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Grothendieck’s mathematics and Child Born of Water

Saturday, December 13th, 2014

[ by Charles Cameron — two approaches to mathematics, two types of heroism, and their respective complementarities ]
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I wish to propose a clear analogy between the mathematician Grothendieck‘s two styles of approach to a problem in mathematics, and the Navajo Twin Gods, Monster-Slayer and Child-Born-of-Water.

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Twins

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Steve Landsburg‘s post, The Generalist, compares two approaches to mathematics, as practiced by two eminent mathematicians:

If there was a nut to be opened, Grothendieck suggested, Serre would find just the right spot to insert a chisel, he’d strike hard and deftly, and if necessary, he’d repeat the process until the nut cracked open. Grothendieck, by contrast, preferred to immerse the nut in the ocean and let time pass. “The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough.”

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In the paras leading up to this one, Landsburg gives us the insight that these two approaches can be generalized as “zooming in” and “zooming out”:

Imagine a clockmaker, who somehow has been oblivious all his life to many of the simple rules of physics. One day he accidentally drops a clock, which, to his surprise, falls to the ground. Curious, he tries it again—this time on purpose. He drops another clock. It falls to the ground. And another.

Well, this is a wondrous thing indeed. What is it about clocks, he wonders, that makes them fall to the ground? He had thought he’d understood quite a bit about the workings of clocks, but apparently he doesn’t understand them quite as well as he thought he did, because he’s quite unable to explain this whole falling thing. So he plunges himself into a deeper study of the minutiae of gears, springs and winding mechanisms, looking for the key feature that causes clocks to fall.

It should go without saying that our clockmaker is on the wrong track. A better strategy, for this problem anyway, would be to forget all about the inner workings of clocks and ask “What else falls when you drop it?”. A little observation will then reveal that the answer is “pretty much everything”, or better yet “everything that’s heavier than air”. Armed with this knowledge, our clockmaker is poised to discover something about the laws of gravity.

Now imagine a mathematician who stumbles on the curious fact that if you double a prime number and then halve the result, you get back the number you started with. It works for the prime number 2, for 3, for 5, for 7, for 11…. . What is it about primes, the mathematician wonders, that yields this pattern? He begins delving deeper into the properties of prime numbers…

Like our clockmaker, the mathematician is zooming in when he should be zooming out. The right question is not “Why do primes behave this way?” but “What other numbers behave this way?”. Once you notice that the answer is all numbers, you’ve got a good chance of figuring out why they behave this way. As long as you’re focused on the red herring of primeness, you’ve got no chance.

Now, not all problems are like that. Some problems benefit from zooming in, others from zooming out. Grothendieck was the messiah of zooming out — zooming out farther and faster and grander than anyone else would have dared to, always and everywhere. And by luck or by shrewdness, the problems he threw himself into were, time after time, precisely the problems where the zooming-out strategy, pursued apparently past the point of ridiculousness, led to spectacular, unprecedented, indescribable success. As a result, mathematicians today routinely zoom out farther and faster than anyone prior to Grothendieck would have deemed sensible. And sometimes it pays off big.

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I no longer have — alas — a copy of Where the Two Came to their Father, the first volume in the Bollingen Series, with its suite of 18 sand paintings beautifully rendered in silkscreen by Maud Oakes, but their respective black and blue colorations lead me to suppose that the illustration at the head of this post, taken rom that series, shows the twin heroes, Monster Slayer (black) and Child Born of Water (blue) whose journeys and initiation are the subject of the rituasl “sing” recorded in that book.

The theme of two male hero twins is central to the mythologies of the American continent, according to Jospeh Campbell, who contributed a commentary to Oakes’ recording of Jeff King‘s performance of this ceremony, and lacking both the King > Oakes > Campbell book and Gladys Reichard‘s two volumes on Navaho Religion, I must draw on brief quotes from miscellaneous web sources to dramatize the differences between the twins.

