I Am a Strange Loop (34 page)

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, launching 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

T
O 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.