"fractions of the Third World?"
Although places like Papua New Guinea are "small"...
There are approximately 1 billion people in India, and their culture hasn't gone through the diminution of the supernatural, and rejection of theories involving supernatural intervention in the natural world, that Western culture undertook during the Enlightenment.
Can we call the Enlightenment the one thing that makes our world different from the world inhabited by 17th Century Puritans? (and Anglicans, Quakers, Anabaptists, Lutherans, Catholics, etc. )
Admittedly, Mather was showing signs of the burgeoning of Enlightenment-era thinking. He wanted proof that he could put his finger on, proof that was equal in authority to Scripture, but unimpeachable as "this sect's interpretation of Scripture". He wanted evidence that would convince the Sadducees of his day.
I'm not sure I'd say that all 1 billion of India's people believe in witchcraft, but the point is well taken. There are quite a few African villages, throughout the continent, where witchcraft is believed in as well.
So is there merely a thin veneer of enlightenment thinking, with most of the world mired in superstition? I don't know, but I suspect it's not nearly as bad as that.
As for Cotton Mather -- it is sort of fascinating to see him approach his argument with the Sadducees on their terms, rather than resorting to authority. But then, Jonathan Edwards, in addition to being a fire and brimstone Puritan preacher, also dabbled in science, and I believe some of his observations of natural phenomena are as rigorous as anything his contemporaries were doing.