Monster Slayer is the doer of deeds, similar in nature to other masculine, not to say macho, heroes — while Child Born of Water is the contemplative of the pair:

The Sun [Jóhonaa’éí] gave them prayersticks and then told them that the younger of the two (Born for Water) would sit watching these prayersticks while the older (Monster Slayer) went out to kill the monsters. If these prayersticks began to burn, this would signal that his brother was in danger and that he should go to him to help.

Reichard explains:

Monster Slayer (na’ye’ ne’zyani) (I) represents impulsive aggression, whereas Child-of-the-water represents reserve, caution, and thoughtful preparation.

A measure of their respective strategies, and of the ways in which the insights of Child Born of Water can succeed where the brute force tactics of Monster SLayer fail, can be gleaned from this section of their story, also I believe taken from Reichard:

When The Twins visited Sun the second time, he said he was willing to help them, but this time he wanted them to return the favor: “I wish you to send your mother to the west that she may make a new home for me.” Whereupon Monster Slayer, believing himself equal to any task, replied, “I will do so.I will send her there.” Then Child-of-the-water reminded them both: “No, Changing Woman is subject to no one? we cannot make promises for her. She must speak for herself? she is her own mistress. But I shall tell her your wishes and plead for you.”

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One commentator glibly suggests that the joint presentation of the hero as twins is “a clever reminder that progress depends upon cooperation between our mind and our heart” — but the psychologist Dr Howard Teich offers a far more depthful interpretation: that the two twins represent two forms of masculine heroism, one the familiar macho hero of war movies, and the other wiser and subtler, the possessor of traits commonly attributed to the feminine — and hugely undervalued — in our culture.

Dr Teich suggests we must (urgently) abandon the division of virtues into “male” and “female” types, reognize that these types are complementary rather than rivalrous, that both are necessary functions of both males’ and females’ psyches, and begin to integrate the wholeness that both strategies together represent, in our own approaches to our lives in general, to the natural world around us, and indeed to warfare — unsurprisingly, since we first encounter the twins in the ceremonial specifically devised by the Navajo to protect young warriors on their way to battle, and to reintegrate them in harmony and balance on their return.

As Teich puts it:

Monster Slayer and Child Born of Water, as these Twin Heroes are called, are the most sacred of all the legendary heroes in Navaho mythology. It is rare for the Navaho even to speak of the twins; their presence is to be felt rather than observed, and their lessons absorbed rather than applied. Although the lessons the twins hold may be countless, their particular manifestation of a deeper, more complex image of masculinity deserves the reader’s especial attention.

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I’d like to suggest that in the same way that there are “zooming in” and “zooming out” styles in mathematics, and “monster-slayer” and “born of water” styles of heroism, there are in fact twin traditions of understanding the world which we might term scientific and poetic, or in Teich’s terms — and those of the alchemists — solar and lunar.

A unified or “solunary” vision will encompass the virtues of both.

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Dr Teich’s review of the King > Oakes > Campbell book under the title A Dual Masculinity was irst piublished in The San Francisco Jung Institute Library Journal, Vol. 13, No. 4, 1995. He now has a book out treating these themes: Solar Light, Lunar Light.

Oh, and please don’t expect me to know anything more about Grothendieck’s mathematics than I read in Landsburg’s article.

Creating a web-based format for debate and deliberation: discuss?

Friday, December 12th, 2014

[ by Charles Cameron — Talmud, hypertext, spider webs, Indra’s net, noosphere, rosaries, renga, the bead game, Xanadu, hooks-and-eyes, onward! ]
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Let me firmly anchor this post and its comments, which will no doubt shift and turn as the wind wishes, in discussion of the possibility of improving on current affordances for online deliberation.

Let’s begin here:

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There are a variety of precursor streams to this discussion: I have listed a few that appeal to me in the sub-head of this post and believe we will reach each and all of them in some form and forum if this discussion takes off. And I would like to offer the immediate hospitality of this Zenpundit post and comment section to make a beginning.

