Gödel, Escher, Bach: An Eternal Golden Braid

“It is an inherent property of intelligence that it can jump out of the task which it is performing, and survey what is done; it is always looking for, and often finding, patterns.”

Hofstadter asserts that human intelligence distinguishes itself from machine learning by its ability to work outside of formal systems. Finding these patterns requires both Connection and Openness Through Interdisciplinary Approach. By drawing on multiple disciplines, humans can see how the patterns in some structures overlap with other structures through isomorphisms, a mathematical term Hofstadter applies to other fields to illustrate how human cognition functions.

“Perhaps by then I’ll have thought of the right answer to YOUR puzzle, using your figure-ground hint, relating it to MY puzzle.”

In this allegorical dialogue, Achilles and the Tortoise exchange riddles that each is seeking an answer to. Their conversation leads to the revelation that the solution to one may lead to a solution to the other. Hofstadter uses this narrative to exemplify the theme of Self-Reference and Strange Loops. The hint for each riddle is self-referential, occurring across multiple tiers and returning in an infinite spiral.

“Heavens, you certainly have an admirable boomerang collection!”

Hofstadter uses Lewis Carroll’s technique of stacking riddles and hints to introduce references to the concepts explored in each chapter. Here, a comment by Achilles alludes to recursive figures, a mathematical example of strange loops. By using the dialogues to draw attention back to the chapters, Hofstadter creates a formatted structure of recursion within the text itself by positioning both narrative and math as reflection.

“Just as a fish’s DNA is contained inside every tiny bit of flesh, so a creator’s ‘signature’ is contained inside every tiny section of his creations.”

Hofstadter asserts that the interdisciplinary evidence for strange loops reveals the creator’s mark of uniquely human intelligence. Bach’s music and Escher’s art both utilize self-reference and recursion. Unlike artificial intelligence, human cognition can turn the eye on itself, reflecting and innovating. This interdisciplinary approach highlights Connection and Openness Through Interdisciplinary Approach.

“The issue we are broaching is whether meaning can be said to be inherent in a message, or whether meaning is always manufactured by the interaction of a mind of a mechanism with a message.”

In Chapter 6, Hofstadter examines the nature of meaning and whether it is found within the symbol or the interpreter's mind. He determines that meaning is a process of human intelligence, which connects symbolic pattern-making with the real world through isomorphism.

“A contradiction in mathematics was found, mathematicians would immediately seek to pinpoint the system responsible for it, to jump out of it, to reason about it, and to amend it. Rather than weakening mathematics, the discovery and repair of a contradiction would strengthen it.”

Throughout the text, Hofstadter employs contradictions and paradoxes to illustrate the idea that meaningful interpretation can be found within error. His proposal mirrors a practice in his own life; Hofstadter diligently records his own verbal slip-ups and malaphors. Hofstadter argues that meaningful interpretation specifically occurs because of contradiction, not in spite of it.

“Gödel showed that in order to pull the heavy rope across the gap, you can’t use a lighter rope; there just isn’t a strong enough one.”

In this passage, Hofstadter uses Gödel’s incompleteness theorem to reveal the limiting and incomplete nature of formal systems. Gödel’s theorem expresses the idea that formal systems cannot reveal all truths, and the metaphor of the rope demonstrates that there is no way to bypass this reality within formal systems.

“Zen is a system and cannot be its own metasystem; there is always something outside of Zen, which cannot be fully understood or described within Zen.”

In Zen Buddhism, enlightenment has many tiers. Hofstadter includes a kõan in which a Buddhist monk tells his disciples that there is a higher level to Buddhism. When they ask him what higher Buddhism is, the monk responds by saying what it is not. Hofstadter uses this kõan to show how meaning is found outside the limitations of a formal system: There is always a more complex truth to be discovered. Just as the principles and rules of Zen Buddhism cannot be used to understand the hidden meaning and truth that exists just outside of it, formal systems cannot be used to assess their own completeness.

“Enlightenment always comes after the road of thinking is blocked. If you do not pass the barrier of the patriarchs or if your thinking road is not blocked, whatever you think, whatever you do, is like a tangling ghost.”

This quote by Mumon equates a block in thinking with the limitations of a formal structure. By embracing Connection and Openness Through Interdisciplinary Approach and isomorphism, Hofstadter equates the discovery of statements of truth outside of a formal system to enlightenment.

“Are there laws of thinking which are ‘sealed off’ from the lower laws that govern the microscopic activity in the cells of the brain? Can mind be ‘skimmed’ off of brain and transplanted in other systems?”

In the closing of Chapter 10, Hofstadter raises important questions about how systems manage and move along levels of description and meaning. He recognizes that humans employ a system of chunking that allows them to create generalized meaning without getting bogged down in lower processes and thoughts. To create an artificial intelligence that can match the processing of human intelligence, sealing off tiers to test out statements within a structure of contradiction is necessary.

“We make the following observation: despite the complexity of its input, a single neuron can respond only in a very primitive way—by firing, or not firing.”

Like formal systems, human intelligence has tiered capabilities. Hofstadter compares the work of a single neuron to an ant in a colony. On its own, it is meaningless, one symbol in a long string. However, working in tandem with others, a neuron can form a chain that can be interpreted.

