2002/10/11

The inspiration for the subtitle of this blog.

[Gödel, Escher, Bach: An Eternal Golden Braid, by Douglas Hofstadter]

A few days ago, I succumbed to temptation and opened up Gödel, Escher, Bach: An Eternal Golden Braid, by Douglas Hofstadter. It is one of the most absolutely amazing books I have ever read. Very difficult to follow though without a pretty well-rounded education, which has made me appreciate my high school’s curriculum all the more. You need a basic grounding in fugues and counterpoint, in set theory and logic, and in visual arts (so far) to appreciate the first three chapters.

It’s really impossible for me to describe what the book is about and why it is so amazing without forcing you to read it yourselves, so let me just provide some brief paraphrases of moments in the book.

You are given the string MI and you want to obtain MU with these four rules:
1) if you have any string of the form Mx, where x is any string of characters, you can obtain the string Mxx
2) if you have I anywhere in a string, you can add U, i.e. if you have MI, you can get MIU
3) if you have III anywhere in a string, you can replace it with U, i.e. if you have MIII, you can get MU
4) if you have UU anywhere in a string, you can replace it with U, i.e. if you have MUU, you can get MU
Using any combination of these four rules, obtain MU from MI.

The game above represents a formal system, albeit a very simple one. You can see the parallels to mathematics: the strings are theorems, the laws are permissible methods of reasoning or logic, and the very first given string MI is your axiom. Hofstadter presents ever more complicated examples of these formal systems in order to explore the question of whether mathematics represents reality, what is consciousness, etc. Specifically, he is looking at “Strange Loops,” which is his non-technical term for certain types of recursive paradoxes that “loop” into infinity. He uses Escher’s art as a good example.

As a further demonstration of his ingeniousness, he uses dialogues between Achilles and Mr. Tortoise (the characters in Zeno’s motion paradox, which essentially states that when Achilles races with the tortoise, he can never win the race if he gives the tortoise a head start). (The paradox is wrong by the way, because an infinite geometric series with a ratio whose absolute value is less than 1 converges.) In the third dialogue, called “Contracrostipunctus,” Achilles and Mr. Tortoise’s conversation reflects Gödel’s Incompleteness Theorem using an analogy of a phonograph player. They then continue on to Bach, who used all of the notes in his name in the last Contrapunctus in the Art of Fugue (of which, by the way, I have a recording, performed by Glenn Gould). Mr. T comments that this is similar to an acrostic, where the first letters of all the lines spell out some hidden message. At that point, I accidentally stared at the first letters of the four paragraphs I was reading and realized they spelled out “BACH.” I then started from the beginning again and realized that the first letters of all the paragraphs spelled out “HOFSTATDER’S CONTRACROSTIPUNCTUS ACROSTICALLY BACKWARDS SPELL ‘J.S. BACH.’ ” Isn’t that witty? Perhaps a bit too obvious for your tastes. But I thought it was pretty amazing.

And these kinds of message revealing hidden message types of tricks are scattered all throughout the dialogues. Each dialogue basically presents metaphorically the mathematical/musical/artistic ideas to be discussed in the following chapter, except the metaphors themselves are metaphoric and other such “Strange Loops” of self-referentiality, which in turn is another huge self-reference, neh? I know that sounds confusing, but this book is a work of genius, so I suggest you all read it.