Loops, Goals, and Loopholes - I Am a Strange Loop - Douglas R. Hofstadter

I Am a Strange Loop - Douglas R. Hofstadter (2007)

Chapter 4. Loops, Goals, and Loopholes


The First Flushes of Desire

WHEN the first mechanical systems with feedback in them were designed, a set of radically new ideas began coming into focus for humanity. Among the earliest of such systems was James Watt’s steam-engine governor; subsequent ones, which are numberless, include the float-ball mechanism governing the refilling of a flush toilet, the technology inside a heat-seeking missile, and the thermostat. Since the flush toilet is probably the most familiar and the easiest to understand, let’s consider it for a moment.

A flush toilet has a pipe that feeds water into the tank, and as the water level rises, it lifts a hollow float. Attached to the rising float is a rigid rod whose far end is fixed, so that the rod’s angle of tilt reflects the amount of water in the tank. This variable angle controls a valve that regulates the flow of water in the pipe. Thus at a critical level of filling, the angle reaches a critical value and the valve closes totally, thereby shutting off all flow in the pipe. However, if there is leakage from the tank, the water level gradually falls, and of course the float falls with it, the valve opens, and the inflow of water is thereby turned back on. Thus one sometimes gets into cyclic situations where, because a little rubber gizmo didn’t land exactly centered on the tank’s drain right after a flush, the tank slowly leaks for a few minutes, then suddenly fills for a few seconds, then again slowly leaks for a few minutes, then again fills for a few seconds, and so on, in a cyclic pattern that somewhat resembles breathing, and that never stops — that is, not until someone jiggles the toilet handle, thus jiggling the rubber gizmo, hopefully making it land properly on the drain, thus fixing the leak.

Once a friend of mine who was watching my house while I was away for a few weeks’ vacation flushed the toilet on the first day and, by chance, the little rubber gizmo didn’t fall centered, so this cycle was entered. My friend diligently returned a few times to check out the house but he never noticed anything untoward, so the toilet tank kept on leaking and refilling periodically for my entire absence, and as a result I had a $300 water bill. No wonder people are suspicious of feedback loops!

We might anthropomorphically describe a flush toilet as a system that is “trying” to make the water reach and stay at a certain level. Of course, it’s easy to bypass such anthropomorphic language since we effortlessly see how the mechanism works, and it’s pretty clear that such a simple system has no desires; even so, when working on a toilet whose tank has sprung a leak, one might be tempted to say the toilet is “trying” get the water up to the mark but “can’t”. One doesn’t truly impute desires or frustrations to the device — it’s just a manner of speaking — but it is a convenient shorthand.

A Soccer Ball Named Desire

Why does this move to a goal-oriented — that is, teleological — shorthand seem appealing to us for a system endowed with feedback, but not so appealing for a less structured system? It all has to do with the way the system’s “perceptions” feed back (so to speak) into its behavior. When the system always moves towards a certain state, we see that state as the system’s “goal”. It is the self-monitoring, self-controlling nature of such a system that tempts us to use teleological language.

But what kinds of systems have feedback, have goals, have desires? Does a soccer ball rolling down a grassy hill “want” to get to the bottom? Most of us, reflexively recoiling at such a primitive Aristotelian conception of why things move, would answer no without hesitation. But let’s modify the situation just a tiny amount and ask the question again.

What about a soccer ball zipping down a long, narrow roadside gutter having a U-shaped cross-section — is it seeking any goal? Such a ball, as it speeds along, will first roll up one side of the gutter and then fall back to the center, cross it and then roll up the other side, then again back down, and so forth, gradually converging from a sinusoidal pathway wavering about the gutter’s central groove to a straight pathway at the bottom of the gutter. Is there “feedback” here or not? Is this soccer ball “seeking” the gutter’s mid-line? Does it “want” to be rolling along the gutter’s valley? Well, as this example and the previous one of the ball rolling down a hill show, the presence or absence of feedback, goals, or desires is not a black-and-white matter; such things are judgment calls.

The Slippery Slope of Teleology

As we move to systems where the feedback is more sophisticated and its mechanisms are more hidden, our tendency to shift to teleological terms — first the language of goals and then the language of “wishing”, “desiring”, “trying” — becomes ever more seductive, ever harder to resist. The feedback doesn’t even need to be very sophisticated, as long as it is hidden.

