Douglas Hofstadter - I Am a Strange Loop

Let us thus imagine that we can give a specific numerical “input” to the chainium by taking any positive integer we are interested in — 641, say — and placing exactly that many dominos end to end in a “reserved” stretch of the network. Now, when we tip over the chainium’s first domino, a Rube Goldberg–type series of events will take place in which domino after domino will fall, including, shortly after the outset, all 641 of the dominos constituting our input stretch, and as a consequence various loops will be triggered, with some loop presumably testing the input number for divisibility by 2, another for divisibility by 3, and so forth. If ever a divisor is found, then a signal will be sent down one particular stretch — let’s call it the “divisor stretch” — and when we see that stretch falling, we will know that the input number has some divisor and thus is not prime. By contrast, if the input has no divisor, then the divisor stretch will never be triggered and we will know the input is prime.

Suppose an observer is standing by when the domino chainium is given 641 as input. The observer, who has not been told what the chainium was made for, watches keenly for while, then points at one of the dominos in the divisor stretch and asks with curiosity, “How come that domino there is never falling?”

Let me contrast two very different types of answer that someone might give. The first type of answer — myopic to the point of silliness — would be, “Because its predecessor never falls, you dummy!” To be sure, this is correct as far as it goes, but it doesn’t go very far. It just pushes the buck to a different domino, and thus begs the question.

The second type of answer would be, “Because 641 is prime.” Now this answer, while just as correct (indeed, in some sense it is far more on the mark), has the curious property of not talking about anything physical at all. Not only has the focus moved upwards to collective properties of the chainium, but those properties somehow transcend the physical and have to do with pure abstractions, such as primality.

The second answer bypasses all the physics of gravity and domino chains and makes reference only to concepts that belong to a completely different domain of discourse. The domain of prime numbers is as remote from the physics of toppling dominos as is the physics of quarks and gluons from the Cold War’s “domino theory” of how communism would inevitably topple country after neighboring country in Southeast Asia. In both cases, the two domains of discourse are many levels apart, and one is purely local and physical, while the other is global and organizational.

Before passing on to other metaphors, I’d just like to point out that although here, 641’s primality was used as an explanation for why a certain domino did not fall, it could equally well serve as the explanation for why a different domino did fall. In particular, in the domino chainium, there could be a stretch called the “prime stretch” whose dominos all topple when the set of potential divisors has been exhausted, which means that the input has been determined to be prime.

The point of this example is that 641’s primality is the best explanation, perhaps even the only explanation, for why certain dominos did fall and certain other ones did not fall. In a word, 641 is the prime mover. So I ask: Who shoves whom around inside the domino chainium?

The Causal Potency of Collective Phenomena

My next metaphor was dreamt up on an afternoon not long ago when I was caught in a horrendous traffic jam on some freeway out in the countryside, with several lanes of nearly touching cars all sitting stock still. For some reason I was reminded of big-city traffic jams where you often hear people honking angrily at each other, and I imagined myself suddenly starting to honk my horn over and over again at the car in front of me, as if to say, “Get out of my way, lunkhead!”

The thought of myself (or anyone) taking such an outrageously childish action made me smile, but when I considered it a bit longer, I saw that there might be a slim rationale for honking that way. After all, if the next car were magically to poof right out of existence, I could fill the gap and thus make one car-length’s worth of progress. Now a car poofing out of existence is not too terribly likely, and one car-length is not much progress, but somehow, through this image, the idea of honking became just barely comprehensible to me. And then I remembered my domino chainium and the silly superlocal answer, “That domino didn’t fall because its neighbor didn’t fall, you dummy!” This myopic answer and my fleeting thought of honking at the car just ahead of me seemed to be cut from the same cloth.

As I continued to sit in this traffic jam, twiddling my thumbs instead of honking, I let these thoughts continue, in their bully-like fashion, to push my helpless neurons around. I imagined a counterfactual situation in which the highway was shrouded in the densest pea-soup fog imaginable, so that I could barely make out the rear of the car ahead of me. In such a case, honking my horn wouldn’t be quite so blockheaded. For all I know, that car alone might well be the entire cause of my being stuck, and if only it would just get out of the way, I could go sailing down the highway!

If you’re totally fog-bound like that, or if you’re incredibly myopic, then you might think to yourself, “It’s all my neighbor’s fault!”, and there’s at least a small chance that you’re right. But if you have a larger field of view and can see hordes of immobilized cars on all sides, then honking at your immediate predecessor is an absurdity, for it’s obvious that the problem is not local. The root problem lies at some level of discourse other than that of cars. Though you may not know its nature, some higher-level, more abstract reason must lie behind this traffic jam.

Perhaps a very critical baseball game just finished three miles up the road. Perhaps it’s 7:30 on a weekday morning and you’re heading towards Silicon Valley. Perhaps there’s a huge blizzard ten miles ahead. Or it may be something else, but it’s surely some social or natural event of the type that induces large numbers of people all to do the same thing as one another. No amount of expertise in car mechanics will help you to grasp the essence of such a situation; what is needed is knowledge of the abstract forces that can act on freeways and traffic. Cars are just pawns in the bigger game and, aside from the fact that they can’t pass through each other and emerge intact post-crossing (as do ripples and other waves), their physical nature plays no significant role in traffic jams. We are in a situation analogous to that in which the global, abstract, math-level answer “641 is prime” is far superior to a local, physical, domino-level answer.

Neurons and Dominos

The foregoing down-to-earth images provide us with helpful metaphors for talking about the many levels of causality inside a human brain. Suppose it were possible to monitor any selected neuron in my brain. In that case, someone might ask, as I listened to some piece of music, “How come neuron #45826493842 never seems to fire?” A local, myopic answer might be, “Because the neurons that feed into it never fire jointly”, and this answer would be just as correct but also just as useless and uninformative as the myopic answers in the other situations. On the other hand, the global, organizational answer “Because Doug Hofstadter doesn’t care for the style of Fats Domino” would be much more on target.

Of course we should not fall into the trap of thinking that neuron #45826493842 is the sole neuron designated to fire whenever I resonate to some piece of music I’m listening to. It’s just one of many neurons that participate in the high-level process, like voters in a national election. Just as no special voter makes the decision, so no special neuron is privileged. As long as we avoid simplistic notions such as a privileged “grand-music neuron”, we can use the domino-chainium metaphor to think about brains, and especially to remind ourselves of how, for a given phenomenon in a brain, there can be vastly different explanations belonging to vastly different domains of discourse at vastly different levels of abstraction.