Moulton Lava

Moultonic Musings

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Sunday, October 30, 2005

Disjunction Dysfunction and the Error Function

The polisphere is flush with characters (pejoratively known as Control Freaks) who are enamored of the wonderfulness of the fabled Heaviside Switch Functions, who, alas, don't grok the mathematical subtleties of their true dynamics.

But there is a huge cognitive leap to get to where Poincare and Lorentz got us when they uncovered the essential elements of Chaos Theory.

The Building Blocks of Chaotic Dramas

There is a story to be told here, but I don't quite see how to tell it. The young heroes of Harry Potter know instinctively that to solve the problems at hand in a functional manner, they are going to have to break every rule in the book. And Dumbledore knows that too. But not the other kids or adults.

No one is a better storyteller than JK Rowling. And yet this point seems to be lost on most people.

A useful mathematical insight into terrorism can be found in the angular crookedness of the elementary function known as the Heaviside Switch Function.

For reasons unbeknownst to me, humans since the days of Adam and Eve have become foolishly enamored of the wonderfulness of such dreadful Heaviside Switch Functions.

Perhaps the Heaviside Switch Function is the only mathematical function that a politician can wrap his nimrodic brain around.

But such functions are graceless. As we learn from the study of Nonlinear Dynamics, they typically yield erratic, chaotic, dysfunctional systems. After all, a tiny perturbation in the vicinity of the switching point yields an abrupt and dramatic switch in the value of the function. This sudden reversal is the key feature of systems which produce chaotic, catastrophic, tragic, and disastrous outcomes.

None of this is a new observation. The story of Adam and Eve and the Apple is perhaps the oldest story in our culture about this ubiquitous problem which exemplifies the functional learning disability which has plagued Homo Schleppians since the dawn of civilization.

There may not exist a prayer of a chance of exploring such issues in political venues which are themselves ensnared in the selfsame perennial error in reasoning.

Here, by way of comparison, is the more enlightened, unsung and obscure alternative, known to mathematicians as the Error Function (shown in blue) ...

Error Function (Blue) vs Heaviside Switch Function (Yellow)

As you can see, the Error Function (shown in blue) is continuous and gracefully transitions from one extreme to the other. That's what makes it more functional than Heaviside's graceless and abrupt Switch Function.

Now, how could one work that distinction into a story that could be understood by ordinary readers who are not all that keen on math?

The storytelling problem begins with the observation that the graceful Error Function cannot be expressed with any precision in any text language. One is obliged to turn to mathematics or graphic art. So for starters, one would be obliged to employ the cartoon expressiveness of the graphic novel.

In business, leadership amounts to offering something (of value) to trade at a tempting price. If the price is right, trade takes place by mutual consent.

The Error Function suggests that leadership in business and trade includes the opportunity to adjust the shape of the price function, so as to produce the most vital and vibrant trade economy.

In short, the Error Function is an Engine of Trade, since it establishes the conditions of free trade along an entire axis of consumption (the axis spanning Saintly Good to Demonic Evil).

By comparison, the Heaviside Switch Function is not an Engine of Trade, but an Engine of Drama. That's because there occurs a dramatic switch when some 'player' crosses the boundary (or is merely adjudged to have crossed the imaginary boundary between Saintly Good Conservation of Resources and Demonic Evil Gluttonous Consumption). A variety of classical pathologies arise in the wake of the Heaviside Switch Function, including disagreements and conflict, violence and oppression, injustice and rebellion, corruption and scandal, alienation and inequality, hopelessness and suffering.

Perhaps the enlightened solution is to traffic in stories which present simplified transaction examples arising from the erratic dysfunctionality of the Heaviside Regulatory Model. After all, JK Rowling made a fortune trafficking in such stories. There is clearly a market for them.

Thursday, October 27, 2005

Chaotic Flights of Fancy

System Doom

Theodore Vail and the Bell System

A century ago, a forward-looking entrepreneur named Theodore Vail crafted one of the most successful technology systems ever to serve the American public. Starting from Alexander Graham Bell's invention of the telephone, Vail created a business venture to market the new telecommunications marvel. Eventually, Vail's visionary genius gave rise to the Bell System — an entirely self-contained operation that delivered end-to-end telecommunications services to residential and business customers, as well as to the government.

