Thoughts on Control and Proportion

In the common usage, a system is generally said to be "under control" whenever there is some way to act on the system to produce a predictable outcome. This is a perfectly sensible definition, but it makes clear that control is really a matter of degree. Does "outcome" refer to the ultimate behavior of the system, or only certain of its moving parts? Does "predictable" mean that outcomes are foreseeable one second into the future, or one year into the future, or indefinitely into the future?

There are other interesting questions one can ask that may be particular to a system or a class of systems: When control-actions put together serially or in parallel (or in that awkward hybrid of the two sometimes known as "concurrency") are their cumulative results foreseeable, so that large, structured actions can be composed of smaller ones? Do control-actions have an outcome that is constant with time, or does their behavior change, albeit in a predictable way? Do what degree is control of the system susceptible to irregularities of the environment or noise in the input?

Degree, though, seems to be an essential aspect of the notion of control. The things we are typically concerned with controlling are macroscopic and complicated. In those rare instances where we unequivocally succeed in controlling a physical phenomenon (a working machine is one instance of such a success) behavior of the thing may be quite steady and predictable, but still show susceptibility to abrupt, unexpected failures or malfunctions. Friction, cross-talk, ambient vibrations, waste-heat, leaky gates, freak-accidents, and all their like loom threateningly in the background of any working order. Though such entropic can never be eliminated, a successful machine (at least in all current conventional senses) has a design that somehow subsumes these forces.

The world is a vast and inconceivably complicated place. That anything is predictable or understandable at all is something of a miracle. That humans can produce even dim understanding or very modest instances of control is more miraculous still. In this sense, any given thing in the world, from a pebble to a space shuttle to a low-pressure trough to a working farm to an ocean, analyzed in full, contains a volume of information that is completely beyond the comprehension of even the most brilliant human mind. In essence, the information density of even the tiniest, simplest objects renders them wholly immovable to the human mind.

The idea of control thus hearkens back to Archimedes using a lever to move the whole world. Control is a particular arrangement that gives an agent (i.e. something with a minded purpose) leverage enough to move to the world from one understandable condition to another. This suggests a (very) slightly more formal idea of control as a specific kind of proportion: a thing is "controllable" when there is some arrangement by which a relatively low bandwidth input yields a comparably (very) high bandwidth output, i.e. a state of the system that is foreseen.

This is not necessarily a new way of looking at things, by any stretch. (After all, we can even give old Archimedes some credit, not to mention such big names as Boltzmann or Weiner, who first began asking the modern versions of such questions.) Control is essentially a means of mapping some relatively coarse vision of a complex phenomenon onto its extremely fine-grained reality, and doing so in a way that is suitably robust and structure-preserving. The transistor is the classic example of this, whereby something as complex as a semiconductor can be made to act like a trivial logic function. This view is suggestive of certain interesting avenues of investigation. One has to wonder, for instance, if establishing a regime of control, i.e. designing a machine or proving its properties, is something like playing Michael Barnsley's "Chaos Game": the design ask and answer, over and over again, how do the small things resemble the big things in this picture?

Amped Up Over Tasers: When Technical Details Don't Mix

Adoption of the electrical neuromuscular incapacitation device commonly known as the "Taser" is a contentious issue in law enforcement policy. As with many contemporary issues, the adoption and deployment of the Taser appears to incite a spirited public debate that is, in actuality, not so much a debate as a vicious clash of two different sets of vague and incommensurable intuitions. As should not be surprising, this quagmire of ill-formed but irreconcilable feelings is even more severely exacerbated whenever the works of science and late technology are involved.

In recent local news, an action group calling itself "People for a Taser-Free Columbia" hosted a public discussion on the police department's issuance of Tasers to beat officers. At that event, opponents raised frequent objection to the Taser: it subjects its victims to an electrical potential of 50,000 volts. This quantity, opponents contend, is manifestly unsafe, whence the Taser's status as a non-lethal weapon is questionable. The Columbia Daily Tribune reports that a certain city councilman dismissed this objection by noting that "it's not the voltage that matters, it's the amperage." Exchanges such as these have become typical in American civic discourse: one side glibly cites what appears to be a fact, and the other side, without considering any particular features of the problem, dismisses that fact as irrelevant. Because such exchanges have become typical it is easy to mistake them for serious policy debate, when in fact they are little more than hysterics and posturing on either side. This recent exchange by Taser opponents and the city councilman is a striking illustration of how much worse matters become when technical information is involved, and is often unwittingly used to perpetuate an unproductive argument.

A closer examination reveals how little the participants in this debate are actually saying. (Sadly, it's worth even less than the few sentences we've given it here.) Let's first address the councilman's inane rebuttal.

Electrical current is commonly measured in amperes. On this definition, the councilman seems to assert that the voltage figure quoted by opponents is irrelevant on the basis that it is the quantity of electrical current that actually determines the magnitude of injury. The councilman's objection superficially appears erudite: he correctly distinguishes between current, which measures the rate at which electrical charge flows through a medium, versus voltage, which is a measure of potential energy. Opponents, however, are clearly objecting to the Taser on the basis that its sheer energy output seems to be frighteningly high. Does the councilman's remark actually address the spirit of this concern? According to Ohm's Law, a basic physical principle familiar to anyone who has ever taken a college physics course,

electrical current = electrical potential / electrical resistance

which means in particular that current, and thus 'amperage', is proportional to voltage. Thus a small voltage will produce a small current when applied to a given conductor (e.g. a human subject), and a very large voltage will produce a very large current when applied to the same conductor. The number of amperes experienced by a person shot with a Taser dart thus depends upon the number of volts of electrical potential generated by the device. What the councilman seems to be saying is: "It doesn't matter how many volts the suspect is subjected to, as long as he doesn't suffer too many amps." In light of simple physical laws, such an argument makes no sense. Moreover, it does nothing to address generic fears that Thomas A. Swift's Electric Rifle shoots out a quantity of electricity that is simply too large.

