> Godel explored this exhaustively and concluded (though did not prove) that this is not possible

While I agree that Aaronson's take on philosophy is not very original or profound, I don't think that Gödel concluded anything of the sort, and if he said something in that vein (assuming you're referring to the discussion on mechanism in the link you're provided), it is more an expression of his faith that any problem can be solved (and since not every problem can be solved by a machine, the mind is more powerful than any machine) -- hardly a conclusion.