Elsewhere around the globe, the telephone and telegraph services were operated by the same government bureaucrats who ran the post office. For the longest time, the US was unique in the way telephone service was provided by means of the ubiquitous Bell System.

The jewel of the Bell System was Bell Telephone Laboratories — arguably the premier research and development organization on the face of the planet for much of the Twentieth Century. The Bell System became a model for the application of systems theory, systems engineering, and systems operation. The American telecommunications network had become the envy of the world. Americans enjoyed both the best and the cheapest phone system of any nation.

Then, around 1980, the US government decided that the Bell System had become too successful and too monopolistic. The Justice Department moved to break up the Bell System in the biggest anti-trust case since Standard Oil. And in 1984, AT&T consented to the dismantling of the Bell System. The rule of law was dominant over the concept of system integration.

What is it about high-functioning systems that the government finds so threatening? To gain some further insight into the destiny of large integrated systems, let us turn to another system model — this one from the natural world. Let us examine the solar system.

Henri Poincaré and the Solar System

By 1900, the educated world had become familiar with Newton's laws of gravitational mechanics. The Copernican model of the solar system had become accepted. The calculus was in place, and scientists now had the tools to compute the orbits of the planets around the sun. In this climate, a question arose.

Can it be shown that the solar system is stable, under the assumption of Newtonian gravitational mechanics? If you run a model of the solar system (such models are called orreries), can you show that the orbits of the planets are forever stable, or does the solar system "blow up" by flying apart or spiraling into the sun?

The King of Sweden was sufficiently intrigued by this question that he offered a handsome cash prize to anyone who could rigorously prove that the solar system was stable, under the assumption that the motions of the heavenly bodies is indeed governed by Newton's laws.

Enter Henri Poincaré, who decided to take a crack at this problem and the prize.

Computing the orbits of all the planets and their satellites is a daunting task, so Poincaré began his work with a much simpler model. He began with a trivial solar system that contained one large sun at the center and but one planet, in a nice circular orbit. It was fairly easy for Poincaré to show that after one orbit, the planet returned to exactly the same position. This two-body solar system was periodic, hence stable.

Then Poincaré added a third body. But he didn't just add another planet in a different orbit. Nope nope nope. He did something you might not have expected. He added a comet in an elongated elliptical orbit — one that swung way out to the outer reaches, far from the sun, and then swooped in close to the sun, crossing the orbit of the planet along the way.

And then Poincaré discovered something astounding.

For the longest time, the planet, in its circular orbit, and the comet, in its elongated elliptical orbit, remained in their original orbits. But then, on one pass, something different happened.

Eventually, as the comet crossed the orbit of the planet, it did so while the planet was nearby. When the comet passed close to the planet, the gravitational attraction between them kicked into action, and the orbit of the comet was bent. Thereafter, it assumed a new and different elliptical orbit. And the planet was affected, too. It was deflected a tiny amount from its circular orbit, which now became slightly elliptical, too.

Nothing much interesting happened after that for a very long time, until the comet made another close pass to the planet. And then, once again, its path was bent, sending it into yet another orbit. And the planet's orbit was perturbed again, as well.

So the first thing that Poincaré found was that a three-body solar system isn't periodic. The orbits are not permanent, but change due to the interactions between the bodies.

But wait. There's more...

Poincaré discovered something even more unexpected.

He went back to his three-body solar system and ran the orrery again. Only this time, he started the comet in just a slightly different orbit than the first time. And what he found amazed him. While the same basic phenomenon happened, the history of the solar system was markedly different. After a few close passes, the two versions differed wildly from each other. Even the tiniest, most infinitesimal change eventually leads to dramatically different histories.

Poincaré had discovered what we today call "exquisite sensitivity to initial conditions." What that meant was that the future of the solar system was not predictable, because the tiniest error in specifying the initial conditions eventually leads to completely different outcomes.

But wait. There's more.

