I Am a Strange Loop

be a theorem of PM! To him, this would have been a disaster of the highest order. Thus he had to concede that there was meaning to be found in his murky-looking tomes (otherwise, why would he have spent long years of his life writing them, and why would he care which strings were theorems?) — but that meaning depended on using a mapping that linked shapes on paper to abstract magnitudes (e.g., zero, one, two…), operations (e.g., addition), relationships (e.g., equality), concepts of logic (e.g., “not”, “and”, “there exists”, “all”), and so forth.

Russell’s dependence on a systematic mapping to read meanings into his fortress of symbols is quite telling, because what the young Turk Godel had discovered was simply a different systematic mapping (a much more complicated one, admittedly) by which one could read different meanings into the selfsame fortress. Ironically, then, Godel’s discovery was very much in the Russellian spirit.

By virtue of Godel’s subtle new code, which systematically mapped strings of symbols onto numbers and vice versa (recall also that it mapped typographical shunting laws onto numerical calculations, and vice versa), many formulas could be read on a second level. The first level of meaning, obtained via the old standard mapping, was always about numbers, just as Russell claimed, but the second level of meaning, using Godel’s newly revealed mapping (piggybacked on top of Russell’s first mapping), was about formulas, and since both levels of meaning depended on mappings, Godel’s new level of meaning was no less real and no less valid than Russell’s original one — just somewhat harder to see.

Extra Meanings Come for Free, Thanks to You, Analogy!

In my many years of reflecting about what Godel did in 1931, it is this insight of his into the roots of meaning — his discovery that, thanks to a mapping, full-fledged meaning can suddenly appear in a spot where it was entirely unsuspected — that has always struck me the most. I find this insight as profound as it is simple. Strangely, though, I have seldom if ever seen this idea talked about in a way that brings out the profundity I find in it, and so I’ve decided to try to tackle that challenge myself in this chapter. To this end, I will use a series of examples that start rather trivially and grow in subtlety, and hopefully in humor as well. So here we go.

Standing in line with a friend in a cafe, I spot a large chocolate cake on a platter behind the counter, and I ask the server to give me a piece of it. My friend is tempted but doesn’t take one. We go to our table and after my first bite of cake, I say, “Oh, this tastes awful.” I mean, of course, not merely that my one slice is bad but that the whole cake is bad, so that my friend should feel wise (or lucky) to have refrained. This kind of mundane remark exemplifies how we effortlessly generalize outwards. We unconsciously think, “This piece of the cake is very much like the rest of the cake, so a statement about it will apply equally well to any other piece.” (There is also another analogy presumed here, which is that my friend’s reaction to foods is similar to mine, but I’ll leave that alone.)

Let’s try another example, just a tiny bit more daring. There’s a batch of cookies on a plate at a party and I pick one up, take a bite, and remark to my children, “This is delicious!” Immediately, my kids take one each. Why? Because they wanted to taste something delicious. Yes, but how did they jump from my statement about my cookie to a conclusion about other cookies on the plate? The obvious answer is that the cookies are all “the same” in some sense. Unlike the pieces of cake, though, the cookies are not all parts of one single physical object, and thus they are ever so slightly “more different” from one another than are the pieces of cake — but they were made by the same person from the same ingredients using the same equipment. These cookies come from a single batch — they belong to the same category. In all relevant aspects, we see them as interchangeable. To be sure, each one is unique, but in the senses that count for human cookie consumption, they are almost certain to be equivalent. Therefore if I say about a particular one, “My, this is delicious!”, my statement’s meaning implicitly jumps across to any other of them, by the force of analogy. Now, to be sure, it’s a rather trivial analogy to jump from one cookie to another when they all come from the same plate, but it’s nonetheless an analogy, and it allows my specific statement “This is delicious!” to be taken as a general statement about all the cookies at once.

