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What is the impact of Mathematics in social science today?. That is to say, what are the mathematics that a social scientist is using and, from the point of view of a mathematician, what are the mathematics they should use more or start out using.

Let me give a particular example to start with: Social Inequality. Every day I can see governments trying to eradicate the social inequality in terms of wealth. Assuming wealth can be measure, Is there a theoretical result saying that it is possible (or impossible) to work out such a thing exactly (mathematically speaking)?

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Please read tea.mathoverflow.net/discussion/44/… about "soft questions" on mathoverflow. We're considering being more restrictive about them. –  Scott Morrison Nov 16 '09 at 6:39
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17 Answers

up vote 11 down vote accepted

Right now, the idea of actually applying the results of Social Choice theory (aka voting theory) is gaining steam. There are plenty of mathematical results that show that plurality voting method (what US has) is one of the least effective ways to measure the will of the electorate, and something like Borda count or Approval voting would result in all kinds of benefits for promoting meaningful democracy.

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Do you have any reference that support what you are saying? I strongly believe that what you are saying here is sort true. However, I have never seen or read anything serious about it. –  Csar Lozano Huerta Nov 14 '09 at 21:29
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There is Saari's "Chaotic elections! - A mathematician looks at voting", which a friend recommended to me (I never read it). I think it contains a proof sketch of "No voting system can be perfect". Here's the link: books.google.com/… –  Konrad Voelkel Nov 15 '09 at 0:45
    
Thanks Anna, the books is in my library! I'll take a look at it tomorrow :) –  Csar Lozano Huerta Nov 15 '09 at 7:58
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Here's a nice article on voting theory from the SIAM web site! siam.org/news/news.php?id=1443 –  Vectornaut Nov 15 '09 at 19:56
    
The Borda count is too easy to manipulate. The Condorcet methods are good, and range voting isn't bad (strategically, it degenerates into approval voting, or a hybrid of approval voting under incomplete information). –  Charles Jul 25 '10 at 18:50
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I'm not sure if he's asking the wrong audience. We could address a slightly different question, namely what are uses of mathematics in social science, or potential uses, that mathematicians would endorse.

One example that comes to mind is the site, FiveThirtyEight. For those who don't follow this as closely, 538 is the number of electors in a US presidential election. The main person at this site, Nate Silver, has hit 50 home runs in the subject of American political polling. (He previously did baseball statistics.) He did this by using first-year graduate statistics, which of course amounts to certain parts of 18th and 19th century analysis. More importantly than the specific formulas, Silver had a disciplined mindset. He also has his ideology (he's liberal), but he wasn't slave to his ideology. His reasoning is, here is a big set of measurements, here are the results of multivariate regression, here is the uncertainty in the results, here are possible systematic errors, here is how we could learn more from different measurements. In the last election, this proved superior to more typical polling analysis; the usual standard is undergraduate statistics at best.

I'm sure that the same could be done for many other social science topics. Of course Nate Silver's lesson is not new in economics, but in other social sciences more could probably be done.

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This is very interesting, and Nate Silver looks like an extraordinary guy and certainly a good statistcian. (See also en.wikipedia.org/wiki/Nate_Silver .) Of course, we need to be cautios about this story and possible interpretations. (For example, quite a few academic statisticians are involved in polling planning and analysis and they probably are aware of graduate level statistics.) Also for role models of using mathematics in social sciences, I would still prefer the standard academic avenues of papers in professional journals etc. –  Gil Kalai Nov 20 '09 at 9:26
    
It often happens that experts are asked to consult on a project that cannot make full use of their expertise. By acting as a consultant to himself rather than to a standard polling organization, Silver found something new. Part of the issue may also be that public efforts, such as publicly released polls, are often less sophisticated than private paid efforts. –  Greg Kuperberg Nov 20 '09 at 14:52
    
I asked a fellow statistician and this was the reply: at first glance it seems that Silver does not provide enough information on his methods, so that I cannot say if I see new ideas or not. Nearest neighbor analysis, and similar methods are old (these days "small area estimation" is another common title for borrowing information from "neighbors"), and it is not clear how he uses "demographic data". He seems to combine different polls, this is also a common procedure (meta analysis). –  Gil Kalai Feb 11 '10 at 14:06
    
(continued) One could just judge him by how much better are his predictions than others'. For this you would need to have not just his success rate, but that rates of others. Even getting success rates is not easy, he and others may be selective and not post now some of their past failures. In short, it is not easy to say if there is anything new here. Pollsters often do not publish their methods, so I can't blame him. –  Gil Kalai Feb 11 '10 at 14:07
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If anything, I would be slightly reassured if Silver does not have any "new" methods. This is a situation where good methods are more important than new methods. Are Silver's predictions better than those of others? I don't know. His site is vastly better than anything else conveniently available to the public. Indeed, he has changed the public question from success "rate" in the sense of who wins, to quantitative accuracy in the sense of predicting the share of votes. Moreover, since it is entirely a public service, he can't be selective about his past performance. –  Greg Kuperberg Feb 11 '10 at 16:22
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Here's a relevant book, written by a mathematics professor at Cambridge: Naive Decision Making: Mathematics Applied to the Social World

