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Revision as of 13:58, 20 February 2006
Gödel, Escher, Bach - An Eternal Golden Braid
This wiki is for conversations and thoughts on the book. Below is a list of chapters of the book where you can enter your own thoughts about each section.
PART I: GEB
INTRODUCTION: A MUSICO-LOGICAL OFFERING
- Author
- Bach
- Canons and Fugues
- An Endlessly Rising Canon
- Escher
- Gödel
- Mathematical Logic: A Synopsis
- Banishing Strange Loops
- Consistency, Completeness, Hilbert's Program
- Babbage, Computers, Artificial Intelligence...
- ...and Bach
- Gödel Escher Bach
- Three-Part Invention
CHAPTER 1: THE MU-PUZZLE
- Formal Systems
- Theorems, Axioms, Rules
- Inside and Outside the System
- Jumping out of the System
- M-Mode, I-Mode, U-Mode
- Decision Procedures
- Two-Part Invention
CHAPTER 2: MEANING AND FORM IN MATHEMATICS
- The pq-System
- The Decision Procedure
- Bottom-up vs. Top-down
- Isomorphisms Induce Meaning
- Meaningless and Meaningful Interpretations
- Active vs. Passive Meanings
- Double-Entendre!
- Formal Systems and Reality
- Mathematics and Symbol Manipulation
- The Basic Laws of Arithmetic
- Ideal Numbers
- Euclid's Proof
- Getting Around Infinity
- Sonata for Unaccompanied Achilles
CHAPTER 3: FIGURE AND GROUND
- Primes vs. Composites
- The tq-System
- Capturing Compositeness
- Illegally Characterizing Primes
- Figure and Ground
- Figure and Ground in Music
- Recursively Enumerable Sets vs. Recursive Sets
- Primes as Figure Rather than Ground
- Contracrostipunctus
CHAPTER 4: CONSISTENCY, COMPLETENESS, AND GEOMETRY
- Implicit and Explicit Meaning
- Explicit Meaning of the Contracrostipunctus
- Implicit Meanings of the Contracrostipunctus
- Mapping Between the Contracrostipunctus and Gödel's Theorem
- The Art of the Fugue
- Problems Caused by Gödel's Result
- The Modified pq-System and Inconsistency
- Regaining Consistency
- The History of Euclidean Geometry
- The Many Faces of Noneuclid
- Undefined Terms
- The Possibility of Multiple Interpretations
- Varieties of Consistency
- Hypothetical Worlds and Consistency
- Embedding of One Formal System in Another
- Layers of Stability in Visual Perception
- Is Mathematics the Same in Every Conceivable World?
- Is Number Theory the Same in All Conceivable Worlds?
- Completeness
- How an Interpretation May Make or Break Completeness
- Incompleteness of Formalized Number Theory
- Little Harmonic Labyrinth
CHAPTER 5: RECURSIVE STRUCTURES AND PROCESSES
- What Is Recursion?
- Pushing, Popping, and Stacks
- Stacks in Music
- Recursion in Language
- Recursive Transition Networks
- Bottoming Out" and Heterarchies"
- Expanding Nodes
- Diagram G and Recursive Sequences
- A Chaotic Sequence
- Two Striking Recursive Graphs
- Recursion at the Lowest Level of Matter
- Copies and Sameness
- Programming and Recursion: Modularity, Loops, Procedures
- Recursion in Chess Programs
- Recursion and Unpredictability
- Canon by Intervallic Augmentation
CHAPTER 6: THE LOCATION OF MEANING
- When is One Thing Not Always the Same?
