Gödel, Escher, Bach: An Eternal Golden Braid

by Douglas R. Hofstadter

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 the heart of cognitive science: meaning, reduction, recursion, and much more.

Genres: Science, Nonfiction, Philosophy, Mathematics, Music, Psychology, Computer Science

756 pages, Paperback

First published April 1, 1979

About the author

Douglas Richard Hofstadter is an American scholar of cognitive science, physics, and comparative literature 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.

Hofstadter is the son of Nobel Prize-winning physicist Robert Hofstadter. Douglas grew up on the campus of Stanford University, where his father was a professor. Douglas attended the International School of Geneva for a year. He graduated with Distinction in Mathematics from Stanford in 1965. He spent a few years in Sweden in the mid 1960s. He continued his education and received his Ph.D. in Physics from the University of Oregon in 1975.

Hofstadter is College of Arts and Sciences Distinguished Professor of Cognitive Science at Indiana University in Bloomington, where he directs the Center for Research on Concepts and Cognition which consists of himself and his graduate students, forming the "Fluid Analogies Research Group" (FARG). He was initially appointed to the Indiana University's Computer Science Department faculty in 1977, and at that time he launched his research program in computer modeling of mental processes (which at that time he called "artificial intelligence research", a label that he has since dropped in favor of "cognitive science research"). In 1984, he moved to the University of Michigan in Ann Arbor, where he was hired as a professor of psychology and was also appointed to the Walgreen Chair for the Study of Human Understanding. In 1988 he returned to Bloomington as "College of Arts and Sciences Professor" in both Cognitive Science and Computer Science, and also was appointed Adjunct Professor of History and Philosophy of Science, Philosophy, Comparative Literature, and Psychology, but he states that his involvement with most of these departments is nominal.

In April, 2009, Hofstadter was elected a Fellow of the American Academy of Arts and Sciences and a Member of the American Philosophical Society.
Hofstadter's many interests include music, visual art, the mind, creativity, consciousness, self-reference, translation and mathematics. He has numerous recursive sequences and geometric constructions named after him.

At the University of Michigan and Indiana University, he co-authored, with Melanie Mitchell, a computational model of "high-level perception" — Copycat — and several other models of analogy-making and cognition. The Copycat project was subsequently extended under the name "Metacat" by Hofstadter's doctoral student James Marshall. The Letter Spirit project, implemented by Gary McGraw and John Rehling, aims to model the act of artistic creativity by designing stylistically uniform "gridfonts" (typefaces limited to a grid). Other more recent models are Phaeaco (implemented by Harry Foundalis) and SeqSee (Abhijit Mahabal), which model high-level perception and analogy-making in the microdomains of Bongard problems and number sequences, respectively.

Hofstadter collects and studies cognitive errors (largely, but not solely, speech errors), "bon mots" (spontaneous humorous quips), and analogies of all sorts, and his long-time observation of these diverse products of cognition, and his theories about the mechanisms that underlie them, have exerted a powerful influence on the architectures of the computational models developed by himself and FARG members.

All FARG computational models share certain key principles, among which are: that human thinking is carried out by thousands of independent small actions in parallel, biased by the concepts that are currently activated; that activation spreads from activated concepts to less activated "neighbor concepts"; that there is a "mental temperature" that regulates the degree of randomness in the parallel activity; that promising avenues tend to be explored more rapidly than unpromising ones. F

Reviews

[Mark Lawrence · Rating 5 out of 5]

Expand your mind! Not for the faint of heart & yet by no means dry.

Hofstadter makes some fascinating observations about emergent properties (such as intelligence) and diverts us into the extremely heavy mathematics of Godel via the self referencing systems that are Bach's fugues and Escher's 'optical illusion' style artwork.

Before too many chapters have passed though you'll be firmly in number theory land, albeit doled out as painlessly as is possible with such stuff, leavened with imagined philosophical debates between ancient Greeks and other proxies. I seem to remember Achilles spends a lot of time talking to a tortoise...

Number theory requires no great resource of mathematical knowledge - just an extremely agile and open mind. If you let him Hofstadter will show you how Godel destroyed Betrand Russell's Principa Mathematica - his attempt to logically deduce all of mathematics from a set of axioms. Godel shows us that (I paraphrase drastically) that all logical systems allow statements about natural numbers that are true but unprovable within the system.

And somehow this isn't even what the book's about...

As the pages turn you will be steadily more tested and at some point it will become apparent you've not been paying close enough attention. However, even without taking pen to paper and labouring through the instructive exercises you can get a pretty decent glimpse at some exciting and fundamental thinking.


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[Manny · Rating 4 out of 5]

This is a nice book if you want to understand the Gödel incompleteness proof, and get an account that is both accessible and reasonably rigorous. There's a lot of other fun stuff as well, but it's the Gödel proof that's the core of the book, and if that doesn't turn you on then you aren't really going to think GEB is worth the effort.

Personally, I would say that this is one of the most amazing things ever. The more you think about it, the more bizarre it gets... there are mathematical theorems that are true, but which you can't prove. And not only can you can prove that that is so, you can even construct examples of such theorems! It sounds about as possible as eating your own head, but it really works... Hofstadter shows you the machine, takes it to pieces, and then puts it back together again and runs the engine. Vroom!

PS I remember, not long after GEB came out, leafing through an interview with Sylvester Stallone. The interviewer asked him what he was reading at the moment. "Godel, Escher, Bach," said Stallone. "It's really hard."

Probably Rambo is in real life a smart, well educated person, and this is deeply unfair to him, but I couldn't help finding it funny.

[Aloha · Rating 5 out of 5]

As I work my way through this dense book, I am reminded of the Zen tale of 4 blind men and an elephant. To settle a dispute between townspeople over religion, the Zen master had 4 blind men and an elephant led in. With the men not knowing it’s an elephant, the Zen master had each feel a part of the elephant. Each blind man gave a varying but inaccurate guess of what it was he felt. In conclusion, the Zen master exclaimed that we are all like blind men. We have never seen God, but can only guess based on our subjective feeling.

In much the same way, each chapter in GEB is like feeling a part of an elephant. Hopefully, by the time we touched each part, we have a good idea of what the book is about. Here is my layman’s take on what that elephant is, filtered by my interest in human cognition.

Gödel, Escher and Bach

The heart of this book is these Strange Loops that represent the activities inside our brains that turn into consciousness. GEB uses art and music, in combination with math and computing, to illustrate these self-referential loops. The mechanic of the loops is represented by the works of the mathematician Kurt Gödel, the artist M.C. Escher, and the musician J.S. Bach. Kurt Gödel’s Incompleteness Theorem shows that a formula is unprovable within its axiomatic system. Gödel’s usage of mathematical reasoning to analyze mathematical reasoning resulted in self-referential loopiness, basically saying a formula cannot prove itself. M.C. Escher creates visual presentations of this loopiness in his Waterfall and Drawing Hands.

Finally, J.S. Bach’s Musical Offering were complex puzzles offered to King Frederick the Great in the form of canons and fugues. A simple description of a canon would be a theme that played against itself, such as in “Row, Row, Row Your Boat.”

J.S. Bach - The Musical Offering:
http://www.youtube.com/watch?v=ZQWsOG...

Escher’s visual endless loops, Gödel’s incomplete self-referential theorem, and Bach’s canons and fugues in varying levels help to illustrate the characteristics of consciousness. The book alternates between Chapters and Dialogues. The Dialogue is between Achilles and the Tortoise inspired by Lewis Carroll’s “What the Tortoise Said to Achilles”, which in turn was inspired by Zeno of Elea’s dialogue between Achilles and the Tortoise. The purpose of the Dialogue is to present an idea intuitively before it is formally illustrated in the following Chapter. GEB presents varying ways of explaining about systems and levels that create these self-referential infinite loops.

Systems

To discuss intelligence, GEB starts off explaining the playground in which this takes place. We’re introduced to the idea of a formal system by the MU-puzzle. In a formal system, there are two types of theorems. In the first type, theorems are generated from the rules within the system. The second type is theorems about the system. This puzzle contains the string MIU. This system tells us to start with the string MI and transform it to MU by following certain rules. After going through the process, we find that we cannot turn MI into MU following these steps no matter how long we try. We would merely be generating countless strings. To stop endlessly generating strings requires the second type of theorem in which we analyze the system itself. This requires intelligence in which we gauge that this will be an endless task. We then guess at the answer intuitively. If a computer was told to try to generate the answer, it would go on ad infinitum. We humans, however, would soon realize that this is a hopeless situation and stop. We, the intelligent system, critiques ourselves, recognizes patterns, and jump out of the task it is assigned to do. It is difficult, however, for us to jump out of ourselves. No matter how much we try, we cannot get out of our own system. We, as a self-referential system, can talk about ourselves, but cannot jump out of ourselves. Thus, it is impossible to know all there is to know about ourselves. The countless self-help techniques are testaments to that.

Formal systems are often built hierarchically, with the high-level meaning where consciousness lies building from the low-level primitive functions. The most interesting example of levels is in the typogenetics of the DNA. GEB gives a detailed account of how enzymes work on the strands, with typographical manipulations creating new strands. The new strands in turn act as programs that define the enzymes. The enzymes again work on the strands. This system of enzymes causing the creation of new strands, strands defining the enzymes, creates a change of levels as new information are created from the process. Even readers who don’t like math would find it interesting to see how the coding of our DNA works, as chemicals help to turn simple codes into us. GEB gives further details on the complex process of chemicals and codes, but this is the basic idea.

