This is so true. If the problem domain is so small that you can get away with no tests (only small programs or small teams can indeed get away with it, IMHO), then it cannot be a system of interest. For large systems (i.e. systems so large that any one human cannot fully grasp them), testing is required. Testing does not prove much about the system, it only encodes the assertions of the programmers about how it is meant to work. Testing is merely a harness that constrains the flight domain of the code, it does not guarantee that the code within the flight domain is correct, only that it presents a certain level of certainty about it.

Above some system size, the certainty of correctness is merely probabilistic, and automated testing is what gives you assurance. Programming without testing is like driving with a seatbelt: sure it is feasible, but it's insane to do it.