## 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.

Ok. Convinced. Where can I sign up?

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AmitMarch 2, 2011 at 1:06 am

@Amit: There will be a follow up soon.

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fabiorojasMarch 2, 2011 at 1:09 am

Hats off. This is a great, simple explanation of the usefulness of formal models in social science. It will help many to unlearn the “I suck at math, therefore math sucks” crap most of us are fed in sociology schools.

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GuillermoMarch 2, 2011 at 2:20 am

I think a transition to a more mathematized sociology will be very different. Though I’m an economist, I’ve sat in on PhD sociology courses at Northwestern, for instance, and the level of mathematical knowledge among students is very, very low.

Consider even basic theory results in economics. The proof of Nash existence in finite games, as far as I know, requires the use of a fixed point theorem. Just saying that Nash has an equilibrium because the requirements of Kakutani are satisfied surely wouldn’t be convincing to a lay sociologist. But the math behind why Kakutani works, though only at the level of a first year graduate course in mathematics, is far from trivial. And note that NE existence is a 60 year old, one-page-long paper: the really nice modern mathematical results in economics are often quite a bit more complicated.

Fields like economics essentially weed out the mathematically unprepared before graduate programs even begin…but it’s tough to imagine any sociology program rejecting a promising student because of their lack of Real Analysis. If that is the case, then, good journals in sociology need to decide between printing papers using a methodology inaccessible to much of the profession, or maintaining the fairly-unmathematical status quo. How could sociology get around this problem?

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afinetheoremMarch 2, 2011 at 5:27 am

@afinetheorem: I agree it’s not easy, but it’s also not as hard as you make it out to be because there are multiple models of disciplinary development.

The economic model is one way. In that model, you eject all work that is not mathematical and select students based on math ability. Then you get a group of people who are essentially doing applied math.

But there are other models. Demography is a good example. It’s a field where almost everybody knows some models and the models are low tech. It’s a field where even qualitative demographers are expected to have basic fluency in things like life tables and Markov processes. Some specialists do high tech, but most don’t. Since the models are low tech, you can teach them, people accept them, and they can see how they are motivated by real examples.

So there’s a successful model of formalization that is not the econ model. Now the question is professional. That’s the real stickler, in my view. Sociology is a field that people flee to because they are bad at math or scared of math.

Despite that problem, there has been success on some fronts. All students must take multiple regression. A lot of students take network analysis. Descriptive? Sure, but it shows that if properly packaged and connected to real world concerns, a lot of people are willing to give it a shot.

Also, graduate committees at most soc programs look at quantitative scores. Northwestern is atypical in that it’s a bastion of qualitative culture studies (#1 according to US News rankings). Most top 20 programs have a small, but noticable, trickle of math, econ, and engineering folks (I was one). Many leading sociologists have physical science degrees, such as James Coleman (chemistry) and Harrison White (PhD in physics and sociology). The potential is there.

Ultimately, what you need is a cohort of scholars who are willing to fight the good fight. It starts with building a language for formalization and then fighting it out in journals. I’ve read old economics journals. It wasn’t all fixed point theorems in 1910. It took decades to build the field. The same with sociology. We’ve got cool ideas, now it’s time for a new generation to build the tools. And if they can show success, it’ll be respected and built into the system.

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fabiorojasMarch 2, 2011 at 5:45 am

[…] « Math and sociology » – […]

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Les liens du matin (85) « Rationalité LimitéeMarch 2, 2011 at 6:59 am

Don’t sell verbal theory short! For what it’s worth, I think that academic analytic philosophy is an example of a tradition of verbal theory that strives to meet many of these desiderata. I’m thinking specifically of (1), (3), (4), and especially (6) — the discipline has tried to model its research culture off the culture of mathematics (and the two cultures have also historically intersected).

Of course not all analytic philosophy that is published meets these criteria, and certainly most of it is not mathematically precise, though results in logic, decision theory, and formal epistemology are. The discipline as a whole does strive to meet these criteria, though.

