^ was half-asleep when I wrote that. The math/programming stuff is simply a hobby, partly because I enjoy it and it relaxes me, and partly because I find working on unrelated but challenging problems enhances my performance at work. Nothing to do with a possible career-shift, which at this point is extremely speculative.
If you have any recommendations regarding books, that'd be very welcome. Keep in mind that I'm well below your level of expertise. The calc book is by a guy named Spivak. I enjoy it, but it's much more thorough than I remember my course having been, with a far greater emphasis on proofs.
My progress may also have something to do with reading it in the early morning or late night/evening with the blackberry sitting on the open page.
Ah, gotcha (I can just see the expressions of people reading about math helping to relax you!). Spivak's
Calculus isn't a bad book at all. I mean I'd say there are a few better ones out there, but again if you find it comfortable (or should I say "relaxing"
), then I'd stick with it. It's so important in mathematics to be comfortable with your textbooks.
Regarding other books, well if it's for a hobby (as well as a benefit for your daily life), the path I'd suggest would be that of set-theory and mathematical logic. From talking with you in S&T, I know that you have a decent grasp on logical themes and the how it's applied to everyday life. What you stand to gain through ST&ML is, like with Spivak &
Calculus, a bottom-up construction of the logic that you know, a philosophical appreciation for what it means, and full mathematical rigor about how our common system of logic can be tweaked to form new logics and such. Going on this, Herbert B. Enderton is the author you want.
Elements of Set Theory
A Mathematical Introduction to Logic
These should be read in this order (but don't
have to be). Both start off at a 2nd or 3rd year undergraduate pace (you'd be fine), and progress to rigorous proofs of the Incompleteness Theorems and such. Basically a "get as far as you can" type o effort. I cannot stress enough, though, that Enderton is an amazing author, these books tickle the mathematical fancy, and IMPORTANTLY they teach you a lot about how to think about mathematical proofs. Meaning that if you want to go beyond calculus (let's say to measure theory, which is really the most profound area of math IMHO), you'll be better prepared to understand what comes next if you get a lot from the Enderton books.
In terms of a "Putnam-style" book, I would suggest
Mathematical Miniatures. Don't tell
Binge Artist, but I actually use it as a resource for coming up with some of those puzzlers in the S&T thread! But it contains a lot of problems form contests worldwide, and isn't too technical or anything.