Eventually, the comet, when it crossed the orbit of the planet doesn't have a close call. Nope. Eventually it collides with the planet. Not unlike the asteroid that plummeted into the Yucatan some 65 million years ago, thereby killing off the dinosaurs.

So the solar system, is not stable, but chaotic, unpredictable, and doomed.

And all the objects — in this case just three of them — are religiously obeying Newton's laws of celestial mechanics: the inverse square law of gravity, and F = ma.

What Poincaré had discovered in the early years of the Twentieth Century were the roots of Chaos Theory. But it would take another 50 years before that branch of mathematics came of age.

Poincaré presented his results to the King of Sweden, who was so astonished that he awarded Poincaré the prize. Of course Poincaré had proven the opposite of what the King had anticipated. The solar system was not inherently stable under the assumption of Newtonian gravitational mechanics, but unpredictable, chaotic, and doomed.

Poincaré's most astonishing result was that a rule-based system, in which the rules were rigorously followed, was not an orderly system as everyone imagined, but a chaotic system.

Is there no hope? Are we hopelessly trapped in a rule-based system, hopelessly trapped in an unpredictable, chaotic, and doomed system?

Is there is a solution? We turn now to CNN Headline News...

Imagine there is a traveler aboard that ill-fated comet — a trenchant CNN news reporter who is going to keep us informed of the unfolding drama. And who better to do that but our favorite CNN field reporter, Christianne Amanpour.

So here is Christianne, decked out in her flak jacket and fly-away hair, gripping her microphone and breathlessly reporting the trajectory of the ill-fated comet from the depths of space.

In between live reports, Christianne is chatting offline with her colleague and occasional news rival, Wolf Blitzer, safely stationed (as usual) at the Pentagon in his well-pressed suit. Wolf is envious of Christianne's plum assignment — clearly the capstone of her fabulous career. But Christianne is troubled, since this story has a disturbing (if predictable) ending.

"I hope you have your make-up kit so you can look good for your final transmission," says Blitzer.

"Very funny," retorts Christianne, "but I really wish there were some way to change the ending. I can't imagine you having to don a flak jacket and take over for me after I meet my maker."

But Wolf has pity for young Christianne, so he offers a glimmer of hope. "Let me call up this guy at Jet Propulsion Laboratory and see if he has any bright ideas." Wolf places the call and explains the situation to his buddy in rocket science.

"What resources does Christianne have?" inquires the space jock.

"She has her flak jacket, her microphone, her makeup kit that she never uses..."

"What's in the makeup kit, Wolf?" asks the guy at Jet Propulsion Lab.

"Oh, the usual... Face powder, eye shadow, and the aerosol can of hair spray that she never uses."

"Did you just say that Christianne has an aerosol can of hair spray she never uses?"

"Well yeah," says Wolf, "haven't you ever seen her, with her fly-away hair?"

"She's saved," announces the rocket guy.

How is Christianne saved?

The rocket scientist advises Christianne to fire her aerosol can sideways to her direction of travel. Yes, sideways. Christianne does this, thus producing a reactionary thrust, in accordance with Newton's Second Law of Motion. She's got the Force with her, now. By changing the direction of her trajectory, she can steer around the looming earth and avoid a direct collision.

Our CNN news crew has discovered the joys of rocket science.

Christianne lives to report another day, and Wolf can continue to wear his suit at the Pentagon.

The solution involved an understanding of the model of the system in which Christianne is embedded. And then she had to solve that model (with the aid of some rocket scientists at JPL who know how to solve system models). By solving the model, they supplied her with the best practice that was concordant with the laws of celestial mechanics, to change her fate from doom to victorious survival.

Note that the solution did not involve adding more laws to the books of celestial mechanics. It did not involve any legislative or political or judicial acts. It just involved comprehending and solving a system model.

And that is the moral of my story. Rule-based systems are unpredictable, chaotic, and doomed. The solution is to advance to model-based reasoning, and to learn to solve system models for the best practice to avoid the Apocalypse.

Functional Systems vs. Rule-Based Systems

So... How does this help us understand the error in the architecture of human culture?

Our civil culture is rule-based. Most people believe, without question, that rule-based systems are orderly and predictable.

