Word and Flux The Discrete and the Continuous In Computation, Philosophy, and Psychology Volume I From Pythagoras to the Digital Computer The Intellectual Roots of Symbolic Artificial Intelligence with a Summary of Volume II Continuous Theories of Knowledge Bruce J. MacLennan Copyright c 2021 (version of January 2, 2021) This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/4.0/. 2 3 Preface Things, then, both those properly so called and those that sim- ply have the name, are some of them unified and continuous, for example, an animal, the universe, a tree, and the like, which are properly and peculiarly called \magnitudes"; others are discon- tinuous, in a side by side arrangement, and, as it were, in heaps, which are called \multitudes," a flock, for instance, a people, a heap, a chorus, and the like. Wisdom, then, must be considered to be the knowledge of these two forms. | Nicomachus of Gerasa (c. 60 { c. 120 CE) We live at the end of an era. For 2500 years philosophy and science have been dominated by the view that knowledge can be represented in discrete symbol structures and that thought is the formal manipulation of these structures. This view has been so pervasive that it has rarely been recognized as an assumption worthy of criticism. Its dominance has been strengthened by the fact that the occasional criticisms of it have seemed too indefinite to merit scientific consideration. Nevertheless, as a consequence of its wide acceptance, the implications of the traditional view have been extensively explored, so that now its weaknesses have become apparent. Crucial to this process has been a technological development: the digital computer, for the digital computer is the formal symbol manipulator par excellence. Therefore the availability of high-speed computers has permitted the empirical investigation of the traditional view of knowledge; in particular, the modest success, until recently, of artificial intelligence has exposed the limitations of the traditional view. Throughout its 2500 year history the tradition has had its dissenters, but there are two reasons that it has not had significant competition until now. First, the tradition had to be worked through to its conclusion, which has cul- minated in the theory of computation and \symbolic" artificial intelligence. Second, the alternative views seemed inherently nonscientific, and so they have been generally unacceptable to our increasingly scientific culture. This situation changed abruptly in the 1980s with the emergence of connectionism and neural network theory, which provide the basis for a scientific account of 4 an alternative theory of knowledge. Thus two related events, the final work- ing out of the traditional theory and the emergence of a scientific alternative, have combined to precipitate a revolution in the theory of knowledge. It is perhaps the most significant change in our understanding of the fabric of knowledge in two and a half millennia; we indeed live at a historic moment. This book and its intended companion are simply the detailed presenta- tion and justification of the preceding claims. They originated in a graduate course, \Epistemology for Computer Scientists," which I taught in the mid to late 1980s to address a general lack of up-to-date philosophical knowledge among AI researchers. In the early 1990s this developed into courses and seminars intended to explain the radical reorganization of epistemology im- plied by connectionism and artificial neural network approaches to cognitive science and AI. The title of the two volumes, Word and Flux, derives from a fundamen- tal tension between discrete and continuous phenomena that has permeated our culture's view of knowledge from its earliest attempts at philosophical thinking. This first volume is divided in two parts. Part I traces the tradi- tional view of knowledge from its origin in ancient Pythagoreanism, where we first find the attempt to reduce continua to discrete symbol structures | to arithmetize geometry. The issue of this attempt includes formal logic and the notion that thought is (digital) computation. Part II investigates the acceleration of this process in the nineteenth and twentieth centuries, includ- ing the apparently successful arithmetization of geometry, the attempts to formalize mathematics and science, and the computational theories of mind that have dominated cognitive science and artificial intelligence. This ac- celeration also brought with it the first signs of weakness in the traditional view through the discovery, in the first half of the twentieth