This is a simpleminded approach to dealing with
divergences.

We are trying to calculate some Feynman diagrams and there are
a finite number of divergent integrals. Presumably the high energy
behavior of the theory is not being handled properly.
We generally believe in the theory so we believe that if it properly
handled the high energy behavior the integrals would turn out finite.
Since we can't calculate the integrals we simply treat them as
additional parameters to be determined by observation.
So e.g. instead of \(\{e,m\}\), we have \(\{e,m,I_1,I_2,I_3\}\).

We then proceed to calculate. Things are very simple. We don't have to
worry about overlapping divergences. There are no infinities at all.
We don't have to prove that the integrals can be absorbed by the
original parameters. If they can be it will appear in the calculations.
If they can't it's not fatal.

Will this work?