But that belief appears to be in question. The entire branch of mathematics known as Chaos Theory studies the production of chaotic and disorderly systems from the repeated application of a few simple rules. Scientists now know perfectly well that repeated application of rules generates chaos. It's a mathematical fact.

Just as the heavenly bodies obey the discoverable laws of celestial mechanics (which have only been known since the time of Kepler, Copernicus, Galileo, and Newton), the earthly bodies also obey the discoverable laws of human socio-cultural mechanics.

At the time of Copernicus, few souls knew of his work. Indeed, he did not publish his tome until after his death. And I can understand why. Systems thinkers have not been well received by the Powers That Be. After all, look at how the Justice Department viewed the creation of Theodore Vail.

Today there is a new Copernican Revolution afoot. Today there are emerging models of human socio-cultural dynamics that apply to the motions (and emotions) of the earthly bodies. If you run these models forward, the way Poincaré did with Newton's Laws, you find the system is chaotic, unpredictable, and doomed.

But if you solve the system models for best practices (the way the scientists and engineers did at Bell Telephone Laboratories), you find a computable solution. But the solution is generally not rule-based. The solutions are functional and operational, in the sense that they are constructed by solving the underlying system models.

Rule-based systems are too weak to compute the "rocket science" needed to avoid the apocalyptic doom of rule-based cultures. This is a theoretical notion for which there are insufficient stories to illustrate the point. Umberto Eco said, "Whereof we cannot express a theory, we must narrate a story instead." And even if there is a theory, if the theory is arcane, we still need a story to convey the idea to those whose eyes glaze over when theory gets too dense.

The first systems thinkers who gained any insight into the models of human socio-cultural mechanics, and who solved them for best practices ended up founding all the world's major religions. But their followers only got the recommended solution for the times, not the general model itself, nor the methods of reasoning to solve it.

Now we are getting the underlying model itself. And thanks to a handful of brilliant thinkers, we also have the associated methods of reasoning.

In the case of the solar system, and in the case of the Bell System, one can bother to learn the underlying system models, the associated methods of reasoning to compute and solve the models, and the resulting policies and practices that emerge from that analytical process.

In the case of human socio-cultural systems, there is a similar program of research afoot. Much of it is arcane and difficult to access. But a few geniuses have figured out how to tell stories about it. From Fyodor Dostoevsky to Lewis Carroll to JK Rowling, there are some remarkable political allegories which begin to make these abstractions accessible to a general audience. Alas, most theoreticians are lousy story tellers. Where is Umberto Eco when we need him?

Here is an excellent video from BBC Four on Chaos Theory.

Further Reading:

Ivars Peterson, Newton's Clock: Chaos in the Solar System, Freeman, 1993.

Chaos and the Solar System

Chaos — A Mathematical Adventure

More on Henri Poincaré.


Among all critters, there are some who are more skittish than others, and have a more active fear response.

Among the many mechanisms of fear response, the one that I find most intriguing is the Bombesin-mediated fear response. Bombesin is a neuropeptide originally discovered in the skin of the Fire-Bellied Toad, Bombina Bombina. It's a vaso-constrictor which mediates such responses as goose-bumps (goose-flesh), horripilation (hair-raising or bristling), that spine-tingling sensation, and the feeling that one's skin is crawling. Bombesin is the neuropeptide of the cold sweat.

Bombesin is also a Gastrin-Releasing Peptide that acts on the gut to suppress appetite and to suppress the urge to urinate or defecate. Skinny kids (especially those exhibiting Anorexia) are likely to be reacting to Bombesin-induced or Bombesin-mediated reactions to stress.

All of these responses are designed to aid the critter in hiding out from predators and avoiding detection by any and all measures.

By shutting down the normal food-processing metabolic processes, the metabolic wastes are not passed out of the body by normal means, and so the toxins build up in the blood. In Bombina and related amphibian species, these toxins are concentrated and exuded through glands and pores in the skin, giving the creature a noxious fetid odor and nasty, repugnant taste. In you ingest this slime, you get 'Unkenschnupfen' -- an uncontrollable sneezing fit.

