1. Zeno’s Paradoxes

2. What is a paradox? Anything that is/leads to a logical contradiction –A “square circle” –The notion of a “set of all sets”

3. Who is Zeno? Greek philosopher 5 th Century BCE Elea (Greek colony) Student of Parmenides

4. What are Zeno’s Paradoxes? Arguments to support the teachings of Parmenides –Everything that exists is one –Change is illusory –Motion is illusory Based largely on limiting processes and definitions

5. Example 1: Unity/Plurality Paradox of the “heap of grain”

6. Example 2: Motion The Dichotomy Achilles and the Tortoise

7. Mathematics? Zeno’s Paradoxes were not developed as mathematical problems Some of them (e.g., dichotomy) seem to be able to be resolved with modern tools (e.g., infinite series) But is this the point?

8. Modeling Reality Logic: “Truth” as system-relative In order to resolve the paradoxes with modern math, we have to make certain assumptions about physical reality. –Cf. “Zeno processes” –Is matter discrete or continuous? Your model may “work”, but does it accurately reflect reality? –(Does this even matter?) –Ex. John& Jane, Diogenes

9. Berkeley & the Calculus Newton/Leibniz: Fluxions & infinitesimals “And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities? “ -George Berkeley (ca. 1734)