I Am a Strange Loop

CHAPTER 8

Embarking on a Strange-Loop Safari

Flap Loop, Lap Loop

I’VE already described, in Chapter 4, how enchanted I was as a child by the brazen act of closing a cardboard box by folding down its four flaps in a cyclic order. It always gave me a frisson of delight (and even today it still does a little bit) to perform that final verboten fold, and thus to feel I was flirting dangerously with paradoxicality. Needless to say, however, actual paradox was never achieved.

A close cousin to this “flap loop” is the “lap loop”, shown on the facing page. There I am with a big grin (I’ll call myself “A”), front and center in Anterselva di Mezzo, sitting on the lap of a young woman (“B”), also grinning, with B sitting on C’s lap, C on D’s lap, and so forth, until one complete lap has been made, with person K sitting on my lap. One lap with lots of laps but no collapse. If you’ve never played this game, I suggest you try it. One feels rather baffled about what on earth is holding the loop up.

Like the flap loop, this lap loop grazes paradoxicality, since each of its eleven lap-leaps is an upwards leap, but obviously, since a lap loop can be realized in the physical world, it cannot constitute a genuine paradox. Even so, when I played the “A” role in this lap loop, I felt as if I was sitting, albeit indirectly, on my own lap! This was a most strange sensation.

Seeking Strange Loopiness in Escher

And yet when I say “strange loop”, I have something else in mind — a less concrete, more elusive notion. What I mean by “strange loop” is — here goes a first stab, anyway — not a physical circuit but an abstract loop in which, in the series of stages that constitute the cycling-around, there is a shift from one level of abstraction (or structure) to another, which feels like an upwards movement in a hierarchy, and yet somehow the successive “upward” shifts turn out to give rise to a closed cycle. That is, despite one’s sense of departing ever further from one’s origin, one winds up, to one’s shock, exactly where one had started out. In short, a strange loop is a paradoxical level-crossing feedback loop.

One of the most canonical (and, I am sorry to say, now hackneyed) examples is M. C. Escher’s lithograph Drawing Hands (above), in which (depending on where one starts) one sees a right hand drawing a picture of a left hand (nothing paradoxical yet), and yet the left hand turns out to be drawing the right hand (all at once, it’s a deep paradox).

Here, the abstract shift in levels would be the upward leap from drawn to drawer (or equally, from image to artist), the latter level being intuitively “above” the former, in more senses than one. To begin with, a drawer is always a sentient, mobile being, whereas a drawn is a frozen, immobile image (possibly of an inanimate object, possibly of an animate entity, but in any case motionless). Secondly, whereas a drawer is three-dimensional, a drawn is two-dimensional. And thirdly, a drawer chooses what to draw, whereas a drawn has no say in the matter. In at least these three senses, then, the leap from a drawn to a drawer always has an “upwards” feel to it.

As we’ve just stated, there is by definition a sharp, clear, upwards jump from any drawn image to its drawer — and yet in Drawing Hands, this rule of upwardness has been sharply and cleanly violated, for each of the hands is hierarchically “above” the other! How is that possible? Well, the answer is obvious: the whole thing is merely a drawn image, merely a fantasy. But because it looks so real, because it sucks us so effectively into its paradoxical world, it fools us, at least briefly, into believing in its reality. And moreover, we delight in being taken in by the hoax, hence the picture’s popularity.

The abstract structure in Drawing Hands would constitute a perfect example of a genuine strange loop, were it not for that one little defect — what we think we see is not genuine; it’s fake! To be sure, it’s so impeccably drawn that we seem to be perceiving a full-fledged, true-blue, card-carrying paradox — but this conviction arises in us only thanks to our having suspended our disbelief and mentally slipped into Escher’s seductive world. We fall, at least momentarily, for an illusion.

Seeking Strange Loops in Feedback

Is there, then, any genuine strange loop — a paradoxical structure that nonetheless undeniably belongs to the world we live in — or are so-called strange loops always just illusions that merely graze paradox, always just fantasies that merely flirt with paradox, always just bewitching bubbles that inevitably pop when approached too closely?

