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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 ]

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.




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.”


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.


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.”


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.


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.


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.

Heavy breathing on the line: The ghost of databases past

Thursday, July 18th, 2013

[dots connected by Lynn C. Rees]



What did Lucius Aemilius Paullus know and when did he know it?

Section 2, Amendment XIV:

Section 2. Representatives shall be apportioned among the several States according to their respective numbers, counting the whole number of persons in each State, excluding Indians not taxed. But when the right to vote at any election for the choice of electors for President and Vice President of the United States, Representatives in Congress, the Executive and Judicial officers of a State, or the members of the Legislature thereof, is denied to any of the male inhabitants of such State, being twenty-one years of age, and citizens of the United States, or in any way abridged, except for participation in rebellion, or other crime, the basis of representation therein shall be reduced in the proportion which the number of such male citizens shall bear to the whole number of male citizens twenty-one years of age in such State.

Article I, Section 2, U.S. Constitution:

Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other Persons. The actual Enumeration shall be made within three Years after the first Meeting of the Congress of the United States, and within every subsequent Term of ten Years, in such Manner as they shall by Law direct.

For working on the paternal genealogy of Howard Ira Milligan, my mother’s father, U.S. Census records have proved to be an important primary source.


Dajjal and Antichrist: the family resemblance

Sunday, May 26th, 2013

[ by Charles Cameron — on the assignment of archetypal roles to members of the British Royal Family ]

For those having trouble distinguishing the Dajjal from the Antichrist, I thought I’d post two screen-caps from a video of the Royal Wedding of Prince William and Kate Middleton, one of whom is identified as the Dajjal:

together with the cover of a book identifying the Antichrist — by his coat of arms — as Prince Charles:

In true conspiracist connect-the-dots fashion, then, the Antichrist is the Dajjal’s father.


Dajjals, Antichrists, Messiahs and Mahdis all function as Rorschach blots on which people project their hopes and fears, associating celebrities and leaders they despise and admire with archetypal instances of the final evil and the final savior.

By now, we should surely have figured out that this tells us more about those making the attributions than it does about the supposed, dreaded or hoped for end of days…

Fire walking and the intensity of apocalyptic arousal

Monday, April 22nd, 2013

[ by Charles Cameron — in response to Steve Engel ]

Phpto credit: MDeeDubroff


My friend Steve Engel wrote regarding the Boston bombers and my elucidation of a Mahdist video that Tamerlan Tsarnaev had seen and “liked”:

Thank you, Charles, for your close examination of nuances that may underlie the actions of people who envision themselves as warriors for the sake of ancient prophecy. Those among us who feel that they dwell in meaningless sorrow make likely customers for purveyors of self-hypnosis–whether of this brand or some other flag-waving, self-justifying cruelty.

I’ve been pondering how to express my reasons for paying particular attention to religious and a fortiori eschatological motives for terror for some time now. The varieties of end times thinking have been an interest of mine for decades, to be sure, and both religion and its specifically end times variants tend IMO to be easily ignored in our so rational post-Enlightenment and high-tech times — so I have both personal and analytic reasons to be keenly interested. But there’s more, and I believe StevE’s comment may be just the thing to pry loose a better explanation than I have given up till now.

I’ll use the well-worn phrase, “work expands to fill the time available” as my starting point.


Turning to StevE’s point about potential recruits to terrorism or other crimes…

It’s easy, it seems to me, to think that just any old ideology would do, that the disgruntled simply pick one and use it as a cover or rationalization — but I suspect that emotions can “intensify to fill the ideology available” to paraphrase the other phrase, and that certain ideologies have structural features equivalent to high ceilings in an architectural space, so that “intensifying to fill the ideology available” can have a certain fierce purity when the ideology is a religious one and pious self-dedication a possibility — even more so when “martyrdom” can be aspired to — and yet more so again where one perceives oneself under divine sanction in the culminating battle of all time, immediately prior to judgment.

I’ve been to two “fire walkings” in my life — the first at Mt Takao, where crowds gather for a yearly ceremony in which one writes one’s sins on a sliver of wood and cast it into the fire, the coals of which which the Yamabushi mountain monks then walk across (see image above), followed by intrepid amateur ascetics…

The second — ah, the second was pitched as an occasion where you could “prove the power of mind over matter” for yourself, and come away from the experience “knowing you had achieved the impossible”. And when the instructor went around the room afterwards and asked people, “Now you know you can do the impossible, what’s next for you?” he got answers like, “I’ll have the courage to ask my boss for a $25 a month raise…”

Times are hard for many of us, and I don’t want to knock either the courage it takes to ask or the value of a $25 monthly raise — but if you’ve just “done the impossible”, is this the most you can ask?