It is possible the councilman actually meant something else by his objection, but I cannot think of any other sensible (or favorable) way to construe the remark. It is also worth noting that the councilman's remark is similar to an aphorism oft quoted in the electrician's trade. In the context of creating an electrical current, an electrician or engineer certainly is concerned with the problem of creating a current using as little voltage as possible, but this is quite a different concern than determining how much current can be applied to a human being without causing injury or death. Let this be a lesson about the danger of repeating aphorisms without clearly understanding their meaning and application. Be careful not to use pre-packaged phrases to cover up a lack of understanding.

The councilman's technical flub was only made worse when the Deputy Chief of Police asserted, according to the same Tribune article, that the Taser induces a current that is "much less than a standard wall socket's output." Considering that, in the United States, electricity is transmitted to homes at 120 volts, it is impossible that a correctly functioning wall outlet could induce a current greater than the peak current produced by a successful Taser deployment. According to Taser International Inc., the device's manufacturer, the TASER X26 delivers a maximum effective voltage of 1200 V across the body of the subject. All things being equal, Ohm's Law entails that the peak current delivered by a relatively low-voltage taser is at least ten times what a wall outlet could deliver.

This flub, however, leads into one of the subtle difficulties of citing figures: there are many to choose from, and it often matters which you choose. One should note that I said the X26 delivers a peak voltage of 1200 V; the one-second baseline average reported by the manufacturer is only 0.76 volts. There is probably a substantial and pertinent debate to be had on whether the peak or the average current output is more meaningful to the issue of the Taser's ability to do mortal harm, or how long a peak must be in order to be physiologically relevant. However, none of these issues were raised by any participants that I am aware of, nor is there any evidence that the Deputy Chief took these matters into account. One could blame the genre of news reporting: there is only so much space, and reporters favor simple questions with short answers. Moreover, the Deputy Chief is a peace officer, not an electrical engineer; he may have little or no knowledge of the Taser's technical details, beyond those necessary to its operation, and is mostly likely citing the nearest available figure.

Nonetheless, newspapers need to be able report news concisely, and police officers need to be able to do their jobs without worrying about a lot of scientific ephemera. What could have been done differently? The reporter could have troubled his or her self to formulate a somewhat more specific question -- and to follow up on fact-checking the answer. The Deputy Chief, for his part, could have given a more generic expression of his confidence in the device's safety. This would sufficiently express his position without giving the misleading and all too often conjured appearance that "this is all very scientific and things are completely under control." Don't cite specifics unless you are certain of specifics.

(Specifically Deputy Chief claimed the Taser's current output to be 0.014 amperes, without specifying this as average or a peak, or giving a source. I could not locate this figure on the manufacturer's website.)

Now, on to the question of the Taser opponents. The attentive reader may have noted that the peak voltage I cited above for the TASER X26 (1200 V) is substantially below the sensational 50,000 V figure. This is because, while the electrical circuitry of the Taser does in fact produce an internal potential of 50,000 V, this voltage is not the voltage applied to the body of a person shocked by a Taser. According to a Taser International fact sheet the Taser's effectiveness is partially due to its ability to administer a shock even in the case that the terminal probes do not make skin contact with a subject, e.g. in the instance that the probes are embedded in exterior clothing and do not reach the skin. The Taser accomplishes this by steadily increasing its internal voltage, up to 50,000 V, until the potential difference is sufficient to produce an electrical arc from the probes to the body of the subject. Much of the current produced by the 50,000 V potential difference, however, is lost in the process of "jumping" across the gap between the probes and the body of the subject. As soon as the potential difference built up within the device is released (by shocking the unfortunate person on the other end), the voltage rapidly drops. The extra voltage within the Taser is thus built up only far enough to overcome any electrical resistance on the probe end and thus to produce a current sufficient to subdue the human subject. (The basic idea at work here is also very succinctly described by Ohm's law.) What this means it that no person subjected to a shock from a correctly functioning Taser experiences anything close to 50,000 volts.

(That is not to say, however, that their experience is a pleasant one.)

Though I respect opponents' interest in protecting the public safety, they would know that the figure they so often quote is actually not pertinent if they had troubled themselves to learn about the issue they're debating. Throwing out a figure as an emotional artifact ("Look how big it is!") fails both rational discourse and impassioned elqoeunce. When facts in general, and numbers in particular, are used this way, it is usually results either from someone's lazy skimming of the available information for the first apparently supporting technical fact, or from the widespread repeating of such a carelessly disjointed fact. The end product is a clumsy admixture of fevered pleading and rote recital. The fact that Taser opponents often cite a technical fact of no relevance to their legitimate concerns diminishes their apparent gravity and opens the door to glib dismissals like the councilman's 'amperage' remark.

