I Am a Strange Loop (15 page)

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
two
copies 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.