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4.32 AVERAGE
4.32 AVERAGE
Have you ever seen or heard something you didn’t understand? Did you dismiss it as nonsense? Did you search for a context against which it might make sense?
What did you do to find context? Have you ever been unable to find it? Did you give up, or bookmark it for later? Or did you somehow know that the search would be endless?
Is anything “nonsense” or is there always some possible context? Or is everything nonsense at a small enough scale? How many steps does it take to understand a simple thing like “water?” Do all of those steps make sense?
This book is highly readable utter nonsense. Think Lewis Carroll cubed.
What did you do to find context? Have you ever been unable to find it? Did you give up, or bookmark it for later? Or did you somehow know that the search would be endless?
Is anything “nonsense” or is there always some possible context? Or is everything nonsense at a small enough scale? How many steps does it take to understand a simple thing like “water?” Do all of those steps make sense?
This book is highly readable utter nonsense. Think Lewis Carroll cubed.
challenging
informative
slow-paced
This behemoth is essentially about paradox in the spheres of mathematics and art. Even the consideration of whether or not to read GEB is ridden with paradox. 1) You probably need to have some skeletal knowledge of number theory as a prerequisite, but math majors would probably be annoyed. 2) In order to fully appreciate GEB you'd need to feel cohesion between chapters and in order to get this big-picture connectedness you'd probably have to finish the book in less than 2 months – but tackling it in such a time frame would be a chore and strip away the fun, thus minimizing appreciation.
Still, my year-and-a-half journey definitely had its delights. The author has both a childlike wonder and firm intellectual grasp on the material, and both are apparent. The dialogues are super witty and jam-packed with connections to other parts of the book. Hofstadter gets you to play around with formal systems, marvel at the human brain, contemplate the concept of infinity. His own fascination with the topics is contagious.
The downside is the frustration. I struggled most with Bloop/Gloop/Floop and then it turns out they were barely even integral to the thesis, or anything really. After wading through Gödel's Theorem (which was somehow both fun and grueling) I felt the whole AI / brain mystery thing was kind of anti-climactic. Which is a shame. Both are deeply interesting concepts, but the tenuous connection between them is only made at the very end.
If you have some time on your hands for a project and these sound like concepts you'd enjoy pondering, GEB would probably be worthwhile.
Still, my year-and-a-half journey definitely had its delights. The author has both a childlike wonder and firm intellectual grasp on the material, and both are apparent. The dialogues are super witty and jam-packed with connections to other parts of the book. Hofstadter gets you to play around with formal systems, marvel at the human brain, contemplate the concept of infinity. His own fascination with the topics is contagious.
The downside is the frustration. I struggled most with Bloop/Gloop/Floop and then it turns out they were barely even integral to the thesis, or anything really. After wading through Gödel's Theorem (which was somehow both fun and grueling) I felt the whole AI / brain mystery thing was kind of anti-climactic. Which is a shame. Both are deeply interesting concepts, but the tenuous connection between them is only made at the very end.
If you have some time on your hands for a project and these sound like concepts you'd enjoy pondering, GEB would probably be worthwhile.
words cannot do this book justice. this book is incredible. it takes clever writing to a whole other level and is enthralling. however, you're going to need a niche set of interests to properly enjoy this book (think maths, comp sci, philosophy and a bit of cellular biology) but if you find those interesting, you will pour over it. going straight in at the number one spot on my list of favourite books.
me encanto muchito, me hizo sentir como si ahora sabia mas del munod, y me cambio mi vista erronea que finalmente las matematicas podran encompass todo. el godels theorem basicamente enseña que cualquier sistema formal que sea suficientemente powerful, nunca puede llegar a ser complete. tambien habla de conciousness y human intelligence soportando el conjecture que es posible que nuestros cerebros no son nada mas que las neurones que en higher level forman estos symbols que nos habilitan a ver el mundo de above y observar tambien nuestros propios pensamientos. habla de como ai, finalmente no hay razon por pensar de que no podran alcanzar los capabilidades de pensar y actuar como humanos. en conclusion, me encanto, y me hizo sentar to buaaa. aqui dejo un resumen escrito de forma simplona describiendo de que va el libro y los conceptos mas imporatnates.
