Yeah, that makes a lot of sense... I never ever studied anything before sophomore year of college. And then when faced with serious large-scale programming problems, and 300 level math, I realized I don't know how to study. It was definitely a struggle, but I'm glad it happened to me early, and not when I entered a "real world".
Doesn't matter. He'll run across a problem that doesn't just fall down and surrender in the face of his blinding intellect, and at that point, his studying and thinking skills (not the math trivia) will kick in and allow him to slay the beast.
A trivial tangent: oddly enough I find that, whether I'm programming for distributed applications or animations, it is the number theory, and not (surprisingly) the linear algebra, that I am most grateful for having learned. Boutade's Law: do not underestimate the algorithmic utility of knowing your way around the integers.
This is intriguing. Care to elaborate on a problem you've had and how number theory helped? I did the typical math courses required for a computer science degree (discrete math, linear algebra, diffeq, probability and statistics), but stopped short of the serious math courses (analysis). I've been meaning to start up with analysis (mainly for fun) but I'm intrigued to see how it might help in my day to day work.
Well, unless you're terribly interested in moving on to things like differential geometry, I'd put analysis off a bit longer were I you. If you'd like something stimulating and potentially useful, more advanced combinatorics, number theory, or algebraic geometry might all be good choices. The first two tend to overlap a bit when they are presented in textbooks (with some group theory tossed in as well). If you do crazy graphics programming, or if you program robots' spatial reasoning, algebraic geometry might pay off - though, AG doesn't have as easy an entry. (One exception might be via the book Computational Algebraic Geometry by Hal Schenck).
As far as number theory "paying off": 1) Many proofs in number theory are algorithmic in nature; 2) Computers understand integers with greater facility than they do the psuedo-reals we call floats - often times efficient integer approximations will be more appropriate than slower floating point solutions, and knowing the integers will help you develop/understand these approximations; 3) If cryptography is your bag, number theory is a must; 4) Martin Davis, Yuri Matiyasevich, Julia Robinson, Hilbert's 10th problem, Turing Machines, computability theory, and (the number theory bit) diophantine equations (sorry to be cryptic, this is getting rather long winded).
I hold a bachelor's in math. In my current job, I use basic algebra and trigonometry on a regular basis, and occassionally basic calculus.
I have never had to use the theoretical information or complex analysis directly, but the practice gained in studying the theory and the proofs has helped me indirectly many times in programming. For most people. in depth knowledge of mathematics is of limited use, but in depth knowledge of problem solving gained by studying math can be invaluable.
As you mentioned, it varies wildly by your career path of course.
Actually, for the last two years before striking out on my own, I ended up working on a project involving complex calculations with credit derivatives. And before, I created a side project implementing risk models such as Black Scholes, Monte Carlo, etc. So the math ended up being useful after all.
But yeah, I think even none of the things you learn in college end up being useful to you in "real world", it's all about problem solving techniques you master that apply in any situation.
