On what GEB is really all about (twenty years later)

So what is this book, Gödel, Escher, Bach: an Eternal Golden Braid — usually known by its acronym, “GEB” — really all about?

That question has hounded me ever since I was scribbling its first drafts in pen, way back in 1973. Friends would inquire, of course, what I was so gripped by, but I was hard pressed to explain it concisely. A few years later, in 1980, when GEB found itself for a while on the bestseller list of The New York Times, the obligatory one-sentence summary printed underneath the title said the following, for several weeks running: “A scientist argues that reality is a system of interconnected braids.” After I protested vehemently about this utter hogwash, they finally substituted something a little better, just barely accurate enough to keep me from howling again.

Many people think the title tells it all: a book about a mathematician, an artist, and a musician. But the most casual look will show that these three individuals per se, august though they undeniably are, play but tiny roles in the book’s content. There’s no way the book is about these three people!

Well, then, how about describing GEB as “a book that shows how math, art, and music are really all the same thing at their core”? Again, this is a million miles off — and yet I’ve heard it over and over again, not only from nonreaders but also from readers, even very ardent readers, of the book.

And in bookstores, I have run across GEB gracing the shelves of many diverse sections, including not only math, general science, philosophy, and cognitive science (which are all fine), but also religion, the occult, and God knows what else. Why is it so hard to figure out what this book is about? Certainly it’s not just its length. No, it must be in part that GEB delves, and not just superficially, into so many motley topics — fugues and canons, logic and truth, geometry, recursion, syntactic structures, the nature of meaning, Zen Buddhism, paradoxes, brain and mind, reductionism and holism, ant colonies, concepts and mental representations, translation, computers and their languages, DNA, proteins, the genetic code, artificial intelligence, creativity, consciousness and free will — sometimes even art and music, of all things! — that many people find it impossible to locate the core focus.

The Key Images and Ideas that Lie at the Core of GEB

Needless to say, this widespread confusion has been quite frustrating to me over the years, since I felt sure I had spelled out my aims over and over in the text itself. Clearly, however, I didn’t do it sufficiently often, or sufficiently clearly. But since now I’ve got the chance to do it once more — and in a prominent spot in the book, to boot — let me try one last time to say why I wrote this book, what it is about, and what its principal thesis is.

In a word, GEB is a very personal attempt to say how it is that animate beings can come out of inanimate matter. What is a self, and how can a self come out of stuff that is as selfless as a stone or a puddle? What is an “I” and why are such things found (at least so far) only in association with, as poet Russell Edson once wonderfully phrased it, “teetering bulbs of dread and dream” — that is, only in association with certain kinds of gooey lumps encased in hard protective shells mounted atop mobile pedestals that roam the world on pairs of slightly fuzzy, jointed stilts?

GEB approaches these questions by slowly building up an analogy that likens inanimate molecules to meaningless symbols, and further likens selves (or “I”’s or “souls” if you prefer — whatever it is that distinguishes animate from inanimate matter) to certain special swirly, twisty, vortex-like, and meaningful patterns that arise only in particular types of systems of meaningless symbols. It is these strange, twisty patterns that the book spends so much time on, because they are little known, little appreciated, counterintuitive, and quite filled with mystery. And for reasons that should not be too difficult to fathom, I call such strange, loopy patterns “strange loops” throughout the book, although in later chapters, I also use the phrase “tangled hierarchies” to describe basically the same idea.

This is in many ways why M. C. Escher — or more precisely, his art — is prominent in the “golden braid”: because Escher, in his own special way, was just as fascinated as I am by strange loops, and in fact he drew them in a variety of contexts, and wonderfully disorienting and fascinating.

[…] GEB was inspired by my long-held conviction that the “strange loop” notion holds the key to unraveling the mystery that we conscious beings call “being” or “consciousness.”

(GEB: Twentieth-Anniversary Edition, Preface, pp. P1-P2)

On self-reference

Before going further, I should explain the term “self-reference.”

