Do great mathematicians understand all areas of math?
Historically, yes. Currently, not really. Over the past couple of centuries, mathematics has become so incredibly diverse in it’s subdivisions that it’s really impossible for any one person to have expertise in all areas of math. Research mathematicians typically specialize in one (maybe two) fields, which is enough to occupy all of their time. There may be some out there who have at least a decent understanding of most, if not all branches of mathematics, but none who could be considered an expert in all fields. To give you an idea of how deep of a subject math actually is, a professor at my old university once told me that if you rate one’s mathematical knowledge on a scale of 1 to 100, 1 being completely ignorant, and 100 representing a full knowledge of all that is currently known in math, a person with a four-year degree in math (that would be me) would only score about a 3 or a 4.
