“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 à 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 café 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 Gödel’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 Gödel’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 Külot 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 Gödel 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”) Dalí. Indeed, some years after Gödel, such self-affirming formulas were concocted and studied by logicians such as Martin Hugo Löb 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 Gödel’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 Gödel’s discovery to be a naïve 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 Gödel, 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 Gödel came along? It cropped up because Principia Mathematica is in essence about the natural numbers, and what Gödel 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 Gödel 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 destruction were already hinted at by the seemingly innocent fact that PM had enough power to talk about arbitrarily subtle properties of whole numbers.
People of earlier eras had intuited much of this richness when they had tried to embed the nature of many diverse aspects of the world around us — stars, planets, atoms, molecules, colors, curves, notes, harmonies, melodies, and so forth — in numerical equations or other types of numerical patterns. Four centuries ago, laun
ching this whole tendency, Galileo Galilei had famously declared, “The book of Nature is written in the language of mathematics” (a thought that must seem shocking to people who love nature but hate mathematics). And yet, despite all these centuries of highly successful mathematizations of various aspects of the world, no one before Gödel had realized that one of the domains that mathematics can model is the doing of mathematics itself.
The bottom line, then, is that the unanticipated self-referential twist that Gödel found lurking inside Principia Mathematica was a natural and inevitable outcome of the deep representational power of whole numbers. Just as it is no miracle that a video system can create a self-referential loop, but rather a kind of obvious triviality due to the power of TV cameras (or, to put it more precisely, the immensely rich representational power of very large arrays of pixels), so too it is no miracle that Principia Mathematica (or any other comparable system) contains self-focused sentences like Gödel’s formula, for the system of integers, exactly like a TV camera (only more so!), can “point” at any system whatsoever and can reproduce that system’s patterns perfectly on the metaphorical “screen” constituted by its set of theorems. And just as in video feedback, the swirls that result from PM pointing at itself have all sorts of unexpected, emergent properties that require a brand-new vocabulary to describe them.
CHAPTER 12
On Downward Causality
Bertrand Russell’s Worst Nightmare
TO MY mind, the most unexpected emergent phenomenon to come out of Kurt Gödel’s 1931 work is a bizarre new type of mathematical causality (if I can use that unusual term). I have never seen his discovery cast in this light by other commentators, so what follows is a personal interpretation. To explain my viewpoint, I have to go back to Gödel’s celebrated formula — let’s call it “KG” in his honor — and analyze what its existence implies for PM.
As we saw at the end of Chapter 10, KG’s meaning (or more precisely, its secondary meaning — its higher-level, non-numerical, non-Russellian meaning, as revealed by Gödel’s ingenious mapping), when boiled down to its essence, is the whiplash-like statement “KG is unprovable inside PM.” And so a natural question — the natural question — is, “Well then, is KG indeed unprovable inside PM?”
To answer this question, we have to rely on one article of faith, which is that anything provable inside PM is a true statement (or, turning this around, that nothing false is provable in PM). This happy state of affairs is what we called, in Chapter 10, “consistency”. Were PM not consistent, then it would prove falsities galore about whole numbers, because the instant that you’ve proven any particular falsity (such as “0=1”), then an infinite number of others (“1=2”, “0=2”, “1+1=1”, “1+1=3”, “2+2=5”, and so forth) follow from it by the rules of PM. Actually, it’s worse than that: if any false statement, no matter how obscure or recondite it was, were provable in PM, then every conceivable arithmetical statement, whether true or false, would become provable, and the whole grand edifice would come tumbling down in a pitiful shambles. In short, the provability of even one falsity would mean that PM had nothing to do with arithmetical truth at all.
What, then, would Bertrand Russell’s worst nightmare be? It would be that someday, someone would come up with a PM proof of a formula expressing an untrue arithmetical statement (“0 = s0” is a good example), because the moment that that happened, PM would be fit for the dumpster. Luckily for Russell, however, every logician on earth would give you better odds for a snowball’s surviving a century in hell. In other words, Bertrand Russell’s worst nightmare is truly just a nightmare, and it will never take place outside of dreamland.
Why would logicians and mathematicians — not just Russell but all of them (including Gödel) — give such good odds for this? Well, the axioms of PM are certainly true, and its rules of inference are as simple and as rock-solidly sane as anything one could imagine. How can you get falsities out of that? To think that PM might have false theorems is, quite literally, as hard as thinking that two plus two is five. And so, along with all mathematicians and logicans, let’s give Russell and Whitehead the benefit of the doubt and presume that their grand palace of logic is consistent. From here on out, then, we’ll generously assume that PM never proves any false statements — all of its theorems are sure to be true statements. Now then, armed with our friendly assumption, let’s ask ourselves, “What would follow if KG were provable inside PM?”
