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Physicists (since Feynman) have computed things by defining certain infinite sums which (unlike ones you might have met in Calc II) always diverge when interpretted in an ordinary mathematical sense, then finding a way to interpret them differently ('regularize' them) to get finite quantities.

The best one used outside of string theory--in the context of Feynman diagrams, where one's sums depend on all embeddings of graphs (finite bunched of paths that can branch and rejoin)--is to realize that the dimension of the space-time shows up in the sum, turn the dimension into a variable, d, write the sum as a sum ofmultiples of powers of (d-4). It is infinite when d = 4 because you get some negative powers. Just throw away all the terms with negative powers of (d-4) and add up the rest (as in Calc II). Mirable dictu the result agrees with experiment to 14 places after the decimal.

There is no correponding procedure in string theory, where the sums depend on embedding Riemann surfaces (surfaces glued together out of copies disks from the complex plane so that you can tell which complex-valued functions on them have derivatives in the complex sense).

Physicists (since Feynman) have computed things by defining certain infinite sums which (unlike ones you might have met in Calc II) .... (snip) .... together out of copies disks from the complex plane so that you can tell which complex-valued functions on them have derivatives in the complex sense).

Huh?

I really appreciate the attempt to educate me, and if I had the time to put the brainpower to it I think I could absorb it ok, ... but - D@mn that just hurts too much to think about it.

"Physicists (since Feynman) have computed things by defining certain infinite sums which (unlike ones you might have met in Calc II) always diverge when interpretted in an ordinary mathematical sense, then finding a way to interpret them differently ('regularize' them) to get finite quantities."

Renormalization used in quantum field fiels theories is an incredibly clever trick.