Well, what about our old friend video feedback as a candidate for strange loopiness? Unfortunately, although this modern phenomenon is very loopy and flirts with infinity, it has nothing in the least paradoxical to it — no more than does its simpler and older cousin, audio feedback. To be sure, if one points the TV camera straight at the screen (or brings the microphone right up to the loudspeaker) one gets that strange feeling of playing with fire, not only by violating a natural-seeming hierarchy but also by seeming to create a true infinite regress — but when one thinks about it, one realizes that there was no ironclad hierarchy to begin with, and the suggested infinity is never reached; then the bubble just pops. So although feedback loops of this sort are indisputably loops, and although they feel a bit strange, they are not members of the category “strange loop”.

Seeking Strange Loops in the Russellian Gloom

Fortunately, there do exist strange loops that are not illusions. I say “fortunately” because the thesis of this book is that we ourselves — not our bodies, but our selves — are strange loops, and so if all strange loops were illusions, then we would all be illusions, and that would be a great shame. So it’s fortunate that some strange loops exist in the real world.

On the other hand, it is not a piece of cake to exhibit one for all to see. Strange loops are shy creatures, and they tend to avoid the light of day. The quintessential example of this phenomenon, in fact, was only discovered in 1930 by Kurt Gödel, and he found it lurking in, of all places, the gloomy, austere, supposedly paradox-proof castle of Bertrand Russell’s theory of types.

What was a 24-year-old Austrian logician doing, snooping about in this harsh and forbidding British citadel? He was fascinated by paradoxes, and although he knew they had supposedly been driven out by Russell and Whitehead, he nonetheless intuited that there was something in the extremely rich and flexible nature of numbers that had a propensity to let paradox bloom even in the most arid-seeming of deserts or the most sterilized of granite palaces. Gödel’s suspicions had been aroused by a recent plethora of paradoxes dealing with numbers in curious new ways, and he felt convinced that there was something profound about these tricky games, even though some people claimed to have ways of defusing them.

Mr Berry of the Bodleian

One of these quirky paradoxes had been concocted by an Oxford librarian named G. G. Berry in 1904, two years before Gödel was born. Berry was intrigued by the subtle possibilities for describing numbers in words. He noticed that if you look hard enough, you can find a quite concise description of just about any integer you name. For instance, the integer 12 takes only one syllable to name, the integer 153 is pinpointable in but four syllables (“twelve squared plus nine” or “nine seventeens”), the integer 1,000,011 is nameable in just six syllables (“one million eleven”), and so forth. In how few syllables can you describe the number 1737?

In general, one would think that the larger the number, the longer any description of it would have to be, but it all depends on how easily the number is expressible in terms of “landmark” integers — those rare integers that have exceptionally short names or descriptions, such as ten to the trillion, with its extremely economic five-syllable description. Most large numbers, of course, are neither landmarks nor anywhere near one. Indeed, by far most numbers are “obscure”, admitting only of very long and complex descriptions because, well, they are just “hard to describe”, like remote outposts located way out in the boondocks, and which one can reach only by taking a long series of tiny side roads that get ever narrower and bumpier as one draws nearer to the destination.

Consider 777,777, whose standard English name, “seven hundred seventy-seven thousand seven hundred seventy-seven”, is pretty long — 20 syllables, in fact. But this number has a somewhat shorter description: “777 times 1001” (“seven hundred seventy-seven times one thousand and one”), which is just 15 syllables long. Quite a savings! And we can compress it yet further: “three to the sixth plus forty-eight, all times ten cubed plus one” or even the stark “the number whose numeral is six sevens in a row”. Either way, we’re down to 14 syllables.

Working hard, we could come up with scads of English-language expressions that designate the value 777,777, and some of them, when spoken aloud, might contain very few syllables. How about “7007 times 111” (“seven thousand seven times one hundred eleven”), for instance? Down to 13 syllables! And how about “nine cubed plus forty-eight, all times ten cubed plus one”? Down to 12! And what about “thrice thirty-nine times seven thousand seven”? Down to just 11! Just how far down can we squeeze our descriptions of this number? It’s not in the least obvious, because 777,777 just might have some subtle arithmetical property allowing it to be very concisely expressed. Such a description might even involve references to landmark numbers much larger than 777,777 itself!

