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November 10, 2015

Weil, Venting

Posted by Tom Leinster

From the introduction to André Weil’s Basic Number Theory:

It will be pointed out to me that many important facts and valuable results about local fields can be proved in a fully algebraic context, without any use being made of local compacity, and can thus be shown to preserve their validity under far more general conditions. May I be allowed to suggest that I am not unaware of this circumstance, nor of the possibility of similarly extending the scope of even such global results as the theorem of Riemann–Roch? We are dealing here with mathematics, not theology. Some mathematicians may think they can gain full insight into God’s own way of viewing their favorite topic; to me, this has always seemed a fruitless and a frivolous approach. My intentions in this book are more modest. I have tried to show that, from the point of view which I have adopted, one could give a coherent treatment, logically and aesthetically satisfying, of the topics I was dealing with. I shall be amply rewarded if I am found to have been even moderately successful in this attempt.

I was young when I discovered by harsh experience that even mathematicians with crashingly comprehensive establishment credentials can be as defensive and prickly as anyone. I was older when (and I only speak of my personal tastes) I got bored of tales of Grothendieck-era mathematical Paris.

Nonetheless, I find the second half of Weil’s paragraph challenging. Is there a tendency, in category theory, to imagine that there’s such a thing as “God’s own way of viewing” a topic? I don’t think that approach is fruitless. Is it frivolous?

Posted at November 10, 2015 11:41 PM UTC

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Re: Weil, Venting

For what reason or omission I do not know, when I think of Erdős and his imagined Book, categorical questions do not come to mind.

Posted by: Jesse C. McKeown on November 11, 2015 4:59 PM | Permalink | Reply to this

Re: Weil, Venting

Tom wrote:

I was young when I discovered by harsh experience that even mathematicians with crashingly comprehensive establishment credentials can be as defensive and prickly as anyone. I was older when (and I only speak of my personal tastes) I got bored of tales of Grothendieck-era mathematical Paris.

How does the second sentence tie in to the first? Are you referring to tales involving prickly defensiveness? There are also lots of interesting tales of how people figured out sheaves, schemes, abelian categories, stacks, the etale topology, topoi and other things. I always find those interesting.

Posted by: John Baez on November 11, 2015 5:37 PM | Permalink | Reply to this

Re: Weil, Venting

How does the second sentence tie in to the first?

It doesn’t. They were separate thoughts. I was trying to say, briefly, two things:

  • That although Weil seems to be rather defensive here, it’s no longer a surprise to me that someone so garlanded could be like that. So that’s not my reason for being interested in this passage. Though his prickliness is still kind of funny, as is what seems to me to be a terribly unconvincing attempt at humility (“I shall be amply rewarded if I am found to have been even moderately successful”).

  • That although Weil’s dig at “theologists” could be construed as criticism of Grothendieck and pals, that’s not my reason for being interested in this passage either.

    (I was talking about personality-based stories of that era, of which a lot have been written. I was fascinated by them for a while, but eventually the fascination wore off. Mathematical tales are different.)

As I said in the post, what mostly does interest me is whether some people (e.g. category theorists (e.g. me)) do go around imagining that there’s such a thing as God’s own viewpoint. And if they do, whether they should stop.

Posted by: Tom Leinster on November 11, 2015 5:55 PM | Permalink | Reply to this

Re: Weil, Venting

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.”

Posted by: John Baez on November 11, 2015 10:36 PM | Permalink | Reply to this

Re: Weil, Venting

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!”

Posted by: Blake Stacey on November 13, 2015 12:52 AM | Permalink | Reply to this

Re: Weil, Venting

I think God is probably a pluralist. Why would there be so many different ways of viewing so many different things in mathematics, if God didn’t like to look at things from more than one point of view?

Posted by: Mike Shulman on November 11, 2015 6:35 PM | Permalink | Reply to this

Re: Weil, Venting

Maybe those other viewpoints are the work of the devil?

Posted by: John Baez on November 11, 2015 10:16 PM | Permalink | Reply to this

Re: Weil, Venting

Some evidence:

In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics. — Hermann Weyl, 1939

Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine. — Sir Michael Atiyah, 2002

Are any algebraists as eager to slam the opposition as these guys?

