Gödel, Escher, Bach: An Eternal Golden Braid
by Douglas Hofstadter
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Description
Douglas Hofstadter's book is concerned directly with the nature of "maps' or links between formal systems. However, according to Hofstadter, the formal system that underlies all mental activity transcends the system that supports it. If life can grow out of the formal chemical substrate of the cell, if consciousness can emerge out of a formal system of firing neurons, then so too will computers attain human intelligence. Gödel, Escher, Bach is a wonderful exploration of fascinating ideas at show more the heart of cognitive science: meaning, reduction, recursion, and much more show lessTags
Recommendations
Member Recommendations
Zaklog Cryptonomicon strikes me as the kind of book that Hofstadter would write if he wrote fiction. Both books are complex, with discursive passages on mathematics and a positively weird sense of humor. If you enjoyed (rather than endured) the explanatory sections on cryptography and the charts of Waterhouse's love life (among other, rarely charted things) you should really like this book.
111
EerierIdyllMeme An obvious suggestion (surprised it's not here already). Both are creative and fictional riffing off of formal logic and incompleteness.
Also recommended by tomduck
70
heidialice GEB is a thousand times as intense, but if you enjoyed the parts about self-referentiality it's worth a skim. Conversely, if GEB is just too much, Paulos' concise introduction to the theme is very accessible.
20
EerierIdyllMeme A few similar themes (Bach, human cognition) come up in similar ways.
01
P_S_Patrick Arturo Perez-Reverte has recieved inspiration for his excellent mystery thriller from Hofstadter's Godel Escher Bach, even without some of the chapter introduciton quotes, that much is clear. He uses the bewildering Escherian theme of worlds within a world, Godels incompleteness theorum is alluded to in the monologue of one character, and Bach is discussed in relevance to the mystery too, along with a few miscellaneous paradoxes which are also slipped in, in a similar spirit in which they permeate the more complex non-fictional work. Non-fiction readers who have enjoyed GEB should be amused by the Flanders panel, and I think they should enjoy it even if they do not often dip into fiction. It would be harder to recommend GEB to fans of the Flanders Panel, due to its sheer length, but if you were intrigued by the themes in the story then it should at least be worth finding GEB in a library and dipping into it.
04
Lorem Things in 4D I consider a more accessible version of GEB in its breadth and how it does get to complex topics. If you enjoyed the more complicated parts of 4D, definitely look at GEB and if GEB was a little too much, 4D might remind you why math(s) are never boring
Member Reviews
Can human thought processes be emulated artificially? This is very much a "how" question and Hofstadter explores it as such, looking for the most promising way forward. He believes it lies in the study of self-reference and tackles this from multiple angles across a variety of disciplines (music, art, mathematics, computer programming, molecular biology, etc.), on the theory that without self-reference you can't arrive at self-awareness and full consciousness or actual thinking. He makes the discussion a fascinating one in how ably he ties things together that I wouldn't have thought had much in common.
Jumping to the end, much of the concluding chapters demonstrate the extremely complicated and multi-layered processes that emerge when show more you break down even the most simple mental processes in humans, demonstrating how very difficult it is to emulate these processes artificially. Hofstadter seems wedded to the idea that artificial intelligence means closely emulating human thought and how it is arrived at. Getting to the same (or improved) results by any different means doesn't seem to count as true AI in his opinion. He tries to prognosticate what artificial intelligence will and will not be capable of in future and you could say that he still isn't wrong, pointing to his own definition. The AI that he envisions is not GenAI like ChatGPT but something that still lies well beyond us, a computer that really does think (and feel?) the same independent way that a person does, and consequently has many of our same flaws as well. Whether computer science today or in future sees any utility in trying to realize the AI of Hofstadter's vision is doubtful. To me it seems fair to weigh his 1979 vision against the direction things are actually going in 2026 and say that he has already missed the mark in several respects, all stemming from his missing the "why" that industrial science has actually been motivated by.
