I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless.

My idea is to amend my article with some theories that seemed useless when they are created but found use after some time.

I came with some ideas like the Turing machine but I think I'm not grasping the right examples.

Can someone point me some theories that seemed like the Lychrel numbers and then become 'useful'?

Edit: As some people pointed out that I've published this on MSE I present a code here to find some candidates as Lychrel numbers.

def reverseNum(n):
    st = str(n)
    return int("".join([st[i] for i in xrange(len(st)-1,-1,-1)]))

def isPalindrome (n):
    st = str(n)
    rev = str(reverseNum(st))
    return st==rev

def isLychrel (n, num_interations):
    p = n
    for i in xrange(num_interations):
            if isPalindrome(p):
                    return i
            p = p + reverseNum(p)
    return -1

for i in xrange(1000):
    p = isLychrel(i,100)
    if (p < 0):
            print i,p
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    What about math that was once useful but now useless? Like all of the tricks engineers had to use to multiply using slide rules... – Brian Rushton Dec 17 '12 at 19:12
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    This sort of appendix seems contrary to the nature of mathematics. The argument isn't countered by providing a list of other ideas that people might have said were useless. Instead, why not focus on the education aspects? According to the Wikipedia article, the search has led a few computer programmers into what is ostensibly number theory, and may have introduced many young people to a fundamental idea behind proofs - even if you haven't found a palindrome by $10^9, there might still be one. Sounds a lot like Skewes' number, also probably called useless. – Zack Wolske Dec 17 '12 at 19:13
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    I think that "usefulness" is probably not the correct measure for mathematics. Other properties, such as beautiful results, the occurrence of complicated structure, or the use of unexpected techniques are also good ways of judging math. – André Henriques Dec 17 '12 at 21:39
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    It is interesting to look for things that turned out to be much more useful than initially thought, but I think you ought to look for reasons that you know for studying Lychrel numbers instead of hoping that more will come in the future. It seems to me like the primary motivation is that this is a simple question that seems like it should be easy to answer, but apparently isn't, so by searching for the answer, we may come to understand the integers better. – Selim Dec 18 '12 at 7:05
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    I agree with Zack. If you don't see a way of arguing directly for the usefulness of Lychrel numbers research (arguing for the usefulness of other allegedly useless results in mathematics is really no argument at all for the case in question), then don't go into it. Focus on other types of payoff. – Todd Trimble Jan 15 '13 at 13:18

15 Answers 15

up vote 40 down vote accepted

Number theory, in particular investigations related to prime numbers, was famously considered as useless (cf Hardy), for practical matters. Now, since "everybody" needs some cryptography it is quite useful to know how to generate primes (eg, for an RSA key) and alike, sometimes involving prior 'useless' number theory results.

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    This answer being given, let me add that I am not convinced your idea regarding the article is a good one. – user9072 Dec 17 '12 at 18:20
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    Number theory may have been regarded as "useless" in this sense, but was it ever considered to be as useless as Lychrel numbers? – Franz Lemmermeyer Dec 17 '12 at 18:23
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    It is also true that in a perfect world --say, a world ruled by mathematicians, cryptography itself would be useless: why to hide things? – Pietro Majer Dec 17 '12 at 19:08
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    Pet peeve: "cf" stands for "conferre", which means "to compare"; you are using it as reference or a "see for example". Though an extremely common usage, it is incorrect. "cf" should be used for "compare with", and you don't want to compare the writings of Hardy with the statement that Number Theory was considered useless; rather, you want to use Hardy's writings as a reference to the assertion that Number Theory was considered useless... – Arturo Magidin Dec 17 '12 at 22:20
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    @Arturo Magidin: yes, the usage is a bit odd here; also but perhaps not only as I changed the phrase after the reference was typed. I might sometimes use it strangely or even in a wrong way, but I can assure you I do know the meaning. Incidentally, since we are discussing such matters, it seems to me it actually does not stand for (the infinitive) 'conferre.' :) – user9072 Dec 17 '12 at 23:54

The Radon transform, when introduced by Johann Radon in 1917, was useless, until Cormack and Hounsfield developed Tomography in the 60's (Nobel prize for medicine 1979).