Isomorphism

Isomorphism is a process of change that preserves information. As intelligent beings, we are able to detect isomorphism and thus recognize patterns. This allows a system to be interpreted in varying ways without losing important information. This is illustrated by Bach’s canons and fugues. A canon can vary in complexity, in which the “copies” can vary in time, pitch, and speed. Also, the “copy” of the theme can be inverted, in which the melody jumps down whenever the original jumps up. The “copy” can also be played backwards, such as in the crab canon. However the “copy” modifies itself, it still contains all of the information of the original theme.

Isomorphism is mathematically illustrated in the author’s pq-system invention. In this system, we are able to perceive that the string --p---q----- means “2 plus 3 equals 5”, with the dashes representing numbers, p representing plus, and q representing equals. The recognition of an isomorphism leads to more isomorphisms, such as in the development of language. This pattern recognition occurs countless times as part of our intelligence process such that we don’t even notice it. We regularly see patterns in our daily lives. The lower level isomorphisms are so simple, that we only see explicit meanings. However, the lower level isomorphism helps us to create the higher level isomorphisms.

From our experiences, we all have lower level explicit isomorphisms from which we deduce new patterns. These are our “conceptual skeletons”. When we see new patterns, we create higher level isomorphisms until the system is consistent to us. This process involves interplay and comparisons of our conceptual skeletons, seeing similarities and differences. Our conceptual skeletons can even exist in different dimensions that enables us to comprehend the multiple meaning of this statement, “The Vice President is the spare tire on the automobile of government.” When two ideas match in their conceptual skeleton, the mind is forced to link and create subideas from the match. While this is an important function of cognition, it also can create erroneous beliefs. This was illustrated visually with M.C. Escher’s painting, Relativity.

When you look at this, do you see a puzzling world that does not follow the physical laws? Most of us who are familiar with building structures expect some sort of an organization with stairs, gravity, and other physical laws. If you are familiar with building structures, you would start off identifying the lower or established isomorphisms, the staircases, the people, etc. From the lower isomorphisms, you create higher level isomorphisms with the new bizarre patterns that defy the physical laws. Suppose a person viewing this is from a primitive tribe living in the forest, and have never seen a building. What do you think that person would see when looking at Escher’s art piece? Perhaps that person would only see geometric shapes and nothing else, since there are no lower level isomorphisms of building structures, etc. The Dreaming in Aboriginal art adds a further dimension to interpretation of geometric shapes.

In much the same way, we build language based on isomorphisms. Children increase their word count by identifying matches to words they already know. Interesting problems with meaning comes when translating words from one language to the next, especially in literature and poetry, which often relies on implicit meaning to understand the content. This implicit meaning can change according to a society’s culture and history. The author’s book, Le Ton Beau De Marot: In Praise of the Music of Language, seeks to analyze that by featuring the work of the French poet Clément Marot.

Figure and Ground

There are two types of figure/ground. The first one is cursive, in which the ground is only a by product or negative space of the figure, and is of less importance than the figure. The second one is recursive, in which the ground is as important as the figure. This idea is also compared to theorems and nontheorems, or provability and nonprovability, nonprovability being key to the Strange Loops that is at the core of this book.

The chapter Figure and Ground starts with a set of rules for typographical operations which were used in the MU-puzzle and the pq-system, which is the mechanical process of the Turing machine, the parent of what we now know as computer intelligence. Basically, the process involves reading and processing of symbols, writing it down, copying a symbol from one place to another, erasing the symbol, checking for sameness, and keeping a list of generated theorems. This process of generating theorems is reliant on the sifting out of nontheorems. The parallel to this is the idea of figure and ground, and the idea of recursion with figure and ground holding equal importance. This is aesthetically explained using Escher’s art, Tiling of the Plane Using Birds, and a discussion on melody and accompaniment.

Figure and ground form the basis for the idea of recursive and recursively enumerable (or r.e.). A recursive set is one in which figure and ground holds equal importance. That is, its r.e. and the complement of its r.e. are equal. However, GEB showed that there exists formal systems in which the figure and ground are not recursive, do not carry the same weight, and are not complementary. Basically, this is saying that there are systems in which its nontheorems cannot be generated via a typographical decision procedure. A typographical decision procedure sifts out nontheorems from theorems by performing tests that use the logic of the figure/ground. Hence, “there exist formal systems for which there is no typographical decision procedure.”

Recursion

We are led to the process of recursion. Recursion is the process of building up from a block of structure. The simplest explanation of recursion would be the visual imagery of the Russian Maruscha dolls, in which an item is nested within an item within an item. However, this doesn’t mean that a process is simply a replication of itself. For example, in language, we start with smaller components such as words and phrases, and build up complex sentences from there.

The process is explained in GEB as “push, pop and stack” of Artificial Intelligence. When you “push”, you are temporarily stopping what you are doing to do something else. When you “pop”, you return to it but starting from where you left off, at one level higher. To remember where you left off, you store the information in a “stack.” The example given in the book is of someone answering multiple phone calls. We use the “push, pop and stack” process especially in our usage of language. The most complex example of recursion is in the genetic mechanism of DNA, in which the DNA molecules are formed from the smaller building blocks.

The defining characteristic of recursion is the change in levels, so that it is recursive instead of being circular. Neurologically, this is illustrated in the process of how symbols interact with each other. At its minimal are the bare particles that do not interact with others. They are nonexistent since all particles interact with each other. The process of interaction creates entanglement and a hierarchy of entanglements, a “6 degree of separation” of infinite loops. Recursion is a part of this entanglement.

Recursion is reliant on sameness/differentness. The same thing happens with slight modifications and at a different level. This is visually represented in M.C. Escher’s Butterflies.



A rule that is a product of the recursion process is the fantasy rule. The fantasy rule states that fantasies can be nested within fantasies, with differing levels of reality. The carry-over rule states that “inside a fantasy, any theorem from the reality level higher can be brought in and used.” However, the reverse cannot be true. You cannot bring something inside the fantasy out to the reality level higher. An example of this is when an writer finds inspiration from real life and brings it into the writing. But the writer cannot bring an imagined character out of the book into real life.

Messages

The process of entanglement involves the exchange of messages. This brings up the question of meaning in messages. Is meaning implicit in the message, or does meaning come about via interaction? A profound example is the genetic information in DNA. Our cells contain the genotype in our DNA which holds critical messages that triggers the manufacture of proteins, which triggers more reactions such as replication, until we have our physical manifestation or phenotype. There are varying thoughts as to the meaning of DNA. One view says that the DNA is meaningless out of the chemical context if there is no trigger to stimulate its production into the phenotype. The other view says that the structure of DNA is powerful implicitly. This goes to the heart of the question as to whether the value of information is dependent on whether it is usable to the environment. If we are not able to interpret or sense the message, does the message have any less value?

There are three levels of information, the frame message, the outer message, and the inner message. A frame message is implicit in the structure. It’s just there. The inner message is the transmitted message, content that is understood. Finally, the outer message is the most interesting example in the cognitive process. The outer message has several layers. It is the information that tells you how to decode the frame message to get the inner message that is implicit in the frame message.

However, in order to get at the frame message, we need to “recognize” that there is a need for an outer message as a decoding mechanism. Paradoxically, in order to “understand any message, you have to have a message which tells you how to understand that message.” This seems like it can go on infinitely with the messages never successfully acquired. Yet somehow, messages are often transmitted. This is because the human brain comes with the ability to recognize when there is a message. Thus, the outer message starts as a set of triggers that sets us to develop a decoding mechanism. Once the outer message is fully understood, there is no need for the inner message, since the inner messages can be reconstructed once we have fully developed the outer message.

It seems that the frame message would be useless without the outer message that includes the triggers, and there is no need for the inner message once we have the full outer message. This seems to be saying that the most important part of the messaging process is the recognition or consciousness part. This is similar to the fact that computer memory is not the same as computer computational power. A computer may contain countless data, but without the procedure with which to retrieve and process it, the data is useless.

Possibilities of AI

The discussions in the book on levels and hierarchy of systems and recursion lays out the fact that at the lowest level is a simple formal system which leads to the highest level, our informal system, the brain. This idea of a formal system being at the core of a flexible, self-referential informal system leads to the possibility of consciousness in inanimate objects, or artificial intelligence. However, we cannot logically and mathematically duplicate the informal system of the brain from the formal system. As was previously laid out in the book, the process of moving to higher levels to a complex system involves so many rules and unprovable elements, that AI researchers are currently unable to simulate the working of the human brain.

The computer can easily have deductive reasoning, in which it can logically come to a conclusion based on known facts. However, human intelligence includes analogical awareness, which involves complicated processes of nested meanings, comparison, and jumping of levels. Furthermore, there is the added self-referential element of how we “decide” to use our knowledge.

Even how information storage in the brain points out the sheer difficulty of emulating human intelligence. Our brains function via overlapping and tangled symbols such that each neuron could be identified with the whole of the brain instead of having information stored locally. It seems a symbol cannot be isolated from other symbols in the brain. Neurologist Karl Lashley, in his experiment, had rats learn to navigate mazes. After the training, parts of the rats’ brains were removed. Even with increasing removal of their brains, the rats were still able to navigate the mazes, although they had some motor impairment. However, neurosurgeon Wilder Penfield showed that memory is localized. He inserted electrodes into various parts of patients’ brains. These electrodes emit pulses similar to those emitted by neurons. When certain neurons were stimulated, memories and impressions of specific events were recalled. These two opposing experiments indicate that memory is not only coded locally, but spread throughout the brain. This is to safeguard against loss of information in case of brain damage.

