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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.

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! ]

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:


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


    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?

    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? ]

    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.


    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.


    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.


    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.


    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.

    Ronfeldt’s In-Depth Review of America 3.0

    Monday, September 23rd, 2013


     David Ronfeldt, RAND strategist and theorist has done a deep two-part  review of America 3.0 over at his Visions from Two Theories blog. Ronfeldt has been spending the last few years developing his TIMN analytic framework (Tribes, Institutions [hierarchical], Markets and Networks) which you can get a taste from here  and here or a full reading with this RAND paper.

    David regards the familial structure thesis put forward by James Bennett and Michael Lotus in America 3.0 as “captivating”  and “compelling” for  “illuminating the importance of the nuclear family for America’s evolution in ways that, in my view, help validate and reinforce TIMN”. Both reviews are detailed and should be read in their entirety, but I will have some excerpts below:

    America 3.0 illuminates significance of nuclear families — in line with TIMN (Part 1 of 2) 

    ….Bennett and Lotus show at length (Chapter 2, pp. 29-45) that the nuclear family explains a lot about our distinctive culture and society:

    “It has caused Americans to have a uniquely strong concept of each person as an individual self, with an identity that is not bound by family or tribal or social ties. … Our distinctive type [of] American nuclear family has made us what we are.” (p. 29)And “what we are” as a result is individualistic, liberty-loving, nonegalitarian (without being inegalitarian), competitive, enterprising, mobile, and voluntaristic. In addition, Americans tend to have middle-class values, an instrumental view of government, and a preference for suburban lifestyles. 

    As the authors carefully note, these are generally positive traits, but they have both bright and dark sides, noticeable for example in the ways they make America a “high-risk, high-return culture” (p. 38) — much to the bane of some individuals. The traits also interact in interesting ways, such that Americans tend to be loners as individuals and families, but also joiners “who form an incomprehensibly dense network of voluntary associations” — much to the benefit of civil society (p. 39). 

    In sum, the American-style nuclear family is the major cause of “American exceptionalism” — the basis of our freedom and prosperity, our “amazing powers of assimilation” (p. 53), and our unique institutions:

    “It was the deepest basis for the development of freedom and prosperity in England, and then in America. Further, the underlying Anglo-American family type was the foundation for all of the institutions, laws, and cultural practices that gave rise to our freedom and prosperity over the centuries.” (p. 52)The authors go on to show this for America 1.0 and 2.0 in detail. They also reiterate that Americans have long taken the nuclear family for granted. Yet, very different marriage and family practices are the norm in most societies around the world. And the difference is profoundly significant for the kinds of cultural, social, economic, and political evolution that ensue. Indeed, the pull of the nuclear model in the American context is so strong that it has a liberating effect on immigrants who come from societies that are organized around extended families and clans (p. 55) — an important point, since America is a land of immigrants from all over, not just from Anglo-Saxon nuclear-family cultures.

    ….As for foreign policy, the authors commend “an emerging phenomenon we call “Network Commonwealth,” which is an alignment of nations … who share common ties that may include language, culture and common legal systems.” (p. 260) Above all, they’d like to see the “Anglosphere” take shape. And as the world coalesces into various “global networks of affinity” engaged in shifting coalitions (p. 265), America 3.0 would cease emphasizing democracy-promotion abroad, and “reorient its national strategy to a primary emphasis on maintaining the freedom of the global commons of air, sea, and space.” (p. 263) [UPDATE: For more about the Network Commonwealth and Anglosphere concepts, see Bennett’s 2007 paper here.]

    Read the whole thing here.

    America 3.0 illuminates significance of nuclear families — in line with TIMN (Part 2 of 2)  

    ….Overlaps with TIMN themes and propositions

    Part 1 discussed America 3.0’s key overlap with TIMN: the prevalence and significance of the nuclear family in the American case. This leads to questions about family matters elsewhere. Furthermore, it should be pointed out that there is more to TIMN’s tribal form than the nature of the family. I also spotted several additional thematic overlaps between America 3.0 and TIMN, and I want to highlight those as well. Thus, in outline form, this post addresses:

    • Seeking a fuller understanding of family matters beyond the American case.
    • Gaining a fuller understanding of the tribal/T form.
    • Anticipating the rise of the network/+N form.
    • Recognizing that every form has bright and dark sides.
    • Recognizing the importance of separation among the forms/realms.
    • Recognizing that balance among them is important too.
    • Cautioning against the exportability of the American model.

    After these points, the post ends by summarily noting that America 3.0 is more triformist than quadriformist in conception — but a worthy kind of triformist plus, well worth reading.

    My discussion emphasizes the T and +N forms. Bennett and Lotus also have lots to say about +I and +M matters — government and business — and I’ll squeeze in a few remarks along the way. But this post mostly skips +I and +M matters. For I’m more interested in how America 3.0 focuses on T (quite sharply) and +N (too diffusely). 

    By the way, America 3.0 contains lots of interesting observations that I do not discuss — e.g., that treating land as a commodity was a feature of nuclear-family society (p. 105), and so was creating trusts (p. 112). Readers are advised to harvest the book’s contents for themselves.

    ….Caution about the exportability of the American model: TIMN sharpens — at least it is supposed to sharpen — our understanding that how societies work depends on how they use four cardinal forms of organization. This simplification leaves room for great complexity, for it is open to great variation in how those forms may be applied in particular societies. Analysts, strategists, and policymakers should be careful about assuming that what works in one society can be made to work in another. 

