Main Page
From Geb
(Difference between revisions)
| Line 3: | Line 3: | ||
This wiki is for conversations and thoughts on the book. This is the | This wiki is for conversations and thoughts on the book. This is the | ||
| - | + | ||
| - | + | ==[[PART I: GEB]]== | |
| - | *[[INTRODUCTION: A MUSICO-LOGICAL OFFERING]] | + | *[[INTRODUCTION: A MUSICO-LOGICAL OFFERING]]== |
| - | + | ||
*[[Author]] | *[[Author]] | ||
*[[Bach]] | *[[Bach]] | ||
| Line 19: | Line 18: | ||
*[[...and Bach]] | *[[...and Bach]] | ||
*[[Gödel Escher Bach]] | *[[Gödel Escher Bach]] | ||
| - | *[[ | + | *[[Three-Part Invention]] |
| - | + | ==[[CHAPTER 1: THE MU-PUZZLE]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[Formal Systems]] | *[[Formal Systems]] | ||
*[[Theorems, Axioms, Rules]] | *[[Theorems, Axioms, Rules]] | ||
| Line 30: | Line 26: | ||
*[[M-Mode, I-Mode, U-Mode]] | *[[M-Mode, I-Mode, U-Mode]] | ||
*[[Decision Procedures]] | *[[Decision Procedures]] | ||
| - | *[[ | + | *[[Two-Part Invention]] |
| - | + | ==[[CHAPTER 2: MEANING AND FORM IN MATHEMATICS]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[The pq-System]] | *[[The pq-System]] | ||
*[[The Decision Procedure]] | *[[The Decision Procedure]] | ||
| Line 48: | Line 41: | ||
*[[Euclid's Proof]] | *[[Euclid's Proof]] | ||
*[[Getting Around Infinity]] | *[[Getting Around Infinity]] | ||
| - | *[[ | + | *[[Sonata for Unaccompanied Achilles]] |
| - | + | ==[[CHAPTER 3: FIGURE AND GROUND]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[Primes vs. Composites]] | *[[Primes vs. Composites]] | ||
*[[The tq-System]] | *[[The tq-System]] | ||
| Line 61: | Line 51: | ||
*[[Recursively Enumerable Sets vs. Recursive Sets]] | *[[Recursively Enumerable Sets vs. Recursive Sets]] | ||
*[[Primes as Figure Rather than Ground]] | *[[Primes as Figure Rather than Ground]] | ||
| - | *[[ | + | *[[Contracrostipunctus]] |
| - | + | ==[[CHAPTER 4: CONSISTENCY, COMPLETENESS, AND GEOMETRY]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[Implicit and Explicit Meaning]] | *[[Implicit and Explicit Meaning]] | ||
*[[Explicit Meaning of the Contracrostipunctus]] | *[[Explicit Meaning of the Contracrostipunctus]] | ||
| Line 87: | Line 74: | ||
*[[How an Interpretation May Make or Break Completeness]] | *[[How an Interpretation May Make or Break Completeness]] | ||
*[[Incompleteness of Formalized Number Theory]] | *[[Incompleteness of Formalized Number Theory]] | ||
| - | *[[ | + | *[[Little Harmonic Labyrinth]] |
| - | + | ==[[CHAPTER 5: RECURSIVE STRUCTURES AND PROCESSES]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[What Is Recursion?]] | *[[What Is Recursion?]] | ||
*[[Pushing, Popping, and Stacks]] | *[[Pushing, Popping, and Stacks]] | ||
| Line 107: | Line 91: | ||
*[[Recursion in Chess Programs]] | *[[Recursion in Chess Programs]] | ||
*[[Recursion and Unpredictability]] | *[[Recursion and Unpredictability]] | ||
| - | *[[ | + | *[[Canon by Intervallic Augmentation]] |
| - | + | ==[[CHAPTER 6: THE LOCATION OF MEANING]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[When is One Thing Not Always the Same?]] | *[[When is One Thing Not Always the Same?]] | ||
*[[Information-Bearers and Information-Revealers]] | *[[Information-Bearers and Information-Revealers]] | ||
