Yes he was indeed a sensitive genius. He made the identical mistake Georg cantor did, they both forgot to breathe, and get out into the wilderness. Nietztche, both father Carl and son Friedrich suffered the same fate.
NOW John Muir the same type of person but no madness. He would get a kennel cough on occasion and his wife would send him home to recuperate. Home for Muir, was the Sierras, culture was just a house.
May be. But that does not matter. They had different work to do and Godel's work is as breathtaking as beauty of nature, IMO. I will also suggest a reading of the linked Scientific American article.