Self-reference is ubiquitous. It happens every time any one says “I” or “me” or “word” or “speak” or “mouth”. It happens every time a newspaper prints a story about reporters, every time someone writes a book about writing, designs a book about book design, makes a movie about movies, or writes an article about self-reference. Many systems have the capability to represent or refer to themselves somehow, to designate themselves (or elements of themselves) within the system of their own symbolism. Whenever this happens, it is an instance of self-reference.

Self-reference is often erroneously taken to be synonymous with paradox. This notion probably stems from the most famous example of a self-referential sentence: the Epimenides paradox. Epimenides the Cretan said, “All Cretans are liars.” I suppose no one today knows whether he said it in ignorance of its self-undermining quality or for that very reason. In any case, two of its relatives, the sentences “I am lying” and “This sentence is false”, have come to be known as the Epimenides paradox or the liar paradox. Both sentences are absolutely sell-destructive little gems and have given self-reference a bad name down through the centuries. When people speak of the evils of self-reference, they are certainly overlooking the fact that not every use of the pronoun “I” leads to paradox.

Let us use the Epimenides paradox as our jumping-off point into this fascinating land. There are many variations on the theme of a sentence that somehow undermines itself. Consider these two:

This sentence claims to be an Epimenides Paradox, but it is lying.

This sentence contradicts itself — or rather — well, no, actually it doesn’t!

What should you do when told, “Disobey this command”? In the following sentence the Epimenides quality jumps out only after a moment of thought: “This sentence contains exactly threee erors.” There is a delightful backlash effect here.

Kurt Gödel’s famous Incompleteness Theorem in metamathematics can be thought of as arising from his attempt to replicate as closely as possible the liar paradox in purely mathematical terms. With marvelous ingenuity. he was able to show that in any mathematically powerful axiomatic system S it is possible to express a close cousin to the liar paradox, namely, “This formula is unprovable within axiomatic system S.” In actuality, the Gödel construction yields a mathematical formula, not an English sentence: I have translated the formula back into English to show what he concocted. However, astute readers may have noticed that, strictly speaking, the phrase “this formula” has no referent. since when a formula is translated into an English sentence, that sentence is no longer a formula!

[...]

When a word is used to refer to something, it is said to be being used. When a word is quoted, though, so that one is examining it for its surface aspects (typographical, phonetic. etc.), it is said to be being mentioned The following sentences are based on this famous use-mention distinction:

You can’t have your use and mention it too.

You can’t have “your cake” and spell it “too”.

“Playing with the use-mention distinction” isn’t “everything in life, you know”.

In order to make sense of “this sentence” you will have to ignore the quotes in “it”.

This  is a sentence with “onions”, “lettuce”, “tomato” and “a side of fries to go”.

This is a hamburger with vowels, consonants, commas, and a period at the end.

The last two are humorous flip sides of the same idea. Here are two rather extreme examples of self-referential use-mention play:

Let us make a new convention: that anything enclosed in triple quotes — for example, ‘‘‘No, I have decided to change my mind; when the triple quotes close, just skip directly to the period and ignore everything up to it’’’— is not even to be read (much less paid attention to or obeyed).

A ceux qui ne comprennent pas l’anglais, la phrase citée ci-dessous ne dit rien: “For those who know no French, the French sentence that introduced this quoted sentence has no meaning.”

(Metamagical Themas, pp. 7-10)

from “Who Shoves Whom Around Inside the Careenium?
or, What Is the Meaning of the Word ‘I’?”

The Achilles symbol and the Tortoise symbol encounter each other
inside the author’s cranium.

ACHILLES: Fancy meeting you here! I’d thought that our dialogue in Paris was the last one we’d ever have.

TORTOISE: You can never tell with this author. Just when you think he’s done with you, he drags you out again to perform for his readers.

ACHILLES: I don’t see why we should have to perform at his whim.

TORTOISE: Just try resisting. Then you’ll see why. You don’t have any choice in the matter!

ACHILLES: I don’t?

TORTOISE: Look — to refuse to perform is tantamount to suicide. Let’s face it, Achilles — you and I (at least in these Hofstadterian versions of ourselves) come to life only when Hofstadter writes dialogues about us. We had it good in Gödel, Escher, Bach, but now that that’s over and done with, I have a feeling the pickings are going to be pretty slim. Hofstadter knows he can’t live off us forever! So we’d better take what we can get!