The most famous example is conic sections. Conic sections were of great interest to Greek mathematicians, and their theory was highly developed in the 2-nd century BC.

However I don't know of any application until Kepler's discovery that the celestial bodies move on conic sections. Thus 18 centuries passed between math research and the first application!

EDIT. There is a conjecture discussed in the paper The Astronomical Origin of the Theory of Conic Sections by O. Neugebauer, Proc. Amer. Phil. Soc., Vol. 92, No. 3, jstor, reprint - doi: 10.1007/978-1-4612-5559-8_21

that conic sections appeared for the first time in the theory of sundials. But this is only a conjecture, and Apollonius does not mention sundials. Thanks to user Miles who brought this fact to my attention.

EDIT2. However most histories of Greek mathematics say that conic sections were invented/discovered by Menaechmus, as a tool for doubling the cube, which is of course useless.

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    This actually seems to be a non-example. Conic sections were apparently first studied by Menaechmus in the 4th centure BCE. We're not sure what his motivation was, but he definitely used them in his method of doubling the cube. Some speculate that this problem led him to discover conics; others suggest that he was prompted by the fact that the tip of a sundial traces a hyperbola on any given day (outside the Arctic circles, anyway). In any case, it looks like conic sections had applications as soon as people knew about them. – Selim Dec 18 '12 at 6:57
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    en.wikipedia.org/wiki/Menaechmus (cites Boyer's and Cooke's history of math texts) www-history.mcs.st-andrews.ac.uk/Biographies/Menaechmus.html www-history.mcs.st-and.ac.uk/HistTopics/Sundials.html W.W. Dolan: Early Sundials and the Discovery of the Conic Sections, Mathematics Magazine, 1972. 45(1): p. 8-12. – Selim Dec 18 '12 at 16:02
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    Sorry about the formatting there---I tried to make it nice . . . – Selim Dec 18 '12 at 16:08
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    Conic sections were apparently used by the Greeks (possibly Archimedes) in real life: en.wikipedia.org/wiki/Parabolic_reflector – YangMills Jan 16 '13 at 0:53
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    Doubling the cube is an absolutely useless problem? I guess you don't care whether the citizens of Delos are able to defeat the plague sent by Apollo! en.wikipedia.org/wiki/Doubling_the_cube#History – Gerry Myerson Aug 24 '17 at 22:40

Divergent series, anyone?

It was devil's work, just a curiosity, unorthodox idea for Euler and a strange concept for Abel, Ramanujan (Abel claiming that it can't and mustn't be used for serious calculations)... but today, we use it for "real" things.

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    Useless concept for Euler? Is my sarcasm detector broken? – Franz Lemmermeyer Dec 17 '12 at 18:21
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    Done. Interesting enough, our both answers are related to Hardy, champion of "mathematics without practical use" (or at least what he hoped to be without practical use). – Harun Šiljak Dec 17 '12 at 18:22
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    Previous comment goes for quid's (now deleted) comment. @Franz: strange goes for Euler, useless goes for Abel and Ramanujan, I should have phrased it better. – Harun Šiljak Dec 17 '12 at 18:23
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    @Harun: that's bettet. Even if the idea was very orthodox for Euler, who defended using divergent series at each and every opportunity. It wasn't orthodox for the Bernoullis, however. – Franz Lemmermeyer Dec 17 '12 at 18:29
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    The more I think about it, the more I think that divergent series are rather useful math that became useless than the other way round. Nevertheless I like your answer better than the question. – Franz Lemmermeyer Dec 17 '12 at 21:17

Fast Fourier transform: Originally developed by Gauss in early 19th century. Gauss thought it is unworthy of publication, because there were better computational techniques. It only appeared in his collected works after his death, where nobody noticed it. Rediscovered by Cooley and Tukey, and was instantly recognized as important. See e.g. http://www.math.ethz.ch/education/bachelor/seminars/fs2008/nas/woerner.pdf

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    FFT was inented by Gauss as a computational tool in an applied problem (in astronomy). So it was certainly useful (though perhaps too shallow for Gauss to pubish). – Alexandre Eremenko Aug 25 '17 at 0:08

Investigations on the independence of Euclid's 5th axiom: about 2000 years of fruitless research, until Bolyai and Lobachevski resolved the issue (cf e.g. http://en.wikipedia.org/wiki/Non-Euclidean_geometry) and Gauss also raised his hand, but still not useful for practical problems, until Einstein developed a non-euclidean explanation of the universe.

