414 of 441 people found the following review helpful
Format: PaperbackVerified Purchase
I'm here to witness that even people as seriously math-challenged as I am can participate in this wonderful book. It took me a *long* time to read-- I flipped back and forth, beat the pages up, asked my more math-oriented friends for help. I spent forever trying to solve the MU exercise. It was worth it. I still feel like I understood parts of it only in intuitive flashes, but those flashes showed me a room more interesting than most of the well-lit chambers ordinary books provide.
Reading Godel, Escher, Bach is like joining a club. People who see you reading it will open spontaneous conversations and often gift you with unexpected insights. (I had a fascinating conversation with a total stranger about Godel's theorem.)
Wish I could give more than five stars.
200 of 213 people found the following review helpful
on October 1, 2000
Gödel, Escher, Bach: An Eternal Golden Braid debates, beautifully, the question of consciousness and the possibility of artificial intelligence. It is a book that attempts to discover the true meaning of "self."
As the book introduces the reader to cognitive science, the author draws heavily from the world of art to illustrate the finer points of mathematics. The works of M.C. Escher and J.S. Bach are discussed as well as other works in the world of art and music. Topics presented range from mathematics and meta-mathematics to programming, recursion, formal systems, multilevel systems, self-reference, self-representation and others.
Lest you think Gödel, Escher, Bach: An Eternal Golden Braid, to be a dry and boring book on a dry and boring topic, think again. Before each of the book's twenty chapters, Hofstadter has included a witty dialogue, in which Achilles, the Tortoise, and friends discuss various aspects that will later be examined by Hofstadter in the chapter to follow.
In writing these wonderful dialogues, Hofstadter created and entirely new form of art in which concepts are presented on two different levels simultaneously: form and content. The more obvious level of content presents each idea directly through the views of Achilles, Tortoise and company. Their views are sometimes right, often wrong, but always hilariously funny. The true beauty of this book, however, lies in the way Hofstadter interweaves these very ideas into the physical form of the dialogue. The form deals with the same mathematical concepts discussed by the characters, and is more than vaguely reminiscent of the musical pieces of Bach and printed works of Escher that the characters mention directly in their always-witty and sometimes hilarious, discussions.
One example is the "Crab Canon," that precedes Chapter Eight. This is a short but highly amusing piece that can be read, like the musical notes in Bach's Crab Canon, in either direction--from start to finish or from finish to start, resulting in the very same text. Although fiendishly difficult to write, the artistic beauty of that dialogue equals Bach's music or Escher's drawing of the same name.
As good as all this is (and it really is wonderful), it is only the beginning. Other topics include self-reference and self-representation (really quite different). The examples given can, and often do, lead to hilarious and paradoxical results.
In playfully presenting these concepts in a highly amusing manner, Hofstadter slowly and gently introduces the reader to more advanced mathematical ideas, like formal systems, the Church-Turing Thesis, Turing's Halting Problem and Gödel's Incompleteness Theorem.
Gödel, Escher, Bach: An Eternal Golden Braid, does discuss some very serious topics and it can, at times, be a daunting book to handle and absorb. But it is always immensely enjoyable to read. The sheer joy of discovering the puns and playful gems hidden in the text are a part of what makes this book so very special. Anecdotes, word plays and Zen koans are additional aspects that help make this book an experience that many readers will come to feel to be a turning point in their lives.
Like every other book written by Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid, has an index and a bibliography that must be noted as exceptionally well done.
Although filled with English wordplay, this book is in no way tied to the American origin of its author. For years, it was thought that Gödel, Escher, Bach: An Eternal Golden Braid, would be impossible to translate, but so far, it has successfully been translated into French, German, Spanish, Chinese, Swedish, Dutch and Russian.
A profound and beautiful meditation on human thought and creativity, this book is indescribably gorgeous and definitely one of a kind.