It would be nice if policy decisions were made the same way that a scientific question is studied or an engineering problem is solved: a collection of experts would assemble, collect and analyze as much data as possible, render a decision or propose a course of action, and publish the findings for public scrutiny. Even if such a system of governance were possible, however, I think it would be at best naive to expect it at this stage in history. Moreover, human societies have to take into account human passions; these can't be swept under the rug if any kind of peace or social harmony is to be maintained. A mode of governance more scientific than those in existence today would still have to account first for the hopes and fears of the governed, else it would be nothing but a brutal, mechanical autocracy. Putting aside dreams of later and elsewhere, what could be done differently right here and right now?

The episode at People for a Taser-Free Columbia's forum is problematic because both parties are clearly talking past one another, and using misconstrued technical details to do so. The effective output of the Taser and its short- and long-term physiological effects are essential points to understand and take into account. However, neither party seems to be seriously considering these issues, so much as cherry-picking bits and pieces to suit their existing biases. Both parties, however, have valid concerns that do not necessarily lie within the bounds of engineering details or known laws of electricity. Police serve a useful and necessary function any society, but citizens do have a legitimate interest in checking police powers and in dictating what is and what is not acceptable police action. This is a natural source of tension. The idea of a new kind of weapon is viscerally scary to any normal human being, and it is natural that some citizens would be concerned that the police, to whom they have granted a substantial measure of power, use these new weapons carefully and responsibly. On the other hand, early evidence suggests the Taser is a highly effective, non-lethal means for officers to subdue aggressive persons, with the promise to greatly reduce the incidence of serious injury to either police or citizens during arrest scenarios. These are two points of view that surely can be brought to some satisfactory reconciliation; Taser opponents surely would not want to see a greater number of suspects or police officers harmed during arrests, nor would Taser proponents want to see the police issued a weapon that posed unacceptable dangers to public.

To emphasize this last point, consider that in the most controversial Taser episodes in recent history (some of which are also local) are controversial not because an electrical neuromuscular incapacitator was deployed, but because it was deployed under highly questionable circumstances. In one sensationally publicized episode last year, Columbia police used a Taser on a man threatening to jump from the I-70 overpass at Providence Road, causing him to fall 15 feet to the highway median below. A Taser was used in spite of the fact that the man in question threatened nothing more than suicide, in spite of the fact that he was only passively uncooperative in his refusal to move himself to safety, and in spite of the fact that the officer's action quite foreseeably caused the man to fall from this perch, thus causing the very injuries police had sought to prevent the man from visiting on himself. In another episode on August 28 of last year, police in Moberly, Missouri used a Taser against one Stanley Harlan during a routine traffic stop; Mr. Harlan died shortly thereafter. The event has come under scrutiny because Mr. Harlan was Tased at once while lying on the ground. The city of Moberly has since agreed to pay the Harlan family $2.4 million.

Episodes such as these underscore the legitimacy opponents' concerns that the new weapon may be irresponsibly or recklessly deployed. At the same time, it is clear that the harm in these episodes was not essential to the Taser itself, but resulted from improper police conduct, which, it can be convincingly argued, could be remediated by better training and more stringent department policies on Taser use. By contrast, these salient points are lost behind shrill assertions about tens of thousands of volts, or the supercilious amperes that somehow matter more.

In closing, I would like to point out that I have assumed that Taser International Inc.'s technical data is all complete and correct, and that I have no reason to believe otherwise. However, this is a crucial assumption. The skeptical citizen should remember that every manufacturer has a large material interest in the perceived safety of their product and so is not necessarily an impartial judge of potential dangers said product may pose to the public.

The lesson in all of this is that we would do well to find a better way to express and legitimize the seemingly vague and unquantifiable hopes and fears from which our views originate. It's more communicative to say, "I'm afraid the police," than, "According to my calculations, the Gizmotron 3000 shoots out too many volts!" We need to be willing to acknowledge that we're human beings, with human hopes and human fears that need to be openly acknowledged and should be respected without qualification. At the same time, we also need to consider hard facts and well analyzed data in making decisions, being careful not to misuse them to give trappings of legitimacy to arguments that more emotional than rational. It's ultimately hopes and fears that bring us together as human beings; we should use the facts to harmonize rather than to obfuscate our deeper motives.

Is Truth-Value A Strange Attractor?

"What is Truth?"

Formal logic is interesting because it captures both the essential qualities and the essential deficiencies of how human beings think. We have a natural tendency to draw sharp lines and draw (frequently binary) distinctions, while nature tends force us to revise the boundaries we use to draw human-navigable maps of the world. Logic is pragmatic: it makes sense, and yields results. However, like all practical expedients, logic is quite fallible, and often entails subtle complexities even in the pursuit of relatively simple goals. Some authors [3] have characterized logic as a way of thinking about thinking. This is a very interesting view. It means that if we soberly and seriously attend to what goes on in our logical constructions, we may learn something about how we think and what our thinking can and can't and tell us. This, however, means we must understand logic as a human construct with both human relevance and human imperfections.

There has been some excellent writing by some superb minds on the subject of formal logic, its relevance and its connection to informal logic, and I leave it to better experts than to elucidate this matter [5]. Instead, because I try to write as much as possible to the level of the lay-person, I would like to give a few simple and informally constructed examples to give the reader a flavor of basic logic, as a lead-in to a somewhat surprising, and unexpectedly colorful analogy between the abstractions of logic and the deceptively simple behavior of a certain class of phenomena that are both intuitively sensible and concretely physical.

