Analogy, Once Again, Does its Cagey Thing
Okay, okay, enough’s enough. The jig’s up! Let me confess. For the last several pages, I’ve been playing a game, talking about strangely named plays by strangely named playwrights as well as a strangely titled review by a strangely named reviewer, but the truth is (and you knew it all along, dear reader), I’ve really been talking about something totally different — to wit, the strange loop that Austrian logician Kurt Godel (Gerd Kulot) discovered and revealed inside Russell and Whitehead’s Principia Mathematica.
“Now, now,” I hear some voice protesting (but of course it’s not your voice), “how on earth could you have really been talking about Whitehead and Russell and Principia Mathematica if the lines you wrote were not about them but about Y. Ted Enrustle and Prince Hyppia: Math Dramatica and such things?” Well, once again, it’s all thanks to the power of analogy; it’s the same game as in a roman a clef, where a novelist speaks, not so secretly, about people in real life by ostensibly speaking solely about fictional characters, but where savvy readers know precisely who stands for whom, thanks to analogies so compelling and so glaring that, taken in their cultural context, they cannot be missed by anyone sufficiently sophisticated.
And so we have worked our way up my ladder of examples of doubly-hearable remarks, all the way from the throwaway cafe blurt “This tastes awful” to the supersophisticated dramatic line “The number g is not prim”. We have repeatedly seen how analogies and mappings give rise to secondary meanings that ride on the backs of primary meanings. We have seen that even primary meanings depend on unspoken mappings, and so in the end, we have seen that all meaning is mapping-mediated, which is to say, all meaning comes from analogies. This is Godel’s profound insight, exploited to the hilt in his 1931 paper, bringing the aspirations embodied in Principia Mathematica tumbling to the ground. I hope that for all my readers, understanding Godel’s keen insight into meaning is now a piece of cake.
How Can an “Unpennable” Line be Penned?
Something may have troubled you when you learned that Prince Hyppia’s famous line about the number g proclaims (via analogy) its own unpennability. Isn’t this self-contradictory? If some line in some play is truly unpennable, then how could the playwright have ever penned it? Or, turning this question around, how could Prince Hyppia’s classic line be found in Y. Ted Enrustle’s play if it never was penned at all?
A very good question indeed. But now, please recall that I defined a “pennable line” as a line that could be written by a playwright who was tacitly adhering to a set of well-established dramaturgical conventions. The concept of “pennability”, in other words, implicitly referred to some particular system of rules. This means that an “unpennable” line, rather than being a line that could never, ever be written by anyone, would merely be a line that violated one or more of the dramaturgical conventions that most playwrights took for granted. Therefore, an unpennable line could indeed be penned — just not by someone who rigorously respected those rules.
For a strictly rule-bound playwright to pen such a line would be seen as extremely inconsistent; a churlish drama critic, ever reaching for cute new ways to snipe, might even write, “X’s play is so mega-inconsistent!” And thus, perhaps it was the recognition of Y. Ted Enrustle’s unexpected and bizarre-o “mega-inconsistency” that invariably caused audiences to gasp at Prince Hyppia’s math-dramatic outburst. No wonder Gerd Kulot received kudos for pointing out that a formerly unpennable line had been penned!
“Not” is Not the Source of Strangeness
A reader might conclude that a strange loop necessarily involves a self-undermining or self-negating quality (“This formula is not provable”; “This line is not pennable”; “You should not be attending this play”). However, negation plays no essential role in strange loopiness. It’s just that the strangeness becomes more pungent or humorous if the loop enjoys a self- undermining quality. Recall Escher’s Drawing Hands. There is no negation in it — both hands are drawing. Imagine if one were erasing the other!
In this book, a loop’s strangeness comes purely from the way in which a system can seem to “engulf itself ” through an unexpected twisting-around, rudely violating what we had taken to be an inviolable hierarchical order. In the cases of both Prince Hyppia: Math Dramatica and Principia Mathematica, we saw that a system carefully designed to talk only about numbers and not to talk about itself nonetheless ineluctably winds up talking about itself in a “cagey” fashion — and it does so precisely because of the chameleonic nature of numbers, which are so rich and complex that numerical patterns have the flexibility to mirror any other kind of pattern.
Every bit as strange a loop, although perhaps a little less dramatic, would have been created if Godel had concocted a self-affirming formula that cockily asserted of itself, “This formula is provable via the rules of PM”, which to me is reminiscent of the brashness of Muhammad (“I’m the greatest”) Ali as well as of Salvador (“The great”) Dali. Indeed, some years after Godel, such self-affirming formulas were concocted and studied by logicians such as Martin Hugo Lob and Leon Henkin. These formulas, too, had amazing and deep properties. I therefore repeat that the strange loopiness resides not in the flip due to the word “not”, but in the unexpected, hierarchy-violating twisting-back involving the word “this”.
I should, however, immediately point out that a phrase such as “this formula” is nowhere to be found inside Godel’s cagey formula — no more than the phrase “this audience” is contained in Cagey’s line “Anyone who crosses the picket line to go into Alf and Bertie’s Posh Shop is scum.” The unanticipated meaning “People in this audience are scum” is, rather, the inevitable outcome of a blatantly obvious analogy (or mapping) between two entirely different picket lines (one outside the theater, one on stage), and thus, by extension, between the picket-crossing members of the audience and the picket-line crossers in the play they are watching.
The preconception that an obviously suspicion-arousing word such as “this” (or “I” or “here” or “now” — “indexicals”, as they are called by philosophers — words that refer explicitly to the speaker or to something closely connected with the speaker or the message itself) is an indispensable ingredient for self-reference to arise in a system is shown by Godel’s discovery to be a naive illusion; instead, the strange twisting-back is a simple, natural consequence of an unexpected isomorphism between two different situations (that which is being talked about, on the one hand, and that which is doing the talking, on the other). Bertrand Russell, having made sure that all indexical notions such as “this” were absolutely excluded from his formal system, believed his handiwork to be forever immunized against the scourge of wrapping-around — but Kurt Godel, with his fateful isomorphism, showed that such a belief was an unjustified article of faith.
Numbers as a Representational Medium
Why did this kind of isomorphism first crop up when somebody was carefully scrutinizing Principia Mathematica? Why hadn’t anybody thought of such a thing before Godel came along? It cropped up because Principia Mathematica is in essence about the natural numbers, and what Godel saw was that the world of natural numbers is so rich that, given any pattern involving objects of any type, a set of numbers can be found that will be isomorphic to it — in other words, there are numbers that will perfectly mirror the objects and their pattern, numbers that will dance in just the way the objects in the pattern dance. Dancing the same dance is the key.
Kurt Godel was the first person to realize and exploit the fact that the positive integers, though they might superficially seem to be very austere and isolated, in fact constitute a profoundly rich representational medium. They can mimic or mirror any kind of pattern. Like any human language, where nouns and verbs (etc.) can engage in unlimitedly complex dancing, the natural numbers too, can engage in unlimitedly complex additive and multiplicative (etc.) dancing, and can thereby “talk”, via code or analogy, about events of any sort, numerical or non-numerical. This is what I meant when I wrote, in Chapter 9, that the seeds of PM’s
