mathematicians get paid to teach and bring in grant money.
but to be able to rest easy at night, they try to advance the knowledge of mathematics.
odd as it may seem, there are people who enjoy thinking, about the same puzzles that have intrigued scholars for thousands of years, such as what is the structure of the set of prime numbers?
or how many different geometric shapes can be formed by setting polynomials equal to zero, and how can one classify them?
the question of whether these problems add to the ability of the US govt to kill foreigners, or to enhance the profit margin of Wall street, are not of primary concern to them.
there are however extremely few people or even organizations with the foresight to fund this kind of free flowing intellectual activity, which only adds to the intrinsic worth of the society, and not to its practical inevntory of goods, and so again it is prudent to have a second source of income.
I had a lawyer ask me the other day what was the use of knowing that there are infinitely many prime numbers, what good did it do anyone to know that? If that isn't ironic. I suppose the idea was that it is more useful to society to be able to cheat someone out of his property through clever writing of a contract, than to open his mind to the beauty of geometry or number theory.
in the other direction, it does seem as if mathematicians are rather good at solving all kinds of problems in any direction they turn their attentiton. I.e. they are often successful in applied areas that interest them. my wife changed from math major to physician and is a really good physician, especially skilled at diagnosis of cases, occasionally saving lives by doing so.
My son changed from mathematics to working in the development and maintenance of software used over the internet to help companies accomplish whatever business goals they have, and he is extremely good at it. he is still a problem solver, but in a different area, as is my wife.
I still like to mull over and over the same old useless questions about the structure and behavior of differential equations, polynomials, cohomology groups, matrices, Riemann surfaces, ...
a mathematician is some one who is curious, and bright, and willing to work hard at understanding, really deeply understanding things, and at using that understanding to explain previously mysterious phenomena. These qualities seem to be of value in many areas.
So many people come to us for answers, like you for instance.
