As recognizers of patterns and manipulators of abstract concepts, we are all mathematicians. We differ in our levels of skill and passion, but we all can benefit by sharing ideas, questions, problems and solutions with one another.
Mathematicians are less concerned with getting correct "answers" in a hurry than they are with the careful and interesting thinking that they do when they tackle an open problem. Often, thinking about problems is just as interesting as finding the solutions, and asking questions is important as answering them.
Mathematicians are concerned with finding true statements that can be made about mathematical objects. They use the and rules of logic to write proofs about these statements.
