I Am a Strange Loop - Douglas R. Hofstadter [66]
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