You may find these examples too childish for words. The first one involves an “analogy” between several slices of the same cake, and the second one an “analogy” between several cookies on the same plate. Are these banalities even worthy of the label “analogy”? To me there is no doubt about it; indeed, it is out of a dense fabric of a myriad of invisible, throwaway analogies no grander than these that the vast majority of our rich mental life is built. Yet we take such throwaway analogies so much for granted that we tend to think that the word “analogy” must denote something far more exalted. But one of my life’s most recurrent theme songs is that we should have great respect for what seem like the most mundane of analogies, for when they are examined, they often can be seen to have sprung from, and to reveal, the deepest roots of human cognition.

Exploiting the Analogies in Everyday Situations

As we’ve just seen, a remark made with the aim of talking about situation A can also implicitly apply to situation B, even if there was no intention of talking about B, and B was never mentioned at all. All it takes is that there be an easy analogy — an unforced mapping that reveals both situations to have essentially the same central structure or conceptual core — and then the extra meaning is there to be read, whether one chooses to read it or not. In short, a statement about one situation can be heard as if it were about an analogous — or, to use a slightly technical term, isomorphic — situation. An isomorphism is just a formalized and strict analogy — one in which the network of parallelisms between two situations has been spelled out explicitly and precisely — and I’ll use the term freely below.

When an analogy between situations A and B is glaringly obvious (no matter how simple it is), we sometimes will exploit it to talk “accidentally on purpose” about situation B by pretending to be talking only about situation A. “Hey there, Andy — take your muddy boots off when you come into the house!” Such a sentence, when shouted at one’s five-year-old son who is tramping in the front door with his equally mud-oozing friend Bill, is obviously addressed just as much to Bill as to Andy, via a very simple, very apparent analogy (a boy-to-boy leap, if you will, much like the earlier cookie-to-cookie leap). Hinting by analogy allows us to get our message across politely but effectively. Of course we have to be pretty sure that the person at whom we’re beaming our implicit message (Bill, here) is likely to be aware of the A/B analogy, for otherwise our clever and diplomatic ploy will all have been for naught.

Onward and upward in our chain of examples. People in romantic situations make use of such devices all the time. One evening, at a passionate moment during a tender clinch, Xerxes queries of his sweetie pie Yolanda, “Do I have bad breath?” He genuinely wants to know the answer, which is quite thoughtful of him, but at the same time his question is loaded (whether he intends it to be or not) with a second level of meaning, one not quite so thoughtful: “You have bad breath!” Yolanda answers his question but of course she also picks up on its potential alternative meaning in a flash. In fact, she suspects that Xerxes’ real intent was to tell her about her breath, not to find out about his own — he was just being diplomatic.

Now how can one statement speak on two levels at once? How can a second meaning lie lurking inside a first meaning? You know the answer as well as I do, dear reader, but let me spell it out anyway. Just as in the muddy- boots situation, there is a very simple, very loud, very salient, very obvious analogy between the two parties, and this means that any statement made about X will be (or at least can be) heard as being about Y at the same time. The X/Y mapping, the analogy, the partial isomorphism — whatever you wish to call it — carries the meaning efficiently and reliably from one framework over to the other.

Let’s look at this mode of communication in a slightly more delicate romantic situation. Audrey, who is not sure how serious Ben is about her, “innocently” turns the conversation to their mutual friends Cynthia and Dave, and “innocently” asks Ben what he thinks of Dave’s inability to commit to Cynthia. Ben, no fool, swiftly senses the danger here, and so at first he is wary about saying anything specific since he may incriminate himself even though talking “only” about Dave, but then he also realizes that this danger gives him an opportunity to convey to Audrey some things that he hasn’t dared to raise with her directly. Accordingly, Ben replies with a calculated air of nonchalance that he can imagine why Dave might be hesitant to commit himself, since, after all, Cynthia is so much more intellectual than Dave is. Ben is hoping that Audrey will pick up on the hint that since she is so much more involved in art than he is, that’s why he’s been hesitant to commit himself as well. His hint is carried to her implicitly but