we all study mathematics , and all of us learn mathematical methods to solve problems , we learn how to prove , how to think mathematically
but the question is, what is mathematics ? how can we define it as a branch of science ?
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You might be interested in this post about classifications of mathematics, its vastness and branches, etc., and also the links available in the answers.
In particular, visit the Mathematical Atlas's website, and explore its links and its Maths Map
I would say the "space" of mathematics has no "limit points"! And math is definitely not a "subset" of science. Their intersection is certainly not empty, indeed, math, in one way or another, has non-empty intersections with just about any domain of study you can think of.
The term mathematics is defined by usage: mathematics comprises those things that people call mathematics. Thus, the definition changes over time, and even at any one time the term means different things to different people. In this it is no different from many other terms, e.g., science fiction.
To address one part of your question,
how can we define it as a branch of science ?
many would argue that mathematics is not a branch of science at all, although it does have a close relationship to the sciences. As Einstein said, "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." For further discussion, see Wikipedia and the essay "Is Mathematics a Science?" by Arturo Magidin.
Sam sat with his eyes closed for several minutes, then said softly:
"I have many names, and none of them matter." He opened his eyes slightly then, but he did not move his head. He looked upon nothing in particular.
"Names are not important," he said. "To speak is to name names, but to speak is not important. A thing happens once that has never happened before. Seeing it, a man looks on reality. He cannot tell others what he has seen. Others wish to know, however, so they question him saying, 'What is it like, this thing you have seen?' So he tries to tell them. Perhaps he has seen the very first fire in the world. He tells them, 'It is red, like a poppy, but through it dance other colors. It has no form, like water, flowing everywhere. It is warm, like the sun of summer, only warmer. It exists for a time on a piece of wood, and then the wood is gone, as though it were eaten, leaving behind that which is black and can be sifted like sand. When the wood is gone, it too is gone.' Therefore, the hearers must think reality is like a poppy, like water, like the sun, like that which eats and excretes. They think it is like to anything that they are told it is like by the man who has known it. But they have not looked upon fire. They cannot really know it. They can only know of it. But fire comes again into the world, many times. More men look upon fire. After a time, fire is as common as grass and clouds and the air they breathe. They see that, while it is like a poppy, it is not a poppy, while it is like water, it is not water, while it is like the sun, it is not the sun, and while it is like that which eats and passes wastes, it is not that which eats and passes wastes, but something different from each of these apart or all of these together. So they look upon this new thing and they make a new word to call it. They call it 'fire.'
"If they come upon one who still has not seen it and they speak to him of fire, he does not know what they mean. So they, in turn, fall back upon telling him what fire is like. As they do, they know from their own experience that what they are telling him is not the truth, but only a part of it. They know that this man will never know reality from their words, though all the words in the world are theirs to use. He must look upon the fire, smell of it, warm his hands by it, stare into its heart, or remain forever ignorant. Therefore, 'fire' does not matter, 'earth' and 'air' and 'water' do not matter. 'I' do not matter. No word matters. But man forgets reality and remembers words. The more words he remembers, the cleverer do his fellows esteem him. He looks upon the great transformations of the world, but he does not see them as they were seen when man looked upon reality for the first time. Their names come to his lips and he smiles as he tastes them, thinking he knows them in the naming. The thing that has never happened before is still happening. It is still a miracle. The great burning blossom squats, flowing, upon the limb of the world, excreting the ash of the world, and being none of these things I have named and at the same time all of them, and this is reality--the Nameless.
Mathematics is what mathematicians do and its relationship to science, as Galileo said is the language with which God has written the universe.
[Edit by A. Caicedo: This is by no means a silly answer. In fact, one could argue it is the only sensible "definition": Mathematics is what mathematicians do qua mathematicians. This view is put forward in several places, emphasizing its place as a social activity over other views that describe it in more abstract or more utilitarian terms. For a lengthy account of it, see What is mathematics, really?, by Reuben Hersh. (Of course, it is a view not all mathematicians or philosophers of mathematics agree on, but that's another story.)]
I now feel obligated to give the standard college dictionary definition of mathematics: The study of quantity, form, arrangement, and magnitude; especially, the methods and processes for disclosing, by rigorous concepts and self-consistent symbols, the properties and relations of quantities and magnitudes, whether in the abstract, pure mathematics, or in their practical connections, applied mathematics.
According to Wikipedia, mathematics is the study of quantity, structure, space and change, as abstract and circular as that sounds (what do we mean by these words anyway?) However, classifying "what is math" in more concrete terms may be impossible, since it is too far-reaching and depends on who you ask.
Mathematics is done by inferring new theorems from old knowledge using well-defined jungement rules. What these rules are, and why they are accepted, varies between mathematicians. The study of these issues is refered to as the philosophy of mathematics.
Insofar as mathematics is a branch of science, it is not, as it doesn't rely on observational evidence (though some may contest this, see philosophy of mathematics again).