Undergraduate approach to learning math I am going into my sophomore year as an undergraduate and I would like to ask the more experienced folks a couple questions about learning math and related things. What are your experiences and advice concerning the following dilemmas?
Being limited to a rate of 4-5 courses per semester, I realize that I am certainly not going to be able to take all of the courses that I am interested in. I would like to get a build a broad and solid base of knowledge by studying all areas of math at least a bit, but this comes at a cost of being able to take the more advanced, deeper courses. My plan was to self-study measure theory/Banach spaces and topology this year so that I'll be able to immerse myself in the graduate-level courses, which I expect to be more challenging and interesting and rewarding. I was wondering if people had experiences/regrets/wisdom about whether or not this is a good idea? Do you think it's better to build up a broad foundation thoroughly or throw yourself beyond your comfort zone?
On a similar note, what is your advice concerning specialization versus developing a broad taste? In my very limited experience, I have enjoyed representation theory, algebraic number theory, and complex analysis a lot. But there are still so many areas that I've yet to sample: algebraic topology, differential geometry, more advanced real analysis, algebraic geometry, analytic number theory, combinatorics... What's a good balance between trying all the different fields of math and trying to quickly become an expert in one?
Stepping back a bit, let me pose this question for a broader context. I am rather interested in philosophy, psychology, computer science, physics, and economics in addition to mathematics. I would like to take courses in these subjects as well but I am worried that this will put me at a disadvantage should I choose to ultimately devote myself to math. To people who chose either path -- regrets? hindsight? And of course, to anybody -- opinions on this issue? 
 A: I think you should choose classes based on who is teaching them.  If you go on to become a grad student you're going to learn all this material eventually anyway, and as you know more math you can learn more math quickly.  So the reason to take one class instead of another class is because it has a teacher who you learn well from!
A: My best advice you any beginning student:
1) Learn as much as you can, as broadly as you can.  Take as many hard core mathematics courses-especially proof-oriented ones-as you can. "As you can" is the most important part of this statement here,because mathematics takes a lot of time and discipline to learn.As a result,there are very real time constraints on what you can learn at one time in one semester.You also want to sample many diverse fields where mathematics has a role,which is just about everything. You want to be as well-rounded and flexible as possible in your base knowledge because that's what ultimately becomes your foundation as a researcher. 
2) Know thyself and plan accordingly. It's a lot better to take only 3 hard core mathematics courses in one semester and get B's and above in them then to try and take 8 courses and barely squeak by. Maybe you're the type that can go days at a time without sleep running on caffiene and/or....other chemical assistance and fear without landing in a hospital and ace 9 courses.Even if you are of that priviledged few-it'll catch up with you sooner or later,trust me. I've buried a few friends that it caught up with just as they finished thier PHDs at Stanford or Oxford.It's better to do less and do it better for not only your grades,but your health. Quality not quantity. If you've got health problems and personal issues that will get in the way of a total commitment to your studies and undermine your performance-as was the case for me-you should consider taking a few semesters off to take care of that. Otherwise,you could be a very sorry student later. Trust me. 
3) Grades Matter. I know,you're probably like,duh? But this really hits on your question about the comfort zone-because throwing yourself outside it to prove yourself a badass is risking your GPA. Graduate level mathematics is serious buisness. Indeed,a number of such classes will be Moore method type courses where you're basically on your own and will have to prove basically all results yourself and a good portion you'll be graded on. And if you do poorly-it could yank your GPA down destructively. Getting into as prestigious a graduate school as you can manage determines how successful you'll be early in your career and whether or not you'll be banished to obscurity in some community college. My point is you don't want to risk that being a badass,it's not worth it.I would strongly advise you audit such courses at first.That way,you learn the high-level material and your performance doesn't affect your grades.Once you've gotten your feet wet-by all means,try and take ONE graduate course.Then depending on how it goes-take more.
4) Talk to People. Develop good bonds with willing professors and graduate students.This way,you learn about the field from many other people's perspectives as well as learning about thier experiences. It's also good to hang out with people who enjoy what you do! 
5) Start Reading Journals. This is probably something you're not really going to be able to do until your last year before graduate school simply because you don't have enough background. But there are a few journals that are written at a low enough level for undergraduates to understand-like the American Mathematical Monthly. But you definitely should try and get a feel for active,living mathematics,even if most of it goes over your head at first.Go to seminars as often as you can. 
Good luck!      
A: In my country, future mathematicians usually take only mathematics courses (with a bit of physics and computer science thrown in) from the beginning of their undergraduate studies. Some universities are tougher than others, and students end up learning advanced material pretty early. Some of those students burn out, some have great fun and learn a lot of stuff. On the other hand, a respectable part of working mathematicians comes a longer way, learning something else first (physics, engineering, economics,...).
If you decide to work hard, a good rule to avoid burnout is to check your choice at regular intervals: if you get bad marks and/or need large amounts of coffeine you're doing it wrong. Make sure you always have a social network around you.
