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Bach also wrote fifteen two part inventions. This two-part Dialogue was written not by me [Hofstadter], but by Lewis Carroll in 1895. Carroll borrowed Achilles and the Tortoise from Zeno, and I in turn borrowed them from Carroll. The topic is the relation between reasoning, reasoning about reasoning, reasoning about reasoning about reasoning, and so on. It parallels, in a way, Zeno’s paradoxes about the impossibility of motion, seeming to show, by using infinite regress, that reasoning is impossible. It is a beautiful paradox, and is referred to several times later in the book. (p. viii)

While you read[]


Bach - Inventions & Sinfonias (Evgeni Koroliov)

Bach's Inventions and Sinfonias, performed by Evgeni Koroliov.


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

A performance and video by Salvatore Nicolosi, showing the sheet music.

If at first you missed the fact that Lewis Carroll wrote this dialogue, the slightly archaic writing may have caught you by surprise. The style of Carroll's dialogue was Hofstadter's inspiration in writing the other dialogues.

Hofstadter put the Three-Part Invention before this dialogue, providing the context about Zeno's paradox that Lewis Carroll assumed his audience had already heard.


  1. Tortoise asks Achilles if he would win a race where the distances were constantly increasing, instead of decreasing. What is he referring to?
  2. What does "Solvitur ambulando" mean?

Curry and Kelleher wrote brief lecture notes for the class where they discussed this dialogue.


The title that Hofstadter gave this dialogue refers to Bach's fifteen two-part Inventions (BWV 757-771). Each Invention establishes a theme that the two parts will frequently repeat to each other. The theme can be transposed so that it begins at different points in the musical scale.

You could think of many moments in this dialogue as Achilles stating a theme, and then the Tortoise repeating it, but one step higher.

Keep in mind, of course, that both the music and the dialogue already existed. Hofstadter's re-titling of the dialogue points out a way that you can connect Carroll's dialogue to Bach's music within the context of GEB.

Because the two-part Inventions and the three-part Inventions (or Sinfonias) are often performed together, the video links are the same as they were for the Three-Part Invention. This time, start listening at the beginning.


Axioms inside and outside the system[]

Tortoise may sound like he's being unreasonable, but he (and therefore Lewis Carroll) is pointing out a difficult problem in the philosophy of mathematics. It has to do with when it's okay to "jump out of the system" as described in Chapter 1.

Tortoise introduces a Euclid-style logical system that has axioms inside the system, such as this one:

(A) Things that are equal to the same are equal to each other.

But also depends on unstated logical axioms outside the system:

(Modus ponens) If A is true, and A implies Z, then Z is true.

Could Achilles have a way out by simply asking Tortoise to accept modus ponens itself as an axiom? Not quite!

Modus ponens turns the logical statements themselves into mathematical objects to be manipulated, like triangles and their sides. Here's why this matters. Tortoise could suggest replacing modus ponens with this statement, which is equally valid as an axiom:

(Colorful modus ponens) If A is blue, and A implies Z, then Z is blue.

If Tortoise accepts that A could be "blue", he'd then ask Achilles why he should believe statement Z simply because it's blue.

We don't actually have to replace "modus ponens" with "colorful modus ponens". Given real modus ponens, Tortoise could ask why he should believe Z simply because modus ponens labels it with the symbol "true", which is as arbitrary as "blue". Achilles could ask him to accept it as an axiom, and then the same infinite dialogue would occur.

Achilles can only win if he gets Tortoise to agree that modus ponens is a rule that is outside the system, and that you have to believe its conclusions because the rule says so. Tortoise will never agree to this.

It's good to question rules that say "you have to do X because this rule says so". Rules of this form are the basis of many cults, as well as self-perpetuating memes such as chain letters, the sig virus, and The Game. However, in order to resolve Tortoise's paradox, mathematicians have to accept that kind of rule at least once. Then, they have to be careful to distinguish statements that they can manipulate inside the system from unquestionable rules outside the system.

Carroll would have been delighted to learn what Gödel would later demonstrate -- that it's not even enough for mathematicians to accept an unquestionable rule from outside the system once. It merely creates a new system that you can be outside. Gödel's theorem shows that, no matter how many times you jump out of the system, there will always be true statements that you cannot prove within your new system.

For other descriptions of this paradox, see:

Easter eggs[]

Lewis Carroll ends his dialogue with two groaner puns, and one of them is also a reference to his own Alice's Adventures in Wonderland. In chapter 9 of Alice, the Mock Turtle reminisces:

'...we went to school in the sea. The master was an old Turtle--we used to call him Tortoise--'
'Why did you call him Tortoise, if he wasn't one?' Alice asked.
'We called him Tortoise because he taught us,' said the Mock Turtle angrily: 'really you are very dull!'

The fact that Carroll already had a character named Tortoise may have inspired him to make characters out of the Tortoise and Achilles from Zeno's paradox.

Cultural references[]

Lauren Ipsum, a book that introduces children to theoretical computer science, reuses the characters of Tortoise and Achilles yet again in What the Tortoise Said to Laurie.


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