math and sociology

Last week, I opined that maybe sociology needs more formal modelling. Not surprisingly, this drew many heckles. But the orghead raised a good point, I didn’t provide much justification in that brief post. Consider this a follow up.

First, let’s be clear about what I mean. By “math” I mean models and proofs, not statistics. That’s an important distinction. Statistics is using math to test hypotheses (verbal or otherwise) with quantitative data. Math is used to express statistical ideas and prove things about them. However, math can also be used to express sociological ideas and derive ideas through logical proof. By “math in sociology,” I mean “writing down equations describing social processes (the models) and proving new things about the models.”

Now, let me offer a few arguments about why we might need more math in sociology.

1. Clarity: Math usually requires you to boil down your argument to a few variables and the express a relationship between them. You simply can’t do math unless you start with some assumptions and variables. That is a very useful and clarifying exercise.

2. Surprises: Sometimes, you derive a conclusion that is truly surprising and that you didn’t quite expect. My favorite example is the existence of the Nash equilibrium. In every finite game – ANY finite game – there’s an equilibrium. If you told me that before I took game theory 101, I would have been suspicious. But it’s true and I can even explain the proof.

3. Avoiding redundancy: You can do it with verbal theory, but I find it easier to spot old theory if it’s expressed in some sort of model. Is it really true that DiMaggio and Powell said everything about institutional theory in 1983? Well, that might be easier to figure out if they had laid out a few models. Maybe Brayden might like institutional theory if there was some sort of formal model.

4. Integration of results: One nice thing about math is that it allows you to spot the relationships between theories, which is often hard in verbal theories. For example, Teppo recently linked to a video about evolutionary theory and anthropology. Turn’s out there’s an argument – is there natural selection of groups? If so, how does that relate to natural selection of individuals and their genes? Verbally, seems tough to figure out. But if you write down the models, you learn that group selection entails individual selection and vice versa. They are “isomorphic.” Here is an example where math was used to make a simple, but powerful point about competing views on evolution.

5. Existence proofs: One very nice feature of formal reasnoning is that it logically allow to say things like “whenever you have X, you get Y.” That’s very hard in many verbal models. But knowing that phenomena Y is a logical consequence of X is pretty handy and math is really, really good at generating these claims.

6. Culture: Math oriented people bring a particular intellectual style, one that emphasizes tool building and puzzle solving. That’s not all of sociology by any means, but it can’t hurt to have more people with those skill sets in the profession.

Ok, let’s turn to qualifications and counter-arguments:

1. Are you saying that verbal theory can’t do these things? Not at all. I am making a division of labor argument. Math is good for some sociological activities, but not others. For example, models could help distill core ideas, or occasionally derive the cool existence result. But it’s useless for observing the real social world, which is ultimately what sociology all about.

2. Are you saying we saying we should become like economists? Dear Lord, no. As a group, economists have committed the scientific method fallacy. They assume that one really good tool for science accounts for all of science. They have essentially abolished field studies, history, ethnography, and other important tools. Sociology should not engage in petty debates that end up dumping our best work. Instead, we should create a social science that strives to combine important different types of research.

3. Don’t we already have enough math in sociology? No. Remember what I said above. I am not talking about statistics or network theory. I am talking about models and logical proofs. This isn’t taught in the core of ANY graduate program in sociology and mathematical sociology courses are rare. The major journals will have a few proofs per year, but these aren’t integrated into the core of sociological thinking.

Overall, it’s a matter of relative use. I think math is severely underutilized in sociology. It’s not a cure all, but there’s a huge growth potential, especially if it’s properly and reconciled with other types of sociological research.