Gödel, Escher, Bach: An Eternal Golden Braid

In previous chapters, Hofstadter explores how the human mind engages in multiple levels of meaning. Similarly, Hofstadter shows that computer programs contain diverse levels of description and self-reference. Humans move along these various levels with ease: “We can just shut one out, and pay attention to the other—which is what all of us do” (285). For example, they understand that their bodies are both comprised of molecules and made up of cells. Humans do not find it difficult to hold these two statements of truth in their minds at once.

Levels of description explain a system’s model, which contains different tiers of processes. An image on a television screen has multiple layers of meaning. Watching the actor Shirley MacLaine laughing is one level; the narrative she participates in is another. While watching the film, the viewer thinks about how the screen is presenting a series of dots that come together to create a symbol. This is another level of description.

Hofstadter uses the example of a car engine to further explain this concept. The engine has a mechanical level of description and a chemical one. The human mind toggles among levels of meaning but is not required to hold multiple levels at one time. Sometimes levels can mirror one another, and Hofstadter asserts that the confusion caused by recursion often impacts how humans understand themselves. However, self-reference, and even confusion, are important parts of cognition and intelligence. When computers can engage in self-reference, they can perform more complex tasks that more closely align with human intelligence.

Computer systems and artificial intelligence use the same hierarchical levels to organize their applications. The higher the levels go, the more abstract their processes become. Abstraction is necessary to access deeper levels of description. However, unlike interpretation of the mind, artificial intelligence is limited because it requires instruction defining the perimeters of its actions. When computer systems utilize self-reference, they can apply abstraction across multiple levels of description.

...Ant Fugue

Achilles, the Tortoise, the Crab, and the Anteater discuss how different levels reveal different layers of meaning. The Anteater describes his Aunt Hillary, an ant colony that has collective consciousness.

As computer scientists began to develop models that could mimic the thinking of human intelligence, new insights into how brains work emerged. Hofstadter devotes Chapters 11 and 12 to breaking down the simple and complex processes of the human mind, showing how machine learning can offer information about how people think. Hofstadter refers to high-level thoughts as a type of calculus of thinking. Like the ants in the colony of Aunt Hillary, the brain uses singular neurons and brings them together to form complex thought.

A single neuron cannot, on its own, achieve the levels of meaning that are derived from the interaction of neurons in the larger structures of the brain: “Despite the complexity of its input, a single neuron can respond only in a very primitive way—by firing, or not firing” (340). When these neurons are packaged together, they form symbols, concepts like rattlesnakes or pancakes. Hofstadter shows how thoughts move from simple symbols to classes to consciousness.

English French German Suite

Hofstadter juxtaposes Lewis Carroll’s poem “Jabberwocky” with two French translations of the same poem by Frank L. Warrin and Robert Scott. Hofstadter uses the three versions to illustrate how different minds can produce different interpretations.

Now that Hofstadter has examined an individual human brain and how it organizes various levels of thoughts, he moves forward with his analogy of the ant colony, examining how different minds can be mapped using isomorphisms. A true isomorphism cannot be mapped out using two minds, because their memories and experiences are too varied. The translations of Lewis Carroll’s poem “Jabberwocky” included in the previous chapter demonstrates the problems that occur when trying to find patterns across multiple minds.

However, communication and semantic networks reveal partial isomorphisms which contribute to greater understanding. Partial isomorphisms reveal themselves in how humans think. Patterns are found in the process and have important implications for developing machine learning: “These stable, reliable pathways are what constitute knowledge” (378). Hofstadter refers to a person’s sense of self—the “I”—has a kind of subsystem which manages a complex constellation of symbols.

Aria with Diverse Variations

The Tortoise visits his friend Achilles, who has been unable to sleep. The Tortoise tells Achilles about a Count who could not sleep, so he commissioned a composer to develop an aria with many variations to help him pass his time at night. To thank the Tortoise for keeping him company, Achilles gifts him a rare gold box. At the end of the dialogue, a Cop arrives to arrest Achilles for stealing the box, but Achilles tells him that the Tortoise had it all along.

The title represents three types of computer languages that illustrate recursion. A BlooP is a bounded loop that represents a model that does not allow for recursion and cannot pass the Turing test. A FlooP, on the other hand, is a free loop with recursion. This means that it can move beyond the primary restrictions to achieve upper bounds of computation and utilize if-statements. A GlooP is a further level of unbinding beyond the FlooP. Many scientists, like Alan Turing, believe that computational models can never move beyond the FlooP tier. However, Hofstadter argues that just as formal systems cannot be used to determine all true statements and evaluate their own consistency, FlooP models cannot speak to what lies outside their borders.

Air on G’s String

After touring a porridge factory, the Tortoise and Achilles discuss a recent prank phone call. A caller shouted a single line twice to the Tortoise: “Yields falsehood when preceded by its quotation!” (431). This liar paradox leads the pair to strange loops while physically representing the idea as they walk along a looped flight of stairs like the one represented in Escher’s lithograph Above and Below.