Is there any branch of Mathematics which has no applications in any other field or in real world ?
for instance , maybe : number theory ? mathematical logic ?
is there something like this ?
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Lots of branches of mathematics currently have no application in any other field or the real world. As you get higher up the ivory tower, the object that you're studying becomes so esoteric, that there might not be relevance to other things.
However, that does not preclude the possibility that someone eventually finds a relevance for it. Before the 20th century, Number Theory was considered recreations 'useless' math. It has since spawned a huge industry of security.
Of course, someone might come along and say "Hey, there's this connection between (this esoteric field) and (that esoteric field)", like what Andrew Wiles did (Andrew Wiles proved Fermat's last theorem using many techniques from algebraic geometry and number theory [Source:Wikipedia]).
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That depends on the resolution you look at things. But essentially everything ends up with some application to something else.
It may be tempting to say things like "modern set theory is not useful to other branches of mathematics other than set theory". But this is not true at all.
Set theory is useful for model theory and general topology; and model theory is very useful for algebra, and general topology is useful in analysis; both algebra and analysis are useful in real world problem-solving.
So set theory ends up as being very useful. One can look closer and ask, "Why does research about infinite and bizarre sets whose existence is negated by the axiom of choice - a common assumption nowadays - is useful?" The answer, of course, is similar to the above, with an additional twist: even if we may not know right now what are the applications, in research new methodologies and ideas are developed, and those trickle and drizzle slowly from one field to another. Eventually things become useful.
For example number theory, which a century ago was considered without real world application, is now a key theory in cryptography which is a very important tool in the modern world.
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Mathematics, at least Mathematics that the human do(because I am also keeping in mind Plato's idea of mathematics), has its roots from nature as the initial point. We first observe nature and note down the facilities that we are able to recognize on a paper, and then generalize and abstract as much as possible. This is Mathematics that we do, which I am sure what you mean in your question. As a result of this, I believe anything in this system has a root somewhere deep or deeper inside nature. (I am not sure of your "application" usage)
Edit: Suggesting mathematical logic surely indicates that our "real world" definitions completely differ from each other.
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Being a matter that deals more with quality and less with experiment than any other; maths is an avant-garde needing a delay for it to be applied. If i did not believe in the necessity of ethics i would say it´s one of the most aesthetical disciplines; but also when you prove a theorem you open a new road ar at least a path and make the task of discovering new formal landscapes easier to successors; are truth and beauty and harmony and structures in the Universe immediately useful?
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