Godel, Escher, and Bach: An Eternal Golden Braid By Douglas Hofstadter Book Review By Grace Gilles CS110

Introduction Bach was known for using canons, or repetitions of a theme, which include things like rounds, and fugues, repetitions of a theme that are similar to canons, but less rigid, in his musical compositions. Escher was known for using paradoxes and illusions, including strange loops in his intellectually and mathematically stimulating graphic art. Godel used mathematical logic to explore mathematical reasoning itself. He looked at such problems as the paradoxical statement “This statement is a lie.”

Formal Systems: the MU puzzle Hofstadter talks a great deal about formal systems, which he explains using the MU puzzle. You begin with the string MI and manipulate it using these four rules, in order to get the string MU: – If your string ends with I, you can add U to the end. – If your string is Mx, you can also have Mxx. – If you have III, you can replace it with U. – If you have UU, you can drop it. Hofstadter discusses the differences between humans and computers in solving this problem: – Humans will automatically make observations, such as realizing that the string U is impossible to derive from MI, while computers require complex programming to make observations about problems – Humans can “jump out of the system” and use knowledge and reasoning outside of the system to help them solve a problem within the system, while computers cannot.

Formal Systems: the pq- System Another formal system that Hofstadter discusses is the pq- system, which uses the characters p, q, and -. – This system contains only one rule: suppose x, y, and z all stand for particular strings containing only hyphens. If xpyqz is a theorem then xpy-qz- is a theorem. – In order to be a theorem, the number of hyphens in the first two groups must add up the number of hyphens in the third group. Hofstadter uses this system to discuss isomorphism, such as that between the system and traditional addition, meaningful and meaningless interpretations of the system, and active vs. passive meanings.

Formal Systems: the tq- System With the tq- system, Hofstadter seeks to develop a system with the ability to distinguish between prime and composite numbers in its operation. The strings are of the form xtyqz, with x, y, and z being strings of X, Y, and Z number of hyphens, where X*Y=Z. – Hofstadter captures the idea of composite numbers with the rule, if x-ty-qz is a theorem, then Cz is a theorem. – However, the rule “if Cx is not a theorem then Px is a theorem,” cannot be used to find prime numbers, because it requires one to reason outside the system. Hofstadter uses the tq- system to discuss the figure and ground, or positive and negative space, idea, in both math/logic and music.

Zen in Godel and Escher’s Work In this chapter, Hofstadter ponders Zen, and its relation to the book’s title characters. He discusses the Zen master Mumon and his Mumonkan, which is comprised of 48 koans, each with an explanation and a poem, and how the work is riddled with paradoxes. He also compares Escher’s art to Mumon’s koans, because they are filled with logical paradoxes. Mumon’s logic solves the MU puzzle with this proof: – The I count begins at 1 (not a multiple of 3). – Two of the rules do not affect the I count. – The other two rules cannot influence the I count to contain a multiple of 3 if the original string does not contain a multiple of 3. – Therefore the I count cannot be a multiple of 3 and it is impossible to derive MU. He also discusses MIU producible numbers and using TNT to find producible numbers. He relates this to the Zen topic by showing that the TNT-Numbers are a recursively enumerable set.