240 of 270 people found the following review helpful
on December 14, 1999
It seems highly appropriate that Douglas Hofstatder should re-release his epic work now. His central theme plays so eloquently in this place and time: Every system folds in on itself, be it physics, mathematics, or any form of language. All these systems are inherently self-referential, and as such, take on a life of their own. A life their creators could never imagine. Many reviewers have focused on the explicit messages of the book, their likes or dislikes, but the great beauty of this work lies within the realm of what it does not say. It is, no doubt, the most difficult book I have ever read, and I have to admit it took me several false starts to finally get through the thing. It is so incredibly deep - one cannot simply wade through it like a sci-fi novel. But if you take your time, spend, say about a year on it - work through the TNT exercises, discover the hidden messages the author has left, read the bibliography - and at some point it will strike you; the incredible richness of the message. The book, you, the world, all of it IS open. The pages of this universe are blank, unwritten. Dr. Hofstadter has woven a message of eternal optimism, one that transcends even the infinite depth to the tapestry of topics spread before us: The great freedom that we, nature's most remarkable matrix, are part of a future without destiny. Even if we were created, any purpose impressed upon us is lost in a cacophany of unexpected relationships. Deterministic, yet infinitely complex and unpredictable. We can never understand anything completely, and thus every life can experience the magic of observing that which cannot be explained. This is a book of wonders, and you will never regret the time you spent on it.
82 of 90 people found the following review helpful
on October 7, 2002
When this book first came out, I, along with probably most mathematically and scientifically minded people of my generation, would certainly have considered it one of the best books ever written. Hofstadter has refined the task of writing a book into almost an art form. Drawing on the central theme of "strange loops" (ideas that loop back on themselves in a paradoxical manner, as might be seen in the art of M.C. Escher), Hofstadter successfully draws together ideas from a large variety of different human pursuits. An important idea--shown to be connected to other ideas in artificial intelligence, music, and art--is Godel's incompleteness theorem, which shows that there are limits on our ability to prove concepts that may, nevertheless, be true. This, too, is based on a "strange loop"--these loops seem to crop up everywhere and Hofstadter spends a lot of the book showing how they are pretty much fundamental to human knowledge.
However, after reading the new preface in this 20th anniversary edition, I'm left with the sense that this once great book is now merely good. For one thing, Hofstadter seems to have evolved from a brilliant young man with a lot of great ideas into a somewhat cantakerous middle-aged man. He seems angry at the New York Times, and his readers, for not fully understanding the central message of the book. Yet he also excuses himself from making any attempt to update the book or bring the ideas in line with many of the enormous changes that have happened over the last 20+ years. It seems surprising to me that Hofstadter would constrain his own book to having only one central message--surely he should understand that a book of this complexity will mean many things to many different people, and that indeed is the reason for its popularity.
So, I still highly recommend this book, but I'm left just a little disappointed that Hofstadter seems somewhat at war with his readers and as a result, won't attempt to update the book or try to help us reconcile the many events of the last 20 years with the themes of his book.
31 of 32 people found the following review helpful
on January 23, 2004
I give this book high marks. The read is difficult, I concede. However, remember that in order to make progress, oftentimes we must take a leap of faith. The book even argues that proving something to be true requires you to "just believe" because logic eventually runs out upon deconstruction. See chapter VII.
I have had similar trouble that others report. I have had to re-read parts to make sure I get his points, whether I agree or not. And yes, he conveys his ideas in what some may consider an offhand way. There is much value in the saying, "To be great is to be misunderstood."
You dont have to like this book. Just make sure you're certain why you do or don't like it. Is it because the Hof doesn't know what he is talking about, or because he "wastes" your time with his lingo and fictional prancing about? Or is it because there's a chance that you don't understand? I am not condescending readers who don't like GEB, but we too often rate someone's ideas based on our inability to understand and yes, sometimes be entertained immediately. Don't expect him to do all the work. What are you bringin' to the party?