ACHILLES: Yes... I remember those good old days. Sometimes we had such wonderful lines. Like that one you had, something how the “Achillean flash” swoops about my brain “in shapes stranger than the dash of a gnat-hungry swallow.” Isn’t that how it went?

TORTOISE: Something like that. Hofstadter like that one well enough that he had me say it in at least two dialogues! Pretty strange, eh?

ACHILLES: The way you talk about all this is so bizarre, to my mind. I mean, granted that we’re figments of someone else’s imagination; but still, you know how characters in a novel are supposed to “come alive” and have “wills of their own”.... Surely it’s not just a cliché?

TORTOISE: I wouldn’t know, I’m not a novelist. Nor is Hofstadter.

ACHILLES: I mean, am I really just a tool of Hofstadter (however benevolent he is), or am I genuinely exerting my own free will here (as I feel I am doing)? What it comes down to is: Who pushes whom around inside this cranium?

TORTOISE: Now there’s a planted line, if I ever heard one. That’s a direct quote from GEB, page 710, where Hofstadter is quoting from Roger Sperry of split-brain fame. It’s where Sperry’s giving his mind-brain-free-will philosophy, which Mr. H evidently espouses. But let’s get on with the subject matter of this dialogue. I think we’ve done enough introduction. You must have something on your mind, Achilles, which Mr. H wants to bring up through you.

ACHILLES: I wish you’d quit putting it in that upside-down way, Mr. T.

TORTOISE: All right. But am I right? Isn’t there something you’re just itching to tell me?

(Metamagical Themas, pp. 604-605)

On “poetic lie-sense” and translating Pushkin

I would propose an alternate name for the art of compromise in poetry translation — I would say that poetry translation is the art of “poetic lie-sense.” Yes, one is always lying, for to translate is to lie. But even to speak is to lie, no less. No word is perfect, no sentence captures all the truth and only the truth. All we do is make do, and in poetry, hopefully, do so gracefully.

I do not, I freely though ruefully admit, have a mastery of all those subtle nuances of the Russian words I was translating. I have, rather, a basic sense of what each one means — I know the ballpark it’s in. Thus благородный, for example, which occurs in a few of the stanzas that I’ve memorized, means to me “noble,” and I can also see inside it to its roots, which tell me that it originally meant “well-born” (and [...] so does the name “Eugene”).  But I don’t feel, when I hear it, the rich resonances that a native speaker of Russian must feel; I just think to myself, “noble,” and then let any synonym or even roughly related word spring to mind.  “Aristocratic”?  Fine.  “High-born”?  Fine.  “Fine”?  Perhaps.  And so forth.

What matters is not getting each and every word to match perfectly in connotations, but getting the overall sense and the overall tone of a line across, and doing so with an elegant rhythm and a high-quality rhyme, to boot.  That’s what matters.  Rhythm, rhyme, sense, and tone — all of them together are what Eugene Onegin is about, and not just literal meaning.  To throw any of these overboard is to destroy the poem utterly.

I have exploited poetic lie-sense so many times in making this translation that it’s almost silly to try to pick examples — just take any line whatsoever!  For instance, line 1 of stanza I.1.  In the original, it runs as follows: Мой дядя самых честных правил, which could be literally rendered as “My uncle, of most honest principles,” and phonetically rendered as Moj dyádya sámykh chéstnykh právil.  But my translation’s opening line runs this way: “My uncle, matchless moral model.”  As you see, already in line 1 of stanza I.1 I have introduced alliteration where there is none, I have used concepts like “morality” and “role model” that are not spelled out explicitly in the original, and with my choice of the word “matchless” I have perhaps wound up somewhat overstating the uniqueness of the speaker’s uncle’s admirable character traits.  Compromise lies everywhere.

