Bill Totten's Weblog

Friday, July 23, 2010

Thinking in Straight Lines

by Dmitry Orlov

ClubOrlov (July 17 2010)

Let's face it, we, the civilized, educated, enlightened part of humanity like things to be straight. Let primitive tribesmen live in picturesque and practical round huts - we require abstract boxes of steel and concrete clad in plate glass, with plenty of nice straight lines, true vertical and horizontal planar surfaces and lots of ninety-degree angles to please the eye. Let these tribesmen spend their days meandering up and down picturesque winding paths laid down by grazing animals - when we build a road, we take a map and apply a ruler to it, and anything that's in the way of that ruler, picturesque or not, must be dynamited and bulldozed because everyone knows that traveling in straight lines is more efficient.

This is good enough for most of us, and so we have come to regard straight lines as natural. In fact, in our world there are just two types of natural phenomena that give rise to straight lines: objects drop or hang down in straight vertical lines, and light beams travel in straight lines; beyond plumb lines and lines of sight everything is either a curve or a squiggle. But since most of our environment is artificial - and crammed full of straight lines and flat horizontal and vertical surfaces - we hardly ever have to confront this fact. Of course, the more scientifically astute among us know that straight lines are but a convenient fiction. We start with a conceptual framework of space that consists of x, y and z axes, and proceed to coerce our observations to fit this framework until the mismatch becomes too obvious to ignore, as with objects dropped from orbit, or with light from far-away galaxies that's so warped by nearby galaxies that the image looks like a smear.

But the fiction is indeed very convenient. To start with, all straight lines are interchangeable and compatible. When we build, we tend to put things either on top of or next to other things, and if they involve straight lines, then no intricate fitting is involved - we can just slap it together any which way and efficiently move on to our next box-building exercise. When we go to a lumberyard, what we buy is not so much wood as straight lines cut through wood. Trees know a lot more than we do about constructing maximally efficient structures out of wood, but we like straight lines, and so we cut through the strongest part of the tree - the concentric rings of wood that make up the trunk - for the sake of making a perfectly straight stick. We could build beautiful, strong, long-lasting structures using round timbers grown to order (as some of us do) but generally we don't because we are mentally lazy, always in too much of a hurry, and have made a fetish of straight lines.

Quite unsurprisingly, our preference for straight lines carries over into the way we think about relationships between things - the mental models we construct of our world. For instance, we consider it a matter of moral rectitude and straight dealing that the price be linearly proportional to the amount of stuff we get: if you pay twice as much, you should get twice as many potatoes. Quantity discounts are acceptable and sometimes expected, but pricing on a curve is generally seen as underhanded. We mistrust curves. Stepwise functions are fine, though, because they are made up of straight line segments. We can put up with having tax brackets, but try taxing people based on a nonlinear formula, and there is sure to be a tax revolt. Were the potato market a product of biological evolution rather than of human artifice, it would perhaps work like this: the price would be some nonlinear function that's directly proportional to the customer's net worth, and the number of potatoes dispensed would be some nonlinear function that's inversely proportional to his net girth. Place your moneybags on one sliding scale, your flab-bags on the other, and some potatoes come out. Such a natural regulatory mechanism would prevent fat, rich gluttons from out-eating the rest of us, but it cannot be, for we have a very strong cultural preference for a simple linear relationship between price and quantity.

Straight lines are popular with grocers and their customers, but nobody loves a straight-edge more than the technocrat. Real-world data generally look like a collection of unique artifacts described by a multitude of qualitatively dissimilar properties and inferred relationships, all fluctuating unpredictably over time in a way that resists the direct application of the straight-edge. Therefore, the first step is to quantify the properties and, if at all possible, ignore the relationships. The next step is to choose just two parameters and to plot these artifacts as points on a piece of graph paper. Then, finally, a technocrat can grab a straight-edge, slap it down on the piece of paper, move it around a bit to find what looks like a good fit, and draw a straight line. Voilà: a linear relationship between two complex phenomena has been found, which can now be treated as real and objective - something that can be shared with one's colleagues and be used as a basis for setting policy - because it involves a straight line which tells that one thing is proportional to some other thing, so that we know what result to expect when we perturb one or the other.

Straight lines are popular with engineers as well. Engineers work hard to design linear, time-invariant systems in which the output is directly proportional to the input any time you like. To them, deviations from linear behavior are defects. They are to us as well: we can hear it if the audio amplifier has nonlinear effects because it distorts the sound, and we can see it if the optics distorts the image. We can tell a straight line from a crooked one without any tools. But the mathematical tools which engineers use when they design these linear time-invariant systems are particularly good, as mathematical tools go. Mathematics can be quite fun as a sort of advanced parlor game for philosophers, but most math is rather problematic from an engineer's point of view. You can describe just about anything using a set of differential equations, but most of the interesting phenomena - the behavior of an airfoil in an airstream, for instance, or the behavior of high-temperature gases in a combustion chamber - produce equations that can't be solved analytically, and can only be approached using numerical methods, using a computer. A mathematical model is constructed, and random numbers are thrown at it to see what comes out. But linear time-invariant systems are described using a singularly well-behaved class of differential equations which do have closed-form, analytical solutions that directly provide answers to design questions, and so engineering students are drilled in them ad naseam and then go on to design and build all kinds of machinery that behaves as linearly as possible, from humble volume knobs to complex aircraft control surfaces. In turn, this well-behaved, predictable machinery allows us to achieve linear effects within the economy: build more stuff - get proportionally more money; spend more money - get proportionally more stuff. But, just as one might suspect, this only works up to a point.

