An old joke, but one that ran through my mind as I read the article The Last Days of the Polymath in the Autumn issue of Intelligent Life. That's polymath, not in the Tim Gower's sense of a massive collaborative effort to solve math problems, but the more traditional sense of people who know a lot about a lot, people like Leonardo da Vinci and Ben Franklin. But the article reminisces about an earlier time, when one could learn "a lot" about an area, such as physics, without needing to know all that much, basically what's covered in a first-year college sequence today. As Bill pointed out, we don't even have many polymaths in sense of knows a lot about a lot of math either.
Advances in knowledge have made it impossible to know a lot about a lot. Advances in communication and travel have made polymaths less important since we can now better pool our talents. I might only know a lot about a little, but there isn't much I can't find a lot about quickly.
Richard Posner's quote in the article caught my eye.
“Even in relatively soft fields, specialists tend to develop a specialized vocabulary which creates barriers to entry,” Posner says with his economic hat pulled down over his head. “Specialists want to fend off the generalists. They may also want to convince themselves that what they are doing is really very difficult and challenging. One of the ways they do that is to develop what they regard a rigorous methodology—often mathematical.
“The specialist will always be able to nail the generalists by pointing out that they don’t use the vocabulary quite right and they make mistakes that an insider would never make. It’s a defense mechanism. They don’t like people invading their turf, especially outsiders criticising insiders. So if I make mistakes about this economic situation, it doesn’t really bother me tremendously. It’s not my field. I can make mistakes. On the other hand for me to be criticizing someone whose whole career is committed to a particular outlook and method and so on, that is very painful.” [Spelling Americanized]
We monomaths develop specialized vocabularies and mathematical tools and models because it helps use deeply understand an area. But much of what Posner says rings true. For example I've seen these "defense mechanisms" kick in quite often from both computer scientists and economists towards those trying to cross into each other's fields.