Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter
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Book Reviews of Gödel, Escher, Bach: An Eternal Golden Braid
Book Review: A whole new world
Summary: 5 Stars
Just like other reviewers of this book, this one is seriously math-challenged. I thought I'd never make it to the end. I had to work hard sometimes. But it was one of the most worthwhile reading experiences of my life, indeed it was fascinating and I ended up recommending it to everybody (almost). This is an eccentric book, which at the beginning seems to be about everything and nothing. The author describes "self-referential systems" and wonders whether they may come to think about themselves and, ultimately, about the possibility of Artificial Intelligence (AI). This is one of the books that have taught me the most and made me think about things that for me are little or not familiar at all, like Typographic Number Theory (TNT) or propositional calculus. I don't pretend to having understood all of it perfectly, but definitely my intellectual horizon and my knowledge widened a lot. The title comes from three people whose work illustrates the wide field of self-referential systems. One is Kurt Godel, a mathematician who formulated the famous and complex Theorem of Incompletitude, which says something like in every formulation of Number Theory there are one or several propositions which are undecidable, that is, it is impossible to affirm if they are theorems of that formulation, or not. What is a formulation of Number Theory? Well, starting from an axiom (a given equation), and according to some precise rules of addition, substraction, or substitution, one develops the equation until finding (or not) some preestablished outcome. These formulations are self-referential because they turn in on themselves, that is, they take the sources of their subsequent development from the very elements present in the original axiom.
OK, where do Bach and Escher enter the picture? Simple: Escher's engravings and illustrations (of which the book offers many beautiful examples), and Bach's music, are self-referential. They present an initial theme, and then they develop by turning in on themselves according to some rules or patterns. This is also how DNA chains and the resulting organisms grow, and even some poetry. The book is written with great sense of humor and didactic skill; it intersperses "technical" chapters with funny and seemingly absurd dialogues between cartoon-like characters, which illustrate with good clarity subjects previously exposed. Mathematics, biology, computers, AI, music, painting, and language are part of the subjects taken on.
For example, this book helped me to understand better how computers and software languages work. In one of the most interesting parts, the author explains how our brains function at different levels: the strictly neuronal, the cellular, the chemical, and the symbolic. In the same way, computers work at different levels, from that which occurs between components of the microcircuits, passing through successive levels of programming, until the "symbolic" level, which is represented by what we see on the screen. Just as people don't need to know or screen what is happening with their pancreas or stomach in order to go about, but they limit themselves to eating, breathing, walking, thinking, computer users don't need to know what is going on with the circuits while they use them. We only take notice of organs or programs ewhen something's wrong: we either feel bad or the computer paralyzes. Finally, the other debate is about the possibility or not of AI, which is far from being resolved. An enormous book.
Book Review: A wonderful read for all aspiring thinkers
Summary: 5 Stars
The Atlanta Journal Constitution describes Gödel, Escher, Bach (GEB) as "A huge, sprawling literary marvel, a philosophy book, disguised as a book of entertainment, disguised as a book of instruction." That is the best one line description of this book that anybody could give. GEB is without a doubt the most interesting mathematical book that I have ever read, quickly making its place into the Top 5 books I have ever read.The introduction of the book, "Introduction: A Musico-Logical Offering" begins by quickly discussing the three main participants in the book, Gödel, Escher, and Bach. Gödel was a mathematician who founded Gödel's Incompleteness Theorem, which states, as Hofstadter paraphrases, "All consistent axiomatic formulations of number theory include undecidable propositions." This is what Hofstadter calls the pearl. This is one example of one of the recurring themes in GEB, strange loops.
Strange loops occur when you move up or down in a hierarchical manner and eventually end up exactly where you started. The first example of a strange loop comes from Bach's Endlessly rising canon. This is a musical piece that continues to rise in key, modulating through the entire chromatic scale, ending at the same key with which he began. To emphasize the loop Bach wrote in the margin, "As the modulation rises, so may the King's Glory."
The third loop in the introduction comes from an artist, Escher. Escher is famous for his paintings of paradoxes. A good example is his Waterfall; Hofstadter gives many examples of Escher's work, which truly exemplify the strange loop phenomenon.