Let us recall: straight lines are but a convenient fiction. There is no physical analogue of a mathematical straight line that goes from minus infinity to plus infinity. The best we can do is use all of our artifice to create relatively short straight line segments. Truth be told, the engineers can't create linear systems; they can only create systems that exhibit linear behavior in their linear region. Outside of that region, nature does what it always does: make crazy curves and squiggles and generally behave in random and unpredictable ways. An example of what happens when we exceed the limits of the linear region from our everyday experience is the phenomenon of overloading an audio amplifier. The resulting effect is called clipping, and it sounds like a particularly unpleasant, harsh, grating noise. There are only two solutions: turn down the volume (return to the linear region), or get a more powerful amplifier.

In the economic realm, the effects of exceeding the limits of the linear region can be even more unpleasant. While within that region, building more houses generates more wealth, but just outside of that region strange things begin to happen rather quickly: house prices crash, mortgages go bad, and building any more houses becomes a singularly bad idea. In the linear region, having more money makes you richer, in the sense of being able to buy more stuff, but outside of that region one is forced to realize that since most money has been loaned into existence, it is in fact composed of debt, and once this debt goes bad, no matter how good your net worth looks on paper you are still facing destitution, greatly exacerbated by the fact that you are out of practice when it comes to being poor. In the linear region, investing more money in energy production produces more energy, but just outside that region it produces less energy, and may also inadvertently destroy entire industries and ecosystems.

If linearity is a fiction that is only useful up to a point, then what about time-invariance? Clearly, it too must have its limits. Stepping on the accelerator may produce the same acceleration every time, but the amount of fuel in the tank decreases monotonically until there is none left. When it comes to more complex, dynamic systems - industries, economies, societies - they may continue to respond to external stimuli in a linear and time-invariant manner up to a point, but behind this stable façade their capabilities erode, their resources dwindle, their complexity increases, and beyond a certain point an entirely different process begins: the process of collapse. Such systems generally do not become smaller, spontaneously become less complex or reduce their resource use while continuing to respond to external stimuli in a controlled, linear manner.

But so strong and so deeply ingrained is our habit of thinking in straight lines that often we cannot imagine that we can ever leave the linear region, or, once we do, that we have done so, even when the evidence is staring us in the face. Forensic analyses of airplane crashes have revealed that sometimes, as his last act, the pilot ripped the control console off the cockpit floor - an act that requires superhuman strength - so hard was he pulling back on the yoke to bring up the airplane's nose. I am sure that there are plenty of pilots - in all walks of life - who will prefer to crash, gripping the controls with all their might, gaze fixed on the distant, irrelevant or fictional horizon, than to push the eject button. Their entire life's experience has been confined to the linear region, and so they cannot imagine that it can ever end.

One particularly significant example of this thinking is the belief in Peak Oil, generally expressed as the idea that global oil production already has or will soon reach an all-time peak, and will then gradually decrease over a time span of several decades. Oil depletion is being modeled as a linear function of oil production: a few percent a year, holding more or less steady from one year to the next. At the same time, the use of oil by industrialized societies is often quite usefully characterized as an addiction. Let us exercise this metaphor a bit and see where it takes us. Suppose you have a junkie who has an ever-increasing heroin habit and who has to go out and hustle harder and harder to score his next fix. Now, suppose global heroin production peaks, prices go up, supply dwindles, and our junkie has to start cutting the dose. Not too far along what you then have is a sick junkie, in withdrawal, who cannot go out and hustle for his next fix. And very soon after that you have a collapse of the heroin market because the junkies have all been forced to kick the habit to one extent or another. This disruption of the heroin market, even if temporary, causes heroin production to decrease even faster, production costs and associated risks to go up, and so forth. Beyond a certain point, the heroin market would no longer be characterized as a linear, time-invariant system where the more you pay the more of it you get any time you like, because there would be so little of it around.

Similarly with oil. Right after Hurricane Katrina there was some disruption of gasoline supplies in some of the southern US states. People have written to me to tell me that the result was instant mayhem: society at all levels swiftly stopped functioning. The shortage was temporary and was quickly forgotten, but were it a long-term, systemic shortage, we would no doubt observe all the usual effects: much extra fuel evaporated from topping off fuel tanks and burned from driving around with a full tank and full jerrycans in the trunk, much fuel wasted from driving around looking for gasoline and from idling in long lines at filling stations, a lot of siphoning of gas from tanks and motorists left stranded as a result, a lot of people unable to get to work, and, shortly after that, hoarding, looting and rioting, commerce at a standstill, use of federal troops to restore public order, curfews and limitations on all travel, bank holidays and a balance of payments crisis, and, finally, the general inability to pay for further oil production or imports. All of these disruptions cause oil production to fall even faster, along with all other economic activity, until there is simply not that much demand for the stuff. As much of the global oil industry is idled, drilling rigs, refineries and pipelines fall into disuse and become inoperable. Instead of a nice few-percent-a-year gradual decline, we would have what Douglas Adams would have described as a "spontaneous existence failure".

I am sure that some people would like me to whip out my straight-edge, plot some straight lines and make some projections: What is my price forecast? What production numbers are we talking about, ten or twenty years out? Well, that to me feels like a complete waste of time. I'd rather spend time learning how to train trees for round timber construction. The future is certain to be nonlinear, but I am quite sure that there will be trees in it. The reason I bring this up is that there are a few of pilots out there who I hope will have the presence of mind to push the eject button instead of clutching at the controls with their eyes locked on the artificial horizon.

Bill Totten


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