Apocalyptic arousal hopes for more than $25 a month — in most cases it longs for the sudden and immediate reversal of all the good fortune that appears to befall “bad” people right now, and the no less sudden reapportionment of all those blessings on the heads of the “good” people — oneself prominent among them. It shakes the world to its foundations, and it cleanses it.


I’ll let Richard Landes give you a sense of how believing oneself a participant in apocalypse can make the everyday moment deeply significant, and give the “end times we live in” importance beyond measure — with an excerpt from his great book, Heaven on Earth: The Varieties of the Millennial Experience:

For people who have entered apocalyptic time, everything quickens, enlivens, coheres. They become semiotically aroused — everything has meaning, patterns. The smallest incident can have immense importance and open the way to an entirely new vision of the world, one in which forces unseen by other mortals operate. If the warrior lives with death at his shoulder, then apocalyptic warriors live with cosmic salvation before them, just beyond their grasp.


Image source:


Thanks again to Steve Engel for prompting these reflections.

About today, a normal and highly significant Thursday

Thursday, March 28th, 2013

[ by Charles Cameron — remembering that there are two calendars, two times, the secular and the sacred, and that there’s more poetry, more depth of heart and thus intellect too, in the sacred calendar — a truth made known also to popes and monarchs ]

Cardinal Bergoglio, now Pope Francis, washes & kisses the feet of residents of a shelter for drug users, Maundy Thursday, 2008


It’s an extraordinary day, today, for myself as it happens, and more significantly for the world. Specifically, it is significant for poor and powerful alike, for it sets them in a proper relation to one another.

Today is Maundy Thursday, the day on which Christ’s Last Supper, which became and remains the prototype of the Catholic Mass, is remembered. His Last Supper would be closely followed by his crucifixion the next day, Good Friday, and resurrection on Easter Sunday.

After that final supper, Christ made a gesture which initially disturbed his close followers — he washed their feet, then instructed them to do the same.

His gesture repeats itself to this day.


Pope Francis, who as an archbishop had followed this practice by washing and kissing the feet of young people in drug shelters (image above) and AIDS clinics, is breaking with Vatican tradition — the pope washing the feet of priests in some major basilica in Rome — this year, by going to a youth prison today for the ceremony:

The Mass of the Lord’s Supper that Pope Francis will celebrate on Holy Thursday in the chapel of the Casal del Marmo Penitential Institute for Minors will be, by his express desire, very simple. Concelebrating with the Holy Father will be Cardinal Agostino Vallini, vicar general of the Diocese of Rome, and Fr. Gaetano Greco, chaplain of the Institute.

Around 10 girls and 40 boys will take part in the Mass. The Pope will wash the feet of 12 of them, who will be chosen from different nationalities and diverse religious confessions. The youth will also say the readings and the prayers of the faithful. Given the intimate nature of the pastoral visit, journalists will be restricted to the area outside the building and no live coverage will be transmitted.

That’s a slightly formal Vatican announcement of the occasion: “diverse religious confessions” in this case almost certainly includes Muslims and atheists or agnostics. Father Greco, the prison chaplain, is quoted as saying:

Only eight of our residents are Italian: six boys and two girls … The others are all foreigners. And most of them are Muslim. Then there are some who have no religious belief at all. Therefore many of them don’t even know who the Pope is. For this reason too, it was far from easy to explain to them the importance of the Pope’s visit.

A young Neapolitan who has been here for a while came to my help. He gathered them all together, to try to make them understand above all what the Pope’s act, which is an act of love for them, actually meant. I was upset for a moment by the first looks, that were either blank or only faintly curious about my enthusiasm. Then our friend broke the silence with that most classic of Neapolitan expressions: “Maronna mia, o Papa accà!” [good heavens! The Pope here!] and he ran his hand through his hair, his face betraying emotions mingled with happiness. At that very instant all the others, seeing his amazement, realized that it must really be something very special and began to question me. Little by little, I saw their enthusiasm growing.

Father Greco said of his young charges that the Pope’s visit “will make them see that their lives are not bound by a mistake, that forgiveness exists and that they can begin to build their lives again.”


This symbolic gesture, this washing of the feet, does indeed have enormous imaginative power, if we will allow it, to touch the heart and transform our behavior.

