Gödel, Escher, Bach
Encyclopedia
Gödel, Escher, Bach: An Eternal Golden Braid (commonly GEB) is a book by Douglas Hofstadter
, described by his publishing company as "a metaphorical fugue
on minds and machines in the spirit of Lewis Carroll
".
On its surface, GEB examines logic
ian Kurt Gödel
, artist
M. C. Escher
and composer
Johann Sebastian Bach
, discussing common themes in their work and lives. At a deeper level, the book is an exposition of concepts fundamental to mathematics
, symmetry
, and intelligence.
Through illustration and analysis, the book discusses how self-reference and formal rules allow systems to acquire meaning despite being made of "meaningless" elements. It also discusses what it means to communicate, how knowledge can be represented and stored, the methods and limitations of symbolic representation, and even the fundamental notion of "meaning" itself.
In response to confusion over the book's theme
, Hofstadter has emphasized that GEB is not about mathematics, art, and music but rather about how cognition and thinking emerge from well-hidden neurological mechanisms. In the book, he presents an analogy about how the individual neurons of the brain
coordinate to create a unified sense of a coherent mind by comparing it to the social organization displayed in a colony
of ants.
and later by Lewis Carroll
in "What the Tortoise Said to Achilles
". These origins are related in the first two dialogues, and later ones introduce new characters such as the Crab. These narratives frequently dip into self-reference
and metafiction
.
Word play
also features prominently in the work. Puns are occasionally used to connect ideas, such as "the Magnificrab, Indeed" with Bach's Magnificat in D; "SHRDLU
, Toy of Man's Designing" with Bach's Jesu, Joy of Man's Desiring
; and "Typographical Number Theory
", or "TNT", which inevitably reacts explosively when it attempts to make statements about itself. One Dialogue contains a story about a genie (from the Arabic "Djinn") and various "tonics" (of both the liquid
and musical
varieties), which is titled "Djinn and Tonic
".
One dialogue in the book is written in the form of a crab canon
, in which every line before the midpoint corresponds to an identical line past the midpoint. The conversation still makes sense due to uses of common phrases that can be used as either greetings or farewells ("Good day") and the positioning of lines which double as an answer to a question in the next line. Another is a sloth canon, where one character repeats the lines of another, but slower and negated.
and self-reference
, where objects and ideas speak about or refer back to themselves. For instance, there is a phonograph that destroys itself by playing a record titled "I Cannot Be Played on Record Player X" (an analogy to Gödel's incompleteness theorems), an examination of canon
form in music
, and a discussion of Escher's lithograph of two hands drawing each other
. To describe such self-referencing objects, Hofstadter coins the term "strange loop
", a concept he examines in more depth in his follow-up book I Am a Strange Loop
. To escape many of the logical contradictions brought about by these self-referencing objects, Hofstadter discusses Zen
koans. He attempts to show readers how to perceive reality outside their own experience and embrace such paradoxical questions by rejecting the premise — a strategy also called "unasking"
.
Call stack
s are also discussed in GEB, as one dialogue describes the adventures of Achilles and the Tortoise as they make use of "pushing potion" and "popping tonic" involving entering and leaving different layers of reality. Subsequent sections discuss the basic tenets of logic, self-referring statements, ("typeless") systems, and even programming.
and contrapunctus (counterpoint
). In a dialogue between Achilles and the Tortoise, the author hints that there is a contrapunctal acrostic in the chapter that refers both to the author (Hofstadter) and Bach. This can be found by taking the first word of each paragraph, to reveal: Hofstadter's Contracrostipunctus Acrostically Backwards Spells "J. S. Bach". The second acrostic is found by taking the first letters of the first (in bold) and reading them backwards to get "J. S. Bach" (just as the first acrostic claims).
in 1980.
For Summer 2007, the Massachusetts Institute of Technology created an online course for high school students built around the book.
