After Socrates

The First Generation

  • Antisthenes and Diogenes
  • Plato's Academy
  • Aristotle's Lyceum


General Background

Rising power of Macedonia:


Diogenes: "A Socrates gone mad"

The Academy

Plato's Inheritance

  • Protected by a sacred olive grove.
  • Co-Ed, unlike the Lyceum---Axiothea and Lasthenia.
  • Path to Academia was lined by the graves of dead Athenians.

The First Academy

  • Eudoxus (Greatest of classical Greek mathematicians---first adequate definition of a ratio)
  • Archytas (Pythagorean, first mathematical physicist)
  • Theaetetus (Proved that there are exactly five platonic solids)

The Lyceum

More diverse topics than the Academy: History, music, psychology, biology, medicine...

Unlike the Academy, the Lyceum was concerned with making sense of this world.

Motion and Change

Zeno's Paradoxes

  1. Achilles and the Tortoise
  2. First and Second Dichotomy Paradox
  3. The Arrow
  4. The Stadium

Second Dichotomy Paradox

Motion is impossible, since there's no way to begin a motion.

The Arrow

Motion is nonexistent, since everything is, at every moment, stationary.

The Stadium

Motion cannot be atomic.

Aristotle's Solution

The Continuum

Time is not composed of moments.

The Jumper Puzzle

At the moment you jump off a roof, are you standing on the roof?

No, because if you're still on the roof, you haven't jumped yet.

Yes, because, if you're not standing on the roof any more, you've already jumped.


Moments are a way of talking about the boundaries of changes.

Your jumping of the roof was the beginning of your motion to the ground. But you weren't jumping while you were already moving to the ground. And you weren't jumping before you started moving to the ground.

Time is not divided

While we can consider any finite number of moments, by variously describing the world's motions, we cannot coherently consider an infinite number of moments.

The continuum is "potentially" but not "actually" infinite.

This resolves the Second Dichotomy Paradox.

The Arrow

But what about the Arrow? Even if time isn't made out of moments, are things really motionless at moments? Constantly starting and stopping?