Gödel, Escher, Bach: An Eternal Golden Braid (commonly
GEB) is a
Pulitzer PrizeThe Pulitzer Prize is a U.S. award for achievements in newspaper journalism, literature and musical composition. It was established by Hungarian-American publisher Joseph Pulitzer and is administered by Columbia University in New York City....
-winning book by
Douglas HofstadterDouglas Richard Hofstadter is an American academic whose research focuses on consciousness, thinking and creativity. He is best known for his book Gödel, Escher, Bach: an Eternal Golden Braid, first published in 1979, for which he was awarded the 1980 Pulitzer Prize for general non-fiction...
, described as "a metaphorical
fugueIn music, a fugue is a type of contrapuntal composition or technique of composition for a fixed number of parts, normally referred to as "voices". In the Middle Ages, the term was widely used to denote any works in canonic style; by the Renaissance, it had come to denote specifically imitative works...
on minds and machines in the spirit of
Lewis Carroll Charles Lutwidge Dodgson , better known by the pen name Lewis Carroll , was an English author, mathematician, logician, Anglican deacon and photographer...
".
On its surface,
GEB examines
logicLogic, from the Greek λογική is the art and science of reasoning. More specifically, it is defined by the Penguin Encyclopedia to be "The formal systematic study of the principles of valid inference and correct reasoning". As a discipline, logic dates back to Aristotle, who established its...
ian
Kurt GödelKurt Gödel was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N...
,
artistThe definition of an artist is wide-ranging and covers a broad spectrum of activities to do with creating art, practicing the arts and/or demonstrating an art. the worlds best artist is a man named mitchell peter lay who is often loved by the ladies. The common useage in both everyday speech and...
M. C. EscherMaurits Cornelis Escher , usually referred to as M.C. Escher , was a Dutch-Frisian graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...
and
composerA composer is a person who creates music, usually by musical notation, for interpretation and performance. The level of distinction between composers and other musicians varies, which affects issues such as copyright and the deference given to individual interpretations of a particular piece of...
Johann Sebastian BachJohann Sebastian Bach was a German composer and organist whose ecclesiastical 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 a detailed and subtle 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.
Gödel, Escher, Bach: An Eternal Golden Braid (commonly
GEB) is a
Pulitzer PrizeThe Pulitzer Prize is a U.S. award for achievements in newspaper journalism, literature and musical composition. It was established by Hungarian-American publisher Joseph Pulitzer and is administered by Columbia University in New York City....
-winning book by
Douglas HofstadterDouglas Richard Hofstadter is an American academic whose research focuses on consciousness, thinking and creativity. He is best known for his book Gödel, Escher, Bach: an Eternal Golden Braid, first published in 1979, for which he was awarded the 1980 Pulitzer Prize for general non-fiction...
, described as "a metaphorical
fugueIn music, a fugue is a type of contrapuntal composition or technique of composition for a fixed number of parts, normally referred to as "voices". In the Middle Ages, the term was widely used to denote any works in canonic style; by the Renaissance, it had come to denote specifically imitative works...
on minds and machines in the spirit of
Lewis Carroll Charles Lutwidge Dodgson , better known by the pen name Lewis Carroll , was an English author, mathematician, logician, Anglican deacon and photographer...
".
On its surface,
GEB examines
logicLogic, from the Greek λογική is the art and science of reasoning. More specifically, it is defined by the Penguin Encyclopedia to be "The formal systematic study of the principles of valid inference and correct reasoning". As a discipline, logic dates back to Aristotle, who established its...
ian
Kurt GödelKurt Gödel was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N...
,
artistThe definition of an artist is wide-ranging and covers a broad spectrum of activities to do with creating art, practicing the arts and/or demonstrating an art. the worlds best artist is a man named mitchell peter lay who is often loved by the ladies. The common useage in both everyday speech and...
M. C. EscherMaurits Cornelis Escher , usually referred to as M.C. Escher , was a Dutch-Frisian graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...
and
composerA composer is a person who creates music, usually by musical notation, for interpretation and performance. The level of distinction between composers and other musicians varies, which affects issues such as copyright and the deference given to individual interpretations of a particular piece of...
Johann Sebastian BachJohann Sebastian Bach was a German composer and organist whose ecclesiastical 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 a detailed and subtle 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 themeI 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 coordinate to create a unified sense of a coherent mind by comparing it to the social organization displayed in a colony of ants.
Structure
GEB takes the form of an interweaving of various narratives. The main chapters alternate with dialogues between imaginary characters, inspired by Lewis Carroll's "
What the Tortoise Said to Achilles"What the Tortoise Said to Achilles" is a brief dialogue by Lewis Carroll which playfully 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...