In its February 19, 2010 investigative summary on the 2001 anthrax attacks
, the Federal Bureau of Investigation
revealed that Bruce Edwards Ivins
was inspired by GEB to hide secret codes based upon nucleotide sequences in the anthrax-laced letters he allegedly sent in September and October 2001. He used bold letters, as suggested on page 404 of the book. He attempted to hide the book from investigators by throwing it in the trash.
" dialogue, which reads almost exactly the same forwards as backwards).
Hofstadter gives one example of translation trouble in the paragraph "Mr. Tortoise, Meet Madame Tortue", saying translators "instantly ran headlong into the conflict between the feminine gender of the French noun tortue and the masculinity of my character, the Tortoise". Hofstadter decided to translate the French character as "Madame Tortue", and the Italian version as "Signorina Tartaruga". Because of other troubles translators might have retaining the meaning of the book, Hofstadter "painstakingly went through every last sentence of GEB, annotating a copy for translators into any language that might be targeted".
Translation also gave Hofstadter a way to add new meaning and puns. For instance, in Chinese
, the subtitle is not a translation of an Eternal Golden Braid, but a seemingly unrelated phrase Jí Yì Bì (集异璧, literally "collection of exotic jades"), which is homophonic
to GEB in Chinese. Some material regarding this interplay is to be found in Hofstadter's later book Le Ton beau de Marot
, which is mainly about translation.
Douglas Hofstadter
Douglas Richard Hofstadter is an American academic whose research focuses on consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics...
, described by his publishing company as "a metaphorical fugue
Fugue
In music, a fugue is a compositional technique in two or more voices, built on a subject that is introduced at the beginning in imitation and recurs frequently in the course of the composition....
on minds and machines in the spirit of Lewis Carroll
Lewis Carroll
Charles Lutwidge Dodgson , better known by the pseudonym Lewis Carroll , was an English author, mathematician, logician, Anglican deacon and photographer. His most famous writings are Alice's Adventures in Wonderland and its sequel Through the Looking-Glass, as well as the poems "The Hunting of the...
".
On its surface, GEB examines logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
ian Kurt Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
, artist
Artist
An artist is a person engaged in one or more of any of a broad spectrum of activities related to creating art, practicing the arts and/or demonstrating an art. The common usage in both everyday speech and academic discourse is a practitioner in the visual arts only...
M. C. Escher
M. C. Escher
Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...
and composer
Composer
A composer is a person who creates music, either by musical notation or oral tradition, for interpretation and performance, or through direct manipulation of sonic material through electronic media...
Johann Sebastian Bach
Johann Sebastian Bach
Johann Sebastian Bach was a German composer, organist, harpsichordist, violist, and violinist whose sacred and secular works for choir, orchestra, and solo instruments drew together the strands of the Baroque period and brought it to its ultimate maturity...
, discussing common themes in their work and lives. At a deeper level, the book is an exposition of concepts fundamental to mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
, and intelligence.
Through illustration and analysis, the book discusses how self-reference and formal rules allow systems to acquire meaning despite being made of "meaningless" elements. It also discusses what it means to communicate, how knowledge can be represented and stored, the methods and limitations of symbolic representation, and even the fundamental notion of "meaning" itself.
In response to confusion over the book's theme
I Am a Strange Loop
I Am a Strange Loop is a 2007 book by Douglas Hofstadter, examining in depth the concept of a strange loop originally developed in his 1979 book Gödel, Escher, Bach....
, Hofstadter has emphasized that GEB is not about mathematics, art, and music but rather about how cognition and thinking emerge from well-hidden neurological mechanisms. In the book, he presents an analogy about how the individual neurons of the brain
Brain
The brain is the center of the nervous system in all vertebrate and most invertebrate animals—only a few primitive invertebrates such as sponges, jellyfish, sea squirts and starfishes do not have one. It is located in the head, usually close to primary sensory apparatus such as vision, hearing,...
coordinate to create a unified sense of a coherent mind by comparing it to the social organization displayed in a colony
Ant colony
An ant colony is an underground lair where ants live, eat and mate. Colonies consist of a series of underground chambers, connected to each other and the surface of the earth by small tunnels. There are rooms for nurseries, food storage, and mating...
of ants.