Librarian Berry, after ruminating about the subtle nature of the search for ever shorter descriptions, came up with a devilish characterization of a very special number, which I’ll dub b in his honor: b is the smallest integer whose English-language descriptions always use at least thirty syllables. In other words, b has no precise characterization shorter than thirty syllables. Since it always takes such a large number of syllables to describe it, we know that b must be a huge integer. Just how big, roughly, would b be?

Any large number that you run into in a newspaper or magazine or an astronomy or physics text is almost surely describable in a dozen syllables, twenty at most. For instance, Avogadro’s number (6×10²³) can be specified in a very compact fashion (“six times ten to the twenty-third” — a mere eight syllables). You will not have an easy time finding a number so huge that no matter how you describe it, at least thirty syllables are involved.

In any case, Berry’s b is, by definition, the very first integer that can’t be boiled down to below thirty syllables of our fair tongue. It is, I repeat, using italics for emphasis, the smallest integer whose English-language descriptions always use at least thirty syllables. But wait a moment! How many syllables does my italicized phrase contain? Count them — 24. We somehow described b in fewer syllables than its definition allows. In fact, the italicized phrase does not merely describe b “somehow”; it is b’s very definition! So the concept of b is nastily self-undermining. Something very strange is going on.

I Can’t Tell You How Indescribably Nondescript It Was!

It happens that there are a few common words and phrases in English that have a similarly flavored self-undermining quality. Take the adjective “nondescript”, for instance. If I say, “Their house is so nondescript”, you will certainly get some sort of visual image from my phrase — even though (or rather, precisely because) my adjective suggests that no description quite fits it. It’s even weirder to say “The truck’s tires were indescribably huge” or “I just can’t tell you how much I appreciate your kindness.” The self-undermining quality is oddly crucial to the communication.

There is also a kind of “junior version” of Berry’s paradox that was invented a few decades after it, and which runs like this. Some integers are interesting. 0 is interesting because 0 times any number gives 0. 1 is interesting because 1 times any number leaves that number unchanged. 2 is interesting because it is the smallest even number, and 3 is interesting because it is the number of sides of the simplest two-dimensional polygon (a triangle). 4 is interesting because it is the first composite number. 5 is interesting because (among many other things) it is the number of regular polyhedra in three dimensions. 6 is interesting because it is three factorial (3×2×1) and also the triangular number of three (3+2+1). I could go on with this enumeration, but you get the point. The question is, when do we run into the first uninteresting number? Perhaps it is 62? Or 1729? Well, no matter what it is, that is certainly an interesting property for a number to have! So 62 (or whatever your candidate number might have been) turns out to be interesting, after all — interesting because it is uninteresting. And thus the idea of “the smallest uninteresting integer” backfires on itself in a manner clearly echoing the backfiring of Berry’s definition of b.

This is the kind of twisting-back of language that turned Bertrand Russell’s sensitive stomach, as we well know, and yet, to his credit, it was none other than B. Russell who first publicized G. G. Berry’s paradoxical number b. In his article about it in 1906, Gödel’s birthyear (four syllables!), Russell did his best to deflect the paradox’s sting by claiming that it was an illusion arising from a naïve misuse of the word “describable” in the context of mathematics. That notion, claimed Russell, had to be parceled out into an infinite hierarchy of different types of describability — descriptions at level 0, which could refer only to notions of pure arithmetic; descriptions at level 1, which could use arithmetic but could also refer to descriptions at level 0; descriptions at level 2, which could refer to arithmetic and also to descriptions at levels 0 and 1; and so forth and so on. And so the idea of “describability” without restriction to some specific hierarchical level was a chimera, declared Russell, believing he had discovered a profound new truth. And with this brand-new type of theory (the brand-new theory of types), he claimed to have immunized the precious, delicate world of rigorous reasoning against the ugly, stomach-turning plague of Berry-Berry.

Blurriness Buries Berry

While I agree with Russell that something fishy is going on in Berry’s paradox, I don’t agree about what it is. The weakness that I focus in on is the fact that English is a hopelessly imprecise medium for expressing mathematical statements; its words and phrases are far too vague. What may seem precise at first turns out to be fraught with ambiguity. For example, the expression “nine cubed plus forty-eight, all times ten cubed plus one”, which earlier I exhibited as a description of 777,777, is in fact ambiguous — it might, for instance, be interpreted as meaning 777 times 1000, with 1 tacked on at the end, resulting in 777,001.

