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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.

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This should really be CW. –  Asaf Karagila Jan 16 '11 at 19:04

17 Answers 17

up vote 38 down vote accepted

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.

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Good answer; I agree that application to areas where it's not naturally suited (e.g. other than physics, other sciences, engineering) is risky. There is of course logic... though many say that's a part of maths anyway. –  Noldorin Jan 16 '11 at 18:46
Logic can/should be applied while discussing any topic. –  Raphael Jan 16 '11 at 19:29
Dear Qiaochu, One could also mention that, among other things, the applications of mathematics to physics played a large role in the development of nuclear weapons, which could also be regarded as quite dangerous. Regards, –  Matt E Jan 16 '11 at 19:35
@Matt: right, but in that case the danger did not come from taking the models too seriously instead of looking at reality, but from the fact that the models were only too accurate... –  Qiaochu Yuan Jan 16 '11 at 21:22
@picakhu: "fields where its use is well-established like physics" includes engineering. –  Qiaochu Yuan Jan 19 '11 at 5:45

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).

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Some quite sophisticated maths is required for all modern cryptography and cryptanalysis, but whether that's an answer to the question depends on whether you think encrypting your web banking is a "non-scientific" problem. –  Tynam Jan 16 '11 at 23:49
@Tynam: Dear Tynam, I agree. Regards, –  Matt E Jan 17 '11 at 1:45

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.

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Arrow's Theorem actually does say that you can't have a voting system that everyone is "reasonably happy with" in all circumstances; essentially it sets some extremely mild conditions that we'd intuitively want a 'good' voting system to have, then proves you can't have all of them. (Of course, if the hypothetical users have utility functions that don't mind certain aberrant behaviours of the system, they can still be "happy" with this.) –  Tynam Jan 16 '11 at 23:44
your parenthetical comment was my point, yes. There will always be people who would rather be a dictator, or have a voting system that puts elections in their favor. For example, in the US, the two major parties have a common incentive for voting systems that discourage and hinder third parties which might split their own votes. –  Harry Stern Jan 17 '11 at 3:17
That is not to say that I don't think the theorem is extremely striking and interesting, of course. –  Harry Stern Jan 17 '11 at 3:31
Arrow's theorem doesn't use utility functions, just the notion of preferences. Arrow imposes a number of fairly reasonable axioms on the voting system and shows that dictatorship is the only system that satisfies them. –  johnny Jul 1 '11 at 9:27
I don't think I'd consider IIA "extremely mild" or "reasonable"; more like "apparently reasonable". –  Charles Jul 5 '11 at 14:26

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.
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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").

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That's not really mathematics, is it? It's statistics. So any conclusions that it draws are, naturally, disputed by everybody who isn't a computational stylometrist. (Which includes most of us here, I would think.) –  TonyK Jan 16 '11 at 19:36
Actually, strictly speaking it is machine learning. Call it by whatever name you wish, it is still, in large part, mathematics. –  Yuval Filmus Jan 16 '11 at 20:05
But machine learning isn't mathematics either! It's engineering. –  TonyK Jan 16 '11 at 20:11
So is cryptography (referring to Matt E's answer). –  Yuval Filmus Jan 16 '11 at 21:11
@TonyK The OP's question is about applications of mathematics, not examples of mathematics. –  isomorphismes Jul 1 '11 at 17:49

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

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+1 for the link. Thanks! –  jericson Jan 17 '11 at 0:21

Gödel's theorem had a great impact in logic and, hence, in philosophy.

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Could you please be more specific? –  Yuval Filmus Jan 16 '11 at 19:25
Sadly, almost all philosophers get it wrong. –  Raphael Jan 16 '11 at 19:30
@Raphael: And also sadly, has had little impact outside of the narrow area of mathematical logic itself. –  Mitch Apr 8 '11 at 20:56
@Mitch I wouldn't be so sure. Though misunderstood, Gödel's theorem is often cited in popular writing. It's often cited alongside relativity (and sometimes evolution) as an academic finding that transformed the way we must think about life-in-general (though this juxtaposition may belie non-logicians' misunderstanding of the theorem). –  isomorphismes Jul 1 '11 at 17:45
@Lao Tzu: Good point. What I meant was that Goedel's incompleteness theorems have had very little impact in mathematics outside of mathematical logic. –  Mitch Jul 1 '11 at 18:41

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.)

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See also his wonderful "A mathematical dream and its interpretation". –  Yuval Filmus Jan 16 '11 at 21:12

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.

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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.

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Wolfram Alpha wolframalpha.com is perhaps an even better example –  magma Feb 24 '12 at 17:04

Lacan applied Topology to Psychoanalysis, to great acclaim (at least of his multitudinous followers).

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Care to elaborate or give any references? What Google turns up is... interesting. –  Fernando Martin Jan 17 '11 at 0:31
I hope this answer is sarcastic, because the samples I've seen of Lacan's use of mathematics are… interesting. –  ShreevatsaR Jul 1 '11 at 8:39

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.

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The general method is known as "Grottochronology". –  Yuval Filmus Jan 16 '11 at 21:14
Are you talking about simulated annealing? –  timur Jan 18 '11 at 6:05

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.

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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.

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True, but the "disorder" (etc..) interpretation is being abandonded (disorder with respect to what and to whom?). It uses a very subjective view of what could be disorder (why a uniform gas is disorded according to you or according to the gas itself?), but indeed 2nd law, has great generality –  Nikos M. May 27 at 21:45
Indeed there are formulations of the 2nd law, with even more generality and without this subjectivity. –  Nikos M. May 27 at 21:46

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?

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(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.

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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.

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I disagree with everything except the third application. 1) The behavior of abstract sets tells us nothing about the universe. Linear algebra, on the other hand, tells us plenty about the universe (via quantum mechanics): this is because the universe behaves much more like a vector space than a set. 2) Moral truth and mathematical truth are unrelated. 4) I have no idea what this means. –  Qiaochu Yuan Jan 16 '11 at 21:26
I didn't mean to say set theory says anything about the universe but that there is limit to scientific knowledge. Needing to keep track of relationships between every object in a Newtonian universe ( not having even considered the Quantum one ), it is impossible to track everything. 2)But moral truth is even less definable than mathematical truth, when there is no hope for absolute mathematical truth, what chance is there that the complex moral truth can be achieved? 4)I'll clean it up, maybe I can get closer to what I am trying to say –  Arjang Jan 16 '11 at 22:18
@QiaochuYuan disagree as much as you like, unfortunately for you, what the answer states is indeed happening and for some people it is seen as a valid application. Whether this is so, is another matter. –  Nikos M. May 27 at 21:33
Regarding moral truth, there is indeed what is called moral realism –  Nikos M. May 27 at 21:37

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