In San Francisco’s Exploratorium museum, there is an enclosure where people can stand and watch a spot of red light dancing about on the walls and floor. If anyone tries to touch the little spot, it darts away at the last moment. In fact, it dances about in a way that seems to be teasing the people chasing it — sometimes stopping completely, taunting them, daring them to approach, and then flitting away just barely in time. However, despite appearances, there is no hidden person guiding it — just some simple feedback mechanisms in some circuitry monitoring the objects in the enclosure and controlling the light beam. But the red spot seems for all the world to have a personality, an impish desire to tease people, even a sense of humor! The Exploratorium’s red dot seems more alive than, say, a mosquito or a fly, both of which attempt to avoid being swatted but certainly don’t have any detectable sense of humor.

In the video called “Virtual Creatures” by Karl Sims, there are virtual objects made out of a few (virtual) tubes hinged together, and these objects can “flap” their limbs and thus locomote across a (virtual) flat plane. When they are given a rudimentary sort of perception and a simple feedback loop is set up that causes them to pursue certain kinds of resources, then the driven manner in which they pursue what looks like food and frantically struggle with “rivals” to reach this resource gives viewers an eerie sensation of witnessing primitive living creatures engaged in life-and-death battles.

On a more familiar level, there are plants — consider a sunflower or a growing vine — which, when observed at normal speed, seem as immobile as rocks and thus patently devoid of goals, but when observed in time-lapse photography, seem all of a sudden to be highly aware of their surroundings and to possess clear goals as well as strategies to reach them. The question is whether such systems, despite their lack of brains, are nonetheless imbued with goals and desires. Do they have hopes and aspirations? Do they have dreads and dreams? Beliefs and griefs?

The presence of a feedback loop, even a rather simple one, constitutes for us humans a strong pressure to shift levels of description from the goalless level of mechanics (in which forces make things move) to the goal-oriented level of cybernetics (in which, to put it very bluntly, desires make things move). The latter is, as I have stressed, nothing but a more efficient rewording of the former; nonetheless, with systems that possess increasingly subtle and sophisticated types of feedback loops, that shorthand’s efficiency becomes well-nigh irresistible. And eventually, not only does teleological language become indispensable, but we cease to realize that there could be any other perspective. At that point, it is locked into our worldview.

Feedback Loops and Exponential Growth

The type of feedback with which we are all most familiar, and probably the case that gave it its name, is audio feedback, which typically takes place in an auditorium when a microphone gets too close to a loudspeaker that is emitting, with amplification, the sounds picked up by the microphone. In goes some sound (any sound — it makes no difference), out it comes louder, then that sound goes back in, comes out yet louder, then back in again, and all of a sudden, almost out of nowhere, you have a loop, a vicious circle, producing a terrible high-pitched screech that makes the audience clap their hands over their ears.

This phenomenon is so familiar that it seems to need no comment, but in fact there are a couple of things worth pointing out. One is that each cycling-around of any input sound would theoretically amplify its volume by a fixed factor, say k — thus, two loops would amplify by a factor of k2, three loops by k3, and so on. Well, we all know the power of exponential growth from hearing horror stories about exponential growth of the earth’s population or some such disaster. (In my childhood, the power of exponentials was more innocently but no less indelibly imprinted on me by the story of a sultan who commanded that on each square of a chessboard there be placed twice as many grains of rice as on the previous square — and after less than half the board was full, it was clear there was not nearly enough rice in the sultanate or even the whole world to get anywhere close to the end.) In theory, then, the softest whisper would soon grow to a roar, which would continue growing without limit, first rendering everyone in the auditorium deaf, shortly thereafter violently shaking the building’s rafters till it collapsed upon the now-deaf audience, and then, only a few loops later, vibrating the planet apart and finishing up by annihilating the entire universe. What is specious about this apocalyptic scenario?

Fallacy the First

The primary fallacy in this scenario is that we have not taken into account the actual device carrying out the exponential process — the sound system itself, and in particular the amplifier. To make my point in the most blatant manner, I need merely remind you that the moment the auditorium’s roof collapsed, it would land on the amplifier and smash it to bits, thus bringing the out-of-control feedback loop to a swift halt. The little system contains the seeds of its own destruction!