But that little ambiguity is just the tip of the iceberg. The truth of the matter is that it is far from clear what kinds of English expressions count as descriptions of a number. Consider the following phrases, which purport to be descriptions of specific integers:

• the number of distinct languages ever spoken on earth

• the number of heavenly bodies in the Solar System

• the number of distinct four-by-four magic squares

• the number of interesting integers less than 100

What is wrong with them? Well, they all involve ill-defined notions.

What, for instance, is meant by a “language”? Is sign language a language? Is it “spoken”? Is there a sharp cutoff between languages and dialects? How many “distinct languages” lay along the pathway from Latin to Italian? How many “distinct languages” were spoken en route from Neanderthal days to Latin? Is Church Latin a language? And Pig Latin? Even if we had videotapes of every last human utterance on earth for the past million years, the idea of objectively assigning each one to some particular “official” language, then cleanly teasing apart all the “truly distinct” languages, and finally counting them would still be a nonsensical pipe dream. It’s already meaningless enough to talk about counting all the “items” in a garbage can, let alone all the languages of all time!

Moving on, what counts as a “heavenly body”? Do we count artificial satellites? And random pieces of flotsam and jetsam left floating out there by astronauts? Do we count every single asteroid? Every single distinct stone floating in Saturn’s rings? What about specks of dust? What about isolated atoms floating in the void? Where does the Solar System stop? And so on, ad infinitum.

You might object, “But those aren’t mathematical notions! Berry’s idea was to use mathematical definitions of integers.” All right, but then show me a sharp cutoff line between mathematics and the rest of the world. Berry’s definition uses the vague notion of “syllable counting”, for instance. How many syllables are there in “finally” or “family” or “rhythm” or “lyre” or “hour” or “owl”? But no matter; suppose we had established a rigorous and objective way of counting syllables. Still, what would count as a “mathematical concept”? Is the discipline of mathematics really that sharply defined? For instance, what is the precise definition of the notion “magic square”? Different authors define this notion differently. Do we have to take a poll of the mathematical community? And if so, who then counts as a member of that blurry community?

What about the blurry notion of “interesting numbers”? Could we give some kind of mathematical precision to that? As you saw above, reasons for calling a number “interesting” could involve geometry and other areas of mathematics — but once again, where do the borders of mathematics lie? Is game theory part of mathematics? What about medical statistics? What about the theory of twisting tendrils of plants? And on and on.

To sum up, the notion of an “English-language definition of an integer” turns out to be a hopeless morass, and so Berry’s twisty notion of b, no less than Escher’s twisty notion of two mutually drawing hands, is an ingenious figment of the imagination rather than a genuine strange loop. There goes a promising candidate for strange loopiness down the drain!

Although in this brief digression I’ve made it sound as if the idea Berry had in 1904 was naïve, I must point out that some six decades later, the young mathematician Greg Chaitin, inspired by Berry’s idea, dreamt up a more precise cousin using computer programs instead of English-language descriptions, and this clever shift turned out to yield a radically new proof of, and perspective on, Gödel’s 1931 theorem. From there, Chaitin and others went on to develop an important new branch of mathematics known as “algorithmic information theory”. To go into that would carry us far afield, but I hope to have conveyed a sense for the richness of Berry’s insight, for this was the breeding ground for Gödel’s revolutionary ideas.

A Peanut-butter and Barberry Sandwich

Bertrand Russell’s attempt to bar Berry’s paradoxical construction by instituting a formalism that banned all self-referring linguistic expressions and self-containing sets was not only too hasty but quite off base. How so? Well, a friend of mine recently told me of a Russell-like ban instituted by a friend of hers, a young and idealistic mother. This woman, in a well-meaning gesture, had strictly banned all toy guns from her household. The ban worked for a while, until one day when she fixed her kindergarten-age son a peanut-butter sandwich. The lad quickly chewed it into the shape of a pistol, then lifted it up, pointed it at her, and shouted, “Bang bang! You’re dead, Mommy!” This ironic anecdote illustrates an important lesson: the medium that remains after all your rigid bans may well turn out to be flexible enough to fashion precisely the items you’ve banned.

