Free I Am a Strange Loop by Douglas R. Hofstadter Page B

Relative timing, of course, will be of the essence, but once again, details do not concern us. The basic idea is just that we can imagine a network of precisely timed domino chains that amounts to a computer program for carrying out a particular computation, such as determining if a given input is a prime number or not. (John Searle, so fond of unusual substrates for computation, should like this “domino chainium” thought experiment!)
    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