This is a series on the book Gödel, Escher, Bach: An eternal golden braid by Douglas Hofstadter. To catch up, you should read the first in the series, the second diary, the third diary and the fourth diary and you should be up to page 102 in the book itself.
Things start getting complex, so this week I am just going to write about chapter 4, and leave the dialogue for next week.
From the Overview
Chapter IV: Consistency, completeness and geometry. The preceding dialogue is explicated to the extent possible at this stage. This leads back to the question of how and when symbols in a formal system acquire meaning. The history of Euclidean and non-Euclidean geometry is given, as an illustration of the elusive notion of "undefined terms". This leads to ideas about the consistency of different and possibly "rival" geometries. Through this discussion the notion of undefined terms is clarified, and the relation of undefined terms to perception and thought processes is considered
Consistency and completeness are at the center of Godel's theorem, because he says that, in a formal theory powerful enough to be worthwhile (we'll get to what that means, exactly) you can't have both. There are formal systems that are complete and consistent, but they are boring - like the pq- system.
This chapter is also all about isomorphisms. Here are some questions to spur thinking and comments:
On page 83, he talks about not being able to avoid the boomerang; level one 'forces' the meaning of level two. Indeed, if we listen to a language we understand, this seems to be so. But what if we are listening to a language we do not understand? Or understand partially? If I listen to people speaking, say, Mandarin, I know it's a language, but I can't understand it at all. With Hebrew or Spanish, I understand a little. Here's another question for those who are reasonably fluent in at least two languages: If you go to a movie in the foreign with subtitles in your native language, do you listen, or read?
On page 87, he introduces a new formal system, which seems really weirdat first. Is this like a pun, where the same symbols have two meanings? Or like slang? Or like music that stops sounding discordant as you get used to it?
On page 91, he gets into nonEuclidean geometry. Is this like a different kind of music or art? Why did it take so long to develop nonE geometery? Especially once people knew the world was spherical?
On page 97-98, he talks about the Escher painting "Relativity" and says one choice is to just see it as lines. Can you do this? I can't. But what about someone who had never seen a staircase?
On page 99, he asks whether math has to be the same on all worlds ... but another possibility is that, while nothing that is true here is false there, or vice versa, the type of math we know is different. What if they never came up with the idea of prime numbers?