I don’t know much about sociology theory, so I don’t know whether there is a body of verbal theory in sociology that is alive to these concerns. But I think it’s important to keep in mind that verbal theory can strive for conceptual clarity, logical precision, and argumentative rigor.

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SamMarch 2, 2011 at 7:15 am

“Mathematics” is a broad field. I am reading Von Neumann and Morgenstern’s

Theory of Games and Economic Behaviorand one of their points is that the traditional mathematics of calculus used in mechanics may not be appropriate for economics. In sociology, we prefer statistics. Boolean algebra might help. In a graduate course in criminology theory, I had read that crime is a Lacan toroid verging to a strange attractor. Such “fashionable nonsense” was long ago parodied by Tom Lehrer’s songSociology– “with one little matrix/they really can do great tricks/all in the name of sociology.” Humor helps. Mathematics is more than a tool: it is a toolset. The one tool that Sears Craftsman never warrantees is the screwdriver because so many of us use them as chisels.LikeLike

Michael E. MarottaMarch 2, 2011 at 10:02 am

Minor quibble in regard to #4, but one that proves your point about the usefulness of math.

Trivers doesn’t say that Group Selection is isomorphic to Individual Selection. Trivers said that group selection – the idea that evolution operates at a higher level than the individual or their genes- is essentially wrong.

When he does mention the word isomorphic, he is disputing theorists who do claim group selection is isomorphic to individual selection. See part 2, 7 minutes 30 seconds. “Just because two languages are isomorphic – or can be made isomorphic – does not mean that they are equally useful in thinking about things.”

He then uses Kin selection to explain genetic imprinting, something which he claims group selection would take all sorts of exceptions and premises.

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JamesMarch 2, 2011 at 10:10 am

If Maths is instilled into the Sociologists education from day one I don’t think it should be a serious problem.

From my experience, I think the problem will be the opposition to this within sociology itself. There are too many academics that are maths-phobic, and resort to arrogant, yet incoherent arguments against modelling in sociology.

What if sociology looked at other (sub-)disciplines such as behavioural economics and institutional economics for insipiration? Aren’t these doing a lot of exciting ‘mixed’ methods work right now?

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JonMarch 2, 2011 at 10:22 am

My worry is that sociology will become like economics even if we don’t want it to do so. What’s to say you can control your mathematical leviathan? Early economics couldn’t seem to do it. What makes us different?

And besides, I suck at math.

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Josh McCabeMarch 2, 2011 at 1:26 pm

“My worry is that sociology will become like economics even if we don’t want it to do so.”

It could happen. But other disciplines – such as biology – have a much more balanced approach to mathematical formalization, so I don’t see so great a danger in the horizon.

Economics, on the other hand, suffers from a severe case of physics envy.

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GuillermoMarch 2, 2011 at 1:54 pm

Well said, Fabio. My feeling is that a lot of resistance against the use of formal models could be overcome if people would realize that it is in no way opposed to the empirical methods like field studies, history or ethnography (as you mention). One is about building theoretical ideas, the other about inspiring or testing those ideas – they don’t get in each other’s way.

@Amit: you can sign up here: http://www.sscnet.ucla.edu/soc/groups/mathsoc/index.php

Seriously, all those who expressed support for Fabio’s post: the ASA Math Soc section could really use some more members. Join the club…

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RenseMarch 2, 2011 at 5:47 pm

[…] Convergence in the Social Sciences In a post over at orgtheory.net, Fabio Rojas makes the case for more math in […]

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Methodological Convergence in the Social Sciences « Marc F. BellemareMarch 2, 2011 at 7:41 pm

Just to follow up on where to build this work; the math soc section of ASA (other ASA problems ignored!) is a great community. There’s a nice overlap/core of math soc at INSNA as well.

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James MoodyMarch 2, 2011 at 7:50 pm

Please clarify how these sociological models will be a relevant analytical tool to any particular organization given that the organization exists in actual moments (I should say, not in variables). Even the best model can only approximate one moment in time by using a previous (different) moment or set of moments in time as, well, models.

I politely accuse you of advocating a method that is inherently academic (that is, wrong).