I’ve quoted this before, I know: it’s the account given by the man I know who, more than any other, prayed, lived, worked, and saw his great dream and hope accomplished in his lifetime — the overturning of the apartheid regime in S. Africa. For myself, it’s the heart of what he taught me: here, in his own words, Fr Trevor Huddleston CR describes how his own role in that drama began:

On Maundy Thursday, in the Liturgy of the Catholic Church, when the Mass of the day is ended, the priest takes a towel and girds himself with it; he takes a basin in his hands, and kneeling in front of those who have been chosen, he washes their feet and wipes them, kissing them also one by one. So he takes, momentarily, the place of his Master. The centuries are swept away, the Upper Room in the stillness of the night is all around him: “If I, your Lord and Master, have washed your feet, ye ought also to wash one another’s feet.” I have knelt in the sanctuary of our lovely church in Rosettenville and washed the feet of African students, stooping to kiss them. In this also I have known the meaning of identification. The difficulty is to carry the truth out into Johannesberg, into South Africa, into the world.


Earlier today, Queen Elizabeth II will have celebrated “the Royal Maundy” in the chapel of my old College and that of my mentor, Trevor Huddleston, at Oxford, Christ Church — our college chapel is also the cathedral of that great city.

The current Dean Of Christ Church, the Very Reverend Christopher Lewis, gave the BBC this historical detail:

The last time it happened here was in 1644 when Charles I was thrown out of London and welcomed in Oxford.

Here’s the glorious building where the ceremony will have taken place:

The chapel of Christ Church, Oxford, and Cathedral of the Oxford Diocese


The Royal Maundy is celebrated by the monarch giving two purses to local pensioners — 83 men and 83 women this year since HM is 83 years old — a white purse containing sterling silver 1p, 2p, 3p and 4p pieces, and a red purse contained a £5 and a 50p coin commemorating the 60th anniversary of the Queen’s coronation in 1953.

Maundy Thursday commemorates the day of the Last Supper of Jesus Christ with the Apostles. The word ‘Maundy’ comes from the command or ‘mandatum’ by Christ at the Last Supper, to love one another.

The tradition of the Sovereign giving money to the poor dates from the thirteenth century. The Sovereign also used to give food and clothing, and even washed the recipients’ feet. The last monarch to do so was James II.

The recently retired Archbishop of Canterbury, Dr Rowan Williams, gives us a sense of the rich symbolism of the occasion:

It’s a curious survival, the Royal Maundy, but a touching one, looking back to the days when the monarch really was expected to be a bit like a priest for the nation – acting out the great symbols of faith on behalf of everyone. …

And that’s very much what the Royal Maundy is about. What we see today is only a shadow of what used to be done hundreds of years ago, when the monarch would actually do what Jesus did at the Last Supper and wash the feet of a number of poor people. Back in the Middle Ages, this meant that the King was just doing what priests and bishops often did, not only on Maundy Thursday but on many other occasions.

They didn’t all do it because they were lovely humble people – some were, and some definitely weren’t – but because they accepted one great truth that needed repeating over and over again, the one big thing that Christianity had brought into the world of human imagination.

And that was – and is – the truth that power constantly needs to be reminded of what it’s for. Power exists, in the Church or the state or anywhere else, so that ordinary people may be treasured and looked after, especially those who don’t have the resources to look after themselves. The Bible is crystal clear that this is the standard by which the gospel of Jesus judges the powerful of this world.

Not everyone would agree — but I believe the current Queen and the current Pope would — as would his predecessor, Benedict XVI. It bears repeating:

Power exists, in the Church or the state or anywhere else, so that ordinary people may be treasured and looked after, especially those who don’t have the resources to look after themselves.


Here is the text from St John’s gospel — the gospel that focuses its attention at the symbolic level — describing the original event [John 13.1-15]:

He riseth from supper, and laid aside his garments; and took a towel, and girded himself. After that he poureth water into a bason, and began to wash the disciples’ feet, and to wipe them with the towel wherewith he was girded.

Then cometh he to Simon Peter: and Peter saith unto him, Lord, dost thou wash my feet?

Jesus answered and said unto him, What I do thou knowest not now; but thou shalt know hereafter.

Peter saith unto him, Thou shalt never wash my feet.

Jesus answered him, If I wash thee not, thou hast no part with me.

Simon Peter saith unto him, Lord, not my feet only, but also my hands and my head.

Jesus saith to him, He that is washed needeth not save to wash his feet, but is clean every whit: and ye are clean, but not all. For he knew who should betray him; therefore said he, Ye are not all clean.

So after he had washed their feet, and had taken his garments, and was set down again, he said unto them, Know ye what I have done to you? Ye call me Master and Lord: and ye say well; for so I am. If I then, your Lord and Master, have washed your feet; ye also ought to wash one another’s feet. For I have given you an example, that ye should do as I have done to you.


Here finally, for sheer beauty, is the great 16th century Spanish composer Tomás Luis de Victoria‘s First Lamentation for Maundy Thursday, sung by the Tallis Scholars in the chapel of Merton College, Oxford — just around the corner from Christ Church:

Via New Liturgical Movement

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