I Am a Strange Loop

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