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Gdel, escher, Bach
Introduction to the 20th anniversary edition
[to do: insert outline]
Overview viii
list of illustrations xiv
words of thanks xix
part i: geb
introduction: a musico-logical offering 3
Author 3
Bach 3
Canons and Fugues 8
An Endlessly Rising Canon 10
Escher 10
Gdel 15
Mathematical Logic: A Synopsis 19
Banishing Strange Loops 21
Consistency, Completeness, Hilberts Program 23
Babbage, Computers, Artificial Intelligence 24
and Bach 27
Gdel, Escher, Bach 27
Three-Part Invention 29
chapter 1: the mu-puzzle 33
Formal Systems 33
Theorems, Axioms, Rules 35
Inside and Outside the System 36
Jumping out of the System 37
M-Mode, I-Mode, U-Mode 38
Decision Procedures 39
Two-Part Invention 43
CHAPTER 2: Meaning and form in mathematics 46
The pq-System 46
The Decision Procedure 47
Bottom-up vs. Top-down 48
Isomorphisms Induce Meaning 49
Meaningless and Meaningful Interpretations 51
Active vs. Passive Meanings 51
Double-Entendre! 52
Formal Systems and Reality 53
Mathematics and Symbol Manipulation 54
The Basic Laws of Arithmetic 55
Ideal Numbers 56
Euclids Proof 58
Getting Around Infinity 59
Sonata for Unaccompanied Achilles 61
CHAPTER 3: figure and ground 64
Primes vs. Composites 64
The tq-System 64
Capturing Compositeness 65
Illegally Characterizing Primes 66
Figure and Ground 67
Figure and Ground in Music 70
Recursively Enumerable Sets vs. Recursive Sets 71
Primes as Figure Rather than Ground 73
Contracrostipunctus 75
chapter 4: consistency, completeness, and geometry 82
Implicit and Explicit Meaning 82
Explicit Meaning of the Contracrostipunctus 82
Implicit Meanings of the Contracrostipunctus 84
Mapping Between the Contracrostipunctus and Gdels Theorem 85
The Art of the Fugue 86
Problems Caused by Gdels Result 86
The Modified pq-System and Inconsistency 87
Regaining Consistency 88
The History of Euclidean Geometry 88
The Many Faces of Noneuclid 91
Undefined Terms 92
The Possibility of Multiple Interpretations 94
Varieties of Consistency 94
Hypothetical Worlds and Consistency 95
Embedding of One Formal System in Another 97
Layers of Stability in Visual Perception 97
Is Mathematics the Same in Every Conceivable World? 99
Is Number Theory the Same in All Conceivable Worlds? 100
Completeness 100
How an Interpretation May Make or Break Completeness 102
Incompleteness of Formalized Number Theory 102
Little Harmonic Labyrinth 103
chapter 5: recursive structures and processes 127
What Is Recursion? 127
Pushing, Popping, and Stacks 128
Stacks in Music 129
Recursion in Language 130
Recursive Transition Networks 131
Bottoming Out and Heterarchies 133
Expanding Nodes 134
Diagram G and Recursive Sequences 135
A Chaotic Sequence 137
Two Striking Recursive Graphs 138
Recursion at the Lowest Level of Matter 142
Copies and Sameness 146
Programming and Recursion: Modularity, Loops, Procedures 149
Recursion in Chess Programs 150
Recursion and Unpredictability 152
Canon by Intervallic Augmentation 153
chapter 6: the location of meaning 158
When is One Thing Not Always the Same? 158
Information-Bearers and Information-Revealers 158
Genotype and Phenotype 159
Exotic and Prosaic Isomorphisms 159
Jukeboxes and Triggers 160
DNA and the Necessity of Chemical Context 161
An Unlikely UFO 162
Levels of Understanding of a Message 162
Imaginary Spacescape 163
The Heroic Decipherers 164
Three Layers of Any Message 166
Schrdingers Aperiodic Crystals 167
Languages for the Three Levels 167
The Jukebox Theory of Meaning 170
Against the Jukebox Theory 170
