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"What is Mathematics -- really?"
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Undergraduate Mathematics
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In theory what is it?
In a mathematics degree, one studies mathematical theories, proofs, and methods. If you concentrate on the theories and proofs, then you are doing a 'pure' math degree. If you concentrate on the methods, then you are doing an 'applied' math degree. For example, if you study numerical methods for optimizing shipping networks, then you're studying applied mathematics. However, if you study the theories and proofs of numerical optimization and try to create new methods, then you are studying 'pure' math. In reality, mathematicians argue about what, if any difference exist between pure math and applied math. (Read "A Mathematician's Apology" by G.H. Hardy.) Personally, I think some math theories have practical applications and some don't, so you could distinguish pure from applied by that criteria. Beyond that, there's no point in arguing which is 'better' or 'worse'.
In a book called Mathematics : Its Content, Methods and Meaning (by A. D. Aleksandrov (Editor), A. N. Kolmogorov (Editor), The MIT Press, 1969), I learned where mathematics originated. The book said that math originates from the repetitive observations of our daily life. For example, we see that if we have four tables in a hall, each seating 10 people, then we will need 40 chairs. Hence, multiplication is born (4 x 10 = 40). All the basic mathematical ideas, called axioms, come from the abstraction of such observations to general rules. By thoroughly learning such foundations, one gains the ability to notice when something is awry in one's everyday life, and when something is correct. If you are interested in gaining this ability, then you will enjoy math. Math classes typically don't teach you where math comes from.
What is it used for?
Above all, mathematics makes you understand how logical thought processes behave; there variations, pitfalls, and strengths. You will learn to distinguish between what is logically true, and what is a logical fallacy. This is why mathematics is sometimes called the mother of all sciences. If you grasp the principles of mathematics and can relate them to other fields, then you can basically take a math degree anywhere you want, e.g. Law, Medicine, Business, Politics, etc. If you study the other sciences, you will undoubtedly learn these principles as well, but they will be less rigorous and will be very specific to the field. It may be harder to recognize that the idea you have learned applies in other areas. Whereas if you learn the idea from the mathematical perspective, then you know that the idea only has application in other areas and you will be able to apply them there.
What does the major actually entail -- work-wise
Taking math classes at the university level is challenging, rewarding, and fun, especially at the higher levels. Math professors are usually very interesting people, because they love what they do. Obviously, if you don't have a positive attitude towards the classes and material, like most people, then you will despise the classes. Therefore, if you are considering a math degree, one important thing you should do is look at the course requirements for the math degree. Then, go sit in on some of the classes and read the syllabi for the courses. This will tell you if the material in those classes seems interesting to you. This suggestion actually applies to any major you are interested in.
What kind of jobs do you get with it?
I consider a math degree as an infinitely adaptable foundation, which you can use towards any field that interests you. Most higher paying jobs require mathematical aptitude, but not necessarily a degree. In any case, a math degree is particularly suited for graduate studies in Business, Law, and Medicine. It will complement and reinforce your natural skills. The commonly held belief is that people with math degrees become math teachers and professors, actuaries (insurance industry), and computer programmers. In reality, many also work in consulting, management, and just about every other field. See the links below for more information.
What are the fellow students like (personalitywise) in it?
Math seems to attract a higher proportion of shy, introspective, quiet types than other majors. But, by no means is that a rule. Many students will be outgoing, and most have a good sense of humour. They will tend to be logical thinking types, but still flexible. Being in a field that's hundreds of thousands of years old creates a more relaxed atmosphere than other majors.
Common Misconceptions
A common misconception is the limited occupational scope of a math degree. It is a great foundation for graduate studies or entering the work force in any field of interest.
Author's Note
This is the personal opinion of the author. In the author's ideal world, everyone would have a math degree and then a graduate degree in the specialty of their choice. The author is a math and computer science major from the University of Michigan at Ann Arbor, at this time finishing a Master's degree in Urban Planning at MIT.
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