Non-Scientific questions solved by mathematics I have a general question about the applications of mathematics.
What are some applications of mathematics that are not scientific, perhaps maybe literary or philosophical, or political. 
I am basically asking for the scope of mathematics, and looking for concrete examples. 
 A: Gödel's theorem had a great impact in logic and, hence, in philosophy.
A: As a simple example I would cite the Seven Bridges of Königsberg problem solved by Euler. At that time, Euler was mailed by a politician of the city of Königsberg asking him to help in a problem of finding the "touristically" best walk through some bridges of the city. Euler, at first, argued that such a problem was not a mathematical problem. But the politician insisted writing back to Euler saying that no engineer that was consulted had a conclusive answer. So, after reviewing the problem, Euler realized that the problem was asking for a solution that had nothing to do with geometry (I mean, a solution that is not metric related) as everybody was expecting. At the end, Euler proofed topologically the existence of a solution and gave rise to a new branch of mathematics: topology.
A: Michael Harris's article An automorphic reading of Thomas Pynchon's Against the Day describes some applications of mathematics to literature.  (On the same page, the article Do android's prove theorems in their sleep? gives some applications of literature to mathematics.)
A: In my opinion, the application of mathematics outside of fields where its use is well-established like physics is dangerous. 
Mathematics cannot prove anything about the world: it can only prove things about models of the world.  Some people take their models too seriously. Sometimes, mathematics which approximate reality or some portion thereof—with caveats—are interpreted (by mistake, or willfully) as authoritative and correct descriptions of reality. For example, the Gaussian copula approximation of David X. Li was putatively abused by mortgage-bond traders in investment banks antecedent to the financial-market crash of 2008.
By literary applications,  do you mean applications to the study of literature, or to its creation?  And does the latter mean inspiration or something else?  Certainly art draws its inspiration from many sources.
A: Through computer science and finally software develeopment, almost all questions people post these days are solved "by" (rather "with the help of") mathematics. Take Google as a prominent example.
A: Breaking the Enigma code was an application of mathematics to warfare, which would probably not be regarded as a scientific application.  My understanding is that mathematics continues to play a large role in modern cryptanalysis (such as in the work of the NSA in the United States).
A: One example of the influence of mathematics on politics would be Arrow's Impossibility Theorem, which states something like "no voting system is perfectly fair".
Similarly, game theory has applications to economics and politics.
A: Mathematics expands one's imagination. As such, while it might not solve philosophical, literary, or political "problems", it can give one a different perspective on questions from those fields.
Here are some examples of applications of mathematics outside science. I blog about this topic, so several of the links will be to short essays I have written that directly address your question.


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*Christopher Alexander's essay A City Is Not A Tree relates graph theory, posets, lattice theory, and order theory to the design of cities.

*C. Alexander's book Notes on the Synthesis of Form more generally relates the mathematical structure of objects to design problems and architecture. Alexander's book was the basis for a movement in software programming (another kind of architecture).

*An application of Andrei Kolmogorov's axioms of probability has been to literary analysis. One can do probabilistic analysis of texts to bolster or attack theories about a text. I think Kolmogorov actually developed his axioms so that they would be applicable to text (Tolstoy was a big deal in Kolmogorov's time, as well as ours).

*Generalisations about groups of people ("men", "feminists", "vegetarians") are often better understood using a distribution or probability theory. (E.g., "women sprint slower than men" should be understood to mean "the average woman sprints slower than the average man", or perhaps a statement about the relative skew of the two populations, etc.)

*The post-structural "critique of binary opposition" meets an alternative within group theory. $SU(3)$ is an example of a "trichotomy" and finite group theory offers many other relational alternatives besides binary opposition.

*Some postmodern philosophers are interested in applying topology to cultural analysis.

*There is a branch of political science called spatial voting theory which models voting patterns of citizens and legislators using mathematics.

*As mentioned in another answer, Arrow's impossibility theorem bears on politics as well. Other applications of mathematics to politics include fair division problems and optimal gerrymandering.

*Economics and psychology are not really sciences and both have adapted mathematical modelling.

*As mentioned elsewhere, some 20th-century authors (Borges, Pynchon, and Neal Stephenson come to mind) incorporated mathematics into their fiction.

*Finance is not a science either and the geometric series is daily applied there to compute net present value.

*The geometric series / analysis of $\sum_0^{\infty} {1 \over 2^n}$ also resolves Zeno's Paradox.

*Ancient (e.g. Pythagorean) and modern (e.g. La Monte Young's) theories & practices of music have involved mathematics.

*Stephen Wolfram, John Rhodes, and Kenneth Krohn have all applied semigroup theory (finite state automata) to philosophical problems. The Church-Turing thesis has implications for philosophy as well.

*Epistemology and the problem of causality is addressed by statistics and probability, as well as graph theory.

*John Gottman famously applied dynamical systems theory to the question of love -- which is far from scientific!

*The philosophical problem of mutual causation ("which came first, the chicken or the egg?") is addressed by dynamical systems theory.

