Let's say that the doctor tells you that you have a medical condition that is difficult to diagnose and it could be life threatening. Would you take the advice of the best doctors, or would you ask your neighbors for their diagnoses?
The weight of public opinion is a political concern, but if someone has no training in a particular subject of science, then their opinion on the subject matters very little. What matters is that they get the best information from the most informed experts. At that point they can decide what if any political action should be taken.
things you can't prove are speculation, and fall dangerously close to religious doctrine. therefore, you should apply these things in your personal life if you like, but don't try to force or coerce them on others. to do so is immoral.
In principle I agree with Ivan. If someone doesn't have scientific training, then their scientific opinion is of very limited value. They don't just hand out PhDs, as those of us in grad school are well-aware.
But I'm troubled by your phrase "difficult to prove." If a theory is difficult to prove, then it's not really a theory, is it? Scientific theories have to be testable and falsifiable. They aren't "proven" in the mathematical sense. But an idea has to have a good deal of scientific data in support of it before it can be called a theory. And if a scientific hypothesis is resting on shaky ground, then I wonder why we are allowing it to have such significant global impact. If we weren't sure about the effect of a certain vaccine, would we want to supply it to the public? If we weren't able to place definite constraints on the mass of the Higgs' Boson, would we want to spend billions of dollars on a collider that will operate in the appropriate energy regime? Certainly not! Scientific theories have to have a solid backbone in order to be afforded that label. So if a theory rests on shaky ground, then it seems to me that it's the responsibility of us as scientists, and not the general public, to voice concern.
It is difficult to imagine a scenario where public opinion should count, try this-----
A maned mission to Mars is planed ,the scientists in control of the mission give it a 95% chance of success.
But a small group of scientists out side the project protest, and say x,y,x has not been taken into account, the project is doomed to fail.
Of course we have not sent people to Mars before.
The mission will cost a $trillion .
The mission is planed to carry 8 people.
The mission has to launch on a set date.
if it's the public's money, then their opinion is the only one that does count. if a bunch of scientists want to fund such an expedition out of their own pocket, then it's the scientists' opinion which counts.