Mathematizing Mechanisms: The Key to a Future of Linking Theory and Empirics, Individual and Society

• In Event: Recent Advances in Mathematical Sociology

Abstract

When a theory’s premises include an input-outcome relation, denoted X → Y, where X is a personal characteristic and Y a personal outcome, its mathematization is immediate, and links theory and empirics as well as individual and society.

1. From X → Y to general function to specific function. Example:
“actual reward X and just reward X* produce justice evaluation J” to “as X increases, J increases at a decreasing rate, and as X* increases, J decreases at an increasing rate” to J = ln(X/X*)
Note: In general, the specific function may emerge in empirical work or be obtained by imposing additional conditions on the general function (or both).

2. If the probability distribution of X is known or can be ascribed, the probability distribution of Y is immediate. Example:
If X is ordinal, lognormal, Pareto, or power-function, J is negatively-skewed exponential, normal, positively-skewed exponential, and negatively-skewed exponential, respectively.
Graphs of the distributions provide pictures of society; e.g., the quantile function depicts both the location of every person and major aspects of the society – minimum, maximum, inequality, proportion below the mean, etc.
Further Y distributions emerge if there are multiple Xs or, in the case of J, if X and X* both vary. For example, status S can become Erlang or Mirror-Exponential, and J can become Equal, Erlang, Laplace, Logistic, etc.

3. Two types of empirical work follow:
3.1. Testable propositions can be generated from both the specific function and the Y distribution. These include classical deductive implications (including novel predictions) and Toulmin-type propositions.
3.2. Terms from the specific functions can be approximated or estimated.
The justice evaluation J, the actual reward X, and the just reward X* are routinely estimated. Moreover, the determinants of the actual and just rewards can also be estimated and contrasted, as can their distributions.

Author

• Guillermina Jasso, New York University