Non-Scientific questions solved by mathematics

Solution 1:

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.

Solution 2:

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).

Solution 3:

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.

Solution 4:

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.

• 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.