But there is something specious about this scenario, too, because as we all know, things never get that far. The auditorium never collapses, nor are the audience members deafened by the din. Something slows down the runaway process far earlier. What is that thing?

Fallacy the Second

The other fallacy in our reasoning also involves a type of self-destruction of the sound system, but it is subtler than being smashed to smithereens. It is that as the sound gets louder and louder, the amplifier stops amplifying with that constant factor of k. At a certain level it starts to fail. Just as a floored car will not continue accelerating at a constant rate (reaching 100 miles per hour, then 200, 300, 400, soon breaking the sound barrier, etc.) but eventually levels out at some peak velocity (which is a function of road friction, air resistance, the motor’s internal limits, and so forth), so an amplifier will not uniformly amplify sounds of any volume but will eventually saturate, giving less and less amplification until at some volume level the output sound has the same volume as the input sound, and that is where things stabilize. The volume at which the amplification factor becomes equal to 1 is that of the familiar screech that drives you mad but doesn’t deafen you, much less brings the auditorium crashing down on your head.

And why does it always give off that same high-pitched screeching sound? Why not a low roar? Why not the sound of a waterfall or a jet engine or long low thunder? This has to do with the natural resonance frequency of the system — an acoustic analogue of the natural oscillation frequency of a playground swing, roughly once every couple of seconds. An amplifier’s feedback loop has a natural oscillation frequency, too, and for reasons that need not concern us, it usually has a pitch close to that of a high-frequency scream. However, the system does not instantly settle down precisely on its final pitch. If you could drastically slow down the process, you would hear it homing in on that squealing pitch much as the rolling soccer ball seeks the bottom of the gutter — namely, by means of a very rapid series of back-and-forth swings in frequency, almost as if it “wanted” to reach that natural spot in the sonic spectrum.

What we have seen here is that even the simplest imaginable feedback loop has levels of subtlety and complexity that are seldom given any thought, but that turn out to be rich and full of surprise. Imagine, then, what happens in the case of more complex feedback loops.

Feedback and Its Bad Rap

The first time my parents wanted to buy a video camera, sometime in the 1970’s, I went to the store with them and we asked to see what they had. We were escorted to an area of the store that had several TV screens on a shelf, and a video camera was plugged into the back of one of them, thus allowing us to see what the camera was looking at and to gauge its color accuracy and such things. I took the camera and pointed it at my father, and we saw his amused smile jump right up onto the screen. Next I pointed the camera at my own face and presto, there was I, up on the screen, replacing my father. But then, inevitably, I felt compelled to try pointing the camera at the TV screen itself.

Now comes the really curious fact, which I will forever remember with some degree of shame: I was hesitant to close the loop! Instead of just going ahead and doing it, I balked and timidly asked the salesperson for permission to do so. Now why on earth would I have done such a thing? Well, perhaps it will help if I relate how he replied to my request. What he said was this: “No, no, no! Don’t do that — you’ll break the camera!”

And how did I react to his sudden panic? With scorn? With laughter? Did I just go ahead and follow my whim anyway? No. The truth is, I wasn’t quite sure of myself, and his panicky outburst reinforced my vague uneasiness, so I held my desire in check and didn’t do it. Later, though, as we were driving home with our brand-new video camera, I reflected carefully on the matter, and I just couldn’t see where in the world there would have been any danger to the system — either to the camera or to the TV — if I had closed the loop (though a priori either one of them would seem vulnerable to a meltdown). And so when we got home, I gingerly tried pointing the camera at the screen and, mirabile dictu, nothing terrible happened at all.

The danger I suppose one could fear is something analogous to audio feedback: perhaps one particular spot on the screen (the spot the camera is pointing straight at, of course) would grow brighter and brighter and brighter, and soon the screen would melt down right there. But why might this happen? As in audio feedback, it would have to come from some kind of amplification of the light’s intensity; however, we know that video cameras are not designed to amplify an image in any way, but simply to transmit it to a different place. Just as I had figured out in the calm of the drive home, there is no danger at all in standard video feedback (by the way, I don’t know when the term “video feedback” was invented, nor by whom; certainly I had never heard it back then). But danger or no danger, I remember well my hesitation at the store, and so I can easily imagine the salesperson’s panic, irrational though it was. Feedback — making a system turn back or twist back on itself, thus forming some kind of mystically taboo loop — seems to be dangerous, seems to be tempting fate, perhaps even to be intrinsically wrong, whatever that might mean.

