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==[[CHAPTER 17: CHURCH, TURING, TARSKI, AND OTHERS]]== | ==[[CHAPTER 17: CHURCH, TURING, TARSKI, AND OTHERS]]== | ||
*[[Formal and Informal Systems]] | *[[Formal and Informal Systems]] | ||
Revision as of 19:22, 16 June 2007
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.
Red links are pages that have no comment, but you can add your own comments. Blue links contain comments.
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
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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 ω-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
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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