“Yet even in this pathologically difficult case of translation, there seems to be some rough equivalence obtainable. Why is this so, if there really is no isomorphism between the brains of people who will read the different versions?”

Hofstadter uses the idea of isomorphic mapping across multiple brains to explain how humans use partial isomorphisms to make meaning—something that machine learning has not yet mastered. Hofstadter suggests that partial isomorphisms uncovered across brains may reveal truths about how humans create and process thought.

“The interesting thing about a subsystem is that, once activated and left to its own devices, it and work on its own.”

The brain is referred to as a subsystem that operates independently and can perform its own adaptations. For artificial intelligence to reach the level of human intelligence, it must be able to self-replicate and make its own altercations.

“Chaos, part of perfection? Order and chaos make a pleasing unity? Heresy!”

In this dialogue, Hofstadter reiterates the idea that intelligence requires a blending of hard and soft concepts. This relates to his assertion that human cognition transcends formal systems and embraces paradox. In the passage, Achilles and the Tortoise discuss how both chaos and perfection play a role in understanding how systems work. By embracing both rules and rule-breaking, a more complete picture is formed.

“Yields falsehood when preceded by its quotation! Yields falsehood when preceded by its quotation!”

In this dialogue, Achilles recounts an unusual phone call he received earlier in the day. The liar paradox presented by a prank caller adds to the recursive structure of the text, solidifying Hofstadter’s themes into the format of the work itself.

“If the basic trouble with TNT is that it contains a ‘hole’—in other words, a sentence which is undecidable, namely G—then why not simply plug up the hole?”

As Hofstadter explores the limitations and functions of TNT, he considers the question of why programmers are unable to encode an axiom that removes the limitations of previous systems. Hofstadter returns to Gödel’s incompleteness theorem, which shows that a system cannot be used to evaluate itself for completeness. While artificial intelligence can engage in self-referential behavior and fix its own problems, it is unable to account for what is outside of itself.

“No matter how a program twists and turns to get out of itself, it is still following the rules inherent in itself. It is no more possible for it to escape than it is for a human being to decide voluntarily not to obey the laws of physics.”

These sentences illustrate Gödel’s incompleteness theorem. Although a formal system can produce true statements, it is impossible for the system to produce all true statements, including those that lie outside the confines of its own limitations. Furthermore, because a system cannot see outside of its own axioms, it is unable to evaluate its own completeness. Although humans must abide by this same incompleteness theorem, they are able to draw upon a more comprehensive structuring of consciousness that has yet to be exhibited my machine learning. This comprehensive structuring depends upon Connection and Openness Through Interdisciplinary Approach.

“A Zen person is always trying to understand more deeply what he is, by stepping more and more out of what he sees himself to be, by breaking every rule and convention which he perceives himself to be chained by.”

As part of Hofstadter’s modeling of interdisciplinary approaches, Zen Buddhism is used to explain how higher-tiered intelligence jumps outside of the rules of a formal system. This is connected to The Recursive Nature of Being. By breaking down one’s thoughts, examining them, and trying them in different spaces, humans gain a self-referential knowledge of how systems rule their lives and how they can break free of them.

“Music is not a mere linear sequence of notes. Our minds perceive pieces of music on a level far higher than that.”

Hofstadter uses music to show how humans engage with different tiers of meaning. He argues that meaning does not lie within the object itself, contrary to a popular philosophical concept of essences. Instead, meaning is found through interpretation. Humans do not separate individual notes and extract meaning from them; it is only when the notes are paired together that they form a larger idea.

“A natural and fundamental question to ask, on learning of these incredibly intricately interlocking pieces of software and hardware is: ‘How did they ever get started in the first place?’”

Although Hofstadter does not answer his own question, he feels it is important to raise it. The Recursive Nature of Being naturally leads to questions about the origin of life and how humans began to form their unique style of thinking and consciousness. He relates this idea to jumping outside of a formal system and the Zen Buddhist notion of transcendence. Hofstadter explains that humans cannot know what they do not know, because they are confined to the axioms of their own system.

“The only way to understand such a complex system as a brain is by chunking it on higher and higher levels, and thereby losing some precision at each step.”

Here, Hofstadter develops an argument for the blending of hard and soft concepts to create an informal system. Part of understanding this system requires embracing its unknowable complexity. Embracing Connection and Openness Through Interdisciplinary Approach allows for the blending of complex ideas because it reveals patterns that make meaning.

“Another of those confounded hypotheticals! Why are the rest of you always running off into your absurd worlds of fantasy?”

This question by the Sloth speaks to Hofstadter’s concept that sophisticated intelligence requires the holding of contradictions, paradoxes, and hypotheticals. The Sloth, known by his companions to be dim-witted, cannot keep up with the way the other characters postulate different scenarios. This relates to the way formal systems often cannot account for contradictions and paradoxes.

“The world is a giant heap of randomness; when you mirror some of it inside your head, your head’s interior absorbs a little of that randomness.”

Hofstadter asserts that it is an insult to human creativity to imply that it is the result of chance. The randomness that is produced in the human mind as creativity is a type of strange loop or conceptual revelation, one which can be mechanized for artificial intelligence so long as there is room for both hard and soft principles.