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Index
Part II: EGB Introduction: A Musico-Logical Offering Three-Part Invention Formal Systems Theorems, Axioms, Rules Inside and Outside the System Jumping out of the System M-Mode, I-Mode, U-Mode Decision Procedures or, What the Tortoise Said to Achilles by Lewis Carroll' 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 The telephone rings; Achilles picks it up. Primes vs. Composites The tq-System Capturing Compositeness Illegally Characterizing Primes Figure and Ground in Music Recursively Enumerable Sets vs. Recursive Sets Primes as Figure Rather than Ground 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 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? Completenes How an Interpretation May Make or Break Completeness Incompleteness of Formalized Number Theory . . . . . . . . . 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 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 Schrodinger'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? 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 Semi-Interpretations 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 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 Translation Puzzles for You How to Distinguish True from False? The Rules of Well-Formedness A Few More Translation Exercises A Non typographical 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 -Incomplete Systems and Undecidable Strings Non-Euclidean TNT The Last Rule Tension and Resolution in TNT Formal Reasoning vs. Informal Reasoning Number Theorists Go out of Business Hilbert's Program 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 Mumon Has the Last Word À Levels of Description Chunking and Chess Skill Similar Levels Computer Systems Instructions and Data Superconductivity: A "Paradox" of Renormalization "Sealing-off" . . . . then, one by one, the four voices of the fugue chime in.) New Perspectives on Thought Intensionality and Extensionality The Brain's "Ants" Larger Structures in the Brain Mappings between Brains A "Grandmother Cell"? Classes and Instances The Prototype Principle 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 Minds and Thoughts CHAPTER XI11 Self-Awareness and Chaos Representability and Refrigerators Ganto's Ax in Metamathematics 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 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 XIV 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 The Last Straw TNT Is É-Incomplete Multifurcation The Passion According to Lucas Jumping Up a Dimension The Limits of Intelligent Systems There Is No Recursive Rule for Naming Ordinals. Self-Transcendence-A Modern Myth Advertisement and Framing Devices Simplicio, Salviati, Sagredo: Why Three? Zen and "Stepping Out" CHAPTER XVII 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 Parallel Progress in Al and Brain Simulation? Beauty, the Crab, and the Soul Irrational and Rational Can Coexist on Different Levels More Against Lucas An Underpinning of Al Church's Theorem Tarski's Theorem The Impossibility of the Magnificrab Two Types of Form Beauty, Truth, and Form The Neural Substrate of the Epimenides Paradox Turing The Turing Test Turing Anticipates Objections Samuel's Checker Program When Is a Program Original? Theorem Proving and Problem Reduction Shandy and the Bone Changing the Problem Space Applying Al to Mathematics The Crux of Al: 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 "Almost" Situations and Subjunctives Layers of Stability Frames and Nested Contexts 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 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 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 Bibliography INDEX
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