Posted by: John Baez on November 11, 2015 10:22 PM | Permalink | Reply to this

Re: Weil, Venting

… so, when starts the Contravariance Witch-Hunt?

Posted by: Jesse C. McKeown on November 12, 2015 2:27 AM | Permalink | Reply to this

Re: Weil, Venting

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.

Posted by: Mike Shulman on November 12, 2015 3:53 AM | Permalink | Reply to this

Re: Weil, Venting

… so many different ways of viewing so many different things in mathematics

Are there different ways to see that there are different things in mathematics?

Are there different ways to see the differences in the ways of viewing something?

Posted by: David Corfield on November 12, 2015 8:51 AM | Permalink | Reply to this

Re: Weil, Venting

Yes. No.

Posted by: Gejza Jenča on November 13, 2015 7:42 AM | Permalink | Reply to this

Re: Weil, Venting

I think Weil may be engaging in a bit of a straw man argument here, or at least conflating two different arguments. Part of what he writes I whole-heartedly agree with:

My intentions in this book are more modest. I have tried to show that, from the point of view which I have adopted, one could give a coherent treatment, logically and aesthetically satisfying, of the topics I was dealing with.

In other words, he tried to write a book which tells a coherent story about a subject. That’s a laudable goal, which isn’t always compatible with doing things in the most general possible context. (Sometimes because the most general possible context changes from one topic to another!) There may be people who object that you can and therefore should do things in a more general context, even in a textbook with “Basic” in its title, but I think the sentences I quoted do a good job of answering that.

But not everything is textbook exposition. There are times when it is very fruitful to find more general contexts and broader perspectives. Weil chooses the most grandiose possible language to describe that impulse (“full insight into God’s own way of viewing their favorite topic”), but somehow I doubt that all the mathematicians he was criticizing would have described their pursuits the same way. (Personally, I like John’s “tao of mathematics”, though I don’t claim to be very good at following it myself.)

I can’t speak to the prevalence, fruitfulness, or frivolity of such a tendency in category theory, but I’m reminded of some of the work of the probabilist Michel Talagrand. In his books he is fond of recalling the advice he got from his PhD advisor, Gustave Choquet, always to consider any problem in the most general setting in which it makes sense. This sounds to me like a more sober version of exactly the approach Weil is criticizing. But doggedly following that advice has led Talagrand to make significant advances, like his concentration inequality and the “generic chaining” technique, that have become extremely important tools in probability, stochastic processes, theoretical computer science, machine learning, convex geometry… Fruitless indeed!

Posted by: Mark Meckes on November 12, 2015 3:56 PM | Permalink | Reply to this

Re: Weil, Venting

It’s certainly true that there is too much excommunication, and denigration of other styles, in mathematics (this is trivially true, because any amount of this is too much). I’m not sure that it’s specific to mathematics, and I’m also not sure that it’s inevitable (I know the philosophy community quite well, and there is a certain amount of excommunication there, but not nearly as much as there was about 30 years ago, when there was a lot). So the remedy might be as prosaic as just getting people to talk to each other.

If you think it may have roots in intellectual issues, here’s one: foundationalism, that is, the belief that our knowledge relies on axioms which are based on insight into fundamental phenomena. (The “insight into fundamental phenomena” is the crucial point here: merely deriving things from axioms is harmless if the axioms aren’t supposed to be privileged parts of our mathematical knowledge.)

Myself (being an anti-foundational pragmatist, and a category theorist) I find it natural to regard category theory as an account of the practice of mathematics, and I’m always a bit bemused by the way that category theorists want it to be a new foundations for mathematics. This is especially true of homotopy type theory: it’s a wonderful body of work, and yields insight into a lot of other mathematics. But foundational? Why? How? That (if you want to point a finger, which I don’t) would be a good place to start an excommunication (this is, of course, one of those topics on which any attempt to formulate an opinion ends up undermining itself).

Posted by: Graham White on November 16, 2015 12:10 PM | Permalink | Reply to this

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