I Am a Strange Loop (41 page)

Simmballs filled with meaning are now seen to be doing a stately dance in a blurry soup that they don’t suspect for a split second consists of small interacting magnetic marbles called “simms”. And the reason I say the simmballs are “filled with meaning” is not, of course, that they are oozing some mystical kind of sticky semantic juice called “meaning” (even though certain meat-infatuated philosophers might go for that idea), but because their stately dance is deeply in synch with events in the world around them.

Simmballs are in synch with the outer world in the same way as in
La Femme du boulanger,
the straying cat Pomponnette’s return was in synch with the return of the straying wife Aurélie: there was a many-faceted alignment of Situation “P” with Situation “A”. However, this alignment of situations at the film’s climax was just a joke concocted by the screenwriter; no viewer of
La Femme du boulanger
supposes for a moment that the cat’s escapades will continue to parallel the wife’s escapades (or vice versa) for months on end. We know it was just a coincidence, which is why we find it so humorous.

By contrast, a careenium’s dancing simmballs
will
continue tracking the world,
will
stay in phase with it,
will
remain aligned with it. That (by fiat of the author!) is the very nature of a careenium. Simmballs are
systematically
in phase with things going on in the world just as, in Gödel’s construction, prim numbers are systematically in phase with
PM’
s provable formulas. That is the only reason simmballs can be said to have meaning. Meaning, no matter what its substrate might be — Tinkertoys, toilet paper, beer cans, simms, whole numbers, or neurons — is an automatic, unpreventable outcome of reliable, stable alignment; this was the lesson of Chapter 11.

Our own brains are no different from careenia, except, of course, that whereas careenia are just my little fantasy, human brains are not. The symbols in our brains truly
do
do that voodoo that they do so well, and they do it in the electrochemical soup of neural events. The strange thing, though, is that over the eons that it took for our brains to evolve from the earliest proto-brains, meanings just sneaked ever so quietly into the story, almost unobserved. It’s not as if somebody had devised a grand plan, millions of years in advance, that high-level meaningful structures — physical patterns representing abstract categories — would one day come to inhabit big fancy brains; rather, such patterns (the “symbols” of this book) simply came along as an unplanned by-product of the tremendously effective way that having bigger and bigger brains helped beings to survive better and better in a terribly cutthroat world.

Just as Bertrand Russell was blindsided by the unexpected appearance of high-level Gödelian meanings in the heart of his ultraprotected bastion,
Principia Mathematica,
so someone who had never conceived of looking at a brain at any level other than that of Hans Moravec’s squirting chemicals would be mightily surprised at the emergence of symbols. Much as Gödel saw the great potential of shifting attention to a wholly different level of
PM
strings, so I am suggesting (though I’m certainly far from the first) that we have to shift our attention to a far higher level of brain activity in order to find symbols, concepts, meanings, desires, and, ultimately, our selves.

The funny thing is that we humans all
are
focused on that level without ever having had any choice in the matter. We
automatically
see our brains’ activity as entirely symbolic. I find something wonderfully strange and upside-down about this, and I’ll now try to show why through an allegory.

In which the Alfbert Visits Austranius

Imagine, if you will, the small, lonely planet of Austranius, whose sole inhabitants are a tribe called the “Klüdgerot”. From time immemorial, the Klüdgerot have lived out their curious lives in a dense jungle of extremely long
PM
strings, some of which they can safely ingest (strings being their sole source of nutrition) and others of which they must not ingest, lest they be mortally poisoned. Luckily, the resourceful Klüdgerot have found a way to tell apart these opposite sorts of
PM
strings, for certain strings, when inspected visually, form a message that says, in the lilting Klüdgerotic tongue, “I am edible”, while others form a message in Klüdgerotic that says “I am inedible”. And, quite marvelously, by the Benevolent Grace of Göd, every
PM
string proclaiming its edibility has turned out to be edible, while every
PM
string proclaiming its inedibility has turned out to be inedible. Thus have the Klüdgerot thrived for untold öörs on their bountiful planet.

