You Can't Measure Fun

The question often arises, at a precise moment in time, which party goer is a greater asset to the general ambiance: the modest slow burn or the fashionably late social butterfly? The decision is particularly convoluted exactly when the two appear to be socially on par with each other. How shall we characterize their respective states in this crucial slice of time? First, some units with which to conduct an empirical analysis...

Let's define the dimensionless unit Gregariousness (G) of party goer x to be the amount of joy that x contributes to himself and his peers through his presence. As Gregariousness inevitably varies over time, due to influences in the social environment, it will also be critical for us to define the unit of Sociability (S) as the time-derivative of Gregariousness - that is, the instantaneous rate of change of joy-providing with respect to time.



Although party goer 1 is currently more gregarious than party goer 2, PG1 has aggregated this charisma over the course of the afternoon and evening, while PG2 has just stepped in the door. But, here we must consult our derived unit to see how they really measure up! PG2 seems to be aclimating to the firendly company at a much higher rate than PG1. We can say that, although PG1 has greater total gregariousness, PG2 has significantly higher sociability.

Since we are dealing with scalar valued data, these individual measurements allow us to make some quick calculations about the party at large:



We have earlier established that gregariousness varies according to "influences in the social environment." This is an important generalization that swings the door wide open to some very interesting analysis.
Consider modeling sociability in n-dimensional space as a function of n "social environment variables."



The resulting manifold is everywhere continuous, differentiable, and orientable, and thus integrable as a non vector-valued surface. For the example above:



Conceptually, we can resolve an individual's net gregariousness by observing how his or her gregariousness has varied according to several party variables, modeling this variation by a function, and integrating over the measurement interval. Simple in principle, but horrendous in practice, as the time component of S must either be directly parametrized by (or otherwise coerced into) the SI units of the control variables.