[…]

For one last example, let’s look at the concluding line of the novel’s second stanza: Но вреден север для меня (No vréden séver dlya menyá — “But harmful is the North to me”).  Here, Pushkin is subtly (or not so subtly) alluding to the fact that it was from the northern town of Petersburg that he was sent by the czar into exile in southern Russia, for nothing more serious than having written a few slightly irreverent poems.  Falen says here, “But found it noxious in the north,” thus using poetic lie-sense by introducing alliteration where there was none, and also — if you want to be nitpicky — by having the chutzpah to change present into past.  Arndt says, “The North, though, disagrees with me.”  Johnston: “but I’m allergic to the North…”  Elton/Briggs: “But baneful is the North to me…”  And finally, here is Deutsch: “But find the North is not my style.”

By contrast, my translation says: “The North was, shall I say, ‘severe.’”  By golly, I don’t just toy around with tenses; I also sin in a big-time way by playing on the fact that the Russian word for “north” is pronounced “séver.”  To some readers, this flippancy of mine will come across as so irreverent towards Pushkin that they would exile me to Bessarabia if they had the chance; to others it will merely seem amusing.  As for me, I see it as just another typical example of poetic lie-sense, and a quite Pushkinesque one, if I don’t say so myself.

My translation abounds in this kind of thing....

(Eugene Onegin, Translator’s Preface, pp. xxxiii-xxxv)

A Few Stanzas from Eugene Onegin, in Hofstadter’s Translation

A une damoyselle malade Clément Marot Ma mignonne, Je vous donne Le bon jour; Le séjour C’est prison. Guérison Recouvrez, Puis ouvrez Votre porte Et qu’on sorte Vitement, Car Clément Le vous mande. Va, friande De ta bouche Qui se couche En danger Pour manger Confitures; Si tu dures Trop malade, Couleur fade Tu prendras, Et perdras L’embonpoint. Dieu te doint Santé bonne, Ma mignonne.

Chickadee Carol Hofstadter Chickadee, I decree A fine day. Dart away From your cage And engage In brave flight, So you might Flee the croup. Hope you swoop Into ham, Apple jam, And French bread, Or instead You will lose The bright hues Of your plumes. Flu consumes Scrawny birds; Heed my words And take care. Slip the snare That does pinch My wee finch. Hopes abound That aground You won’t be, Chickadee.

“Chickadee” was Carol’s sole foray into Marot territory. She was uncertain whether she could do a job that would meet my approval, and hence put off doing it for ages. This drove me crazy and in my heavy-handed way, I kept on prodding her to try — and that of course made her less inclined to do it, rather than more so. A typical marital interaction.

But one pretty spring day, not long after I had written “Carol Dear” for her in the hospital, I went into her study in Bloomington and chanced to see a lined notebook lying open on her desk, with a penciled-in poem on the page. I read the poem and was enormously touched: it was called “My Chickadee” and was very beautifully rendered. Carol was out of the house at the time, but as soon as she got back I told her what I had seen, and how beautiful I thought it was. She couldn’t believe I liked it so well, and I assured her I was sincere. My only critical comment was that she might improve it a little by thinning it down from four syllables to three, which she immediately did, and having done so, she agreed with me that that way it was better.

“Chickadee” is a lovely exploration of the “bird” conceit, from beginning to end. The idea of replacing the metaphorical prison by a “cage,” for example, is charming and elegant, as is the transfer of the loss of color from skin to feathers. The bird swooping along, picking up bits of food in midcourse, is another pretty image, a frame blend par excellence, and it reminds me of a similar image she once suggested...

It was early May of 1987, and Carol and I were visiting Spain for the first time, playing cassettes of wonderful music by de Falla, Albéniz, and Granados wherever we drove, and steeping ourselves in the craggy wildness of Spanish landscapes. One evening, we were sitting together on the balcony of our hotel, the Hotel Alhambra Palace, savoring the spectacular view of the city of Granada and the distant Sierra Nevada mountains as the sun slowly sank in the west. The city was spread out beneath us, and swarming all through the vast chasms of warm air between us and the houses far below were uncountably many swallows, all of them swooping and darting after invisible bugs, their sunset-time meal, which they no doubt were enjoying as much as we were enjoying the delicious tapas we had already made a ritual out of, after just a few days in Spain. Popping a green olive into my mouth, I said to Carol, “If I could be any kind of bird, I’d be a swallow... only I wouldn’t like eating insects.” Smiling, she replied, “There are trade-offs... Now if you could be a Thai-food-eating swallow, that would be ideal!” I took up her image, embellishing it a little: “Yeah, with little tiny specks of Thai food darting around in the sky like insects...”

