This video begins with Pat Churchland saying that deeming aspects of the world inherently "mysterious" is anti-scientific and unjustified.
Noam Chomsky responds that any being with a cognitive structure is limited by that structure, so there will necessarily be some facts that we are incapable of knowing because of our limitations; these facts are "mysteries," and contrast with "problems," which are knowable facts we just haven't found out about.
Chomsky suggests that consciousness may be a mystery rather than a problem, and he has said similar things about free will. I'm not sure if Churchland is referring to Chomsky in her references to "old-fashioned philosophers."
I think Chomsky is clearly right about the existence of mysteries, though it's an open question (as he says) whether consciousness specifically is a mystery or merely a problem.
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