Gödel, Escher, Bach Wiki


Bach wrote fifteen three-part inventions. In this three-part Dialogue, the Tortoise and Achilles – the main fictional protagonists in the dialogues – are “invented” by Zeno (as in fact they were, to illustrate Zeno’s paradoxes of motion). Very short, it simply gives the flavor of the Dialogues to come. (p. viii)

While you read[]


Bach - Inventions & Sinfonias (Evgeni Koroliov)

Bach's Inventions and Sinfonias, performed by Evgeni Koroliov. The Sinfonias begin at 25:00.


Bach-Inventions and Sinfonias (BWV 772-801) with sheet music

A performance and video by Salvatore Nicolosi, showing the sheet music. The Sinfonias begin at 25:48.


  1. Assuming you've read the following chapter (Chapter 1), how are Zeno's "theorems" connected to theorems of the MIU-system?
  2. What's the difference between the "Achilles paradox" and the "dichotomy paradox"?
  3. How would you resolve Zeno's paradoxes using modern knowledge?

Curry and Kelleher also asked some detailed questions about this dialogue for their class. If you're motivated to answer them, you'll find them in this PDF.


The title of this dialogue refers to Bach's Sinfonias (BWV 772-786), which are also sometimes called his "Three-Part Inventions" in English, with the other Inventions being called the "Two-Part Inventions".

Despite the sound of the name, these are not symphonies, they're short keyboard pieces. Each piece has three melodic lines that imitate each other on a common theme.

In many cases, Hofstadter uses the Bach music he's referring to as inspiration for the dialogue, using the different characters as the different voices of the music. Some of the dialogues have very strong connections to the music, while others have only oblique references. This is one of the oblique ones.


Zeno's paradox[]

Zeno of Elea was a Greek philosopher, best known for the paradoxes attributed to him and his use of "reductio ad absurdum" -- an argument that a claim is false because the things that logically follow from it are absurd.

This dialogue illustrates his "Achilles paradox", which is in fact described using a parable of Achilles in a footrace with a tortoise.

Escher's Möbius strip[]

Escher's "Moebius Strip 1", as it appears on WikiArt.

This is one of many instances where ➟ M. C. Escher was inspired by the intersection of visual art and mathematics. The cut-out strip helps to illustrate a surprising property of the surface known as the ➟ Möbius strip: you can cut it in half down the middle and still end up with one continuous strip.

The cut-out holes, meanwhile, make this quite improbable as a real-world object. They also serve to let you see more of the strip and to give the strip a sort of "face". In the dialogue, one of these holes is described as appearing in "Zeno's flag".

Easter eggs[]

"The sixth patriarch is Zeno" (p. 30)
Achilles claims that Zeno is a Zen master. The real ➟ Zeno of Elea, of course, couldn't have been a Zen master, because he was Greek and he lived about a millennium before Zen Buddhism existed.
Achilles' confusion stems from a pun: he repeats "the sixth patriarch is Zeno" to himself. The sixth patriarch is Enō, at least when named in Japanese. (His Wikipedia article is under his Chinese name, ➟ Huineng.) Hofstadter playfully lets the dialogue proceed as if Zeno and Enō are the same person.


(This section is for adding your thoughts about the chapter. Sign what you write with your user name. Others may edit this section for length later. More free-form, unedited discussion can take place in the comment section below.)

GEB points out many surprising connections between ideas, but there's no direct connection between the Möbius strip and Bach's music, and the book never claims there is one. Really, you shouldn't expect there to be, as the Möbius strip was devised long after Bach's lifetime. There's a misleading YouTube video out there -- which I won't do the favor of linking to -- that uses a computer animation to convince many people that there is such a connection between Bach's Crab Canon and a Möbius strip. It mostly accomplishes this by ignoring both the structure of the Crab Canon and the properties of the Möbius strip, and hammering them together until they appear to fit.
As you read GEB, you may start to see connections in everything, which is great until the point where you connect so many things that the connections become meaningless. I think the seductive falsehood presented by that video can be taken as a reminder to keep yourself grounded, to sometimes take a step back and ask "Wait, is this real"? --rspeer
responding to rspeer - "there's no direct connection between the Möbius strip and Bach's music" - maybe not, though I thought there was a music piece (don't know if it is Bach) that can "loop" from the end of the piece and go back to the front - is that in a sense a "musical Möbius strip"? --prayingmantises