Mathematical Sociology: Models, Structures, and Open Problems

Mathematical sociology has developed through partially independent programs concerned with exchange, status, mobility, organizations, social influence, collective action, and institutional trust. These traditions often share matrices, hazard rates, games, and differential equations while representing non-equivalent social objects. This survey reconstructs four foundational programs by substantive mechanism. Each model is written as a tuple comprising units, a state space with its social semantics, a structure, a transition law, an observation operator, parameters, and a source of identifying variation. The central result is a mechanism indistinguishability theorem: two transition laws are locally indistinguishable to first order at a common latent state exactly when their difference lies in the kernel of the differential of the observation operator, and equality of whole observed paths follows when the observation is projectable through the dynamics. No increase in sample size under one observation design removes such non-identification. A corollary on aggregation explains why the mean of a population of logistic actors is not logistic, and a worked example shows that founding counts cannot separate organizational legitimation from competition. A worked cross-scale case study carries one chain from discrete adoption through continuum propagation to a first-passage reduction, with the status of every claim stated explicitly.