Binary distinctions are everywhere in human thought: yes and no, up and down, light and dark, before and after, inside and outside, present and absent. They're a basic staple of how we see the world. It turns out, they're also a very efficient way to encode and store a lot of information. Consider the familiar game "Twenty Questions", where one player thinks of something and the other players try to determine what it is the first player is thinking of by asking him or her a series of no more than twenty yes-no questions. A few years back, some enterprising folks manufactured and marketed a electronic version of this game, packaged in a unit small enough to fit in the palm of one's hand. The game, and its clones, seemed to be astonishingly skilled at guessing what human players were thinking of -- provided, of course, they did not change the thought-of object midway through the game, or choose something extremely specific or idiosyncratic. The secret to the little gadget's success, however was no different than the strategy commonly used by human players: ask very broad questions at the start (e.g. "Is it an animal?"), and gradually narrow the scope until the set of possibilities is small enough to allow successful guessing (e.g. "Is it a cat?").

It appears that a large class of familiar things (and even many unfamiliar things) can be identified by a series of yes-no questions. This is, for instance, why Twenty Questions is not too hard to win, and not even too hard, with the help of modern technology, to implement as an electronic circuit. This observation is also at the heart of classical logic. In such traditional systems of logic, every proposition is either true or false (No exceptions!) and propositions can be connected using a few simple operators to express the truth or falsity of more complex expressions. In this context, 'operator' is perhaps an over-glorified word. The operations I am referring to correspond (tellingly) to very common words that are staples of ordinary reasoning about everyday things: 'and', 'or', and 'not'. Letting '1' stand for 'true' and '0' stand for 'false', we can succinctly express 'and', 'or', and 'not' in the following tables:

AND	0	1
0	0	0
1	0	1

---

OR	0	1
0	0	1
1	1	1

---

NOT
0	1
1	0

These correspond to basic intuition: if p and q are statements about something, p AND q is true only if the truth of p coincides with the truth of q. For instance, suppose p stands for 'eats grass' and q stands for 'says moo'. If we apply p AND q to a cow, then p AND q = 1 certainly, since we've seen cows eating grass, whence p = 1, and since everybody knows the cow says "moo", whence q = 1 as well. On the other hand, if we apply p AND q to a sheep, we have p AND q = 0, since sheep eat grass but generally have other things to say.

We don't, however, have to restrict ourselves to just conjunctions, disjunctions, and negations of simple true-false statements; we can use 'and', 'or', and 'not' to connect formulas to other formulas. Recursively, if P is any (arbitrarily complex!) formula, and Q is any other formula, we can connect them using any of our operators to get a new formula whose value depends on the respective values of P and Q in a way that respects our simple truth-tables above. The basic construction of logical formulae thus uses only very simple building-blocks; the formulae themselves, however, can become huge and immensely complicated.

An example is instructive. Suppose I come to you and say "I'm a secret super-spy!" Your first inclination might be, "Well, if it looks like a spy and it acts like a spy, it's a spy." That is, you might represent your belief that I'm a spy by the formula:

let
p = "looks like a spy"
q = "acts like a spy"
in
p AND q

However, after a while, you might think to youself, "Gee, I've never met a real spy before, so don't really know how a spy looks or acts", so you decide to refine your idea of the situation a little further. Spies keep a lot of secrets, so you decide that if I don't act secretively enough, I'm probably not a spy, or I'm not a very good spy, or I must really trust you to keep my secrets. You also decide that spies are pretty busy working for someone, and so I need to be off doing spy things as frequently as possible, and not ordinary stuff, and so your idea grows:

let
p = "looks like a spy"
q = "acts like a spy"
r = "is secretive"
s = "really trusts you"
t = "is busy"
in
(p AND q AND (r OR s) AND t)

After some further reflection, though, you realize that it's also possible that I'm deep undercover, and so even though I might be part of some super-secret operation, I might be going to great lengths to appear as if I'm leading an ordinary life so I don't blow my cover That means that either I'm deep undercover or I'm not telling the truth about being a spy, and so things get even more complicated:

let
p = "looks like a spy"
q = "acts like a spy"
r = "is secretive"
s = "really trusts you"
t = "is busy"
u = "is deep under cover!"
v = "is telling a tall tale ..."

in
(p AND q AND (r OR s) AND t AND (u OR (NOT v)))

If I regale you with tales of super-spy exploits, you'll have even more information that you'll have to take into account: if I say I was part of a secret plot to blow up Professor Nightmare's death-ray on Flaming Death Island, then that means that either I was part of a super-awesome adventure and it didn't make the news and death-rays exist, or I'm telling you a tall tale, in which case maybe I'm not trustworthy and I'm making the whole thing up:

let
p = "looks like a spy"
q = "acts like a spy"
r = "is secretive"
s = "really trusts you"
t = "is busy"
u = "is deep under cover!"
v = "is telling a tall tale ..."
x = "super awesome adventure!"
w = "none of the exciting news is ever fit to print"

in
(p AND q AND (r OR s) AND t AND (u OR (NOT v)) AND (x OR (w AND (NOT v))))

As you can see, things may get arbitrarily complicated, and as your idea of me as the super-spy depends upon more and more variables, you find your belief pulled ever more chaotically back and forth between amazement and incredulity. However, I don't have to cook up an incredible story in order to exhibit an instance of a phenomenon with simple parts and simple rules that nonetheless behaves in strange and wildly unpredictable ways.

