What I Have Learned About Mathematics in Modeling in Social Science
By the way, what I say here is for the most part well known nowadays, but not well practiced.
1. Patterns in space [agglomerations, central places] or in time [abrupt changes, stickyness to change] can be emergent effects of large numbers of otherwise undistinguished interactions, typically two body. Not always, and often they are a matter of device, planning, and conspiracy.
2. While there is a good deal to be said about unpredictability, I suspect that nonlinearity and the butterfly effect are less important than are deliberate actions by historically identifiable actors.
3. While economics is often presented in terms of optimization and smooth changes, those marginals, keep in mind that like #1 above it can be seen as a sum of discrete individual transactions plus conspiracy and control. Hence economics' marginality may be artifactual rather than a matter of derivatives of an objective function. Of course, one more transaction is effectively marginal, and the market can be a convergent effect of many such (ala Aumann).
4. Statistics that focuses on means/variance analysis, and most of what is taught and done, needs to catch up to modern data analysis techniques, with their concern with skewed and polluted distributions, graphical presentations of data clouds. Also, ala P. Levy, spend time on big- or fat-tailed distributions that are as well such that adding them up gives you the same sort of distribution, with some sort of scaling constant, much as in adding up gaussians and the square root of N.
5. Almost always someone will write a model and then test on data using some sort of regression. The data needs to be cleaned up, and double blind techniques (as used nowadays in particle physics) help you avoid playing with the data to get a result.
6. A number of ideas from finance, such as portfolio management and real options and random walk ideas as in Black-Scholes, should be of value to other parts of social science--either as ideas or as mathematical models.
7. Similarly, the time value of money ideas and discounting needs better modeling in trying to figure out present values and also the range of such values.
8. Almost no numbers in social science can be known to better than two significant figures (budgets, accounting, and demographics are different, some of the time). Hence when people quote much more, something is likely fishy. On the other hand those two significant figures need estimates of the range that is likely to be the case. I don't know the right probability range, but numbers always need something like error bars. If someone is using some sort of modeling program and out comes 8 figures (as in billions of dollars to the dollar), most of those are fake. Moreover if you do modeling, sensitivity analysis is essential.
9. Network analysis, currently popular in much of social science, should make use of the deep work of mathematicians on graphs and on queing. Keep in mind its foundations in telephone networks.
10. Notions such as catastrophe theory, fuzzy sets, nonlinear, neural nets, chaos, fractals, agent-based models, and cellular automata, only are useful if they are instantiated in formal models. And then you want to ask if the meaning of the modeling device (say chaos theory), and whether there are other accounts of what you see. If they just inspire reflection on what you are doing, almost always the historians and sociologists have written insightfully about these notions, totally outside of the mathematical realm and more about narratives supported by archival evidence, with rich cases without the mathematics and with no loss.
10a. Scaling is a pervasive phenomenon, but not always. That things look the same over a wide range of scales is deeply important, but over the shortest scales that breaks down. Over the longest scales, new phenomena arise. A deep insight of mathematics is often when you are doing counting up you end up with scaling-type phenomena (zeta functions and automorphic functions).
11. Lots of modeling is less about mathematics and more about institutions. Adam Smith on the pin factory, the fable of the bees/Mandeville, kinship in anthropology and rules in society... Think too of Coase's papers--no math, deep ideas, poignant examples.
12. Analogies need to be worked out as best they can be, rather than casually employed. Analogy is destiny only if it is quite rigorous.
13. Always begin with a toy model, with fewer variables and a smaller data set to find out if what you are trying to do makes sense and might even be fruitful.
14. Models should lead to an understanding of the mechanism in the actual situation you are studying.
15. It is often the case that some phenomena allow for several apparently distinct modeling procedures. There are two possibilities: the phenomena are insensitive to how they are modeled, the procedures are connected by deep mathematical facts. [In physics, the Ising lattice in two dimensions of simply interacting individuals can be modeled and solved by counting or combinatorial analysis, by symmetries, by scaling, by matrix symmetries and commutative matrices, and by procedures derived from quantum mechanics (this is a classical regime)--the Bethe Ansatz. All of these are connected mathematically, and the mathematicians' goal is to understand why.]