From Pythagoras to the Digital Computer: The Intellectual Roots of Symbolic Artificial Intelligence Volume I of Word and Flux: The Discrete and the Continuous In Computation, Philosophy, and Psychology Bruce MacLennan Copyright c 2006 Draft of June 12, 2009
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Contents Preface 2 1 Overview 3 1.1 Method of Presentation...................... 3 1.2 Artificial Intelligence and Cognitive Science.......... 5 1.3 The Theory of Computation................... 5 1.4 Themes in the History of Knowledge.............. 7 1.5 Alternative Views of Cognition................. 9 1.6 Connectionism.......................... 10 2 The Continuous and the Discrete 13 2.1 Word Magic............................ 13 2.2 Pythagoras: Rationality & the Limited............. 15 2.2.1 Discovery of the Musical Scale.............. 16 2.2.2 The Rational....................... 18 2.2.3 The Definite and the Indefinite............. 25 2.2.4 The Discovery of the Irrational............. 29 2.2.5 Arithmetic vs. Geometry................. 31 2.3 Zeno: Paradoxes of the Continuous & Discrete......... 32 2.3.1 Importance of the Paradoxes............... 32 2.3.2 Paradoxes of Plurality.................. 33 2.3.3 Paradoxes of Motion................... 35 2.3.4 Summary......................... 39 2.4 Socrates and Plato: Definition & Categories.......... 39 2.4.1 Background........................ 39 2.4.2 Method of Definition................... 40 2.4.3 Knowledge vs. Right Opinion.............. 41 2.4.4 The Platonic Forms.................... 41 5
6 CONTENTS 2.4.5 Summary: Socrates and Plato.............. 46 2.5 Aristotle: Formal Logic...................... 46 2.5.1 Background........................ 46 2.5.2 Structure of Theoretical Knowledge........... 46 2.5.3 Primary Truths...................... 47 2.5.4 Formal Logic....................... 50 2.5.5 Epistemological Implications............... 51 2.6 Euclid: Axiomatization of Continuous & Discrete....... 52 2.6.1 Background........................ 52 2.6.2 Axiomatic Structure................... 53 2.6.3 Theory of Magnitudes.................. 54 2.6.4 Summary......................... 56 3 Words and Images 59 3.1 Hellenistic Logic.......................... 59 3.1.1 Modal Logic........................ 60 3.1.2 Propositional Logic.................... 60 3.1.3 Logical Paradoxes..................... 62 3.2 Medieval Logic.......................... 63 3.2.1 Debate about Universals................. 64 3.2.2 Language of Logic.................... 67 3.3 Combining Images and Letters.................. 75 3.3.1 The Art of Memory.................... 75 3.3.2 Combinatorial Inference................. 77 3.3.3 Kabbalah......................... 78 3.4 Lull: Mechanical Reasoning................... 81 3.4.1 Background........................ 83 3.4.2 Ars Magna........................ 83 3.4.3 From Images to Symbols................. 93 3.4.4 Significance........................ 94 3.4.5 Ramus and the Art of Memory............. 95 4 Thought as Computation 99 4.1 Hobbes: Reasoning as Computation............... 99 4.2 Wilkins: Ideal Languages..................... 103 4.3 Leibniz: Calculi and Knowledge Representation........ 112 4.3.1 Chinese and Hebrew Characters............. 112 4.3.2 Knowledge Representation................ 115
CONTENTS 7 4.3.3 Computational Approach to Inference.......... 118 4.3.4 Epistemological Implications............... 122 4.4 Boole: Symbolic Logic...................... 123 4.4.1 Background........................ 124 4.4.2 Class Logic........................ 125 4.4.3 Propositional Logic.................... 130 4.4.4 Probabilistic Logic.................... 131 4.4.5 Summary......................... 132 4.5 Jevons: Logic Machines...................... 134 4.5.1 Combinatorial Logic................... 134 4.5.2 Logic Machines...................... 138 4.5.3 Discussion......................... 144 5 The Arithmetization of Geometry 147 5.1 Descartes: Geometry and Algebra................ 147 5.1.1 Arabian Mathematics................... 147 5.1.2 Analytic Geometry.................... 152 5.1.3 Algebra and the Number System............ 158 5.1.4 The Importance of Informality.............. 161 5.2 Magic and the New Science................... 