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Gödel, Escher, Bach: An Eternal Golden…
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Gödel, Escher, Bach: An Eternal Golden Braid (1979)

by Douglas R. Hofstadter

Other authors: See the other authors section.

MembersReviewsPopularityAverage ratingConversations / Mentions
11,05698375 (4.35)2 / 217
  1. 101
    Cryptonomicon by Neal Stephenson (Zaklog)
    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.… (more)
  2. 60
    Logicomix: An Epic Search for Truth by Apostolos Doxiadis (tomduck, EerierIdyllMeme)
    EerierIdyllMeme: An obvious suggestion (surprised it's not here already). Both are creative and fictional riffing off of formal logic and incompleteness.
  3. 50
    Metamagical Themas: Questing for the Essence of Mind and Pattern by Douglas R. Hofstadter (JFDR)
  4. 40
    Incompleteness: The Proof and Paradox of Kurt Gödel by Rebecca Goldstein (michaeljohn)
  5. 20
    A Mathematician Reads the Newspaper by John Allen Paulos (heidialice)
    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.
  6. 00
    Things to Make and Do in the Fourth Dimension: A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More by Matt Parker (Lorem)
    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… (more)
  7. 00
    The Gold Bug Variations by Richard Powers (hippietrail)
  8. 33
    A New Kind of Science by Stephen Wolfram (Anonymous user)
  9. 01
    Dirk Gently's Holistic Detective Agency by Douglas Adams (EerierIdyllMeme)
    EerierIdyllMeme: A few similar themes (Bach, human cognition) come up in similar ways.
  10. 03
    The Flanders Panel by Arturo Pérez-Reverte (P_S_Patrick)
    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.… (more)
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English (92)  Hebrew (1)  Spanish (1)  Danish (1)  Swedish (1)  All languages (96)
Showing 1-5 of 92 (next | show all)
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 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, no 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. ( )
  TFleet | Jun 17, 2019 |
Notes whilst reading it.
The first part on logic is a bit tough going in places. Take home message for me was: is order innate (normal) or depending on one's perspective? i.e. will supposed other worlds be able to make sense of a the record man sent into space. Will Bach be order for them too, or will they see order in Cage's creations?
A bit dated as the author uses examples of records and jukeboxes to illustrate his stories. I get it; does a teenager?
  robeik | May 10, 2019 |
This is certainly the greatest book of popular science I've encountered or heard of yet--it's accessible, engaging, playful, but also very deep and original in its analysis. It gives brief intellectual history where necessary, and repeats the same arguments in many surprisingly different ways for comprehension.

But its ultimate ignorance/dismissal of the social world, assertion of objective meaning, and computational theory of mind force me to give it a lower rating. ( )
1 vote alexanme | Dec 9, 2018 |
(Original Review, 1980-09-24)

Thoughts on dolphins.

To Danny Weinreb: check out John Lilly's books on dolphins, particularly the one I mentioned recently, "The Mind of the Dolphin". Lilly has spent years trying to answer just those questions you have raised.

To ICL.REDFORD, read the book mentioned above. You will find the arguments for dolphin intelligence much more clearly stated and documented than I was able to do in one short paragraph. It also shows clearly why there is a difference between the 'understanding' of a dog or cat and that of a dolphin or chimpanzee. The conjecture that man's ability to use tools is a major cause of his evolutionary success has a lot of merit. However, to assume that this is the only way intelligence (as we know it or otherwise) can develop is a rather parochial way of viewing things. Communication is also an arguable measure of intelligence, and many animals have this ability to varying degrees. Before we ask "Are dolphins intelligent?" we must ask "What is intelligence?" Doug Hofstadter, in his book "Gödel, Escher, Bach: The Eternal Golden Braid", presents a way of looking at intelligence that is not as restrictive as most current definitions. (I highly recommend this book to anyone, by the way. It is published by Basic Books in hardback, and is worth whatever you may pay for it.) He makes a good argument for the claim that intelligence is a consequence of the complexity of organization of the nervous system of an organism. This would imply that dolphins and whales may be (in some sense) MORE intelligent than we are (although - read on) since it has been shown that these mammals have brains that are more complex than our own. In fact, most current definitions of intelligence tend to describe it in such a way that only humans have it. This I believe to be due to an inability to "step outside the system" and be truly objective about what it is that separates us (if there is indeed anything) from the other inhabitants of this planet. In my view, "intelligence" is a continuum, and not necessarily one-dimensional at that. Looking at dolphins from this perspective, it is ludicrous to compare them to us and say "Are they intelligent?" That's like asking Flipper to take the Stanford-Binet. It ignores the possibility of a universe outside our own in which values may not match our own.