Sitting on the Fence

"Chaos theory" broadly refers to a large area of research in mathematical physics that originated in the 1960s and flowered in the 1970s and 1980s as science turned its attention to physical systems that exhibit large changes in response to small variations. Like logic, chaos and non-linear dynamics are an active area of study with their own deep and fascinating literature, and so I leave it to those more accomplished to exposit their virtues and mysteries. (The interested reader might consult [4] for an informal but very readable overview of the field, and [6] for a more formal but equally readable presentation of the basic mathematics.) Instead, I would like to borrow one very simple device from the field, in the hopes that perhaps it leads us to a interesting analogy.

A physical system is typically said to be bistable if always tends toward one of two stable states as time passes. Such systems are interesting to non-linear dynamicists because, although they exhibit stability after enough time has elapsed, it is often very difficult to predict which state the system will ultimately end in. A classic example is a ball perched on a very thin divider, as in:





Common experience should be enough to convince the reader that the ball will always fall to one side of the divider, or the other. If the divider is relatively wide -- almost but not quite wide enough to allow the ball to be balanced -- the experimenter should be able to make fairly reliable predictions about which side the ball will fall to when placed. If the ball is sufficiently off-center to cause it to roll to the right or to the left, the imbalance will be visible at the outset. However, if the divider is narrow enough relative to the diameter of the ball, it will be very difficult to predict to which side the ball will fall, no matter how much is taken in placing. (The reader is encouraged to go play with some blocks, and so become really thoroughly convinced.) In this case, the very same physical forces are acting on the ball (e.g. gravity, the normal force exerted by the divider), but slight variations in how the ball is placed will be much harder to detect. As if that didn't make predictions hard enough, the many tiny irregularities in the ambient air currents or in the surface of the ball and the divider have a much larger proportional effect on the motion of the ball than they did when it rested upon a relatively wide divider. By making the divider very narrow relative to the diameter of the ball, a huge number of almost invisibly small variables become relevant to the final outcome, and much smaller inaccuracies in the initial placement of the ball may have a much larger impact on its motion and hence its final state, i.e. whether it comes to rest to the right or the left of the divider.

One interesting thing about this example is that predictions are easy when the relevant variables are few and the forces at work are large and easy to observe, but hard when many variables must be accounted-for and the forces at work obscure. Phrased this way, it doesn't seem excessively imaginative to note the wide-versus-narrow comparison made in the ball example is somewhat like the difference between judging the truth of "x is a cow", which requires relatively little information about relatively few features, and judging the truth of "x is a secret super-spy", which seems to require a great deal of information about very hard to discern features.

Getting the Ball Rolling

Suppose I wanted to construct oracle that answers simple yes-no questions. (Think "Magic 8-Ball", not "Delphi") The ball-and-divider gizmo described above is one very good candidate. All we need to do is let "left" stand for "yes" and "right" stand for "no"; if the ball rolls off the divider to the left, that means the oracle says "yes" to our question, whereas if the ball rolls to the right, that means "no". If we want the oracle's advice (in "yes-no" format, of course), we just utter the appropriate incantation, ask our question, then perch the ball atop the divider and see which side it rolls to. In keeping with the venerable old tradition of superstitious parlor games, we could keep the divider very thin, which would give the ball's motion an appropriately oracular irregularity. (Nobody likes an oracle that always say the same thing.) On the other hand, we could make the divider wide enough that the ball's motion would be easy to predict. (As long as we "clear our minds" sufficiently before playing, getting what we expect might make our oracle more suited to the company of the popular Ouja Board and the Magic 8-Ball, human psychology being what it is.) If our construction was precise and careful enough that we could control which way the ball rolled according to its initial placement, we would have something less like a Magic 8-Ball and more like a transistor.

The transistor is the textbook case of a bistable system. Very loosely speaking, a transistor acts as a material whose overall conductivity is "balanced atop" a semiconductor in a way that can be pushed either to conduct or resist an electrical current. This allows the transistor to be used as a two-state switch. (A light switch is a two-state switch, in that it is generally only "up" or "down".) The importance of semiconducting technology to the development of modern technology cannot be overstated; the invention of the transistor set in motion the explosive advance of the digital computer, which, at its most basic, is nothing more than a very complicated assemblage of two-state switches. Thus, our simple ball-and-divider oracle actually shares and interesting (And not coincidental!) kinship with a basic building block of the computer as we know it.

In its present, very simple state, the ball-and-divider oracle can be used as a machine that computes the answers to exceedingly simple yes-no questions. If we "ask" the oracle to check a statement we know to be false, we place the ball slightly to the right, so that it rolls of the divider onto the "no" side; if our statement is true, we place the ball slightly to the left. At this level, of course, the exercise seems silly: the little gizmo only does what you expect it to do. At the same time, in a world that's full of seemingly random occurrences and surprising things, it is actually extremely noteworthy when something behaves the way we expect it to behave. We can thus use our little oracle as a an external model of our internal judgements.