These are primal, irrational intuitions, and who knows where they come from. One might speculate that fear of any kind of feedback is just a simple, natural generalization from one’s experience with audio feedback, but I somehow doubt that the explanation is that simple. We all know that some tribes are fearful of mirrors, many societies are suspicious of cameras, certain religions prohibit making drawings of people, and so forth. Making representations of one’s own self is seen as suspicious, weird, and perhaps ultimately fatal. This suspicion of loops just runs in our human grain, it would seem. However, as with many daring activities such as hang-gliding or parachute jumping, some of us are powerfully drawn to it, while others are frightened to death by the mere thought of it.

God, Gödel, Umlauts, and Mystery

When I was fourteen years old, browsing in a bookstore, I stumbled upon a little paperback entitled “Gödel’s Proof”. I had no idea who this Gödel person was or what he (I’m sure I didn’t think “he or she” at that early age and stage of my life) might have proven, but the idea of a whole book about just one mathematical proof — any mathematical proof — intrigued me. I must also confess that what doubtlessly added a dash of spice to the dish was the word “God” blatantly lurking inside “Gödel”, as well as the mysterious-looking umlaut perched atop the center of “God”. My brain’s molecules, having been tickled in the proper fashion, sent signals down to my arms and fingers, and accordingly I picked up the umlaut-decorated book, flipped through its pages, and saw tantalizing words like “meta-mathematics”, “meta-language”, and “undecidability”. And then, to my delight, I saw that this book discussed paradoxical self-referential sentences like “I am lying” and more complicated cousins. I could see that whatever Gödel had proved wasn’t focused on numbers per se, but on reasoning itself, and that, most amazingly, numbers were being put to use in reasoning about the nature of mathematics.

Although to some readers this next may sound implausible, I remember being particularly drawn in by a long footnote about the proper use of quotation marks to distinguish between use and mention. The authors — Ernest Nagel and James R. Newman — took the two sentences “Chicago is a populous city” and “Chicago is trisyllabic” and asserted that the former is true but the latter is false, explaining that if one wishes to talk about properties of a word, one must use its name, which is the expression resulting from putting it inside quotes. Thus, the sentence “ ‘Chicago’ is trisyllabic” does not concern a city but its name, and states a truth. The authors went on to talk about the necessity of taking great care in making such distinctions inside formal reasoning, and pointed out that names themselves have names (made using quote marks), and so on, ad infinitum. So here was a book talking about how language can talk about itself talking about itself (etc.), and about how reasoning can reason about itself (etc.). I was hooked! I still didn’t have a clue what Gödel’s theorem was, but I knew I had to read this book. The molecules constituting the book had managed to get the molecules in my head to get the molecules in my hands to get the molecules in my wallet to… Well, you get the idea.

Savoring Circularity and Self-application

What seemed to me most magical, as I read through Nagel and Newman’s compelling booklet, was the way in which mathematics seemed to be doubling back on itself, engulfing itself, twisting itself up inside itself. I had always been powerfully drawn to loopy phenomena of this sort. For instance, from early childhood, I had loved the idea of closing a cardboard box by tucking its four flaps over each other in a kind of “circular” fashion — A on top of B, B on top of C, C on top of D, and then D on top of A. Such grazing of paradoxicality enchanted and fascinated me.



Also, I had always loved standing between two mirrors and seeing the implied infinitude of images as they faded off into the distance. (The photo was taken by Kellie Gutman.) A mirror mirroring a mirror — what idea could be more provocative? And I loved the picture of the Morton Salt girl holding a box of Morton Salt, with herself drawn on it, holding the box, and on and on, by implication, in ever-tinier copies, without any end, ever.

