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By virtue of Gödel’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 Gödel’s newly revealed mapping (piggybacked on top of Russell’s first mapping), was about formulas, and since both levels of meaning depended on mappings, Gödel’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 Gödel 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 café, 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