Negative numbers and complex numbers were regarded as absurd and useless by many mathematicians prior to $15^{th}$ century. For instance, Chuquet referred negative numbers as "absurd numbers." Michael Stifel has a chapter on negative numbers in his book "Arithmetica integra" titled "numeri absurdi". And so too were complex/imaginary numbers. Gerolamo Cardano in his book "Ars Magna" calls the square root of negative numbers as a completely useless object.

I guess the same attitude towards Quaternions and Octonions would have been prevalent, when they were initially discovered.

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    Quaternions were designed as a way of representing positions in space (like complex numbers do in the plane). They were superseded by vectors, only to find use recently for efficient computation in graphics. – vonbrand Feb 19 '14 at 11:18
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    Indeed, also Maxwell used the language of quaternions for electrodynamics. I think, that quaternions were deemed quite an essential topic until the more general vector calculus appeared. – Lennart Meier Apr 20 '16 at 12:15

The answer to this question depends on what one means by "useful". We have to distinguish between "useful" for other branches of mathematics or useful for Theoretical Physics or even Physics in general. Being useful in actually describing the laws of the Universe can be understood as being several orders of magnitude more important than just being useful in other areas of mathematics. For example, Generalized Complex Geometry has extensively been used in String Theory and Supergravity (for example in the classification of certain flux compactification backgrounds), and as a result Gualtieri's PhD. Thesis is close to have 900 citations in ten years. I think this would have never happened if Generalized Complex Geometry was only interesting in Mathematics. On the other hand, essentially every piece of "reasonable" Mathematics finds its place and application in Physics.

Dealing with Mathematics' applications to Physics, there are many examples of mathematical theories that in the beginning seemed useless to Physics and only years later were found to have remarkable applications in physics. To name a few:

  • Lie groups: I once heard that when Sophus Lie introduced Lie groups, he said that finally mathematicians had created something that would never be used by physicists. Modern Physics uses Lie groups at so many levels that I cannot even begin to describe it. Let me just say that the Standard Model that describes all the known the fundamental interactionx and particles is based on the Lie group $SU(3)\times SU(2)\times U(1)$.

  • Holonomy theory of Riemannian manifolds. To the best of my knowledge, when this theory was developed in the 50's it was completely unrelated to physics. However, in the 80's it found its realization in Physics through String Theory: the simplest compactification manifolds are six, seven, and eight-dimensional Riemannian manifolds of special holonomy. In addition, the scalar manifold associated to the non-linear sigma model of the effective action of certain String compactifications is again of special holonomy (tipically Kahler, Hyper-Kahler and Quaternionic-Kahler). This manifold encodes, in a way which is not completely understood, the local moduli space of the corresponding compactification.

  • Kodaira's classification of singularities in elliptic fibrations. This is really a shocking example, again from the 50's. In principle completely unrelated to physics, it found its realization again through String Theory, and in particular through F-theory, which requires an elliptically fibered Calabi-Yau manifold as a compactification space space. The singularities of the fibration crucially inform on the matter content of the theory.

  • Gerbes. A gerbe is a particularly abstract construction introduced by Jean Giraud in the 70's. Surprisingly, it has recently found its realization in Physics through again String Theory (who else): many Supergravity solutions, like the self-dual string, are in fact particular instances of gerbes.

By the way, there also various examples of the opposite: mathematical theories first found by physicists and then formalized by mathematicians.

Perhaps homology and the simplicial methods of algebraic topology (and algebraic geometry) which are just now finding applications in topological data analysis. I think algebraic geometry and topology have often been thought to be the pinnacle of math for math's sake, but their machinery is finding uses in very modern technology.