Years later, when I took my children to Holland and we visited the park called “Madurodam” (those quote marks, by the way, are a testimony to the lifelong effect on me of Nagel and Newman’s insistence on the importance of distinguishing between use and mention), which contains dozens of beautifully constructed miniature replicas of famous buildings from all over Holland, I was most disappointed to see that there was no miniature replica of Madurodam itself, containing, of course, a yet tinier replica, and so on… I was particularly surprised that this lacuna existed in Holland, of all places — not only the native land of M. C. Escher, but also the home of Droste’s famous hot chocolate, whose box, much like the Morton’s Salt box, implicated itself in an infinite regress, something that all Dutch people grow up knowing very well.

The roots of my fascination with such loops go very far back. When I was but a tyke, around four or five years old, I figured out, or was told, that two twos made four. This catchy phrase — “two twos” — sent thrills up and down my spine, because I realized that it involved applying the notion of “two” to itself. It was a kind of self-referential operation, the twisting-back of a concept on itself. Just like a daredevil pilot or rock-climber, I craved more such experiences and riskier ones as well, so I quite naturally asked myself what three threes made. Being too small to figure this mystery out for myself (by making a square with three rows of three dots each, for instance), I asked my mother, that Font of Wisdom, for the answer, and she calmly informed me that it was nine.

At first I was delighted, but it didn’t take long before vague worries started setting in that I hadn’t asked her the right question. I was troubled that both my new phrase and the old phrase contained only twocopies of the number in question, whereas my goal had been to transcend twoness. So I pushed my luck and invented the more threeful phrase “three three threes” — but unfortunately, I didn’t know what I meant by it. And so I naturally turned once again to the All-Wise One for help. I remember we had a conversation about this matter (which, at that tender age, I was convinced was surely beyond the grasp of anyone on earth), and I remember she assured me that she fully understood my idea, and she even told me the answer, but I’ve forgotten what it was — surely 9 or 27.

But the answer is not the point. The point is that among my earliest memories is a relishing of loopy structures, of self-applied operations, of circularity, of paradoxical acts, of implied infinities. This, for me, was the cat’s meow and the bee’s knees rolled into one.

The Timid Theory of Types

The foregoing vignette reveals a personality trait that I share with many people, but by no means with everyone. I first encountered this split in people’s instincts when I read about Bertrand Russell’s invention of the so-called “theory of types” in Principia Mathematica, his famous magnum opus written jointly with his former professor Alfred North Whitehead, which was published in the years 1910-1913.

Some years earlier, Russell had been struggling to ground mathematics in the theory of sets, which he was convinced constituted the deepest bedrock of human thought, but just when he thought he was within sight of his goal, he unexpectedly discovered a terrible loophole in set theory. This loophole (the word fits perfectly here) was based on the notion of “the set of all sets that don’t contain themselves”, a notion that was legitimate in set theory, but that turned out to be deeply self-contradictory. In order to convey the fatal nature of his discovery to a wide audience, Russell made it more vivid by translating it into the analogous notion of the hypothetical village barber “who shaves all those in the village who don’t shave themselves”. The stipulation of such a barber’s existence is paradoxical, and for exactly the same reason.

When set theory turned out to allow self-contradictory entities like this, Russell’s dream of solidly grounding mathematics came crashing down on him. This trauma instilled in him a terror of theories that permitted loops of self-containment or of self-reference, since he attributed the intellectual devastation he had experienced to loopiness and to loopiness alone.

In trying to recover, then, Russell, working with his old mentor and new colleague Whitehead, invented a novel kind of set theory in which a definition of a set could never invoke that set, and moreover, in which a strict linguistic hierarchy was set up, rigidly preventing any sentence from referring to itself. In Principia Mathematica, there was to be no twisting-back of sets on themselves, no turning-back of language upon itself. If some formal language had a word like “word”, that word could not refer to or apply to itself, but only to entities on the levels below itself.

When I read about this “theory of types”, it struck me as a pathological retreat from common sense, as well as from the fascination of loops. What on earth could be wrong with the word “word” being a member of the category “word”? What could be wrong with such innocent sentences as “I started writing this book in a picturesque village in the Italian Dolomites”, “The main typeface in this chapter is Baskerville”, or “This carton is made of recyclable cardboard”? Do such declarations put anyone or anything in danger? I can’t see how.

