thoughts on Gödel, Escher, Bach

thoughts

I admire Douglas Hofstadter's work, or at least the four books that I've actually read. This page focuses on Gödel, Escher, Bach: an Eternal Golden Braid, since it is the most famous, most interactive, and most difficult to understand. I found a helpful overview of the book here.

Preface

The Latin in Hofstadter's "Preface to GEB's Twentieth-anniversary Edition" states some broad ideas about the book.

"Quærendo Invenietis": seek and ye shall find, or "By seeking, you will discover" (Hofstadter 1979, 9)

"REQVIESCAT IN CONSTANTIA, ERGO, REPRÆSENTATIO CVPIDI AVCTORIS RELIGIONIS": may this rest in constancy, therefore, a representation of the avid author of a religion

Introduction

On page 9, Hofstadter claims that when the original melody of "Good King Wenceslas" "and its inversion are sung together, starting an octave apart and staggered with a time-delay of two beats, a pleasing canon results." This resulting canon is pleasant, although dissonant at points. While this version keeps both voices in the same key, slightly changing the intervals, my version maintains the original intervals, with both parts moving up or down by the same number of semitones.

Figure and Ground

1    3    7    12    18    26    35    45    56    69
The differences between consecutive integers in this sequence (Hofstadter 1979, 73) are the missing integers. Alternately, the negative space in the set of integers is made up of the differences of consecutive integers in the set. Like the FIGURE-FIGURE Figure, the set and its negative space define each other.

Consistency, Completeness, and Geometry

The two isomorphisms in Figure 20, on page 84, are in turn isomorphic to Gödel's Theorem.

Isomorphism 1
groove-patterns to sounds <—> axioms and rules of formal system to true statements of number theory

Isomorphism 1
sounds to vibrations of the phonograph <—> true statements of number theory to reference to the system

If we have a record that cannot be played in a certain record player, but both the record and record player pass their respective tests, attempting to play the record leads to vibrations that specifically affect our record player, not others.

The Propositional Calculus

Hofstadter mentions a "record of David Oistrakh and Lev Oborin playing Bach's sonata in F Minor for violin and clavier". (Hofstadter 1979, 162) Such a recording is surprisingly hard to find online, but most of the sonata can be found in the following album:

The Location of Meaning

As Hofstadter suggests on page 193, I attempted to code a further exchange between Achilles and the Tortoise in the Carroll Dialogue into TNT.

Achilles: Now that you accept <<A^B>^<<A^B>⊃>> and <<<A^B>^<<A^B>⊃Z>>⊃Z>, OF COURSE you accept Z.
Tortoise: <<<<<A^B>^<<A^B>⊃Z>>⊃Z>⊃Z>^<<A^B>>⊃Z>

Typographical Number Theory

On page 210, Hofstadter prefers the two alternate translations he gives of the TNT sentence because they more explicitly represent 1729 as a sum of two cubes.

Four of the translation puzzles on pages 212-213 are true, and two are false. Translations and truth values are given below.

 ~∀𝑐:∃𝑏:(𝑆𝑆0⋅𝑏)=𝑐
 It is not the case that for all numbers c, there exists a number b such that 2b = c.
 It is not the case that [all numbers are even].
 TRUE

 ∀𝑐:~∃𝑏:(𝑆𝑆0⋅𝑏)=𝑐
 For all numbers c, it is not the case that there exists a number b such that 2b = c.
 All numbers are [not even].
 FALSE

 ∀𝑐:∃𝑏:~(𝑆𝑆0⋅𝑏)=𝑐
 For all numbers c, there exists a number b for which it is not the case that 2b = c.
 TRUE

 ~∃𝑏:∀𝑐:(𝑆𝑆0⋅𝑏)=𝑐
 It is not the case that there exists a number b for which for all numbers c, 2b = c.
 TRUE

 ∃𝑏:~∀𝑐:(𝑆𝑆0⋅𝑏)=𝑐
 There exists a number b for which it is not the case that for all numbers c, 2b = c.
 TRUE

 ∃𝑏:∀𝑐:~(𝑆𝑆0⋅𝑏)=𝑐
 There exists a number b for which for all numbers c, it is not the case that 2b = c.
 FALSE

I also attempted the five practice exercises on page 215, with some help from the internet on "b is a power of 2".

-All natural numbers are equal to 4.
 ∀𝑎:𝑎=𝑆𝑆𝑆𝑆0

-There is no natural number which equals its own square.
 ~∃𝑏:𝑏=(𝑏⋅𝑏)

-Different natural numbers have different successors.
 ∀𝑐:∀𝑑:~(𝑆𝑐=𝑆𝑑)

-If 1 equals 0, then every number is odd.
 <𝑆0⊃∀𝑐:~∃𝑏(𝑏+𝑏)=c>

-b is a power of 2.
 ∀𝑐:(∃𝑑:(𝑐⋅𝑑=𝑏)→(𝑐=𝑆0∨∃𝑎:(𝑐=𝑆𝑆0⋅𝑎)))