About book Gödel, Escher, Bach: An Eternal Golden Braid (1999)
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.SystemsTo 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.IsomorphismIsomorphism 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 GroundThere 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.” RecursionWe 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.MessagesThe 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 AIThe 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.
If you open up the "20th Anniversary Edition" of GEB, you'll see that the first thing Douglas Hofstadter does in the introduction - the very first thing - is grouse that nobody seems to understand what his book is about. Not even its publishers or readers who just absolutely love it. A quick glance at the back cover will give you the same impression - even the glowing, two-sentence blurbs are hilariously vague, all of them variations on the theme of "Well, that certainly was ... something! Yes, quite a wonderful something indeed."So how are you supposed to know whether to pick it up? Or put less delicately, how are you supposed to know whether reading all 740 dense, sprawling pages is worth your while? The short answer is: "Read this book if you like to think about thinking, as well as to think about thinking about thinking." The long answer makes me nervous - since the typical review of this book apparently misses the point entirely, I feel like I'm starting out on thin ice. Oh well, I'll take a crack at it anyway.At its heart, this book is about whether you can start with simple parts and from them, build a system which is so complicated that it becomes more than the sum of its parts in a significant sort of way. For example, scientists have a very clear understanding of how a single neuron functions. They even have a fairly good understanding of how neurons operate in groups to take on specific tasks, like wiggling your pinkie finger. But there are around a hundred billion neurons in a human brain and the structure quickly becomes preposterously complicated - groups of groups of groups of neurons, all acting in interconnected ways to produce conscious thought. How do we get something as complex as human consciousness out of something as simple and well-understood as a neuron?The answer Hofstadter likes is that the brain operates on many different interacting levels, and that conscious thought is a product of the complex interaction between all these levels. So in order to understand something you're reading, you depend on individual neurons operating in basically deterministic ways to move signals around your brain, but you also depend on groups of neurons in your vision centers to recognize text, as well as other groups of neurons on other levels to understand that text, and other groups of neurons on other levels to fit that new understanding into the context of the previous sentence, and so on. All of this applies equally well to artificial intelligence, which is Hofstadter's field. It's just that an electronic brain would be built from transistors and subroutines instead of brain tissue.The title is a little misleading - this book is not at all about how when you get right down to it, Kurt Godel, M.C. Escher, and J.S. Bach are totally interrelated, man. Their work is just useful in getting deeper down into that idea of interacting layers that produce complexity. For example, Kurt Godel was a mathematician who proved that in any self-consistent formulation of number theory, you could generate theorems that, while "true", were not provable in within that formulation. Basically, he showed that any formal mathematical system is necessarily incomplete in specific ways. Here's the part where things start to get craaaaazy: If you build a "well-formed" number theory labeled X, then X can be used to generate a proof of X's self-consistency only if X is inconsistent. The reverse is also true. And all this relates back to how a system can be more than the sum of its parts.These are definitely interesting ideas and very worth reading about, but whether GEB is worth reading is a harder question. It's a very well-written, well-researched book. I love that the author goes (way, way, way) out of his way to spend time explaining difficult ideas, rather than to assume a dull or disinterested readership. But sometimes that tendency to dig deeper can start to obscure the central point of a chapter. I think that's why so many people lose track of what the book is actually about - there really are a ton of fascinating ideas that are all given equal weight.The book hops between two different formats. The first is your standard, well-written, popular discussion of complex scientific, artistic, or philosophical ideas. In fact, Hofstadter is very good at this part. He excels at getting the reader interested in - and even excited about - some traditionally inaccessible stuff. The second format is a series of short dialogs between fictional characters, interspersed between every chapter, that help to allegorically enforce the ideas in whatever chapter. Overall, this approach is very good at getting you to understand the complicated ideas Hofstadter is getting at. I found that my problems with the book weren't with the subject matter, which was fascinating and enjoyable, but with the author. Ol' Dougie H. loves this material. He loves it so much that he tries to infect you with his own personal sense of wonder and whimsy at how complex and beautiful art and life and science are. And of course he's right, but that's not the point. If he trusted you to feel these things for yourself, the book would be maybe 200 pages shorter. As it is, his constant pedagogical wordplay and artful brain teasers started out fun but after page 400 they started making me tired. Also, those forced injections of wonder and whimsy start to take on the flavor of little plugs for the personal fantasticness of Douglas Hofstadter. For example, his discussion of the language processing functions of the brain is interesting, but did he really have to bring up the fact that he's fluent in Russian and translated Eugene Onegin? In a short book or a movie, cleverness can be fun and exciting. In a 740-page tome, not so much.I strongly recommend this book to a very narrow set of people. If you think you'd be interested in the subject matter AND you wouldn't mind playing simple word or math games in the service of understanding it AND the inner workings of a computer scientist's marvelous brain seem interesting to you, then definitely read this book. I enjoyed it and found it very fun and informative, overall. But if you read this review and you get the feeling you probably won't like this book, you're probably right.
Do You like book Gödel, Escher, Bach: An Eternal Golden Braid (1999)?
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.
—Manny
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.
—Cassandra Kay Silva
A friend of mine calls this a book for "pretentious teens and people who are too busy reflecting on their own existence to do anything productive" -- with a bit of self-mockery, I'm sure. My early, tentative take on GEB is that it's decidedly unpretentious, almost certainly written to be as accessible as its subject matter will allow. (If anything, it's a little corny at times.) The subject matter is artificial intelligence, a field which I suppose could turn out to be a dead end in the long run, but which will hardly have been unproductive (think of the historical relationship between alchemy and chemistry). Hofstadter here lays out the terrain of AI, its cruxes and its prospects, and in doing so roams through the fields of music, art, analytic philosophy, math, and genetics, all the while binding the threads into a theory of mind.It's quite a bit to absorb. I read GEB over the course of the summer, dividing my attention between it and all the other books that I read during that time. I wish that I had read it with a pencil in hand; every new section sent me down a rabbit-hole of ideas and unanticipated connections, and you should never trust your ideas to your memory. I look forward to reading this again in a couple of years; I hope that by then I'll be able to talk about it more lucidly.
—Jacob
