Atheist VS Naturalist - the latter sounds more appealing to me...

[Belacqua]

How do we determine 1. what something’s nature is

That's what science does. It tells us what a thing is made of, how it functions, what its limits are, etc. All those things together are its nature. 

2. what constitutes “over and above” it? What do you mean by over and above?

Science gives us high confidence to say what a thing isn't and can't do. A frog isn't made of solid lead, isn't a million miles long, and can't live in the center of the sun. If we observed any of those things, we'd have to conclude that all of biology is wildly wrong, or, more likely, what we saw was incorrect. What we saw isn't really a frog.

Our colleague polymath is sure that science follows any observation. So that if we saw a frog at the center of the sun, all the frog scientists would say, "Gee, I guess they really can live there. How about that. It must actually be in the nature of frogs to live in the center of the sun." I don't believe that extreme cases would be accepted in that way. They'd have to posit that the observation was wrong, or something other than a terrestrial frog was involved. 

"Over and above" just means that an event is outside of the limits that science tells us exist. (But I accept that the nuance of superiority in that phrase might be ethnocentric on my part. I happen to be an opera fan and jump to the impression that frogs who can sing Mozart would be "superior." But I should be more accepting of different values.   Instead of "over and above" I could say "wildly differing from.") 

As I've said before, if it turns out that science has been wrong before and there is a good solid explanation for why frogs can live at the center of the sun, then such a thing would be natural. If, on the other hand, we think it's pretty certain that they can't do that, then we can be comfortable in ruling that out as a natural habitat.

Well, if we’re considering the supernatural as a possible cause, then it must be something. We need to have a rough idea of what it actually is before we can attempt to evaluate its probability as a cause of some event versus an alternative cause.

I haven't thought about that yet.

So far I've been talking about the supernatural as a kind of event.

I really don't know how to think about a supernatural realm, or something like that. If it means "outside of our material world," there is some precedent in theology. But even that immaterial world operates according to its own nature, and so isn't supernatural according to the definition I've been using. 

If there were such things as supernatural events, I don't know that we could assume they have causes. Cause and effect applies to the natural world, but since by definition the supernatural world doesn't work that way I don't know how we'd identify regularities there. 

And again, I'm not saying supernatural events occur. I'm only saying what it is that would constitute a supernatural event if it did occur.

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No worries, I think it was just a glitch on the forum!

I will NOT suggest that it was a supernatural event!