This book is challenging. Once you have spent enough time with it, you might see that it requires you to challenge your understanding of things, take that leap of faith (it's not all about logic), suspend judgment, then see what you think when you get to the other side. Consider the section devoted to the topic of Euclidean vs. non-Euclidean geometry:
Euclid of Alexandria perfected the art of rigor in his Elements, becoming arguably the most influential mathematician in times of antiquity. He made a most convincing case for the accuracy and truthfulness of much of the fundamental geometry we know today. He did so by using five principals upon which to base the remainder of his volumes of assertion. Four of the five principles were based on truths quite simple and so understandable, for the most part we hold them to be self-evident. One of those (the first) was the notion of a straight line, as simple and direct as connecting point A to point B.
His work seemed universal, truthful, and beyond reproach, especially considering the painstaking efforts he went to prove the seemingly most basic of concepts. This all seemed well and good, until others, implicitly or otherwise, began to question the notion or suggest what a different version of what a straight line is. In other words: What if there was more than one type of straight line? How could this be?
To make a long story only slightly longer, we find that there in fact IS more than one type of straight line (what's the difference between a straight line drawn on a piece of paper and a straight line drawn on a basketball? hmmmm....), which spawned elliptical and spherical geometries. Turns out that Euclidean geometry is actually a subset of geometry, not the entire geometry. All these years we thought that a piece of the pie was the whole pie.
The point here is that you must endeavor to see outside what you know to be true. It's not always comfortable or seemingly conceivable, but we must accept a degree of uncertainty before we can realize a new level of certainty.
Give the book a shot. Maybe two. Suspend your judgment and take the hit. You'll see. Regards.
42 of 46 people found the following review helpful
on January 9, 2001
I first read GEB some 20 years ago as a high school senior/college freshman. Even though I was a mathematically inclined physics major, an amateur classical musician, and a lightning-fast reader, the book still took me a year to finish. This is the sort of weighty tome where one reads a chapter, and then sets the book aside for awhile to let things settle in. It's no wonder that a poll by New Scientist magazine of highly-regarded scientists had to be rephrased as "EXCEPT for Godel Escher Bach, what scientific or technical book would you take to an uninhabited island?"
I will cheerfully confess that I cannot remember all of the details of the book, and that there were times when I simply couldn't get at what Hofstadter was trying to explain. Still, some of Hofstadter's writing has stayed with me the past two decades--his classic analogy of Godel's theorem with a stereo system, his discussion of the difficulties of creating an "accurate" translation (using the beginning of "Crime and Punishment"), his wondrous tying-together of math, music, and art. The totally math-phobic will find these, and many other concepts, readily accessible and even symbol-free. Wish I could say as much for some "general audience" philosophy books!
31 of 34 people found the following review helpful
on May 18, 2007
This is a difficult book.
Difficult to read. Difficult to understand. And, I'm finding, difficult to review. What's it about? Good question. The author, himself, isn't very clear on this point, describing it as "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll." I'm not sure I can do better than that. I will tell you this, however: if the book has a "point," it does seem to be that man's consciousness is ultimately mechanical and, therefore, that there is no reason that machines cannot finally be intelligent in the same sense that man is. (And, in fact, be as man in just about every internal way.)
While I take issue with this conclusion, and some of Hofstadter's reasoning along the way, I don't think that my debating his points is the basis on which a prospective reader should decide whether or not to pick up this book. Instead, the prospective reader should know: that this is a lengthy and deep work. It will take a *long* time to read properly, and most readers should not read more than a chapter a day. Many of the sections, and especially the various dialogues that preface the chapters, are quite clever. (These dialogues are usually between Achilles and the Tortoise, of Zeno's paradoxes, and their friends.) Some of the chapters grow incredibly technical. The subject matters vary, wildly and rapidly, and there will be points in reading where you will question your investment.
In the end, you will feel good for having pushed through the hard bits. It will coalesce, more or less, into a whole. Whether you finally agree with Hofstadter's conclusions or not, you'll have learned much and thought about important topics you might otherwise not have.
A good book, certainly not for everyone... but, if you're the "right" audience--someone deeply interested in questions of intelligence, mathematics, computer science and free will, and possessed of a bit of an ironic sense of humor--then this book cannot be recommended highly enough.
Five stars, for the work it represents, and the doors it opens to the reader.