And indeed, Russell’s dismissal of Berry had little effect, for more and more paradoxes were being invented (or unearthed) in those intellectually tumultuous days at the turn of the twentieth century. It was in the air that truly peculiar things could happen when modern cousins of various ancient paradoxes cropped up inside the rigorously logical world of numbers, a world in which nothing of the sort had ever been seen before, a pristine paradise in which no one had dreamt paradox might arise.

Although these new kinds of paradoxes felt like attacks on the beautiful, sacred world of reasoning and numbers (or rather, because of that worrisome fact), quite a few mathematicians boldly embarked upon a quest to come up with ever deeper and more troubling paradoxes — that is, a quest for ever more powerful threats to the foundations of their own discipline! This sounds like a perverse thing to do, but they believed that in the long run such a quest would be very healthy for mathematics, because it would reveal key weak spots, showing where shaky foundations had to be shored up so as to become unassailable. In short, plunging deeply into the new wave of paradoxes seemed to be a useful if not indispensable activity for anyone working on the foundations of mathematics, for the new paradoxes were opening up profound questions concerning the nature of reasoning — and thus concerning the elusive nature of thinking — and thus concerning the mysterious nature of the human mind itself.

An Autobiographical Snippet

As I mentioned in Chapter 4, at age fourteen I ran across Ernest Nagel and James R. Newman’s little gem, Gödel’s Proof, and through it I fell under the spell of the paradox-skirting ideas on which Gödel’s work was centered. One of the stranger loops connected with that period in my life was that I became acquainted with the Nagel family at just that time. Their home was in Manhattan, but they were spending the academic year 1959–60 “out west” at Stanford, and since Ernest Nagel and my father were old friends, I soon got to know the whole family. Shortly after the Nagels’ Stanford year was over, I savored the twisty pleasure of reading aloud the whole of Gödel’s Proof to my friend Sandy, their older son, in the verdant yard of their summer home in the gentle hills near Brattleboro, Vermont. Sandy was just my age, and we were both exploring mathematics with a kind of wild intoxication that only teen-agers know.

Part of what pulled me so intensely was the weird loopiness at the core of Gödel’s work. But the other half of my intense curiosity was my sense that what was really being explored by Gödel, as well as by many people he had inspired, was the mystery of the human mind and the mechanisms of human thinking. So many questions seemed to have been suddenly and sharply brought into light by Gödel’s 1931 article — questions such as…

What happens inside mathematicians’ heads when they do their most creative work? Is it always just rule-bound symbol manipulation, deriving theorems from a fixed set of axioms? What is the nature of human thought in general? Is what goes on inside our heads just a deterministic physical process? If so, are we all, no matter how idiosyncratic and sparkly, nothing but slaves to rigid laws governing the invisible particles out of which our brains are built? Could creativity ever emerge from a set of rigid rules governing minuscule objects or patterns of numbers? Could a rulegoverned machine be as creative as a human? Could a programmed machine come up with ideas not programmed into it in advance? Could a machine make its own decisions? Have its own opinions? Be confused? Know it was confused? Be unsure whether it was confused? Believe it had free will? Believe it didn’t have free will? Be conscious? Doubt it was conscious? Have a self, a soul, an “I”? Believe that its fervent belief in its “I” was only an illusion, but an unavoidable illusion?

Idealistic Dreams about Metamathematics

Back in those heady days of my youth, every time I entered a university bookstore (and that was as often as possible), I would instantly swoop down on the mathematics section and scour all the books that had to do with symbolic logic and the nature of symbols and meaning. Thus I bought book after book on these topics, such as Rudolf Carnap’s famous but forbidding The Logical Syntax of Language and Richard Martin’s Truth and Denotation, not to mention countless texts of symbolic logic. Whereas I very carefully read a few such textbooks, the tomes by Carnap and Martin just sat there on my shelf, taunting and teasing me, always seeming just out of reach. They were dense, almost impenetrably so — but I kept on thinking that if only someday, some grand day, I could finally read them and fully fathom them, then at last I would have penetrated to the core of the mysteries of thinking, meaning, creativity, and consciousness. As I look back now, that sounds ridiculously naïve (firstly to imagine this to be an attainable goal, and secondly to believe that those books in particular contained all the secrets), but at the time I was a true believer!