", in which Achilles and the Tortoise discuss a paradox related to
modus ponensIn classical logic, modus ponendo ponens is a valid, simple argument form sometimes referred to as affirming the antecedent or the law of detachment...
. Hofstadter bases the other dialogues on this one, introducing characters such as a Crab, a Genie, and others. These narratives frequently dip into
self-referenceSelf-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
metafictionMetafiction is a type of fiction that self-consciously addresses the devices of fiction, exposing the fictional illusion. It is the literary term describing fictional writing that self-consciously and systematically draws attention to its status as an artifact in posing questions about the...
.
Word playFor the rapper see WordplayWord play is a literary technique in which the words that are used become the main subject of the work. Puns, phonetic mix-ups such as spoonerisms, obscure words and meanings, clever rhetorical excursions, oddly formed sentences, and telling character names are common...
also features prominently in the work. Puns are occasionally used to connect ideas, such as "the Magnificrab, Indeed" with Bach's
Magnificat in D; "
SHRDLUSHRDLU was an early natural language understanding computer program, developed by Terry Winograd at MIT from 1968-1970. It was written in the Micro Planner and Lisp programming language on the DEC PDP-6 computer and a DEC graphics terminal...
, Toy of Man's Designing" with Bach's
Jesu, Joy of Man's Desiring"Jesu, Joy of Man's Desiring" is the 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 TheoryTypographical Number Theory is a formal axiomatic system describing the natural numbers that appears in Douglas Hofstadter's book Gödel, Escher, Bach...
", or "
TNTTrinitrotoluene , or more specifically, 2,4,6-trinitrotoluene, is a chemical compound with the formula C
6H
23CH
3. This yellow-coloured solid is a reagent in chemistry but is best known as a useful explosive material with convenient handling properties...
", 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
liquidTonic water is a carbonated soft drink flavoured with quinine, which gives it a distinctively bitter taste.The drink has garnered its name effects of this bitter flavouring...
and
musicalThe tonic is the first note of a musical scale in the tonal method of musical composition. The triad formed on the tonic note, the tonic chord, is thus the most significant chord. More generally, the tonic is the pitch upon which all other pitches of a piece are hierarchically referenced...
varieties), which is of course entitled "
Djinn and TonicA gin and tonic is a highball cocktail made with gin and tonic water, usually garnished with a slice of lime or lemon and often served over ice...
".
One dialogue in the book is written in the form of a
crab canonA crab canon is an arrangement of two things 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 is found in J. S...
, 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 which can be used as either greetings or farewells ("Good day") and the positioning of lines which, upon close inspection, double as an answer to a question in the next line.
Themes
GEB contains many instances where objects and ideas speak about or refer back to themselves (cf.
recursionRecursion, in mathematics and computer science, is a method of defining functions in which the function being defined is applied within its own definition. The term is also used more generally to describe a process of repeating objects in a self-similar way...
and
self-referenceSelf-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...
). For instance,
TNTTypographical Number Theory is a formal axiomatic system describing the natural numbers that appears in Douglas Hofstadter's book Gödel, Escher, Bach...
is an illustration of Gödel's incompleteness theorem. There is also a phonograph which destroys itself by playing a record entitled "I Cannot Be Played on Record Player X" (this being an analogy to Gödel's incompleteness theorem), an examination of
canonIn 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
musicMusic is an art form whose medium is sound. Common elements of music are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
, and a discussion of Escher's
lithograph of two hands drawing each otherDrawing 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 rise, from wrists that remain flat on the page, two hands, 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 loopA 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 LoopI 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
ZenZen is a school of Mahāyāna Buddhism, translated from the Chinese word Chán. This word is in turn derived from the Sanskrit dhyāna, which means "meditation" ....
koanA kōan is a story, dialogue, question, or statement in the history and lore of Zen Buddhism, generally containing aspects that are inaccessible to rational understanding, yet may be accessible to intuition...
s. He attempts to show the reader how to perceive reality outside the normal confines of their own experience and embrace such paradoxical questions by rejecting the premise — a strategy also called
"unasking"Mu , and Wu is a word which has been roughly translated as "no", "none", "null", "without", "no meaning"...
.