- Information-Bearers and Information-Revealers
- Genotype and Phenotype
- Exotic and Prosaic Isomorphisms
- Jukeboxes and Triggers
- DNA and the Necessity of Chemical Context
- An Unlikely UFO
- Levels of Understanding of a Message
- Imaginary Spacescape
- The Heroic Decipherers
- Three Layers of Any Message
- Schrödinger's Aperiodic Crystals
- Languages for the Three Levels
- The Jukebox" Theory of Meaning"
- Against the Jukebox Theory
- Meaning Is Intrinsic If Intelligence is Natural
- Earth Chauvinism
- Two Plaques in Space
- Bach vs. Cage Again
- How Universal Is DNA's Message?
- Chromatic Fantasy, and Feud
CHAPTER 7: THE PROPOSITIONAL CALCULUS
- Words and Symbols
- Alphabet and First Rule of the Propositional Calculus
- Well-Formed Strings
- More Rules of Inference
- The Fantasy Rule
- Recursion and the Fantasy Rule
- The Converse of the Fantasy Rule
- The Intended Interpretation of the Symbols
- Rounding Out the List of Rules
- Justifying the Rules
- Playing Around with the System
- Semi-Interpretations
- Ganto's Ax
- Is There a Decision Procedure for Theorems?
- Do We Know the System Is Consistent?
- The Carroll Dialogue Again
- Shortcuts and Derived Rules
- Formalizing Higher Levels
- Reflections on the Strengths and Weaknesses of the System
- Proofs vs. Derivations
- The Handling of Contradictions
- Crab Canon
CHAPTER 8: TYPOGRAPHICAL NUMBER THEORY
- The Crab Canon and Indirect Self-Reference
- What We Want to Be Able to Express in TNT
- Numerals
- Variables and Terms
- Atoms and Propositional Symbols
- Free Variables and Quantifiers
- Translating Our Sample Sentences
- Tricks of the Trade
- Translation Puzzles for You
- How to Distinguish True from False?
- The Rules of Well-Formedness
- A Few More Translation Exercises
- A Nontypographical System
- The Five Axioms and First Rules of TNT
- The Five Peano Postulates
- New Rules of TNT: Specification and Generalization
- The Existential Quantifier
- Rules of Equality and Successorship
- Illegal Shortcuts
- Why Specification and Generalization Are Restricted
- Something Is Missing
- \\omega-Incomplete Systems and Undecidable Strings
- Non-Euclidean TNT
- \\omega-Inconsistency Is Not the Same as Inconsistency
- The Last Rule
- A Long Derivation
- Tension and Resolution in TNT
- Formal Reasoning vs. Informal Reasoning
- Number Theorists Go out of Business
- Hilbert's Program
- A Mu Offering
CHAPTER 9: MUMON AND GÖDEL
- What is Zen?
- Zen Master Mumon
- Zen's Struggle Against Dualism
- Ism, The Un-Mode, and Unmon
- Zen and Tumbolia
- Escher and Zen
- Hemiolia and Escher
- Indra's Net
- Mumon on MU
- From Mumon to the MU-puzzle
- Mumon Shows Us How to Solve the MU-puzzle
- Gödel-Numbering the MIU-System
- Seeing Things Both Typographically and Arithmetically
- MIU-Producible Numbers
- Answering Questions about Producible Numbers by Consulting TNT
- The Dual Nature of MUMON
- Codes and Implicit Meaning
- The Boomerang: Gödel-Numbering TNT
- TNT-Numbers: A Recursively Enumerable Set of Numbers
- TNT Tries to Swallow Itself
- G: A String Which Talks about Itself in Code
- G's Existence Is What Causes TNT's Incompleteness
- Mumon Has the Last Word==
PART II: EGB
CHAPTER 10: LEVELS OF DESCRIPTION, AND COMPUTER SYSTEMS
- Levels of Description
- Chunking and Chess Skill
- Similar Levels
- Computer Systems
- Instructions and Data
- Machine Language vs. Assembly language
- Programs That Translate Programs
- Higher-Level Languages, Compilers, and Interpreters
- Bootstrapping
- Levels on Which to Describe Running Programs
- Microprogramming and Operating Systems
- Cushioning the User and Protecting the System
- Are Computers Super-Flexible or Super-Rigid?