| Line 120: | Line 101: | ||
*[[An Unlikely UFO]] | *[[An Unlikely UFO]] | ||
*[[Levels of Understanding of a Message]] | *[[Levels of Understanding of a Message]] | ||
| - | *[[Imaginary Spacescape | + | *[[Imaginary Spacescape]] |
*[[The Heroic Decipherers]] | *[[The Heroic Decipherers]] | ||
*[[Three Layers of Any Message]] | *[[Three Layers of Any Message]] | ||
| Line 132: | Line 113: | ||
*[[Bach vs. Cage Again]] | *[[Bach vs. Cage Again]] | ||
*[[How Universal Is DNA's Message?]] | *[[How Universal Is DNA's Message?]] | ||
| - | *[[ | + | *[[Chromatic Fantasy, and Feud]] |
| - | + | ==[[CHAPTER 7: THE PROPOSITIONAL CALCULUS]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[Words and Symbols]] | *[[Words and Symbols]] | ||
*[[Alphabet and First Rule of the Propositional Calculus]] | *[[Alphabet and First Rule of the Propositional Calculus]] | ||
| Line 158: | Line 136: | ||
*[[Proofs vs. Derivations]] | *[[Proofs vs. Derivations]] | ||
*[[The Handling of Contradictions]] | *[[The Handling of Contradictions]] | ||
| - | *[[ | + | *[[Crab Canon]] |
| - | + | ==[[CHAPTER 8: TYPOGRAPHICAL NUMBER THEORY]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[The Crab Canon and Indirect Self-Reference]] | *[[The Crab Canon and Indirect Self-Reference]] | ||
*[[What We Want to Be Able to Express in TNT]] | *[[What We Want to Be Able to Express in TNT]] | ||
| Line 193: | Line 168: | ||
*[[Number Theorists Go out of Business]] | *[[Number Theorists Go out of Business]] | ||
*[[Hilbert's Program]] | *[[Hilbert's Program]] | ||
| - | *[[ | + | *[[A Mu Offering]] |
| - | + | ==[[CHAPTER 9: MUMON AND GÖDEL]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[What is Zen?]] | *[[What is Zen?]] | ||
*[[Zen Master Mumon]] | *[[Zen Master Mumon]] | ||
| Line 220: | Line 192: | ||
*[[G: A String Which Talks about Itself in Code]] | *[[G: A String Which Talks about Itself in Code]] | ||
*[[G's Existence Is What Causes TNT's Incompleteness]] | *[[G's Existence Is What Causes TNT's Incompleteness]] | ||
| - | *[[Mumon Has the Last Word]] | + | *[[Mumon Has the Last Word]]== |
| - | + | =[[PART II: EGB]]= | |
| - | + | *[[Prelude...]] | |
| - | *[[ | + | ==[[CHAPTER 10: LEVELS OF DESCRIPTION, AND COMPUTER SYSTEMS]]== |
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
*[[Levels of Description]] | *[[Levels of Description]] | ||
*[[Chunking and Chess Skill]] | *[[Chunking and Chess Skill]] | ||
| Line 248: | Line 216: | ||
*[[From Tornados to Quarks]] | *[[From Tornados to Quarks]] | ||
*[[Superconductivity: A Paradox" of Renormalization"]] | *[[Superconductivity: A Paradox" of Renormalization"]] | ||
| - | *[[Sealing-off | + | *[[Sealing-off]] |
*[[The Trade-off between Chunking and Determinism]] | *[[The Trade-off between Chunking and Determinism]] | ||
| - | *[[Computers Can Only Do What You Tell Them to Do | + | *[[Computers Can Only Do What You Tell Them to Do]] |
*[[Two Types of System]] | *[[Two Types of System]] | ||
*[[Epiphenomena]] | *[[Epiphenomena]] | ||
*[[Mind vs. Brain]] | *[[Mind vs. Brain]] | ||
| - | *[[ | + | *[[...Ant Fugue]] |
| - | + | ==[[CHAPTER 11: BRAINS AND THOUGHTS]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[New Perspectives on Thought]] | *[[New Perspectives on Thought]] | ||