GEB used the concept of “beauty” to come to the conclusion of the possibility of AI. At its lowest level, it is a logical concept. Beauty on the higher level is an illogical, unprovable concept that evolved from the recursive process and chunking of the information from the lower level. Although at the upper level, our consciousness is an unprovable system, at the base level, the neurons are performing logically. Thus, it is possible that the “irrational and rational can coexist on different levels.” This means that it is possible that the same process that is in the human brain can be achieved in AI. In order to achieve human intelligence, AI researchers will have to work on the lower levels such that the upper level is comparable to human intelligence.

GEB was written in 1979. It was reissued as 20 year anniversary edition with a 23 page preface, but little update in the content of the book. It is still valid today, although it did not mention the controversial possibilities of mind uploading mentioned by Marvin Minsky and Ray Kurzweil. Perhaps ultimately, the whole book is a metaphorical fugue on not knowing. All proofs are unproven in the outermost system. Hence, we can never really know ourselves.

[Andrew Breslin · Rating 5 out of 5]

I could not with a clear conscience recommend this book to everyone, because I'm simply not that cruel. It would be like recommending large doses of LSD to everyone: some small minority will find the experience invaluably enlightening, but for most people it's just going to melt their brain.

While you do not need to be a professional mathematician to appreciate this, you really have to like math a lot. You can't just sort of like it. You can't just differ with the masses in not hating mathematics. You can't just find it mildly interesting rather than utterly abstruse and inaccessible. For example, you pretty much have to find the following joke to be hilarious:

There are 10 kinds of people in the world.

Those who understand binary, and those who don't.

If you are slapping your knee right now, then you might like this book. If, during the course of slapping said knee, all the pens fell out of your pocket protector and landed scattered across the piece of paper you were using to make Venn diagrams to help you decide what to have for breakfast, that, of course, is even better.

If you really like math, then this is going to be one of the best books you've ever read. Go get it now! But if you really like math, then you've almost certainly already read it. If you haven't read it already, then you can't possibly like math enough to enjoy it. Hmmmmm.

There's a recursive paradox in there somewhere. Best not to think about it. It might melt your brain.

[notgettingenough]

from Randall Munroe. Mouseover says: 'This is the reference implementation of the self-referential joke.'

------------------------

I know, I know, I know. I'm just kidding myself. I'm as likely to read this as a book on string theory. (Please don't. Please don't tell me I have read a book on string theory, I'm trying to forget the whole sordid story.) But. I hope you like this.

A friend of mine established The Harvester Press in the 1970s. He did it on a wing and a prayer, he was a young teaching academic who couldn't find in print the old literary books he wanted to use as texts and so he set about publishing them. He was probably as surprised as anybody when the idea quickly became viable. He put together a list of books, sold them as a subscription to libraries and away he went. He wasn't an academic any more, he was a proper publisher with a strong reputation for intellectually high end output.

At some point he got sent a completely insane looking ms, ridiculously long, bits of paper stuck on bits of paper, all these pictures which hadn't any copyright permission, and as for the title...well, who was going to buy a book called that....he sent it back with a polite letter.

Some years later he was in NY lunching with the boss of Basic Books, a US academic publisher. He wanted to publish this strange ms. he'd been given. As he was describing it, John interrupted with 'Godel, Escher, Bach I presume?' Evidently Hofstadter had gotten lucky and had on loan a very early word processor. The whole thing was no longer the shambles it once was. Basic Books was keen.

John got talked into taking some thousands of copies. This turned out well for him, but. What he had lost. Ouch. Godel, Escher, Bach in English and in translation would have made him many millions. I won't say he cried about it, but he did ask for a discount on the books he was buying. After such a sad tale it was impossible to say no.

[Barbara · Rating 1 out of 5]

This book told me something about intelligence - the smartest thing to do is to avoid this book's overly lengthy babblings of a self-important graduate student who is way too impressed with himself. It took this guy over 700 pages to illustrate by analogy his not-particularly novel theory which he sums up (finally) as follows:

"My belief is that the explanations of 'emergent' phenomena in our brains --for instance, ideas, hopes, images, analogies, and finally consciousness and free will--are based on a kind of Strange Loop, an interaction between levels in which the top level reaches back down towards the bottom level and influences it, while at the same time being itself determined by the bottom level."

Duh.

[Tara · Rating 5 out of 5]

This book was so metal. Gödel’s incompleteness theorem, which states that all consistent axiomatic formulations of number theory include undecidable propositions, is certainly a large part of what made the book so fascinating and addictive. The issues of self-reference and self-awareness, and how they relate to both human and potential artificial intelligence, were likewise extremely compelling. But the magic is in the math.

Here is a brief summary of the Gödel in the book:



The above image knocked my socks off when I first saw it, and I’m still running around barefoot. The longer I think about it, the more astounding it becomes. No matter how many new branches you form on the trees, no matter how small they might be, there will always be some unreachable truth. Wow. No socks.



The preceding picture by Escher demonstrates another key point of the book, to wit: “…such twisting-back, such looping-around, such self-enfolding, far from being an eliminable defect, was an inevitable by-product of the system’s vast power.” Gödel’s second theorem states that “neither G nor its negation can be a theorem. We have found a ‘hole’ in our system—an undecidable proposition.” (You can even see the corresponding hole in the drawing by Escher—it is in fact necessary for the image to “make sense.”) “The fascinating thing is that any such system digs its own hole; the system’s own richness brings about its own downfall…Once the ability for self-reference is attained, the system has a hole which is tailor-made for itself; the hole takes the features of the system into account and uses them against the system.” No truly robust system can be consistent, and no consistent system can be truly robust.

Thus far, I have attempted to summarize the main thematic elements of the book as succinctly as possible. Now for some of the pros and cons of the book:

Pros: It was conceptually packed full of food for thought. The book had everything you could ask for: logic, mathematics, philosophy, music, art, psychology, genetics, recursive paradoxes, bad jokes about recursive paradoxes, bad jokes about bad jokes about recursive paradoxes…the list goes on and on. Sorry, had to go there. Anyway, I also thought that the examination of the nature of intelligence was a worthwhile portion of the book. The Bongard problems were a particularly neat way of looking at meaning, pattern recognition, and how to program critical thinking/analysis/reasoning.

My personal favorite part, math-wise (other than Gödel’s insanity of course), was Cantor’s Diagonal Argument. The infinitude of the real numbers never ceases to blow me away. Just think of the all the real numbers contained between 0 and 1, for instance: there is an infinite universe contained therein. Infinite. And that is only between two integers. How many of those integer guys are running around again? Oh yeah, quite a few. Thus your mind is blown, or at least mine always is...it’s too beautiful to imagine. Then you have Cantor’s Diagonal Argument, which demonstrates this amazing property, and the proof smacks you right in the face with its elegance and its simplicity, its sheer genius. (By the way, I highly recommend looking at Cantor’s argument on its own if you’re that way inclined.) Basically, Cantor showed that “…no exhaustive table of reals can be drawn up after all—which amounts to saying that the set of integers is just not big enough to index the set of reals.” “The insidious repeatability of the diagonal argument” indeed! Goddammit math! The field of real numbers is just so badass and beautiful.

Cons: The book was a bit repetitive at times, and much more long-winded than it needed to be. Hofstadter’s personality also definitely started to wear on me. I love math puns and groaners as much as the next nerd, but one can be too cute, too clever. The sense that the author was a little too satisfied with all his tricks and puns and overly witty structuring was often irritating. In fact, it bordered on feeling smug and obnoxious more often than not.

Overall, however, no matter how annoying the author’s too-clever cleverness could be, the fact remains that the book explored some genuinely fascinating, complex conceptual realms, and did so in quite a bit of detail. For that reason, it was an excellent read. I’ll leave you with two philosophical nuggets from the book (the second is clearly humorous in tone, but fun to think about nonetheless):

“From the balance between self-knowledge and self-ignorance comes the feeling of free will.”

“Everyone knows that the insane interpret the world via their own peculiarly consistent logic; how can you tell if your own logic is ‘peculiar’ or not, given that you have only your own logic to judge itself? I don’t see any answer. I am just reminded of Gödel’s second Theorem, which implies that the only versions of formal number theory which assert their own consistency are inconsistent…”

[Forrest · Rating 5 out of 5]

If I were clever enough, I would write this review as a fugue. This is the formal structure that Hofstadter uses throughout Gödel, Escher, Bach. Whether the whole book is a fugue, I'm not smart enough to tell. But the fugue is used as a metaphor for layers of brain activity, thoughts, superimposed over the “hardware” of the brain, the neurons.

In fact, though I would recommend starting at the beginning of the book, I suppose one might begin anywhere and read through and back again, a'la Finnegan's Wake. No, the book isn't designed this way, but considering that I couldn't discern a solid central idea until page 302 of the book, and that this was only one of several theses in the book, I wouldn't be surprised if it proved possible to begin anywhere.

The idea presented there is “To suggest ways of reconciling the software of mind with the hardware of brain is a main goal of this book.”

The question is, does it succeed?

I would argue that it does not.

And it does not matter.

There are some works, such as Giorgio De Santilliana's Hamlet's Mill or Daniel Schacter's Searching for Memory that are so vast and all-encompassing that it is difficult to pin down one central thesis. These are the kind of works that you might not understand in your lifetime, the thoughts of a genius transposed directly to paper that, unless you are an equally-gifted person or a savant, you cannot hope to fully comprehend. Still, the threads and nuggets of gold that are spread throughout make it worth the time spent in the dark mines of incomprehension, if only to find that one fist-sized chunk of precious metal and appreciate its beauty set against the background of your own ignorance.

