I Am a Strange Loop

And yet Russell was also careful to point out that all these curious patterns of horseshoes, hooks, stars, and squiggles could be interpreted, if one wished, as being statements about numbers and their properties, because under duress, one could read the meaningless vertical egg ‘0’ as standing for the number zero, the equally meaningless cross ‘+’ as standing for addition, and so on, in which case all the theorems of PM came out as statements about numbers — but not just random blatherings about them. Just imagine how crushed Russell would have been if the squiggle pattern “ss0 + ss0 = sssss0” turned out to 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 Gödel 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, Gödel’s discovery was very much in the Russellian spirit.

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.)