Also, the theory of graphs and its applications to networks and network and systems analysis.

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    People are also looking at applications of knot theory to protein folding. Knot theory belongs to algebraic and geometric (low-dimensional) topology. – Todd Trimble Aug 3 '16 at 19:37
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    I have to disagree with this. Algebraic topology has long had a history intertwined with physics. Simplicial homology tells you plenty about de Rham cohomology and to quote Tao "The integration on forms concept is of fundamental importance in differential topology, geometry, and physics, and also yields one of the most important examples of cohomology, namely de Rham cohomology, which (roughly speaking) measures precisely the extent to which the fundamental theorem of calculus fails in higher dimensions and on general manifolds." – Dan Piponi Aug 24 '17 at 15:38
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    Graph theory? Useless? It was developed to solve the bridges of Koenigsberg problem! :-) – Wojowu Aug 24 '17 at 16:05
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    I disagree with this evaluation of early topology: early topologists like Poincare and Bohl had very specific applications to differential equations and mechanics in mind. – Alexandre Eremenko Aug 25 '17 at 0:10

Binary numbers: discovered by Leibniz and of no use until the advent of computers.

The theory of finite fields, introduced by Galois (but as often, using former ideas) was considered useless and as a mathematical curiosity for almost 2 centuries, until they found a use in error-correcting codes for telecommunications.

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    I think I must have missed something here. Obviously finite fields were useful to mathematicians far earlier than the 1940's (class field theory, curves and varieties over finite fields, the Frobenius, etc.). Since when did this question become a forum on practical applications in the usual sense? – Todd Trimble Jan 4 '14 at 18:39
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    -1, which shows that negative numbers can be useful. – Joël Sep 12 '17 at 16:46
  • I can explain the -1. 1830-1940 is not "almost two centuries". Moreover, finite fields have had a central place in geometry, algebra, and number theory even before Galois invented them -- with Gauss for instance. – Joël Sep 12 '17 at 16:48

Real numbers: Kroneckers "God created the integers, everything else is man made" is a prototypical continuation of the way the Pythagoreans wished the world to be, namely that everything can be measured in integers or their ratios.

Fourier transforms were not useful in the 19th century, save as a tool for solving differential equations or for obtaining other theoretic results. Today of course they are ubiquitous.

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    I find this answer quite surprising. In particular a quotation of Jacobi comes to mind 'M. Fourier avait l'opinion que le but principal des mathématiques était l'utilité publique et l'explication des phénomènes naturels ; mais un philosophe comme lui aurait dû savoir que le but unique de la science, c’est l'honneur de l'esprit humain, et que sous ce titre, une question de nombres vaut autant qu'une question du système du monde.' – user9072 Jan 15 '13 at 16:31
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    The first part of which roughly translates to: "Mr. Fourier held the opinion that the principal goal of mathematics was the public good and the explication of natural phenomena;" Thus, and in general, I am unconvinced that it ever was mainstream opinion that this type of mathematics was 'useless'. – user9072 Jan 15 '13 at 16:35
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    "save as a tool for solving differential equations" - some might say this is was a pretty important and heavy use! – Yemon Choi Jan 15 '13 at 19:01

Non-Euclidean and especially Riemannian geometry, which became the language of General Relativity (and group theory).

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    This is similar to Weis's answer. – Ben McKay Sep 12 '17 at 16:17

My favorite example---admittedly kind of historical fiction, since I'm not an expert on the history---is Apollonius' On Conics. He proves all sorts of theorems that could be described as interesting only to those with the right mindset things of the form: if you draw this line and then this other line, and connect it to this other thing then it crosses through this other point that you might agree is an interesting point.

But two thousand years later, Newton pulled a great many of these things together in his demonstration that the force of gravity being proportional to the inverse square of distance results in orbits that are conic sections, thereby explaining Kepler's laws and giving really compelling evidence for his theory.

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    This is similar to Eremenko's answer. – Ben McKay Sep 12 '17 at 16:16

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