From Pythagoras to the Digital Computer: The Intellectual Roots of Symbolic Artificial Intelligence

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1 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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3 Contents Preface 2 1 Overview Method of Presentation Artificial Intelligence and Cognitive Science The Theory of Computation Themes in the History of Knowledge Alternative Views of Cognition Connectionism The Continuous and the Discrete Word Magic Pythagoras: Rationality & the Limited Discovery of the Musical Scale The Rational The Definite and the Indefinite The Discovery of the Irrational Arithmetic vs. Geometry Zeno: Paradoxes of the Continuous & Discrete Importance of the Paradoxes Paradoxes of Plurality Paradoxes of Motion Summary Socrates and Plato: Definition & Categories Background Method of Definition Knowledge vs. Right Opinion The Platonic Forms

4 6 CONTENTS Summary: Socrates and Plato Aristotle: Formal Logic Background Structure of Theoretical Knowledge Primary Truths Formal Logic Epistemological Implications Euclid: Axiomatization of Continuous & Discrete Background Axiomatic Structure Theory of Magnitudes Summary Words and Images Hellenistic Logic Modal Logic Propositional Logic Logical Paradoxes Medieval Logic Debate about Universals Language of Logic Combining Images and Letters The Art of Memory Combinatorial Inference Kabbalah Lull: Mechanical Reasoning Background Ars Magna From Images to Symbols Significance Ramus and the Art of Memory Thought as Computation Hobbes: Reasoning as Computation Wilkins: Ideal Languages Leibniz: Calculi and Knowledge Representation Chinese and Hebrew Characters Knowledge Representation

5 CONTENTS Computational Approach to Inference Epistemological Implications Boole: Symbolic Logic Background Class Logic Propositional Logic Probabilistic Logic Summary Jevons: Logic Machines Combinatorial Logic Logic Machines Discussion The Arithmetization of Geometry Descartes: Geometry and Algebra Arabian Mathematics Analytic Geometry Algebra and the Number System The Importance of Informality Magic and the New Science Pythagorean Neoplatonism Hermeticism Alchemy Renaissance Magic and Science The Witches Holocaust Belief and the Practice of Science Reduction of Continuous to Discrete The Problem of Motion Berkeley: Critique of Infinitesmals The Rational Numbers Plato: The Monad and the Dyad Cantor: The Real Line Infinities and Infinitesmals Summary Technical Progress Psychology and Sociology of Science

6 8 CONTENTS 6 Theory of Computation Philosophies of Mathematics Peano and the Move Toward Rigor Logicism Intuitionism Formalism Interpretations and Models Metamathematics Models of Computation Introduction Markov s Normal Algorithms General Assumptions of the Discrete Invariants of Calculi Atomic Constituents Texts and Schemata Processes Formal Semantics Limitations of the Discrete Undecidable Propositions Gödel s Incompleteness Theorem Corollaries to Gödel s Theorem The Undecidable and the Uncomputable Introduction Undecidability of the Halting Problem General Undecidability The Löwenheim-Skolem Theorem Background The Theorem Epistemological Implications Limitations of the Discrete Transcending the Discrete Logical Positivism Historical Background Logical Atomism Russell: Knowledge by Acquaintance and Description Wittgenstein: The Tractatus

7 CONTENTS The Vienna Circle and Verifiability Background Verifiability and Meaning Reductionism and the Unity of Science The Collapse of Logical Positivism Nature of Atomic Facts Verifiability Problems External Criticism Summary Influence of Logical Positivism Quantum Mechanics Behaviorism The Turing Test Conclusions The Computational Theory of Mind Historical Background Gestation Birth Growth Characteristics of Cognitive Science Chomsky: Formal Linguistics Phrase-Structure Grammars Transformational Grammars Government-Binding Theory The Problem of Language Acquisition Symbolic Knowledge Representation Newell & Simon: The Physical Symbol System Hypothesis Schank: Conceptual Dependency Theory Abelson: Scripts Expert Systems The Frame Problem BIBLIOGRAPHY 341

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