In other words, please speak to the practical side of your argument.

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AustenMarch 3, 2011 at 12:26 am

[…] thought this was spot on. I think psychology would also benefit from more […]

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Math in the social sciences « the skepticMarch 3, 2011 at 3:52 pm

Austen: models are, in a sense, supposed to be wrong. They are simplifications of reality, that we design because reality itself is too complex for us to understand. In that way, models help us to make sense of the complexities of reality. If a model (or any theory, for that matter) becomes as complex as reality itself, we are back to square one…

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RenseMarch 3, 2011 at 10:58 pm

[…] orgtheory.net, Fabio Rojas breaks a lance for more formal modeling in sociology. A message worth […]

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Rense Corten , Archive » Mathematical Sociology again at OrgtheoryMarch 3, 2011 at 11:44 pm

[…] makes his case here, but the debate began in the comments of this […]

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Rojas Makes the Case for Formal Models in Sociology « PermutationsMarch 4, 2011 at 6:26 pm

[…] math and sociology @ orgtheory.net by Fabio Rojas […]

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Math in the social sciences, with discussion « Brokering the ClosureMarch 8, 2011 at 6:23 pm

[…] math and sociology (orgtheory.wordpress.com) […]

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Negotiating Careers Outside Academia « Brain ThingAugust 7, 2011 at 10:13 pm

[…] math and sociology (orgtheory.wordpress.com) british sociological association bsa pgf postgraduate forum soa sociologists outside academia Sociology TwitterDiggFacebookDeliciousStumbleUpon […]

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Brain Thing » Negotiating Careers Outside AcademiaOctober 30, 2011 at 7:46 pm

To afinetheorem and fabiorojas:

I’m a sociologist by training who has studied as much economics as sociology as well as a fair amount of physical science/applied mathematics. It seems to me that if it’s deemed that sociology ought to become more mathematical economics may not be the best model, at least not at the beginning. Perhaps the mathematics background one gets as a result of an undergraduate degree in physics or engineering might be the better way to go. Such students are required to take three semesters of calculus, differential equations, and linear algebra (often differential equations and linear algebra are part of the same course where the linear algebra part is focused on differential and difference equations). The math courses taken by these students may focus on derivations but tend not to focus very much (if at all) on the kinds of proofs found in pure mathematics and even theoretical economics. If we could get sociologists close to this type of mathematical background, either by requiring applicants to graduate programs to already have it or, perhaps more realistically, trying to provide as much of it as possible once students start graduate study in sociology programs, I think this relatively modest step would be a vast improvement. A way to do this once students start graduate study would be to require all PhD students in sociology to take at least a semester and perhaps a year of mathematics. The first semester could be a course in Mathematics for Sociologists and an ideal book for such a course might be Alpha C. Chiang’s Fundamental Methods of Mathematical Economics. Even though it has Economics in the title it covers areas of mathematics that are quite useful for sociologists too. I suspect that afinethorem might regard Chiang as not rigorous enough but remember I’m proposing something, at least as a start, that is more modest than what economics programs are trying to do. Thus, a book like Simon and Blume’s Mathematics for Economists, which I understand is often used for PhD students in economics. although a very good book, would, I suspect, be too difficult for what I have in mind. The second semester, if there is one, would be a course on mathematical modeling and a book like Giordano et al.’s A First Course in Mathematical Modeling would be an ideal text. Once students had the background they would obtain from these courses, then sociology programs could begin to teach their statistics courses differently, including more mathematical statistics and probability theory (intuitive not measure theoretic). I suspect that what I’m proposing would be controversial among sociologists. But it seems to me that economists, engineers, physicists, and others are doing a lot of work these days (consider the work on the small world problem and social networks more generally) on what, arguably, are sociological problems. Many of us (perhaps most of us) in the discipline don’t have the mathematical background required to understand this work and, perhaps, contribute to it, which I think is kind of shame. The proposal I’ve made won’t get us to a point where we will understand all of this work and other work similar to it—but I think it would be a start.

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Michael LewisDecember 7, 2011 at 4:23 pm