Cover of Naive Decision Making

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Tom Korner's book is excellent: I'd recommend it to any strong undergraduate. But, the title is (deliberately) misleading. It's really a tour of analysis, probability theory and game theory: a beautiful excuse to motivate lots of mathematics which an undergraduate sees, but perhaps doesn't apply to anything. This book is certainly not a textbook on how to apply maths to the Social Sciences. But you should still read it! –  Matthew Daws Nov 15 '09 at 9:27
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There's nothing wrong with this question per se, but you're asking the wrong audience. Mathematics is used all over the sciences, and mathematicians themselves have relatively little to do with most of its uses. You'd do better to ask this question of social scientists. I don't know if there's a "socialscienceoverflow.net".

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It is not the wrong audience because I wanted to know whether the mathematicians know what are the tools that the Social Scientists are using. On the other hand, what could a mathematician say to the a social scientist?. Which are the Mathematical facts that might impact SS. –  Csar Lozano Huerta Nov 14 '09 at 20:43
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OK. But I suspect that social scientists can answer the question "What is the impact of Mathematics in social science today?" better than mathematicians can. –  HJRW Nov 15 '09 at 2:39
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One should always be careful in (ab)using mathematics in the context of social science, lest we forget Lang's denunciation of S.P. Huntington's "political opinions masquerading as science."

There's a real and present danger in trying to quantify things that can't necessarily be quantified. Mathematical arguments are considered precise because they are grounded in formal logic, and when the softer sciences try to make use of a mathematical style of argumentation, they often do so with not the best of intentions. That is, they try to misappropriate the certainty of mathematics by camouflaging their premises as mathematical statements.

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I once heard someone claim that it's impossible to prove racism is bad because of Godel's incompleteness. –  Aaron Mazel-Gee Nov 15 '09 at 5:19
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I'm not really old enough to remember in detail but there was a time when catastrophe theory was seen as a great hope for applications of mathematics to the social sciences and popular articles abounded. People I speak to today don't seem to know much about it so I'll try to give a quick summary:

Some dynamical systems are played out on a manifold but are characterised by having "fast" variables and "slow" variables. Along some axes the dynamics play out too quickly to observe so that all we see is a projection of the dynamical system onto the slow axes. The projection of the manifold typically has a bunch of singularities and these manifest themselves as apparent sudden shifts in the dynamics - catastrophes. For smallish dimensions, Rene Thom classified these singularites finding a fairly small number up to diffeomorphism. The conclusion then is that sudden shifts in the dynamics of a variety of dynamical systems should emerge from this small number of catastrophes and have behaviour predictable from this classification without knowing the details of the system. This universality meant that people could try to tackle systems that were hitherto impossible or difficult to analyse: like prison riots and international relations.

Needless to say, catastrophe theory is no longer popular in the social sciences, but the mathematics is sound with a nice appearance of the ADE classification due to Arnold. It was the "chaos theory" of its time.

There may be some useful lessons to learn from this.

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Mathematics is of great use in many social sciences, as it is the sciences. There was a time when people argued that biology was too "complex" to use mathematics to get significant insights into the subject. I think people no longer think this. Unfortunately, many pieces of mathematics as they "stray" directly from mathematics itself into applications in other fields get "written off" by some people in the mathematics community. For example, Game Theory was pioneered by John Von Neumann and many mathematicians have made contributions to this beautiful subject. However, most of the research in game theory is now done by mathematically trained people (sometimes mathematicians) who teach in economics departments, political science departments, etc. rather than in mathematics departments.

There are many questions some of which arise in game theory but also in areas which have come to be called social choice theory, etc. which are very mathematical which deal with fairness questions of a variety of kinds. Examples include fair apportionment, fair division, fair games, bankruptcy, weighted votings, voting in general, etc. I have been trying to encourage mathematics departments to partner with interested parties to offer an interdisiplinary degree program which I call "fairness studies" which builds on what has been accomplished as of now and to train people to do more work in this area. So far, not a lot of interest.

Best,

Joe Malkevitch

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I am not surprised by this lack of interest. My department offers a service course "Mathematics in the Real World", consisting of different modules dealing with voting, elementary statistics, finance, etc. In a recent revamp, the module on apportionment has been dropped. The apparent reason: many of the faculty members teaching the course found it too hard for their own comprehension. The book that the course follows for the most part also has a chapter on fair division that was never even considered for inclusion. –  Victor Protsak Jan 5 at 19:50
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Social scientists have long looked at the connections between people and studied "social networks", e.g. the famous paper The Strength of Weak Ties. There is a huge push currently to infuse this area of social science with more mathematics, and there have been several recent articles in the Notices of the AMS about this "network science".

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In both psychology and economics, they use statistics as well as game theory. As a simple solution to the problem you mentioned, the government could confiscate all wealth and destroy it, leaving us socially equal.

Also, as Anna points our there's social choice theory where one should mention Arrow's theorem[http://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem].

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Agent-Based Models are also used in the social sciences, and the analysis and design of these is deeply mathematical.

