Gödel, Escher, Bach: an Eternal Golden Braid
I have been wanting to read Douglas R. Hofstadter's book Gödel, Escher, Bach: an Eternal Golden Braid for several years. I finally bought it earlier this year and started the task of reading it. Hofstadter, in the preface, describes the book as "a very personal attempt to say how it is that animate beings can come out of inanimate matter." It can be classified as being about metamathematics, consciousness, or artificial intelligence. It uses the concept of "strange loops" to describe how different levels of a system interact to produce unexpected properties. These comments can only be taken as a high level introduction to these concepts, as the book is very dense.
The structure of the book consists of chapters divided by dialogues, which are between the characters of Achilles and the Tortoise, and others who join in later on. the dialogues illustrate a concept that is expanded in the following chapters. For example, in the dialogue "...Ant Fugue", the Anteater tells Achilles and the Tortoise about how he interacts with an ant colony as if it were a sentient thing, even though the ants themselves are simple beings incapable of larger concepts of self or complex communication. The ants carry signals via chemicals, which in turn produce symbols, which create an end result such as "get food here" or "build there". In the following chapter, this is related to the brain and its concept of self, against the fact of neurons and the signals they carry, and the symbols they create, all the way to the super-symbol of self.
The three men of the title are chosen because each created things that were somehow strange loops. Bach created canons and fugues, which are musical notes arranged in a tightly controlled but elaborate pattern. The mathematician Gödel invented the Incompleteness Theorem, which uses a system's symbols to describe itself. And Escher created art that creates illusions on different levels.
The author first delves into a version number theory which he calls Typographical Number Theory (TNT), because it describes how to turn axioms into theorems by using typographical rules. The first big concept is the difference between I-mode and M-mode. M-mode is machine mode, where one mechanically applies different rules. I-mode is intelligent mode, where one can make a leap of intuition, to create an abstraction out of other rules. For example, a machine can transform different strings to make theorems until it reaches a certain one, but an intelligent person can step back and, using I-mode, intuit that either there is an easier way, or that the goal is unreachable.
He also talks about figure and ground, and how some things are best expressed as lacking a well-defined properties. One of his examples is prime and composite numbers. He discusses meaning and form, and introduces the concept of isomorphism, which is a map or translation to a different meaning.
He then gets into a description of recursion. The dialogue preceding this is an interesting one where Achilles and the Tortoise read a story about themselves, wherein they have adventures and find a book about themselves, which they in turn read, etc. The discussion of recursion moves into musical patterns, linguistic structures, and of course math and computer science. An interesting chapter centers on meaning and how it exists in a text, whether meaning is intrinsic to a text or whether it only has meaning in the proper context. It turns out context is crucial, of course, and is described as the writing on a bottle which contains the actual message. The most interesting point in this discussion is that a message in Japanese might have to have an explanatory note saying that the text is in Japanese, but that note would have to be in a different language, or it would be pointless. The fact that the message is in Japanese is in effect the bottle, or context, and understanding this is the key to understanding the text itself.
Hofstadter then gets into a discussion of levels of meaning, specifically inside a computer. He describes the difference between transistors, registers, machine code, assembly language, and higher level languages. He then relates this to the brain and neurons, using the ant colony analogy. Then the discussion segues into finite searches and infinite searches, and the more abstract question of how to determine whether a search is infinite or finite. (I must admit I got a little lost around this point.)
Then he starts to tie it all together, using the TNT to talk about itself. This point about self-reflection is a key point in the whole book. Examples abound in Escher's work, such as the picture of the dragon trying to escape from a two-dimensional space, though it is in fact always two-dimensional, or his picture of two hands drawing each other (kind of a mutual reflection). Other examples are sentences which comment on themselves, like "This sentence is true." Or, more mysteriously, "This sentence is false." The truth/false nature of this last sentence is ambiguous, and in a way it transcends meaning. Another crucial sentence is "This sentence has not proof." He comes up with a method, following Gödel, of letting TNT talk about itself, and creates a sentence in TNT which as an ambiguous meaning like the one above. This is a bit of a paradox, since part of TNT's claim (like the Principia Mathematica) is that it is complete, and the lack of proof for this statement, G, or it's negation, ~G, means that it is really incomplete.