One feature of GEB, which I was particularly fond of, is the 'little stories' in between each chapter of the book. These stories which star Achilles and the Tortoise of Lewis Carroll fame, are illustrations of the points which Hofstadter brings out in the chapters. They also serve as a guidepost to the careful reader who finds clues buried inside of these sections. Hofstadter introduces these stories by reproducing "What the Tortoise Said to Achilles" by Lewis Carroll. This illustrates Zeno's paradox, another example of a strange loop.
In GEB Hofstadter comments on the trouble author's have with people skipping to the end of the book and reading the ending. He suggests that a solution to this would be to print a series of blank pages at the end, but then the reader would turn through the blank pages and find the last one with text on it. So he says to print gibberish throughout those blank pages, again a human would be smart enough to find the end of the gibberish and read there. He finally suggests that authors need to write many pages more of text than the book requires just fooling the reader into having to read the entire book. Perhaps Hofstadter employs this technique.
GEB is in itself a strange loop. It talks about the interconnectedness of things always getting more and more in depth about the topic at hand. However you are frequently brought back to the same point, similarly to Escher's paintings, Bach's rising canon, and Gödel's Incompleteness theorem. A book, which is filled with puzzles and riddles for the reader to find and answer, GEB, is a magnificently captivating book.
Book Review: Big Book of Interesting Points
Summary: 5 Stars
It's difficult for me to decide just exactly how much I like Gödel, Escher Bach. At 740+ pages, it certainly has a lot to say. And since it's not a novel, there are clearly some sections I can say I liked better than others. Yet even after reading the entire book, including authors preface explicitly describing GEB's meaning, I must still say I'm unsure exactly what the book is about.
Hofstadter's thesis is an attempt to explain in a personal way how "animate beings can come out of inanimate matter." This quest begins mainly with an analysis of Gödel's Theorem, which essentially states (please forgive me for any scientific inaccuracies) that it is impossible for any mathematical system to be complete.
Hofstadter works us through this theorem rather slowly and gently, at first looking at simple, crude mathematical systems, and examining their successes and failures to depict our real number system. Interspersed with this examination are fascinating dialogues between fictional characters. These dialogues are at times odd, witty, clever, deep, philosophical, and are the icing of GEB. Hofstadter also from time to time looks at the artwork of M.C. Escher and the music of J.S. Bach. He notes the patterns found in their works and how they loop back on each other, or contain elements of self-reference, important to Hofstadter because it is this act of self-reference which makes Gödel's Theorem possible.
Hofstadter then gradually shifts from looking at mathematical systems and levels of systems to our own thought and our separate levels. He examines cell structure and DNA, and moves outward to artificial intelligence and its relation to our intelligence. This of course raises fascinating questions. What do we think of when we think of the number "two?" We certainly don't have a "10" in binary code sitting somewhere in our heads. Hofstadter attempts to link these two realms of mathematical theory and intelligence theory and believes that intelligence is comprised of hierarchical levels, many independent and unconscious of levels above or below.
Though written in 1979, this book covers ground which we have yet to traverse. Although a computer can now beat the best human chess players, the world is seeking computers with more "intelligence." Where do thoughts come from?
This book is not for those with a light interest in math or numbers. Although you do not have to mathematically derive any of Hofstadter's proofs or understand everything (I didn't), it helps to be familiar with what Hofstadter is trying to say and to have an interest in puzzles, number, and the like. Familiarity with Escher or Bach can help, but is not as requisite as the first.
I am not a computer programmer, nor am I computer myself, so I have no idea whether Hofstadter's observations and assertions are true or false. His book, however, I found very interesting and thought provoking, and quite rewarding. I recommend this book to all those interested in math/computers, and for experts, it is probably required reading.
Book Review: Transfiguration in Print
Summary: 5 Stars
I read this book in 1980, and each chapter of it still stands out in my mind in glittering bas-relief against all I read before or have read since. What stands out the most, however, is that my thinking changed after reading this book. Everything since reading it has been inevitably processed through its filter, which has quite simply changed my life. Perhaps a first-reading today would not have an equal effect; in 1980 it was timely. Things we take for granted now were difficult to conceptualize then--the internet, for instance. But I'll leave it for contemporary readers to decide whether Gödel, Escher, Bach: An Eternal Golden Braid is as eternal as the title claims. For me, the answer is a resounding yes. I can say without doubt that my life would have been different if I had not read it at that time.