Meaning Is Intrinsic If Intelligence is Natural 171
Earth Chauvinism 171
Two Plaques in Space 173
Bach vs. Cage Again 174
How Universal Is DNAs Message? 175
Chromatic Fantasy, and Feud 177
chapter 7: the propositional calculus 181
Words and Symbols 181
Alphabet and First Rule of the Propositional Calculus 181
Well-Formed Strings 181
More Rules of Inference 183
The Fantasy Rule 183
Recursion and the Fantasy Rule 184
The Converse of the Fantasy Rule 185
The Intended Interpretation of the Symbols 186
Rounding Out the List of Rules 187
Justifying the Rules 188
Playing Around with the System 188
Semi-Interpretations 189
Gants Ax 189
Is There a Decision Procedure for Theorems? 190
Do We Know the System Is Consistent? 191
The Carroll Dialogue Again 192
Shortcuts and Derived Rules 193
Formalizing Higher Levels 194
Reflections on the Strengths and Weaknesses of the System 195
Proofs vs. Derivations 195
The Handling of Contradictions 196
Crab Canon 199
chapter 8: typographical number theory 204
The Crab Canon and Indirect Self-Reference 204
What We Want to Be Able to Express in TNT 204
Numerals 205
Variables and Terms 206
Atoms and Propositional Symbols 207
Free Variables and Quantifiers 207
Translating Our Sample Sentences 209
Tricks of the Trade 210
Translation Puzzles for You 212
How to Distinguish True from False? 213
The Rules of Well-Formedness 213
A Few More Translation Exercises 215
A Nontypographical System 215
The Five Axioms and First Rules of TNT 215
The Five Peano Postulates 216
New Rules of TNT: Specification and Generalization 217
The Existential Quantifier 218
Rules of Equality and Successorship 219
Illegal Shortcuts 220
Why Specification and Generalization Are Restricted 220
Something Is Missing 221
w-Incomplete Systems and Undecidable Strings 221
Non-Euclidean TNT 222
w-Inconsistency Is Not the Same as Inconsistency 223
The Last Rule 223
A Long Derivation 225
Tension and Resolution in TNT 227
Formal Reasoning vs. Informal Reasoning 228
Number Theorists Go out of Business 228
Hilberts Program 229
A Mu Offering 231
chapter 9: mumon and Gdel 246
What is Zen? 246
Zen Master Mumon 246
Zens Struggle Against Dualism 251
Ism, The Un-Mode, and Unmon 254
Zen and Tumbolia 255
Escher and Zen 255
Hemiolia and Escher 257
Indras Net 258
Mumon on MU 259
From Mumon to the MU-puzzle 259
Mumon Shows Us How to Solve the MU-puzzle 260
Gdel-Numbering the MIU-System 261
Seeing Things Both Typographically and Arithmetically 262
MIU-Producible Numbers 264
Answering Questions about Producible Numbers by Consulting TNT 265
The Dual Nature of MUMON 266
Codes and Implicit Meaning 267
The Boomerang: Gdel-Numbering TNT 267
TNT-Numbers: A Recursively Enumerable Set of Numbers 269
TNT Tries to Swallow Itself 270
G: A String Which Talks about Itself in Code 271
Gs Existence Is What Causes TNTs Incompleteness 271
Mumon Has the Last Word 272
part ii: egb
Prelude 275
chapter 10: levels of description, and computer systems 285
Levels of Description 285
Chunking and Chess Skill 285
Similar Levels 287
Computer Systems 287
Instructions and Data 289
Machine Language vs. Assembly language 290
Programs That Translate Programs 291
Higher-Level Languages, Compilers, and Interpreters 292
Bootstrapping 293
Levels on Which to Describe Running Programs 294
Microprogramming and Operating Systems 295
Cushioning the User and Protecting the System 296
Are Computers Super-Flexible or Super-Rigid? 297
Second-Guessing the Programmer 298
AI Advanced Are Language Advances 299
The Paranoid and the Operating System 300
