37

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.

isomorphismes
  • 4,542
picakhu
  • 4,906

17 Answers17

39

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.

Qiaochu Yuan
  • 419,620
  • 2
    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
  • 6
    Logic can/should be applied while discussing any topic. – Raphael Jan 16 '11 at 19:29
  • 7
    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: Exactly what I was thinking! – George Lowther Jan 16 '11 at 21:13
  • 2
    @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
  • 1
    @Noldorin; I would argue that many of the world's problems are due to a failure to apply maths outside the other sciences. If only public officials were required to understand elementary logic, game theory and statistics... – Tynam Jan 16 '11 at 23:47
  • @Raphael: You miss my point. I'm saying mathematics should be applied when discussing logic. It needn't be of course (the early Greek logicians did not), but these days it's almost essential. – Noldorin Jan 16 '11 at 23:51
  • 1
    @Tynam: True, though it's also a fault of (some) mathematicians to over-simplify subjects outside of their domain and try to make rigorous what are inherently very holistic and non-mathematical subjects. – Noldorin Jan 16 '11 at 23:52
  • 1
    @Noldorin.1: I fail to see how you can use logik properly without (explicitly or implicitly) using its formal properties. But, of course, even/especially mathematicians are often sloppy in favor of brevity/readability. @Noldorin.2: I think what Tynam meant is rather that officials should have some intuition about what a study really implies, what makes an argument valid, what the issues of multicriteria optimisation are, ..., since these are things they use every day without knowing. And thus, often failing miserably. – Raphael Jan 17 '11 at 11:27
  • 1
    Having an engineering background, I disagree that applying math outside physics is risky. Having said that, I still think this is the best answer given there! – picakhu Jan 19 '11 at 04:57
  • 2
    @picakhu: "fields where its use is well-established like physics" includes engineering. – Qiaochu Yuan Jan 19 '11 at 05:45
  • @Qiaochu Yuan Yyour characterisation of the causes of the financial crisis is misleading. People who didn't really understand the mathematics (traders) made decisions that loosely related to mathematics (is there a risk measure which has some maths in it which appears prima facie to back up what I'm trying to do anyway?) and ignored the parts of the model that didn't agree with what they wanted to believe (e.g., the copula being used assumed many correlations to be zero, which are not zero) -- then risk managers were steamrolled. – isomorphismes Jul 01 '11 at 07:37
  • @Lao: well, this is why I generally try not to speculate about things that aren't mathematics. Feel free to edit the statement into something more accurate. I think the point stands that mathematics was abused in some way. – Qiaochu Yuan Jul 01 '11 at 12:47
  • @Qiaochu Yuan I'm not sure how to edit the answer because I don't see how it answers the OP's question. OP asked for examples of applications of mathematics to X and this answer says something like "I think such applications are bad" ? I'm also not sure what the word "dangerous" means in this answer. As someone else pointed out, nuclear bombs are dangerous (the meaning is clear there). Picking apart this response further: what does "This is as far as philosophical, political, or sociological applications go" mean? – isomorphismes Jul 01 '11 at 16:50
  • @Lao: yes, I know it doesn't really answer the OP's question, but I thought it was worth expressing skepticism about a certain genre of possible answers. "Dangerous" means that people making decisions based on misapplying mathematical models might make terrible decisions even if the mathematics is sound. "Philosophical, political, or sociological" means "not literary," since that was another category described by the OP and I did not understand what it meant. – Qiaochu Yuan Jul 01 '11 at 17:01
  • @Qiaochu: Gotcha. I edited the answer, but I'm not sure how much of an improvement (if an improvement at all) the result is. – isomorphismes Jul 01 '11 at 17:13
  • 1
    I draw a different conclusion from the fact that mathematical models must inaccurately represent the world: Proceed with caution, but do proceed. Trygve Haavelmo described the flaws of non-mathematical economics: http://nobelprize.org/nobel_prizes/economics/laureates/1989/haavelmo-lecture.html Mathematical theories are more disprovable than non-mathematical ones. Furthermore, certain ideas (like a Jacobian) are impossible to think up in a natural language. So although mathematical models are imperfect, they're better than the alternative: leaving useful tools in the toolbox, unused. – isomorphismes Jul 05 '11 at 07:02
  • 1
    @Lao: well, you put words in my mouth with that last edit. By "dangerous" I did not mean "bound to be inaccurate": I meant what I said. It's not as if one shouldn't do dangerous things, but just that one should recognize the danger. – Qiaochu Yuan Jul 05 '11 at 14:03
  • @Qiaochu Sorry about that. I figured my best wouldn't be very good in this case. – isomorphismes Jul 07 '11 at 05:59
  • Nice answer, but setting the record straight, just like statistics, it is not the math that are flawed, but rather applied in cases where the underlying assumptions and conditions are not met in order for the results to follow – Nikos M. May 27 '14 at 21:25
  • "Mathematics cannot prove anything about the world: it can only prove things about models of the world" - sounds like "penicillin doesn't help people, it only kills microbes". Is there a way to prove something about the world, other than suggesting a model? And this example about traders is not convincing: the theories of market are not axiomatized yet. So they do not have logical fundament. As a corollary, there is nothing strange that "specialists" there abuse what they present as "logical reasoning" when making decisions. This is the level of alchemy in comparison with modern chemistry. – Sergei Akbarov May 28 '14 at 14:44
28

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).

Matt E
  • 123,735
  • 1
    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 01:45
25

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.

Harry Stern
  • 861
  • 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 03:17
  • 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 03:31
  • 1
    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 01 '11 at 09:27
  • I removed the things that I said which were irrelevant to the question. – Harry Stern Jul 01 '11 at 22:51
  • 2
    I don't think I'd consider IIA "extremely mild" or "reasonable"; more like "apparently reasonable". – Charles Jul 05 '11 at 14:26
20

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.
Community
  • 1
isomorphismes
  • 4,542
14

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").

isomorphismes
  • 4,542
Yuval Filmus
  • 57,157
  • 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
  • 4
    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
  • 1
    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 01 '11 at 17:49
  • Now that this reappeared at the top of the list I was hunting through to see if anyone had listed this. It was the first thing that popped into my head. – Matt Jul 02 '11 at 03:17
10

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

6

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

  • Could you please be more specific? – Yuval Filmus Jan 16 '11 at 19:25
  • 8
    Sadly, almost all philosophers get it wrong. – Raphael Jan 16 '11 at 19:30
  • 1
    @Raphael: And also sadly, has had little impact outside of the narrow area of mathematical logic itself. – Mitch Apr 08 '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 01 '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 01 '11 at 18:41
6

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.

isomorphismes
  • 4,542
Júlio César
  • 672
5

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.)

Matt E
  • 123,735
3

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.

Raphael
  • 2,720
2

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

Yuval Filmus
  • 57,157
2

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.

J126
  • 17,451
2

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.

Asaf Karagila
  • 393,674
1

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.

Tpofofn
  • 4,771
  • 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 '14 at 21:45
  • Indeed there are formulations of the 2nd law, with even more generality and without this subjectivity. – Nikos M. May 27 '14 at 21:46
1

(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.

Nikos M.
  • 2,158
0

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.

jimjim
  • 9,675
  • 7
    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 – jimjim 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 '14 at 21:33
  • Regarding moral truth, there is indeed what is called moral realism – Nikos M. May 27 '14 at 21:37
0

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?

GEdgar
  • 111,679