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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 ?

Asaf Karagila
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FNH
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    Number theory is central to cryptography, and mathematical logic is at the heart of theoretical computer science. – user7530 Jan 26 '13 at 22:35
  • Why the downvoting? – MJD Jan 26 '13 at 22:36
  • @user7530 , great ! ok , is there is any branch has no applications ? or ever mathematical branch has some applications ? – FNH Jan 26 '13 at 22:37
  • @MJD , I don't know ! but i noticed that my question is downvoted always ! – FNH Jan 26 '13 at 22:38
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    @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. – Git Gud Jan 26 '13 at 22:46
  • In A Mathematician's Apology, the two fields he mentioned were Number Theory and Relativity. (This was only a few years before Hiroshima/Nagasaki.) – André Nicolas Jan 26 '13 at 22:57
  • @AndréNicolas, it's not clear if you're drawing a directly relationship b/w Number Theory and Relativity to Atomic Weapons? – alancalvitti Jan 27 '13 at 06:24
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    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. – Elchanan Solomon Jan 27 '13 at 06:33
  • I am saying that Hardy (a committed pacifist) mentioned these two as subjects whose purity meant they could not be used for military purposes. – André Nicolas Jan 27 '13 at 06:36
  • Relativity is certainly related to atomic weapons. Number Theory, not so far as I know. Also, I don't think Hardy lived long enough to see any applications of Number Theory --- he was certainly gone long before the RSA public-key cryptography system came on the scene. – Gerry Myerson Jan 27 '13 at 06:52
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    This should be at least community wiki. – Michael Greinecker Jan 27 '13 at 10:23
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    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. – emory Jan 27 '13 at 12:07
  • @GerryMyerson, Re relativity & atomic weapons: "The analysis made by Cockcroft and Walton of the energy relations in a transmutation is of particular interest, because a verification was provided by this analysis for Einstein's law concerning the equivalence of mass and energy." (-- 1951 Nobel speech). Relativity as a mathematical model played no part in C&W's research. Nor is it mentioned in Bohr & Wheeler "Mechanism of nuclear fission" Phy Rev 1939. Where do you see the relation? – alancalvitti Jan 27 '13 at 15:25
  • I don't know about "Relativity as a mathematical model," but it says "a verification was provided by this analysis for Einstein's law concerning the equivalence of mass and energy," and that law, $E=mc^2$, was in the 1905 paper on special relativity, right? – Gerry Myerson Jan 27 '13 at 22:28
  • Yes of course, that's why I included that quote. But special relativity was not applied to the development of atomic energy (mass = energy doesn't tell you how to convert one to the other). Rather, the converse: the latter helped confirm relativity. – alancalvitti Jan 28 '13 at 02:55
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    If not, we really should try to invent one. – Will Jagy Jan 29 '13 at 01:10
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    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. – Camilo Arosemena-Serrato Jan 29 '13 at 20:56
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    @Camilo, but we need useful fields! In order to make rockets and yachts! And we need rockets and yachts so that we can... uh... –  Jan 30 '13 at 10:16
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    @IsaacSolomon Hilarious!!! but SO TRUE! – Rustyn May 03 '14 at 18:00
  • Mathematics describes the world. That said, you can never be a true mathematician if you never venture into the applied field. It would be like claiming to be a boxer without ever stepping into a ring. All greatest mathematicians in history proved their worth via application, from theory to practice, anything less than that makes you incomplete. – Digio Jan 03 '17 at 22:53

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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 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]).

Apurv
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Calvin Lin
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    Jim Simons applying algebraic geometry to finance, the list can go on.. – pyCthon Jan 27 '13 at 02:17
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    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 (-: – hippietrail Jan 27 '13 at 06:32
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    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. – Chan-Ho Suh Jan 27 '13 at 08:55
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    @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? – alancalvitti Jan 27 '13 at 15:34
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    @Chan-HoSuh Cardano called the square root of negative numbers as a completely useless object in his book "Ars Magna". – Calvin Lin Jan 27 '13 at 19:00
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    Cardano's remark about uselessness is more of a reflection of his uneasiness with the concept than a factual statement; after all, he did include them in the book and show how to manipulate them to arrive at solutions in particular cases of "casus irreducibilis". There is quite a readable account of the whole matter in Dunham's Journey Through Genius; no need to repeat the usual myths. – Chan-Ho Suh Jan 28 '13 at 02:37
  • @Chan-HoSuh Thanks, I'd look at that. I was always under the impression that complex numbers were thought to be pointless when first introduced. – Calvin Lin Jan 28 '13 at 04:57
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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.