This can only make so much sense in the limited way I'm describing it, and there are so many other levels. Hofstadter then goes on to see if TNT can be superseded, but it turns out that there is no system that can be described as complete. Even if you made a system called TNT+G, it would suffer from the same limitations, and likewise TNT^G, or TNT^G^G.
He then describes how DNA works, and the process that turns it into RNA, and into proteins, which then perform functions. Some of these functions include making the DNA replicate itself, so that part of the message of DNA is a message to create more DNA. One question is, is DNA data, or is it program, or is it both?
The last chapters talk about the implications for artificial intelligence, how an intelligence might be formed out of circuits and software instead of neurons and impulses. He questions whether the seemingly infinite nature of recursion and self-reflection that is consciousness means that intelligence is not able to be produced in a limited system. He then wraps it up by talking about systems that are self-referential and the implications.
This book is a definite A+. I have always been interested in intelligence and AI. Hofstadter clearly brought a lot of different theories together in this book. All the concepts build on each other, creating a symbol linkage that is very complex in the mind. One of the most fascinating illustrations is of Bongard problems, puzzles of two different sets of six boxes. The sets are different, but not easily differentiated. One might have a certain number of shapes, and the other one might have another number of shapes, but the boxes have different sizes and shapes, so it is difficult to tell exactly what the difference is. These puzzles require deep abstract thinking and problem solving skills, the type of skills which are difficult to program into software.
Hofstadter describes these curious connections and references as strange loops. How does an intelligent being make sense of such abstractions and use them to solve problems? How are concepts related? How does the mind think about itself? How does a camera look at itself? I must admit that my mind started clouding over at some of the concepts. The math is just on the edge of my abilities, but I don't think it's necessary to be a math whiz to appreciate the book, especially the language centric parts. Anybody who has thought about thinking will enjoy this book. This one will stay on my bookshelf, and I hope to read it again to see if I understand it any better in the future. In the meantime, Hofstadter has a new book coming out, called I Am a Strange Loop.
The structure of the book consists of chapters divided by dialogues, which are between the characters of Achilles and the Tortoise, and others who join in later on. the dialogues illustrate a concept that is expanded in the following chapters. For example, in the dialogue "...Ant Fugue", the Anteater tells Achilles and the Tortoise about how he interacts with an ant colony as if it were a sentient thing, even though the ants themselves are simple beings incapable of larger concepts of self or complex communication. The ants carry signals via chemicals, which in turn produce symbols, which create an end result such as "get food here" or "build there". In the following chapter, this is related to the brain and its concept of self, against the fact of neurons and the signals they carry, and the symbols they create, all the way to the super-symbol of self.
The three men of the title are chosen because each created things that were somehow strange loops. Bach created canons and fugues, which are musical notes arranged in a tightly controlled but elaborate pattern. The mathematician Gödel invented the Incompleteness Theorem, which uses a system's symbols to describe itself. And Escher created art that creates illusions on different levels.
The author first delves into a version number theory which he calls Typographical Number Theory (TNT), because it describes how to turn axioms into theorems by using typographical rules. The first big concept is the difference between I-mode and M-mode. M-mode is machine mode, where one mechanically applies different rules. I-mode is intelligent mode, where one can make a leap of intuition, to create an abstraction out of other rules. For example, a machine can transform different strings to make theorems until it reaches a certain one, but an intelligent person can step back and, using I-mode, intuit that either there is an easier way, or that the goal is unreachable.
He also talks about figure and ground, and how some things are best expressed as lacking a well-defined properties. One of his examples is prime and composite numbers. He discusses meaning and form, and introduces the concept of isomorphism, which is a map or translation to a different meaning.