162 5.2.1 Pythagorean Neoplatonism................ 162 5.2.2 Hermeticism........................ 164 5.2.3 Alchemy.......................... 168 5.2.4 Renaissance Magic and Science............. 172 5.2.5 The Witches Holocaust................. 174 5.2.6 Belief and the Practice of Science............ 176 5.3 Reduction of Continuous to Discrete.............. 180 5.3.1 The Problem of Motion................. 180 5.3.2 Berkeley: Critique of Infinitesmals............ 186 5.3.3 The Rational Numbers.................. 195 5.3.4 Plato: The Monad and the Dyad............ 199 5.3.5 Cantor: The Real Line.................. 204 5.3.6 Infinities and Infinitesmals................ 207 5.4 Summary............................. 211 5.4.1 Technical Progress.................... 211 5.4.2 Psychology and Sociology of Science.......... 212
8 CONTENTS 6 Theory of Computation 215 6.1 Philosophies of Mathematics................... 215 6.1.1 Peano and the Move Toward Rigor........... 215 6.1.2 Logicism.......................... 219 6.1.3 Intuitionism........................ 224 6.1.4 Formalism......................... 231 6.2 Interpretations and Models.................... 232 6.3 Metamathematics......................... 237 6.4 Models of Computation...................... 241 6.4.1 Introduction........................ 241 6.4.2 Markov s Normal Algorithms.............. 243 6.5 General Assumptions of the Discrete.............. 256 6.5.1 Invariants of Calculi................... 256 6.5.2 Atomic Constituents................... 258 6.5.3 Texts and Schemata................... 260 6.5.4 Processes......................... 262 6.5.5 Formal Semantics..................... 265 7 Limitations of the Discrete 269 7.1 Undecidable Propositions..................... 270 7.1.1 Gödel s Incompleteness Theorem............ 270 7.1.2 Corollaries to Gödel s Theorem............. 274 7.2 The Undecidable and the Uncomputable............ 284 7.2.1 Introduction........................ 284 7.2.2 Undecidability of the Halting Problem......... 285 7.2.3 General Undecidability.................. 286 7.3 The Löwenheim-Skolem Theorem................ 291 7.3.1 Background........................ 291 7.3.2 The Theorem....................... 292 7.4 Epistemological Implications................... 295 7.4.1 Limitations of the Discrete................ 296 7.4.2 Transcending the Discrete................ 297 8 Logical Positivism 303 8.1 Historical Background...................... 303 8.2 Logical Atomism......................... 306 8.2.1 Russell: Knowledge by Acquaintance and Description. 306 8.2.2 Wittgenstein: The Tractatus............... 308
CONTENTS 1 8.3 The Vienna Circle and Verifiability............... 311 8.3.1 Background........................ 311 8.3.2 Verifiability and Meaning................ 311 8.3.3 Reductionism and the Unity of Science......... 314 8.4 The Collapse of Logical Positivism............... 315 8.4.1 Nature of Atomic Facts.................. 315 8.4.2 Verifiability Problems................... 317 8.4.3 External Criticism.................... 319 8.4.4 Summary......................... 320 8.5 Influence of Logical Positivism.................. 321 8.5.1 Quantum Mechanics................... 321 8.5.2 Behaviorism........................ 323 8.5.3 The Turing Test...................... 326 8.5.4 Conclusions........................ 328 9 The Computational Theory of Mind 331 9.1 Historical Background...................... 331 9.1.1 Gestation......................... 331 9.1.2 Birth............................ 334 9.1.3 Growth.......................... 336 9.1.4 Characteristics of Cognitive Science........... 337 9.2 Chomsky: Formal Linguistics.................. 340 9.2.1 Phrase-Structure Grammars............... 340 9.2.2 Transformational Grammars............... 340 9.2.3 Government-Binding Theory............... 340 9.2.4 The Problem of Language Acquisition......... 340 9.3 Symbolic Knowledge Representation.............. 340 9.3.1 Newell & Simon: The Physical Symbol System Hypothesis.......................... 340 9.3.2 Schank: Conceptual Dependency Theory........ 340 9.3.3 Abelson: Scripts..................... 340 9.3.4 Expert Systems...................... 340 9.3.5 The Frame Problem................... 340 BIBLIOGRAPHY 341