Structure
GEB takes the form of an interweaving of various narratives. The main chapters alternate with dialogues between imaginary characters, usually Achilles and the tortoise, first used by Zeno of EleaZeno of Elea
Zeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".- Life...
and later by Lewis Carroll
Lewis Carroll
Charles Lutwidge Dodgson , better known by the pseudonym Lewis Carroll , was an English author, mathematician, logician, Anglican deacon and photographer. His most famous writings are Alice's Adventures in Wonderland and its sequel Through the Looking-Glass, as well as the poems "The Hunting of the...
in "What the Tortoise Said to Achilles
What the Tortoise Said to Achilles
"What the Tortoise Said to Achilles", written by Lewis Carroll in 1895 for the philosophical journal Mind, is a brief dialogue which problematises the foundations of logic. The title alludes to one of Zeno's paradoxes of motion, in which Achilles could never overtake the tortoise in a race...
". These origins are related in the first two dialogues, and later ones introduce new characters such as the Crab. These narratives frequently dip into self-reference
Self-reference
Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
and metafiction
Metafiction
Metafiction, also known as Romantic irony in the context of Romantic works of literature, is a type of fiction that self-consciously addresses the devices of fiction, exposing the fictional illusion...
.
Word play
Word play
Word play or wordplay is a literary technique in which the words that are used become the main subject of the work, primarily for the purpose of intended effect or amusement...
also features prominently in the work. Puns are occasionally used to connect ideas, such as "the Magnificrab, Indeed" with Bach's Magnificat in D; "SHRDLU
SHRDLU
SHRDLU was an early natural language understanding computer program, developed by Terry Winograd at MIT from 1968-1970. In it, the user carries on a conversation with the computer, moving objects, naming collections and querying the state of a simplified "blocks world", essentially a virtual box...
, Toy of Man's Designing" with Bach's Jesu, Joy of Man's Desiring
Jesu, Joy of Man's Desiring
Jesu, Joy of Man's Desiring is the most common English title of the 10th movement of the cantata Herz und Mund und Tat und Leben, BWV 147 composed by Johann Sebastian Bach. A transcription by the English pianist Myra Hess was published in 1926 for piano solo and in 1934 for piano duet...
; and "Typographical Number Theory
Typographical Number Theory
Typographical Number Theory is a formal axiomatic system describing the natural numbers that appears in Douglas Hofstadter's book Gödel, Escher, Bach...
", or "TNT", which inevitably reacts explosively when it attempts to make statements about itself. One Dialogue contains a story about a genie (from the Arabic "Djinn") and various "tonics" (of both the liquid
Tonic water
Tonic water is a carbonated soft drink in which quinine is dissolved. Originally used as a prophylactic against malaria, tonic water usually now has a significantly lower quinine content and is consumed for its distinctively bitter taste...
and musical
Tonic (music)
In music, the tonic is the first scale degree of the diatonic scale and the tonal center or final resolution tone. The triad formed on the tonic note, the tonic chord, is thus the most significant chord...
varieties), which is titled "Djinn and Tonic
Gin and tonic
A gin and tonic is a highball cocktail made with gin and tonic water poured over ice. It is usually garnished with a slice or wedge of lime, or lemon. The amount of gin varies according to taste...
".
One dialogue in the book is written in the form of a crab canon
Crab canon
A crab canon—also known by the Latin form of the name, canon cancrizans—is an arrangement of two musical lines that are complementary and backward, similar to a palindrome. Originally it is a musical term for a kind of canon in which one line is reversed in time from the other . A famous example...
, in which every line before the midpoint corresponds to an identical line past the midpoint. The conversation still makes sense due to uses of common phrases that can be used as either greetings or farewells ("Good day") and the positioning of lines which double as an answer to a question in the next line. Another is a sloth canon, where one character repeats the lines of another, but slower and negated.