A Strange Land where “Because” Coincides with “Although”
Indeed, reader, let’s posit, you and I, that KG is provable in PM, and then see where this assumption — I’ll dub it the “Provable-KG Scenario” — leads us. The ironic thing, please note, is that KG itself doesn’t believe the Provable-KG Scenario. Perversely, KG shouts to the world, “I am not provable!” So if we are right about KG, dear reader, then KG is wrong about itself, no matter how loudly it shouts. After all, no formula can be both provable (as we claim KG is) and also unprovable (as KG claims to be). One of us has to be wrong. (And for any formula, being wrong means being false. The two terms are synonyms.) So… if the Provable-KG Scenario is the case, then KG is wrong (= false).
All right. Our reasoning started with the Provable-KG Scenario and wound up with the conclusion “KG is false”. In other words, if KG is provable, then it is also false. But hold on, now — a provable falsity in PM?! Didn’t we just declare firmly, a few moments ago, that PM never proves falsities? Yes, we did. We agreed with the universal logicians’ belief that PM is consistent. If we stick to our guns, then, the Provable-KG Scenario has to be wrong, because it leads to Russell’s worst nightmare. We have to retract it, cancel it, repudiate it, nullify it, and revoke it, because accepting it led us to a conclusion (“PM is inconsistent”) that we know is wrong.
Ergo, the Provable-KG Scenario is hereby rejected, which leaves us with the opposite scenario: KG is not provable. Now the funny thing is that this is exactly what KG is shouting to the rooftops. We see that what KG proclaims about itself — “I’m unprovable!” — is true. In a nutshell, we have established two facts: (1) KG is unprovable in PM; and (2) KG is true.
We have just uncovered a very strange anomaly inside PM: here is a statement of arithmetic (or number theory, to be slightly more precise) that we are sure is true, and yet we are equally sure it is unprovable — and to cap it off, these two contradictory-sounding facts are consequences of each other! In other words, KG is unprovable not only although it is true, but worse yet, because it is true.
This weird situation is utterly unprecedented and profoundly perverse. It flies in the face of the Mathematician’s Credo, which asserts that truth and provability are just two sides of the same coin — that they always go together, because they entail each other. Instead, we’ve just encountered a case where, astoundingly, truth and unprovability entail each other. Now isn’t that a fine how-do-you-do?
Incompleteness Derives from Strength
The fact that there exists a truth of number theory that is unprovable in PM means, as you may recall from Chapter 9, that PM is incomplete. It has holes in it. (So far we’ve seen just one hole — KG — but it turns out there are plenty more — an infinity of them, in fact.) Some statements of number theory that should be provable escape from PM’s vast net of proof — they slip through its mesh. Clearly, this is another kind of nightmare — perhaps not quite as devastating as Bertrand Russell’s worst nightmare, but somehow even more insidious and troubling.
Such a state of affairs is certainly not what the mathematicians and logicians of 1931 expected. Nothing in the air suggested that the axioms and rules of inference of Principia Mathematica were weak or deficient in any way. They seemed, quite the contrary, to imply virtually everything that anyone might have thought was true about numbers. The opening lines of Gödel’s 1931 article, quoted in Chapter 10, state this clearly. If you’ll recall, he wrote, speaking of Principia Mathematica and Zermelo-Fraenkel set theory: “These two systems are so extensive that al
l methods of proof used in mathematics today have been formalized in them, i.e., reduced to a few axioms and rules of inference.”
What Gödel articulates here was virtually a universal credo at the time, and so his revelation of PM’s incompleteness, in the twenty-five pages that followed, came like a sudden thunderbolt from the bluest of skies.
To add insult to injury, Gödel’s conclusion sprang not from a weakness in PM but from a strength. That strength is the fact that numbers are so flexible or “chameleonic” that their patterns can mimic patterns of reasoning. Gödel exploited the simple but marvelous fact that the familiar whole numbers can dance in just the same way as the unfamiliar symbolpatterns of PM dance. More specifically, the prim numbers that he invented act indistinguishably from provable strings, and one of PM’s natural strengths is that it is able to talk about prim numbers. For this reason, it is able to talk about itself (in code). In a word, PM’s expressive power is what gives rise to its incompleteness. What a fantastic irony!
Bertrand Russell’s Second-worst Nightmare
Any enrichment of PM (say, a system having more axioms or more rules of inference, or both) would have to be just as expressive of the flexibility of numbers as was PM (otherwise it would be weaker, not stronger), and so the same Gödelian trap would succeed in catching it — it would be just as readily hoist on its own petard.
Let me spell this out more concretely. Strings provable in the larger and allegedly superior system Super-PM would be isomorphically imitated by a richer set of numbers than the prim numbers (hence let’s call them “super-prim numbers”). At this point, just as he did for PM, Gödel would promptly create a new formula KH for Super-PM that said, “The number h is not a super-prim number”, and of course he would do it in such a way that h would be the Gödel number of KH itself. (Doing this for Super-PM is a cinch once you’ve done it for PM.) The exact same pattern of reasoning that we just stepped through for PM would go through once again, and the supposedly more powerful system would succumb to incompleteness in just the same way, and for just the same reasons, as PM did. The old proverb puts it succinctly: “The bigger they are, the harder they fall.”
I Am a Strange Loop Page 23