*[[Intensionality and Extensionality]] | *[[Intensionality and Extensionality]] | ||
| - | *[[The Brain's Ants | + | *[[The Brain's Ants]] |
*[[Larger Structures in the Brain]] | *[[Larger Structures in the Brain]] | ||
*[[Mappings between Brains]] | *[[Mappings between Brains]] | ||
| Line 282: | Line 247: | ||
*[[Procedural and Declarative Knowledge]] | *[[Procedural and Declarative Knowledge]] | ||
*[[Visual Imagery]] | *[[Visual Imagery]] | ||
| - | *[[ | + | *[[English French German Suite]] |
| - | + | ==[[CHAPTER 12: MINDS AND THOUGHTS]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[Can Minds Be Mapped onto Each Other?]] | *[[Can Minds Be Mapped onto Each Other?]] | ||
*[[Comparing Different Semantic Networks]] | *[[Comparing Different Semantic Networks]] | ||
| - | *[[Translations of Jabberwocky | + | *[[Translations of Jabberwocky]] |
*[[ASU's]] | *[[ASU's]] | ||
*[[A Surprise Reversal]] | *[[A Surprise Reversal]] | ||
| Line 305: | Line 267: | ||
*[[The Self-Symbol and Consciousness]] | *[[The Self-Symbol and Consciousness]] | ||
*[[Our First Encounter with Lucas]] | *[[Our First Encounter with Lucas]] | ||
| - | *[[ | + | *[[Aria with Diverse Variations]] |
| - | + | ==[[CHAPTER 13: BlooP AND FlooP AND GlooP]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[Self-Awareness and Chaos]] | *[[Self-Awareness and Chaos]] | ||
*[[Representability and Refrigerators]] | *[[Representability and Refrigerators]] | ||
| Line 341: | Line 300: | ||
*[[Terminology: General and Partial Recursive]] | *[[Terminology: General and Partial Recursive]] | ||
*[[The Power of TNT]] | *[[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 Two Ideas of the Oyster | + | |
*[[The First Idea: Proof-Pairs]] | *[[The First Idea: Proof-Pairs]] | ||
*[[Proof-Pair-ness Is Primitive Recursive...]] | *[[Proof-Pair-ness Is Primitive Recursive...]] | ||
| Line 354: | Line 310: | ||
*[[Arithmoquining]] | *[[Arithmoquining]] | ||
*[[The Last Straw]] | *[[The Last Straw]] | ||
| - | *[[TNT Says Uncle! | + | *[[TNT Says Uncle!]] |
| - | *[[Yields Nontheoremhood When Arithmoquined | + | *[[Yields Nontheoremhood When Arithmoquined]] |
*[[Gödel's Second Theorem]] | *[[Gödel's Second Theorem]] | ||
*[[TNT Is [w]-Incomplete]] | *[[TNT Is [w]-Incomplete]] | ||
| Line 368: | Line 324: | ||
*[[Bifurcations in Number Theory, and Metamathematicians]] | *[[Bifurcations in Number Theory, and Metamathematicians]] | ||
*[[Hilbert's Tenth Problem and the Tortoise]] | *[[Hilbert's Tenth Problem and the Tortoise]] | ||
| - | *[[ | + | *[[Birthday Cantatatata...]] |
| - | + | ==[[CHAPTER 15: JUMPING OUT OF THE SYSTEM]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[A More Powerful Formal System]] | *[[A More Powerful Formal System]] | ||
*[[The Gödel Method Reapplied]] | *[[The Gödel Method Reapplied]] | ||
| Line 385: | Line 338: | ||
*[[Advertisement and Framing Devices]] | *[[Advertisement and Framing Devices]] | ||
*[[Simplicio, Salviati, Sagredo: Why Three?]] | *[[Simplicio, Salviati, Sagredo: Why Three?]] | ||
| - | *[[Zen and Stepping Out | + | *[[Zen and Stepping Out]] |
| - | *[[ | + | *[[Edifying Thoughts of a Tobacco Smoker]] |