The underbelly of these creatures are painted bright red, orange, and yellow. The yellow variety, Bombina Variegata, is called the Yellow-Bellied Toad. The bright colors are a warning to predators that they are nasty tasting and toxic. As a last resort, Bombina will exhibit the so-called Unken Reflex in which it stiffens and curves its spine inward to expose the brightly colored underbelly. It will even flip over on its back to maximize this unmistakeable display.

Humans, of course, inherit all the classic defense mechanisms of our ancestral species, including Bombesin-mediated responses and even the Unken Reflex. If an infant is held by a stranger ("no kin of mine"), it will sometimes arch its back in the Unken Reflex. The obvious solution is to pass the infant back to its own mother, who can comfort and nurse it. A few years ago, I was holding a friend's baby and he exhibited the Unken Reflex. His mommy understood he wanted to be nursed — a function I obviously could not perform.

The alternative solution to the Unken Reflex is to reprise the famous scene with the Pillsbury Dough Boy. You just gently poke the toady little urchin in the belly. This will make it giggle and generate a surge in Oxytocin, which is exactly what the kid wants.

What's odd is when a Pillsbury Belly Poke is perceived as a Act of Terror. In that case, there may be some kind of pathology in the kid that reflects an undiagnosed trauma from early infancy. One way to tell is to smell the kid's skin. If it smells skanky, there is probably a persistant fear-response associated with excess release of Bombesin.

The German word for 'toad' is Unken. It's etymologically related to the English word 'uncanny', meaning "that class of the frightening which leads back to what is known of old and long familiar."

This connection comes to us by way of Sigmund Freud, who wrote a 1919 essay on this subject.

If you translate 'uncanny' back into German, you get 'das unheimliche' which literally means 'unhomely', but which is usually understood to mean the opposite — 'homely'.
We glean this from Freud's 1919 essay, Das Unheimliche...

Freud observes, back in 1919, that almost nothing had been written on the uncanny in relation to aesthetics, although he refered in passing to Ernst Jentsch's 1906 essay "The Psychology of the Uncanny".

Freud mirrors Jentsch's approach to the subject: after an initial concern with the etymology of the uncanny, he collects "all those properties of persons, things, sense impressions, experiences and situations which arouse in us the feeling of uncanniness", then relates these phenomena to the "primary narcissism" of early childhood and "primitive" cultures.

The opening section of the essay examines the etymology of the word 'uncanny', firstly through Greek, Latin, French, Spanish and English definitions, then through a lengthy consideration of the German words 'heimlich' (homely) and 'unheimlich' ('unhomely'). Loosely related to heimisch (native), heimlich can mean familiar, intimate and cherished, but its other definitions shade into apparently opposite significations, such as weird, concealed and secret: "Thus heimlich is a word the meaning of which develops in the direction of ambivalence, until it finally coincides with its opposite." Linguistically, what is heimlich can thus become unheimlich, and for Freud this ambiguity is a constitutive feature of the "special core of feeling" that characterises the uncanny.

The unrevealed secret bears an uncanny resemblance to our own unrevealed secrets. The odd, the weird, the spooky, the mysterious is not so odd or homely after all, for we all know it equally well as the mokita that no one ever speaks of — the squeaking skeleton in the closet.

Listen to the whiny plaintive mating call of the Yellow-Bellied Toad, Bombina Variegata...

And compare that to the slightly more aggressive call of the Fire-Bellied Toad, Bombina Bombina...

Freud goes on in his 1919 essay to provied a lengthy analysis of uncanny experiences drawn from literature, fairy tales, personal experience, clinical cases, metapsychological theory and anthropology.

As above, so below.

Twig Walkingstick advises me that the Toady-Goady Man takes his moniker from the Poison Pen Toad, Dissoglossidae, known for lashing out at its prey with its tart-tipped tongue.

I could have poked you better
    Didn't mean to be Unken
You know that was the last goad
    On my pen.

Wednesday, October 26, 2005

Some Thoughts on Yom Kippur

Speculations on some theological notions about scape-goating rituals.


The Fantastic Flying Scape-Goat for Azazel

Inspiration comes at times from strange and unexpected sources.