What we are popularly defining as AI today is function-oriented, and possibly all it will ever be. While it can be increasingly made to look and sound human, it will never "think" because that isn't its objective. This book remains brilliant for its wonderfully entertaining way of exploring the possibility of something beyond that - like Commander Data from Star Trek - which perhaps some niche areas of AI study are still pursuing, and how they might get there. show less
Jumping to the end, much of the concluding chapters demonstrate the extremely complicated and multi-layered processes that emerge when show more you break down even the most simple mental processes in humans, demonstrating how very difficult it is to emulate these processes artificially. Hofstadter seems wedded to the idea that artificial intelligence means closely emulating human thought and how it is arrived at. Getting to the same (or improved) results by any different means doesn't seem to count as true AI in his opinion. He tries to prognosticate what artificial intelligence will and will not be capable of in future and you could say that he still isn't wrong, pointing to his own definition. The AI that he envisions is not GenAI like ChatGPT but something that still lies well beyond us, a computer that really does think (and feel?) the same independent way that a person does, and consequently has many of our same flaws as well. Whether computer science today or in future sees any utility in trying to realize the AI of Hofstadter's vision is doubtful. To me it seems fair to weigh his 1979 vision against the direction things are actually going in 2026 and say that he has already missed the mark in several respects, all stemming from his missing the "why" that industrial science has actually been motivated by.
What we are popularly defining as AI today is function-oriented, and possibly all it will ever be. While it can be increasingly made to look and sound human, it will never "think" because that isn't its objective. This book remains brilliant for its wonderfully entertaining way of exploring the possibility of something beyond that - like Commander Data from Star Trek - which perhaps some niche areas of AI study are still pursuing, and how they might get there. show less
"Help. My mind has bent, and now I cannot unbend it!"
That was me after I finished reading Godel, Escher, Bach. This book, which is about, well, everything, takes the reader on a journey through mathematics, music, and art, and gives it a little twist just to keep one on one's mental toes.
A major theme is recursion, or self-referencing. If you're at all familiar with any of the GEBs, you'll see this in Godel's Incompleteness Theorem, Escher's Drawing Hands, and Bach's Crab Canon. Other mathematicians, artists, and musicians are introduced as well, providing more on this Eternal Golden Braid.
Not only does Hofstadter give us so much on logical themes, but he also gives the reader some puzzles too, particularly some that require multiple show more steps (though the answers are right in front of the reader's face at times).
This book is a must read for one who considers oneself a student of mathematics, art, or even music, or who has a strong admiration for most of these things. I suppose computer scientists could read it too.
Nevertheless, this is a great book, and a challenge, but definitely worth the read. show less
That was me after I finished reading Godel, Escher, Bach. This book, which is about, well, everything, takes the reader on a journey through mathematics, music, and art, and gives it a little twist just to keep one on one's mental toes.
A major theme is recursion, or self-referencing. If you're at all familiar with any of the GEBs, you'll see this in Godel's Incompleteness Theorem, Escher's Drawing Hands, and Bach's Crab Canon. Other mathematicians, artists, and musicians are introduced as well, providing more on this Eternal Golden Braid.
Not only does Hofstadter give us so much on logical themes, but he also gives the reader some puzzles too, particularly some that require multiple show more steps (though the answers are right in front of the reader's face at times).
This book is a must read for one who considers oneself a student of mathematics, art, or even music, or who has a strong admiration for most of these things. I suppose computer scientists could read it too.
Nevertheless, this is a great book, and a challenge, but definitely worth the read. show less
I started reading this book almost simultaneously with my application for an Artificial Intelligence master at my University. Honestly, I got a little frustrated with the one-sided approach to AI that I read about, the premise always seems to be 'artificial intelligence'=='machine learning'. So I was entirely happy during my read of this book. It gave me a playful introduction and a new look into first-order logic, a subject in which I was already pretty invested, as well as an endless supply of inspiration on which to draw in my further AI adventures. I loved this book for it and I cannot believe that no-one in my first few explorations with AI told me to stop what I was doing and read this tome first.