| - | + | ==[[CHAPTER 16: SELF-REF AND SELF-REP]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[Implicitly and Explicitly Self-Referential Sentences]] | *[[Implicitly and Explicitly Self-Referential Sentences]] | ||
*[[A Self-Reproducing Program]] | *[[A Self-Reproducing Program]] | ||
| Line 445: | Line 395: | ||
*[[Level Mixing in the Cell]] | *[[Level Mixing in the Cell]] | ||
*[[The Origin of Life]] | *[[The Origin of Life]] | ||
| - | *[[ | + | *[[The Magnificrab, Indeed]] |
| - | + | ==[[CHAPTER 17: CHURCH, TURING, TARSKI, AND OTHERS]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[Formal and Informal Systems]] | *[[Formal and Informal Systems]] | ||
*[[Intuition and the Magnificent Crab]] | *[[Intuition and the Magnificent Crab]] | ||
| Line 455: | Line 402: | ||
*[[The Public-Processes Version]] | *[[The Public-Processes Version]] | ||
*[[Srinivasa Ramanujan]] | *[[Srinivasa Ramanujan]] | ||
| - | *[[Idiots Savants | + | *[[Idiots Savants]] |
*[[The Isomorphism Version of the Church-Turing Thesis]] | *[[The Isomorphism Version of the Church-Turing Thesis]] | ||
*[[Representation of Knowledge about the Real World]] | *[[Representation of Knowledge about the Real World]] | ||
| Line 472: | Line 419: | ||
*[[Beauty, Truth, and Form]] | *[[Beauty, Truth, and Form]] | ||
*[[The Neural Substrate of the Epimenides Paradox]] | *[[The Neural Substrate of the Epimenides Paradox]] | ||
| - | *[[ | + | *[[SHRDLU, Toy of Man's Designing]] |
| - | + | ==[[CHAPTER 18: ARTIFICIAL INTELLIGENCE: RETROSPECTS]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[Turing]] | *[[Turing]] | ||
*[[The Turing Test]] | *[[The Turing Test]] | ||
*[[Turing Anticipates Objections]] | *[[Turing Anticipates Objections]] | ||
| - | *[[Parry Encounters the Doctor | + | *[[Parry Encounters the Doctor]] |
*[[A Brief History of AI]] | *[[A Brief History of AI]] | ||
*[[Mechanical Translation]] | *[[Mechanical Translation]] | ||
| Line 507: | Line 451: | ||
*[[PLANNER Facilitates Problem Reduction]] | *[[PLANNER Facilitates Problem Reduction]] | ||
*[[Syntax and Semantics]] | *[[Syntax and Semantics]] | ||
| - | *[[ | + | *[[Contrafactus]] |
| - | + | ==[[CHAPTER 19: ARTIFICIAL INTELLIGENCE: PROSPECTS]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[Almost" Situations and Subjunctives"]] | *[[Almost" Situations and Subjunctives"]] | ||
*[[Layers of Stability]] | *[[Layers of Stability]] | ||
| Line 544: | Line 485: | ||
*[[AI Has Far to Go]] | *[[AI Has Far to Go]] | ||
*[[Ten Questions and Speculations]] | *[[Ten Questions and Speculations]] | ||
| - | *[[ | + | *[[Sloth Canon]] |
| - | + | ==[[CHAPTER 20: STRANGE LOOPS, OR TANGLED HIERARCHIES]]== | |
| - | + | ||
| - | + | ||
| - | + | ||
*[[Can Machines Possess Originality?]] | *[[Can Machines Possess Originality?]] | ||
*[[Below Every Tangled Hierarchy Lies An Inviolate Level]] | *[[Below Every Tangled Hierarchy Lies An Inviolate Level]] | ||
Revision as of 17:57, 6 February 2006
Gödel, Escher, Bach - An Eternal Golden Braid
This wiki is for conversations and thoughts on the book. This is the
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
- [[[w]-Incomplete Systems and Undecidable Strings]]
- Non-Euclidean TNT
- [[[w]-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 [w]-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