Call stackIn 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, function stack, or run-time 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" and "popping" tonics. Entering a picture in a book would count as "pushing", entering a picture in a book within a picture in a book would have caused a double "pushing", and "popping" refers to an exit back to the previous layer of reality. The Tortoise humorously remarks that a friend of his (a weasel) performed a "popping" while in their current state of reality and has never been heard from since; the implied question is, "Did the friend simply cease to exist, or has the friend achieved a higher state of reality?" Also, since the reader is "pushed" into the world of Tortoise and Achilles, would the friend have ascended to the same level of reality in which the readers of
GEB reside? Subsequent sections discuss the basic tenets of logic, self-referring statements, ("typeless") systems, and even programming.
One puzzle (in the dialogue "Aria with Diverse Variations") is a speculation concerning an author who writes a book and chooses to end the story without actually stopping the text. That an author cannot make a sudden ending (with regard to the story) come as a surprise, when the fact that there are only a few pages left in the book is obvious to the reader. Such an author might wrap up the main point, and then continue writing, but drop clues to the reader that the end has already passed, such as wandering and unfocused prose, misstatements, or contradictions.
Puzzles
The book is filled with puzzles. An example of this is the chapter entitled "Contracrostipunctus", which combines the words
acrosticAn acrostic is a poem or other form of writing in an alphabetic script, 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...
and
contrapunctus (
counterpointIn 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 Western 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 contrapuntal 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". This is only the acrostic. The counterpoint acrostic is found by taking the first letters of the acrostic (in bold) and reading them backwards to get "J. S. Bach" (as the acrostic itself claims).
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 CanonA crab canon is an arrangement of two things 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 is found in J. S...
" 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
ChineseChinese or the Sinitic language is a language family consisting of languages mutually unintelligible to varying degrees. Originally the indigenous languages spoken by the Han Chinese in China, it forms one of the two 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
homophonicA 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 MarotLe 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.
People featured in Gödel, Escher, Bach
- Charles Babbage
Charles Babbage, FRS was an English mathematician, philosopher, inventor and mechanical engineer who originated the concept of a programmable computer. Parts of his uncompleted mechanisms are on display in the London Science Museum. In 1991, a perfectly functioning difference engine was...
- Johann Sebastian Bach
Johann Sebastian Bach was a German composer and organist whose ecclesiastical and secular works for choir, orchestra, and solo instruments drew together the strands of the Baroque period and brought it to its ultimate maturity...
- John Cage
John Milton Cage Jr. was an American composer, philosopher, poet, music theorist, artist, printmaker, and amateur mycologist and mushroom collector. A pioneer of chance music, electronic music and non-standard use of musical instruments, Cage was one of the leading figures of the post-war...
- Lewis Carroll
Charles Lutwidge Dodgson , better known by the pen name Lewis Carroll , was an English author, mathematician, logician, Anglican deacon and photographer...
- Georg Cantor
Georg Ferdinand Ludwig Phillip Cantor was a German mathematician, born in Russia. He is best known as the creator of set theory, which has become a fundamental theory in mathematics...
- Alonzo Church
Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem.- Life :Alonzo...
- M. C. Escher
Maurits Cornelis Escher , usually referred to as M.C. Escher , was a Dutch-Frisian graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...
- Frederick the Great
- Kurt Gödel
Kurt Gödel was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N...
- Douglas R. Hofstadter
- Johann Kirnberger
Johann Philipp Kirnberger was a musician, composer , and music theorist. A pupil of Johann Sebastian Bach, he became a violinist at the court of Frederick II of Prussia in 1751. He was the music director to the Prussian Princess Anna Amalia from 1758 until his death. Kirnberger greatly admired J.S...
- René Magritte
René François Ghislain Magritte
[p] was a Belgian surrealist artist. He became well known for a number of witty and thought-provoking images...
- Marvin Minsky
Marvin Lee Minsky is an American cognitive scientist in the field of artificial intelligence , co-founder of Massachusetts Institute of Technology's AI laboratory, and author of several texts on AI and philosophy.-Biography:...
- Johann Joachim Quantz
Johann Joachim Quantz was a German flautist, flute maker and composer.-Biography:Quantz was born in Oberscheden, near Göttingen, Germany, and died in Potsdam....
- Willard Van Orman Quine
Willard Van Orman Quine was an American philosopher and logician in the analytic tradition...
- Srinivasa Ramanujan
Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was an Indian mathematician and self taught genius who, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued...
- Alfred Tarski
Alfred Tarski was a Polish logician and mathematician...
- Alan Turing
Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was influential in the development of computer science and provided an influential formalisation of the concept of the algorithm and computation with the Turing machine...
- Terry Winograd
Terry Allen Winograd is an American professor of computer science at Stanford University, and co-director of the Stanford Human-Computer Interaction Group...
- Zhaozhou Congshen
See also
External links