*Dynamical systems are also used in philosophy to address questions about the mind, the body, and the environment within which a mind/body finds itself.

*The field of computational linguistics has implications for literature and philosophy and is mathematically posed.

*The philosophy of language and philosophy of logic are both strongly influenced by mathematics (graph theory, category theory, boolean algebra, heyting algebra, ...)

*The religious and philosophical implications and meanings of "infinity" have been addressed in mathematics (calculus, cardinality of sets, transfinite arithmetic, the continuum hypothesis, ...).

*Quantum mechanics has many implications for philosophy and is quite mathematical (you may object that it's science, but there is a philosophical side which is less scientific).

*Nietzsche proposed (I think in The Eternal Return) that after an infinite amount of time, a complex system must repeat itself. Cantor disproved this with a dynamical system that evolves according to $(\exp i \cdot \omega t, \exp i \cdot 2 \omega t, \exp i \cdot {\omega \over \pi} t )$ (such a system, despite having only three parts, will never exactly repeat itself).

*Dmitri Tymoczko has written a book about how the harmonic value system of Western music (typical theory of music for Western Europe 17th-19th centuries) corresponds to an orbifold.

A: Lacan applied Topology to Psychoanalysis, to great acclaim (at least of his multitudinous followers).
A: A professor gave a talk in our seminar about using math to find a proto-language, a language that is an earlier ancestor of many modern languages.  Among other techniques they used the "killing fish" method to try to find correlations between two now distinct languages.  The method is described as such:  If you want to find the deepest part of the ocean you can place a single fish in the water and slowly drain the water.  As the water level lowers, the fist will seek out deeper and deeper places.  Eventually the fish will get stuck in a local minimum.
A: I believe that Tarski's truth undefinability theorem has uses in philosophy, as the mentioned incompleteness theorems, and that's besides cranks attributing it to proofs that god exists, that mathematics is just a bunch of crap etc etc.
I have a friend who's studying cognitive science (as well as a math degree) and he says that there's some good math modeling there, and we all saw the financial Nobel prizes that were given to mathematicians.
However, I have a quarrel with the idea of using mathematics outside the ideal realm of mathematical objects. This is because mathematics is inherently precise and perfect, you can define notions that capture exactly what you want them to capture, while our "real" world is inherently imperfect and imprecise - and we can never judge what's true in the real world, as Tarski's theorem tells us.
The above argument means that taking mathematics into the real world is to allow imperfect and imprecise definitions and "latitude" for things to change beyond our original meaning. This is not mathematics anymore, in my eyes anyway.
In the very first math class I had in the academia, the teacher came in and said "Mathematics is the science of deducing from certain assumptions." and three and a half years later, I only grow to understand deeper and deeper how true this is. In the real world, i.e. outside of mathematical idealism, you can't prove anything - only find evidence to supposedly support a claim, or disprove it. So the ability to assume things and deduce things with absolute certainty becomes problematic. Which is non-mathematical in my eyes anyway.
A: Some literary works are attributed to famous writers but doubtfully so. One way of checking whether they're genuine is by extracting and comparing statistical "fingerprints". The field is called "computational stylometry". One of the standard challenges is identifying which works are genuine Shakespeare and which should be attributed to Fletcher (as if it mattered who wrote the piece, rather than its "artistic worth").
A: Interestingly, I saw this video a couple of days back and I think this is relevant to the question though it may not directly answer your question.
"L'importance des mathématiques" by Timothy Gowers at The Millennium Meeting
A: Although from physics (not necessarily mathematics), the second law of thermodynamics is often applied to topics outside of its strict application domain (heat and work) to state that chaos and disorder generally increase.
A: (adding the comment as answer)
If it can be formulated in the symbolic language of mathematics, it becomes scientific parlance, hey even if sth is not formulated in symbolic language it can still be scientific.
A: These are some (intuitive?) ideas, by no means I advocate them as they lack any type of rigor/sound philosophical reasoning that might justify them, read with extreme suspicion/scepticism/disapproval/disbelief : (This is a non scientific drawing of analogies/hand waving post between math and non scientific things)
Philosophy : Set theory (when dealing with infinities), shows the limits of human knowledge.
e.g. even if the universe was infinite and we could use every single particle to store information, then the relationship between every object in the universe is in the power set and Just like the real line is beyond any description in a list/book/linear manner. 
Philosophy/Religion Gödel's theorem : Many religions use the selling point : "ultimate/absolute truth and complete set of rules to live by" , where as Gödel's theorem shows mush simpler set of rules are incomplete,inconsistent or both.
Philosophy/Religion : Game of life is a good example of determinism does not imply predictability, give very few simple rules,  there is no way to figure out what will happen by the rules alone.
A: APPORTIONMENT of seats in a legislature to districts.  I don't mean drawing the districts.  For example, given that the 50 states are what they are, and the House of Representatives shall have 535 members, how many representatives should each state be given?
