My example will be the old suspicion, which the likes of B. F. Skinner struggled so hard to prove justified, that the scientific attitude turns human beings into machines. A person's "spirit" needs attention as well. Just to say it this way already assumes there is something to a human being that is not the mechanistic body. This is an overcomplication.
Everything called "spirit" and that is worthwhile (friendship, fraternity, reverance, joy of life, etc.) is simply the mechanism of a human being operating as it is best fit to do. To call this "spiritual" overcomplicates the situation by forcing us to divide the "spirit" from the mechanism. I don't know how many people realize they would have to present such a division for inspection. I don't have to do it -- instead I point at someone very "spiritual" and say, "There's a human being, and look how wonderfully she is operating!" Who is going to disagree with me? And who in not going to agree that an "un-spiritual" person like B. F. Skinner isn't quite the best arrangement of food, water, and air?
In short, non-spirituality is the view that "spirituality" means the same as fine tuning of the human biological mechanism.
I am calling it non-spirituality by analogy to non-euclidean geometry, which does not deny euclidean geometry, but rather expands and (in a sense) simplifies it.