This is one of my most favorite books of all time. Tough to read, even harder to summarize. But worth to read, and then re-read. And re-read. The dialogues where the turtoise discusses important riddles and philosophical questions alone, make this book worthwhile. How they are self referentially tied into the whole picture, makes it even better.
At it's core Gödel Escher Bach is a book about self-reference and it's applications in all kinds of different fields. Music, Art, Science, Mathematics, and Cognition. The general idea here is that self-reference is abundant, and that systems that are capable of self-reference can produce very interesting, and essentially unpredictable patterns.
And that there is a sort of beauty inherent in self-reference, which is why we use it so much in art.
Furthermore it's about how self-reference breaks mathematics and trying to formalize it, by step by step deconstructing Gödels Proof, playing with toy mathematical systems and deriving self-referential machines, that break any kind of system thrown at them. This whole idea is then further explored from different angles in the dialogues between the turtoise and the crab, where the crab builds better and better machines to play records, while the turtoise comes along creating records that break any machine the crab tries to devise. This shows beautifully, the connections between Turings Halting problem and Gödels Incompleteness theorem. And ties it to the grander scheme of the book. Recursion. Self-Reference.
Loops that can reason about themselves, that contain a self-referential copy to themselves, is pretty much what brains are as well, to a certain extent. We can model ourselves, acting in a world, thinking about the world, acting out in a certain way, which influences the state of the world, which influences ourselves within it and so on, recursively, ad infinitum.
Strange Loops, i.e. loops that form a part of themselves, that are self-referential in nature, can form hierarchies interacting with other strange loops. This is essentially what evolution does, and what gives rise to ecosystems, bodies, brains, multicellular life, civilizations, economies and cultures. It's the nestling of loops, interacting with one another, producing emergent properties along the way, that couldn't be predicted by the single parts forming them. Very much related to the concepts discussed in Complexity. And also related to many ideas from Michael Levin.
The only way that I can describe this book is as a brain-teaser. I am not sure if I will ever understand it in it's entirety, but I understand enough to appreciate it as the masterpiece it is. And whenever I read it again, I will understand more things, that I didn't notice or understand before. And that, is the true measuring stick of a book, worth re-reading.