This is a well-known issue. Basically, a mathematical problem tends to involve several non-trivial steps. If you are too pessimistic, it is impossible to see all these steps (because you get bogged down in proving details and lose track of the point of the problem.) On the other hand, if you are too optimistic, you will take too long to debunk an incorrect sequence of steps, leading to the problem you describe.
One solution is to work with someone else, and take turns being optimistic. (E.g., one person proposes a solution, then the other tries to shoot it down; it's much easier to be pessimistic about other people's ideas.) Another solution is what Mr. Weissman proposes: just investigate the problem, look at similar problems, try to falsify the problem, try to prove something stronger, etc.
I'm sure that professional mathematicians deal with this issue all the time, so you might want to ask one of them as well.
This is a well-known issue. Basically, a mathematical problem tends to involve several non-trivial steps. If you are too pessimistic, it is impossible to see all these steps (because you get bogged down in proving details and lose track of the point of the problem.) On the other hand, if you are too optimistic, you will take too long to debunk an incorrect sequence of steps, leading to the problem you describe.
One solution is to work with someone else, and take turns being optimistic. (E.g., one person proposes a solution, then the other tries to shoot it down; it's much easier to be pessimistic about other people's ideas.) Another solution is what Mr. Weissman proposes: just investigate the problem, look at similar problems, try to falsify the problem, try to prove something stronger, etc.
I'm sure that professional mathematicians deal with this issue all the time, so you might want to ask one of them as well.