Greg’s tweet shows us a page of the Talmud, which is interesting to me for two reasons:

  • it presents many voices debating a central topic
  • it does so using an intricate graphical format
  • The script of a play or movie also records multiple voices in discourse, as does an orchestral score — but the format of the Talmudic score is more intricate, allowing the notation of counterpoint that extends across centuries, and provoking in turn centuries of further commentary and debate.

    What can we devise by way of a format, given the constraints of screen space and the affordances of software and interface design, that maximizes the possibility of debate with respect, on the highly charged topics of the day.

    We know from the Talmud that such an arrangement is possible in retrospect (when emotion can be recollected in tranquility): I am asking how we can come closest to it in real time. The topics are typically hotly contested, patience and tolerance may not always be in sufficient supply, and moderation by humans with powers of summary and editing should probably not be ruled out of our consdierations. But how do we create a platform that is truly polyphonic, that sustains the voices of all participants without one shouting down or crowding out another, that indeed may embody a practic of listening..?

    Carl Rogers has shown us that the ability to express one’s interlocutor’s ideas clearly enough that they acknowledge one has understood them is a significant skill in navigating conversational rapids.

    The Talmud should be an inspiration but not a constraint for us. The question is not how to build a Talmud, but how to build a format that can host civil discussion which refines itself as it grows — so that, to use a gardening metaphor, it is neither overgrown nor too harshly manicured, but manages a carefully curated profusion of insights and —

    actual interactions between the emotions and ideas in participating or observing individuals’ minds and hearts

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    Because polyphony is not many voices talking past one another, but together — sometimes discordant, but attempting to resolve those discords as they arrive, and with a figured bass of our common humanity underwriting the lot of them.

    And I have said it before: here JS Bach is the master. What he manages with a multitude of musical voices in counterpoint is, in my opinion, what we need in terms of verbal voices in debate.

    I am particularly hoping to hear from some of those who participated in tweeted comments arising from my previous post here titled Some thoughts for Marc Andreessen & Adam Elkus, including also Greg Loyd, Callum Flack, Belinda Barnet, Ken (chumulu) — Jon Lebkowsky if he’s around — and friends, and friends of friends.

    What say you?

    An English game

    Wednesday, December 10th, 2014

    [ by Charles Cameron — image and Sotheby’s cataloguing via Michael Robinson ]
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    Carl Jung told his friend Sir Laurens van der Post:

    One of the most striking testimonies to the quality of the English spirit is the English love of sport and games in a classical sense and their genius for inventing games.

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    Exempli gratia:

    Pooh Stcks

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    Catalog entry:

    Shepard, E.H.
    “FOR A LONG TIME THEY LOOKED AT THE RIVER BENEATH THEM…”

    Illustration for Chapter six of ‘The House at Pooh Corner,’ the episode “in which Pooh invents a new game and Eeyore joins in.” The game is ‘Poohsticks.’ Sold this morning: £314,500 (US$492,727)

    Shepard, Ernest H. (10 December 1879 – 24 March 1976)
    188 by 148mm, original ink drawing,
    signed “EHShepard” lower left.

    **

    For those who don’t know the game, here’s the relevant excerpt from The House at Pooh Corner:

    Pooh had just come to the bridge; and not looking where he was going, he tripped over something, and the fir-cone jerked out of his paw into the river. ‘Bother,’ said Pooh, as it floated slowly under the bridge, and he went back to get another fir-cone which had a rhyme to it. But then he thought that he would just look at the river instead, because it was a peaceful sort of day, so he lay down and looked at it, and it slipped slowly away beneath him, and suddenly, there was his fir-cone slipping away too.