Themes
GEB contains many instances of recursionRecursion
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...
and self-reference
Self-reference
Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
, where objects and ideas speak about or refer back to themselves. For instance, there is a phonograph that destroys itself by playing a record titled "I Cannot Be Played on Record Player X" (an analogy to Gödel's incompleteness theorems), an examination of canon
Canon (music)
In music, a canon is a contrapuntal composition that employs a melody with one or more imitations of the melody played after a given duration . The initial melody is called the leader , while the imitative melody, which is played in a different voice, is called the follower...
form in music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
, and a discussion of Escher's lithograph of two hands drawing each other
Drawing Hands
Drawing Hands is a lithograph by the Dutch artist M. C. Escher first printed in January 1948. It depicts a sheet of paper out of which, from wrists that remain flat on the page, two hands rise, facing each other and in the paradoxical act of drawing one another into existence...
. To describe such self-referencing objects, Hofstadter coins the term "strange loop
Strange loop
A strange loop arises when, by moving up or down through a hierarchical system, one finds oneself back where one started.Strange loops may involve self-reference and paradox...
", a concept he examines in more depth in his follow-up book I Am a Strange Loop
I Am a Strange Loop
I Am a Strange Loop is a 2007 book by Douglas Hofstadter, examining in depth the concept of a strange loop originally developed in his 1979 book Gödel, Escher, Bach....
. To escape many of the logical contradictions brought about by these self-referencing objects, Hofstadter discusses Zen
Zen
Zen is a school of Mahāyāna Buddhism founded by the Buddhist monk Bodhidharma. The word Zen is from the Japanese pronunciation of the Chinese word Chán , which in turn is derived from the Sanskrit word dhyāna, which can be approximately translated as "meditation" or "meditative state."Zen...
koans. He attempts to show readers how to perceive reality outside their own experience and embrace such paradoxical questions by rejecting the premise — a strategy also called "unasking"
Mu (negative)
or Wu , is a word which has been translated variously as "not", "nothing", "without", "nothingness", "non existent", "non being", or evocatively simply as "no thing"...
.
Call stack
Call stack
In computer science, a call stack is a stack data structure that stores information about the active subroutines of a computer program. This kind of stack is also known as an execution stack, control stack, run-time stack, or machine stack, and is often shortened to just "the stack"...
s are also discussed in GEB, as one dialogue describes the adventures of Achilles and the Tortoise as they make use of "pushing potion" and "popping tonic" involving entering and leaving different layers of reality. Subsequent sections discuss the basic tenets of logic, self-referring statements, ("typeless") systems, and even programming.
Puzzles
The book is filled with puzzles. An example of this is the chapter titled "Contracrostipunctus", which combines the words acrosticAcrostic
An acrostic is a poem or other form of writing in which the first letter, syllable or word of each line, paragraph or other recurring feature in the text spells out a word or a message. As a form of constrained writing, an acrostic can be used as a mnemonic device to aid memory retrieval. A famous...
and contrapunctus (counterpoint
Counterpoint
In music, counterpoint is the relationship between two or more voices that are independent in contour and rhythm and are harmonically interdependent . It has been most commonly identified in classical music, developing strongly during the Renaissance and in much of the common practice period,...
). In a dialogue between Achilles and the Tortoise, the author hints that there is a contrapunctal acrostic in the chapter that refers both to the author (Hofstadter) and Bach. This can be found by taking the first word of each paragraph, to reveal: Hofstadter's Contracrostipunctus Acrostically Backwards Spells "J. S. Bach". The second acrostic is found by taking the first letters of the first (in bold) and reading them backwards to get "J. S. Bach" (just as the first acrostic claims).
Impact
Gödel, Escher, Bach won the Pulitzer PrizePulitzer Prize
The Pulitzer Prize is a U.S. award for achievements in newspaper and online journalism, literature and musical composition. It was established by American publisher Joseph Pulitzer and is administered by Columbia University in New York City...
in 1980.
For Summer 2007, the Massachusetts Institute of Technology created an online course for high school students built around the book.
