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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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Why do you keep tagging those questions as [contest-math]? – Asaf Karagila Jan 26 at 23:21
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Also, I voted against this question and voted to close it because there is no "canonical" definition, and this is really just a matter of opinion. Questions without a possible clear answer should not be asked on the site. – Asaf Karagila Jan 26 at 23:22
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This kind of question has no answer and everyone who works with math know this. – Sigur Jan 26 at 23:22
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@Sigur: What is the harm? In any case, There is no well-defined answer is a useful answer. – Brian M. Scott Jan 26 at 23:26
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People - this was asked in earnest. There is no need to deride the asker...or act arrogant: "everyone knows (math) knows that the question is absurd..." If we declare ourselves to be mathematicians, we should at least be able to say how so, and in what way, and what counts as math, at least insofar as what distinguishes the field. Else, one could say "what I do is undefined". – amWhy Jan 26 at 23:47
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5 Answers

up vote 11 down vote accepted

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.

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I think, this question should have been asked from Hilbert. :D +1 – Babak S. Jan 27 at 15:04
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Thanks. ;-) I see @Asaf voted it to be closed, so I want it to be closed either. When I were writing my last comment, I didn't aware that the OP's icon is Hilbert. I feel his soul alive somewhere in this question. :D – Babak S. Jan 27 at 15:13
@amWhy: Deserves nice answer! :-) +1 I had better watch it!!! – Amzoti May 1 at 3:08
up vote 7 down vote

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.

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+1: thanks for this answer. Math is in many respects closer to philosophy than it is to physics. Even though it gets many inputs from science (e.g., calculus was invented for the purpos of science) is not primarily driven to understand to "laws" of nature but more to understand how far the human brain can take us. – Fabian Jan 27 at 7:25
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+1 for reminding me missed Arturo Magidin. – Babak S. Jan 27 at 15:05
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There is no such definition in wide use, and it is hard to conceive of one that would be fit or useful or interesting or worth making or agreeable.

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up vote 5 down vote

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.

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Since the answer is in fact correct and useful (as are the others), I take the downvote to be a display of pique. – Brian M. Scott Jan 26 at 23:53
upvote from me + 1 for taking the question seriously!! – amWhy Jan 27 at 0:01
@amWhy: Is that all someone needs to do to get your votes? Neat! :-) (For what it's worth I haven't voted on any answer, nor I intend to.) – Asaf Karagila Jan 27 at 0:03
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@amWhy: I thought it pretty clearly a serious question, at worst a bit naïve. And I’ve seen it asked many, many times, so it’s clearly a natural question. – Brian M. Scott Jan 27 at 0:03
@Asaf - no, it takes more than taking a question seriously to get an upvote from me. But not taking a question seriously (and answering superficially, or sarcastically) is certainly grounds for me to seriously consider downvoting! – amWhy Jan 27 at 0:06
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up vote 2 down vote

A rigorous way of solving problems.

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