The Border between Software and Hardware 301
Intermediate Levels and the Weather 302
From Tornados to Quarks 303
Superconductivity: A Paradox of Renormalization 304
Sealing-off 305
The Trade-off between Chunking and Determinism 306
Computers Can Only Do What You Tell Them to Do 306
Two Types of System 307
Epiphenomena 308
Mind vs. Brain 309
Ant Fugue 311
chapter 11: brains and thoughts 337
New Perspectives on Thought 337
Intensionality and Extensionality 337
The Brains Ants 339
Larger Structures in the Brain 340
Mappings between Brains 341
Localization of Brain Processes: An Enigma 342
Specificity in Visual Processing 343
A Grandmother Cell? 344
Funneling into Neural Modules 346
Modules Which Mediate Thought Processes 348
Active Symbols 349
Classes and Instances 351
The Prototype Principle 352
The Splitting-off of Instance from Classes 352
The Difficulty of Disentangling Symbols from Each Other 354
Symbols Software or Hardware? 356
Liftability of Intelligence 358
Can One Symbol Be Isolated? 359
The Symbols of Insects 360
Class Symbols and Imaginary Worlds 361
Intuitive Laws of Physics 362
Procedural and Declarative Knowledge 363
Visual Imagery 364
English French German Suite 366
Chapter 12: minds and thoughts 369
Can Minds Be Mapped onto Each Other? 369
Comparing Different Semantic Networks 371
Translations of Jabberwocky 372
ASUs 373
A Surprise Reversal 374
Centrality and Universality 374
How Much Do Language and Culture Channel Thought? 376
Trips and Itineraries in ASUs 377
Possible, Potential, and Preposterous Pathways 378
Different Styles of Translating Novels 379
High-Level Comparisons between Programs 380
High-Level Comparisons between Brains 382
Potential Beliefs, Potential Symbols 382
Where is the Sense of Self? 384
Subsystems 385
Subsystems and Shared Code 386
The Self-Symbol and Consciousness 387
Our First Encounter with Lucas 388
Aria with Diverse Variations 391
Chapter 13: Bloop and Floop and Gloop 406
Self-Awareness and Chaos 406
Representability and Refrigerators 406
Gants Ax in Metamathmatics 407
Finding Order by Choosing the Right Filter 407
Primordial Steps of the Language BlooP 409
Loops and Upper Bounds 410
Conventions of BlooP 410
IF-Statements and Branching 411
Automatic Chunking 412
BlooP Tests 413
BlooP Programs Contain Chains of Procedures 413
Suggested Exercises 415
Expressibility and Representability 417
Primitive Recursive Predicates Are Represented in TNT 417
Are There Functions Which Are Not Primitive Recursive? 418
Pool B, Index Numbers, and Blue Programs 418
The Diagonal Method 420
Cantors Original Diagonal Argument 421
What Does a Diagonal Argument Prove? 422
The Insidious Repeatability of the Diagonal Argument 423
From BlooP to FlooP 424
Terminating and Nonterminating FlooP Programs 425
Turings Trickery 425
A Termination Tester Would Be Magical 426
Pool F, Index Numbers, and Green Programs 427
The Termination Tester Gives Us Red Programs 427
GlooP 428
Is a Myth 428
The Church-Turing Thesis 429
Terminology: General and Partial Recursive 429
The Power of TNT 430
Air on Gs String 431
Chapter 14: On Formally Undecidable Propositions of TNT and Related Systems 438
The Two Ideas of the Oyster 438
The First Idea: Proof-Pairs 438
Proof-Pair-ness Is Primitive Recursive 440
And Is Therefore Represented in TNT 441
The Power of Proof-Pairs 441
Substitution Leads to the Second Idea 443
Arithmoquining 445
The Last Straw 446
TNT Says Uncle! 448
Yields Nontheoremhood When Arithmoquined 449
Gdel s Second Theorem 449
TNT Is w-Incomplete 450
Two Different Ways to Plug Up the Hole 451
