Gödel, Escher, Bach. An Eternal Golden Braid

It covers math and philosophy, and also much more, from biology, AI, music, programming. Probably the longest and most difficult book I have finished; I struggled, and in many parts I wanted to skip to the end, yet I am impressed by how the author explored fascinating corners of philosophy with real skill. I drew on Nat Eliason’s excellent notes and recommend his precise analysis of the first five chapters. Even writing this took considerable effort. I would not reread it, but it feels revolutionary, especially considering its age.

Introduction

It analyzes Bach’s music, which is fascinating and full of hidden meanings. In the “Musical Offering” there is an early example of a strange loop, where canons circle back on themselves. The Epimenides paradox (“this statement is false”) also belongs to loops. To escape these paradoxes, we would have to eliminate self-reference. But a complete system requires it, so we keep it.

1. MU

Introduction to the MU puzzle, which has no solution if we stay within its rules, which is what a machine will do endlessly. Humans, instead, recognize patterns.

2. Meaning and Form in Mathematics

The concept of isomorphism (often present in Escher), when two structures map onto each other.

3. Figure and Ground

We often focus on the figure, but sometimes there is more meaning hidden in the background… there is the famous dialogue about the Tortoise’s record that always breaks the Crab’s phonograph, showing that there is meaning “above” the discussion.

4. Consistency, Completeness, Geometry

Some basic concepts, such as Consistency -> every theorem becomes true in some world, and Completeness -> when all statements that are true are theorems.

Interesting dialogue where Achilles and the Tortoise go in and out of story levels (like reading a book about themselves), very meta, but in the end they do not get out.

5. Recursive structures and processes

Here we reach the core of the book, recursion. The idea of “pop” for moving between levels, as in the dialogue.

6. The Origin of Meaning

A record is a carrier of information and the phonograph a revealer of information.

Can a text have an internal logic that can be restored when an intelligence comes into contact with it? Meaning is part of an object that interfaces with intelligence in a predictable way. There are therefore 3 levels for each message

• Understand the inner message (what the sender intended)
• Understand the frame, the context of the message
• Understand the outer message (build the mechanism to decode it) Note that you can end up in an infinite loop of assuming there is always something beyond the message.

7. Propositional Calculus

Some rules of logic, arranged in a structure called “TNT”. There is also the famous crab canon where they speak in a crisscrossing way.

8. Typographical Number Theory

Introduction to Gödel’s circularity, namely that any system capable of proving the consistency of TNT is at least as strong as the language structure itself… So a loop is inevitable. Introduction to koans.

9 . Mumon and Gödel

More koans, monks, and the idea of enlightenment. A lot on Zen; I understood little, even with the notes.

10. Levels of description and computer systems

Dialogue about the ant colony, which makes a strong point. We are a system, but we do not need to know everything about all levels to function (atoms, molecules, cells, etc.), just as the ants in a colony work independently. So we do not need to know everything about a system, or rather, each level reasons at its own level.

11. Brains and thoughts

Basics of neurons, which may or may not fire an electrical impulse. It explores whether you can have intelligence on any hardware if the concept is all that matters. Jabberwocky dialogue by Carroll in multiple languages, very challenging.

12. Minds and thoughts

We are a bundle of contradictions, including our brains. I did not grasp much more.

13. BlooP, Floop, GlooP

Even Eliason struggled here; his note says that if you have a complete formalization of number theory, then Gödel’s method is applicable, otherwise it is incomplete. I think this is where he explores functions.

14. On the undecidable propositions of TNT

Another chapter that lost both Eliason and me, full of mathematical and logical concepts.

15. Stepping out of the system

A program cannot step out of itself, while in Zen there is an idea of transcendence.

16. Self-ref and self-rep

It is difficult to make a self-reference without saying “this sentence”. A biology section with the idea of DNA and RNA and their incompleteness as a system.

17. Church, Turing, Tarski, and others

The whole idea of thinking can be seen as a system where the lowest level is many rules -> key takeaway. It also discusses SHRDLU and other early AI systems.

18. A retrospective on AI

The idea of the Turing test to see whether a machine reasons, and a dialogue about CongiunTV.

19. A perspective on AI

I got lost. There is a conceptual outline, and programs can be good at specific things, but putting concepts together is hard.

20. Strange loops

The grand finale, with the idea of loops. The example of three authors who create one another can exist only if, at a higher level, they are in another author’s story (or see Drawing Hands or Magritte). Understanding these loops and self-reference is at the core of the idea of AI.

We, on the outside, can know that the print gallery image is incomplete in a way that the man in it can never know. We can step outside and see the hole in the system, which he can never observe.