Nobody's going to be too impressed at a ball that rolls off a wall, but suppose we gave our construction a little more refinement and complexity. Suppose we have at our disposal a team of master craftsmen, and we ask them to modify our oracle as follows:



Essentially, our oracle now has a small replica of itself built onto its left and right sides. Instead of controlling our device by placing the ball, let's also ask our craftsmen to give us some way to modify at will the slope on top of the various dividers, so that we know which side the ball will roll to when it encounters a divider. (Perhaps each divider has a sloped piece that can be snapped on and off the top, so that the direction of motion can be reversed by turning the piece around.) Obviously, the measurements must be (excruciatingly) precise, and the device very carefully constructed, but if all goes according to plan, we can now ask our oracle more complicated questions. But how?

In the original construction, we assigned a truth-value (that is, 'true' or 'false') to each side of the divider. I chose 'left' for 'true' and 'right' for false, but we could have easily chosen the other way. In essence, the original construction corresponds to the simplest logical formula of all, namely, the formula with one variable and no connectives, e.g. just p. However, our new oracle now has two smaller copies of the original constructed into it. We can use this! Suppose I arrange the device so that the ball rolls to the left of the center-most (that is, highest) divider. After it rolls to left, it will fall a short distance and (if the device is correctly constructed) encounter another divider. If this second divider can also be arranged to direct the motion of the ball, we can make it stand for a second statement whose truth depends upon the first. For instance, suppose that we have a statement '(x eats grass) and (x says moo)'. We let the first divider stand for 'x eats grass' and let the two other dividers stand for 'x says moo' (we do need to use them both). If we adopt the same semantics for our two secondary dividers as we did for the first, i.e. if the left side of each stands for 'true' and the right side 'false', our device now computes not just p but p AND q:



It is no mistake that the labels along the bottom of the device correspond to the truth table given for 'and' in the above. We can similarly arrange devices that behave as OR and as NOT:






Now things will really start to take off, provided of course that we have the continued support of our craftsmen. Suppose that our team is able to construct machines with as many dividers as we please -- even up to very huge numbers --- and suppose that these machines have the same stable, predictable, reproducible behavior as the simple ball-and-divider construction we began with. The tremendous importance and difficulty of this stability to the function of the overall machine should not be underestimated and cannot be overstated. Small influences cannot be overlooked, and we must ensure that every possible force is very precisely accounted for in the design and construction of the machine. If we don't, there is no way that our thoughts can follow the bouncing ball -- its motion will be chaotic and random! (This challenge is not unlike tremendous effort that has been required -- and continues to be required -- to design and construct reliable solid-state electronics.) If, however, we can overcome the difficulties of physical construction, our little oracle will have grown into a programmable machine that, given a set of truth-judgments, can evaluate the truth of arbitrarily complex logical formulae. How might this be done?

Suppose we've overcome the construction challenges, and we can add as many dividers as we please. Suppose also that we have a logical formula with N many propositions (e.g. "x eats grass", "x says moo") and a collection of N many truth-judgments about our respective propositions (e.g. "it's true that x eats grass", "it's false that x says moo"). Starting with just one divider in the center of the board, we add 2^k additional dividers for each kth additional proposition. That is, we add 2 additional dividers for the first additional variable, 4 for the second, 8 for the third, and so on. We add the dividers in tiers of the same height, and associate to each divider a collection of dividers with the same height. Thus, for example, if p, q and r are propositions in our formula, we arrange our dividers as:



Note that our first tier, which consists of only a single divider, splits the plane of the device in half. The second tier, consisting of two additional dividers, taken together with the first, splits the plane in quarters. The third tier, consisting of four additional dividers, and taken together with the first and second tiers, splits the plane into eighths. Thus, for each proposition, we take a set of dividers all of which stand lower on the board than the last tier, and place them on the board so that split each segment of free space in half.

Now we want our machine to express some relation between the variables. That is, we want to program our machine with a chosen logical formula. This is done by labeling the slots in the board that lie between each pair of dividers. For instance, we obtained 'p and q' as well as 'p or q', both of which have two propositional variables, by changing the labels along the bottom of the board. If the ball falls into a slot labelled '0', this indicates that our formula expresses something false; if it falls into a slot labelled '1', this indicates something true. The machine is now configured in a way that models some logical formula with N propositional variables. For example:



Our machine is now programmed and ready to go. How do we give it input, that is, how do we ask it to compute the truth of our logical formula given a set of true-false judgments about the variables? This is accomplished by tilting the dorsal surfaces of our dividers, in order to govern which way the ball should roll when it encounters a particular divider. If we stick to our true-left, false-right convention (which we might as well, to keep things simple), we slope a given divider to the left or to the right according to whether the proposition corresponding to its tier is true or false. For example, if 'p' and 'r' are true but 'q' is false:



Now we're ready to go! Once we've gathered all of our information and set up the machine, we set the ball in the middle and just let it roll; the label attached to the ball's final state corresponds to the truth or falsity of our original statement.

(It should be noted how strongly the operation of our machine resembles that of the famous and very real "Plinko" device, which was prominently featured on the popular game show "The Price Is Right". The important difference between Plinko and our machine is that Plinko seemed to purposefully admit a fairly high degree of random behavior, as evidenced by the conspicuous bounciness of the pegs. This seemed to be an important part of both its appeal and its challenge.)