As far as I can tell, the book is really about intelligence, both human and artificial. Hofstadter does a lot of preliminary work priming the reader's brain with assumptions taken from theoretical mathematics and computer programming. But don't let that scare you off! I'm no math whiz, but I found most of the logical puzzles at least comprehensible after a few careful reads. Hofstadter also gives the occasional exercise, leaving the reader without an answer to his question. Like all good teachers, Hofstadter understands that the students who work things out on their own are the best prepared students. That doesn't mean that you won't understand many of the book's salient points if you can't successfully answer his questions. You can. But in order to understand the finer points, I suppose one would have to have a pretty good grasp on the answers to those questions.

I don't.

And it didn't matter.

What did matter, for me, was having a little bit of a background in the idea of nested hierarchies and a smidgen of knowledge in non-linear dynamics (aka “chaos theory”). For the former, I'd recommend Valerie Ahl's seminal Hierarchy Theory: A Vision, Vocabulary, and Epistemology . For the latter, just do what you were going to do anyway and look it up on Wikipedia. I won't tell anyone.

The idea of nested hierarchies is central to the understanding of what makes human intelligence different from machine intelligence. The short story is this: human thought is structured from the ground up according to the basic laws of physics, in particular, electricity, because it is through electricity that neural networks . . . well, network. The issue is that the layers interceding between neural electrical firings and human thought are tangled. They are explainable, or ought to be explainable, by a series of “tangled” layers that lead up to the higher functioning of thought. Again, this is one of the central points of the book.

And this is the point where Hofstadter utterly fails.

And it doesn't matter.

You see, Hofstadter never convincingly shows those transitional layers between neural activity and thought, though he claims they must be there. He claims that it should be possible to create an Artificial Intelligence (AI) that is every bit as human as human intelligence. The problem is, how do you define human intelligence?

Hofstadter presents the problem like this:

Historically, people have been naïve about what qualities, if mechanized, would undeniably constitute intelligence. Sometimes it seems as though each new step towards AI, rather than producing something which everyone agrees is real intelligence, merely reveals what real intelligence is not. If intelligence involves learning, creativity, emotional responses, a sense of beauty, a sense of self, then there is a long road ahead, and it may be that these will only be realized when we have totally duplicated a living brain.

One of the big issues in identifying whether an AI is actually intelligent is the notion of “slipperiness”. The concept here is that human thoughts can deal in a larger possibility space (my words) than machine “intelligence”. Hofstadter quotes from an article in The New Yorker, in which two statements are made that, while possible, would constitute lunacy on the part of anyone who actually believed them. They are:

If Leonardo da Vinci had been born a female the ceiling of the Sistine Chapel might never have been painted.

And if Michelangelo had been Siamese twins, the work would have been completed in half the time.

Then he points out another sentence that was “printed without blushing”:

I think he [Professor Philipp Frank] would have enjoyed both of these books enormously.

Hofstadter comments: “Now poor Professor Frank is dead; and clearly it is nonsense to suggest that someone could read books written after his death. So why wasn't this serious sentence scoffed at? Somehow, in some difficult-to-pin-down sense, the parameters slipped in this sentence do not violate our sense of 'possibility' as much as in the earlier examples.”

This allowable playfulness is something so complex and multi-layered, that an AI would be hard-pressed to correctly parse an “appropriate” reaction.

This is just one case portraying the difficulty inherent in trying to define and understand intelligence and the connection between brain hardware and mind-thought. The book is rife with them. I'm not convinced that Hofstadter was fully convinced that there will ever be a machine so “intelligent” as to completely mirror human thought.

And, one last time, it doesn't matter.

This book has set me to thinking, thinking hard, about what it means to be human. Not merely as an intellectual exercise, but deep in my emotional breadbasket, if you will, I feel human in a way that I can't explain when I think about the difficulty of trying to translate my hopes, fears, love, creativity, wordplay, happiness, sadness, and ambitions into machine language. There has been a lot of talk lately about “singularity,” that moment when machines become self-aware. I'm beginning to think that it will never happen. And I'm fine with that.

Besides, Hofstadter gives an implicit warning when quoting Marvin Minsky, who said:

When intelligent machines are constructed, we should not be surprised to find them as confused and as stubborn as men in their convictions about mind-matter, consciousness, free will, and the like.

In other words, if we do somehow construct true Artificial Intelligence, with the same capacity for thought and feeling as human beings, whose to say the “person” we create isn't going to turn out to be a real douchebag?

Terminator, anyone?

[Infinite Jen · Rating 5 out of 5]

From the vault of James O. Incandenza.

The Balkanization of Disciplinary Fields Via Academic Fiat and Truncated Lifespans. Year of the Eric Clipperton Action Figure With Frighteningly Realistic Gun Shot Mediated Demapping Accessories. Post-irony-sincerity (Stephen Fry; voiced by Stephen Fry; appearing as Stephen Fry) claymation superimposed on background of ubiquitous irony, represented by (Aleister Crowley; voiced by Ozzy Osborne; as vast as the horizon and undergoing various geometrical manipulations along median, saggital, coronal, and transverse planes w/r/t the emergence of childlike sincerity w/ narration by Thulsa Doom (James Earl Jones; voiced by James Earl Jones with disruptively pregnant pauses and gravid affect leading viewers to suspect solar plexical gong ringing induced dyspnea for reasons inscrutable but nonetheless deeply concerning; 8.5”x11 Dora The Explorer Maximum Security Journal, variously inscribed by scintillating buffet of crayola and mashed potato cuneiform.

Entry One: “Over the blackly implacable telos of nature’s flawed unfolding”

Diplomatic relations between Jen and Reality have pretty much broken down. The bromides which compose my therapist’s lingua franca, while never palliative, at least tokened my smug regard as necessary professional fluff, no more. I now find each instance (i.e. why should you be an expert on Meteorology? It takes time to become even conversant in complex fields. It’s not even your primary field of study. Even the smartest people are ignorant of 99.9% (some fiendish repeating decimal) of all human knowledge) akin to being struck in the pelvic bone by, (i.e. directly on edge, that is, pierced in the pubis by a steeple of heavy-duty paper based product of great thickness and superior durability to that of average paper products) by a box of envelopes. A perfect hatchet toss. I’m paying 90 bucks an hour to be told that constructing a vast and polymathic intellect capable of keeping up with any conceivable cocktail digression is difficult? No shit, Richard! It is then that I always imagine reaching over the Modern & Affordable IKEA landfill he calls an office. Reaching over to seize him by the cravat, dislocating my jaws and slowly pillowing his baldpate with the moist tissues of my mouth, with my oral mucosa expanding all around his face like pink, glistening airbags, undulating like a python working a gazelle deep into its bowels. A giant condom with teeth unrolling down the shaft of an enormous zucchini. It’s only after I’ve engulfed him up to the shoulders that he begins the terrified murmurings of Jonah into the strobe of my obstructed glottis as it closes and opens upon him like puzzled elevator doors. The upper esophageal sphincter parts to accept his eulogy, the sounds spiraling down my gullet and striking the waiting marsh of digestive enzymes, producing strange acoustics as broken prayers and uxorious prattle discloses itself involuntarily. Suction being insufficient to draw up his entire being, with a mighty effort I lift him parallel to the ground, then tip my head backwards until his legs are bicycling against the ceiling as gravity forces him deeper into this hell. I walk out of his office shockingly distended. Satiated and sleepy.

But I have procured a book which will ameliorate these serpentine tensions. It covers such a range of topics that I’ll always be poised (in conversation) to spring (with unnerving bellicosity) into the most esoteric of discussions. Like Gödel’s monstrous progeny (Gödel’s incompleteness theorem) this universal acid (i.e. the theorem) which can’t be contained (i.e. will prove corrosive to whatever container it is placed inside), I will dissolve the foundational axioms of any system with a viciousness which will prove my own self negation. I consumed the entire text (i.e. waterlogged its bulk within my shower over the course of a day, rendering a slurry of bloated pulp stripped with diluted ink and bits of toxic adhesives, which I ate/drank in order to become enlightened, and, it must be said, gastrointestinally distressed to the point of requiring emergency irrigation.) and had to be shuttled to the hospital forthwith. I lay in bed delirious and sympathetic to Odin having thrown himself on Gungnir (i.e. his spear) in the name of wisdom. I will observe any changes in my cognition over the following days, provided the gastric lavage has quelled my garrulous stomach.

Entry Two: “Dita Von Teese in Voluptuous Recumbency an Infinite Recursion of Pulchritude.”

I now see the metafictions which form the connective tissues of my life. Homologies and Isomorphisms are ubiquitous in the deep physical principles of reality which in turn are reflected in these infinite record players and sassy crustaceans. When this struck me, (after the residual contents of my stomach were propelled several feet away from me), I couldn’t help but think of what I would do with my favorite Dita poster when I got home. I would use a simple camera and a mirror to produce an infinite hallway of my couchant goddess soaping her cleavage with a giant foam olive. Which would assist me in reaching such lofty conceptual heights that I would gaze into this non-terminating martini of morphean passivity and minister sapphically unto it thus;

ME: “Empress, how might I know everything? How can I surmount this logical problem whose circumference will continue expanding as each solution discloses further inconsistencies?”

DITA: “MIU, PQ, TNT, Typogenetics, B/F/Gloop. I am a Strange Loop.”

ME: “You are a Strange Loop.”

DITA: “J.S. Bach’s endlessly rising canon. A cauldron of book soup. I am a Strange Loop.”

ME: “You are a Strange Loop.”

DITA: “Logic. Art. Human thought. Here’s the scoop; You are a Strange Loop.”