I did have some issues with the show more book. While I loved the first few dialoges and I was thoroughly impressed with the underlying themes of the dialogs, they did become somewhat stale and forced after a few of them. Also, the constant meta-ness that Hofstadter supplies is very interesting and part of what makes the book great, but at some times this also seemed a little forced and it undermined the credability of the story just a little.
Even so, writing (and reading for that matter) this book has been an amazing feat and I cannot imagine that I will not read it again some time and take even more insight away from it. This book should be mandatory reading for anyone doing something or another in the field of AI, or conciousness or anything related. show less
I did have some issues with the show more book. While I loved the first few dialoges and I was thoroughly impressed with the underlying themes of the dialogs, they did become somewhat stale and forced after a few of them. Also, the constant meta-ness that Hofstadter supplies is very interesting and part of what makes the book great, but at some times this also seemed a little forced and it undermined the credability of the story just a little.
Even so, writing (and reading for that matter) this book has been an amazing feat and I cannot imagine that I will not read it again some time and take even more insight away from it. This book should be mandatory reading for anyone doing something or another in the field of AI, or conciousness or anything related. show less
This was the second book I bought from Amazon when it launched in the UK: has taken me a decade (and a philosophy degree) to finish it.
It's not a quick read, then. Partly because it assumes nothing and happily digresses down to first principles, but also because you spend half the time staring into space and thinking about the ideas it raises.
(The digressions are something else, mind. Had to laugh about 4/5s of the way in when he says "now we can come to the main theme".)
It's not a quick read, then. Partly because it assumes nothing and happily digresses down to first principles, but also because you spend half the time staring into space and thinking about the ideas it raises.
(The digressions are something else, mind. Had to laugh about 4/5s of the way in when he says "now we can come to the main theme".)
Pensare il pensiero
Posseggo questo libro dal mese di gennaio 1985. Un saggio lungo quasi mille pagine che parte dal principio di verificare se vi sia un confine tra il cervello umano e l'intelligenza artificiale.
Questo libro, considerato una pietra miliare delle riflessioni contemporanee e del discorso sulle due culture, quella scientifica e quella umanistica, è un libro davvero importante e intelligente, uno dei più difficili che abbia mai letto. A dire il vero questo è un libro che va letto e riletto in continuazione. Ogni volta che l'intelligenza cerca di osservare con distacco se stessa crea un circolo vizioso, una "eterna ghirlanda brillante" come dice il sottotitolo: pensare il pensiero comporta inevitabilmente ripensare il show more pensiero e così via.
Nascono così una serie di riflessioni sulle quali occorre riflettere perchè ognuna contiene un limite proprio in questo riflettersi in se stessa. E' un gioco teorico e strutturale che l'autore ritrova nel teorema matematico di Godel, nei geometrici, logici disegni dalle prospettive impazziate dell'incisore olandese Escher e nello svilupparsi dell'arte della fuga di Bach. Il queste pagine riflessioni sul dna e i programmi per i computer, la matematica moderna e classica i paradossi come quello di Achille e la tartaruga convivono con la filosofia zen, la musica di John Cage e un pò tutte le branchie della conoscenza umana in un lavoro di sintesi che continuamente si apre e si chiude. Esattamente ciò che faccio io da circa 30 anni! show less
Posseggo questo libro dal mese di gennaio 1985. Un saggio lungo quasi mille pagine che parte dal principio di verificare se vi sia un confine tra il cervello umano e l'intelligenza artificiale.
Questo libro, considerato una pietra miliare delle riflessioni contemporanee e del discorso sulle due culture, quella scientifica e quella umanistica, è un libro davvero importante e intelligente, uno dei più difficili che abbia mai letto. A dire il vero questo è un libro che va letto e riletto in continuazione. Ogni volta che l'intelligenza cerca di osservare con distacco se stessa crea un circolo vizioso, una "eterna ghirlanda brillante" come dice il sottotitolo: pensare il pensiero comporta inevitabilmente ripensare il show more pensiero e così via.
