I was discussing philosophy with a theist friend recently and the argument "this only applies to natural things and God is supernatural so this doesn't apply" came up. I've seen this argument in other debates as well, but I have to confess that I don't completely understand it. What does "supernatural" actually mean? The dictionary definition that seems to best apply is "unexplainable by natural law or phenomena".
There's two possible meanings to that. One is "unexplainable by the laws of physics as they are currently known" and the other is "unexplainable by the laws of physics even if we knew them all".
The existence of supernatural things of the first type is not denied by any sufficiently well-informed scientist - it's no secret that the laws of physics are incomplete. One possible example might be the details of the event horizon around a black hole - at small scales this requires a theory of quantum gravity, which we don't yet have. I think we'll eventually eliminate all such supernatural things by having a complete theory of physics.
I suspect theists would therefore prefer the second definition. But does this definition even make any sense? What would it mean for there to be phenomena in our universe for which no physical theory could be described to explain? Well, supernatural phenomena of that sort of can be said to exist too - when a quantum variable is measured and the wavefunction collapses, the result isn't necessarily determined by anything in the universe. But I don't see any theists suggesting that God acts on the universe by deciding how each and every wavefunction collapse occurs, despite the apparent omnipotence that such power would grant. I suspect this is because this would eliminate any possibility of human free will - quantum wavefunction collapse is part of all physical processes, so controlling quantum wavefunction collapse would mean controlling all our thoughts and actions. There would be no will except the will of God possible, making all religion (and indeed everything) rather pointless.
It seems to me there are philosophical reasons to reject the concept of a non-random supernatural process - if something is non-random there is some sort of (at least partially) predictable pattern to it, which means one could come up with a law of physics to describe that pattern, which means it's no longer supernatural. "Predictable" only means predictable in principle, though, not predictable in practice. Chaitin's constant (the probability that a random program will halt if run for long enough) for example isn't random but is uncomputable in the sense that only a finite number of its digits could be determined by any finite algorithm. Curiously, this number could be thought of as omniscient - it encompasses all mathematical knowledge (since it can be used as an oracle to solve the halting problem) but a number (even a real, uncomputable one) doesn't seem like it could match the theists' descriptions of God as having certain properties like compassion and goodness.
As well as the gaps in our knowledge of physics and the gaps caused by quantum improbability, there are also gaps due to the fact that there are some real world phenomena which we just can't do experiments on for one reason or another. We can't do experiments on UFOs because we can't predict when and where they will show up (though I'm sure that if one did show up in a suitably equipped science lab, laws of physics could be found to describe it).
Another thing we can't do physics on is subjective experience, simply because it's subjective. We don't currently have any technology by which one person can experience what it's like to be another person (and even if we did, there is no objective way to be sure that it's the same experience for both people - one can't compare subjective things with objective things). All we can do is ask people to report on their subjective experiences, and a personal report isn't as reliable a piece of evidence as a repeatable experiment.
Each one of us can't even be truly sure that other human beings actually *have* subjective experiences - maybe they're just p-zombies who say they do. It's a useful working hypothesis to assume that they do, though (and the opposite assumption would be rather dangerous for all concerned).
So perhaps God is Himself a subjective experience. That certainly dovetails with some things that theists say, like "I know God is in my heart and I've experienced His love, but I have no way to prove that to you". And objective evidence of God does seem to be rather thin on the ground, to put it mildly. If this is what God is then I am a teeny bit jealous of theists for having that experience that I never had (even when I was a theist, went to church, prayed regularly etc.).
If neuroscientists are able to generate religious feelings in others by stimulation of their temporal lobes with magnetic fields then, objectively, that suggests that religious experiences (at least these ones) are "fake" in the sense that there is no "objective God" causing them. But on the other hand, why should we treat subjective experiences as less "real" than objective reality? What could be more real to someone than something they have experienced first hand? By that logic, hallucinations caused by drugs or schizophrenia should also be considered "real" in this subjective sense.
Objective reality, on the other hand, consists of the things that we can in principle (given sufficient experimental data and suitable application of logic) convince other rational human beings of - in other words the things that we can in principle (if we're honest) all agree upon. As such, only objectively verifiable things should be used as a basis for public policy.