When I was sixteen, I had the unusual experience of teaching symbolic logic at Stanford Elementary School (my own elementary-school alma mater), using a brand-new text by the philosopher and educator Patrick Suppes, who happened to live down the street from our family, and whose classic Introduction to Logic had been one of my most reliable guides. Suppes was conducting an experiment to see if patterns of strict logical inference could be inculcated in children in the same way as arithmetic could, and the school’s principal, who knew me well from my years there, one day bumped into me in the school’s rotunda, and asked me if I would like to teach the sixth-grade class (which included my sister Laura) symbolic logic three times a week for a whole year. I fairly jumped at the chance, and all year long I thoroughly enjoyed it, even if a few of the kids now and then gave me a hard time (rubber bands in the eye, etc.). I taught my class the use of many rules of inference, including the mellifluous modus tollendo tollens and the impressive-sounding “hypothetical syllogism”, and all the while I was honing my skills not only as a novice logician but also as a teacher.

What drove all this — my core inner passion — was a burning desire to see unveiled the secrets of human mentation, to come to understand how it could be that trillions of silent, synchronized scintillations taking place every second inside a human skull enable a person to think, to perceive, to remember, to imagine, to create, and to feel. At more or less the same time, I was reading books on the brain, studying several foreign languages, exploring exotic writing systems from various countries, inventing ways to get a computer to generate grammatically complicated and quasi-coherent sentences in English and in other languages, and taking a marvelously stimulating psychology course. All these diverse paths were focused on the dense nebula of questions about the relationship between mind and mechanism, between mentality and mechanicity.

Intricately woven together, then, in my adolescent mind were the study of pattern (mathematics) and the study of paradoxes (metamathematics). I was somehow convinced that all the mysterious secrets with which I was obsessed would become crystal-clear to me once I had deeply mastered these two intertwined disciplines. Although over the course of the next couple of decades I lost essentially all of my faith in the notion that these disciplines contained (even implicitly) the answers to all these questions, one thing I never lost was my intuitive hunch that around the core of the eternal riddle “What am I?”, there swirled the ethereal vortex of Gödel’s elaborately constructed loop.

It is for that reason that in this book, although I am being driven principally by questions about consciousness and self, I will have to devote some pages to the background needed for a (very rough) understanding of Gödel’s ideas — and in particular, this means number theory and logic. There won’t be heavy doses of either one, to be sure, but I do have to paint at least a coarse-grained picture of what these fields are basically about; otherwise, we won’t have any way to proceed. So please fasten your seat belt, dear reader. We’re going to be experiencing a bit of weather for the next two chapters.

Post Scriptum

After completing this chapter to my satisfaction, I recalled that I owned two books about “interesting numbers” — The Penguin Dictionary of Curious and Interesting Numbers by David Wells, an author on mathematics whom I greatly admire, and Les nombres remarquables by François Le Lionnais, one of the two founders of the famous French literary movement Oulipo. I dimly recalled that both of these books listed their “interesting numbers” in order of size, so I decided to check them out to see which was the lowest integer that each of them left out.

As I suspected, both authors made rather heroic efforts to include all the integers that exist, but inevitably, human knowledge being finite and human beings being mortal, each volume sooner or later started having gaps. Wells’ first gap appeared at 43, while Le Lionnais held out a little bit longer, until 49. I personally was not too surprised by 43, but I found 49 surprising; after all, it’s a square, which suggests at least a speck of interest. On the other hand, I admit that squareness gets a bit boring after you’ve already run into it several times, so I could partially understand why that property alone did not suffice to qualify 49 for inclusion in Le Lionnais’s final list. Wells lists several intriguing properties of 49 (but not the fact that it’s a square), and conversely, Le Lionnais points out some very surprising properties of 43.

So then I decided to find the lowest integer that both books considered to be utterly devoid of interest, and this turned out to be 62. For what it’s worth, that will be my age when this book appears in print. Could it be that 62 is interesting, after all?