On the face of it, most people assume the positive, since there are religious scientists. Or because there are many scientific discoveries that were uncovered by religious people. Et cetera. But this is a massively simplistic view of the problem.
First of all, this phenomenon of religious scientists is, in my opinion, adequately explained by thinking of science, or the scientific method, as a sort of social ritual. In this view, the scientific method is just another social forum, like playing football. You don’t use the rules of football for religion, duh! That would make no sense; you don’t bring footballs to church and you don’t bring Jesus to the football field (unless you’re Tim Tebow…). Just like football has a bunch of rules designed only for playing football, religious scientists also believe that there are social rules for doing science. And again, you don’t bring the social rules, the agreed upon social contract for “doing science”, to church anymore than you bring the social rules for churchgoing to the lab.
Play the rules of the game — whether the scientific method or football — according to the letter, and you will be successful in that arena. Falsifiability is restricted to the domain of science just like not going offsides before the snap is restricted to the football field. So in this sense, religion is compatible with science just like religion is compatible with football.
Secondly, this papers over the pretty uncontroversial human activity of rationalization. Much like a vegetarian who eats meat but uses some rationalization to explain away why their meat eating isn’t inconsistent with their vegetarianism.
If my reasoning above is true, then they both explain why science and religion are “compatible”. Of course, this means “compatible” in only the most superficially coincidental ways. Much like being a nice guy is compatible with being a rapist. Of course, if my reasoning above is true, then this in and of itself means that science and religion are not compatible. Why? Because when we talk about compatibility, we means occurring at the same time at the same place. Religious scientists do not practice both religious thought and scientific thought at the same time because they consider them to be two separate social contracts; they’re no more compatible than oil and water. Just because you can have both oil and water in the same glass this doesn’t mean they’re compatible.
The pithy saying (and equivocation fallacy) “science tells us how the heavens go, religion tells us how to go to heaven” is itself a ringing endorsement of the bullshit idea that science and religion are compatible. Not only that, but as an aside, “heaven” is the translation of the word οὐρανός (ouranos) which was transliterated into Latin as Uranus. Which is how I feel about the whole thing ;-).
So really, if you want to bypass all of this argumentation when we get to the question of whether science and religion are compatible, instead of asking that question one should ask whether religious claims are falsifiable. If you say yes, then that answers your question of whether science and religion are compatible. If you say no, then this is also the answer to whether science and religion are compatible.
But if we look at the fundamental premises behind the epistemology of religiosity and the epistemology of the scientific method, there is a massive, irreconcilable contradiction.
Science, more than just assuming the natural, assumes reductionism. Big things can be broken down into their constituent parts and analyzed bit by bit. And then those constituent parts themselves can be further broken down, and those parts can be analyzed. The budding engineer who takes apart a radio or computer is assuming the necessary precursor for scientific thinking. Religion, on the other hand, is vehemently not reductionist. The fundamental particle in religious thought is not the quark, gluon, or Higgs-Boson. It’s the mind. Of course, this doesn’t apply wholesale to modern religions like Christianity but for older religious thought like animism. But Christianity still has the mind as the fundamental particle for living things.
So not only are science and religion incompatible, they will never be compatible due to reductionism. Unless and of course the mind actually is a fundamental particle instead of being an emergent property of the reducible brain.
Of course, I don’t think that science is just some sort of social ritual. I think things like falsifiability follows necessarily from the laws of logic. So religion isn’t just incompatible with science. It is incompatible with logic.