- Second-Guessing the Programmer
- AI Advanced Are Language Advances
- The Paranoid and the Operating System
- The Border between Software and Hardware
- Intermediate Levels and the Weather
- From Tornados to Quarks
- Superconductivity: A Paradox" of Renormalization"
- Sealing-off
- The Trade-off between Chunking and Determinism
- Computers Can Only Do What You Tell Them to Do
- Two Types of System
- Epiphenomena
- Mind vs. Brain
- ...Ant Fugue
CHAPTER 11: BRAINS AND THOUGHTS
- New Perspectives on Thought
- Intensionality and Extensionality
- The Brain's Ants
- Larger Structures in the Brain
- Mappings between Brains
- Localization of Brain Processes: An Enigma
- Specificity in Visual Processing
- A Grandmother Cell"?"
- Funneling into Neural Modules
- Modules Which Mediate Thought Processes
- Active Symbols
- Classes and Instances
- The Prototype Principle
- The Splitting-off of Instance from Classes
- The Difficulty of Disentangling Symbols from Each Other
- Symbols -- Software or Hardware?
- Liftability of Intelligence
- Can One Symbol Be Isolated?
- The Symbols of Insects
- Class Symbols and Imaginary Worlds
- Intuitive Laws of Physics
- Procedural and Declarative Knowledge
- Visual Imagery
- English French German Suite
CHAPTER 12: MINDS AND THOUGHTS
- Can Minds Be Mapped onto Each Other?
- Comparing Different Semantic Networks
- Translations of Jabberwocky
- ASU's
- A Surprise Reversal
- Centrality and Universality
- How Much Do Language and Culture Channel Thought?
- Trips and Itineraries in ASU's
- Possible, Potential, and Preposterous Pathways
- Different Styles of Translating Novels
- High-Level Comparisons between Programs
- High-Level Comparisons between Brains
- Potential Beliefs, Potential Symbols
- Where is the Sense of Self?
- Subsystems
- Subsystems and Shared Code
- The Self-Symbol and Consciousness
- Our First Encounter with Lucas
- Aria with Diverse Variations
CHAPTER 13: BlooP AND FlooP AND GlooP
- Self-Awareness and Chaos
- Representability and Refrigerators
- Ganto's Ax in Metamathmatics
- Finding Order by Choosing the Right Filter
- Primordial Steps of the Language BlooP
- Loops and Upper Bounds
- Conventions of BlooP
- IF-Statements and Branching
- Automatic Chunking
- BlooP Tests
- BlooP Programs Contain Chains of Procedures
- Suggested Exercises
- Expressibility and Representability
- Primitive Recursive Predicates Are Represented in TNT
- Are There Functions Which Are Not Primitive Recursive?
- Pool B, Index Numbers, and Blue Programs
- The Diagonal Method
- Cantor's Original Diagonal Argument
- What Does a Diagonal Argument Prove?
- The Insidious Repeatability of the Diagonal Argument
- From BlooP to FlooP
- Terminating and Nonterminating FlooP Programs
- Turing's Trickery
- A Termination Tester Would Be Magical
- Pool F, Index Numbers, and Green Programs
- The Termination Tester Gives Us Red Programs
- GlooP...
- ...Is a Myth
- The Church-Turing Thesis
- Terminology: General and Partial Recursive
- The Power of TNT
- Air on G's String
CHAPTER 14: ON FORMALLY UNDECIDABLE PROPOSITIONS OF TNT AND RELATED SYSTEMS
- The Two Ideas of the Oyster
- The First Idea: Proof-Pairs
- Proof-Pair-ness Is Primitive Recursive...
- ...And Is Therefore Represented in TNT
- The Power of Proof-Pairs
- Substitution Leads to the Second Idea
- Arithmoquining
- The Last Straw
- TNT Says Uncle!
