What's the goal of mathematics? Are we just trying to prove every theorem or find theories which lead to a lot of creativity or what?
I've already read G. H. Hardy Apology but I didn't get an answer from it.
 A: Besides Hardy's, another wonderful essay on this and related topics is "The Study of Mathematics" by Bertrand Russell, available online for example here. I recommend the whole essay, but the part that addresses your question most directly is maybe this:  "mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the actual world, but every possible world, must conform." The goal of mathematics, then, is to understand the most general laws of nature possible.
So 2+2=4 is more general than the law of gravity because in a possible world gravity could work backward, but 2+2 can't equal anything else. Of course there are logical difficulties with statements such as this, but the goal is to understand truths of the greatest possible generality.
In practice it tends to be easier to work with special cases and examples, and so that is what mathematicians do. But the goal is always to identify the general underlying principle which ties together the special cases.
A: There was a book I read at one point, can't remember which, that had the following parable in the introduction (paraphrasing from memory):
"There was once a group of people who wanted to get to a mountain far in the distance and climb it. Between them and the mountain, was a great desert they would have to cross. It was clear that the people had neither the knowledge to cross the desert nor to climb the mountain once they got there. So they began to study and work. At first, everyone worked on a bit of everything: how to carry water; how to avoid sand traps; how to climb safely; etc. Slowly, some people began devoting more of their time to studying the desert, and some to studying mountain climbing. And so the people studying the desert began to develop theories of sand dunes, wind, erosion, and so on; these theories were not really necessary in order to cross the desert, but the people studying the desert found them interesting nonetheless. Those studying mountain climbing were only interested in the desert-studies in so far as it allowed them to get closer to the mountain, and often complained that the desert-study people were wasting time considering theoretical questions that were unnecessary for the purposes of crossing the desert. The desert study people didn't care: they had fallen in love with the desert and were intent on studying it, regardless of whether it helped them all get to the mountain or not."
In the analogy, the mountain was the Theory of Everything from Physics; the Desert was mathematics. There was a time when mathematics' main concern was really to help study the world (help us cross the desert), but it has moved on from that.
This doesn't really answer your question. Perhaps the following won't either: when I was in my early second year in grad school, and looking for an advisor, Hendrik Lenstra told me: "Ask yourself why you do mathematics. If the answer is not 'Because I have to' then quit grad school now." Meaning: unless you would be doing mathematics in your free time, then don't try it as your profession.
A: My try:
Finding, understanding and characterising models that can (eventually) be used to 
describe and solve less abstract problems.
A: I've always viewed pure mathematics as somewhat like the R in R&D. We like to determine what is possible, not what we can develop into something practical (useful within mathematics can be a goal, but not useful in the real world). Someday, someone may come along and say "Oh, by the way, do you have ... ?", and we might be able to say "As a matter of fact, we do", but we won't be hurt if that day never comes.
I was a dancer for a while, I did it because "I had to". There was no earthly value to what I did, but the creative and physical challenge was tremendous (and irresistible). Picking up on something Arturo said in his response, when two of my children were looking at colleges to study musical theatre, a frequent question asked by parents at Q&A sessions was "Should my child also major in something to fall back on?", and the answer always was "If they feel the need for a safety net, they shouldn't be doing theatre." I think anyone in a creative discipline has to have the "need" to do it. We as mathematicians are no different. 
A: It's probably cheaper to let us do mathematics, than to pay for mental care.
A: As a kid I loved maths. Just because it was challenging my mind to work in a logical fashion.
There's no goal to maths. Mostly nobody who studies it deep enough expect to become a John Nash and get anything back in a life time. Mathematicians do it just for the heck of it. Nothing else.
A: I think the answer to this question depends on your philosophy of mathematics. Hilbert said that 

"Mathematics is a game played according to certain rules with meaningless marks on paper."

while Arnold said that

"Mathematics is the part of physics where experiments are cheap."

I think most pure mathematicians think like Hilbert does (this is called formalism), and just want to enjoy the creative process behind making these marks on paper. There is no "goal". Certainly, if we have a conjecture we've decided is particularly nice or fundamental, we might work extra hard to prove it, but it still is not an "external" goal of mathematics.
I'm sure that applied mathematicians take pleasure in attacking the (obviously quite difficult and important) problem of modeling reality with mathematics. Thus for them, math does have a goal, and consequently the aesthetic is quite different - as Hardy said,

"... most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts."

