As an amateur, I'd say yes, just for the fun of it (and perhaps to provide new potentially useful though rather sketchy ideas to professionals). I think one can't make breakthroughs in any area if he or she doesn't find it cool enough to spend most of their (free or not) time on it. On the other hand, I can easily understand that people who got involved in a long and difficult career can't take too high a risk trying to tackle a long-standing open problem that might be intrinsically unsolvable (like the Continuum Hypothesis).

My advice would thus be: if you have no real career issues, no family to feed and find math extremely exciting, just go for it!