Ok, if it's so great, then what is it that makes it so? Trying to describe what is great about this book is like trying to say what is great about a particular Bach Fugue, or an Escher print. A fugue doesn't typically have a great melody. The rhythm can be monotonous and predictable. You know where it begins and where it will end up, and the subjects will enter on cue and not deviate from their lines. Escher's prints, similarly, are composed of mediocre representations of their subjects. Shading is not sublime, some of the features may be grotesque. What is it, then, that makes that fugue or that print great? Ahhh... that's the subject of this book, and in beautifully crafted recursion, its own principles apply equally to itself.
You will come away with an understanding of the underlying principles of intelligence, beauty, craft, logic, and universal principles of creation. A deeper appreciation for those things does not necessarily mean a simpler or easier means of describing same. In fact, it may be the opposite, sort of "the more you know, the more you know you don't know." This confrontation with, and participation in, the infinite seems to be the root of our longing--a sense that Bach, Escher, and Gödel weaved intuitively into their own work without feeling compelled to explain.
So, rather than expect to come away from this book with the answers to the big quiz of life, I would say what one is more likely to find is a deepening of "mu," a rendering of knowledge into its proper place where "answers" in the Western sense dissolve into the questions in a deepening spiral of association and metaphor to the point at which one sees so many possibilities that the original question loses its significance. So is it nirvana? Not this book. But one may get caught up in the craft of the author and his inspirations to the point that one might feel lifted up and deposited at the trailhead of a new path, and the best that one can expect in this life is a new, sublime path. I think Hofstadter would agree that nirvana, should it exist, is the path, not the destination, and this book may well prove the author to be a willing bodhisattva.
Shooshie
Book Review: Still completely original- and totally misunderstood.
Summary: 5 Stars
When GEB was first published,the reviews and enthusiasm were endless. It's a brilliant introduction to recursion, said many. No, it's an introduction to, and demonstration of Godel incompleteness, said others. It's a demonstration of the commonality of art and science, said others. And there's something about ants near the end, but we're not sure why.Readers today echo the same sentiments. They're all right, in their own way- but none of these views really get at what Hofsteader was trying to do. Yes, GEB is a tuorial on Godel, Bach, ants, recursion and a dozen other esoteric topics, and it's a heck of an intellectual entertainment, but Hofsteader didn't just write GEB to show prove what a clever book he could write. At the core, GEB is, first and formost, a theory of Artifical Intelligence; all the bits on Godel, recursion and combinations are just a tutorial to bring the reader up to speed for what's about to follow.
When GEB was first published, the dominant paradigm in AI was top-down; you built inference engines, programmed them with high-level knowledge about systems, and tried to get them to generalize from their. To a small minority- including Hofstader- this begged the really important questions: Where did the ability to make inferences come from in the first place? How was knowledge represented?
A few pioneers then- people like John Holland- were looking at bottom-up models in which one posited the simplest levels of an organization- the individual elements and the rules of interconnection and communication. They reasoned that that's what the brain was, so if you couldn't derive AI from a model that echoed the brain, you weren't really proving anything. It was from this perspective that GEB was written, and given the state of AI at the time, it's not surprising that most readers- even the most enthusiastic among them- totally missed the point.
Today, the bottom-up, or connectionist paradigm is gaining new respectability, and the work over the last few decades in complexity theory has given us more insight into the mechanisms of connectionism. Reading GEB in that context, not only is Hofstader's thesis much clearer, but the book appears that much more brilliant and prescient, given when it was first written.
If you've never read GEB, read it it now, and then read George Dyson's "Darwin Among the Machines", Waldrop's "Complexity", Resnick's "Turtles, Termites and Traffic James", and John Holland's "Hidden Order". If you've read GEB before, take a look at those same books and then go back and reread GEB. You'll see it in an entirely new light.