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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 ?

Andrés E. Caicedo
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FNH
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    Why do you keep tagging those questions as [contest-math]? – Asaf Karagila Jan 26 '13 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 '13 at 23:22
  • I would say the study of maps on sets. –  Jan 26 '13 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 '13 at 23:22
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    Mathematics is whatever people mean by the word. Does that mean that your notion of mathematics may be different from mine? Yes. In this respect mathematics is no different from, say, science fiction. – Brian M. Scott Jan 26 '13 at 23:23
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    I think the question should remain open. The answer could simply be that it is a matter of opinion. The fact is that the OP is interested in knowing about what mathematics really is, and I think a good "answer" can be given. The answer might not tell the OP what mathematics really is, but tell him about how people approach the question. – Thomas Jan 26 '13 at 23:24
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    @Thomas, what is the advantage to have a question having this kind of answer? – Sigur Jan 26 '13 at 23:25
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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 '13 at 23:26
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    @Sigur: the advantage is that apparently people don't know (like the OP) that the question itself can't be answered. The OP doesn't know that this might just be a bout opinion. So why not just write that as an answer? – Thomas Jan 26 '13 at 23:27
  • @AsafKaragila , sorry , it was a msitake maybe ! i will not do this again ! – FNH Jan 26 '13 at 23:30
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    Yeah, but questions with no definitive answer are by definition off topic here. – Tim Seguine Jan 26 '13 at 23:33
  • @Tim: There is no well-defined answer is pretty definitive! – Brian M. Scott Jan 26 '13 at 23:33
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    @BrianM.Scott that's almost language lawyering – Tim Seguine Jan 26 '13 at 23:34
  • @Tim: No, it isn’t. It’s the factually correct answer to the question. – Brian M. Scott Jan 26 '13 at 23:35
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    @BrianM.Scott agreed, but is it on topic? I am not convinced. – Tim Seguine Jan 26 '13 at 23:36
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    @Brian: If the answer is about the definition of the term, this should be asked in a Linguistics site. This is not a linguistics site. – Asaf Karagila Jan 26 '13 at 23:38
  • @Tim: The question is about mathematics and has an unambiguous answer. It is even a short answer, as may be seen from @T97778’s posted answer. I thought that obvious enough to concentrate in mine on how the term is quasi-defined in practice. – Brian M. Scott Jan 26 '13 at 23:40
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    @Asaf: It is primarily a mathematical question. – Brian M. Scott Jan 26 '13 at 23:41
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    @Brian: To quote: "it is hard to conceive of one that would be fit or useful or interesting or worth making or agreeable". This means exactly that it is definitively off topic according to the FAQ – Tim Seguine Jan 26 '13 at 23:41
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    @Brian: Is it? I don't see how. – Asaf Karagila Jan 26 '13 at 23:45
  • @Tim: You do persist in missing the point. That answer is a useful answer to the question asked. It might even save someone else from asking the question later. I fail to understand the desire that some people seem to have to close harmless questions. – Brian M. Scott Jan 26 '13 at 23:46
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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 '13 at 23:47
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    @Asaf: It can’t be answered without a fairly broad knowledge of mathematics $-$ far broader than the small element of linguistic knowledge required, which is merely that the meaning of words other than technical terms of art is determined by usage. – Brian M. Scott Jan 26 '13 at 23:47
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    @Brian: No, I think you are missing the point. Off topic questions are not useful to a site since they decrease the signal to noise ratio. – Tim Seguine Jan 26 '13 at 23:48
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    @Tim: It’s on topic. I’d not be at all surprised to find that something similar had even been asked here before, and I’d be utterly unsurprised if it were asked in the future. Having answers already available is therefore an advantage. – Brian M. Scott Jan 26 '13 at 23:49
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    @amWhy nobody is deriding the asker? who did? – Tim Seguine Jan 26 '13 at 23:49
  • @Tim - how can something be "off-topic" if there's no definitive notion, or broad fuzzy domain, which we call mathematics? – amWhy Jan 26 '13 at 23:50
  • @Tim: "This kind of question has no answer and everyone who works with math know this." – amWhy Jan 26 '13 at 23:51
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    @Brian: I think it's time to take it to meta. – Asaf Karagila Jan 26 '13 at 23:51
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    @amWhy personally didn't seem like derision to me; but okay if you think so. – Tim Seguine Jan 26 '13 at 23:51
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    @amWhy not-definiable=off-topic; If you can't make a notion precise, then it is not mathematical. If it is not mathematical then it doesn't belong on this site. – Tim Seguine Jan 26 '13 at 23:57
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    If I wrote that as an answer, it would have been at least as good as any of those which were already posted – Tim Seguine Jan 26 '13 at 23:58
  • @Tim then mathematics is off-topic? If we can't make the notion of mathematics precise, then mathematics it's not mathematics? Just using your definition... – amWhy Jan 26 '13 at 23:59
  • @Tim: No, it would not: unlike the three posted answers, it would not have answered the question. – Brian M. Scott Jan 26 '13 at 23:59
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    @amWhy, If you want to quote me, then at least do it properly. If we can't make the notion of "mathematics" precise, then "mathematics" is not mathematical. The quotation marks indicate that I am referring to the word, not the concept. As Asaf pointed out, this question is linguistic not mathematical – Tim Seguine Jan 27 '13 at 00:02
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    @Tim: By the way, have you checked the description of the (soft-question) tag? This is probably the archetypal example of an on-topic soft question. – Brian M. Scott Jan 27 '13 at 00:13
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    @Brian I'm out. This discussion is pointless. I concede to make this end. Or at least to make peace – Tim Seguine Jan 27 '13 at 00:21
  • I answered essentially the same question at http://math.stackexchange.com/questions/424926/ . – Andreas Blass Sep 05 '13 at 11:55

9 Answers9

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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.

