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

[Belacqua]

And once again, the problem is that the term 'nature' or 'world' is ill-defined. And when attempts are made to define it (like those you have made), it is quickly found that the whole notion of a supernatural is simply incoherent.

The definitions I've given have been clear and coherent. You don't like them, and of course you don't believe the supernatural exists, but that doesn't mean that the terms are poorly defined. 

You are a man of great and unshakable faith. You have assurance of things hoped for, the evidence of things not seen. In your case, the assurance of things not seen is your confidence that all solvable problems will be solved by the scientific method. 

Your evidence of things not seen is more worrying. I brought up the strong likelihood that there are things in the world which human beings don't and can't understand. About those things, we can't say anything. You agreed that there may well be such things, but then you immediately turned around and told us something about them with great confidence. You said that if more intelligent beings could learn about those things, they would do so with the scientific method. So you are talking about things you don't know anything about. 

I lack your faith. And I am skeptical. I think we shouldn't make confident statements about things we don't know anything about. 

Plato believed that logic alone could give us information about the real world. And that is simply false. In order to find out anything about the real world, we have to actually look at the real world.

After we've looked at the real world, and while we continue to look, logic can tell us a great deal. Nobody ever claimed that logic in isolation was sufficient. (You shouldn't lie about Plato.) All observational input must be interpreted through logic if it is to have any meaning. If math is logical deductions from axioms, yet it is useful in understanding the real world, then that's proof of what I say.

And your attempt to change the subject by bringing up Socrates' theory of learning is a distraction. That isn't relevant to the thread topic. I brought up Neoplatonic thinking because it gives a coherent definition of "natural" -- the portion of the world which is knowable by science. If you don't like that definition, fine, but don't try to cloud things by bringing up irrelevancies.