly confused to see several discouraging "answers" to my question, and to then have it closed simply because a few people with a higher reputation than me said it couldn't be done. As informed as they may be, I found that they can be mistaken.