Nascono così una serie di riflessioni sulle quali occorre riflettere perchè ognuna contiene un limite proprio in questo riflettersi in se stessa. E' un gioco teorico e strutturale che l'autore ritrova nel teorema matematico di Godel, nei geometrici, logici disegni dalle prospettive impazziate dell'incisore olandese Escher e nello svilupparsi dell'arte della fuga di Bach. Il queste pagine riflessioni sul dna e i programmi per i computer, la matematica moderna e classica i paradossi come quello di Achille e la tartaruga convivono con la filosofia zen, la musica di John Cage e un pò tutte le branchie della conoscenza umana in un lavoro di sintesi che continuamente si apre e si chiude. Esattamente ciò che faccio io da circa 30 anni! show less
Never mind the Escher and Bach stuff; that’s just window dressing. This book is about Godel’s Theorem. And wow, what a book.
Imagine a glorious future in which, by means of magic and genetic engineering, the human species is transformed into a better, smarter, faster, more beautiful, more creative, more moral, stronger, happier species, a more alive species. We make Elysium, then we live in the Elysium we’ve created.
In this Arcadia, this Heaven, this Eden, this Platonic Form of the world animated and electrified by benevolent intelligence, you walk across grassy fields and you see the whole thing, The Dream:
Everyone is wearing flowing white robes. (Why? Just because.)
Over there athletic people engage in athletic contests, their show more good-natured competition embodying grace, fluidity, and the confidence of a well-disciplined, healthy body.
Over here, mathematicians use sticks to draw in the dirt on a river bank, proving astoundingly beautiful and useful new theorems.
In another direction a young man or woman lounges, back against a tree, releasing sweet strains of melody into the air by means of some sort of elegant string instrument.
Are you with me?
Okay.
In that universe, every non-fiction book is this good.
What’s it about?
It’s about, principally, Godel’s Theorem. The other stuff, at least in the first part (Escher, Bach, etc.), is just add-ons. Godel’s Theorem is often mis-characterized as “disproving all of mathematics!” or some similar nonsense. No. It says something about formal mathematical systems, systems of clearly stated axioms with clearly stated rules of inference for deriving implications of the axioms. The theorem essentially says that any formal system sophisticated enough to be used for number theory - reasoning about integers - either has internal inconsistencies or is unable to prove every truth in number theory.
This does not “Undercut all of mathematics” or whatever. It simply means that a consistent formalistic approach to mathematics can never derive all mathematical truths. There are some truths that can only be proven in other ways. Indeed, Godel shows how to prove some of those truths by reasoning outside formal systems!
To prove it, Godel had the profound insight that any formal system can be re-interpreted as a set of numbers and arithmetical operations on them, so formal number theory talks about itself! This is so cool.
E.g., suppose your formal system has the symbol string x#@^&?G-!y. (This might mean, say, “x is the largest number in the prime factorization of y.”) We also have a rule that allows us to derive x(=y (x is not greater than y) from the first string. But we also can interpret x as 5, # as 0, @ as 2, and so on, so the initial symbol string can be interpreted as a number. And so the second string is a number that we can derive from the first. So the rules of inference in this interpretation are arithmetic operations on numbers. Thus we can apply mathematical reasoning to the system and derive conclusions about the symbol strings it will generate and those that it won’t generate.
A simplified analogy: Suppose that we can prove - by reasoning outside the formal system - that the system will never produce a string whose number is prime. What Godel proved, in this analogy, was there is always a symbol string that asserts “N is a prime number” (in the first interpretation) whose number was N (in the second interpretation). Thus, if the statement is true, the formal system will never prove it!
(It is possible to verify that a well-designed system will never "prove" a false statement, so you can avoid that problem.)
In fact, not only do such true-but-formally-unprovable statements exist, in any formal system complex enough to be useful, but an infinity of them exists!
It was the idea of reinterpreting the symbols as numbers that was Godel’s real stroke of freakin’ genius. The theorem is based on that.