26 of 28 people found the following review helpful
on June 5, 2006
GEB: an Eternal Golden Braid is a difficult book to explain. It's a book about strange loops, recursivity, paradox, number theory, formal systems, molecular biology, Zen Buddhism, impressionism, and fugues. These concepts are introduced through the works of mathematician Kurt Gödel, artist M.C. Escher, and composer J.S. Bach, as well as some other supporting characters, like Charles Babbage (the first one to think of an Analytical Engine, a mechanical device for churning out algebraic theorems) and Alan Turing (of Turing Test fame). And then, of course, there are the dialogues, populated by the Greek warrior Achilles, a tortoise, a crab, and an anteater. Out of this confusing mess of concepts Hofstadter attempts to grapple with a truth he feels lies at the heart of Artificial Intelligence and Human Consciousness--that it forms from the same tangled hierarchies as Gödel's Incompleteness proof or Escher's "Print Gallery" or Bach's "Canon per Tonos" (a theme that changes notes according to a fixed system that somehow always returns to its starting note, one octave higher in pitch).
Much of the book deals with formal systems--meaningless symbol-shunting procedures for producing theorems from axioms--and the way they are mapped on to "truths" about the world (what Hofstadter calls "isomorphisms"). One of the most extensively used formal system in the book is called TNT, for `typographical number theory' (also one in a series of Hofstadter puns, as TNT, when joined with a process called Gödel Numbering, tends to self-destruct), which is just a new way of expressing simple number-theoretical truths (such as the commutativity and associativity of addition; i.e., b+c=c+b and b+(c+d)=c+(b+d)). On the surface, formal systems seem utterly trivial. Hofstadter introduces them as a theatre on which strange loops emerge. Strange loopiness enters formal systems when they can express Epimenide's paradox, a single sentence that reads "This sentence is untrue." Hofstadter explains how this realization came to pass when a German mathematician named Kurt Gödel discovered inconsistencies in Bertrand Russell and Alfred North Whitehead's "Principia Mathematica", a treatise meant to banish self-reference in set and number theories. Any formal system capable of expressing all number-theoretical truths can also be used to represent itself through a system of Gödel Numbering, which is just a way of interpreting symbols in the formal system as large numbers. Any formal system powerful enough to represent itself through Gödel Numbering can make the statement "There is no theorem with Gödel Number G'", where G' is the Gödel number for that statement. In other words, a powerful formal system will inevitably make claims that are paradoxical, inconsistent whether you call them true or false.
Hofstadter combines the strange loopiness of formal systems with the concept of isomorphisms to come to some conclusions about human consciousness. First, he claims that the brain has a formal system for representing concepts in the world that exhibits self-reference and self-modification in a tangled hierarchy, just like the simplified formal systems he introduces in the book, Escher illustrates, and Bach incorporates into his music. He takes his time making his case, ending each chapter with a dialogue between Achilles and a Tortoise, a convention Zeno used to prove the impossibility of motion, and Lewis Carroll burrowed in his Two-Part Invention. These dialogue's are usually esoteric and highly amusing, including a series on Achilles' record player and one of the Tortoise's records designed explicitly to create vibrations that destroy the record player. This is a parallel to the explosive self-repudiation of TNT--any record player that can produce a high fidelity representation of the magnetic strips on the record will destroy itself, and any record player that can't is useless as a record player. Another dialogue introduces an anteater who converses with an ant colony that is collectively cognizant, even if each individual ant isn't (a parallel to meaning arising from meaningless formal systems of neuron representations in the human mind). In the main text, Hofstadter introduces the reader to the computer languages of Bloop, Floop, and Gloop (Gloop is just theoretical, a self-altering program reminiscent of the tangled hierarchy of the human mind), simple programs designed to reproduce themselves (an analogue to strings of DNA that encode for DNA synthesizing enzymes), and the Zen concept of MU (where neither `yes` or `no` suffice, say MU, or `unask the question`).