If all goes well (Does it ever?) our machine can successfully automated a potentially very elaborate arrangement of logical judgments. Assuming that the work in setting up the machine is not too difficult or time consuming (which, in our example, it almost certainly would be for all but the most trivially simple formulae), we can now model our basic truth-judgments as we understand them and apply them to very complex situations that would otherwise be humanly impossible to reason through. This is effectively what makes the modern digital computer so powerful and useful: it can apply our own basic "and/or" and "not" intuitions to arrangements that are complicated vastly beyond our own human cognitive abilities. Unlike our Plinko-like machine, however, an electronic computer is much easier to program and will execute much more quickly, though, again, this difference is relative and not at all absolute. Any computer programmer or engineer of even modest experience will most certainly agree. (Despite the computer comparisons, I cannot resist a technical note that our deterministic Plnko-machine is not a Turing Machine, since it lacks a random-access memory.) These differences aside, our imaginary machine is now much more than silly toy: it is a very complicated system that behaves the way we expect it to. Again, given the dear scarcity of things in life that we can expect, this very noteworthy.

In the same way that a bridge allows us to use ordinary human ambulation to cross expanses that we could not traverse on foot, our machine is a tool that allows us to apply our ordinary reasoning to systems whose complexity is far beyond human comprehension. This is all good and fine but (again inviting the reader to go play with some blocks), the bigger bridges get, the harder they are to construct, and the same law of increasing difficulty applies to the size and complexity of the ball-and-stick logic machine.

"Truth is stranger than fiction."

If Newton stood on the shoulders of giants to see as far as he did, it would seem that none of us are in a position to refuse the invitation for a boost above eye-level by someone of greater stature. So, let's push our little logic machine its logical conclusion, using a venerable old trick: limits at infinity.

In the preceding section, we saw that we could use our little logic machine to compute logical statements in as many variables as we pleased, simply by adding enough dividers. As we add variables, a forest of dividers springs up on the plane of our device, becoming (exponentially!) more dense with each variable we add, so that this:



rapidly becomes this:



(Actually, with better artistic skills and more powerful drawing tools, this would
come out something like a collapsed Sierpinski Gasket.)

As the number of dividers increases, the outline of our device rapidly converges to a pair of continuous slopes in opposite directions, both leading down from the same elevated point. A few features, both abstract and concrete, are immediately apparent.

Firstly, the functional details of our machine are now humanly unmanageable. How are we supposed to read infinitely small labels, or manipulate infinitely small parts? The parts do not even need to be infinitely small for the machine to become impractical; they just need to be very small relative to human eyesight. However, the machine is only really interesting if it can handle formulae with lots of variables, which necessarily entails exactly this kind of tortured, inhuman precision. If we wanted badly enough to use the machine, we could construct still other machines to program and use it, etching ball-slots with lasers, or watching the progress through a microscope. Be that as it may, even these methods have limits, and the density of trajectories within the machine grows explosively with each additional variable.

Secondly, the size of the ball that actually runs the machine now matters. An infinitely small ball (that is, one that has zero diameter but somehow "still exists") could be dropped on the machine, somehow falling through the infinitely dense forest of dividers to come to rest in an infinitely small notch, thus evaluating the truth or falsity of a logical statement somehow depending upon an infinite amount of information. Of course, this always mattered, but we glossed over it in the construction. The ball obviously must be large enough to fall through the space between dividers. This means, however, that computing a logical formula with more variables requires adding more dividers and hence requires a small ball. Not only that, the size of the ball must shrink exponentially as the number of variables grows. Basically, there is a vicious dependency between components of our machine. This dependency moreover tightens (Worsens?) as the machine grows in power. The ball we drop will rapidly shrink to the size of a dust-speck as we increase the informational output, no matter whether it starts out the size of an orange or the size of a planet.

There's more. Notice that any sufficiently large ball (really, any ball with a diameter greater than zero), when dropped into our infinitely powerful logic-engine, will appear to roll all the way to the left or all the way to the right. Moreover, because our dividers must be very (i.e. infinitely) small in order to fit onto a finite board, the starting divider which sits at the center will always be very small relative to the diameter of the ball. But that means that our infinite logic-engine, when used with a finite-diameter ball, will actually behave exactly like the very chaotic ball-and-divider construction we started with!

Suppose our board has finitely many dividers, but the space between them is sufficiently small relative to the diameter of the ball. When we try to run our logic engine, the ball will eventually reach a stable state but will fail to fall all the way to the bottom of the board because the spaces below will be too narrow for it to fit. What this means is that, given a ball of a certain spatial extent R, there is a limit to the number variables that a given (finite) logic-machine M can model, and hence to the number of logical formulae it can compute. Moreover, the upper bound on the number of variables in the formulae M can compute is proportional R! Essentially, a finite-variable logic-machine is approximated by an infinite-variable logic machine using a ball with non-zero diameter. What happens in such a case is that the ball simply settles into a rut somewhere between the central divider and one of the edges of the board. Everything below this height corresponds to the "random noise" that is assumed away by the modeled formula.

What does all this mean? Infinite information looks like randomness, and classically logical systems can only model systems with finitely many variables, where the difficulty of model-construction grows exponentially as information about the system is added. Somewhat pretending to understanding, this vaguely resembles a certain theory of Chaitin's [2]. Without pretending, this simple, mechanical analogy greatly resembles familiar human reasoning, in which too many details makes things fuzzy, so that hypotheses that require too much information to evaluate are indistinct from an uninformed assertion of "random" behavior.

In an alternate construction, we might notice that infinitely small labels along the bottom of our infinitely dense board somewhat resemble a Cantor dust or a one-dimensional Julia Set [1]: a ball might roll arbitrarily close to the "true" side of the board but come out false, or arbitrarily close to the "false" side and come out true. Thus life always manages to surprise us, and categorizations always prove intractably blurry about the edges, no matter how logical we choose to be.