Anyway: The next time someone tells you, “Godel’s Theorem proves that all mathematics is invalid,” or whatever, just give them a wedgie and move on. All it proves is that a certain approach to mathematics cannot prove everything. Which, unless you had unrealistic ambitions for it in the first place, is not that surprising. show less
Imagine a glorious future in which, by means of magic and genetic engineering, the human species is transformed into a better, smarter, faster, more beautiful, more creative, more moral, stronger, happier species, a more alive species. We make Elysium, then we live in the Elysium we’ve created.
In this Arcadia, this Heaven, this Eden, this Platonic Form of the world animated and electrified by benevolent intelligence, you walk across grassy fields and you see the whole thing, The Dream:
Everyone is wearing flowing white robes. (Why? Just because.)
Over there athletic people engage in athletic contests, their show more good-natured competition embodying grace, fluidity, and the confidence of a well-disciplined, healthy body.
Over here, mathematicians use sticks to draw in the dirt on a river bank, proving astoundingly beautiful and useful new theorems.
In another direction a young man or woman lounges, back against a tree, releasing sweet strains of melody into the air by means of some sort of elegant string instrument.
Are you with me?
Okay.
In that universe, every non-fiction book is this good.
What’s it about?
It’s about, principally, Godel’s Theorem. The other stuff, at least in the first part (Escher, Bach, etc.), is just add-ons. Godel’s Theorem is often mis-characterized as “disproving all of mathematics!” or some similar nonsense. No. It says something about formal mathematical systems, systems of clearly stated axioms with clearly stated rules of inference for deriving implications of the axioms. The theorem essentially says that any formal system sophisticated enough to be used for number theory - reasoning about integers - either has internal inconsistencies or is unable to prove every truth in number theory.
This does not “Undercut all of mathematics” or whatever. It simply means that a consistent formalistic approach to mathematics can never derive all mathematical truths. There are some truths that can only be proven in other ways. Indeed, Godel shows how to prove some of those truths by reasoning outside formal systems!
To prove it, Godel had the profound insight that any formal system can be re-interpreted as a set of numbers and arithmetical operations on them, so formal number theory talks about itself! This is so cool.
E.g., suppose your formal system has the symbol string x#@^&?G-!y. (This might mean, say, “x is the largest number in the prime factorization of y.”) We also have a rule that allows us to derive x(=y (x is not greater than y) from the first string. But we also can interpret x as 5, # as 0, @ as 2, and so on, so the initial symbol string can be interpreted as a number. And so the second string is a number that we can derive from the first. So the rules of inference in this interpretation are arithmetic operations on numbers. Thus we can apply mathematical reasoning to the system and derive conclusions about the symbol strings it will generate and those that it won’t generate.
A simplified analogy: Suppose that we can prove - by reasoning outside the formal system - that the system will never produce a string whose number is prime. What Godel proved, in this analogy, was there is always a symbol string that asserts “N is a prime number” (in the first interpretation) whose number was N (in the second interpretation). Thus, if the statement is true, the formal system will never prove it!
(It is possible to verify that a well-designed system will never "prove" a false statement, so you can avoid that problem.)
In fact, not only do such true-but-formally-unprovable statements exist, in any formal system complex enough to be useful, but an infinity of them exists!
It was the idea of reinterpreting the symbols as numbers that was Godel’s real stroke of freakin’ genius. The theorem is based on that.
Anyway: The next time someone tells you, “Godel’s Theorem proves that all mathematics is invalid,” or whatever, just give them a wedgie and move on. All it proves is that a certain approach to mathematics cannot prove everything. Which, unless you had unrealistic ambitions for it in the first place, is not that surprising. show less
A very dense book that still manages to have a sense of whimsy and humor. Best absorbed in chunks and then read again at a later date. And then yet again. You will certainly have your share of "homework" to do to get the most out of GEB.
At the risk of sounding snobbish, I have to wonder how accessible GEB is to those, even readers of other books on philosophy, who do not have at least some grounding in the diverse subject matter. While Hofstadter's enthusiasm for the material is infectious, the math alone in this book can be an exercise in frustration.