century, of the theoretical limitations of discrete symbol systems. Volume II, which was never completed, intended to explore the history of an alternative view of knowledge.1 The criticisms that have been made against the tradition over the centuries provide the insight necessary to see both the weaknesses of the tradition and the requirements for an alternative. Volume II was also intended to include a systematic presentation of the al- ternative | at least as systematic as was possible at that early date. The foundation is provided by the theory of artificial neural networks and mas- sively parallel analog computation. Volume II would have concluded with a 1See Chapter 11 for a detailed outline of the unwritten chapters of Volume II. 5 discussion of the implications of this theory for our understanding of knowl- edge, in general, and for our understanding of the mind and of science, in particular. I regret that other research and writing activities distracted me from completing Volume II, but perhaps the completed part of Chapter 10 and the detailed outline in Chapter 11, which together constitute Part III of this book, will compensate to some degree. Ars longa, vita brevis! Bruce MacLennan Douglas Lake Tennessee 6 Contents Preface 3 I The Archaeology of Computation 3 1 Overview 5 1.1 Method of Presentation . 5 1.2 Artificial Intelligence and Cognitive Science . 7 1.3 The Theory of Computation . 7 1.4 Themes in the History of Knowledge . 9 1.5 Alternative Views of Cognition . 11 1.6 Connectionism . 12 2 The Continuous and the Discrete 15 2.1 Word Magic . 15 2.2 Pythagoras: Rationality & the Limited . 17 2.2.1 Discovery of the Musical Scale . 18 2.2.2 The Rational . 20 2.2.3 The Definite and the Indefinite . 27 2.2.4 The Discovery of the Irrational . 31 2.2.5 Arithmetic vs. Geometry . 33 2.3 Zeno: Paradoxes of the Continuous & Discrete . 34 2.3.1 Importance of the Paradoxes . 34 2.3.2 Paradoxes of Plurality . 35 2.3.3 Paradoxes of Motion . 37 2.3.4 Summary . 41 2.4 Socrates and Plato: Definition & Categories . 41 2.4.1 Background . 41 7 8 CONTENTS 2.4.2 Method of Definition . 42 2.4.3 Knowledge vs. Right Opinion . 43 2.4.4 The Platonic Forms . 43 2.4.5 Summary: Socrates and Plato . 48 2.5 Aristotle: Formal Logic . 48 2.5.1 Background . 48 2.5.2 Structure of Theoretical Knowledge . 48 2.5.3 Primary Truths . 49 2.5.4 Formal Logic . 52 2.5.5 Epistemological Implications . 53 2.6 Euclid: Axiomatization of Continuous & Discrete . 54 2.6.1 Background . 54 2.6.2 Axiomatic Structure . 55 2.6.3 Theory of Magnitudes . 56 2.6.4 Summary . 58 3 Words and Images 61 3.1 Hellenistic Logic . 61 3.1.1 Modal Logic . 62 3.1.2 Propositional Logic . 62 3.1.3 Logical Paradoxes . 64 3.2 Medieval Logic . 65 3.2.1 Debate about Universals . 66 3.2.2 Language of Logic . 69 3.3 Combining Images and Letters . 77 3.3.1 The Art of Memory . 77 3.3.2 Combinatorial Inference . 79 3.3.3 Kabbalah . 80 3.4 Lull: Mechanical Reasoning . 83 3.4.1 Background . 84 3.4.2 Ars Magna . 85 3.4.3 From Images to Symbols . 95 3.4.4 Significance . 96 3.4.5 Ramus and the Art of Memory . 97 4 Thought as Computation 101 4.1 Hobbes: Reasoning as Computation . 101 4.2 Wilkins: Ideal Languages . 105 CONTENTS 9 4.3 Leibniz: Calculi and Knowledge Representation . 114 4.3.1 Chinese and Hebrew Characters . 114 4.3.2 Knowledge Representation . 117 4.3.3 Computational Approach to Inference . 120 4.3.4 Epistemological Implications . 124 4.4 Boole: Symbolic Logic . 125 4.4.1 Background . 126 4.4.2 Class Logic . 127 4.4.3 Propositional Logic . 132 4.4.4 Probabilistic Logic . 133 4.4.5 Summary . 134 4.5 Jevons: Logic Machines . 136 4.5.1 Combinatorial Logic . 136 4.5.2 Logic Machines . 140 4.5.3 Discussion . 146 II The Triumph of the Discrete 149 5 The Arithmetization of Geometry 151 5.1 Descartes: Geometry and Algebra . 151 5.1.1 Arabian Mathematics . 151 5.1.2 Analytic Geometry . 156 5.1.3 Algebra and the Number System . 162 5.1.4 The Importance of Informality . 165 5.2 Magic and the New Science . 166 5.2.1 Pythagorean Neoplatonism . 166 5.2.2 Hermeticism . 168 5.2.3 Alchemy . 172 5.2.4 Renaissance Magic and Science . 176 5.2.5 The Witches' Holocaust . 178 5.2.6 Belief and the Practice of Science . 180 5.3 Reduction of Continuous to Discrete . 184 5.3.1 The Problem of Motion . 184 5.3.2 Berkeley: Critique of Infinitesmals . 190 5.3.3 The Rational Numbers . 199 5.3.4 Plato: The Monad and the Dyad . 203 5.3.5 Cantor: The Real Line . 208 10 CONTENTS 5.3.6 Infinities and Infinitesmals . 211 5.4 Summary . 215 5.4.1 Technical Progress . 215 5.4.2 Psychology and Sociology of Science .