[QUOTE=NinjaChick]
2. More math. Euclid is a must, and I would recommend Newton as well (though Newton is hard to work through). I’d also recommend some Descartes for a start on Algebra, Leibniz and Newton for calculus, and Ptolemy for astronomy. If you want to study Newton, you’ll also definitely want a passing familiarity with Apollonius’s Conics.
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All of these books are great for if you’re interested in mathematical history, but of limited value if you actually want to learn math. The modern presentation is just so different that you’ll pretty much have to unlearn everything you’ve read from before the mid-19th century.
If you actually want to learn math, you’ll need a solid background in basic mathematics (high school level stuff through calculus and linear algebra). Stewart’s calculus book is supposed to be pretty good, so you can start there if you need a refresher. For linear algebra, you want Axler’s “Linear Algebra Done Right”. It also doesn’t hurt to have a background in basic combinatorics, and something like Rosen’s “Discrete Mathematics” is a pretty solid introduction. It wouldn’t hurt to have a little more exposure to symbolic logic; check out Barwise & Etchmenedy for a reasonable introduction.
The meat and potatoes of modern mathematics are analysis, algebra and topology. High-level undergraduate/introductory graduate courses will generally use Rudin’s “Principals of Mathematical Analysis” for the first and Munkres’s “Toplogy” for the third. Algebra classes are split a bit more; I used Herstein’s “Topics in Algebra”, but Dummit & Foote’s “Abstract Algebra” is also popular. At the very least, you need to go through the equivalent of those classes to be considered mathematically knowledgeable, although you haven’t even really scratched the surface of modern topics.
There are some other books and subjects that are worth looking at along the way. Knuth’s “Concrete Mathematics” is a great continuation of the basic combinatorics outlined in Rosen. If you want more, van Lint & Rosen’s “A Course in Combinatorics” is a popular graduate-level introduction to combinatorics. Paul Halmos wrote a couple books, “Naive Set Theory” and “Finite-Dimensional Vector Spaces” that are generally considered classics.
Theoretical computer science isn’t generally taught to undergraduates in math, but it really should be, as it contains some of the more important ideas of the 20th century. Sipser’s “Introduction to the Theory of Computation” is one of the best textbooks I’ve ever seen, and is very popular, so it’s definitely worth checking out.
It also doesn’t hurt to have some exposure to probability & statistics. Wackerly et al. is probably the best undergraduate level text.
Finally, I’ll throw in a recommendation for Kellison’s “The Theory of Interest” (but wait for the 3rd edition, due out sometime next year). It’s a rigorous introduction to a topic that should be much more widely taught, and is well worth your time to study.
Most math programs will also include a course in differential equations, but I’m not convinced that it’s worth studying at the the undergraduate level, so I left it off. It’s easy to find books if you’re interested.