What is the goal of mathematics? It is the same as that of any other scholarly discipline: to attain understanding of its object of study. I fairly admit that my answer is trite, so let me expand it and along the way answer some closely related questions.
The principal myth that needs to be shot down is that mathematics is pursued because it is useful. That mathematics is substantially useful, we can all agree; but there is no complete answer to the vexing question of the uses and value of mathematics, let alone practical criteria to measure and evaluate them. Such criteria if available and actually applied and enforced, could only lead to a net loss by restricting the mathematical output so as to agree with an externally derived standard of value. This, with a flair for the dramatic, I can only construe as censorship. Less dramatically, it would lead to a fatal impoverishment of the total structure of mathematics. To take a simple example, quantum mechanics is hardly thinkable without the apparatus of Hilbert space theory; solid state physics is impossible without quantum mechanics which on its turn made possible the advent of the modern computer, a technological revolution pregnant with new and astonishing possibilities. But there is no clear, direct path between Hilbert space theory and computers. It is perfectly possible to imagine an alternative world with the former but without the latter. I will leave to those more capable than me the imagining of mathematics without Hilbert space theory, not because of any central position of the latter but because the edifice of mathematics is somewhat like a house of cards -- do not tamper too much lest it all falls down.
The utilitarian view drives a wedge between pure and applied mathematics, which is most unfortunate for at least two reasons. The first is the implied idea that the application of mathematics is somehow "impure" with all its emotionally charged overtones of vice and barbarism. The second and more substantial one is that there simply is no such thing as applied mathematics, there is only mathematics, period. What some call applied mathematics is more like a two stage process of developing mathematics at just about the point where it can be applied to solve some problem of another discipline. The two stages are necessarily intermingled, coexist within the same person (even in the same brain slice) and cannot be easily separated, but they are conceptually different, for one because they demand a different set of skills, not to mention a different type of knowledge and expertise.
At the root of this confusion is the failure to recognize the autonomous structure of mathematics. Mathematics arose as a comment on that common field of experience we usually call reality, but very early in its history it has shed off those shabby trappings. Its object of study is what we could call, lacking a better word, and with no implied adherence to any version of platonism (not that platonism is bad; it actually has a lot to recommend it), the mathematical universe. It has its own specific conceptual framework and it unfolds and develops according to its own internal laws. Its most far-reaching organizational principle is the axiomatic-deductive method; it was handed down to us by Euclid and there are no signs that it will be replaced by something else. Its standards of value are all its own and not borrowed ready-made from neighboring disciplines. What constitutes good mathematics may very well be bad economics, physics, etc. The standards are different and there is little point in confusing them, unless of course, confusion is the point.
The personal reasons why someone wishes to do mathematics are I guess, no different in kind from the reasons why someone wishes to do research in biology or history or chinese literature, and can equally vary from the pursuit of a precarious sublime to the more prosaic but painfully true fact that there simply may be nothing else that he or she can do with anything approaching a modest competence. Personally, I freely admit that I live in the proverbial Ivory Tower. In its defence, I would like to say that they make up splendid habitations and although the air is somewhat rarefied, it is a damn fine view from up here. But Society as a whole is impervious to these protestations; it really is irrelevant wether a mathematician is a selfish hedonist or The Great Benefactor of Mankind, wishing to fight hunger and ignorance and bring everlasting peace to the world. This vaunted nobility does not make him any the better.
In my experience, the uses and value of mathematics, like of any other scholarly pursuit, can be experienced in practice, but such questions do not have a direct answer, or whatever answers can be rehearsed, while no doubt eloquent and evidence of a touching faith, are only intelligible to those already converted. I am constantly reminded of my own despair at my utter inability to convey to my non-mathematician friends the excitement, the bewilderment, the sense of wide-eyed wonder; only the dearth, the dregs, the drudge. And the answers, if any there are, must always be qualified by the remembrance that nothing is obtained from nothing and the freedom to think about mathematics must be bought at the expense of the community. But freedom is not the same as value and to repeat myself, questions of value in mathematics qua mathematics cannot be decided by any promises of present or future applications, which are largely just the operation of the power of wishful thinking. Dr. Johnson defined a lexicographer as a "maker of dictionaries" and then added with more melancholy than humor "a harmless drudge". Dr. Johnson could very well have barked a similar comment in the direction of all scholars in general and mathematicians in particular, and in all probability with even more propriety as the subject itself can hardly provide a single topic of debate at polite dinner tables. On the other hand, I would like to point out that maintaining a mathematician costs Society very little, especially in comparison with the allowance for colleagues of other professions that will go unnamed such as physics. The needs of a mathematician are very frugal consisting as they mainly do, in time to think, a chair to sit, a generous supply of pen and paper, maybe a small class of five to ten students say, like so many lab rats on which to test one's own understanding.
Since Antiquity, mathematics has its place secured in the course of studia liberalia or liberal studies. These were the studies undertaken not for the very laudable purpose of getting a job but because they were worthy of a free man. "Liberal" has the same root as the verb "to liberate," which can only mean that one of the purposes of education is to free us from the compulsions of habit and prejudice. Are these ideals dead to us now? I surely hope not.
Regards, G. Rodrigues