Logic thus gives us a bridge beyond some of the limitations of working memory and attention span, which mechanism moreover can extend. However, this extent quickly and easily collapses under its own weight when stretched too far, in exactly the way that a real bridge does. Either the ideas we consider abstractions are very much like our supposedly "concrete" perceptions of the "real world", or supposedly abstract constructions, no matter how empyrean or pure, are subject to the same noisy unpredictability of the physical world we actually inhabit.

---

[1] Barnsley, Michael F. Fractals Everywhere. Academic Press Inc., 1988.

[2] Chaitin, G. J. Algorithmic Information Theory. Cambridge University Press, 1987.

[3] Ernst, Zachary. Free Logic Now! Available here.

[4] Gleick, James. Chaos: The Making of a New Science. Penguin Group, 1987.

[5] Quine, W. V. Philosophy of Logic. Harvard University Press, 1950.

[6] Strogatz, Steven H. Nonlinear Dynamics and Chaos. Perseus Books Publishing, 1994.

When Is It Time to Turn Out the Lights?

I happened to be reading "The Illusion of Conscious Will" by Daniel Wegner (which I highly recommend -- it is an excellent book) and in the text I came across the following excerpt from a book by Julian Jaynes:

Consciousness is a much smaller part of our mental life than we are conscious of, because we cannot be conscious of what we are not conscious of ... How simple that is to say; how difficult to appreciate! It is like asking a flashlight in a dark room to search around for something that does not have any light shining upon it. The flashlight, since there is light in whatever direction it turns, would have to conclude that there is light everywhere. And so consciousness can seem to pervade all mentality when it actually does not.[3]

On reading this, I immediately thought to myself that the solution is, in fact, very easy: turn the flashlight off.

This reminds me of a certain other venerable old quotation:

It is said that things coming in through the gate can never be your own treasures. What is gained from external circumstances will perish in the end. However, such a saying is already raising waves when there is no wind. It is cutting unblemished skin. As for those who try to understand through other people's words, they are striking at the moon with a stick; scratching a shoe, whereas it is a foot that itches. What concern have they for the truth?[2]

This is the interesting thing about carrying a light with you: it is an invitation to explore deep and extremely dark places. This makes me think of the last time I ventured into a cave. (Missouri is, after all, "The Cave State". Because it's like living in a cave?), I went deep enough inside that I needed a light, and shining it about I saw all manner of strange and wonderful things. Still, I could not get past the sense that I was in some sense an anomalous occurrence in that place, if not an intruder outright. It's true, there were things before my eyes that I could perceive and understand, but they were made perceptible and understandable by my own planning and device. It's true that there's genuine information in a beam of light, in both the technical and the colloquial sense of "information". Moreover, such information would not be unavailable otherwise. Even so, investigating what's in the dark by getting rid of the inconveniently dark part seems very unlike the fabled objectivity that classical science strives to attain.

Without admitting romanticism or mysticism, we already know that the Universe contains at least two kinds of knowledge: things that are computable, and things that are not. How many other divisions may there be besides? If we admit that the mind is a physical process and not some magic causeless cause that can create information ex nihilo, there surely must be knowledge that it cannot produce, or even that certain of its subfunctions are unable to produce. (Aphasias are one such interesting and highly celebrated case.) This seems like a little explored and very challenging but very interesting domain for scientific investigation and philosophical inquiry. On the other hand, it might turn out that this is a domain wherein scientific investigation cannot possibly answer the questions we are trying to ask. What then?

We acknowledge that 'knowing' is a physical process, caused by physical processes. Still, we intuitively think of it as a "meta-process", that pertains to certain other processes. The time may yet come when we have to collapse all of the "meta-" distinctions in our modes of thought into a unified whole. A science that successfully transcends all barriers to knowledge will have to remove the distinction between the events in our brains that constitute "knowing" and the events elsewhere that seem, perpetually, to confront, confound and challenge us as human beings.

I realize that this is a very bold and very sweeping proposition, requiring a very unwieldy camel to be threaded through a very delicate needle. Even so, there seems to be a deep conceptual incongruity in our macro-scale world knowledge, and even our personal self-knowledge, that we habitually lean upon but seldom acknowledge. If the vivid sense of identity and "there-ness" that each human being experiences is dependent upon and essentially the same as phenomena in the world at large, why is the feeling of difference and identity so strong? This is the essence of the so-called Hard Problem of cognitive science, and of such puzzling questions as Hofstadter's "identical human copy" problem [1], and no doubt of many troubled personal introspections by many people. I would argue, however, that the question has a much broader, transpersonal significance: Are there facts that are scientifically unknowable? And have we reached a stage in the progress of science where such a question may be regarded as serious and respectable? Perhaps, perhaps not. If not, it seems science is moving more slowly hoped. Even so, this isn't a victory for romanticism either. The romantics, for all their rhapsodizing about essences and sneering at analytic problem-solving, have yet to give any clear, convincing, and accessible explanation of of the key practical point:

When and how do we turn the light off?

---

[1] Hofstadter, Douglas. I Am A Strange Loop. Basic Books, 2007.

[2] Mumon's Introduction to The Gateless Gate, Katsuki Sekida (trans.)

[3] Wegner, Daniel. The Illusion of Conscious Will. MIT Press, 2002.