On a fateful döö in the Austranian möönth of Spöö, a strange-looking orange spacecraft swoops down from the distant planet of Ukia and lands exactly at the North Pöö of Austranius. Out steps a hulking whiteheaded alien that announces itself with the words, “I am the Alfbert. Behold.” No sooner has the alien uttered these few words than it trundles off into the Austranian jungle, where it spends not only the rest of Spöö but also all of Blöö, after which it trundles back, slightly bedraggled but otherwise no worse for the wear, to its spacecraft. Bright and early the next döö, the Alfbert solemnly convenes a meeting of all the Klüdgerot on Austranius. As soon as they all have assembled, the Alfbert begins to speak.

“Good döö, virtuous Klüdgerot,” intones the Alfbert. “It is my privilege to report to you that I have made an Austranius-shaking scientific discovery.” The Klüdgerot all sit in respectful if skeptical silence. “Each
PM
string that grows on this planet,” continues the Alfbert, “turns out to be not merely a long and pretty vine but also, astonishingly enough, a message that can be read and understood. Do not doubt me!” On hearing this non-novelty, many Klüdgerot yawn in unison, and a voice shouts out, “Tell us about it, white head!”, at which scattered chuckles erupt. Encouraged, the Alfbert does just so. “I have made the fantastic discovery that every
PM
string makes a claim, in my beautiful native tongue of Alfbertic, about certain wondrous entities known as the ‘whole numbers’. Many of you are undoubtedly champing at the bit to have me explain to you, in very simple terms that you can understand, what these so-called ‘whole numbers’ are.”

At the sound of this term, a loud rustling noise is heard among the assembled crowd. Unbeknownst to the Alfbert, the Klüdgerot have for countless generations held the entities called “whole numbers” to be incomprehensibly abstract; indeed, the whole numbers were long ago unanimously declared so loathsome that they were forever banned from the planet, along with all their names. Clearly, the Alfbert’s message is not welcome here. It is of course wrong (that goes without saying), but it is not merely wrong; it is also totally absurd, and it is repugnant, to boot.

But the whiteheaded Alfbert, blithely unaware of the resentment it has churned up, continues to speak as the mob rustles ever more agitatedly. “Yes, denizens of Austranius, fabulously unlikely though it may sound, in each
PM
string there resides
meaning.
All it takes is to know how to look at the string in the proper way. By using a suitable mapping, one can…”

All at once pandemonium erupts: has the Alfbert not just uttered the despised word “one”, the long-banished name of the most dreaded of all the whole numbers? “Away with the alien! Off with its white head!” screams the infuriated mob, and a moment later, a phalanx of Klüdgerot grabs hold of the declaiming alien. Yet even as it is being dragged away, the pontificating Alfbert patiently insists to the Klüdgerot that it is merely trying to edify them, that it can perceive momentous facts hidden to them by reading the strings in a language of which they are ignorant, and that… But the angered throng drowns out the Alfbert’s grandiose words.

As the brazen alien is being prepared to meet a dire fate, a commotion suddenly breaks out among the Klüdgerot; they have plumb forgotten the age-old and venerated Klüdgerot tradition of holding a Pre-dishing-out-ofdire-fate Banquet! A team is dispatched to pick the sweetest of all
PM
strings from the Principial Planetary Park of Wööw, a sacred sanctuary into which no Klüdgerot has ever ventured before; when it returns with a fine harvest of succulent strings from Wööw, each of which clearly reads “I am edible”, it is greeted by a hail of thunderous applause. After the Klüdgerot have expressed their gratitude to Göd, the traditional Pre-dishing-out-ofdire-fate Banquet begins, and at last it begins to dawn on the Alfbert that it will indeed meet a dire fate in short order. As this ominous fact takes hold, it feels its white head start to spin, then to swim, and then…

Idealistically attempting to save the unsuspecting Klüdgerot, the ever-magnanimous Alfbert cries out, “Listen, I pray, O friends! Your harvest of
PM
strings is treacherous! A foolish superstition has tricked you into thinking they are nutritious, but the truth is otherwise. When decoded as messages, these strings all make such grievously false statements about whole numbers that no one — I repeat, no one! — could swallow them.” But the words of warning come too late, for the
PM
strings from Wööw are already being swallowed whole by the stubbornly superstitious Klüdgerot.