Supernatural Numbers 452
Supernatural Theorems Have Infinitely Long Derivations 454
Supernatural Addition and Multiplication 455
Supernaturals Are Useful 455
But Are They Real? 455
Bifurcations in Geometry, and Physicists 456
Bifurcations in Number Theory, and Bankers 457
Bifurcations in Number Theory, and Metamathematicians 458
Hilberts Tenth Problem and the Tortoise 459
Birthday Cantatatata 461
Chapter 15: Jumping out of the System 465
A More Powerful Formal System 465
The Gdel Method Reapplied 466
Multifurcation 467
Essential Incompleteness 468
The Passion According to Lucas 471
Jumping Up a Dimension 473
The Limits of Intelligent Systems 475
There Is No Recursive Rule for Naming Ordinals 476
Other Refutations of Lucas 476
Self-Transcendence A Modern Myth 477
Advertisement and Framing Devices 478
Simplicio, Salviati, Sagredo: Why Three? 478
Zen and Stepping Out 479
Edifying Thoughts of a Tobacco Smoker 480
Chapter 16: Self-Ref and Self-Rep 495
Implicitly and Explicitly Self-Referential Sentences 495
A Self-Reproducing Program 498
What Is a Copy? 500
A Self-Reproducing Song 500
Epimenides Straddles the Channel 501
A Program That Prints Out Its Own Gdel Number 502
Gdelian Self-Reference 502
A Self-Rep by Augmentation 503
A Kimian Self-Rep 503
What Is the Original? 503
Typogenetics 504
Strands, Bases, Enzymes 505
Copy Mode and Double Strands 506
Amino Acids 508
Translation and the Typogenetic Code 509
Tertiary Structure of Enzymes 510
Punctuation, Genes, and Ribosomes 512
Puzzle: A Typogenetical Self-Rep 512
The Central Dogma of Typogenetics 513
Strange Loops, TNT, and Real Genetics 514
DNA and Nucleotides 514
Messenger RNA and Ribosomes 517
Amino Acids 518
Ribosomes and Tape Recorders 518
The Genetic Code 519
Tertiary Structure 519
Reductionistic Explanation of Protein Function 520
Transfer RNA and Ribosomes 522
Punctuation and the Reading Frame 524
Recap 525
Levels of Structure and Meaning in Proteins and Music 525
Polyribosomes and Two-Tiered Canons 527
Which Came First The Ribosome or the Protein? 528
Protein Function 528
Need for a Sufficiently Strong Support System 529
How DNA Self-Replicates 530
Comparison of DNAs Self-Rep Method with Quining 531
Levels of Meaning of DNA 531
The Central Dogmap 532
Strange Loops in the Central Dogmap 534
The Central Dogmap and the Contracrostipunctus 534
E. Coli vs. T4 537
A Molecular Trojan Horse 538
Recognition, Disguises, Labeling 540
Henkin Sentences and Viruses 541
Implicit vs. Explicit Henkin Sentences 542
Henkin Sentences and Self-Assembly 542
Two Outstanding Problems: Differentiation and Morphogenesis 543
Feedback and Feedforward 544
Repressors and Inducers 544
Feedback and Strange Loops Compared 545
Two Simple Examples of Differentiation 546
Level Mixing in the Cell 546
The Origin of Life 548
The Magnificrab, Indeed 549
Chapter 17: church, Turing, tarski, and others 559
Formal and Informal Systems 559
Intuition and the Magnificent Crab 560
The Church-Turing Thesis 561
The Public-Processes Version 562
Srinivasa Ramanujan 562
Idiots Savants 567
The Isomorphism Version of the Church-Turing Thesis 567
Representation of Knowledge about the Real World 569
Processes That Are Not So Skimmable 570
Articles of Reductionistic Faith 571
Partial Progress in AI and Brain Simulation? 572
Beauty, the Crab, and the Soul 573
Irrational and Rational Can Coexist on Different Levels 575
More Against Lucas 577
An Underpinning of AI 578
Churchs Theorem 579
Tarskis Theorem 580
The Impossibility of the Magnificrab 581
Two Types of Form 581