One night few weeks ago, a handful of friends who occasionally get together for Bible Study were slated to talk about Baptism — the Christian sin-cleansing ritual that signifies acceptance of the tenets of the faith.

It turned out that none of the practicing Christians in the small group knew how the practice of Baptism arose and came to be a sacred ritual in their faith.

The sect of pre-Christians who wrote the Dead Sea Scrolls were called Early Morning Baptisers because of their ritual morning baths. Even to this day, there are Jewish communities that have 'Mikvah' ritual baths. The Halachic Jewish Law spells out the formalities of the Mikvah Ritual Bath, as well as related rituals for washing the hands and the eating utensils to purify them for ceremonial occasions.

The High Priest (Kohen Gadol) took a prescribed ritual bath before beginning the Day of Atonement (Yom Kippur) ceremony to expiate the collective sins of the community (Lev. 16:4).

The Yom Kippur Ceremony of Leviticus 16 is no longer practiced in modern times, but it's inspiring, to say the least.

In that ceremony (which you can read about in Leviticus 16 or on any number of modern discussions on the Web), the High Priest selected by lot a pair of young, unblemished goats to bear the collective sins of the community. The Goat of Expiation was called the Goat for Azazel.

Of all the obscure terms in the Old or New Testament, none are quite so intriguing or controversial as this name: Azazel.

Scholars are divided on how to interpret the name. Some say it just means 'The Goat Departed' — the Goat Sent Away. In the first English translation (in 1530) of the Hebrew text of Leviticus, William Tyndale translated 'Azazel' as 'scapegoat' — the goat that was set free (escaped from ritual slaughter) in the wilderness.

Other scholars claim that 'Azazel' means 'rugged one of the Lord' or 'rocky ledge on high'. The latter interpretation of 'Azazel' as a place name corresponds to the practice of leading the Goat for Azazel up to a rocky bluff and pitching it over the precipice (in contravention of the original Halachic Law of Leviticus which specifies to simply set it free, alive in the desert).

And yet other scholars say that 'Azazel' was the name of an evil spirit who dwelled in the desert wilderness.

The reason for pitching the hapless goat over the side of a mountain is clear from apocryphal stories. On at least one occasion, the goat bearing the collective sins of the community wandered back into the camp a few days later (presumably still bearing the sins of the community). No doubt this unscripted event mortified the elders, who took pains to revise the ceremony to ensure that the goat would not depart from the ritual script. One can imagine the younger children were somewhat less mortified than the elders, and perhaps a few of them might have thought this departure from the standard script was a riot.

But I digress. The physically fit man (think of Arnold Schwarzenegger) who led the Goat for Azazel into the desert wilderness (and who eventually found he had to schlep the hapless critter all the way up to the high bluff to pitch it over the top) was obliged to launder his clothes and bathe his body before returning to camp (Lev. 16:26), so as to wash away any of the 'sinful contamination' of handling the Goat for Azazel. Clearly we have a Baptism Ritual emerging in concert with the Scapegoat Ritual for the Expiation of Sin.

These stories are not lost on Christian scholars, many of whom suggest this story (both the prescribed ritual and the misadventure of the Goat Who Returned) inspired the planned script of the Passion of Christ. Like the Scapegoat for Azazel who departed from the canonical script and returned to camp after a few days wandering in the desert, Jesus planned to return from the ritual sacrifice, too.

But wait. There's more.

Did the goat of the revised ritual ever survive and return to camp, bruised and stigmatized after being pitched over the side of the mountain? The apocryphal stories are less clear on this twist, but there is some evidence that such stories exist, if only in legend.

The Greek or Latin word for goat is 'capra' (as in Capricorn). But consider the word 'caprice' which has the same root. If you look up the etymology of 'caprice' you find something both curious and astounding. 'Caprice' literally means 'fantastic goat leap'.

Thus we get the inspiring character of Caprice the Fantastic Flying Scape-Goat for Azazel who survives the ordeal and comes back, Christlike, to the community.

Whether it ever actually happened that way in history is unclear. But inspiring stories don't have to be historically accurate to be inspiring.