I Am a Strange Loop

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.