One of the reasons that mathematics is so vast and powerful is because people have focused their efforts in specialized areas. If math weren't so broad that a single person could understand it all then it would be pretty unimpressive.

I disagree the characterization that mathematicians don't try to broaden their horizons. Like scholars of any subject, mathematicians follow their interests, which often take them to places never expected to be. To me, that fits the bill for "broadening horizons". And if they don't seem to move far outside of their focus of study, well, I say it has to be this way because mathematical material is often so dense and precise that it takes a very long time to understand what's going on in any one subject.