    ‘That’s funny,’ said Pooh. ‘I dropped it on the other side,’ said Pooh, ‘and it came out on this side! I wonder if it would do it again?’ And he went back for some more fir-cones. It did. It kept on doing it. Then he dropped two in at once, and leant over the bridge to see which of them would come out first; and one of them did; but as they were both the same size, he didn’t know if it was the one which he wanted to win, or the other one. So the next time he dropped one big one and one little one, and the big one came out first, which was what he had said it would do, and the little one came out last, which was what he had said it would do, so he had won twice … and when he went home for tea, he had won thirty-six and lost twenty-eight, which meant that he was – that he had – well, you take twenty-eight from thirty-six, and that’s what he was. Instead of the other way round.

    And that was the beginning of the game called Poohsticks, which Pooh invented, and which he and his friends used to play on the edge of the Forest. But they played with sticks instead of fir-cones, because they were easier to mark.’

    Christian cannibal: first the horror, then the meditation

    Friday, January 17th, 2014

    [ by Charles Cameron — you may not want to watch the video – read the text first, okay? ]
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    Here’s what the BBC-wallah said:

    The Christians were victims; now they’re on top. It’s a dangerous time to be Muslim. A charred and dismembered body is dragged through the streets. Christians have just killed a Muslim passerby. Ouandja “Mad Dog” Magloire was at the head of the mob. He was in a blind fury that day. Muslims killed his pregnant wife, his sister in law, her baby, he tells me. They broke down the door and cut the baby in half. I promised I’d get my revenge. Revenge was an act of cannibalism. First, he stabbed j\his victim. You are Muslim, Muslim, Muslim, he said. I poured petrol over him, I burned him, I ate his leg, right down to the white bone. The victim was just passing through on a bus. Most Christians are horrified, but resigned. No-one tried to help him, say these eyewitnesses. Everyone is so angry with these Muslims. No way anyone was going to intervene.

    This happened at two o’clock in the afternoon, when the streets were crowded with people, just like you see today. Everyone we’ve spoken to is still at a loss to know what to make of it. Was it the act of a madman, was it somebody who’d been pushed by sectarian hatred, was it explained perhaps, by traditional beliefs in magic and sorcery. These fighters are Christians but they also believe in magic. their amulets contain soil from their ancestors’ graves. Some carry the flesh of enemies they’ve killed. These charms are a delicate subject, not often discussed with outsiders. We are bullet-proof, says the commander. Mad Dog Magloire went further. perhaps his crime resulted from his own demons, but to some Christians he’s a hero. That doesn’t bode well for this country’s future.

    If you want to watch him say it, it’s powerful. Here you go:

    Okay, now for the meditation: I want to rescue something out of all this horror.

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    The very first thing I want to note is this:

    We are bullet-proof, says the commander.

    I’ve run across this before, it’s a common motif. Remember the Lakota Ghost Dance shirts? Johnny and Luther Htoo, the cigar-smoking twins who led God’s Army in Myanmar…? Televangelist Wilde Almeda of the Jesus Miracle Crusade in the Philippines?

    This is just to say that in my view, religion with spiritual bullet-proofing is different from religion without it, no matter what name you tag the religion with.

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    Next up:

    Most Christians are horrified, but resigned. … perhaps his crime resulted from his own demons, but to some Christians he’s a hero.

    It could be tribal. It could be magical, maybe. It could be religious, specifically Christian. It could be Mad Dog Magloire‘s “own demons”. It could be, and surely was, that he saw his pregnant wife slaughtered before his own eyes.

    But he projected his thirst for vengeance not on the man — a Muslim — who had butchered them, but on a guy in a passing bus who looked like he was Muslim.

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    Some weeks back, Commander Abu Sakkar of the Farouq Brigades in Syria ate what he took to be the heart of one of his enemies. It turned out to be his enemy’s lung.

  • If you think Mad Dog Magloire doesn’t represent Christianity, maybe Abu Sakkar doesn’t represent Islam.

  • If you think Abu Sakkar is representative of Islam, maybe Magloire is representative of Christianity.
  • I think it is fair to say that any religions with in excess of a billion adherents will find the odd cannibal among them in time of war.

