I find that I have the opposite problem. I think I know less than I actually know. Sometimes I feel (as a mathematician would) that I don't understand something

I find that I have the opposite problem. I think I know less than I actually know. Sometimes I feel (as a mathematician would) that I don't understand something unless I can prove it, assertion by assertion and line by line. I dropped out of grad school after one year, and it seemed like a bigger risk/deal than it was. My sense is that the major benefit of having a PhD (especially from a top school) is the confidence that comes with it. It gives you faith that you know enough to tackle the interesting problems, and an assurance that people (usually uninformed people, often without the educational credentials they overvalue) won't try to peg you down as not up to it. Socially, it gives you the right to be an expert, which is not the same thing as expertise but just as valuable (if not moreso) in the work world. So, even in traditional education, I'd argue that much of the benefit earned (maybe 85%) is confidence, both your own and others' in you. You can get a lot on your own by reading the papers and books, but you never get that official approval or that "proof" that you learned the right stuff. Oh, and the answer is that the water level stays exactly the same. Archimedes' Principle is that a floating object displaces as much water as it weighs. When melted, the iceberg (90% submerged) takes up as much space in the water as the submerged part occupies.