It is precisely this possibility that is addressed by the question "Can we communicate with other 'intelligent' beings?"

Personally, I suspect that we will have common ground with most 'intelligent' species in the area of formal mathematics. That field, more than any other, derives from an attempt to distill the essence of the universe from what we observe. And, more than any other field, it is truly a product of the mind only. ( )
  antao | Nov 10, 2018 |
Amazing. ( )
  georgee53 | Jun 17, 2018 |
Showing 1-5 of 92 (next | show all)
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» Add other authors (24 possible)

Author nameRoleType of authorWork?Status
Douglas R. Hofstadterprimary authorall editionscalculated
尚紀, 柳瀬Translatorsecondary authorsome editionsconfirmed
Feuersee, HermannTranslatorsecondary authorsome editionsconfirmed
Wahlén, JanTranslatorsecondary authorsome editionsconfirmed
Wolff-Windegg, PhilipTranslatorsecondary authorsome editionsconfirmed
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Frederick the Great, King of Prussia, came to power in 1740.
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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 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 is “incomplete”—there are true statements of number theory which its methods of proof are too weak to demonstrate.
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Il libro che ha svelato a una immensa quantità di lettori, in tutto il mondo, gli incanti e le trappole di un’Eterna Ghirlanda Brillante i cui fili si chiamano intelligenza artificiale, macchina di Turing, teorema di Gödel. Una «fuga metaforica» nel variegato mondo che si dispiega fra la mente, il cervello e i computer.

«Ogni due o tre decenni un autore ignoto produce un libro di tale profondità, chiarezza, vastità, acume, bellezza e originalità che subito esso viene riconosciuto come un avvenimento di prima importanza: Gödel, Escher, Bach è un’opera di tal genere… La struttura di questo libro è satura di complicato contrappunto non meno di una composizione di Bach o dell’Ulisse di Joyce»

MARTIN GARDNER, «Scientific American»
(piopas)
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Amazon.com Amazon.com Review (ISBN 0465026567, Paperback)

Twenty years after it topped the bestseller charts, Douglas R. Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid is still something of a marvel. Besides being a profound and entertaining meditation on human thought and creativity, this book looks at the surprising points of contact between the music of Bach, the artwork of Escher, and the mathematics of Gödel. It also looks at the prospects for computers and artificial intelligence (AI) for mimicking human thought. For the general reader and the computer techie alike, this book still sets a standard for thinking about the future of computers and their relation to the way we think.

Hofstadter's great achievement in Gödel, Escher, Bach was making abstruse mathematical topics (like undecidability, recursion, and 'strange loops') accessible and remarkably entertaining. Borrowing a page from Lewis Carroll (who might well have been a fan of this book), each chapter presents dialogue between the Tortoise and Achilles, as well as other characters who dramatize concepts discussed later in more detail. Allusions to Bach's music (centering on his Musical Offering) and Escher's continually paradoxical artwork are plentiful here. This more approachable material lets the author delve into serious number theory (concentrating on the ramifications of Gödel's Theorem of Incompleteness) while stopping along the way to ponder the work of a host of other mathematicians, artists, and thinkers.

The world has moved on since 1979, of course. The book predicted that computers probably won't ever beat humans in chess, though Deep Blue beat Garry Kasparov in 1997. And the vinyl record, which serves for some of Hofstadter's best analogies, is now left to collectors. Sections on recursion and the graphs of certain functions from physics look tantalizing, like the fractals of recent chaos theory. And AI has moved on, of course, with mixed results. Yet Gödel, Escher, Bach remains a remarkable achievement. Its intellectual range and ability to let us visualize difficult mathematical concepts help make it one of this century's best for anyone who's interested in computers and their potential for real intelligence. --Richard Dragan

Topics Covered: J.S. Bach, M.C. Escher, Kurt Gödel: biographical information and work, artificial intelligence (AI) history and theories, strange loops and tangled hierarchies, formal and informal systems, number theory, form in mathematics, figure and ground, consistency, completeness, Euclidean and non-Euclidean geometry, recursive structures, theories of meaning, propositional calculus, typographical number theory, Zen and mathematics, levels of description and computers; theory of mind: neurons, minds and thoughts; undecidability; self-reference and self-representation; Turing test for machine intelligence.

(retrieved from Amazon Thu, 12 Mar 2015 18:17:11 -0400)

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A scientist and mathematician explores the mystery and complexity of human thought processes from an interdisciplinary point of view.

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