Asaf Karagila
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    Some things will always be useless. –  Jan 27 '13 at 10:20
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    @T97778: "Always" is a very strong claim. Name one eternally useless thing. – Jesse Madnick Jan 27 '13 at 10:29
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    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. – Jyrki Lahtonen Jan 27 '13 at 10:33
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    @Jyrki: I'm sorry but I'm negotiating with a major corporate for an exclusive endorsement deal. :-P – Asaf Karagila Jan 27 '13 at 10:46
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    @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. – alancalvitti Jan 27 '13 at 15:03
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    @JyrkiLahtonen, in a dialogue between Turing and Wittgenstein, one of them pointed out that the game of chess is arbitrary but the study of chess is not. A similar dichotomy applies to math: axiom systems might not necessarily be useful even if the study of such systems is logically sound and publishable. Often- but I don't think always - definitions are motivated by applications either in other branches of math (eg, Categories, Matroids just to give 2 examples) or outside of math (going back to the Greeks),. – alancalvitti Jan 27 '13 at 17:37
  • @alan: I don't know about you, but I cannot predict the future. It is a well-known fact that things that we cannot predict may happen. For example in the 1950's some people thought that a home-computer would be a wall-sized monster with a printer for output rather than a monitor. How could they even fathom something like an iPhone? What about graphene which was recently discovered to be a very interesting material with a lot of potential. No one thought that the very same graphite in your pencils would prove so useful again. But still, here we are. [...] – Asaf Karagila Jan 28 '13 at 01:11
  • So what are the uses for amorphous sets, and Dedekind-finite sets? No idea. I don't even care if they will never have an actual practical use that you will deem worthy. I do know that they have a very important use, they keep me in mathematics, and they make sure that I interact with other mathematicians and spread ideas, and I help shaping ideas of others (even outside of set theory, shockingly enough). And my contributions will someday trickle down to the real world, sadly enough. If this contribution is what unlocks the cure for (say) Ebola, does that still make amorphous sets useless? – Asaf Karagila Jan 28 '13 at 01:14
  • That paper by Levi is from 1958. So these theories are mature. Nevermind the physical world - can you name one important pre-existing problem in some other area of math, eg a Clay or DARPA challenge caliber question that has been solved by way of these ideas? Or even a smaller problem? - As opposed to problems solved but which were introduced by the theory itself. – alancalvitti Jan 28 '13 at 01:39
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    @alan: If you keep pretending to know the future please let me know the lottery numbers for this week's draw. – Asaf Karagila Jan 28 '13 at 08:26
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    Who mentioned the future? I'm talking about whether there has been any application so far... "Part of the problem is that we're so concerned with our ideas being simple and beautiful that when we do have a pretty idea, we want so much to believe it... it's the mathematical version of 'rapture of the deep'" -- Lockhart Measurement 2012 – alancalvitti Jan 28 '13 at 14:59
  • I will now mathematically prove that there is a 100% chance that some math thing will always be useless. Imagine an imaginary line that divides currently useless math and currently useful math now imagine a sphere where the imaginary dividing line runs through it. Now using the Borsuk-Ulam Theorem we know that some useful piece of math will have an antipodal useless piece of math. :) However if the math ends up being the only useless thing it might make it useful by showing that a useless thing exists so it will naturally migrate to the useful side, but this can only happen once. – Neil Apr 06 '15 at 12:15
  • @Neil: You're making awfully lots of assumptions about the completeness of the underlying field. What if the line is in fact in $\Bbb Q^n$ or god forbid $\Bbb F_{p^n}^k$ for some large enough $p,n$ and $k$? :-) – Asaf Karagila Apr 06 '15 at 12:17
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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.

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."

Metin Y.
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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.

ex0du5
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  • Have you any sources to read more about that? Just want to give it a try! @ex0du5 – FNH Jun 04 '14 at 06:54
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    Sure. To get a taste for typical work, see http://www.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 http://books.google.com/books?hl=en&lr=&id=Oeh3OQBlHh0C&oi=fnd&pg=PA209 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! – ex0du5 Jun 04 '14 at 07:29
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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?

user55514
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  • You're making the assumption that all math topics will eventually have an application. I don't believe that to be the case. – Timothy Mar 08 '20 at 04:28