I personally spent several years concentrated on mathematics. I had a group of fellow students and we discussed what we were learning in espresso bar patios, during aimless walks and long afternoon teas. I learned a lot and enjoyed it. You should find out what fits you.
A: In my opinion the undergraduate years are good for two things: 1) Developing your abilities to do mathematics with your bare hands and 2) exploring as many different topics (in or out of math) as possible. For the former, make sure you don't shortchange all the standard core courses in analysis, algebra, topology. I would recommend combinatorics and probability, too. In principle, you should be able to learn a lot of this on your own by doing all the problems in a book, but it is also rather important to get feedback from an instructor, both to check your work and to make sure you are presenting your work clearly. Beyond that, I would advise avoiding taking too many specialized graduate math courses. Instead, take all those non-math courses you are interested in. You are going to be totally immersed in math while you're a graduate student, so your undergraduate years are your last chance to explore non-math topics.
ADDED (and inspired by Willie's comment below): And, most importantly, don't worry about any of this too much. It's not as if you have only one chance to do it right. Even if you do it all wrong, you'll learn a lot from your mistakes. It can be argued that those of us who tried too hard to plan carefully and avoid mistakes missed something important.
A: General advice: set a goal, make a plan, follow the plan as far
as is reasonable/comfortable/non self-destructive, take notes
regarding your progress along the way, re-evaluate both your
progress and the sensibility of the goal along the way, take time off to
do the moral equivalent of smell the roses.  Repeat for as long
as feasible and for as long as you have joy of life.
I could talk about taking time to attend lectures of interest
in other fields, go to teas or gatherings in other departments,
and have you set yourself the task of allocating time between
some balance of in-depth focus on a few subjects with making 
a fleeting acquaintance of a wide
array of other subjects that may or may not inspire you to
change subjects or adopt a different view of these subjects.
I'll instead say that I should have attended more special lectures
in subjects outside of mathematics, so that I might have a better
appreciation for different perspectives.
I could talk about how spending a lot of time concerned about
abstractions and reading books can lead to a contemplative life,
possibly with opportunties to engage in discussions that stimulate
and excite the mind, and the thrill that occurs when you embrace
a new perspective and new vistas of intellect are revealed.
I could add the health issues that might arise from an excess
of such behaviour (asthma, less robust immunity, allergies,
less stamina from insufficient exercise, a poor body-mass index).
I will instead say that through trial and error, you will
adjust your life so that you find an appropriate balance of
physical, social, and mental activities.
Something that might be worth considering: volunteerism. 
If you can donate some of your time toward your church or
social group, you may find a way of sharing your math with them.
It may also help you decide what is important.  Then you can
set goals, make plans, follow the plans ...
Gerhard "Ask Me About System Design" Paseman, 2010.07.21
A: Why are you taking courses? Don't take courses if you don't need it. 
Some psychologists match children development to human history. We have a period in which we play with dirt, a period of wars and fights, periods of obscurantism, renaissance,... 
You can let yourself be guided by history (math history) to study math. The most important to study first are those topics appearing first in history, the classics. Take this in a broad sense, after learning about the problem of squaring the circle you can read the proofs of the irrationality of Pi right away without waiting a proportional time to the one human kind waited to know them. 
Don't be too eager about the "hard core" courses. Most of the time, what is hard core about them is an overwhelming number of definitions to learn. eg. Much more useful than an advanced algebra course, in which you learn the (should I least some) huuuge number of definitions that they will give you, is to solve the same number of high school problems in algebra. If you let the definitions come in some osmotic-historical-like way that will be enough and you get a better grasp of them than after a year of being drowned with a list of definitions and theorems that most of them are exercises (and most of them exercises simpler than the ones you would be solving if using the time in a different way). A key point is that what is important is not "what" but "how". It doesn't matter if you run of swim, what is important is to either run of swim a lot and with the finest technique, to keep the muscles trained. It is the same, with math. 
Courses serve as orientation and motivation. They tell you what is important (if it is being taught in the course it should be important then) in the area and motivates you to solve problems (they give you homework) but, there are alternatives to courses to find orientation and motivation. Reading courses. Many of your professor would be willing to point to some sections in some books and tell you some names of theorems and concepts that are important and from that you got all the orientation that a course can give you. 
Join two more friends take a book recommended (maybe by some professor), a book with lots of problems and sit down with a fork and a knife and eat it like a gourmet pizza, solve each and every single problem. 
A: I have a small advice: you need to be really good with the basics, that is, 
the first year of math courses should be crystal clear.
The best way to repeat and learn these courses is to teach, so I suggest
you to do some private tutoring or being a TA. 
Trying to explain concepts for other people will really enlighten you on the subject. 
This has helped me greatly, since all further courses really depends on the basic mathematics.
A good foundation is everything!
