The Concept Of Motion In Ancient Greek: Thought Foundations In Logic, Method, And Mathematics

Cover......Page 1
Half-title page......Page 3
Title page......Page 5
Copyright page......Page 6
Contents......Page 7
Acknowledgements......Page 11
Overview of the Project......Page 13
Methodology, Treatment of Sources, and Relationships of Thinkers Investigated......Page 17
Overview of the Chapters......Page 23
1.1.1 The Concept of Motion (Kinêsis)......Page 29
1.1.2 The Ancient Greek Conceptions of Physis and Natural Philosophy......Page 39
1.1.3 The Concept of Being......Page 42
1.2 Criteria of Inquiry......Page 43
1.2.1 The Principle of Non-Contradiction......Page 44
1.2.2 The Principle of Excluded Middle......Page 49
1.2.3 The Principle of Sufficient Reason......Page 51
1.2.4 Rational Admissibility......Page 58
1.2.5 Saving the Phenomena......Page 61
1.3 The Role of Logic......Page 65
1.3.1 Operators and Operands......Page 67
1.3.2 Negation and Identity as Operators......Page 69
1.4.1 The Use of Mathematics for Science in General......Page 79
1.4.2 How to Do Things with Numbers: Measurement and Countability......Page 85
2.1 Introduction......Page 92
2.2.1 Criteria for Philosophy......Page 95
2.2.2 Logical Operators......Page 104
2.3 Parmenides’ Logical Apparatus as Intimately Tied to His Ontology......Page 115
2.4.1 The Absence of Adequate Basic Concepts for Natural Philosophy......Page 123
2.4.2 No Distinction between Operators and Operands......Page 126
2.4.3 The Indeterminacy of Background Concepts......Page 128
2.4.4 Problems with Relations......Page 129
2.5 Relation to the Doxa Part: The Role of Cosmology......Page 131
3.1 Introduction......Page 136
3.2 The General Aim of Zeno’s Paradoxes......Page 140
3.3.1 Advancing Parmenides’ Criteria......Page 142
3.4 The Fragments, Their Sources, and Their Connection......Page 146
3.5 The Paradoxes of Plurality......Page 148
3.6 The Paradoxes of Motion......Page 155
3.6.1 The Dichotomy: Passing Infinitely Many Segments in a Finite Time......Page 156
3.6.2 Achilles: A Variation of the Dichotomy Paradox......Page 167
3.6.3 The Flying Arrow: Motion as a Sequence of Rests......Page 168
3.6.4 The Moving Rows: Double the Time Is Half the Time......Page 176
3.6.5 The Basic Problems of All Paradoxes of Motion......Page 186
4.1 Introduction......Page 188
4.2 Eleatic Inheritance in the Atomists......Page 190
4.2.1 Rational Admissibility......Page 191
4.2.2 Consistency......Page 194
4.2.3 The Principle of Sufficient Reason......Page 195
4.3.1 What Truly Is Must Explain the Phenomena......Page 196
4.3.2 A Physical Theory......Page 197
4.3.3 Change of Logical Operators38......Page 199
4.3.4 The Atomistic Account of What Is......Page 202
4.3.5 New Physical Features and Their Functions......Page 203
4.4.1 Reply to Eleatic Problems......Page 206
4.4.2 Motion and Changes in the Atomistic Framework......Page 210
4.4.3 Problems that Remain......Page 212
5.1 Introduction: The Investigation of the Natural World as an Eikôs Mythos......Page 214
5.2 The Sophist......Page 222
5.2.1 The Reinterpretation of Negation and the Connection Operator......Page 223
5.2.2 The Reinterpretation of the Criteria for Philosophy 1: The Principle of Non-Contradiction and the Principle of Excluded Middle......Page 237
5.2.3 Widening the Conceptual Possibilities......Page 242
5.2.4 Possible Answers to Parmenides’ Problems......Page 244
5.3 The Timaeus: Logical Advances......Page 247
5.3.1 The Reinterpretation of the Criteria for Philosophy 2: The Principle of Sufficient Reason and Rational Admissibility......Page 248
5.3.2 An Eikôs Mythos......Page 252
6.1 Introduction......Page 257
6.2 Introducing Mathematical Structures......Page 258
6.3.1 Time and Eternity......Page 265
6.3.2 Time as the Measure of Motion......Page 268
6.3.3 Space as Excluded from the Measurement Process......Page 278
6.4 Problems with a Simple Measure......Page 281
6.4.2 Lacking Consistency: The Tortoise Wins the Race......Page 286
7.1 Introduction......Page 289
7.2 Notions of Magnitude Influencing Aristotle’s Concept of a Continuum......Page 296
7.2.1 Parmenides’ Suneches......Page 297
7.2.2 Atomistic Notions of Magnitude......Page 301
7.2.3 A Mathematical Notion of Suneches......Page 303
7.3 Aristotle’s Two Accounts of the Continuum......Page 307
7.3.1 Things Whose Limits Touch and Are One......Page 308
7.3.2 Things Being Divisible without Limits......Page 311
7.4 Implications of Aristotle’s Concept of a Continuum......Page 315
7.4.1 A New Understanding of the Part-Whole Relation......Page 317
7.4.2 A New Twofold Concept of a Limit......Page 323
7.4.3 A New Conception of Infinity......Page 340
8 Time and Space: The Implicit Measure of Motion in Aristotle’s Physics......Page 347
8.1 The General Concept of Measure in Aristotle’s Metaphysics......Page 349
8.1.1 A Simple Measure: Being One-Dimensional and of the Same Kind as What Is Measured......Page 350
8.1.2 Comparison with a Modern Conception and the Relation between Counting and Measuring......Page 356
8.2 The Measure of Movement in Aristotle’s Physics......Page 362
8.2.1 Time as a One-Dimensional Measure and Number of Motion......Page 363
8.2.2 The Search for a Measure of the Same Kind as Motion......Page 368
8.2.3 The Relation of Time and Space......Page 386
9.1 Other Accounts of Speed......Page 397
9.2 Reasons Why Aristotle did not Explicitly Use a Complex Measure......Page 405
9.3 Constructive Developments: A Résumé......Page 415
Bibliography......Page 416
Index Locorum......Page 435
General Index......Page 438