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[ 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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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.

[Mark Safranski a.k.a. “zen“]

Creativity is a subject that has  interested me, going back to the days long, long ago when I was an art student. Creativity is only mildly correlated with IQ, but like “intelligence”, the deeper you delve into the study of  creativity and creative thinking the more “creativity” looks like a multifaceted, multidimensional and diverse set of capacities, habits and circumstances than it does a single, universal, characteristic or ability.

Creativity has been studied from a neuroscientific, psychological, evolutionary,  behavioral, economic and social perspective but what of creativity”s opposite?  What about Ennui?

From a cognitive perspective, the two may be flip sides of the same coin, note the correlation between highly   creative  people and incidence of depression. It may also be a sign of overuse of certain brain functions, like adrenal exhaustion from an excess of physical and mental stress over a long period of time. Creativity, being in “the flow” is intoxicating but it usually involves peak exertion which accumulates weariness. Exemplary performance in one area can also come at the expense of penalties in another area.

Or perhaps ennui is the natural, cyclical ebb and flow between generative conceptual fertility and barrenness, the brain preparing itself for the next creative “surge” to come?

When are you creative and when are you not?

E.O. Wilson 

Last summer, eminent sociobiologist E.O. Wilson published an article in Harvard Magazine:

On the Origins of the Arts 

….By using this power in addition to examine human history, we can gain insights into the origin and nature of aesthetic judgment. For example, neurobiological monitoring, in particular measurements of the damping of alpha waves during perceptions of abstract designs, have shown that the brain is most aroused by patterns in which there is about a 20 percent redundancy of elements or, put roughly, the amount of complexity found in a simple maze, or two turns of a logarithmic spiral, or an asymmetric cross. It may be coincidence (although I think not) that about the same degree of complexity is shared by a great deal of the art in friezes, grillwork, colophons, logographs, and flag designs. It crops up again in the glyphs of the ancient Middle East and Mesoamerica, as well in the pictographs and letters of modern Asian languages. The same level of complexity characterizes part of what is considered attractive in primitive art and modern abstract art and design. The source of the principle may be that this amount of complexity is the most that the brain can process in a single glance, in the same way that seven is the highest number of objects that can be counted at a single glance. When a picture is more complex, the eye grasps its content by the eye’s saccade or consciously reflective travel from one sector to the next. A quality of great art is its ability to guide attention from one of its parts to another in a manner that pleases, informs, and provokes

This is fascinating.  My first question would be how we could determine if the pattern of degree of complexity is the result of cognitive structural limits (a cap on our thinking) or if it represents a sufficient visual sensory catalyst in terms of numbers of elements to cause an excitory response (neurons firing, release of dopamine, acetylcholine etc. ) and a subsequent feedback loop. Great art, or just sometimes interesting designs exhibiting novelty can hold us with a mysterious, absorbing fascination

Later, Wilson writes:

….If ever there was a reason for bringing the humanities and science closer together, it is the need to understand the true nature of the human sensory world, as contrasted with that seen by the rest of life. But there is another, even more important reason to move toward consilience among the great branches of learning. Substantial evidence now exists that human social behavior arose genetically by multilevel evolution. If this interpretation is correct, and a growing number of evolutionary biologists and anthropologists believe it is, we can expect a continuing conflict between components of behavior favored by individual selection and those favored by group selection. Selection at the individual level tends to create competitiveness and selfish behavior among group members—in status, mating, and the securing of resources. In opposition, selection between groups tends to create selfless behavior, expressed in
greater generosity and altruism, which in turn promote stronger cohesion and strength of the group as a whole 

Very interesting.

First, while I am in no way qualified to argue evolution with E.O. Wilson, I am dimly aware that some biological scientists might be apt to take issue with Wilson’s primacy of multilevel evolution. As a matter of common sense, it seems likely to me that biological systems might have a point where they experience emergent evolutionary effects – the system itself has to be able to adapt to the larger environmental context – how do we know what level of “multilevel” will be the significant driver of natural selection and under what conditions? Or does one level have a rough sort of “hegemony” over the evolutionary process with the rest as “tweaking” influences? Or is there more randomness here than process?

That part is way beyond my ken and readers are welcome to weigh in here.

The second part, given Wilson’s assumptions are more graspable. Creativity often is a matter of individual insights becoming elaborated and exploited, but also has strong collaborative and social aspects. That kind of cooperation may not even be purposeful or ends-driven by both parties, it may simply be behaviors that incidentally  help create an environment or social space where creative innovation becomes more likely to flourish – such as the advent of writing and the spread of literacy giving birth to a literary cultural explosion of ideas and invention – and battles over credit and more tangible rewards.

Need to ponder this some more.

Out of our Minds: Learning to be Creative by Sir Ken Robinson 

Wild Bill Donovan by Douglas Waller  

I am about half finished with the first book by creativity in education guru Sir Ken Robinson, who also has a new book out called, The Element. Technically, I have been reading a large volume of books, articles and research regarding creativity and creative thinking lately for a project, but most of those are academic in nature while Robinson is writing for mainstream audiences. I may or may not review it here, but it is clearly argued and Robinson is an effective popularizer.

The biography of Wild Bill Donovan is timely. If the idiosyncratic and at times improvisational OSS, staffed by gifted amateurs, eccentric adventurers and white-shoe, unapologetically elite WASPS, was something that would be impossible to exist in today’s rancid political climate, there are elements in that legacy that are in short supply in today’s modern and highly technological IC.

Ironically, many of the pioneers in creativity research that developed that field within cognitve psychology in America were themselves disproportionately veterans of the OSS.

The Violent Image by Neville Bolt 

Columbia University Press just sent me a review copy of The Violent Image, by Dr. Neville Bolt of King’s College vaunted War Studies Department.  Initially, I was amused by the colorful book jacket, but flipping through, it belies a very weighty, heavily footnoted, academic exploration of the iterative relationship between propagandistic imagery and insurgency. Even a casual perusal indicates that The Violent Image is a book many readers of ZP will  like to  get their hands on.

From the jacket:

….Neville Bolt investigates how today’s revolutionaries have rejuvenated the nineteenth century “ptopaganda of the deed” so that terrorism no longer simply goads states into overreacting, thereby losing legitimacy. Instead the deed has become a tool to highlight the underlying grievances of communities

A small sampling of some of the section titles:

Strategic Communications:the State
Strategic Communications: the Insurgent
Networks in Real and Virtual Worlds
Images as Weapons
POTD as Insurgent Concept of Operations
Anonymity and Leaderless Revolutions
The Arab Uprisings and Liberation Technology
POTD as Metaphor

Endnotes run slightly over 90 pages and the bibliography tips the scales at 50, for those interested in such things.

Looking forward to reading this and seeing how Bolt presents his case.