ME: “I am a Strange Loop.”

Unfortunately, I’m still not out of the woods. The doctor seemed rather confused when I explained to him the logic of my feast. He was especially concerned that I had used an entire jar of vanilla scented bath salts in the tub that would become my cooking vessel. He said I was lucky to be alive, to which I snidely replied; “Luck is for the ill prepared.”

—The tree image! This brilliant depiction of incompleteness, consisting of metaphors on top of metaphors, ostensibly trees, but also clearly denoting the dendritic profusions sprouting from the stalks of axons like broccoli inside the brains of we observing this clever illustration. Within one image, the fractal nature of the Coastal Paradox (i.e. the coastline of any landmass has no well defined length on this level of analysis), the self similarities of nature’s designs, (i.e. recursive growth) the white tree’s twin, a stalk of black with tributary voids subject to patterned outflows (i.e. Bifurcation. Trifurcation. Tetrafurcation. Pentafurcation) like morphological variations in the branching pattern of the coronary arteries which enervate the human heart, symbolizing that which is true yet will not submit to formal rules of inference. The fuzzy boundaries between truth and falsehood, never resolved despite attempts at magnification, like the Mandelbrot Set, exhibiting an elaborate and infinitely complicated boundary.

My epitaph reads;

"Of what wouldst thou ask me?
Why temptest thou me?
Odin! I know all,
where thou thine eye didst sink
in the pure well of Mim."
Mim drinks from mead each morn
from Valfather’s pledge.”

And below, in fine script;

“I would eat it again.”

[Edward · Rating 5 out of 5]

Gödel, Escher, Bach is really something unique and special. The book attempts to put forward the outline of a theory of intelligence, by drawing from an incredibly wide array of disciplines - not just the three (mathematics, art and music), which are implied by the title – but also logical systems, computer science, genetics (there are really too many to list) as well as a considerable amount of literary flair.

The core argument is speculative, and more philosophical than scientific. However the approach and outlook is strictly materialistic. Questions about the nature of intelligence, consciousness and free will do not hinge upon the existence of a soul, or some other mystical “essence”.

GEB constructs the argument from the ground up, detailing each concept in turn, building towards a conclusion. Though the final chapters are perhaps the most rich in terms of speculative ideas, the unifying thesis is ultimately a little weak; a little loosely defined. But the book is so much more about the journey than the destination. Along the way we are treated to a deluge of ideas. Hofstadter is not afraid to get into the weeds – very, very deep into the weeds. This is not a book you can passively absorb. You will be required to think about the problems presented, and work to understand and use the complex logical systems that have been created, or you will quickly lose the thread of the argument. The payoff is there, but the work cannot be avoided.

Aside from the ideas it explores, it is the structure and composition of the book itself that is a wonder. There is so much contained in these pages. It is so deep, so fractally self-referential. Every page creates another link, another loop, another facet for contemplation. This is a book that be can read many times and still offer new ideas with each reading.

[WarpDrive · Rating 5 out of 5]

This is quite a remarkable book: a repository of many brilliant, provocative and insightful ideas (although occasionally not fully developed), and a contributor of much food for thought in disparate areas such as neurosciences, AI, mathematical logic, computer science, molecular biology, even art and music.

A unique endeavor that, while not always successful in the pursuit of a coherent and convincing elucidation of the author's theses, represents something of a classic that must be read for its enriching, wide-ranging, multidisciplinary, hugely entertaining, wonderfully brilliant, highly creative nature.

It is definitely not a great example of conciseness (with its sprawling 800 pages), and not always rigorous (for example, in its treatment of some aspects of mathematical logic and Godel's incompleteness theorems), and also occasionally a bit loose with the terminology; however this is more than compensated by the peculiar and exhilarating way with which the many interconnections and analogies between different disciplines and perspectives are illuminated by the author in his wondrous and sprawling synthesis, architecturally beautifully even if not always based on completely sound foundations.

What is also peculiar with this book is that, while the author does not manage to provide very convincing support to his overall thesis about the ultimate nature of intelligence in self-conscious entities, this does not really matter: what matters, in this book, is not so much the ultimate destination, but the exhilarating intellectual journey that the courageous author bestows on the reader willing to follow him in this adventure: there are so many themes, so many original insights, so many side topics, so many multi-layered threads that it is simply impossible to render justice to this book within the confines of a GR review. I can only invite the discerning reader to embark on this intellectual adventure – it is well-worth it.

The only proviso that I feel the need to emphasize is that some of the points raised by the author (especially in relation to some aspects of mathematical logic), are not completely rigorous, or are just selectively or very partially treated/explained, with the result that the unwary reader, provided with insufficient knowledge of the subject matter in question, might be occasionally mislead into incorrect beliefs.

I have listed here some of the points raised by the author that might have been dealt in a more rigorous way (please note that this is not aimed at conveying a negative impression on the overall quality of this otherwise great book), and also some points that I found particularly interesting and worth mentioning.

This list is, of course, very far from complete:

- pages 20-21: the author states that, in order to banish what he calls “strange loops” (like the famous Russell's Paradox), it was deemed necessary to introduce a “theory of types”, thus adding an artificial-seeming hierarchy to the original theory of sets. This is only a partial view, in my opinion, and this view should also be heavily qualified: in fact, with the ZFC axiomatic system, there is simply no way to define the “set of all sets”, as this would contradict the axiom of regularity. This “artificial hierarchization” the author is complaining about is, actually, a non-problem in ZFC (and similar axiomatic systems).

- Page 40: I really like how the author highlights the difference between the set of axioms and the set of theorems in any formal system, with the former always having a decision procedure, while the latter may have not such a procedure. I would have personally added here the important topic of “axiom schema”: formal systems can have a finite, explicitly defined set of axioms, but also a countably infinite set of axioms defined through an “axiom schema” (defined through a formula in the metalanguage of the formal system). As an example, we can refer to the axiom schema of replacement in ZFC (axioms that asserts that the image of any set under any definable mapping is also a set).

- Pages 71 to 74: the concept of “recursive” versus “recursively enumerable” sets is absolutely fundamental in the understanding of formal systems in general, and of Godel's incompleteness theorems in particular (in fact, one of the main preconditions for a formal system to be subject to Godel's incompleteness is that its set of theorems must be a recursively enumerable set). Here, while the author does a pretty decent job in introducing some form of intuitive understanding of this concept, I do think that a more formal and rigorous definition would have helped greatly.
In layman terms, a recursively enumerable set means that there is a computer program that, in principle, could sequentially enumerate all the theorems of the system without listing any statements that are not theorems (this is why, like Chaitin and others, I prefer to use the term “computably enumerable” or “Turing-recognizable”). This is the case, for example, of PA (Peano Arithmetic) and ZFC. On the other hand, a set is defined “recursive” if there is also a terminating algorithm which can correctly decide whether any given element belongs to the set.
In general, I think that the author should have used, in this instance, more examples and terminology from computational theory; this would have improved the conciseness, conceptual clarity and rigour of the exposition.
On the positive side, the author correctly highlights the critical importance of the fundamental result that there exist recursively enumerable sets which are not recursive (in fact this is, in abstract computational theory terms, the very essence of the first Godel's incompleteness theorem!). Also, this image on page 71 very effectively represents, with considerable visual symbolism, the relationships between various classes of the strings of a “sufficiently powerful” formal system. It is absolutely brilliant and I wanted to show it:



- Pages 86 to 88 (but also as a more general issue throughout the book). The author appears to be highlighting that the structural reason behind the incompleteness theorems is the capability of a sufficiently powerful system to attain self-reference. This is only partially true and it should be heavily qualified too: in reality, while it is true that Godel used a self-referential statement to prove the fact that number theoretical “truth” transcend “theoremhood” in any applicable formal system, this very result has also been separately demonstrated by Turing using computational theory (as an almost immediate consequence of Turing's halting theorem), and most beautifully by Chaitin using algorithmic information theoretical arguments; in either case, no explicit usage of self-reference has been necessary. The problem is not just self-reference (even though it can be said that this is definitely a major reason for this type of issues).

- On a different but related topic, the author shares an all-too-common view that tends see Godel's incompleteness theorems as essentially “limitative” in their nature. I deeply disagree with this perspective – in my view, while these theorems dispel any naive pretension for a 19-century-like (a-la Hilbert) finitistic and absolute achievement of simultaneous consistency and completeness, on the other hand we must bear in mind that:
a) although it is not possible to formalize all mathematics within one single formal system, it is possible to formalize essentially all the mathematics currently used. In particular, ZFC, combined with first-order logic, gives a satisfactory and almost universally accepted formalism for almost all current mathematics.
b) rather than seeing these theoretical results as a limitation to the expressive capability of formal axiomatic systems, what these results actually prove, in my view, are the infinite informational power and content of mathematics.

- Pages 94 to 96: here the author unfortunately gets into a bit of a confused muddle, using terms such as “consistency” in a loose if not cavalier manner, and certainly not in a way fully consistent with the specific notion of consistency of formal system as defined in mathematical logic.
For the broad class of formal systems whose language contains the “negation” symbol, syntactic consistency is simply equivalent to the property that there is no formula F such that both F and “Not F” are both provable within the system. Semantic consistency is, on the other hand, defined for a “theory”, and it refers to theories provided with a “model” (a theory has a model if there exists an “interpretation” under which all theorems in the theory are “true”).
The author unfortunately used spurious terms such as “external” and ��internal” consistency, or “consistency with the external world”, and this confusion is also present in some of his statements (for example, when he states that “consistency is not a property of a formal system per se, but depends on the interpretation which is proposed for it”) - I see what he means and what he says here is not necessarily incorrect, but these statements and terms could be deeply misleading if taken out of context (by the way, the type of consistency analyzed in Godel's first incompleteness theorem is purely syntactic consistency).

