Is there fundamental goal of mathematics? I did not ask this question before scaring of down-voting but could not stop the curiosity and cannot find the answer by searching the web. In physics we are looking for say smallest mass or particle, many many years ago it was electron may be, sometime ago it became quarks and we are still looking for it.Is there anything equivalent in mathematics? I know there are many unsolved mathematical, we can search in the web and find many, but that does not answer the question: are we looking for something fundamental? Are we looking for something fundamental finding which will solve many present problems? 
Sorry if the question was not clear, but the answer provided by @jack and edited by@celtschk, gave me whatever I was looking for. As jack said the concept I was asking is not an active research now and that exactly I was looking for.
 A: In my opinion,
mathematics is an art, not a science.
So, this question is as relevant
as asking
"Is there a fundamental goal
of painting,
or literature,
or movies?"
There are many goals,
some personal,
some societal,
some financial,
some, some, some.
I know I love doing math
(the limited amount that I can do),
and hope to continue doing it
for quite a long time.
A: I don't think your characterization of "the" fundamental goals of physics is accurate. At any given time, there are unsolved problems in physics, and there are unsolved problems in mathematics, and these problems maybe catch the interest of a lot of people, or maybe a smaller number of people, or maybe just a few people.
I think physics and mathematics are actually quite similar: we explore, we try to discover new true things, we try to demonstrate their truth rigorously using appropriate methods (experimental method in physics; logical proof in mathematics).
If there is a difference in this regard between mathematics and physics, it is that the number of interesting and tractable problems in physics is rather more tightly constrained than in mathematics, because a physics problem must comport with physical reality. Mathematicians are much freer in that regard, although they still must comport with logical reality.
A: I think this is an excellent question, and I would hope no one would down vote it.
There was a time when mathematics had a similar goal. There were several issues found in the late 19th century which demonstrated that a naive approach to mathematics led to contradictions. For example, the set of all sets that do not contain themselves is a contradictory object. Does it contain itself? If so, it cannot contain itself. This is Russell's Paradox. This and other things showed that mathematics needed to be founded on a precise foundation that laid out exactly what each mathematical statement meant in full rigour. Such languages are called formal languages.
The effort to resolve these issues was known in the early 20th century as the Foundational Crisis of mathematics. It was hoped that a foundation could be found that covered all that ought to be covered in mathematics, and that was known to be consistent. In particular, David Hilbert published some problems he thought to be important for the development of mathematics into the 20th century, and the second of these problems was part of the ideal foundation: to prove that the axioms of arithmetic were consistent.
It turned out that the axioms of arithmetic cannot be proven consistent in first order logic without assuming the consistency of some other first order system of enough complexity to kind of make a consistency proof not mean much. This is Gödel's Incompleteness Theorem, and it is of course precise, but I just summarized it. Then Turing and Church showed that one cannot solve the question of whether or not an arbitrary computation ever reaches its goal, and this has deep implications for what can be done with foundations. After these discoveries, it was sort of de facto concluded in the mathematical community that an ideal foundation was impossible. Hence there is not much of an active search for one right now.
It should, however, be noted that Kurt Gödel himself recognized that his proof only ruled out one way of having such a foundation. Gödel never said that an ideal foundation for mathematics was impossible, and it seems that he may have even believed the contrary. Nonetheless, mainstream mathematics is not concerned with finding an ideal foundation at the present time.
A: Mathematics is the study of consequences. For instance, what is the consequence of Newton's gravitational inverse-square law? Using mathematics, one works out the theory of celestial mechanics, and more broadly, classical mechanics.
What are the consequences of Einsten's observation that the speed of light in a vacuum is independent of the reference frame? Mathematics shows us that special relativity must be true.
Ultimately, the place of mathematics in the rest of the world, is that we take the observations from other field of study (physics, chemistry, biology, medicine, etc..) and work out the logical consequences of those observations.
The Hawking/Penrose prediction of black holes as a consequence of Einstein's theory of general relativity is one of the great achievements of mathematics, for instance.
