Book Club
Godel, Escher, Bach: The Golden Eternal Braid
by Douglas Hofstadter

Topic: Strange Loops
Godel
Escher
Bach
  • "All Cretans are liars."

  • The following statement is false.
  • The previous statement is true.
metamorfosis.jpg
matamorfosis
paper by Craig S. Kaplan
















Tools for Thinking
  1. Isomorphisms
  2. Recursion
  3. Paradox
  4. Infinity
  5. Formal Systems


Topic: Isomorphism

The word “isomorphism” applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where “corresponding” means that the two parts play similar roles in their respective structures. (G ̈odel, Escher, Bach, p. 49)



Topic: Recursion
A list of instructions that you follow and repeat until you reach a final case.
Example: Fibonacci Series: 1,1,2,3,5,8,13,21,...
f(n) = f(n-2) + f(n-1)
"define the 'thing' in terms of itself"
Example: Fractals

Topic: Infinity



Topic: Paradoxeshttp://en.wikipedia.org/wiki/Paradox
  • Veridical - Zeno and the inability to move (half the distance)
  • Falsidical: 1-1+1-1+1-1 ... = 0 , 1-1+1-1+1-1+1 ... = 0
  • Antinomy: This sentence is not true.
The Barber's Paradox: The barber only shaves people who don't shave themselves. Does the barber shave himself?
Help: Having trouble with the idea of a set that contains "sets of things that don't contain themselves."

Topic: Systems


Of course, there are cases where only a rare individual will have the vision to perceive a system which governs many people’ lives, a system which had never before even been recognized as a system; then such people often devote their lives to convincing other people that the system really is there, and that it ought to be exited from! (pp. 37)


Definitions
  • Axiom - a starting point for reasoning
  • String - ordered sequence
  • Theorem - derivation of ideas
  • Rules of Inference -

Note: GEB dialogues refer to Bach's music. How?
little fugue in G minor


Web Resources: