Mind games - the difference between "brain" and "mind"

There's a poster at my local railway station which never fails to catch my attention. It reads (in classic question and answer, feed and punchline style) :

"What went through the mind of the person who slipped on the platform?" "The floor."

While I admire the wordplay and admit that it is effective as a health and safety notice - the shift from a thought running through someone's head to a piece of concrete smashing into it is as jarring as it is supposed to be - there is something that doesn't quite work.

I think it is to do with the difference between "mind" and "brain". Because "mind" is seldom used in a physical, biological sense, the pun feels like something of a cheat - you simply wouldn't refer to the floor "going through someone's mind" unless you meant that they were thinking about it.

Brains are biological; minds are psychological. A neurosurgeon operates on the brain, not the mind; if someone described themselves as a "mind doctor" you would be more likely to assume they were a psychoanalyst, or even a hypnotist. The meaning of "brain" is somewhat broader - the phrase "to have something on the brain" more often refers to thoughts than to biological conditions - but, as a noun, "mind" almost never refers to the physical brain.

Would "brain", with its double meaning of the seat of thought and the physical organ, have been a better choice? I think so, although, because is is more closely associated with the latter sense than with the former, the shock in the shift of meaning would not have been so great. My choice would have been "head", which works equally well in both contexts.

The wordplay on "running through someone's mind" reminds me of another pun with a far less serious aim. Although I have never heard it used (I must be associating with the wrong people), the phrase "You must be tired, because you've been running through my mind all day" is apparently one of Europe's top ten chat-up lines.

A mind doctor