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    But then consider this, in peacetime:

    In Ireland this week, a man confessed he’d murdered his landlord over a chess game, and eaten his heart. Forensics showed it was a lung that was missing

    **

    We are, after all, human.

    Gaidi Mtaani, the greater scheme of things, I: the story

    Thursday, November 14th, 2013

    [ by Charles Cameron — in a story, in a nutshell, the very different world of al-Shabaab ]
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    The third issue of of the pro-Shabaab magazine Gaidi Mtaani, issued a few months back, contained a long piece titled A Greater Scheme of Things, and in it I’d like to zero in on the extraordinary story which is the heart of the piece — and which offers us a fasacinating insight into jihad:

    it was luminously evident, for me personally, in one memorable incident in Mogadishu’s frontline last year. Just after Dhuhur prayer, one particular day in March, the Mujahideen received news that the enemy was preparing to mount a large offensive in the Northern parts of the city – in an area that was then newly captured by the Mujahideen.

    In no time, the news spread across the entire frontline and the Mujahideen geared up for a protracted battle. New defence posts were erected, the old ones were fortified, trenches were dug, reinforcements were called in, snipers were placed on rooftops and small groups of well-equipped fighters were strategically positioned to guard every street and every alley. Garrisoned in the makeshift base, a short distance behind the frontline, were dozens of young martyrdom seekers, each eager to engage the enemy once called. A flurry of activity engulfed the bullet-battered neighbourhood as sporadic gunfire resonated across the empty streets and a countless number of vehicle-mounted weapons, such as ZU-23 and B-10, Dhshk, Shilka, were all streaming in and out of the base to take their positions. Upon seeing the long line of military vehicles and artillery, one of the fighters erroneously remarked in amazement: ‘today we will defeat these African invaders.’ Materialistically, all possible preparations were made to defeat the encroaching enemy, but were they enough to achieve victory?

    Just after Asr that day, the battle began in deadly earnest, with ear-splitting explosions and exchange of gunfire reverberating from all corners from the city. A salvo of mortars ripped through the fragile rooftops, grenades exploded with alarming ferocity, tanks bulldozed buildings and reduced them to rubble and a hail of bullets cracked into the shell-bestrewn streets. In the backdrop of such a frenzied atmosphere, the Mujahideen maintained remarkable inner tranquillity, for the believer’s heart is an island of santy in such a setting. They’ve put up a sturdy defence in all corners, necessary arrangements were made to outflank the enemy and the chants of Takbeer were gradually rising above the cruel cacophony of gunfire. But something strange was also happening: all the artillery brought by the Mujahideen failed to fire a single shot! The guns were cleaned, lubricated, loaded and reloaded, but they still recoiled without firing a round and with the bullet jammed in the barrel. The field commanders hopped from trench to trench, through tunnels, in order to analyse and resolve the situation, but all their efforts were in vain.

    ‘Today we will defeat these African invaders’ chimed in the memory of one of the commanders and that’s when they had realised that what was happening was in fact a punishment for a grave sin they had committed: trusting in the abilities of their weapons. The certitude with which the statement was expressed proved to be destructive and no sooner had the Mujahideen recognised the mistake and repented than the very same artillery, almost instantaneously, began firing again. The Mujahideen managed to repel the enemy, but those who’ve uttered the statements also learnt a great lesson: that fortifying the bastion of Eeman must take precedence over building a substantial military arsenal; for military arsenal can never bring victory without Eeman.

    Such is the clear manifestation of Allah’s absolute control over all affairs that the Kuffar have failed to comprehend.

    I’ll comment on this story in a follow-up post: for now, I’d just like to let this very revealing tale hang in the air for your consideration…

    Your comments are most welcome.

    **

    Source:

  • Gaidi Mtaani #3, safe copy available from Jihadology, pp. 12-15

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