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Gödel, Escher, Bach - An Eternal Golden Braid

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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

PART II: EGB

• Prelude...

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

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