- Page 127: I partially disagree with the author's definition of recursion, in particular when he states that a recursive definition always refer to a "simpler" version of itself. In reality, a recursive definition always refer to a version of itself that is "closer" to a "base case" (a terminating scenario/definition that does not use recursion), not necessarily to a "simpler" version.

- Page 152: here the age of the book transpires quire clearly. We find a painfully obsolete statement by the author that reflects his deep skepticism about the capability of a computer program to ever achieve world-champion-level skills in chess-playing. This is another example of a all-too typical underestimation of the speed at which scientific and technological progress can deliver impressive results, not just in term of computational power but also in terms of new architectural and algorithmic paradigms. Nowadays, AI can not just play chess at world-champion level, but can also demonstrate outstanding flexibility and deep intuition capabilities in games (such as “GO”) where until a few years ago the task of artificially achieving this type of capability was deemed virtually impossible.

- I really liked how the author introduces, with many examples and with a very gentle progression, the concept of formal systems. From a pedagogical point of view, this is really excellent: the author makes these concepts completely accessible to the layman, and the examples described by the author, while being meaningful and very informative, nevertheless they do not require any significant mathematical nor mathematical logical background. Even concepts such as omega-inconsistency are made easily understandable – a significant achievement. On the other hand, while I do not dislike how the author introduces in a simplistic manner the concept of “Godel-numbering”, I nevertheless think that it is a pity that the author did not explain how Godel used the prime factorization technique to actually pursue his brilliant numbering scheme. But make no mistake: the main message of Godel's approach (that typographical rules for manipulating numerals correspond to arithmetical rules for operating on numbers) is well and effectively explained by the author.

- There are two major actors noticeably missing in the treatment of the many topics addressed in this book: the fist one is epistemology (and the total absence of any reference to Kant is quite disappointing): this is a significant gap, especially considering that a few issues dealt by the author have clear epistemological dependencies and ramifications, for example when the author refers to the perceptual and conceptual division of the world into categories accomplished by the human mind. The second one is information theory – another significant gap, considering that some results in information theory are perfectly applicable to more than a few of the author's items of analysis, starting from the information processing in the brain (which is nothing more than a sophisticated information processing biological machine, after all), all the way down to the information processing in molecular biology and genetics as enabled by the genetic code.

- Page 142 to 146: the author refers to Feynman's diagrams as an example of “recursion at the lowest level of matter”. I think that here the author has significantly misunderstood the “physicality” of such diagrams. In fact, Feynman diagrams are essentially a handy calculational tool designed in accordance with a perturbative approach to the description of particle scattering phenomena. Feynman diagrams are not to be confused with spacetime diagrams, nor with what you would see in a bubble chamber. Feynman diagrams are mathematical idealizations that “represent” intermediate stages of a scattering process: particles do not choose a particular diagram each time they interact. Each Feynman diagram is only a graphical abstraction of a single perturbative contribution to the transition amplitude of an event. Every diagram therefore contributes to the total amplitude for the process, and the sum of all diagrams provide the different amplitudes that "determine" the actual nature of the physical scattering event.

- Chapter XIII (about computability, primitive recursive functions, expressibility and representability) is a real gem. The author's examples of Bloop and Floop programming languages are compelling, accessible and informative. Extremely enjoyable and really well done. Just one small point: in general, primitive recursive functions can be n-ary (can take n arguments), not just one argument. Moreover, primitive recursive functions can be extended to integers and even rational numbers, not just applied only to natural number as it seems implied by the author.

- Chapter XVII is another great gem: several alternative versions of the famous Church-Turing thesis (some general recursive function always exists, capable of sorting numbers into two separate classes according to some predefined criteria, which would give exactly the same answers as a sentient being's method would achieve) are discussed by the author. Provocative, insightful and very enjoyable too.

The above is just a small and extremely partial subset of the many points raised by the author in several areas; there are even deep insights by the author in relation to all sort of related issues, such as a holistic versus a reductionist approach to the study of reality, free will in humans, the theory of meaning, etc.

I can only warmly invite the reader to take the time to enjoy this book - a GR friend of mine stated that this is one of the very few books that he did read twice, and I perfectly understand why.

4.5 stars, rounded up to 5.

[Nathan "N.R." Gaddis]

The reading of a book and its interpretation are determined in part by the cytoplasmic soup in which it is taken up. This reader’s soup consists of a large portion of metaphiction.

This is how Hofstadter apparently intended to structure his work: a Lewis Carroll styled dialogue between Achilles and Tortoise (and friends) introducing a subject followed by a rigorous but popularly accessible explication of that topic.

This is how I read Hofstadter’s book: as a crab canon. A crab canon, as our musicologists will tell you, inverts and reverses the main theme of the canon. I reversed Hofstadter’s organization of the book, reading the dialogues as the primary portion and treating the chapters as their mere explication. The fiction is more compelling than the non-fiction. As it should be.

Is it a difficult book? No. Not if you were in the top 20% of your high school class, paid attention in your biology classes, your math classes, had some chemistry and physics, perhaps took a few 101 courses in college, had a course on logic or mathematical reasoning, know a bit about music (Bach, of course), have some knowledge and interest in writing computer programs in things like BASIC and other computer programing languages popular in the ’70’s and ’80’s, play chess, etc. In other words, an averagely intelligent citizen in an educated nation ought to have no great difficulty with this book. What makes it more than a grab bag is Hofstadter’s setting side-by-side a fairly diverse set of topics and tracing out homologies and isomorphisms and analogies all which ought to culminate in shedding some light on the nature of consciousness and the prospect of Artificial Intelligence.

So but, what is made clear, if the prospect of AI is not, is that metafiction is not just a bunch of intellectual masturbation but is a fictioning which takes real things, ie, metamathematical structures, and uses them in structuring a story or fixing them into narrative metaphors. Metaphictionists are, after all, also Realists. For instance, everybody’s favorite metaphictionist, John Barth, subscribes to popular science journals like Scientific American in order to mine them for metaphor. He writes about his fascination with the coast-line measurement problem and the prospect of structuring his fiction upon principles similar to such things as fractals, Mandelbrot and all that.

Fictionists and novelists deriving their narrative structures from the sciences and philosophy is nothing new. Laurence Sterne borrowed from John Locke the theory of the association of ideas. 19th century social realism of the Zola and Dickens sort is inspired in part from marxist and other socialist movements. Woolf’s studies of experience could not have taken place without Husserl’s phenomenology of consciousness. How much fictional ink has been spent following Freud or feuding with Freud? And you really really have to read Calvino’s fantastical Cosmicomics. Fiction does not simply come out of nowhere. Form and content have always mutually determined each other in any fiction worth reading.

So, shall we say that the form and content of our beloved metafiction of the past half century is in part derived from discoveries in mathematics and metamathematics (Gödel’s contribution)? A mathematical theorem which is about itself? A set which has itself as a member? Etc. etc. . . .

. . . but you get the idea perhaps. I don’t intend to publish a thoroughly researched and exhaustive catalogue of metaphiction with its each and every GEB correlate. I’m happy to just point the way. And make my claim that all these wicked names thrown at metaphiction (‘masturbatory,’ ‘self-indulgent,’ ‘cold,’ ‘pretentious,’ etcetc) are simply wrong headed. Metaphiction is about a real thing, our experience of self-consciousness and what it makes us do. I believe that this wicked name calling in regard to smart fiction is due to a defense formation against the trauma of being the kind of thing we are and the kind of world we make for ourselves, compulsively. And that this defense formation dreams of a time when the recursive habit of thought could come to a comforting end in God’s safe arms; when consistency and completeness would both be simultaneously and within the same system, guaranteed. That when God disappears from our metaphysical world, it disappears from our fictional world; the omniscient, in-control voice of Author-God is gone [replaced with Gödel?]. Author and character become democratic equals providing characters with the metaphysical ladder up which they may escape this terrible mess the author has tried to create.

Examples, gratis, for more meta than you’ll wanna shake your stick at (and all of which occurred to my meta thinking brain whilst GEB-ing): John Barth, clearly, especially that seven story deep frame tale, “Menelaiad” from Lost in the Funhouse and “I’ve Been Told: A Story’s Story” from Where Three Roads Meet; Leyner’s fantastic The Sugar Frosted Nutsack exemplifying the full inclusion of self-within-self including the inclusion, etc; M.J. Nicholls’ A Postmodern Belch although he wouldn’t have thought that he had metamathematical bile in his bones [and we’ll let The Belch stand in for other such character-escapcapades such as Mulligan Stew and At Swim-Two-Birds etc]; Robert Coover’s The Adventures of Lucky Pierre and lot of his other stuff. Should we point out (obligatorily) that DFW was not able to overcome metaphiction for very precise reasons?

But enough about me. How about the book? Read it. It’s fun. Don’t be overwhelmed with Hofstadter’s two irritating verbal tics: treating ‘mind’ and ‘brain’ as interchangeable synonyms and his tendency to characterize his philosophy of mind opponents as ‘soulists.’ Also, it’s a 30 year old text addressing the question of AI; much has changed and much has not changed. Enjoy the damn thing.

[Also, hey, forget the mostly forgettably bland prose. It’s no matter; it merely functions. But do notice that he’s got a novelist’s eye and ear for structure. I dare say, with a few tweaks we’d have a pretty good novel.]



