Lets say there is a theoretical relationship you want to prove over all values of some variable. For example, $F(x)$ increases with $x$.
You are unable to come up with a general theoretical proof. However, you can calculate $F(x)$ for any specific value of $x$, and the expected relationship holds.
Presumably, you can then run some sort of regression on the "data" provided by a whole bunch of the values you specifically calculated. (using $x$ as the IV and $F(x)$ as the DV). Then, you can show that an increasing relationship is supported by the "data" and try to come up with a best fit, etc.
This is sort of like "comparative statics" only taking it one step further -- trying to prove the direction (even shape) of a relationship, rather than just the ordinal relationship of two values.
Is there a name for this type of process -- running statistics on mathematical solutions, vs. measured data? Is this a convincing way to argue a theoretical point? Isasmuch as running an experiment and collecting data values and running statistics on that is?
The final goal is not a theoretical contribution on $F$'s movement with $x$, but rather a model that provides a useful takeaway (e.g. since $F(x)$ increases with $x$, it is advisable to try to minimize $x$).