Fans of M.C. Escher will want to take another look at his "Print Gallery", a picture of a man looking at a picture of a town that contains the gallery the man is in, and the picture he's looking at. This is a tangled bit of self-reference has a blemish at the lower right-hand corner of the picture frame (the picture frame in the print, which is in the center of the print). In this blemish M.C. Escher writes his signature, but Hofstadter points out that the "blemish" is an inescapable side-effect of the self reference. No consistent image could appear in that blemish, just as no consistent interpretation of Epimenide's paradox is correct. Human's don't have privileges access to the formal system of their representations of the world--the inviolate level of human consciousness is off-limits to our perception. "From this balance between self-knowledge and self-ignorance comes the feeling of free will (p. 713)," says Hofstadter. This is the central idea of his book. Formal systems are ubiquitous, and powerful formal systems exhibit tangled hierarchies. The human mind is no exception, and the "blemish" of human consciousness is that inevitable bit of self-ignorance that gives us free-will. By breaking it down to the saliencies of a formal system, Hofstadter has high regard for the prospects of Artificial Intelligence, which ought to be able to build upon a similar edifice.
This is a difficult book to read and understand. It's deeply compelling and reads differently each time. I recommend it to those who have a lot of time on their hands.
69 of 81 people found the following review helpful
I've been reading reviews of GEB for years, and the most fascinating thing about them, aprt from the near-uniform enthusiasm of the readers, is that almost none of the enthusiatic readers have any idea of what the book is actually about! The typical reader seesm to think of GEB as a jouyous romp through any number of fascinating bits of logic, math and science without any idea as to what Hofstader's actually doing.
Yes, it's about Goedel, and recursion, and "strange loops", and linguistics Bach and ants and all that- but only trivially. The bulk of the book is taken up with what amounts to a very entertaining tutorial that sets the reader up for the real thesis of the book. What Hofstadter has attempted in GEB is nothing less than a concise, bottom-up theory of mind. You can read it as a theory of AI, or a theory of human intelligence, but either way he's telling you how to construct an intelligent entity.
True, he doesn't really have a theory of *how* a self-aware being should arise from his metaphorical anthill, but then, neither does anyone else. But he does have a very good story as to how intelligence does arise in such conditions.
If you've read this book before without understanding what his aim was, read it again, with that notion in mind. And if you haven't read it, and you're the sort of person who enjoys mathematic and scientific amusements of any sort, well, read it and discover how much fun a speculative theory can be.
111 of 134 people found the following review helpful
on September 21, 1999
Douglas Hofstadter's imaginative and engaging GEB:EGB asks the question, "Can machines be conscious?" and answers in effect, "Certainly, because we ourselves are such machines." And there is no doubt that he earned his Pulitzer Prize for this fascinating book.
But watch out! His reliance on imagination actually masks the real problem.
The "real problem" is this: mind can't arise _simply_ from self-reference and self-representation, because reference and representation presume the existence of a mind to begin with. Only minds refer and represent; _resemblances_ (even fancy ones like "isomorphisms") aren't references/representations.
And in Hofstadter's undeniably well-presented examples, his reliance on imagination serves to distract from the absolutely crucial fact that the reference and the representation are always provided by a mind _outside_ the system in question: the reader. A formal system complex enough to "represent itself" doesn't become conscious; it takes a mind _outside_ the system to "see" the isomorphisms in question as references/representations. The system _itself_ can't do so unless mind is _already_ there -- so Hofstadter's bootstrapping "explanation" fails.
As an _argument_, then, GEB:EGB is a tremendous begging of the question. Invoking Godel's Theorems and waving one's hands about "strange loops" doesn't alter the fact that Godel's Theorem itself delivers a killing blow to "computational" theories of consciousness: semantics is _not_ reducible to syntax; truth is not reducible to provability within a formal system; reason is not reducible to purely formal logic; meaning is not reducible to isomorphism; and mind is not reducible to computation. (And indeed, this reading has much more in common with what Godel himself thought he had shown than does Hofstadter's attempt to reinterpret Godel's work in favor of strong AI.)
But GEB:EGB is still a remarkable intellectual accomplishment and a joy to read. Just be careful!