In its February 19, 2010 investigative summary on the 2001 anthrax attacks
2001 anthrax attacks
The 2001 anthrax attacks in the United States, also known as Amerithrax from its Federal Bureau of Investigation case name, occurred over the course of several weeks beginning on Tuesday, September 18, 2001, one week after the September 11 attacks. Letters containing anthrax spores were mailed to...
, the Federal Bureau of Investigation
Federal Bureau of Investigation
The Federal Bureau of Investigation is an agency of the United States Department of Justice that serves as both a federal criminal investigative body and an internal intelligence agency . The FBI has investigative jurisdiction over violations of more than 200 categories of federal crime...
revealed that Bruce Edwards Ivins
Bruce Edwards Ivins
Bruce Edwards Ivins was an American microbiologist, vaccinologist, senior biodefense researcher at the United States Army Medical Research Institute of Infectious Diseases in Fort Detrick, Maryland and the key suspect in the 2001 anthrax attacks.On Tuesday, July 29, 2008 he died of an overdose of...
was inspired by GEB to hide secret codes based upon nucleotide sequences in the anthrax-laced letters he allegedly sent in September and October 2001. He used bold letters, as suggested on page 404 of the book. He attempted to hide the book from investigators by throwing it in the trash.
Translation
Although Hofstadter claims the idea of translating his book "never crossed [his] mind" when he was writing it, when approached with the idea by his publisher he was "very excited about seeing [the] book in other languages, especially… French". He knew, however, that "there were a million issues to consider" when translating, since the book relies not only on word-play but "structural puns" as well—writing where the form and content of the work mirror each other (such as the "Crab CanonCrab canon
A crab canon—also known by the Latin form of the name, canon cancrizans—is an arrangement of two musical lines that are complementary and backward, similar to a palindrome. Originally it is a musical term for a kind of canon in which one line is reversed in time from the other . A famous example...
" dialogue, which reads almost exactly the same forwards as backwards).
Hofstadter gives one example of translation trouble in the paragraph "Mr. Tortoise, Meet Madame Tortue", saying translators "instantly ran headlong into the conflict between the feminine gender of the French noun tortue and the masculinity of my character, the Tortoise". Hofstadter decided to translate the French character as "Madame Tortue", and the Italian version as "Signorina Tartaruga". Because of other troubles translators might have retaining the meaning of the book, Hofstadter "painstakingly went through every last sentence of GEB, annotating a copy for translators into any language that might be targeted".
Translation also gave Hofstadter a way to add new meaning and puns. For instance, in Chinese
Chinese language
The Chinese language is a language or language family consisting of varieties which are mutually intelligible to varying degrees. Originally the indigenous languages spoken by the Han Chinese in China, it forms one of the branches of Sino-Tibetan family of languages...
, the subtitle is not a translation of an Eternal Golden Braid, but a seemingly unrelated phrase Jí Yì Bì (集异璧, literally "collection of exotic jades"), which is homophonic
Homophone
A homophone is a word that is pronounced the same as another word but differs in meaning. The words may be spelled the same, such as rose and rose , or differently, such as carat, caret, and carrot, or to, two, and too. Homophones that are spelled the same are also both homographs and homonyms...
to GEB in Chinese. Some material regarding this interplay is to be found in Hofstadter's later book Le Ton beau de Marot
Le Ton beau de Marot
Le Ton beau de Marot: In Praise of the Music of Language , published by Basic Books in 1997, is a book by Douglas Hofstadter in which he explores the meaning, strengths, failings, and beauty of translation....
, which is mainly about translation.
See also
External links
- GEB video lectures, MIT OpenCourseWareMIT OpenCourseWareMIT OpenCourseWare is an initiative of the Massachusetts Institute of Technology to put all of the educational materials from its undergraduate- and graduate-level courses online, partly free and openly available to anyone, anywhere. MIT OpenCourseWare is a large-scale, web-based publication of...
- Partial contents of 20th Anniversary Edition
- Mårten's GEB site
- Class about GEB, at the University of Michigan
- Java 3D game based on the GEB triplets