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.