    ….In retrospect it seems I pulled my punch there. I left out what might/should have come next: TIMN-based counsel to be wary about assuming that the American model, especially its liberal democracy, can be exported into dramatically different cultures. I recall thinking that at the time; but I was also trying to shape a study of just the tribal form, without getting into more sweeping matters. So I must have pulled that punch, and I can’t find anywhere else I used it. Even so, my view of TIMN is that it does indeed caution against presuming that the American model is exportable, or that foreign societies can be forced into becoming liberal democracies of their own design.

    Meanwhile, America 3.0 clearly insists that Americans should be wary of trying to export the American model of democracy. Since so much about the American model depends on the nature of the nuclear family, policies that work well in the United States may not work well in other societies with different cultures — and vice-versa. Accordingly, the authors warn,

    “American politicians are likely to be wrong when they tell us that we can successfully export democracy, or make other countries look and act more like the United States.” (p. 24)

    “A foreign-policy based primarily on “democracy-promotion” and “nation-building” is one that will fail more times than not, … .” (p. 254)TIMN is not a framework about foreign policy. But as a framework about social evolution, it may have foreign-policy implications that overlap with those of America 3.0. In my nascent view (notably herehere, and here), the two winningest systems of the last half-century or so are liberal democracy and patrimonial corporatism. The former is prominent among the more-advanced societies, the latter among the less-developed (e.g., see here). As Bennett and Lotus point out, liberal democracy is most suitable where nuclear families hold sway. And as I’ve pointed out, patrimonial corporatism is more attractive in societies where clannish tribalism holds sway. 

    Read the rest here.

    This discussion about America 3.0 and TIMN seems particularly appropriate in light of the need to process, digest and distill the lessons of more than a decade of COIN and counter-terrorism warfare in Iraq, Afghanistan, Pakistan, Yemen and – increasingly- Africa. One of the more difficult aspects of COIN operations has been for American military and diplomatic to decipher the layered relationships and interplay of family honor, tribe, political institution, emerging market and networks in a nation shattered by dictatorship and war like Iraq or to import modern institutions and  a democratic political system in Afghanistan where they had never existed.

    Many of these aspects were opaque and were understood only through hard-won experience (frequently lost with new unit rotation) or still remain elusive to Americans even after ten years of fighting among alien cultures which were also permeated by the sectarian nuances and conflicts of Islam. A religion to which relatively few Americans adhere or know sufficiently about, yet is a critical psychological driver for many of our adversaries as well as our allies.

    Arguably, the eye-opening response of people to America 3.0 indicates we do not even understand ourselves, much less others

    On Time and timeframes

    Monday, July 1st, 2013

    [ by Charles Cameron — 48 hours, Egyptian time, can mean many things, also DoD foresight, next 48 hrs ]

    We seem to have at least four “times” here, ranking from 12 to 72 hours — or zero to 72 if you take the Army’s announcement itself as a sort of starting pistol — and they’re operating, obviously enough, under different frames. The army likes nicely rounded numbers like “48 hrs”, PERT chart thinking gives you the least time available in which to take the first step towards a desired aim, here “12 hours” — and “24 hours” is the latest target time for serious, visible progress to avert “worst case” response preparations. Or something along those lines.

    And “72 hours”? That may be Egyptian elastic time, and thus roughly comparable to Lakota time

    The Lakota view of time was simple. “Time was never a specific minute, but rather spaces of time,like early morning, just afternoon or just before midnight. The real meaning of time could be summed up by the phrase “nake nula waunyelo” loosely translated it means:

    “I am ready for whatever, any place, any time, always prepared”.

    When work needed to be done, people were prepared to work late inthe fields or stay up until 3 am to finish goods to be sold at market.When no work needed to be done, they didn’t work.

    The irony is in the next comment:

    Policy makers saw an opportunity to improve things by installing awestern time ethic and a respect for the clock.

    I don’t know Egypt — but I have spent time “on Lakota time” and don’t wear a watch or carry a phone these days… The piece I drew those quotes from, Lessons from the Lakota: Time lessons for today’s managers, may or may not be useful advice for managers, but its overview-with-graphic of different time systems is worth a quick look. Anthropologists would be able to tell you more about individual cultures, but my point is that differing timeframes are among the major features of different worldviews, and we need to have a decent sense of them when we interface with them.

    So “72 hours” may be an instance of relaxed but purposive time, okay? Which wouldn’t necessarily fit well with starched and pressed military time.


    And here’s the blockbuster:

    According to the Congressional Budget Office’s Analysis of DoD’s Future Years Defense Program from 2013, military foresight time runs five years ahead, while USG foresight time runs to 2030 at least:

    In most years, the Department of Defense (DoD) provides a five-year plan, called the Future Years Defense Program (FYDP), associated with the budget that it submits to the Congress. Because decisions made in the near term can have consequences for the defense budget well beyond that period, the Congressional Budget Office (CBO) regularly examines DoD’s FYDP and projects its budgetary impact over several decades. For this analysis, CBO used the FYDP provided to the Congress in March 2012, which covers fiscal years 2013 to 2017; CBO’s projections span the years 2013 to 2030.

    That’s confidence of a kind… but consider this:

    I know, I know — whatever happens in Egypt “momentarily” may turn out to be no more than an eddy briefly interrupting a larger time-flow in the “mid-term” — a phenomenon I’d like to map nicely with some river graphics one of these days.


    But time? What was it St Augustine said of it?

    What, then, is time? If no one asks me, I know what it is. If I wish to explain it to him who asks me, I do not know.

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