Meaning Derives from Connections to Cognitive Structures 582
Beauty, Truth, and Form 583
The Neural Substrate of the Epimenides Paradox 584
SHRDLU, Toy of Mans Designing 586
Chapter 18: Artificial Intelligence: Retrospects 594
Turing 594
The Turing Test 595
Turing Anticipates Objections 597
Parry Encounters the Doctor 599
A Brief History of AI 600
Mechanical Translation 603
Computer Chess 603
Samuels Checker Program 604
When Is a Program Original? 606
Who Composes Computer Music? 607
Theorem Proving and Problem Reduction 609
Shandy and the Bone 611
Changing the Problem Space 611
The I-Mode and the M-Mode Again 613
Applying AI to Mathematics 614
The Crux of AI: Representation of Knowledge 615
DNA and Proteins Help Give Some Perspective 616
Modularity of Knowledge 617
Representing Knowledge in a Logical Formalism 618
Deductive vs. Analogical Awareness 619
From Computer Haiku to an RTN-Grammar 619
From RTNs to ATNs 621
A Little Turing Test 621
Images of What Thought Is 623
Higher-Level Grammars 625
Grammars for Music? 626
Winograds Program SHRDLU 627
The Structure of SHRDLU 628
PLANNER Facilitates Problem Reduction 629
Syntax and Semantics 630
Contrafactus 633
Chapter 19: Artificial Intelligence: Prospects 641
Almost Situations and Subjunctives 641
Layers of Stability 643
Frames and Nested Contexts 644
Bongard Problems 646
Preprocessing Selects a Mini-Vocabulary 647
High-Level Descriptions 647
Templates and Sameness-Detectors 650
A Heterarchical Program 651
The Concept Network 653
Slippage and Tentativity 654
Meta-Descriptions 656
Flexibility is Important 657
Focusing and Filtering 657
Science and the World of Bongard Problems 659
Connections to Other Types of Thought 661
Message-Passing Languages, Frames, and Symbols 662
Enzymes and AI 663
Fission and Fusion 664
Epigenesis of the Crab Canon 665
Conceptual Skeletons and Conceptual Mapping 668
Recombinant Ideas 668
Abstractions, Skeletons, Analogies 669
Multiple Representations 670
Ports of Access 670
Forced Matching 671
Recap 672
Creativity and Randomness 673
Picking up Patterns on All Levels 674
The Flexibility of Language 674
Intelligence and Emotions 675
AI Has Far to Go 676
Ten Questions and Speculations 676
Sloth Canon 681
Chapter 20: Strange Loops, Or Tangled Hierarchies 684
Can Machines Possess Originality? 684
Below Every Tangled Hierarchy Lies An Inviolate Level 686
A Self-Modifying Game 687
The Authorship Triangle Again 688
Eschers Drawing Hands 689
Brain and Mind: A Neural Tangle Supporting a Symbol Tangle 691
Strange Loops in Government 692
Tangles Involving Science and the Occult 693
The Nature of Evidence 694
Seeing Oneself 695
Gdels Theorem and Other Disciplines 696
Introspection and Insanity: A Gdelian Problem 696
Can We Understand Our Own Minds or Brains? 697
Gdels Theorem and Personal Nonexistence 698
Science and Dualism 698
Symbol vs. Object in Modern Music and Art 699
Magrittes Semantic Illusions 700
The Code of Modern Art 703
Ism Once Again 704
Understanding the Mind 706
Accidental Inexplicability of Intelligence? 707
Undecidability Is Inseparable from a High-Level Viewpoint 707
Consciousness as an Intrinsic^v
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Strange Loops as the Crux of Consciousness 709
The Self-Symbol and Free Will 710
A Gdel Vortex Where All Levels Cross 713
An Escher Vortex Where All Levels Cross 715
A Bach Vortex Where All Levels Cross 717
Six-Part Ricercar 720
Notes 743
Bibliography 746
Credits 757
Index 759
About the Author 778
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