Gödel, Escher, Bach by Douglas Hofstadter | Notes & Review
Gödel, Escher, Bach: an Eternal Golden Braid by Douglas Hofstadter is his Pulitzer Prize winning contribution to the field of Artificial Intelligence, logic, consciousness, and metamathematics. I worked my way backward through some of his catalogue over the last few years and had hoped to issue this “review” of the book much sooner. As fate would have it, the last few chapters of my notes were lost to the digital void which required me to go back and reread a few chapters to reproduce my thoughts — no small task indeed.
The book can be said to be sensible & nonsensical, I mean that which is perceivable versus those things which appear illogical. That which is perceivable being those things quantifiable and qualifiable. While the illogical classes exist within elusive and intangible spaces.
Many have written on computers’ need of an Oracle, humans which input the spark to action. However, there‘s little mention of timescale regarding the exchange. Can’t “Oracle” become an honorary title from one self-perpetuating species to another once there‘s not a reliance on it?
** Video at the end **
Introduction: A Musico-Logical Offering
This section introduces the trio of Gödel, Escher, and Bach. It addresses what concepts of theirs are focused on by Hofstadter and the underlying thematic continuity/overlap he perceived in their works.
Over the course of this book, Hofstadter uses a variety of intelligence “tests” to outline the progress of AI, leaning on the concept of self-reference heavily to illustrate how intelligences scaffold abilities.
Chapter I: The MU-puzzle
Formally Axiomatic Systems & “exiting the system”
The halting problem
Chapter II: Meaning and Form in Mathematics
Dichotomies; Meaning(ful/less) Symbolic logic & representation
This is a good point to get familiar with Gödel’s Proof
Isomorphism: pattern seeking & inversions
Positive & negative space
Chapter III: Figure and Ground
Recursive intention, meaning emerges from focus
“There exist formal systems whose negative space (set of non-theorems) is not the positive space (set of theorems) of any formal system.”
“There exist recursively enumerable sets which are not recursive.”
“There exist formal systems for which there is no typographical decision procedure.”
“intuitive fluidity”
Chapter IV: Consistency, Completeness, Geometry
Isomorphic lenses for analogical comparisons
Self-reference and Bach’s fugues
Consistency : inconsistency
Chapter V: Recursive Structures and Processes
Embedded media
RTN : recursive transition networks
Feynman Diagrams
Hofstadter’s Law
Chapter VI: The Location of Meaning
Message and interpretation rely on when and where
Layered meaning
Sameness-in-differentness: underlying “universal” language
Chapter VII: The Propositional Calculus
Conflicting axioms/theorems, logic puzzles, and symbolic representation
Rules of Propositional Calculus
Prudence and imprudence
Chapter VIII: Typographical Number Theory
Abstracting abstraction : symbolic representation for modifications, isomorphic relationships
Collisions of meaning disconnected from intentional directionality
Structuring passages of logic
Implications of tension as related to outcomes
Minimal viable reasoning
Chapter IX: Mumon and Gödel
Duality: incompleteness & totality — balanced interpretation requires the unknown
“Ism is an anti philosophy, a way of being without thinking”
Hemiola : temporal inversions
See also: polyrhythms
ATN : augmented transition networks
Algorithmic symbol interpretation and substitutes
Statements that create paradoxical loops - “This statement is false”
Chapter X: Levels of Description, and Computer Systems
Chunking
Machine language : binary
Assembly language : words
Translation
Levels of interpretation
Compartmentalization & leaking
Epiphenomena : “a visible consequence of the overall system organization”
Chapter XI: Brains and Thoughts
“Calculus of descriptions”
Intensionally floating and flexible
Neurons
Brain & regions
Nonlocalizable nature of memory, localizable connectivity of neurons Specialized cells for visual motion perception
A list of symbol/brain interaction types
Consider the brain as an ATN colony
Declarative knowledge: explicitly stored, readable by programmer and program as fact; local
Procedural knowledge: distributed chunks, epiphenomenon; how-to
Retrieved vs assembled
Chapter XII: Minds and Thoughts
What is lost in translation? What isn’t?
“a large portion of every human’s network of symbols is universal”
Etherware : “the pure concepts which lie back of the software”
“Consciousness is that property of a system that arises whenever there exist symbols in the system which obey triggering patterns somewhat like the ones described”
$I is a subsystem, “a constellation of symbols”
Chapter XIII: BlooP and FlooP and GlooP
BlooP : bounded loop
Core truths of N: primitive recursive truths, only predictably terminating calculations order vs chaos
Algorithms: operations, controls
Functions which are BlooP-computable are primitive recursive functions
Diagonal arguments
Incomplete “totalities”
FlooP : free loop, boundless; creates the potential for nontermination Substitution representation
GlooP : mythological, FlooP is the most “unbounded” a computer language can be
Church-Turing Thesis Rules
Chapter XIV: On Formally Undecidable Propositions of TNT
Self-scrutiny and attention to self
Symbol manipulation
Proof pairs
Factoring, compression : isomorphism
Self-referential loops, embedding & abbreviations
Quine
Correlated yet incongruently divergent number theories
Chapter XV: Jumping out of the System
Infinite axiomization : incompleteness
Embedded self-reference & derivative loops
3 Rules
Existing outside of the system is an illusion
Chapter XVI: Self-Ref and Self-Rep
Self-representation
“Reference”: translation, transposition, augmentation
Editing and processing interactions with strands — Directions for interpreting the message of the tape, written on the tape itself — translation
Bonding: pairs, strands
Interconnected parts and layers
Data ← → Program
Chapter XVII: Church, Turing, Tarski, and Others
“every aspect of thinking can be viewed as a high-level description of a system which, on a low level, is governed by simple, even formal, rules.”
