[ben509]

> 1) Learn math as necessary for a field you're interested in

I think this a Rumsfeldian problem: you can address the known unknowns, but what about the unknown unknowns?

There's no great answer to this, and it's why I think the original post was advocating the broad fields of math.

Just knowing math exists, though, doesn't address the depth to which you study it. I've gone at monads about five times now and I think I get it, to where I see why a promises library works the way it does. Same with databases, learned SQL, then stumbled across the wonderfully cranky dbdebunk.com, read Date's book, finally spent some months working on an implementation of the relational algebra and finally grokked it.

The problem with some of these deeper subjects is you'll have many "aha" moments only to look back at your early "ahas" and realize you had no idea what you were doing.