Getting the elephant's trunk to wiggle

The Nature essay, A meeting with Enrico Fermi, by Freeman Dyson is worth digesting. I quote below a few bits that stood out to me:

I am eternally grateful to him for destroying our illusions and telling us the bitter truth.

we could calculate the atomic processes precisely. By 1951, we had triumphantly finished the atomic calculations and were looking for fresh fields to conquer. We decided to use the same techniques of calculation to explore the strong nuclear forces.

[Fermi said:]
"There are two ways of doing calculations in theoretical physics", he said. "One way, and this is the way I prefer, is to have a clear physical picture of the process that you are calculating. The other way is to have a precise and self-consistent mathematical formalism. You have neither...
....To reach your calculated results, you had to introduce arbitrary cut-off procedures that are not based either on solid physics or on solid mathematics....
....my friend Johnny von Neumann used to say, with four parameters I can fit an elephant, and with five I can make him wiggle his trunk."