You are reading a great deal more into Weinberg's statement than is actually there. Would you disagree with the following statement?: "there is nothing apparent at present that requires resort to supernaturalism in the ongoing scientific inquiry into the source and functioning of the human mind."
You ask:
"there is nothing apparent at present that requires resort to supernaturalism in the ongoing scientific inquiry into the source and functioning of the human mind."
I can only ask in return, what would that evidence look like?
Reductionists mock creationists for stipulating that something extremely complex is evidence of supernatural design. I would agree with the creationists, but I bring God to the discussion. You see?
The scientific reductionist brings his lack of spiritual experience to bear on all of his "scientific evidence" and concludes: "there is nothing apparent at present that requires resort to supernaturalism."
OK, but both are merely opinions and as far as I can tell, science cannot be used to get outside of itself.