"The scientist has a lot of experience with ignorance and doubt and uncertainty, and this experience is of very great importance, I think.
When a scientist doesn’t know the answer to a problem, he is ignorant. When he has a hunch as to what the result is, he is uncertain. And when he is pretty damn sure of what the result is going to be, he is still in some doubt.
We have found it of paramount importance that in order to progress, we must recognize our ignorance and leave room for doubt. Scientific knowledge is a body of statements of varying degrees of certainty — some most unsure, some nearly sure, but none absolutely certain.
Now, we scientists are used to this, and we take it for granted that it is perfectly consistent to be unsure, that it is possible to live and not know. But I don’t know whether everyone realizes this is true.
Our freedom to doubt was born out of a struggle against authority in the early days of science. It was a very deep and strong struggle: permit us to question — to doubt — to not be sure.
I think that it is important that we do not forget this struggle and thus perhaps lose what we have gained."
Richard Feynman. "The Value of Science," address to the National Academy of Sciences (Autumn 1955)
Doubt isn't just important, it's crucial.
Thomas Kuhn, in The Structure of Scientific Revolutions, describes how doubt allows science to proceed; old paradigms of how the world works are continually being challenged by new observations which contradict them. New theories are formulated which support new ways of looking at the world. If newer observations at a later date call these new theories into question, then hopefully careful work and reasoning will either affirm or contradict the theory - and the process starts over.
It's neither a neat nor an easy process - Einstein was bitterly criticized by people who believed in a linear, Newtonian universe; in turn, he was critical of probabilistic quantum mechanics - but by 1964, the very quantum entanglement that Einstein (in the Einstein-Podolsky-Rosen paradox) thought pointed to a crucial flaw in probabilistic quantum mechanics was shown by John Bell to be experimentally provable; now we can not only demonstrate quantum entanglement but apply it to cryptography.
If our scientists were happy to accept certainty, and refuse to doubt, we'd still be dealing with Newtonian physics. Nuclear physics, electronics, molecular biology as we know them wouldn't exist. We owe modern civilization to doubt.