- Yields Nontheoremhood When Arithmoquined
- Gödel's Second Theorem
- TNT Is \\omega-Incomplete
- Two Different Ways to Plug Up the Hole
- Supernatural Numbers
- Supernatural Theorems Have Infinitely Long Derivations
- Supernatural Addition and Multiplication
- Supernaturals Are Useful...
- ...But Are They Real?
- Bifurcations in Geometry, and Physicists
- Bifurcations in Number Theory, and Bankers
- Bifurcations in Number Theory, and Metamathematicians
- Hilbert's Tenth Problem and the Tortoise
- Birthday Cantatatata...
CHAPTER 15: JUMPING OUT OF THE SYSTEM
- A More Powerful Formal System
- The Gödel Method Reapplied
- Multifurcation
- Essential Incompleteness
- The Passion According to Lucas
- Jumping Up a Dimension
- The Limits of Intelligent Systems
- There Is No Recursive Rule for Naming Ordinals
- Other Refutations of Lucas
- Self-Transcendence -- A Modern Myth
- Advertisement and Framing Devices
- Simplicio, Salviati, Sagredo: Why Three?
- Zen and Stepping Out
- Edifying Thoughts of a Tobacco Smoker
CHAPTER 16: SELF-REF AND SELF-REP
- Implicitly and Explicitly Self-Referential Sentences
- A Self-Reproducing Program
- What Is a Copy?
- A Self-Reproducing Song
- Epimenides Straddles the Channel
- A Program That Prints Out Its Own Gödel Number
- Gödelian Self-Reference
- A Self-Rep by Augmentation
- A Kimian Self-Rep
- What Is the Original?
- Typogenetics
- Strands, Bases, Enzymes
- Copy Mode and Double Strands
- Amino Acids
- Translation and the Typogenetic Code
- Tertiary Structure of Enzymes
- Punctuation, Genes, and Ribosomes
- Puzzle: A Typogenetical Self-Rep
- The Central Dogma of Typogenetics
- Strange Loops, TNT, and Real Genetics
- DNA and Nucleotides
- Messenger RNA and Ribosomes
- Amino Acids
- Ribosomes and Tape Recorders
- The Genetic Code
- Tertiary Structure
- Reductionistic Explanation of Protein Function
- Transfer RNA and Ribosomes
- Punctuation and the Reading Frame
- Recap
- Levels of Structure and Meaning in Proteins and Music
- Polyribosomes and Two-Tiered Canons
- Which Came First -- The Ribosome or the Protein?
- Protein Function
- Need for a Sufficiently Strong Support System
- How DNA Self-Replicates
- Comparison of DNA's Self-Rep Method with Quining
- Levels of Meaning of DNA
- The Central Dogmap
- Strange Loops in the Central Dogmap
- The Central Dogmap and the Contracrostipunctus
- E. Coli vs. T4
- A Molecular Trojan Horse
- Recognition, Disguises, Labeling
- Henkin Sentences and Viruses
- Implicit vs. Explicit Henkin Sentences
- Henkin Sentences and Self-Assembly
- Two Outstanding Problems: Differentiation and Morphogenesis
- Feedback and Feedforward
- Repressors and Inducers
- Feedback and Strange Loops Compared
- Two Simple Examples of Differentiation
- Level Mixing in the Cell
- The Origin of Life
- The Magnificrab, Indeed
CHAPTER 17: CHURCH, TURING, TARSKI, AND OTHERS
- Formal and Informal Systems
- Intuition and the Magnificent Crab
- The Church-Turing Thesis
- The Public-Processes Version
- Srinivasa Ramanujan
- Idiots Savants
- The Isomorphism Version of the Church-Turing Thesis
- Representation of Knowledge about the Real World
- Processes That Are Not So Skimmable
- Articles of Reductionistic Faith
- Partial Progress in AI and Brain Simulation?
