Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.
Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes.
The purpose of computing is insight, not numbers.
It is better to do the right problem the wrong way than the wrong problem the right way.
If you don't work on important problems, it's not likely that you'll do important work.