[timwiseman]

First, I think Nadam makes a great point.

But it also depends to a large degree what you want to do with your life. A good junior-to-mid grade Infantry officer for instance does not really need a deep knowledge of any one particular thing, but he needs to have a broad knowledge of many things and be able to move seemlessly between them and relate them to each other.

On the other hand, for a mathematician, even saying he wants a deep knowlede of math is far too broad, it would be better to say he wants to focus in on something like Chaos Theory. Even that might be too broad and many fine mathematicians spend their careers working on a handful of closely related problems in a very narrow sub-field of a field of mathematics.