Naturalism is a metaphysical proposition assumed by scientists (at least real scientists!) as they go about their work. Naturalism holds that every phenomenon we observe, and every phenomenon we will observe, is a result of natural processes. Even scientists of faith assume this position, limiting their belief in miracles to their private lives. This keeps us from taking supernatural shortcuts in our scientific investigations. We can never simply shrug and say "God did it!".
For most atheists, this proposition is more than an assumption, it is a belief. But what does this belief actually mean? What differentiates the natural from the supernatural?
If science confirmed the existence of ghosts, would that not simply mean that we have expanded our knowledge of natural law? After all, if you tried to persuade the scientists of a few centuries ago of the truth of some of our modern scientific understanding, then they would probably dismiss you as a superstitious crank.
If this were the nature of the distinction, it would be much the same as that between alternative and conventional medicine. Alternative medicine is simply medicine which has not been shown to work. If it were shown to work, it would become part of the body of conventional medicine.
However I don't think this is what's going on. I don't think scientists would so easily be able to fold the study of gods or vampires into the body of scientific knowledge, and this is because they defy rational comprehension.
Albert Einstein once wrote "The most incomprehensible thing about the world is that it is comprehensible". I think what he is alluding to here is the fact that we can describe pretty much every phenomenon in the universe with simple mathematical formulae. Everything can be broken down into elementary processes. Everything has a reductionist explanation.
Except ghosts, gods and goblins. These things cannot be reduced. Ghosts are not formed of atoms. Gods cannot be modelled by equations. Goblins did not evolve due to natural selection. These ideas make "narrative" sense but do not make mathematical sense.
I believe that mathematics is what distinguishes the possibly real from the definitely unreal. Mathematics is the greatest tool ever discovered by our species in our quest to understand the world around us. It allows us to describe and predict the behaviour of the physical world, and explore realms that we cannot directly observe.
This is why vampires that turn into bats cannot exist. This is not a phenomenon that can be described mathematically. Vampires and bats are two high-level aggregations of matter that make sense on a large scale but not on a small scale.
When a vampire turns into a bat, then some of the atoms of its body must temporarily disappear, while others rearrange themselves. But how does a carbon atom know it is part of the body of a vampire? How is a vampire defined? At what point in the transformation of a human to a vampire does the ability to transform manifest itself? What, at a physical level, is the cause of this transformation?
Unless there are a consistent set of answers to questions like these, comprising an absolutely detailed mathematical description of how it is that vampires turn into bats, then we must conclude that this transformation cannot be consistent with any set of natural laws. It must always remain in the domain of the supernatural. Naturalism predicts that no such creatures can ever be discovered.
If you believe in the truth of naturalism, then you believe that everything that happens is governed by mathematics. No spirits are possible unless they have a mathematical description. God is not possible unless his mind is expressible as an algorithm.
And mathematics just might be the key to understanding reality itself.