And before long, frightful groans are heard resounding far and near; the sensitive Alfbert shields its gaze from the dreadful event. When at last it dares to look, it beholds a sorry sight; on every side, as far as its sole eye can see, lie lifeless shells of Klüdgerot that but moments ago were carousing their silly heads off. “If only they had listened to me!”, sadly muses the kindly Alfbert, scratching its great white head in puzzlement. On these words, it trundles back to its strange-looking orange spacecraft at the North Pöö, takes one last glance at the bleak Klüdgerot-littered landscape of Austranius, and finally presses the small round “Takeoff” button on the craft’s leatherette dashboard, setting off for destinations unknown.

At this point, the Alfbert, having earlier swooned in terror as the banqueters began their ritual reveling, regains consciousness. First it hears shouts of excitement echoing all around, and then, when it dares to look, it beholds a startling sight; on every side, as far as its sole eye can see, masses of Klüdgerot are staring with unmistakable delight at something moving, somewhere above its white head. It turns to see what this could possibly be, just in time to catch the most fleeting glimpse of a thin shape making a strange, high-pitched rustling sound as it rapidly plummets towards —

Brief Debriefing

I offer my apologies to the late Ambrose Bierce for this rather feeble imitation of the plot of his masterful short story “An Occurrence at Owl Creek Bridge”, but my intentions are good. The
raison d’être
of my rather flippant allegory is to turn the classic tragicomedy starring Alfred North Whitehead and Bertrand Russell (jointly alias the Alfbert) and Kurt Gödel (alias the Klüdgerot) on its head, by positing bizarre creatures who cannot imagine the idea of any number-theoretical meaning in
PM
strings, but who nonetheless see the strings as meaningful messages — it’s just that they see only high-level Gödelian meanings. This is the diametric opposite of what one would naïvely expect, since
PM
notation was invented expressly to write down statements about numbers and their properties, certainly
not
to write down Gödelian statements about themselves!

A few remarks are in order here to prevent confusions that this allegory might otherwise engender. In the first place, the length of any
PM
string that speaks of its own properties (Gödel’s string KG being the prototype, of course) is not merely “enormous”, as I wrote at the allegory’s outset; it is inconceivable. I have never tried to calculate how many symbols Gödel’s string would consist of if it were written out in pure
PM
notation, because I would hardly know how to begin the calculation. I suspect that its symbol-count might well exceed “Graham’s constant”, which is usually cited as “the largest number ever to appear in a mathematical proof”, but even if not, it would certainly give it a run for its money. So the idea of anyone directly reading the strings that grow on Austranius, whether on a low level, as statements about whole numbers, or on a high level, as statements about their own edibility, is utter nonsense. (Of course, so is the idea that strings of mathematical symbols could grow in jungles on a faraway planet, as well as the idea that they could be eaten, but that’s allegoric license.)

Gödel created his statement KG through a series of 46 escalating stages, in which he shows that
in principle,
certain notions about numbers
could be
written down in
PM
notation. A typical such notion is “the exponent of the
k
th prime number in the prime factorization of
n
”. This notion depends on prior notions defined in earlier stages, such as “exponent”, “prime number”, “
k
th prime number”, “prime factorization” (none of which come as “built-in notions” in
PM
). Gödel never explicitly writes out
PM
expressions for such notions, because doing so would require writing down a prohibitively long chain of
PM
symbols. Instead, each individual notion is given a name, a kind of abbreviation, which could theoretically be expanded out into pure
PM
notation if need be, and which is then used in further steps. Over and over again, Gödel exploits alreadydefined abbreviations in defining further abbreviations, thus carefully building a tower of increasing complexity and abstractness, working his way up to its apex, which is the notion of prim numbers.