The n-Category Café

Oh, okay.

The idea that one might share “God’s own viewpoint” is remarkably unhumble. About as close as I ever come is referring to the “tao of mathematics”, which makes more sense if you’ve read Taoist stories like this:

Cook Ting was cutting up an ox for Lord Wen-hui. As every touch of his hand, every heave of his shoulder, every move of his feet, every thrust of his knee — zip! zoop! He slithered the knife along with a zing, and all was in perfect rhythm, as though he were performing the dance of the Mulberry Grove or keeping time to the Ching-shou music.

“Ah, this is marvelous!” said Lord Wen-hui. “Imagine skill reaching such heights!”

Cook Ting laid down his knife and replied, “What I care about is the Way, which goes beyond skill. When I first began cutting up oxen, all I could see was the ox itself. After three years I no longer saw the whole ox. And now — now I go at it by spirit and don’t look with my eyes. Perception and understanding have come to a stop and spirit moves where it wants. I go along with the natural makeup, strike in the big hollows, guide the knife through the big openings, and following things as they are. So I never touch the smallest ligament or tendon, much less a main joint.

“A good cook changes his knife once a year — because he cuts. A mediocre cook changes his knife once a month — because he hacks. I’ve had this knife of mine for nineteen years and I’ve cut up thousands of oxen with it, and yet the blade is as good as though it had just come from the grindstone. There are spaces between the joints, and the blade of the knife has really no thickness. If you insert what has no thickness into such spaces, then there’s plenty of room — more than enough for the blade to play about it. That’s why after nineteen years the blade of my knife is still as good as when it first came from the grindstone.

So it’s about avoiding ‘hacks’ — things that require forceful struggle.

Here’s another nice one:

Confucius and his students went on a hike out in the countryside. He was thinking of using the opportunity to engage the students in a discussion about the Tao when one of them approached and asked: “Master, have you ever been to Liu Liang? It is not far from here.”

Confucius said: “I have heard about it but never actually seen it with my own eyes. It is said to be a place of much natural beauty.”

“It is indeed,” the student said. “Liu Liang is known for its majestic waterfalls. It is only about two hours’ trek from here, and the day is still young. Master, if you would like to go there, I would be honored to serve as your guide.”

Confucius thought this was a splendid idea, so the group set off toward Liu Liang. As they were walking and chatting, another student said: “I grew up near a waterfall myself. In summertime, I would always go swimming with the other children from the village.”

The first student explained: “These waterfalls we will see aren’t quite like that. The water comes down from such a great height that it carries tremendous force when it hits the bottom. You definitely would not want to go swimming there.”

Confucius said: “When the water is sufficiently powerful, not even fish and turtles can get near it. This is interesting to ponder, because we are used to thinking of water as their native element.”

After a while, they could see the waterfall coming into view in the hazy distance. Although it was still far away, they could see that it was indeed as majestic as the first student described. Another hour of walking brought them even closer, and now they could clearly hear the deep, vibrating sound it made.

They topped a rise and were able to see the entire waterfall. Then they gasped collectively, because at the bottom of it, they saw a man in the ferociously churning water, being spun around and whipped this way and that by the terrifying currents.

“Quickly, to the waterfall!” Confucius commanded. “He must have fallen in by accident, or perhaps he is a suicide. Either way, we must save him if we can.”

They ran as fast as they could. “It’s useless, Master,” one the students said. “By the time we get down there, he’ll be too far gone for us to do him any good.”

“You may well be right,” Confucius replied. “Nevertheless, when a man’s life is at stake, we owe it to him to make every effort possible.”

They lost sight of the man as they descended the hillside. Moments later, they broke through the forest to arrive at the river, a short distance downstream from the waterfall. They expected to see the man’s lifeless body in the river. Instead, they saw him swimming casually away from the waterfall, spreading his long hair out and singing loudly, evidently having a great time. They were dumbfounded.

When he got out of the river, Confucius went to speak with him: “Sir, I thought you must be some sort of supernatural being, but on closer inspection I see you are an ordinary person, no different from us. We sought to save you, but now I see it is not necessary.”

The man bowed to Confucius: “I am sorry if I have caused you any grave concerns on my behalf. This is merely a trivial recreational activity I enjoy once in a while.”

Confucius bowed back: “You say it is trivial, but to me it is incredible. How can it be that you were not harmed by the waterfall? Are there some special skills that you possess?”

“No, I have no special skills whatsoever,” the man replied. “I simply follow the nature of the water. That’s how I started with it, developed a habit out of it, and derived lifelong enjoyment from it.”

The various translations I find online of Zhuangzi’s parable of the Dexterous Butcher have an extra line at the end, where Lord Wen-hui says something like, “Your words have enlightened me, Cook Ting! Now I know how to nourish life!” However, the first time I heard the story, Lord Wen-hui said, “Amazing! Remind me to tell this to the executioner!”

… so, when starts the Contravariance Witch-Hunt?

I have to admit I feel a certain sympathy for that viewpoint. This semester I’m teaching a class in axiomatic projective geometry, which is an absolutely beautiful subject. I’ve been very sad to find that it’s been essentially completely abandoned — all the people who might have been studying it are now algebraic geometers, and whenever I ask a question about it on MathOverflow they answer me with schemes and divisors.

On the other hand, part of what’s beautiful about the subject is how one can reconstruct algebra from geometry: every Desarguesian plane is (non-canonically) isomorphic to the projective plane over some division ring. So even if algebra is of the devil, well, God created the devil too. In India they speak of maya, delusion, the fundamental animating force of Creation, whose fundamental characteristic is duality: you can’t have good without evil, light without dark… or, I suppose, geometry without algebra.

Yes. No.