Church’s Theorem: There is no infallible method for telling theorems of TNT from nontheorems
Church-Turing Thesis, Tautological Version: Mathematics problems can be solved only by doing mathematics
Church-Turing Thesis, Standard Version:
Church-Turing Thesis, Hardy’s Version: At bottom, all mathematics are isomorphic
Hardy on Ramanujan:
“..imagery and analogical thought processes intrinsically require several layers of substrates…it is precisely at this same point that creativity starts to emerge-which would imply that creativity intrinsically depends upon certain kinds of “uninterpretable” lower-level events”
Church-Turing Thesis Soulist Version: Some kinds of things which a brain can do can be vaguely approximated on a computer but not most, and certainly not the interesting ones. But anyway, even if they all could, that would still leave the soul [for computers] to explain
“Irrational and rational can coexist on different levels”
“Gödel’s theorem must apply to cybernetic machines, because it is of the essence of being a machine, that it should be a concrete instantiation of a formal system.”
“..true on the hardware level — but since there may be higher levels, it is not the last word on the subject.”
Church-Turing Thesis, AI Version: Mental processes of any sort can be simulated by a computer program whose underlying language is of power equal to that of FlooP — that is, in which all partial recursive functions can be programmed
AI Thesis: As the intelligence of machines evolves, it’s underlying mechanisms will gradually converge to the mechanisms of human intelligence
“Our minds contain interpreters which accept notions which are so complex that we cannot consciously describe them. The same can be said about how we respond to music, incidentally.”
Syntactic form & predictably terminating tests, close to the surface
Symantec form & open-ended tests, meaning is not localized
Chapter XVIII: Artificial Intelligence: Retrospects
The Turing Test — an “imitation game”
Objections
Joseph Weizenbaum — Computer Power and Human Reason ~Perceiving a “man behind the curtain”
“One could define AI as coming into existence at the moment when mechanical devices took over any tasks previously performed only by human minds.”
Tesler’s Theorum: “AI is whatever hasn’t been done yet.”
Domains of AI
Meta-author: “author of the author of the result “
“Who composes computer music?
Problem reduction & subgoals
Mechanical mode: fixed framework
Intelligent mode: overview
“The Crux of AI: Representation of Knowledge”
“Modularity of Knowledge”
- How easy is it to insert new?
- How how easy to revise old?
Deductive vs analogical awareness: “stored in memory” ≠ known
Chapter XIX: Artificial Intelligence: Prospects
Thoughts on alternatives & possibilities from George Steiner’s “After Babel”
“Think how immeasurably poorer our mental lives would be if we didn’t have this creative capacity for slipping out of the midst of reality into soft “what if” ‘s!”
Layers of stability: Constant, parameter, variable
Frames and nested contexts
Concept network: recursive and derivative narrative mapping
Meta-descriptions: descriptions of descriptions
Essences of meaning and intention
Malaphor: cross between malapropism & metaphor, a recombinant idea
Focusing: making a description who’s focus is some part of the drawing in the box, to the exclusion of everything else
Filtering: involves making a description which concatenates on some particular ways of viewing the contents of the box, & deliberately ignores all other aspects
Human Visual pattern recognition can occur at the subconscious level
Frame + actor = symbol
Fission & fusion, a balanced and complete analogous model of inversion
Conceptual development
- Metaphase
- Anaphase
- Telophase
“a plural thing made singular and re-pluralized wrongly”
Multiple Representations
Forced Matching
Creativity and Randomness against Intelligence and Emotions
10 Questions (1979)
1) Will a computer program ever write beautiful music?
2) Will emotions be explicitly programmed into a machine?
3) Will a thinking computer be able to add fast?
4) Will there be chess programs that can beat anyone?
5)
6) Could you “tune” an AI program to act like me, or like you — or halfway between us?
7) Will there be a “heart” to an AI program, or will it simply consist of “senseless loops and sequences of trivial operations” (in the words of Marvin Minsky)?
8) Will AI programs ever become “superintelligent”?
9) You seem to be saying that AI programs will be virtually identical to people then. Won’t there be a difference?
10) Will we understand what intelligence and consciousness and free will and “I” are when we have made an intelligent program?
Chapter XX: Strange Loops, Or Tangled Hierarchies
“Can machines possess originality?”
Thoughts from Arthur Samuel (1960)
Revisiting the notion of authorship
Strange Loops
A “prime mover” and “inviolate substrate”
“…as if it were outside the system”
> Klein bottle
> Escher “Drawing Hands”
“Introspection and Insanity: A Godelian Problem” Upside-down face
“Can we understand our own minds and brains?”
Subject-object dichotomy
Acting vs thinking
Doing vs speaking
Hardware & software analogy for brain & mind
Valuing material vs symbolic
A symbol for self: $I
The conundrum of free will
“A Godelian vortex where all levels cross”
“An Escher vortex where all levels cross”
“A Bach vortex where all levels cross”
Shepard tone
Coltrane 5ths
Pitch Axis Theory
A Few Final Thoughts
I must mention that this book is not exclusively academic in a “left brain” sense. Each chapter is preceded by a narrative segments centering on Achilles & Tortoise as well as their encounters with friends that analogically resemble the concepts explained in the main text.
You could read only these conceptual fable-like sections and still take away an excellent collection of insights about consciousness and computation. Although, they are most useful as mechanisms to seed the concepts to you in nontechnical terms.
It’s hard not to like these passages because of how well written and fun the interactions are between all the characters. They’re weird. The Author himself even shows up late into the book. I like them a lot. The segments do a great job of priming your intellect for the difficult analytical lessons on consciousness and computation in the proper chapters.
This book was put into print over forty years ago, some of the science has been eclipsed but much of the philosophy holds up well to this day.
