The distinction between natural and artificial is an artificial one. But it is also a natural one.
For me Marcello's comment resonates, as does the following
from Set Theory with a Universal Set by Thomas Forster. I am basically some kind of atheist or agnostic, but for me the theme is religion in the etymological sense of tying back, from the infinite and paradoxical to the wonder, tedium and frustration of everyday life and the here and now. (I dream of writing a book called Hofstadter, Yudkowsky, Covey: a Hugely Confusing YES!)
"However, it is always a mistake to think of anything in mathematics as a mere pathology, for there are no such things in mathematics. The view behind this book is that one should think of the paradoxes as supernatural creatures, oracles, minor demons, etc. -- on whom one should keep a weather eye in case they make prophecies or by some other means inadvertently divulge information from another world not normally obtainable otherwise. One should approach them as closely as is safe, and from as many different angles as possible."
Somewhere else in the book, he talks about trying to prove a one of the contradictions (of naive set theory) in one of the axiomatic systems presumed to be consistent, and seeing what truths are revealed as the exploded bits of proof spontaneously reassemble. Things like the magic of recursion as embodied in the Y combinator.
Thus I value people like Eliezer trying to ponder the imponderable.
In the sentence "Trying to catch a flying ball, you're probably better off with your brain's built-in mechanisms, then using deliberative verbal reasoning to invent or manipulate probabilities," I think you meant "than" rather than "then"?