______________
Pre-Review Highlights [pertaining to commentaries #1-8]

Review tomorrow unless I say, 'Review tomorrow unless I say,' "'Review tomorrow unless I say,'" '"'Review tomorrow unless I say,'"' "'"'Review tomorrow unless I say,'"'" '"'"'Review tomorrow unless I say,'"'"' "'"'"'Review tomorrow unless I say,'"'"'" '"'"'"'Review tomorrow unless I say,'"'"'" '"'"'"'Review tomorrow unless I say,'"'"'"' . . . . . . .[but you get the idea]

[Andrej Karpathy · Rating 5 out of 5]

This book is a must read or at least must selectively skim for anyone interested in intelligence. Some of the ideas regarding intelligence and how it should be implemented are perhaps slightly outdated (you would see much more statistical reasoning if you asked experts today), which is largely absent in "old AI" approaches to intelligence.

[Xing Chen · Rating 5 out of 5]

Absolutely beautiful. GEB reads like a collection of sparks, produced when the mind is working at its primed, relaxed, hyper-aware and associative best. I read this over numerous nights, curled up in bed, each time feeling as if I was with a wonderful best friend, with whom I could discuss any topic or previously-unformed idea, exercise my memory indexing resources, and unabashedly release the inner infovore. Few things have allowed me to unwind, concentrate, and harness my mental energy as quickly and satisfyingly.

While it requires the reader to think, process, and invest effort in understanding his notations and work through the concepts, it’s written so that a creative young person with basic grounding in math could read through the chapters and be occasionally stumped, but mostly richly rewarded and inspired. I could not think of a more ideal book for a parent/ mentor to go over carefully with a growing child.
If only I could get hold of books that offered a similarly high-quality shot of serotonin more often..

[Annie · Rating 4 out of 5]

I'm going to be the only one who uses a gif in their review of GEB, aren't I? I'm definitely going to be the only one who uses a Legally Blonde gif. Fuck it.



So.

Part 1 deals with catching the reader up to speed on formal logic, number theory, and Godel’s incompleteness theorem. It’s the more tedious part of the book, to be sure. I took formal logic in university, and in Hofstadter’s more recent book, “I Am A Strange Loop,” he gives a pretty good overview of Godel and his relevance, so this wasn’t horrible. If I had no basis I might have struggled intensely.

Part 2 covers applications of part one to more philosophical explorations about consciousness and artificial intelligence and the meta-composition of everything. Hofstadter was in love with the word “meta” before any of us- actually, he’s the reason we use it the way we do today. In fact, that’s essentially the whole point of this book: manipulating mathematical & logical systems into paradoxes by bringing them to extremes through “going meta.”

Probably my favourite part of the book was when he got into discussing applying Godel’s incompleteness theorem to other domains of study than math and logic. Like psychology, or self-identity. Since our own mind is doing the thinking, how can we know what we’re thinking? How can we trust what we conceive of ourselves and our minds when it’s the mind that’s conceiving of itself? Doesn’t that create an infinite loop with no foundation, no proven things to stand on, just like Whitehead & Russell’s work on the “foundations of math” in Principia Mathematica, totally proven to be unprovable and foundation-less by Godel? That’s some acid-trip shit right there.

So. Hofstadter’s more recent book “I Am A Strange Loop” (gonna call this “IAASL” from now on), was written because he didn’t think anyone really understood what he was getting at when he wrote GEB twenty or thirty years earlier. I’m not surprised.

Why? This is probably going to be an unpopular thing to say, but I truly think Hofstadter does a better job of explaining, engaging with, and articulating his project in IAASL- which has the added benefit of being a third the size of GEB. Now, I acknowledge that IAASL is a recent book that’s only building on GEB, and that’s why GEB is more important, generally. It discussed important concepts early on, and was very influential in the 80s and 90s (now very dated and a little less shocking). But it’s unnecessary expanded, and gives you pieces of a jigsaw puzzle without ever really putting them together.

Also, I have this thing where I fucking hate dialogues. I like Hofstadter, just like I like Bishop Berkeley (love him, actually) and Plato and Galileo, but I Hate With A Capital H that they employ dialogues to communicate their thoughts. HATE IT. They're tedious and beat around the bush and make what should be one paragraph into several pages. Hofstadter's pre-chapter dialogues are no exception to my hatred. So GEB gets points subtracted for that.

There’s next to zero Hofstadter himself in GEB. I have no sense of who he is, why he thinks what he thinks. And I value that in a writer. Whereas in IAASL, I got to know Hofstadter very, very well. I suspect it’s just that over time he has became a better writer, thinker, and communicator and is better able to impart his thoughts straightforwardly.

So if you’d like to read the Book That Influenced The World, read GEB. If you’re only interested in understanding and playing with his ideas, read IAASL. If, like me, you say “the fuck with it, I want both” then by all means. Both are worth reading, if dense as all hell, and both gave me some really delightful epiphanies.

But read IAASL first. Trust me on this.

[Kaylee · Rating 2 out of 5]

After an entire tome about the workings of the mind and what it means to be intelligent, you'd think the author would be more self-aware by the end of the book than to say, "indirect self-reference is my favorite topic".

No, Mr. Hofstadter, blatant self-reference is your favorite topic.

I'm notoriously bad at distancing the creation from the creator, so perhaps I was biased from the start -- reading the 20th anniversary intro was like listening to a narcissist who insists he's modest. I didn't find what followed to be original, revolutionary, or brilliant; rather, I found it repetitive, regurgitated, and egotistical. Each chapter, he spent many pages questioning himself and the reader about connections between DNA, Godel's Theorem, fuges, AI, and many other topics from a well-educated mind. Ultimately, he would "prove" himself right -- usually by citing someone else's work with great derision.

Hofstadter has led a very privileged life by somehow accomplishing the task of convincing people that his educated acid trip is something to be read and cherished. Bravo to him. I'd love to see his reaction now that so many of his predictions have proven false (a topic not touched on in the 20th anniversary intro).

[Matt · Rating 5 out of 5]

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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

[Riku Sayuj · Rating 3 out of 5]

Less flourish, more focus. That would have been a better braid.

[Chris Via]

One could write a full dissertation on this book alone. What a monumental masterpiece! Exactly the book I needed to read at this time in my mental life. Thank you, Mr. Hofstadter.

[Koen Crolla · Rating 2 out of 5]

Pretentious crap. Hofstadter is about as interesting and insightful as a 14-year-old stoner who got a hold of some of his dad's reference books. The actual content of this book could fit in under a hundred pages, but Hofstadter feels it necessary to pack on pages upon pages upon pages of barely-relevant filler, much of it apparently just to show off with the fact that he read some classical Greek poetry once.

To be fair, it is a very ambitious book, and one that could have turned out very interestingly, but it's also plainly obvious Hofstadter just wasn't up to the job. The whole thing is a massive (and I do mean that literally) waste of time, though since it does have a knack for making dumb people feel smart, it will undoubtedly appeal to the xkcd crowd.
Even if none of them will ever actually finish the whole thing.

[Jef Sneider · Rating 5 out of 5]

Conversation overheard at a diner in Upstate NY between Rabbit and Dante. They have been arguing about the existence of God. Dante has been arguing against the proposition.

Rabbit: I have been recently reading a book which helps me to counter many of your points Dante. You should take a look at it. Godel, Escher, Bach: an Eternal Golden Braid by Douglas R. Hofstadter carries within it the seed of an answer to your skepticism. Hofstadter argues, using the pictures of Escher, the music of Bach and the mathematical formulas of Godel, that for those inside a system of any kind, be it music, art or mathematics, you can never prove everything which is true about the system! He does this in the most entertaining way, through dialogues and examples that lead even a Rabbit like me to the obvious conclusion.

Dante: it may be obvious to Mr. Hofstadter, and even to you, but to me, there may be an entirely different conclusion from the same information. Can you give me an example?

Rabbit: Of course. Let’s see.... Ah, imagine an Escher drawing, one of those odd drawings where the stairs seem to lead endlessly upward, yet always end back at the bottom.

Dante: I believe that Bach wrote a similar “Endlessly Rising Cantata”.

Rabbit: Exactly Right Comrade! Well, the drawing seems consistent, internally, to the characters in it. They can not imagine the world any other way. Yet for us, standing outside the picture, we can see the obvious tricks that were used to construct it.

Dante: So Mr. Hofstadter somehow argues that this proves there is a God?

Rabbit: Not exactly. He points out that those inside an internally consistent system can never fully understand the forces outside the system. Godel proves it mathematically by showing that there are true equations in mathematics which cannot be proven. (“This statement is false” is an example of a true statement which Godel translates into numbers, then transmutates into a number which cannot exist but does. It is all very confusing to a Rabbit!)

Dante: “Is a statement that is false” is a statement that is false. Is that a true statement? You do have a way with words, Rabbit!

Rabbit: But they are not my words. Hofstadter says that Godel said something like that. And he proves it, I think.

Dante: Whatever.

[Matt · Rating 1 out of 5]

This book was very disappointing, especially after recieving so much hype. I was struggling along through it in a workman like fashion, trying to follow his arguments (which to me often seemed like so much dribble and unnecessary obfuscation and nothing like a fun puzzle), when I got really stuck and so I went to the MIT website and started reading the class notes on this book. That only made me more disgusted with the book, since it turns out that the book is riddled with historical errors where Hofstadter just made things up or ignored actual evidence in favor of making his points.

At that point, I just gave up.

If you are going to read this sort of thing as part of something other than a textbook, you'd be better off wading through the last part of Neil Stephenson's 'The Diamond Age' where it detracted from the text as well, but at least was more amusing than this was after the third chapter or so. Actually, just read Neil Stephenson period, as between his various work he manages to do everything Hofstadter attempts only better.