In her poem, although Carol doesn’t get in a poet’s (or translator’s) self-reference, she makes up for this lack by inserting the phrase “French bread”, delicately hinting at the original poem’s language and culture. [...]

In my judgment, the last seven lines of “Chickadee” are especially well-crafted and beautiful. I must say, as I hear its dolcezza — graced tone — as my eye glides over its elegant form, I can’t help but feel that this poem is among the finest and sweetest of all “Ma Mignonne”s. But then, I’m biased — I loved her so, and still and still I do.

(Le Ton beau de Marot, pp. 72a-72b)

On what “I” am

(From a dialogue between Strange Loop #642, a believer in the ideas of I Am a Strange Loop, and Strange Loop #641, a doubter of the ideas of I Am a Strange Loop)

SL #642: My proposal [...] is to see the “I” as a hallucination perceived by a hallucination, which sounds pretty strange, or perhaps even stranger: the “I” as a hallucination hallucinated by a hallucination.

SL #641: That sounds way beyond strange. That sounds crazy.

SL #642: Perhaps, but like many strange fruits of modern science, it can sound crazy yet be right. At one time it sounded crazy to say that the earth moved and the sun was still....

(I Am a Strange Loop, p. 293 )

In the end, we self-perceiving, self-inventing, locked-in mirages are little miracles of self-reference. […] Our very nature is such as to prevent us from fully understanding its very nature. Poised midway between the unvisualizable cosmic vastness of curved spacetime and the dubious, shadowy flickerings of charged quanta, we human beings, more like rainbows and mirages than like raindrops or boulders, are unpredictable self-writing poems — vague, metaphorical, ambiguous, and sometimes exceedingly beautiful.

To see ourselves this way is probably not as comforting as believing in ineffable other-worldly wisps endowed with eternal existence, but it has its compensations. What one gives up on is a childlike sense that things are exactly as they appear, and that our solid-seeming, marble-like “I” is the realest thing in the world; what one acquires is an appreciation of how enuous we are at our cores, and how wildly different we are from what we seem to be. As Kurt Gödel with his unexpected strange loops gave us a deeper and subtler vision of what mathematics is all about, so the strange-loop characterization of our essences gives us a deeper and subtler vision of what it is to be human. And to my mind, the loss is worth the gain.

(I Am a Strange Loop, p. 363 )

On translators, traitors, and traders

With this, we find ourselves square in the territory staked out by the celebrated and hugely negativistic Italian slogan Traduttore, traditore, which literally means “Translator, betrayer” or “Translator, traducer” – or then again, “Translator, traitor.” The irony residing in the final one of these three possibilities is that it beautifully undermines its own claim. The pithy slogan “Translator, traitor” shows very clearly that a translator need not be a betrayer or traitor, for it beautifully preserves the key quality that makes the original Italian phrase so memorable – namely, its catchiness, which is due to the fact that the two nouns inside it sound so much alike. There is no aspect of the phrase Traduttore, traditore that is missed by “Translator, traitor,” and so this English translation is a checkmate in response to the strong-seeming check tendered by the Italian opponent.

I am not in the least a believer in the extreme pessimism expressed by Traduttore, traditore (or its perfect English counterpart “Translator, traitor”), no matter how catchy this famous phrase may be and no matter how often it is repeated like a clever mantra by supposed literary sophisticates. I hear it as a cute sound bite rather than as a serious thesis about translation. In fact, I am far more inclined to believe in a rival (and also cute) sound bite – namely, “Translator, trader”....

(Translator, Trader, p. 18)

A phrase-trader has to have an intuitive sense of where the pale is – what’s well within it, what’s well beyond it. But the pale is so ethereal, so elusive, so intangible, so impalpable – so pale – that no one can actually see it. The pale is something determined collectively by the masses that speak the language, and no one person can pinpoint it.

(Translator, Trader, p. 50)

Selections and commentary by Glen Worthey
Stanford University Libraries

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