Gödel, Escher, Bach: An Eternal Golden Braid

The Fibonacci sequence, named after 13th-century Italian mathematician Leonardo Fibonacci, describes a series of numbers wherein each number is the sum of the two preceding it, such as in the following progression: 0, 1, 1, 2, 3, 5, 8, 13. When plotted, these numbers form a pattern that is manifested throughout nature, including the shell of a snail and the structure of artichoke. Records of Fibonacci numbers have been found in the works of Indian mathematician and poet Pingula as early as 200 B.C.E. Hofstadter uses the Fibonacci sequence to illustrate his concept of recursion. In this sequence, a simple process (adding the two preceding numbers together) creates an unexpected outcome: a repeated pattern in both nature and art.

Figure-ground perception is a visual-processing technique used by humans to differentiate between objects and their contexts. The object being determined is the figure, and everything surrounding it is ground. Hofstadter argues that figure-ground perception can be challenged within its own formal system, and the negative and positive components can be switched, revealing new truths existing outside the limitations of the initial rules.

Hofstadter uses the term “formal system” to refer to logical reasoning, which applies a set of axioms that all theorems must adhere to. A formal system is a set of symbols, sometimes referred to as an alphabet. Axioms are the rules, or syntax, by which all theorems must abide. Hofstadter argues that, although formal systems present true statements, there are true statements which lie outside their limitations.

A fugue is a type of musical composition, popularized by Bach and other Baroque composers, which layers two or more voices around a singular musical theme. Hofstadter uses fugues to contextualize the concept of strange loops.

Developed by Gödel, the incompleteness theorem is made up of two separate theorems. The first states that there will always be true statements that cannot be proven using a formal system. The second asserts that a formal system cannot be used to establish its own completeness or consistency.

Isomorphism is a mathematical mapping technique wherein one structure is placed over another to reveal a one-to-one correspondence of points. Hofstadter argues that human intelligence employs a similar technique to isomorphism when finding patterns and making meaning.

Zen kõans are stories, paradoxical statements, questions, or statements that are used to explore different concepts and aspects of Zen Buddhism. The Sino-Japanese word describes a metaphor of public record that will connect an idea to real experience by showing the nonlogical nature of existence. A Zen Buddhist principle states that words can never fully represent truth.

Hofstadter uses the liar paradox to illustrate the concept of self-reference. The liar paradox is self-referential, because each statement points to the other, revealing a binary truth. For example, pairing the statements “I am a liar” and “I never lie” forms a self-referential paradox.

Meaningful interpretation describes the convergence of theorem and a reality-based truth. Hofstadter states that meaningful interpretation functions like an isomorphism when two different structures reveal a pattern.

Chapter 7 introduces propositional calculus, which is a branch of mathematical logic that allows for symbols to represent logical connectives, such as “and” or “not.” Hofstadter shows how propositional calculus can be used to develop declarative statements while allowing for contradiction and inconsistency.

Television shows like The Simpsons and Community are often noted for their use of self-referential humor. This means that the television shows frequently break the fourth wall, taking jabs at their creators, structures, or tropes. Hofstadter uses the works of Gödel, Escher, and Bach to show a recurring pattern of self-reference—what happens when a system or concept refers to itself. Hofstadter argues that self-reference creates paradoxes. The concept of self-reference is used as a lens to examine consciousness. The liar paradox illustrates how self-reference challenges the limitations of formal systems.

The singularity is used to refer to a hypothetical point in the future at which artificial intelligence (AI) exceeds the capabilities of human intelligence. In 2005, computer scientist Ray Kurzweil published The Singularity is Near, arguing that this hypothetical date was imminent. In June 2024, Kurzweil published a sequel titled The Singularity is Nearer, following a recent increase of public consciousness about AI. Kurzweil suggests that machine intelligence is unable to employ the complex pattern finding techniques of human cognition.

Hofstadter identifies “strange loops” as the foundation of the text and the center of his theories about consciousness. A strange loop occurs when movement across a tiered, hierarchical structure always leads back to the beginning. Hofstadter uses the terms “strange loop” and “tangled hierarchy” interchangeably.

Typographical Number Theory (TNT) describes a formal system of axioms involving simple numerical statements. TNT is used to show how arithmetic is constructed upon a set of rules within a formal system and how even the most basic formal system is either incomplete, inconsistent, or both.

Zen Buddhism is a branch of Mahayana Buddhism that uses meditation and the interpretation of experience to explore questions about the nature of existence and reality. Zen uses kõans to illustrate illogical ideas and emphasize the parts of existence that cannot be explained through reasoning.