Image Credits - M.C. Escher(from wiki)
Douglass R. Hofstadter famously finds an analogy between the feedeback in M.C. Escher's drawings(one shown above), Johannes Bach's fugal music, and Godel's inconsistency and incompleteness theorems. He tries to argue in his --link here>" Godel, Escher, and Bach " that Godels' theorems show the way towards understanding the Human brain and consequently Artificial Intelligence.
Here's --link here> Bach's musical offering,
Douglass doesn't think Godel's theorem is such an show-stopper to mathematics and science. For him(and yes, me), Godel's theorems just prove that the mathematical adventure is forever - infinite. Godel's theorems can be stated simply as "a consistent finite set of axioms cannot prove an infinity of truths." But, an inconsistent finite set of axioms can prove an infinity of truths; but, at that point, the set of axioms are a bit dubious.
I've pointed out vague thought that people use to not allow consistent but uncomfortable to them knowledge from entering their mind in my Sophie and Silas post, and then, now, many posts since - lots of stuff in movies that are about fear and irrationality - movies like Planet of the Apes/Invasion of the Body Snatchers and so forth. Douglass, in his Godel, Escher, and Bach finds much of these types of evasive thinking against constructive and insightful mathematical thought in Buddhist Koans.
Douglass's first Koan is called Ganto's Axe,
One day Tokusan told his student Ganto, "I have two monks who have been here for many years. Go and examine them." Ganto picked up an ax and went to the hut where the two monks were meditating. He raised the ax, saying, "If you say a word I will cut off your heads; and if you do not say a word, I will also cut off your heads."
People are always using these unreasanable double stances. Here, Ganto and Tokusan are taking a double stance against uttering a word. A word cannot take in the infinit truth of the universe, therefore, we should not think! They take such an extreme stance on this, that they make up a Koan saying they'll chop off the heads of an Monk who hasn't come to this understanding. The Monks are just to clear the mind of all thoughts and not think,
Here's a great Buddhist temple/statue(religion makes for great art!)
The Ganto koan finishes, "Both monks continued their meditation as if he had not spoken. Ganto dropped his ax and said, "You are true Zen students."
Chapter IX - Mumon and Godel is Douglass's main chapter on Buddhists Koans and logical thought comparison.
A famous collection of Koans is from the twelve hundreds A.D.(1200) by a monk Mumon - the Mumonkan (note, Douglass creates a logical axiom game called Mu in the first few chapters)
The first Koan mentioned here is about a teacher who tells students to roll up a blind. Two students roll up the blind. A third person comments, which student was right? The first one was right, and the second was wrong. How do we know? We don't! If you designate one or the other as right, and the other wrong, the teacher is wrong, which can't happen! Then there's a poem about this,
"When the screen is rolled up the great sky opens
Yet the sky is not attuned to Zen
It is best to forget the great sky
And to retire from every wind."
Compare my favorite quote that I've mentioned everywhere from my Gospel of Truth to my Sophie and Silas post,
"People swarm everywhere, talking of incomprehensible matters, in hovels, streets and square, marketplaces, and crossroads. When I ask how many oboloi I have to pay, they answer with hairsplitting arguments about the born and the unborn. If I inquire the price of bread, I am told that the father is greater than the son. I call a servant to tell me whether my bath is ready; he rejoins that the son was created out of nothing."
- Another cute Koan,
"Goso said, "When a buffalo goes out of his enclosure to the ende of the abyss, his horns and his head all pass through, but why cna't the tail also pass?"
The associated poem,
"If the buffalo runs, he will fall into the trench;
If he returns, he will be butchered.
That little tail
Is a very strange thing."
- Here's one Koan a bit more explicitly against learning nature and thinking,
"A monk asked Nansen: "Is there a teaching no master ever taught bewfore?"
Nansen said: "Yes, there is."
"What is is?" asked the monk.
Nansen replied: "It is not mind, it is not Buddha, it is not things."
The associated Poem,
"Nansen was too kind and lost his treasure.
Truly, words have no power.
Even though the mountain becomes the sea,
Words cannot open another's mind."
- a Koan explicitly against words,
"Shuzan held out his short staff and said: "If you call this a short staff, you oppose its reality. If you do not call it a short staff, you ignore the fact. Now what do you wish to call this?"
The associated poem,
"Holding out the short staff,
He gave an oder of life or death.
Positive and negative interwoven,
Even Buddhas and patriarchs cannot escape this attack."
- A big part of what Kurt Godel does is derive the equivalent of the Cretans liar paradox - or the Epimenides paradox starts with "All Cretans are liars." If this is true, then Epimenides is a liar, since he's a Cretan. But, if it's false, then Epimenides is making a true statement that "All Cretans are liars." But, we've already established the falsity of the statement. Round and round it goes!
The Buddhist Monk Koans are always playing this twisted dualities game. Their Koans are this huge elaboration on all this, and as I've shown from my Sophie and Silas post on, a major insight on human behaviors.
Jacob Bronowski explains in his --link here> Origins of Knowledge and Imagination , and I've explained the major points plus my own --link here> James Burke Connections generalisations, both the meaning of Godel's theorems for the nature of mathematics, and human behavior.