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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    $\begingroup$ Number theory is central to cryptography, and mathematical logic is at the heart of theoretical computer science. $\endgroup$
    – user7530
    Jan 26, 2013 at 22:35
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    $\begingroup$ @MathsLover It might interest you knowing about G.H. Hardy's book: A Mathematician's Apology. Throught the book he discusses Pure vs Applied maths. Hardy gloated that his field (number theory) had no application whatsoever and part of the reason he liked that was because having no application meant it couldn't be used for harm. Little did he know that he'd live to find out a harsh truth. $\endgroup$
    – Git Gud
    Jan 26, 2013 at 22:46
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    $\begingroup$ I sometimes like to think of mathematics as serving a social role in keeping so many potentially homeless people off the streets, and in front of blackboards, where they belong. $\endgroup$ Jan 27, 2013 at 6:33
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    $\begingroup$ There may well be a branch of mathematics that has no applications in the real world. Since getting tenure is an application in the real world, these hypothetical branches of mathematics are unlikely to be ever researched in depth. $\endgroup$
    – emory
    Jan 27, 2013 at 12:07
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    $\begingroup$ No offence to the person asking the question. A person who has really experienced maths to a deep level shouldn't care about applications, unless he has a need to feel useful, since we have come to the world to have fun and be happy, and mathematics is a wonderful way one can achieve this, and that's the only useful application I care. $\endgroup$ Jan 29, 2013 at 20:56

5 Answers 5


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 recreational, '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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    $\begingroup$ Jim Simons applying algebraic geometry to finance, the list can go on.. $\endgroup$
    – pyCthon
    Jan 27, 2013 at 2:17
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    $\begingroup$ This answer would be much better with a random sample of even two or three of the alluded-to "lots of" such currently not applied branches of mathematics... Or even one (-: $\endgroup$ Jan 27, 2013 at 6:32
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    $\begingroup$ The complex number example is not a good one. They were introduced to help solve cubic equations, so their relevance was immediately apparent and even a source of great controversy and mystery (cf Cardano, Bombelli, etc.). In the case of Wiles, the person to make that connection was Gerhard Frey. $\endgroup$ Jan 27, 2013 at 8:55
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    $\begingroup$ @pyCthon, Jim Simons has been very successful with Renaissance - but in an interview he also said he's been very lucky. He did not credit mathematical methods for the success. Since they are proprietary we may never know for sure, but wouldn't you think that if the cause of success was the application of algebraic geometry that other hedge funds would adopt the methods? $\endgroup$ Jan 27, 2013 at 15:34
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    $\begingroup$ @Chan-HoSuh Cardano called the square root of negative numbers as a completely useless object in his book "Ars Magna". $\endgroup$
    – Calvin Lin
    Jan 27, 2013 at 19:00

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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    $\begingroup$ Some things will always be useless. $\endgroup$
    – user59761
    Jan 27, 2013 at 10:20
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    $\begingroup$ @T97778: "Always" is a very strong claim. Name one eternally useless thing. $\endgroup$ Jan 27, 2013 at 10:29
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    $\begingroup$ I endorse Asaf's viewpoint here. No published math is created in a vacuum. It is always connected to something else. At the top of the pyramid the connections are to other branches/problems in math, but they are there. So it all depends on the resolution in the sense that the question is more about how many links do you require from a piece of math to an application outside math. $\endgroup$ Jan 27, 2013 at 10:33
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    $\begingroup$ @AsafKaragila, you make a sudden shift from talking about "bizarre sets" eventually being useful without giving an example, to giving the example of the eventual usefulness of number theory, which is not about bizarre sets but rather about nice things like divisor lattices and algebraic curves and Riemann zeta function - fractal and complex but at least we can all agree on its meaning and purpose. $\endgroup$ Jan 27, 2013 at 15:03
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    $\begingroup$ @alan: If you keep pretending to know the future please let me know the lottery numbers for this week's draw. $\endgroup$
    – Asaf Karagila
    Jan 28, 2013 at 8:26

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.

2nd Edit: I had better add this quote from Nikolai Ivanovich Lobachevsky to the entry:

"There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world."

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    $\begingroup$ and what is plato's idea of maths ? $\endgroup$
    – FNH
    Jan 26, 2013 at 23:08
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    $\begingroup$ As a completely independent world from the human mind to say in short. More: en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism $\endgroup$
    – Metin Y.
    Jan 26, 2013 at 23:15
  • $\begingroup$ Now I do believe that some human mathematicians are platonists, yes/no? $\endgroup$
    – Thomas
    Jan 26, 2013 at 23:17
  • $\begingroup$ That's obvious? $\endgroup$
    – Metin Y.
    Jan 26, 2013 at 23:20
  • $\begingroup$ It just sounds like you are saying that the math that humans is contrary to Plato's idea of mathematics. $\endgroup$
    – Thomas
    Jan 26, 2013 at 23:21

The field of nondefinable numbers (which studies a subset of the noncomputable numbers for which no human representation exists or can exist) has at it's topic an ontology for which existence may be asserted and really nothing else. This is about the most meaningless (in the sense of a semantic association) of seriously discussed mathematical topics, and I've heard that the poor souls who brave this field have loose morals and should not be trusted.

  • $\begingroup$ Have you any sources to read more about that? Just want to give it a try! @ex0du5 $\endgroup$
    – FNH
    Jun 4, 2014 at 6:54
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    $\begingroup$ Sure. To get a taste for typical work, see emis.ams.org/journals/LJM/vol16/bov.html where you will note he immediately puts "truths" in quotes to show his propensity to lie. But that's only PA. You get to classical expositions like books.google.com/… and I don't have to tell you what they imply by the "Slaman-Woodin" (sic) results on rigidity. You will notice most results have to do with moving around points you can't distinguish in the first place. These guys are street hustlers with degrees, playing shell games! $\endgroup$
    – ex0du5
    Jun 4, 2014 at 7:29

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?

  • $\begingroup$ You're making the assumption that all math topics will eventually have an application. I don't believe that to be the case. $\endgroup$
    – Timothy
    Mar 8, 2020 at 4:28

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