Godel, Escher, Bach

Kurt Godel, M.C. Escher, and Johann Sebastian Bach, were the three men involved in the “trip-lets” And essentially what a “trip-let” is, it casts a shadow of three letters that have been carved into two blocks that are perfectly aligned above each other. Yet what is very interesting about the shadows that are casted is that top block casts a shadow of “GEB” (Godel, Escher, Bach) but, the bottom block casts a shadow of “EGB” (Eternal Golden Braid). This image of a “Trip-let” was the cover of “Godel, Escher, Bach: an Eternal Golden Braid” (660). The author is known as Douglas R. Hofstdter. Dr. Hofstadter had the idea to write a pamphlet about Godel’s theorem before the book but, he realized that he had to include Bach and Escher because of their work.

“the letters (preferably uppercase) must all be conventionally shaped, and they must fit snugly into the three rectangles that are the orthogonal projections of a rectangular block” (661).

Hofstadter has written in the book in which has both “Achilles” and “Tortoise” start off with the dialogue. And the structure of each dialogue is to mimic a composition constructed by Bach. They are both placed in Zeno’s paradox, “Achilles must catch Tortoise” (663). And yet this is just the first dialogue in Hofstadter’s book but, for the second dialogue it incorporates both Achilles and Tortoise again. What Achilles attempts to prove is “Z”. Yet although he attempts to prove it Tortoise refuses to believe his statements, until he sets rule which tend to prove his proof. This proof is typically known as a theorem of Euclid’s. Another dialogue within the book is know as “Crab-Canon” which has both Achilles and Tortoise’s sentences interspersed. Eventually they both use the same sentences but in reverse order where the Crab then appears to knot the two essential ideas together. The most substantial between these three characters is that the first letter of their names are three of the letters of the four nucleotides of DNA. And two of the nucleotides that are paired together are adenine and thymine; which is essentially just like both Achilles and Tortoise in the book.

Further into the chapter the term “formal system” arises. Hofstadter gives an example in his book using symbols such as M, I, and U, which then construct theorems. Yet there were rules such as:

“1. If the last letter of a theorem is I, U can be added to the theorem. 2. To any theorem Mx, x can be added. (For example, MUM can be transformed into MUMUM, and MU can be transformed into MUU.) 3. If III is in a theorem, it can be replaced by U, but the converse operation is not acceptable. (For example, MIII can be transformed into MU, and UMIIIMU can be transformed into UMUMU.) 4. If UU is in a theorem, it can be dropped. (For example, UUU can be transformed into U, and MUUUIII can be transformed into MUIII.)” (666)

There is also one “axiom” in the system which states that “M” must always be placed in front of the entire theorem.

The book then closes with “Six-Part Ricercar” “which is simultaneously patterned after Bach’s six-part ricercar and the story of how Bach came to write his Musical offering” (668). In this dialogue there is a computer in which pioneers Turing. Babbage another character “improvises at the keyboard of a flexible computer called a “smart-stupid”” (668). And there is a fued between both Babbage and Turing which consisted of figuring out which was real and which one was the program. They both decided to play the “Turing game” which is by asking shrewd questions to figure out who is real. At this point Hofstadter walks into the scene to inform all of the characters that they are all part of his imagination but he also states that he is fake as well.

A reference that that had been included in this chapter is “Exploring the Labyrinth of the Human Mind”. This had written about how the brain naturally makes a decision. It essentially states that when a human is reconstructing a word when they have a jumble of letters. And so the difference between humans and computers is that humans can figure out the word when the letters are jumbled together and also humans can also predict the next number in a sequence. These two actions are considered to be done instinctively and essentially computers cannot do what humans can do naturally. Yet the information on what humans can do compared to computers was interesting I had also found the GEB shadows interesting as well. That’s why when reading this reference I had found the extra information on the GEB shadows interesting.