[E. G. · Rating 4 out of 5]

Preface to GEB's Twentieth-anniversary Edition
Overview
List of Illustrations
Words of Thanks

--Gödel, Escher, Bach: An Eternal Golden Braid

Notes
Bibliography
Credits
Index

[Cassandra Kay Silva · Rating 5 out of 5]

This is an absolutely phenomenal work. Let me break it down for you. Topics covered: DNA and RNA replication, Artificial Intelligence, Zen Buddism, Eschers artwork, Computer programming, Bachs fugues, a whole host of literary paradoxes and critical thinking exercises wow fun! Now let me tell you what all of this great information rests in, the framework of mathematics housed by Godels own theorems and proof. Yikes! Luckily the author understands that not all of us think mathematically. Don't get me wrong its math, there is no getting around it. But he presents the material in so many various forms. He uses Lewis Carols interaction between Achilles and the Tortoise to help make mental connections for those of us who are literary minded (thank you!) and artwork for those of us who are visually minded. And then long strands of proofs in, yes you guessed it, mathematical formulas and the like as the bulk of the work. It is a staggering accomplishment, I was especially impressed by his using Achilles the Tortoise and the Crab (plus Genes) to get ATCG for the DNA portion. Very clever! I will say that I can't comment on how much of math I actually understood. I can say that his mingling of approaches lead me to a great deal of conclusions and just as with everything in math you get those great Ah Hah! moments where it all seems to come together and you make those connections that you have been meandering around for awhile. It is one of the fun things about math that doesn't often duplicate itself in other portions of life. A work worth taking your time over.

[Rob · Rating 5 out of 5]

Deep geekery. Let's build logic from its component parts. And then after by-hand fabricating that nomenclature, we'll use it to talk about intelligence, problem-solving, heuristics, etc. building up to general intelligence (generally) and artificial intelligence (specifically). Deep, heavy, at times extremely fun. Took me five years to read it.

And so somewhat in the spirit of the text:

GEB is like this incredibly attractive, incredibly smart, incredibly funny/witty woman that you meet through a friend. The early part of the relationship is a little tentative—what with both of you trying to get a feel for each other, and both of you not quite knowing what to make of each other—but the time you spend together is lots and lots of fun. And after a little while, you're both very comfortable with each other and the time passes quickly. Perhaps too quickly. You just can't get over how lovely she is, how funny, how brilliant... But then out of the blue she gets heavy. Even your light-hearted conversations end with your head spinning. What happened to the woman you thought you were falling in love with? So you walk away. But she doesn't seem heart-broken in the least. You walk away, and you stay away for a while. Until one night you realize that even if she was getting into deep and heavy subjects that it was YOU who was afraid; she'd asked nothing of you but to listen. And like a coward you walked away. But when you return to her, she takes you back—like nothing ever happened. And before you know it, you really have come to the end of your journey together. But you feel so enriched for it.

And yes that's a terrible and cloying analogy that takes it way too far. But I couldn't help myself.

[bajwa · Rating 5 out of 5]

This was an intellectual and a delightful journey. Quite long. Quite tough. For a person who never liked math and biology (and wasn't good at many others as well), who only loved listening to music (the sort which didn't help with the book at all), who is part of a conservative (not much art and questions I mean) profession and society, this book was sort of a nightmare. Noticed paradox in my statements. You wont mind them much after reading the subject book...

So it all starts with a simple but very interesting question. The question that how a material or physical thing like us can acquire consciousness and such level of sophistication. Can we make a formal system for human intelligence? Now these seem like philosophical questions but the author tried a different approach or perspective to answer them which was a fascinating one. So this book covered music, art, math, biology, AI, computers, psychology, philosophy and some other subjects. It turned out that formal systems are not formal at all. Especially the numbers...

All these long winter nights which I have spent on this book, they were all worth it. Although many things passed way over my head, there was a lot to learn from this book. I like math now. I like biology now. My knowledge about computers and AI has increased. I know some things about music now. I know Godel, Escher, Bach and many other peculiar people throughout the history now. I know about infinities, recursion, incompleteness, self-reference, paradoxes and some other interesting topics...

This book is a masterpiece and intellectually challenging, no doubt it is going to be on my favourite shelf. Now don't really remember how I came to know about this book but I am glad that I did. Also some good background in Mathematics would have helped greatly in the book...

[GoldGato · Rating 3 out of 5]

I have always wanted to be brilliant. So this was the book I chose to make myself brilliant. Not super. Not smart. Not nerdy. Just brilliant.

Alas, it didn't work. It's taken me years (yes, literally years) to get through this tome. If you asked me what it is all about, I couldn't tell you, Alfie. I remain blitheringly stupid. That's why they make British baking shows, for dunces such as I.

Tough read. This should be part of a Marines-type training course for readers. Much admiration for those who understand whatever the bloody hell it's telling you.

Book Season = Year Round (still lost)

[Come Musica · Rating 5 out of 5]

Scelsi di fare il dottorato in logica matematica, grazie a questo libro.

[Ian Scuffling · Rating 5 out of 5]

A mind to marvel at. Douglas R. Hofstadter may be the most brilliant man alive, and his genius is a staggering machine that perceives the infinitesimal operations that are within operations that are within operations. The Eternal Golden Braid that is Gödel, Escher, Bach is as fully realized in its structure, form, and content as any of the greatest novels I’ve ever read in my life—the beauty of a complex simplicity, even in its most inscrutable moments, left me brain drunk on connectivity, recursion, how the mind works, how logic works, how deconstruction works.

I think the core success for Hofstadter in this is how he is able to find conceptual dendrites across multiple disciplines—number theory, art, music theory, genetics, zen, information theory, artificial intelligence/machine learning, literary theory and more—and twist them together like a double-helix, the DNA of a colloidal unconsciousness. In this way, the book is almost like an Akashic Record of 20th century advancements through physics, biology, electronics, art and more. Really, the only absence of scientific philosophy was that of astrophysics and the quantum—but as I say it, it seems possible that it’s in there somewhere.

There’s probably a great case to be made about the density of the work being to its detriment—how impenetrable the number theory work is and therefore unusable the subsequent games are for a non-mathematically-inclined reader who has little to no training in formal logic. But I don’t think these are failings of Hofstadter’s, but rather failings for us as readers to take the time to fully engage the explication, or seize the opportunity to peruse secondary sources for deeper understanding. Truth is, by my measurement, if those things really mattered to you as a reader, and such comprehension were required for “getting” the larger concepts and themes of GEB, then you would dedicate yourself to those moments.

This isn’t to say there aren’t flaws—surely Hofstadter’s fictioneering isn’t as adept and nuanced as some of his arguments, the overlaying structural experimentation, or the intricacies of Bach’s fugues he tries to mimic. In fact, the Dialogues, albeit quite useful for the lay-reader to absorb and prepare for the successive chapter’s density, are a little too precious to be taken seriously as fictions that plumb great depth. That is, they’re doing demanding, deep things by playing with the ideas of the more straight forward science/philosophy chapters, but it amounts to a pretty surface level of experimentation. One can read them and see the cleverness of how they’ve been constructed, but beyond the construction and execution of the idea, there’s not a rigorousness to the fictions that other, similarly styled fictions (ie. metafiction) do. This, I think, gets to a core point people often (and often erroneously) make about metafiction: it’s too cerebral without any of the heart. However, for Hofstadter’s fictions, I do find them cold. It almost didn’t ever matter who was talking, whether the Tortoise or Achilles or the Crab, and the natures of their relationships were utilitarian rather than deeply or vividly wrought.

But, this is really a small quibble to be making with what are ultimately just riddles, puzzles, pieces of a larger thing that does say something deeper about our human selves. In another dimension, Hofstadter is likely another John Barth. At least in this timeline, he’s the Barth of Science Philosophy.

Finally, and kind of as an aside, it seems clear to me that Mark Z. Danielewski read this book and was inspired to create House of Leaves. The house is, like an AI, like a brain, an infinitely recursive structure composed of formal components carrying out rote operations that add up to a larger whole (and larger inside than out). In fact, even the size, shape, length and structure of the book mimics Hofstadter’s. I haven’t taken a chance to revisit if Hofstadter formally appears somehow in MZD’s novel, but he may have wanted to mask the face lurking behind the conceptual bases of his own seminal fiction.

[Colin Murchie · Rating 5 out of 5]

GEB is an astonishing achievement in popularizing mathematical philosophy (!), and among the few truly life-changing books I've read.

The central thesis is that under certain conditions sufficiently complex, recursive self-editing systems can develop arbitrarily complex behavior without reference to external organization - and given an author who spends his days coding AI systems, you can see where he's going.

That's dense, dense stuff, but helped by the author's charming expository style and vastly erudite range of references. (Many concepts are elucidated as comedic dialogues between characters borrowed from fellow author and logician Lewis Carroll or others, and there are many fascinating illustrations.) Many years of operating in the stratosphere of mathematics have made the author perhaps a little too playful in his sense of how many meta-levels a book or argument should contain, but he reiterates the thesis from enough angles that you can skip off the surface of a few pages if you find your brain exploding.

GEB has the potential to put your beliefs about the nature of consciousness and life on a much sounder, but more challenging intellectual footing, and it's a great distillation of the ferment but ever increasing levels of profundity experienced in theoretical science since the 1950s - when we found ourselves with power to explain phenomena we were confident we could leave to the spiritual even post - Enlightenment, just as Godel encountered our first (?) nonnegotiable limit on human knowledge.