Community
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amWhy
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    I think, this question should have been asked from Hilbert. :D +1 – Mikasa Jan 27 '13 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 – Mikasa Jan 27 '13 at 15:13
  • @amWhy , reading your answer again after 7 months is a great deal :D – FNH Sep 13 '13 at 19:54
  • You are Welcome @amWhy! it's a long time since I read your awesome answers :( . – FNH Sep 13 '13 at 20:04
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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.

Brian M. Scott
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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 '13 at 23:53
  • upvote from me + 1 for taking the question seriously!! – amWhy Jan 27 '13 at 00:01
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    @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 '13 at 00:03
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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 '13 at 00:03
  • @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 '13 at 00:06
  • @amWhy: I put a smile there to indicate I am trying to be humorous and diffuse the tension built up in the comments to the main question. I see it didn't work... there's always next time. Except when there's a maximal element, of course. – Asaf Karagila Jan 27 '13 at 00:08
  • @Asaf - I should have put a smile in mine too. I took your comment to be a "smiley" comment - and I know you well enough that your comment DID make me smile. So no offense taken, and no offense intended :-> – amWhy Jan 27 '13 at 00:09
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    Is that how you would explain the meaning of science fiction to a little kid? – Michael Greinecker Jan 28 '13 at 11:17
  • @Michael: I’d explain the term by giving examples and by paraphrasing what I wrote above, possibly at greater length and in simpler terms. In the present context examples are presumably unnecessary: the OP already has a variety of examples of mathematics. – Brian M. Scott Jan 28 '13 at 19:14
  • @BrianM.Scott I agree, of course. A similar answer was for some reason deleted. I added a note to it to indicate that it is not a comment or anything else. Please consider voting to undelete it. – Andrés E. Caicedo Sep 13 '13 at 18:54
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    @Andres: Done. (And I think that the edit really does improve it.) – Brian M. Scott Sep 13 '13 at 19:07
  • @BrianM.Scott Thanks! – Andrés E. Caicedo Sep 13 '13 at 19:11
  • "mathematics comprises those things that people call mathematics"... This seems either wrong if "people" means "anybody", or postponing the problem if "people" means "people who know what mathematics is". Re the "wrong" interpretation, note that many people call "mathematics" what is in fact numerology, although Newton might have been the last mathematician / numerologist. – Did Sep 13 '13 at 19:15
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    @Did: I am taking a linguist’s view. If Joe Bloggs refers to numerology as mathematics, then for him numerology is part of mathematics. If enough people come to share his usage, it becomes part of the language, irrespective of whether those who are generally agreed to be mathematicians accept it. I’ve not encountered this particular usage, but it is a matter of linguistic fact that in U.S. English, at least, one meaning — perhaps the most common meaning — of mathematics is basic computational arithmetic, and there’s not a thing that we mathematicians can do about it. – Brian M. Scott Sep 13 '13 at 19:33
  • What should make one suspicious about what you call the linguist's view is that it answers every definitional question in a single sweep. As is the case with many postmodern postures, this one is seemingly sophisticated but in fact insight-empty. – Did Sep 13 '13 at 20:38
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    @Did: On the contrary, it is a straightforward statement of fact about human language that has nothing to do with ‘postmodern postures’. It requires considerable expansion, of course; not all speakers are equally influential, for instance (though sociolinguistics has shown that the most influential speakers aren’t always the ones that you would expect). My simple statement is nevertheless the guts of it. And yes, it applies to technical terminology as well as to everyday usage: consider, for example, the history of topological senses of the word compact. – Brian M. Scott Sep 13 '13 at 20:53
  • Which would not prevent you from giving a precise definition of compactness, if asked for, and to have a quite definite idea of who are the most influential speakers in this case. So how come this all-encompassing relativism disappears when a concrete question must be dealt with? – Did Sep 13 '13 at 21:11
  • @Did: Nothing disappears; as I indicated in my last comment, the ‘all-encompassing relativism’ is considerably more nuanced than you apparently take it to be. – Brian M. Scott Sep 13 '13 at 21:23
  • A well known problem every advocate of relativism should face (but rarely do) is why limit the relativism to some problems and not apply it to all. In other words, I am quite certain you would never advocate "the linguist's view" if challenged about a specific definitional maths question (and you would be right!). – Did Sep 14 '13 at 07:17
  • @Did: I certainly do advocate it in that case, because it’s correct. In practice this means that in the vast majority of cases I would give the same answer to such questions as you would. That you think otherwise merely demonstrates that you don’t understand what I’ve been saying or what the view is. If you’re actually interested, go read some sociolinguistics; we’re getting too far off topic even for me, I don’t think that you’re making much of an effort, and I’m done with the topic here. – Brian M. Scott Sep 14 '13 at 07:54
  • You say the term mathematics defined by usage - do you mean as opposed to the concept it denotes? This makes me wonder to what you refer to in the second sentence in "the definition" - the one of the concept or the term/name of it? I guess the thing which is defined by usage of the term which denotes it is defined by that property, and then this definition doesn't change over time. Sorry if I was unnecessarily trying to analyze the statement. – Nikolaj-K Apr 23 '15 at 11:13
  • @Nikolaj: I’m saying that there really is no distinction between the two. – Brian M. Scott Apr 26 '15 at 22:40
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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 '13 at 07:25
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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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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.

user 85795
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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.

Will Jagy
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A rigorous way of solving problems.

user60862
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A set of formal rules that manipulate certain strings of symbols .

By adopting this view, I consider playing chess a mathematical activity.

Amr
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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).

Espen Nielsen
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