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Algebraic topology is used in game and social choice theories. There was a question about topology and game theory here is the url for that question:

Are there any interesting connections between Game Theory and Algebraic Topology?

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Let me try to put my 2 cents. Statistics is, of course, very important in social science research (and all other sciences as well). As for mathematical models, they also have become more central in many social sciences. A nice paper about Nash as a role model for mathematical modeling in economics is here http://arielrubinstein.tau.ac.il/papers/47.pdf .

If you want to get an impression of how a mathematical model in social science may look, here is (just as an example) a recent paper on contracts for providers of medical treatements http://www.ratio.huji.ac.il/dp_files/dp516.pdf .

The use of mathematical models is spreading and it is also controversial for various reasons. In many applications of mathematical models in social science, the model can be seen as a rhetoric tool and not as a computational tool.

Regarding equality (and fairness that Joe mentioned,) this is certainly a central theme in several social sciences and other academic areas and there is a vast literature around it.

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Several great mathematicians did some work in economics. Among them are John von Neumann, Stephen Smale, John Milnor, Abraham Robinson, Saharon Shelah, and many others. Quite many mathematicians have looked at how to approach economics in new ways. So I don't think there is immediate need for mathematicians to come and "clean up the mess".

A nice survey of the mathematics commonly used in economics can be found in the talk Mathematical Methods of Economics by Joel Franklin. It is rather old, but still gives a feeling for the kind of mathematics "typically used" (things like optimization theory) in the larger parts of economics. Within the small area of mathematical economics proper, people use tools from algebraic geometry, lattice theory, model theory, measure theory, functional analysis, algebraic topology and many other mathematical areas.

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A negative example. Samuel Huntington's nomination to the National Academy of Sciences was opposed by Serge Lang, who claimed Huntington's work contined bogus mathematics.

http://en.wikipedia.org/wiki/Samuel_P._Huntington

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I will suppose you have an algebraic taste, and that you want to develop theory rather than analyze data.

A starting point is the observation that relationships in a social group can be modeled by a semigroup. Among the leaders of this area, developed around the 70's, you can find Harrison White. He completed a PhD in theoretical physics at MIT, then a PhD in sociology at Princeton. Currently, he is at Columbia. A serious trouble of this approach (from my point of view) is that relationships change over time, so we do not have a stable algebraic structure. There are ways around it... If you want to follow this lead, then I recommend you the book Algebraic Models for Social Networks by Philippa Pattison.

Another observation is that social sciences are full of relationships we don't measure with numbers. So, a categorical-functorial approach has been developed. In that sense Lorrain and White wrote an enjoyable article: http://www.tandfonline.com/doi/pdf/10.1080/0022250X.1971.9989788 I would like to know how to use some category theory in this context...

I must say many sociologists dislike this approach a lot, because they consider it reductionist. I am afraid research in this area may fall in no man's land i.e neither mathematicians nor sociologists read it. However, there are some groups trying to develop this field. In particular, UC Irvine has the Institute for Mathematical Behavioral Sciences: http://www.imbs.uci.edu/. I also recomend to look up for Santa fe institute.

On other hand, a big problem for social sciences is measurement. Suppose, you develop a social theory (for example [1]). To validate the model you need to test it with reality. This is a big issue in social sciences, because you cannot take people inside a box, isolated other variables, and measure a specific behavior. Plus, real life events have many variables to account for. However, these days Google, Facebook, NSA, etc, are collecting a lot of social generated data; and there is not good theory for this sort of data. The void is being filled by computer scientists; for example Carnegie Mellon has a PhD program at its Center for Computational Analysis of Social and Organizational Systems. I attended their main conference some time ago...It impressed me their focus on extract, analyze, and represent relational data rather than understand it (in the sense of constructing theory). That perspective can be very useful, but I dislike it. Hopefully, social generated data can be used one day for constructing theories. (Noam Chomsky argued for developing theory here: http://www.theatlantic.com/technology/archive/2012/11/noam-chomsky-on-where-artificial-intelligence-went-wrong/261637/).

[1] "A New Model of Wage Determination and Wage Inequality." Rationality and Society, 2009 by Guillermina Jasso

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A site like this is a place where real human beings interact with each other in pursuit of some more or less common objective, so what we do here can be viewed in the light of many different social sciences, including several branches of psychology and sociology.

Just by way of a recent application that comes to mind, see the blog of the Augmented Social Cognition research group @ PARC.

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Perhaps I slighted the feelings of some bot or serveral by mentioning only the role of "real human beings" in this enterprise. If so, I humbly e-pologize. –  Jon Awbrey Nov 16 '09 at 15:42
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There was a collaboration between the anthropologist Claude Lévi-Strauss and the Bourbarki's founder André Weil. The latter wrote a appendix to the first part of the former's book, "The Elementary Structures of Kinship".

In this appendix, named as "On the algebraic study of certain types of marriage laws (Murngin system)", André Weil shows how the Murgin's marriage laws can be interpreted algebraically with the tools of group theory. For more information, one can read this featured AMS article.

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