The issues that are actively disputed are not esoteric matters - a great many non-specialists can, and do, understand the issues that are most often challenged in evolution, climatology and medicine, so long as one does not assume a vast yet undetectable conspiracy to falsify data. You are right in saying that there are difficult problems here, but it is probably beyond the capabilities of either science or philosophy to solve them. You can lead a man to reason, but you can't make him think.
But that's the point: thinking alone is no longer sufficient; trust is necessary. How do you know evolution is true (if you haven't yourself done experiments)? Things like Price's equation alone aren't enough, because you have no idea whether they model reality well. And how do you know that Fermat's last theorem is a theorem? Have you checked the proof?
The answer is that you trust experts. This is not "reason" but a certain model of society which you have, that you (and I) believe to be true, but it is not one that is necessarily true in every imaginable society. Some people believe that the society we live in is not as you or I imagine it to be, but an altogether different one.
It may be the case that their model of (a conspiratorial) society is inconsistent with certain definitive observations we can make (e.g., the state of technology), but even this claim is one that is not easy for non-experts to verify, and you must still trust the experts who tell you this is the case. It is not reason that leads you (at least these days) to believe in science, but faith. I am not claiming that this faith is contentually similar to religious faith, but it is epistemologically similar, i.e. it is justified by similar personal experiences that has little to do with "reason."
Fermat's last theorem is an example of my point, in that the people who actively oppose the scientific position are not basing their opposition on a rejection of Fermat's last theorem, or of anything else that can only be understood by a tiny minority of specialists. For the most part, those who oppose the application of science in certain areas of public policy are not doing so on the basis of an epistemological question over how science reached its current position, they do so on the basis of not liking the outcome.
Excellent point! I've been a deeply religious Evangelical until my early twenties, and I find it interesting how large the gap is between the rather conspiratorial world view I used to have, and the one I've spent most of the rest of my life having (academic/secular/urban/liberal).
I feel a big part of the problem is that my 'new' world just truly doesn't get the extent of this difference, so any attempts to bridge this gap, however well-meaning, fail and unfortunately often lead to easy dismissal, or hostility/frustration.
An additional problem is that despite the fact that I've rejected this Evangelical world view, I feel there are so many aspects of it that would be extremely helpful to many of the problems that people in my current world face. I wish I knew how to get both sides to properly get to know each other, so to speak, because it would benefit everyone.
