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 Half-title page Title page Copyright page Contents Acknowledgements Introduction Overview of the Project Methodology, Treatment of Sources, and Relationships of Thinkers Investigated Overview of the Chapters 1 Conceptual Foundations 1.1 The Concepts of Kinêsis, Physis, and Natural Philosophy 1.1.1 The Concept of Motion (Kinêsis) 1.1.2 The Ancient Greek Conceptions of Physis and Natural Philosophy 1.1.3 The Concept of Being 1.2 Criteria of Inquiry 1.2.1 The Principle of Non-Contradiction 1.2.2 The Principle of Excluded Middle 1.2.3 The Principle of Sufficient Reason 1.2.4 Rational Admissibility 1.2.5 Saving the Phenomena 1.3 The Role of Logic 1.3.1 Operators and Operands 1.3.2 Negation and Identity as Operators 1.4 The Role of Mathematics: The Connection between Mathematics and Natural Philosophy 1.4.1 The Use of Mathematics for Science in General 1.4.2 How to Do Things with Numbers: Measurement and Countability 2 Parmenides’ Account of the Object of Philosophy 2.1 Introduction 2.2 Parmenides’ Criteria for Philosophy and His Logical Apparatus 2.2.1 Criteria for Philosophy 2.2.2 Logical Operators 2.3 Parmenides’ Logical Apparatus as Intimately Tied to His Ontology 2.4 Problems for the Very Possibility of Natural Philosophy 2.4.1 The Absence of Adequate Basic Concepts for Natural Philosophy 2.4.2 No Distinction between Operators and Operands 2.4.3 The Indeterminacy of Background Concepts 2.4.4 Problems with Relations 2.5 Relation to the Doxa Part: The Role of Cosmology 3 Zeno’s Paradoxes of Motion and Plurality 3.1 Introduction 3.2 The General Aim of Zeno’s Paradoxes 3.3 Parmenidean Inheritance 3.3.1 Advancing Parmenides’ Criteria 3.3.2 Deepening of the Challenge Parmenides Poses 3.4 The Fragments, Their Sources, and Their Connection 3.5 The Paradoxes of Plurality 3.6 The Paradoxes of Motion 3.6.1 The Dichotomy: Passing Infinitely Many Segments in a Finite Time 3.6.2 Achilles: A Variation of the Dichotomy Paradox 3.6.3 The Flying Arrow: Motion as a Sequence of Rests 3.6.4 The Moving Rows: Double the Time Is Half the Time 3.6.5 The Basic Problems of All Paradoxes of Motion 4 The Atomistic Foundation for an Account of Motion 4.1 Introduction 4.2 Eleatic Inheritance in the Atomists 4.2.1 Rational Admissibility 4.2.2 Consistency 4.2.3 The Principle of Sufficient Reason 4.3 Atomistic Changes 4.3.1 What Truly Is Must Explain the Phenomena 4.3.2 A Physical Theory 4.3.3 Change of Logical Operators38 4.3.4 The Atomistic Account of What Is 4.3.5 New Physical Features and Their Functions 4.4 Consequences of the Atomistic Changes for Natural Philosophy 4.4.1 Reply to Eleatic Problems 4.4.2 Motion and Changes in the Atomistic Framework 4.4.3 Problems that Remain 5 The Possibility of Natural Philosophy According to Plato I: The Logical Basis 5.1 Introduction: The Investigation of the Natural World as an Eikôs Mythos 5.2 The Sophist 5.2.1 The Reinterpretation of Negation and the Connection Operator 5.2.2 The Reinterpretation of the Criteria for Philosophy 1: The Principle of Non-Contradiction and the Principle of Excluded Middle 5.2.3 Widening the Conceptual Possibilities 5.2.4 Possible Answers to Parmenides’ Problems 5.3 The Timaeus: Logical Advances 5.3.1 The Reinterpretation of the Criteria for Philosophy 2: The Principle of Sufficient Reason and Rational Admissibility 5.3.2 An Eikôs Mythos 6 The Possibility of Natural Philosophy According to Plato II: Mathematical Advances and Ultimate Problems 6.1 Introduction 6.2 Introducing Mathematical Structures 6.3 Locomotion and Mathematical Structures 6.3.1 Time and Eternity 6.3.2 Time as the Measure of Motion 6.3.3 Space as Excluded from the Measurement Process 6.4 Problems with a Simple Measure 6.4.1 Restricted Comparability 6.4.2 Lacking Consistency: The Tortoise Wins the Race 7 Aristotle’s Notion of Continuity: The Structure Underlying Motion 7.1 Introduction 7.2 Notions of Magnitude Influencing Aristotle’s Concept of a Continuum 7.2.1 Parmenides’ Suneches 7.2.2 Atomistic Notions of Magnitude 7.2.3 A Mathematical Notion of Suneches 7.3 Aristotle’s Two Accounts of the Continuum 7.3.1 Things Whose Limits Touch and Are One 7.3.2 Things Being Divisible without Limits 7.4 Implications of Aristotle’s Concept of a Continuum 7.4.1 A New Understanding of the Part-Whole Relation 7.4.2 A New Twofold Concept of a Limit 7.4.3 A New Conception of Infinity 8 Time and Space: The Implicit Measure of Motion in Aristotle’s Physics 8.1 The General Concept of Measure in Aristotle’s Metaphysics 8.1.1 A Simple Measure: Being One-Dimensional and of the Same Kind as What Is Measured 8.1.2 Comparison with a Modern Conception and the Relation between Counting and Measuring 8.2 The Measure of Movement in Aristotle’s Physics 8.2.1 Time as a One-Dimensional Measure and Number of Motion 8.2.2 The Search for a Measure of the Same Kind as Motion 8.2.3 The Relation of Time and Space 9 Time as the Simple Measure of Motion 9.1 Other Accounts of Speed 9.2 Reasons Why Aristotle did not Explicitly Use a Complex Measure 9.3 Constructive Developments: A Résumé Bibliography Index Locorum General Index "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