While you are pondering this, I will jump back to the source of it all, which was Gödel’s PM formula that talked about itself. The point is that Gödel numbers, since they can be used as names for formulas and can be inserted into formulas, are precisely analogous to quoted phrases. Now we have just seen that there is a way to use quotation marks and sentence fragments to make a full sentence that talks about itself (or if you prefer, a sentence that talks about another sentence, but one that is a clone to it, so that whatever is true of the one is true of the other).
Gödel, analogously, created a “subjectless formula fragment” (by which I mean a PM formula that is not about any specific integer, but just about some unspecified variable number x). And then, making a move analogous to that of feeding Quine’s Quasi-Quip into itself (but in quotes), he took that formula fragment’s Gödel number k (which is a specific number, not a variable) and replaced the variable x by it, thus producing a formula (not just a fragment) that made a claim about a much larger integer, g. And g is the Gödel number of that very claim. And last but not least, the claim was not about whether the entity in question was a full sentence or not, but about whether the entity in question was a provable formula or not.
An Elephant in a Matchbox is Neither Fish Nor Fowl
I know this is a lot to swallow in one gulp, and so if it takes you several gulps (careful rereadings), please don’t feel discouraged. I’ve met quite a few sophisticated mathematicians who admit that they never understood this argument totally!
I think it would be helpful at this juncture to exhibit a kind of hybrid sentence that gets across the essential flavor of Gödel’s self-referential construction but that does so in Quinean terms — that is, using the ideas we’ve just been discussing. The hybrid sentence looks like this:
“when fed its own Gödel number yields a non-prim number”
when fed its own Gödel number yields a non-prim number.
The above sentence is neither fish nor fowl, for it is not a formula of Principia Mathematica but an English sentence, so of course it doesn’t have a Gödel number and it couldn’t possibly be a theorem (or a nontheorem) of PM. What a mixed metaphor!
And yet, mixed metaphor though it is, it still does a pretty decent job of getting across the flavor of the PM formula that Gödel actually concocted. You just have to keep in mind that using quote marks is a metaphor for taking Gödel numbers, so the upper line should be thought of as being a Gödel number (k) rather than as being a sentence fragment in quote marks. This means that metaphorically, the lower line (an English sentence fragment) has been fed its own Gödel number as its subject. Very cute!
I know that this is very tricky, so let me state it once again, slightly differently. Gödel asks you to imagine the formula that k stands for (that formula happens to contain the variable x), and then to feed k into it (this means to replace the single letter x by the extremely long numeral k, thus giving you a much bigger formula than you started with), and to take the Gödel number of the result. That will be the number g, huger far than k — and lastly, Gödel asserts that this walloping number is not a prim number. If you’ve followed my hand-waving argument, you will agree that the full formula’s Gödel number (g) is not found explicitly inside the formula, but instead is very subtly described by the formula. The elephant’s DNA has been used to get a description of the entire elephant into the matchbox.
Sluggo and the Morton Salt Girl
Well, I don’t want to stress the technical points here. The main thing to remember is that Gödel devised a very clever number-description trick — a recipe for making a very huge number g out of a less huge number k — in order to get a formula of PM to make a claim about its own Gödel number’s non-primness (which means that the formula is actually making a claim of its own nontheoremhood). And you might also try to remember that the “little” number k is the Gödel number of a “formula fragment” containing a variable x, analogous to a subjectless sentence fragment in quote marks, while the larger number g is the Gödel number of a complete sentence in PM notation, analogous to a complete sentence in English.
Popular culture is by no means immune to the delights of self-reference, and it happens that the two ideas we have been contrasting here — having a formula contain its own Gödel number directly (which would necessitate an infinite regress) and having a formula contain a description of its Gödel number (which beautifully bypasses the infinite regress) — are charmingly illustrated by two images with which readers may be familiar.
In this first image, Ernie Bushmiller’s character Sluggo (from his classic strip Nancy) is dreaming of himself dreaming of himself dreaming of himself, without end. It is clearly a case of self-reference, but it involves an infinite regress, analogous to a PM formula that contained its own Gödel number directly. Such a formula, unfortunately, would have to be infinitely long!
Our second image, in contrast, is the famous label of a Morton Salt box, which shows a girl holding a box of Morton Salt. You may think you smell infinite regress once again, but if so, you are fooling yourself! The girl’s arm is covering up the critical spot where the regress would occur. If you were to ask the girl to (please) hand you her salt box so that you could actually see the infinite regress on its label, you would wind up disappointed, for the label on that box would show her holding a yet smaller box with her arm once again blocking the regress.
And yet we still have a self-referential picture, because customers in the grocery store understand that the little box shown on the label is the same as the big box they are holding. How do they arrive at this conclusion? By using analogy. To be specific, not only do they have the large box in their own hands, but they can see the little box the girl is holding, and the two boxes have a lot in common (their cylindrical shape, their dark-blue color, their white caps at both ends); and in case that’s not enough, they can also see salt spilling out of the little one. These pieces of evidence suffice to convince everyone that the little box and the large box are identical, and there you have it: self-reference without infinite regress!
In closing this chapter, I wish to point out explicitly that the most concise English translations of Gödel’s formula and its cousins employ the word “I” (“I am not provable in PM ”; “I am not a PM theorem”). This is not a coincidence. Indeed, this informal, almost sloppy-seeming use of the singular first-person pronoun affords us our first glimpse of the profound connection between Gödel’s austere mathematical strange loop and the very human notion of a conscious self.
CHAPTER 11
How Analogy Makes Meaning
The Double Aboutness of Formulas in PM
IMAGINE the bewilderment of newly knighted Lord Russell when a young Austrian Turk named “Kurt” declared in print that Principia Mathematica, that formidable intellectual fortress so painstakingly erected as a bastion against the horrid scourge of self-referentiality, was in fact riddled through and through with formulas allegedly stating all sorts of absurd and incomprehensible things about themselves. How could such an outrage ever have been allowed to take place? How could vacuously twittering self-referential propositions have managed to sneak through the thick ramparts of the beautiful and timeless Theory of Ramified Types? This upstart Austrian sorcerer had surely cast some sort of evil spell, but by what means had he wrought his wretched deed?
The answer is that in his classic article — “On Formally Undecidable Propositions of Principia Mathematica and Related Systems (I)” — Gödel had re-analyzed the notion of meaning and had concluded that what a formula of PM meant was not so simple — not so unambiguous — as Russell had thought. To be fair, Russell himself had always insisted that PM’s strange-looking long formulas had no intrinsic meaning. Indeed, since the theorems of PM were churned out by formal rules that paid no attention to meaning, Russell often said the whole work was just an array of meaningless marks (and as you saw at the end of Chapter 9, the pages of Principia Mathematica often look more like some exotic artwork than like a
work of math).
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.)
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 clearly via the rather strong couple-to-couple analogy that both Audrey and Ben have built up in their heads over the past several months without ever breathing a word of it to each other. Ben has managed to talk very clearly about himself although without ever talking directly about himself, and what’s more, both he and Audrey know this is so.
I Am a Strange Loop Page 21
