The Concept of Motion in Ancient Greek Thought Foundations in Logic, Method, and Mathematics : Foundations in Logic, Method, and Mathematics

''This book examines the birth of the scientific understanding of motion. It investigates which logical tools and methodological principles had to be in place to give a consistent account of motion, and which mathematical notions were introduced to gain control over conceptual problems of motion. It shows how the idea of motion raised two fundamental problems in the 5th and 4th century BCE: bringing together being and non-being, and bringing together time and space. The first problem leads to the exclusion of motion from the realm of rational investigation in Parmenides, the second to Zeno's paradoxes of motion. Methodological and logical developments reacting to these puzzles are shown to be present implicitly in the atomists, and explicitly in Plato who also employs mathematical structures to make motion intelligible. With Aristotle we finally see the first outline of the fundamental framework with which we conceptualise motion today. Explores the rich conceptual basis on which our understanding of motion and speed is built. Shows the reader which changes in logic, ontology and methodology were crucial for establishing natural science. Provides an overview of the development of the understanding of motion all the way from the Presocratics to Plato and Aristotle.''-- Provides by publisher Cover 1 Half-title page 3 Title page 5 Copyright page 6 Contents 7 Acknowledgements 11 Introduction 13 Overview of the Project 13 Methodology, Treatment of Sources, and Relationships of Thinkers Investigated 17 Overview of the Chapters 23 1 Conceptual Foundations 29 1.1 The Concepts of Kinêsis, Physis, and Natural Philosophy 29 1.1.1 The Concept of Motion (Kinêsis) 29 1.1.2 The Ancient Greek Conceptions of Physis and Natural Philosophy 39 1.1.3 The Concept of Being 42 1.2 Criteria of Inquiry 43 1.2.1 The Principle of Non-Contradiction 44 1.2.2 The Principle of Excluded Middle 49 1.2.3 The Principle of Sufficient Reason 51 1.2.4 Rational Admissibility 58 1.2.5 Saving the Phenomena 61 1.3 The Role of Logic 65 1.3.1 Operators and Operands 67 1.3.2 Negation and Identity as Operators 69 1.4 The Role of Mathematics: The Connection between Mathematics and Natural Philosophy 79 1.4.1 The Use of Mathematics for Science in General 79 1.4.2 How to Do Things with Numbers: Measurement and Countability 85 2 Parmenides’ Account of the Object of Philosophy 92 2.1 Introduction 92 2.2 Parmenides’ Criteria for Philosophy and His Logical Apparatus 95 2.2.1 Criteria for Philosophy 95 2.2.2 Logical Operators 104 2.3 Parmenides’ Logical Apparatus as Intimately Tied to His Ontology 115 2.4 Problems for the Very Possibility of Natural Philosophy 123 2.4.1 The Absence of Adequate Basic Concepts for Natural Philosophy 123 2.4.2 No Distinction between Operators and Operands 126 2.4.3 The Indeterminacy of Background Concepts 128 2.4.4 Problems with Relations 129 2.5 Relation to the Doxa Part: The Role of Cosmology 131 3 Zeno’s Paradoxes of Motion and Plurality 136 3.1 Introduction 136 3.2 The General Aim of Zeno’s Paradoxes 140 3.3 Parmenidean Inheritance 142 3.3.1 Advancing Parmenides’ Criteria 142 3.3.2 Deepening of the Challenge Parmenides Poses 146 3.4 The Fragments, Their Sources, and Their Connection 146 3.5 The Paradoxes of Plurality 148 3.6 The Paradoxes of Motion 155 3.6.1 The Dichotomy: Passing Infinitely Many Segments in a Finite Time 156 3.6.2 Achilles: A Variation of the Dichotomy Paradox 167 3.6.3 The Flying Arrow: Motion as a Sequence of Rests 168 3.6.4 The Moving Rows: Double the Time Is Half the Time 176 3.6.5 The Basic Problems of All Paradoxes of Motion 186 4 The Atomistic Foundation for an Account of Motion 188 4.1 Introduction 188 4.2 Eleatic Inheritance in the Atomists 190 4.2.1 Rational Admissibility 191 4.2.2 Consistency 194 4.2.3 The Principle of Sufficient Reason 195 4.3 Atomistic Changes 196 4.3.1 What Truly