Gödel, Escher, Bach by Douglas Hofstadter
by Parker · 36 things on Twos
- an Eternal Golden Braid
- GEB is in essence a long proposal of strange loops as a metaphor for how selfhood originates, a metaphor by which to begin to grab a hold of just what it is that makes an "I" seem, at one and the same time, so terribly real and tangible to its own possessor, and yet also so vague, so impenetrable, so deeply elusive.
- One of the key qualities that made me so believe in what I was doing is that this was a book in which form was being given equal billing with content - and that was no accident, since GEB is in large part about how content is inseparable from form, how semantics is of a piece with syntax, how inextricable pattern and matter are from each other.
- Thus does sexism silently pervade well-meaning but susceptible brains.
- Our purpose, closely related to that of GEB, was to force our readers to confront, in the most vivid and even jolting manner, the fundamental conundrum of human existence: our deep and almost ineradicable sense of possessing a unique "I"-ness transcending our physical bodies and mysteriously enabling us to exercise something we call "free will", without ever quite knowing just what that is.
- Ricercar was the italian work meaning "to seek"
- A fugue is like a canon, in that it is usually based on one theme which gets played in different voices and different keys, and occasionally at different speeds or upside down or backwards. However, the notion of fugue is much less rigid than that of canon, and consequently it allows for more emotional and artistic expres-sion.
- Kurt Gödel is a mathmetician
- Epimenides paradox: "All Cretans are liars." or "I am lying"; or, "This statement is false".
- The following sentence is false. The proceeding sentence is true.
- MU-puzzle
- Can you produce MU?
- You start with MI
- Rule 1: If you possess a string whose last letter is I, you can add on a U at the end
- Rule 2: Suppose you have Mx. Then you may add Mxx to your collection
- Rule 3: If III occurs in one of the strings in your collection, you may make a new string with U in place of III
- Rule 4: If UU occurs inside one of your strings, you can drop it.
- Theorems are strings that can be produced by rules.
- An axiom is a free theorem.
- Do words and thoughts follow formal rules, or do they not?
- PQ-system
- p, q, and -
- Definition: xp-qx- is an axiom, whenever x is composed of hyphens only
- Rule: Suppose x, y, and z all stand for a particular strings containing only hyphens. And suppose that xpyqz is known to be a theorem. Then xpy-qz- is a theorem.
- Every theorem of the pq-system has three separate groups of hyphens, and the separating elements are one p, and one q, in that order.
- The criterion for theoremhood is that the first two hyphen-groups should add up, in length, to the third hyphen-group.
- Isomorphism is an information-preserving transformation.
- Well-formed strings in any formal system is that they are those strings which, when interpreted symbol for symbol, yield grammatical sentences.
- 1+1=1 with clouds, raindrops running down a window
- A cursively drawable figure is one whose ground is merely an accidental by-product of the drawing act.
- A recursive figure is one whose ground can be seen as a figure in its own right.
- Consistency is not a property of a formal system per se, but depends on the interpretation which is proposed for it.
- A system-plus-interpretation is consistent with the external world if every theorem comes out true when interpreted
- A system-plus-interpretation is internally consistent if all theorems come out mutually compatible when interpreted.
- Consistency: when every theorem, upon interpretation, comes out true (in some imaginable world).
- Completeness: when all statements which are true (in some imaginable world), and which can be expressed as well-formed strings of the system, are theorems.