Zeno’s Paradoxes

What is a paradox?

• Anything that is/leads to a logical contradiction– A “square circle”– The notion of a “set of all sets”

Who is Zeno?

• Greek philosopher• 5th Century BCE• Elea (Greek colony)• Student of

Parmenides

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

Example 1: Unity/Plurality

• Paradox of the “heap of grain”

Example 2: Motion

• The Dichotomy

• Achilles and the Tortoise

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?

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

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)