He then gets into a description of recursion. The dialogue preceding this is an interesting one where Achilles and the Tortoise read a story about themselves, wherein they have adventures and find a book about themselves, which they in turn read, etc. The discussion of recursion moves into musical patterns, linguistic structures, and of course math and computer science. An interesting chapter centers on meaning and how it exists in a text, whether meaning is intrinsic to a text or whether it only has meaning in the proper context. It turns out context is crucial, of course, and is described as the writing on a bottle which contains the actual message. The most interesting point in this discussion is that a message in Japanese might have to have an explanatory note saying that the text is in Japanese, but that note would have to be in a different language, or it would be pointless. The fact that the message is in Japanese is in effect the bottle, or context, and understanding this is the key to understanding the text itself.
Hofstadter then gets into a discussion of levels of meaning, specifically inside a computer. He describes the difference between transistors, registers, machine code, assembly language, and higher level languages. He then relates this to the brain and neurons, using the ant colony analogy. Then the discussion segues into finite searches and infinite searches, and the more abstract question of how to determine whether a search is infinite or finite. (I must admit I got a little lost around this point.)
Then he starts to tie it all together, using the TNT to talk about itself. This point about self-reflection is a key point in the whole book. Examples abound in Escher's work, such as the picture of the dragon trying to escape from a two-dimensional space, though it is in fact always two-dimensional, or his picture of two hands drawing each other (kind of a mutual reflection). Other examples are sentences which comment on themselves, like "This sentence is true." Or, more mysteriously, "This sentence is false." The truth/false nature of this last sentence is ambiguous, and in a way it transcends meaning. Another crucial sentence is "This sentence has not proof." He comes up with a method, following Gödel, of letting TNT talk about itself, and creates a sentence in TNT which as an ambiguous meaning like the one above. This is a bit of a paradox, since part of TNT's claim (like the Principia Mathematica) is that it is complete, and the lack of proof for this statement, G, or it's negation, ~G, means that it is really incomplete.
This can only make so much sense in the limited way I'm describing it, and there are so many other levels. Hofstadter then goes on to see if TNT can be superseded, but it turns out that there is no system that can be described as complete. Even if you made a system called TNT+G, it would suffer from the same limitations, and likewise TNT^G, or TNT^G^G.
He then describes how DNA works, and the process that turns it into RNA, and into proteins, which then perform functions. Some of these functions include making the DNA replicate itself, so that part of the message of DNA is a message to create more DNA. One question is, is DNA data, or is it program, or is it both?
The last chapters talk about the implications for artificial intelligence, how an intelligence might be formed out of circuits and software instead of neurons and impulses. He questions whether the seemingly infinite nature of recursion and self-reflection that is consciousness means that intelligence is not able to be produced in a limited system. He then wraps it up by talking about systems that are self-referential and the implications.
This book is a definite A+. I have always been interested in intelligence and AI. Hofstadter clearly brought a lot of different theories together in this book. All the concepts build on each other, creating a symbol linkage that is very complex in the mind. One of the most fascinating illustrations is of Bongard problems, puzzles of two different sets of six boxes. The sets are different, but not easily differentiated. One might have a certain number of shapes, and the other one might have another number of shapes, but the boxes have different sizes and shapes, so it is difficult to tell exactly what the difference is. These puzzles require deep abstract thinking and problem solving skills, the type of skills which are difficult to program into software.
Hofstadter describes these curious connections and references as strange loops. How does an intelligent being make sense of such abstractions and use them to solve problems? How are concepts related? How does the mind think about itself? How does a camera look at itself? I must admit that my mind started clouding over at some of the concepts. The math is just on the edge of my abilities, but I don't think it's necessary to be a math whiz to appreciate the book, especially the language centric parts. Anybody who has thought about thinking will enjoy this book. This one will stay on my bookshelf, and I hope to read it again to see if I understand it any better in the future. In the meantime, Hofstadter has a new book coming out, called I Am a Strange Loop.
posted by Dylan Peters | 3/29/2007 10:05:00 PM
2 Comments:
It sounds like a pretty slow read, but worth it. I'll have to pick up my copy and give it another try.
I thought you would like that one! I need to finish it now myself since I only got about half way through it before getting distracted by something else.
Chris A - TT
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