Introduction: A Musico-Logical Offering
1.
Listen to Bach’s endlessly rising “Canon Per Tonos” here . Make sure to watch the music while you listen and note the key changes; otherwise, you’ll miss them!
2.
What’s the connection between Bach’s endlessly rising canon and Escher’s waterfall?
3.
Are the following sentences true or false? a.
I always lie (said by degenerate liar) b.
This sentence is false.
4.
In the text, recall that R is the set of all run-of-the-mill sets. Is R run-of-themill or is it self-swallowing?
Three-Part Invention
5.
Here is a link to the Wikipedia page on Bach’s Three Part Inventions (called
“sinfonias”). They were actually written for people to learn how to play the keyboard with two hands.
6.
Why is the title “Three-Part Invention?”
7.
What is a Möbius strip?
8.
What is wrong with Zeno’s argument?
Chapter 1: The MU Puzzle
9.
What string do we obtain when RULE I acts on: a.
MUUUIUI b.
MIIIUIU
10.
What string do we obtain when RULE II acts on MUUUIUI?
11.
What string do we obtain when RULE III acts on: a.
MUUUIUI b.
MIIIUIU
12.
What string do we obtain when RULE IV acts on: a.
MUUUIUI b.
MIIIUIU
13.
What is the difference between a theorem and an axiom: a.
In mathematics? b.
In the sense described in GEB? c.
How does “truth” differ between a theorem and axiom?
14.
Hofstadter suggests that 3 and 2 are important in the MIU-system. How?
15.
Do you think the tree in Figure 11 will produce every string just from MI?
16.
Departing MIU, suppose there was a decision tree generating all true statements (like Figure 11 for all truth). Would all truth be knowable?
17.
Is there a faster way of testing whether a given string is a theorem in MIU than simply trying all possible rules?
Two-Part Invention

18.
What do you think is wrong with the Tortoise’s argument?
Chapter 2
19.
How many axioms are in the pq-system?
20.
Can you provide a procedure for determining whether a string is an axiom in the pq-system?
21.
Is the pq-system the same as addition?
22.
What’s wrong with p <--> horse? What point does Hofstadter make?
23.
What is the difference between meaning in a formal system and in human language?
Sonata for Unaccompanied Achilles
24.
Here is one of Bach’s 6 “sonatas for unaccompanied violin.” This music sounds old now, but it was relatively new at the time for people to write solo pieces for violin.
25.
What’s the (nontrivial) solution to the HE puzzle?
26.
What’s the solution to the ADAC puzzle?
27.
How do these puzzles relate to the dialogue? Bach’s sonatas?
Chapter 3: Figure and Ground
28.
Give the following for the tq-system: a.
Legal symbols b.
Axioms c.
Rule of production
29.
Is the tq-system equivalent to multiplication?
30.
Is there a direct test for whether a given number is prime?
31.
Draw a Venn diagram showing the relationship between “cursively drawable” and “recursive” drawings, as well as the statement “There exist cursively drawable figures that aren’t recursive.”
Contracrostipunctus
32.
Which character is Gödel and which is Hilbert?
33.
What do the records and record players have to do with formal systems?
34.
Lewis Carroll hid messages as acrostics in his writing. Why is that relevant to this dialogue?