What about “This sentence contains eleven syllables” or “The last word in this sentence is a four-letter noun”? They are both very easy to understand, they are clearly true, and certainly they are not paradoxical. Even silly sentences such as “The ninth word in this sentence contains ten letters” or “The tenth word in this sentence contains nine letters” are no more problematical than the sentence “Two plus two equals five”. All three are false or at worst meaningless assertions (the second one refers to something that doesn’t exist), but there is nothing paradoxical about any of them. Categorically banishing all loops of reference struck me as such a paranoid maneuver that I was disappointed for a lifetime with the oncebitten twice-shy mind of Bertrand Russell.

Intellectuals Who Dread Feedback Loops

Many years thereafter, when I was writing a monthly column called “Metamagical Themas” for Scientific American magazine, I devoted a couple of my pieces to the topic of self-reference in language, and in them I featured a cornucopia of sentences invented by myself, a few friends, and quite a few readers, including some remarkable and provocative flights of fancy, such as these:

If the meanings of “true” and “false” were switched, this sentence wouldn’t be false.

I am going two-level with you.

The following sentence is totally identical with this one, except that the words “following” and “preceding” have been exchanged, as have the words “except” and “in”, and the phrases “identical with” and “different from”.

The preceding sentence is totally different from this one, in that the words “preceding” and “following” have been exchanged, as have the words “in” and “except”, and the phrases “different from” and “identical with”.

This analogy is like lifting yourself up by your own bootstraps.

Thit sentence it not self-referential because “thit” it not a word.

If wishes were horses, the antecedent clause in this conditional sentence would be true.

This sentence every third, but it still comprehensible.

If you think this sentence is confusing, then change one pig.

How come this noun phrase doesn’t denote the same thing as this noun phrase does?

I eee oai o ooa a e ooi eee o oe.

Ths sntnc cntns n vwls nd th prcdng sntnc n cnsnnts.

This pangram tallies five a’s, one b, one c, two d’s, twenty-eight e’s, eight f’s, six g’s, eight h’s, thirteen i’s, one j, one k, three l’s, two m’s, eighteen n’s, fifteen o’s, two p’s, one q, seven r’s, twenty-five s’s, twenty-two t’s, four u’s, four v’s, nine w’s, two x’s, four y’s, and one z.

Although I received from readers a good deal of positive feedback (if you’ll excuse the term), I also received some extremely negative feedback concerning what certain readers considered sheer frivolity in an otherwise respectable journal. One of the most vehement objectors was a professor of education at the University of Delaware, who quoted the famous behavioral psychologist B. F. Skinner on the topic of self-referring sentences:

Perhaps there is no harm in playing with sentences in this way or in analyzing the kinds of transformations which do or do not make sentences acceptable to the ordinary reader, but it is still a waste of time, particularly when the sentences thus generated could not have been emitted as verbal behavior. A classical example is a paradox, such as “This sentence is false”, which appears to be true if false and false if true. The important thing to consider is that no one could ever have emitted the sentence as verbal behavior. A sentence must be in existence before a speaker can say, “This sentence is false”, and the response itself will not serve, since it did not exist until it was emitted.

This kind of knee-jerk reaction against even the possibility that someone might meaningfully utter a self-referential sentence was new to me, and caught me off guard. I reflected long and hard on the education professor’s lament, and for the next issue of the magazine I wrote a lengthy reply to it, citing case after case of flagrant and often useful, even indispensable, self-reference in ordinary human communication as well as in humor, art, literature, psychotherapy, mathematics, computer science, and so forth. I have no idea how he or other objectors to self-reference took it. What remained with me, however, was the realization that some highly educated and otherwise sensible people are irrationally allergic to the idea of self-reference, or of structures or systems that fold back upon themselves.

I suspect that such people’s allergy stems, in the final analysis, from a deep-seated fear of paradox or of the universe exploding (metaphorically), something like the panic that the television sales clerk evinced when I threatened to point the video camera at the TV screen. The contrast between my lifelong savoring of such loops and the allergic recoiling from them on the part of such people as Bertrand Russell, B. F. Skinner, this education professor, and the TV salesperson taught me a lifelong lesson in the “theory of types” — namely, that there are indeed “two types” of people in this world.