# Math in the social sciences, with discussion

2011 March 8

Nice discussion on the usefulness, or lack thereof, of mathematics and formal theory building in the social sciences. Make sure you have a look at the comments. More or less chronologically:

- sociolgy needs more… @ orgtheory.net by Fabio Rojas
- math and sociology @ orgtheory.net by Fabio Rojas
- methodological convergence in the social sciences @ Marc F. Bellemare

With some appraisal here, here, and to some extent here.

Somewhat in parallel, a discussion about the death of theoretical (read mathematical) economics at econlog:

- The decline of economic theory @ econlog by Bryan Caplan
- Response on orgtheory.net

All in all, I subscribe to Fabio’s call with both hands.

My subjective list of advantages of formal theory building in social sciences supplementing the one at orgtheory.net:

- If a theory is, among other things, a logically coherent set of propositions then formalizing it is just a translation to a language that makes analyzing it, especially deducing consequences, much easier. And this applies to whatever the subject of the theory is.
- Most of the empirical studies in sociology are analyzed using some form of statistical reasoning, which is mathematical. Given that, building a formal theory of the studied phenomenon should in principle allow for a tighter connection between the theory and empirics (c.f.
*The Theory-Gap in Social Network Analysis*by Mark Granovetter). - I would also add the “accumulativeness”, much in the line of
*Formal Rational Choice Theory: A Cumulative Science of Politics*by David Lalman, Joe Oppenheimer, and Piotr Swistak. Although, I have to admit, after having spent 5 years or so studying mathematical sociology and selective works from mathematical economics, the cumulation is sometimes difficult to observe from a local point of view and local time scale of individual researcher. There are so many specific models (strong assumptions etc.), and it is frequently hard to understand the bigger picture. Perhaps it is just the question of time for a “unification” to arrive, … or a researcher… - ?

4 Responses
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One could add ‘generativity’ as fourth, where generativity is taken to mean the ability to generate, or derive, new predictions from a set of postulates. When relationships among key theoretical concepts are presented mathematically, one is able to derive more predictions that could be tested empirically than if the postulates are formulated in a natural language. Moreover, mathematical formulation of theoretical hypotheses allows for much more rigorous testing. In a natural language, one can say that a relationship between concepts is positive, but in mathematics the relationship can be specified more precisely, so that one is able to say that it is not increasing, but also, say, linear. Thus, there are more conditions that a hypothesis has to satisfy to be supported by data if it is formal. This, in turn, contributes to confidence in hypotheses which pass such refined and more rigorous tests.

As for the cumulation, or growth of sociological knowledge, there are many examples of such growth within sociological social psychology and group processes.

That is true. It is your “generativity” I had in mind in item 1 in “deducing the consequences”. As in here http://www.youtube.com/watch?v=b240PGCMwV0

You said you studied mathematical sociology for several years. Do you have a list of the best articles in this direction? I am a mathematician and now I am investigating which math should be taught to sociologists…