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.
[image error]
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 f
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.
[image error]
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 f
While the ideas espoused throughout the book are extremely thought provoking and insightful the content itself is very dense making it hard to understand what is going on at times and to comprehend the point the author was trying to make. I also greatly enjoyed the different type of writing which each chapter preceded by a dialogue and trying to find and see what the ideas in the chapter and the dialogue had in common, with the authors ability to relate the two very impressive. The puzzles for the reader throughout also made reading the book more engaging though sometimes I wish the answers to those puzzles had been more in depth.
This book was... a challenge. It is either an intricately woven tale, written by a brilliant mind, who tries to guide his readers towards new insights and epiphanies, or, the ramblings of a pretentious graduate student who takes well over 700 pages and uncountable digressions to never actually get to a point. The book did receive a Pulitzer, so I must believe the former to be true. However, I found myself enjoying the book less and less, even resorting to skipping bits and pieces (very unusual for me!) to finally get it over with.
Gödel, Escher, Bach is a book that captures the intricacies of strange loops and tangled hierarchies in science, mathematics, art, and music. The book explores the role of self-referential processes that arise from what may, at first glance, seem like rigid foundations. Godel's incompleteness theorem is a manifestation of this "strange loop" in mathematics; Godel showed definitively that a sufficiently strong formal system will necessarily contain a statement that is summarized by "this statement is not provable in the formal system". In doing so, Godel created a self-referential statement out of seemingly "rigid" mathematical rules. If such rules can arise from rigid mathematical rules, who is to say that the basic building blocks of physics and biology might not create rules which, at each level seem to be deterministic or logical, but create a self-referential system that is capable of thinking of itself independently of those systems? This "higher-level" self reference is what Hofstadter claims is the key to our understanding of consciousness, and it is a compelling argument which shows how consciousness can arise from seemingly "inert" principles. Even if you don't buy the strong version of Hofstadter's argument, the ties he draws to biology, neuroscience, artificial intelligence, Zen Buddhism, and other topics are are delight in themselves.
Interwoven in the book, between chapters of theory and explanation, are dialogues centered around Achilles and the Tortoise from Zeno's Paradox. Sometimes these dialogues serve to provide crucial context for the next chapter. Other times, these dialogues are themselves a demonstration of the context they wish to explain. The book itself is full of charm, wit, and humor, which makes for a much better reading experience than I anticipated going in. There are times when the book gets a bit slow (I found the section on Bongard Problems in AI longer than it probably needed to be), but overall the pace is both patient enough to guide the reader to the conclusions that need to be reached and fast enough to feel like you're building up to something bigger. What results is, in my opinion, a profound discussion about how consciousness (or, in its simpler forms, self-reference) can arise from seemingly "formal" systems - in mathematics, music, and even our own minds.
Interwoven in the book, between chapters of theory and explanation, are dialogues centered around Achilles and the Tortoise from Zeno's Paradox. Sometimes these dialogues serve to provide crucial context for the next chapter. Other times, these dialogues are themselves a demonstration of the context they wish to explain. The book itself is full of charm, wit, and humor, which makes for a much better reading experience than I anticipated going in. There are times when the book gets a bit slow (I found the section on Bongard Problems in AI longer than it probably needed to be), but overall the pace is both patient enough to guide the reader to the conclusions that need to be reached and fast enough to feel like you're building up to something bigger. What results is, in my opinion, a profound discussion about how consciousness (or, in its simpler forms, self-reference) can arise from seemingly "formal" systems - in mathematics, music, and even our own minds.
Kind of interesting, but a little too bogged down in formal logic and little too frenetic structurally to keep me interested... giving up for now. Made it about 50% through.