- Beauty, the Crab, and the Soul
- Irrational and Rational Can Coexist on Different Levels
- More Against Lucas
- An Underpinning of AI
- Church's Theorem
- Tarski's Theorem
- The Impossibility of the Magnificrab
- Two Types of Form
- Meaning Derives from Connections to Cognitive Structures
- Beauty, Truth, and Form
- The Neural Substrate of the Epimenides Paradox
- SHRDLU, Toy of Man's Designing
CHAPTER 18: ARTIFICIAL INTELLIGENCE: RETROSPECTS
- Turing
- The Turing Test
- Turing Anticipates Objections
- Parry Encounters the Doctor
- A Brief History of AI
- Mechanical Translation
- Computer Chess
- Samuel's Checker Program
- When Is a Program Original?
- Who Composes Computer Music?
- Theorem Proving and Problem Reduction
- Shandy and the Bone
- Changing the Problem Space
- The I-Mode and the M-Mode Again
- Applying AI to Mathematics
- The Crux of AI: Representation of Knowledge
- DNA and Proteins Help Give Some Perspective
- Modularity of Knowledge
- Representing Knowledge in a Logical Formalism
- Deductive vs. Analogical Awareness
- From Computer Haiku to an RTN-Grammar
- From RTN's to ATN's
- A Little Turing Test
- Images of What Thought Is
- Higher-Level Grammars...
- Grammars for Music?
- Winograd's Program SHRDLU
- The Structure of SHRDLU
- PLANNER Facilitates Problem Reduction
- Syntax and Semantics
- Contrafactus
CHAPTER 19: ARTIFICIAL INTELLIGENCE: PROSPECTS
- Almost" Situations and Subjunctives"
- Layers of Stability
- Frames and Nested Contexts
- Bongard Problems
- Preprocessing Selects a Mini-Vocabulary
- High-Level Descriptions
- Templates and Sameness-Detectors
- A Heterarchical Program
- The Concept Network
- Slippage and Tentativity
- Meta-Descriptions
- Flexibility is Important
- Focusing and Filtering
- Science and the World of Bongard Problems
- Connections to Other Types of Thought
- Message-Passing Languages, Frames, and Symbols
- Enzymes and AI
- Fission and Fusion
- Epigenesis of the Crab Canon
- Conceptual Skeletons and Conceptual Mapping
- Recombinant Ideas
- Abstractions, Skeletons, Analogies
- Multiple Representations
- Ports of Access
- Forced Matching
- Recap
- Creativity and Randomness
- Picking up Patterns on All Levels
- The Flexibility of Language
- Intelligence and Emotions
- AI Has Far to Go
- Ten Questions and Speculations
- Sloth Canon
CHAPTER 20: STRANGE LOOPS, OR TANGLED HIERARCHIES
- Can Machines Possess Originality?
- Below Every Tangled Hierarchy Lies An Inviolate Level
- A Self-Modifying Game
- The Authorship Triangle Again
- Escher's Drawing Hands
- Brain and Mind: A Neural Tangle Supporting a Symbol Tangle
- Strange Loops in Government
- Tangles Involving Science and the Occult
- The Nature of Evidence
- Seeing Oneself
- Gödel's Theorem and Other Disciplines
- Introspection and Insanity: A Gödelian Problem
- Can We Understand Our Own Minds or Brains?
- Gödel's Theorem and Personal Nonexistence
- Science and Dualism
- Symbol vs. Object in Modern Music and Art
- Magritte's Semantic Illusions
- The Code" of Modern Art"
- Ism Once Again
- Understanding the Mind
- Accidental Inexplicability of Intelligence?
- Undecidability Is Inseparable from a High-Level Viewpoint
- Consciousness as an Intrinsically High-Level Phenomenon
- Strange Loops as the Crux of Consciousness
- The Self-Symbol and Free Will
- A Gödel Vortex Where All Levels Cross
- An Escher Vortex Where All Levels Cross
- A Bach Vortex Where All Levels Cross
- Six-Part Ricercar
