Atomism and Infinite Divisibility

ACKNOWLEDGEMENTS
ABSTRACT

TABLE OF CONTENTS

(The page numbers of the printed text have been retained for easy reference.)

CHAPTER

1. PRELIMINARY NOTIONS 1
   • Introduction  1
     • Mathematical Induction  7
     • Infinity 8
     • Motion 10
   • Developments Leading Toward Atomism 12
     • "Being" or Existence 14
     • Thales and Anaximander 14
     • Pythagoras 16
     • Heraclitus and Parmenides 17
     • Empedocles and Anaxagoras 19
   • Notes and References 21
2. ZENO'S PARADOXES 23
   • The Achilles 23
   • The Dichotomy 28
   • The Arrow 32
   • The Stadium 37
   • The Paradox of Plurality 41
   • Notes and References 44
3. DISCUSSION OF ZENO'S PARADOXES 45
   • On The Achilles 46
   • On The Dichotomy 52
   • A New Resolution of the Paradoxes 53
   • On both the Achilles and the Dichotomy 58
     • (predecessor) 60
   • On The Arrow 63
   • On The Stadium 67
   • About both the arrow and the stadium 72
     • (uniform velocity) 72
   • On The Paradox of Plurality 73
   • Notes and References 82
4. ARISTOTLE (384-322 BC) AND INFINITE DIVISIBILITY 85
   • The Birth of Atomism Proper 85
     • Leukippos (450-420 bc) 86
     • Democritus (460-370 bc) 87
   • Aristotle 89
   • Quantity 90
     • (Quantum) 91
   • The Infinite 92
   • Atomism 96
   • Infinite Divisibility and Matter 98
   • Infinite Divisibility 98
     • (discrete/dense) 100
   • Potential and Actual 104
   • Infinity Times Zero 107
   • The Atomists' Argument Expanded 109
   • "Contact" 114
   • Anywhere is not Everywhere 116
   • Point and "Place" 120
   • Summary of Aristotle's Views 131
   • Notes and References 133
5. ATOMISM AND DIVISIONISM AFTER ARISTOTLE 135
   • Epicurus (341-270 bc) 135
   • Chrysippus (280-207 bc) 144
   • Lucretius (99/94-55/51 bc) 146
   • Arnauld (1612-94) and Infinite Divisibility 147
   • Hume and Infinite Divisibility 152
   • Notes and References 155
6. THE VALIDITY OF BOTH VIEWS 158
   • The Validity of Divisionism 158
   • Developing Numbers 159
     • Negative Numbers 159
     • Fractions 160
   • Pure Numbers 160
     • Multiplication 162
     • Division 162
   • Order 164
   • Extension Without Measure 165
   • The Validity of Atomism 167
     • Aspect Ratio 173
       • (perspective ratio) 174
     • Drawing lines in discrete metric spaces 178
     • Constructing triangles in the atomic plane 181
     • Computing Area of an Atomic Triangle 183
     • Overlapping Atomic Figures 186
   • Notes and References 189
7. CONCLUSIONS 192
   • The Atomic Destruction of the Pythagorean Theorem 192
     • A Classic Proof of the Pythagorean Theorem 192
     • Computing the Area of Atomic Triangles 197
     • The Pythagorean Formula in Atomic Metrics 203
   • Some Final Remarks 208
   • Notes and References 213

BIBLIOGRAPHY 214

Index of notes