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
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,
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
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
(GEB: Twentieth-Anniversary Edition, Preface, pp. P1-P2)
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
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
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)
Shoves Whom Around Inside the Careenium?
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
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
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....
Translator’s Preface, pp. xxxiii-xxxv)
A Few Stanzas from Eugene
in Hofstadter’s Translation
description of the Onegin stanza’s formal characteristics — to which he holds himself in strictest fashion in his translation! — see the final section of his “Analogy as the Core of Cognition” article.
So ran a rakehell’s thoughts, disjointed,
Thick in the dust of trotting steeds,
By Zeus, by Jove, he’d been appointed
Heir to his kinfolk’s trusts and deeds.
Fans of Ruslán
and of Lyudmíla:
Meet my new book! I’ll now reveal a
Few things about its motley crew.
First let me introduce to you
Onegin, my true friend and trusty,
Who by the Neva’s banks was born,
Just as were you, I would have sworn,
Dear reader — but
my memory’s rusty.
There once throve I, but left, I fear;
The North was, shall I say, “severe.”
In this stanza, Hofstadter not only translates the
form and content, but also wonderfully conveys Pushkin’s
own jocular, familiar first- and second-person banter
(both poet and reader are characters here), and the
metaliterary and self-referential aspects of his work (which
were downright Hofstadterian to begin
The verse fairy tale Ruslán and Lyudmíla
was one of Pushkin’s first long works, and was immensely
a brief discussion of the delightful, and very apt, bilingual
pun in this stanza’s final line, see the excerpt from
Translator’s Preface, directly above.
That year, autumnal weather hated
To take its leave from mead and dell;
The world e’er, e’er for winter waited.
’Twas January ere snow fell,
The third, by night. By dawnlight’s waking,
Tatyana, by her sill, was taking
The morn’s white farmyard in: the sheds,
The fence, the roofs, the flowerbeds,
The glass’s faint fantastic tracery,
The trees with wintry silver decked,
The court with merry magpies flecked,
The mountaintops’ light lucid lacery —
Their dazzling, glistening, wintry shawl,
The air was crisp; bright white was all.
This stanza seems particularly poetic and picturesque, reflecting especially
well both Hofstadter’s aesthetics and Pushkin’s original. In addition to the usual constraints of the Onegin stanza,
Hofstadter also imposes another: that of line-initial alliteration.
And even this he takes up a notch: there’s one of these “alliterated” stanzas in each chapter of the novel.
e’er” in line 3 (echoed as “ere” in the next line) is not just a way to squeeze in an extra syllable, but is
a rendering of Pushkin’s own “ждала, ждала” (“waited, waited”).
Finally, the lovely “faint
fantastic tracery” line is borrowed from the James Huneker’s Chopin: The Man and His Music (“At times so delicate is its design that it recalls the faint fantastic tracery
made by frost on glass”) — an analogy for music which Hofstadter re-concretizes into a description of frosted
glass. Pushkin’s original is “легкие узоры” (“faint patterns”); in spite of the unexpected non-Russian source, the degree of “poetic lie-sense” in this translation is really quite minimal.
Dear reader, friend or foe, at present
I’d like — whoever you might be —
To take my leave on terms most pleasant.
And thus farewell. Whate’er from me
You sought in this or that light stanza —
Some boist’rous souvenir bonanza,
Relief from toils and drudgery,
Some lively scenes, some jeux d’esprit —
Perhaps just errors in my grammar! —
God grant that in my modest art,
For entertainment, for your heart,
For dreams, or for the press’s yammer,
You’ve found at least a verse or two.
And on that note, farewell to you!
Blest he who quit life’s celebration
Ne’er having seen its full design,
Nor having drained his cup of wine;
Who shelved the book of life’s narration
Before he’d read its final line,
As I now, with Onegin mine.
These final few stanzas of the work again reflect the metaliterary theme, a favorite
of both Pushkin and his Translator, as they bid farewell
first to their readers, then to their novel’s protagonists, then to the novel itself (conveniently named for one of the protagonists,
enabling the pun — or rather, the double entendre — in the last line: Onegin as protagonist, Onegin as novel). A light, fleeting farewell to life itself adds to the sense of melancholy.
Note also that Hofstadter (almost
alone among the translators of Eugene Onegin) ties up his
translation for us with the same neat bow as Pushkin does:
the first and last words of the novel are identical (though
in different declensions).
A une damoyselle malade
Je vous donne
Le bon jour;
Et qu’on sorte
Le vous mande.
De ta bouche
Qui se couche
Si tu dures
Dieu te doint
A fine day.
From your cage
In brave flight,
So you might
Flee the croup.
Hope you swoop
And French bread,
You will lose
The bright hues
Of your plumes.
Heed my words
And take care.
Slip the snare
That does pinch
My wee finch.
You won’t be,
“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
a penciled-in poem on the page. I read the poem and was enormously
touched: it was called “My Chickadee” and was very beautifully
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
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