53 pages • 1 hour read
Douglas HofstadterGödel, Escher, Bach: An Eternal Golden Braid
Nonfiction | Book | Adult | Published in 1979A modern alternative to SparkNotes and CliffsNotes, SuperSummary offers high-quality Study Guides with detailed chapter summaries and analysis of major themes, characters, and more.
Themes
Self-Reference and Strange Loops
As an example of Self-Reference and Strange Loops, in 1988, children’s performer Shari Lewis released a video of her show Lamb Chop’s Sing-Along, Play-Along, which included the musical track “The Song that Doesn’t End.” The song contained a single verse that could be looped infinitely with the last line of the lyrics leading directly back to the first line. Lewis’s song illustrates Hofstadter’s core theme in Gödel, Escher, Bach: An Eternal Golden Braid. “The Song that Doesn’t End” utilizes self-reference, a concept describing something referring to itself or pointing back to itself, like a snake eating its tail. Shari Lewis’s song uses self-reference in the opening line by referring to itself: “This is the song that never ends.” A strange loop is formed with the last line of the song engaging in self-reference to the first: “You’ll continue singing it forever just because...”
Strange loops are more than just recursive or self-referential: They take a unique shape and move along hierarchies, always returning to the beginning. Many of Escher’s drawings and lithographs visually exemplify the notions of self-reference and strange loops. Sky Castle by M.C. Escher shows a mountainous castle hovering above an ocean where it is reflected. In the reflection, a woman rides atop a swimming turtle’s back, her arms outstretched toward the castle in the sky. The perspective of the image requires the viewer to shift their gaze back and forth between the castle in the sky and the castle in the water, never landing in a single spot. Escher’s 1961 lithograph Waterfall presents a waterfall that flows along multiple tiers and, if followed with the eye, creates an endless loop.
Hofstadter argues for an understanding of consciousness as a strange loop. Thoughts move along tiers of meaning in an unending spiral, seamlessly moving in and out of a formal structure that cannot handle the many contradictions of existence. The layered format of Gödel, Escher, Bach reiterates and enacts this theme. Hofstadter opens with a dialogue by Lewis Carroll, featuring Achilles and the Tortoise just before they engage in a footrace. The initial dialogue in Chapter 1 is used to convey important ideas, such as Zeno’s Theorem. The final dialogue engages in self-reference to the first, discussing Zeno’s Theorem and even incorporating Hofstadter himself into the conversation. This formatting structure gives the entire work the shape of a strange loop. The narrative of the dialogues ends where it began, exemplifying Hofstadter’s ideas on new planes of meaning and embodying both self-reference and the shape of a strange loop.
The Recursive Nature of Being
Understanding recursion requires understanding how it differs from self-reference and strange loops. Although related, recursion occurs when a pattern is repeatedly divided and repeated. Hofstadter shows how recursion works across disciplines. In this way, structures are defined in terms of themselves. In music, a fugue or melody exhibits recursion, replaying the melodic line using different voices or instruments. A strange loop occurs when the line brings the song back to the beginning, such as Shari Lewis’s “The Song that Doesn’t End,” provided in Self-Reference and Strange Loops.
Hofstadter introduces recursion in the beginning of the work with Zeno’s Theorem. Zeno explains that the outcome of a footrace between the Tortoise and Achilles will always end the same way. The Tortoise, given a head start, will move from Point A to Point B. Simultaneously, Achilles will travel to Point A. With this understanding, the action becomes recursive, dividing and repeating itself for infinity. Hofstadter relates the idea of recursion to human intelligence. Symbolic systems require recursive structures to transform symbols into strings. When recursion is paired with self-reference, higher levels of intelligence occur.
In Chapter 2, Achilles wonders if a hint the Tortoise provides him for solving a challenging riddle might prove useful in unlocking a different puzzle the Tortoise is attempting to solve. Hofstadter shows that cognition requires self-reference, the looping back in on itself. To accomplish this, one must move outside the confines of a formal system and apply pattern-finding and meaning-making. By taking a hint from one puzzle and recursively applying it to another, Achilles discovers an isomorphism.
Hofstadter explains isomorphisms as the mapping of patterns by placing one concept over another. A simple example of this is two shapes: a house comprised of a triangle on top of a square and a five-pointed star. When one shape is placed over the others, the five points of the star and the house align. Hofstadter argues that human intelligence employs the mathematical concept of isomorphisms continuously, both across symbols and through a type of collective consciousness similar to that of an ant colony: “There is a lesser ambition which it is possible to achieve: that is, one can certainly jump from a subsystem of one’s brain into a wider subsystem” (477). Humans can also overcome the limitations of formal systems and manage paradoxes and contradictions by using this recursive practice to identify partial patterns. Partial isomorphisms are found when one compares how different minds, or subsystems, function and find partial truths.
Connection and Openness Through Interdisciplinary Approach
After the publication of Gödel, Escher, Bach, Hofstadter worried that his readers took away the wrong message, emphasizing a connection across differences over his theory of strange loops. However, Hofstadter’s use of connections as both a thematic element and point of structure pervades the work, leading to this misdirection. Hofstadter incorporates other disciplines by comparing, for example, strange loops to concepts within Zen Buddhism.
Zen Buddhism rejects duality and concreteness and serves as a jumping-off point for Hofstadter to search for meaning within a world of contradictions. Kõans challenge the meaning and uses of words (symbols) that often have a one-to-one, or dualistic, meaning. They render words so ridiculous that they lose all logicalities. Hofstadter connects how humans make sense of language and life to a use of formal systems:
Relying on words to lead you to the truth is like relying on an incomplete formal system to lead you to the truth. A formal system will give you some truths, but as we shall soon see, a formal system—no matter how powerful—cannot lead to all truths (252).
For Hofstadter, truth is found when one steps outside of a formal system and challenges the rules of duality and simple, prescriptive logic. Sophisticated intelligence requires finding patterns in spaces that seem devoid of connection. Hofstadter models this by incorporating multiple disciplines, finding correlations across academic and artistic spaces to map his ideas in an isomorphic style against the nature of reality. By pulling in Zen Buddhism, Hofstadter adds another layer of meaning to his work on strange loops and recursion.
Isomorphism is a process for exploring this interconnectedness. The mapping of concepts and thoughts to find new patterns comprises human intelligence. Hofstadter claims that human consciousness is derived from the ability to move outside of formal systems—to think outside the box. Hofstadter’s interdisciplinary approach allows him to apply mathematical principles like isomorphism to all aspects of life and disciplines: “‘Isomorphism’ is a word with all the usual vagueness of words—which is a defect but an advantage as well” (50). Hofstadter even connects isomorphism to comparing subsystems and human brains.
Hofstadter’s devotion to openness and the fluidity of disciplines mirrors his assertion that true intelligence occurs when the barriers of formal systems are broken. Hofstadter uses isomorphism to describe the act of finding meaning—something that permeates all aspects of human life. In Chapter 1, he describes an individual who routinely practices leaving formal systems behind. This person discovers how social systems formalize and govern people and tries to rescue others. The format and structure of the work reiterates the breaking of barriers and limitations by drawing on multiple disciplines. Hofstadter incorporates analogies, art, music, dialogues, and puzzles to contextualize his concepts. Bach’s fugues and Escher’s paintings lend concreteness to abstract ideas. By creating an interdisciplinary structure, Hofstadter emphasizes his point that true intelligence requires a blending of hard and soft concepts to jump outside of the confines of a formal system.
