### What I Have Learned About Mathematics in Modeling in Social Science

I have written four books on mathematics and

modeling, and have thought about it since maybe 1966. It seemed

worthwhile to write down what I have learned. The crucial point is that **mathematical
devices bring along ideas, and that something works mathematically
demands that we figure out what idea we have bought**. (E.g., Lagrange multipliers can be prices, or other such.)

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.]