To heck with it. I'm going to remain disorganized. It's more fun.
I think the "new definition" of science is actually better than the old definition. In fact, it isn't a new definition at all. It's an elaboration of what my dictionary says science is.
The OP asked:
I want to know (a) how divine intervention is different from normal naturalistic activity; and (b) how science can study it.
The short answer is that divine intervention, or any supernatural intervention, cannot be described by "laws". While it might follow trends, it cannot be predicted with certainty, because it is the result of intentional action, not "laws".
(For those who haven't followed my previous postings, I don't actually see any evidence that there is a divine, or that it ever intervenes. I'm an atheist, too, but I don't rule out the possibility that such a thing could happen. If it did happen, it could leave evidence.)
For example, if my wife suddenly turned into a pillar of salt, I would take that as evidence of divine intervention. I know a little bit about salt, and about human physiology. I am fairly confident that there are no laws that operate in the real world which would replace the constituent molecules of a human body with an equal volume of salt crystals. If such a thing happened, I would say that a violation of natural law had taken place.
Science cannot find "laws" that will describe under exactly what circumstances spouses will become crystalline. However, science can analyse the effects of that intervention. You could see a before and after photograph that showed a normal female form, followed by a pillar of salt. You could use chemistry to determine the chemical composition of the salt. If you did all that, and convinced yourself that at one time there really was a normal human female present, and that a moment later, there really was a pillar of salt, you would have a very strong case that some sort of supernatural phenomenon had taken place.
Of course, there is always the possibility that you had missed some sort of natural law that allowed for salt substitution, and that in the future, scientific knowledge would increase, and you would find a perfectly natural explanation for spontaneous crystallization. At that point, science would have advanced, and you would realize how naive you had been before. I'll be discussing this later with respect to the scientific theory of Kashrut, but for now, I need to go to work.
as well for someone who believes in a form of idealism as it does for someone who beliefs in a form of materialism. It requires no "objective" underpinnings. It also doesn't require that reality is consistent or follows logic.