To a pure mathematician, the pleasure of mathematics lies entirely in a clever argument, or an insightful definition, etc. - there is nothing external by which we must judge "success" or "failure".
Personally, besides my love for mathematics itself, I actually take pleasure in the idea that there is no goal. If there's no "ultimate purpose" to mathematics, we don't have to fret about whether we're "helping further it" or whatever - we can all just have fun doing whatever we please.
A: Mathematics are a language, a form of thinking that emphasis absolute logic and reasoning, so in a way, mathematicians are explorers of the land of the truth.
The real purpose of mathematics is to see up to where logical thinking can bring us, what we can build with it, what we can simplify, what we can be absolutely sure of.
It's not just a tool you can express physical phenomenons behaviors, make programs runs or build better cars; maths are the expression of the fundamental part of what makes us thinking beings.
It's not just the plain and lame "1+1=2", there are a lot of deeper and more abstract reasoning you can come with in maths, the hard part is always trying to get a grasp of what mathematicians already did.
I don't think math have a goal in particular, a better question is "what goal can we serve by using maths ?", and the answer would be: "just invent a goal, and maths will eventually lead you to an answer.".
Remember physics problem can be solved without maths: most of the time, someone made some maths about something, and later someone found out it has some use in physics. Other times we have a problem in physics, and we still don't have any maths solution.
A: I've been retired for some time. I spent a good deal of effort looking for the meaning of things, and tried more endeavors than I would like to think of thinking doing this or that will resolve all issues. Well, for me anyway, nothing worked.
Then by a stroke of luck, I came across a set of beautiful lecture notes taught by Fields Medal winner Vaughan Jones. I hit the jackpot. Math seems to me to be the only query where there is an answer - indisputable. The joy of getting anything (rare though it might be for me) is undeniable.
Anyone can do it, anywhere, any time. There is a massive supportive community, welcoming to all participants (witness here). 
What better way to know you're alive.
A: This is a really old question, but one I ask myself all the time. (Warning: I'm going to stretch a metaphor so far that the Bard would be worried)
I once heard a description of Grothendieck which said the following: Many mathematicians saw the peak of the mountain and tried to climb it, coming up with the most ingenious ways to summit it, and yet all of them fell short. Grothendieck came along, saw the mountain and instead of trying to climb it he stood back and tried to build the airplane, and when he had succeeded he flew above the mountain from high above, seeing what all the other mathematicians wished, and all without ever trying to climb the mountain. Eventually his airplane revealed so much more about the geography than any mountain climber could have.
Why do people build airplanes? I would argue that planes come not out of seeing the heights of mountains, but out of an infatuation with air itself, with the notion of flight, with the freedom it provides. 
Often I think mathematicians as a whole do what Grothendiek did in this metaphor, they see the mountains which humans wish to summit, and they do not climb them (or at least pure mathematicians do not wish to climb them), they simply try to build better airplanes to fulfill their love of flight, and those same planes reveal more than anyone could ever predict. 
Mathematics can not reveal everything: Sometimes humans wish to explore under the earth, where we see the natural philosophical caves of the earth and sometimes go spelunking in them, and often times those mavericks among us write literature to dig deeper caves, and we must explore those so that one day we might understand the core of humanity and the world, and in this case airplanes and mountain climbers can only get us so far. 
A: Two amazing essays I highly recommend on this subject are Tao's What is good mathematics? and Thurston's On proof and progress in mathematics. 
A: Timothy Gowers has a talk on YouTube http://www.youtube.com/watch?v=BsIJN4YMZZo related to this question.
A: 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
A: The question you've asked is actually a philosophical question, so it requires a philosophical answer. If we look abstractly at what mathematicians do, they are providing "scientific explanations" of pieces of mathematics, so: 
The goal is to provide a satisfactory scientific explanation of mathematics. 
There is a large literature on Scientific Explanation in philosophy. There are three nice articles in the Stanford Encyclopedia of Philosophy to provide a start:
The first, on scientific explanation in mathematics in particular:
http://plato.stanford.edu/entries/mathematics-explanation/
The second, on the philosophy of mathematics generally, which gives a scholarly treatment of many of the concepts in prior posts on this question:
http://plato.stanford.edu/entries/philosophy-mathematics/
The third, on scientific explanation more generally:
http://plato.stanford.edu/entries/scientific-explanation/
A: This is not directly answering your question, but it might be interesting for those asking the question. One reason for doing mathematics are applications in natural science. I expected the article to answere more questions, but I want to recommend it, though:
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
A: Mathematics, indeed, cannot be translated; and, therefore, it is the goal of mathematicians to preserve the language of mathematics; for we would not be at the trouble to learn a new language if we could have all that is written in it just as well in a translation. But as the beauties of mathematics cannot be preserved in any language except that in which it was originally written, we must therefore learn mathematics as a language.
A: It is quite simple. Consider your question. What's the goal of mathematics?
Why Mathematics? This is a simple as the question can get. 
Its about questions. How? what? where? when? why? Think about some of the symbols used for those question words. (t)=when f(x)=what/where... And those are high school math symbols. 
Mathematics is just a tool to answer questions
Mathematics is probably the best tool at solving some problems. 
It does not really get more complicated then that. A tool and a system to solve problems and answer questions. As long as people ask questions, as long as people seek answers there will be maths. 
A: I used to hate math in high school and as an undergrad.  Now in grad school I've discovered that math can help me probe mysteries that until now were beyond my reach.  
Sure mathematicians might be more as Hilbert describes them.  But for scientists and engineers, math is another very powerful tool that we can adapt to our specific needs.  So if it weren't for those mathematicians (past and present), we would be poorer for the lack of an important tool.