Is Must Explain the Phenomena 196 4.3.2 A Physical Theory 197 4.3.3 Change of Logical Operators38 199 4.3.4 The Atomistic Account of What Is 202 4.3.5 New Physical Features and Their Functions 203 4.4 Consequences of the Atomistic Changes for Natural Philosophy 206 4.4.1 Reply to Eleatic Problems 206 4.4.2 Motion and Changes in the Atomistic Framework 210 4.4.3 Problems that Remain 212 5 The Possibility of Natural Philosophy According to Plato I: The Logical Basis 214 5.1 Introduction: The Investigation of the Natural World as an Eikôs Mythos 214 5.2 The Sophist 222 5.2.1 The Reinterpretation of Negation and the Connection Operator 223 5.2.2 The Reinterpretation of the Criteria for Philosophy 1: The Principle of Non-Contradiction and the Principle of Excluded Middle 237 5.2.3 Widening the Conceptual Possibilities 242 5.2.4 Possible Answers to Parmenides’ Problems 244 5.3 The Timaeus: Logical Advances 247 5.3.1 The Reinterpretation of the Criteria for Philosophy 2: The Principle of Sufficient Reason and Rational Admissibility 248 5.3.2 An Eikôs Mythos 252 6 The Possibility of Natural Philosophy According to Plato II: Mathematical Advances and Ultimate Problems 257 6.1 Introduction 257 6.2 Introducing Mathematical Structures 258 6.3 Locomotion and Mathematical Structures 265 6.3.1 Time and Eternity 265 6.3.2 Time as the Measure of Motion 268 6.3.3 Space as Excluded from the Measurement Process 278 6.4 Problems with a Simple Measure 281 6.4.1 Restricted Comparability 286 6.4.2 Lacking Consistency: The Tortoise Wins the Race 286 7 Aristotle’s Notion of Continuity: The Structure Underlying Motion 289 7.1 Introduction 289 7.2 Notions of Magnitude Influencing Aristotle’s Concept of a Continuum 296 7.2.1 Parmenides’ Suneches 297 7.2.2 Atomistic Notions of Magnitude 301 7.2.3 A Mathematical Notion of Suneches 303 7.3 Aristotle’s Two Accounts of the Continuum 307 7.3.1 Things Whose Limits Touch and Are One 308 7.3.2 Things Being Divisible without Limits 311 7.4 Implications of Aristotle’s Concept of a Continuum 315 7.4.1 A New Understanding of the Part-Whole Relation 317 7.4.2 A New Twofold Concept of a Limit 323 7.4.3 A New Conception of Infinity 340 8 Time and Space: The Implicit Measure of Motion in Aristotle’s Physics 347 8.1 The General Concept of Measure in Aristotle’s Metaphysics 349 8.1.1 A Simple Measure: Being One-Dimensional and of the Same Kind as What Is Measured 350 8.1.2 Comparison with a Modern Conception and the Relation between Counting and Measuring 356 8.2 The Measure of Movement in Aristotle’s Physics 362 8.2.1 Time as a One-Dimensional Measure and Number of Motion 363 8.2.2 The Search for a Measure of the Same Kind as Motion 368 8.2.3 The Relation of Time and Space 386 9 Time as the Simple Measure of Motion 397 9.1 Other Accounts of Speed 397 9.2 Reasons Why Aristotle did not Explicitly Use a Complex Measure 405 9.3 Constructive Developments: A Résumé 415 Bibliography 416 Index Locorum 435 General Index 438 "The main object of this book is to study how the understanding of physical motion in ancient Greek thought developed before and up to Aristotle. It investigates which logical, methodological, and mathematical foundations had to be in place to establish a fullyfledged concept of motion that also allows for comparing and measuring speed.1 Given that physical motion is the core concept of natural philosophy, this study thereby also seeks to reconstruct in rough outlines how natural philosophy came to be established as a proper scientific endeavour in ancient Greece.2 According to a prevailing picture, scientific investigation of physical motion and change started properly in the West with Aristotle but only achieved its true form in modern times, with the overthrow of central Aristotelian doctrines. In the early modern period, so runs the narrative, Aristotelianism was rejected and the basis laid for what today we consider the science of physics.3 This interpretation is at least doubly misleading"-- Provided by publisher