Still, the intersection of complex math, music, and art is hardly unheard of. GEB is less a book that you stumble upon, and more one that is passed down to you by a mentor or friend. An show more heirloom, in many ways.
Full of lively discussions about patterns, language, loops, paradoxes, systems, AI, the nature of consciousness... primarily using examples of mathematician Godel, artist M.C. Escher, and composer J.S.Bach, but also dipping into greek philosophy, Zen Buddhism, computer programming, and more.
Ultimately, recommended if you enjoy just... thinking about thinking. show less
At the risk of sounding snobbish, I have to wonder how accessible GEB is to those, even readers of other books on philosophy, who do not have at least some grounding in the diverse subject matter. While Hofstadter's enthusiasm for the material is infectious, the math alone in this book can be an exercise in frustration.
Still, the intersection of complex math, music, and art is hardly unheard of. GEB is less a book that you stumble upon, and more one that is passed down to you by a mentor or friend. An show more heirloom, in many ways.
Full of lively discussions about patterns, language, loops, paradoxes, systems, AI, the nature of consciousness... primarily using examples of mathematician Godel, artist M.C. Escher, and composer J.S.Bach, but also dipping into greek philosophy, Zen Buddhism, computer programming, and more.
Ultimately, recommended if you enjoy just... thinking about thinking. show less
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Douglas Hofstadter is College of Arts and Sciences Professor of Cognitive Science at Indiana University
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Common Knowledge
- Canonical title
- Gödel, Escher, Bach: An Eternal Golden Braid
- Original title
- Gödel, Escher, Bach: An Eternal Golden Braid
- Alternate titles*
- Gödel, Escher, Bach
- Original publication date
- 1979-04
- People/Characters
- Tortoise; Achilles; Zeno (philosopher); Crab; Sloth; Kurt Gödel (show all 30); M. C. Escher; Bach, Johann Sebastian, 1685-1750; Alan Turing; Charles Babbage; Genie; Eta Oin; SHRDLU; Douglas R. Hofstadter; John Cage; Charles Dodgson / Lewis Carroll; Georg Cantor; Alonzo Church; Frederick the Great, King of Prussia; Johann Kirnberger; René Magritte; Marvin Minsky; Johann Joachim Quantz; Willard Van Orman Quine; Srinivasa Ramanujan; Alfred Tarski; Terry Winograd; Zhaozhou Congshen (Jō | shū | ); Wumen Huikai (Mumon); Aunt Hillary
- Dedication
- To M. and D.
- First words
- Frederick the Great, King of Prussia, came to power in 1740. • • Introduction
One of the most central notions in this book is that of a formal system. • • Chapter 1 - The MU-puzzle - Quotations
- In its absolute barest form, Gödel's discovery involves the translation of an ancient paradox in philosophy into mathematical terms. That paradox is the so-called Epimenides paradox, or liar paradox. Epimenides... (show all) was a Cretan who made one immortal statement: “All Cretans are liars.”
Whereas the Epimenides statement creates a paradox since it is neither true nor false, the Gödel sentence G is unprovable (inside P.M.) but true. The grand conclusion? That the system of Principia Mathematica i... (show all)s “incomplete”—there are true statements of number theory which its methods of proof are too weak to demonstrate.
In the human brain, there is gullability. How gullible are you? Is your gullibility located in some "gullibility center" in your brain? Could a neurosurgeon reach in and perform some delicate operation to lower your gullibili... (show all)ty, otherwise leaving you alone? If you believe this, you are pretty gullible, and should perhaps consider such an operation.
Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law. - Last words
- (Click to show. Warning: May contain spoilers.)Tortoise: Reentering Introduction Creates Endlessly Rising Canon, After RICERCAR.
- Publisher's editor
- Kessler, Martin; Bischoff, Maureen; Torre, Vincent; Dorin, Leon; Hoss, Phoebe; Breed, Larry
- Blurbers
- Bernstein, Jeremy; Davis, Ernest; Casti, John L.
- Original language
- English
- Canonical DDC/MDS
- 510.1
*Some information comes from Common Knowledge in other languages. Click "Edit" for more information.
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