It really disturbs me how people demand certain kinds of answers, never thinking that their demands are unreasonable. I was at a party when someone was trying to convince me that there were several problems that science would not be able to resolve. As an example he mentioned the paradox of the universe. Everything must have a beginning, but what happened before the beginning? I told him that the most popular theory in cosmology is that the big bang was the beginning of time itself, and, as such, there was no "before" the big bang. He asked me how this could be so. My physics isn't good enough to provide an answer to this. I told him that it was a consequence of the mathematics of the theory, which I wasn't familiar with. I also noted that it probably would take both of us a lot of math training to even appreciate the explanation.
He and the other person in the conversation would not accept this. My inability to give them an answer to this "paradox" in terms they could understand only supported their position, according to them. It appears that they were using common sense notions of how the world works (e.g. everything has a beginning) and applying it to cosmological events. My inability to explain the paradox in a way they could understand, and my inability to provide an explanation without violating their commonsense notions, supports their notion that there are some problems that science will never resolve.
I recently read something in Gould's Hen's Teeth and Horse's Toes that reminded me of this. Creationists have tried to wrestle with the problem of why cruelty exists in nature. Classic examples are the ichneumons, wasps which paralyze a caterpillar and inject eggs. The caterpillar stays alive while the offspring eat it from the inside out, saving the vital organs for last so that the caterpillar stays alive longer. Creationists came up with different justifications for why God would make something like this, none of which were very satisfying. I like how Gould puts it: "...the answer to the ancient dilemma of why such cruelty (in our terms) exists in nature can only be that there isn't any answer-- and that framing the question 'in our terms' is thoroughly inappropriate in a natural world neither made for us nor ruled by us. It just plain happens."
The lesson here is to understand that when we pose problems and questions, we imply, often tacitly, assumptions and expectations that might turn out to be unreasonable.
P.S.: I do not argue that science can, even potentially, answer all questions. I merely am attacking the example given by my party-mates.
Gould, S. J. (1983). Hen's Teeth and Horse's Toes. W. W. Norton and Company, New York.