"What mathematical characterizations could be given of this notion? See also my question above."

Ultimately none. that is the point of this paper. Or, they can try but they will ultimately fail. Math only describes half the picture.

"How can you know this? The claim is rather strong, but it's not clear what the evidence is."

(The evidence is the science of Complexity. Maybe somebody ought to read a book about it! The one in the 'Work Cited' section of this paper is a good one.)

The likely evolutionary path of the human race:

The future of Man


The 21st century needs its own philosophy; here it is: