A little while back I did some reading on Zeno's paradox (really Zeno's paradoxes --- he had several, not all of which seem to have survived). I had always understood them as an interesting early attempt to grapple with the concept of infinity, but now more or less solved by modern calculus. And from a purely mathematical standpoint this basically holds.
But what surprised me was that Zeno was not fundamentally interested in the mathematical implications of his paradoxes. He was more interested in their philosophical implications. To give a bit of context, Zeno was a student of Parmenides, who was perhaps the purest of the monists. Parmenides held that all things are a single unity and that any change is an illusion. Most other Greek philosophers found this idea absurd since it seems pretty self-evident that things are changing around us --- arrows fly through the air, runners race around a track.
Zeno's purpose in devising these paradoxes was to show that it wasn't so self-evident that things were actually moving at all. And it was a counterpoint to the Pythagoreans, who believed that all things began in the number one, but then proceeded in multiples of the number one (the number two, three, four, etc., and through the numbers, all things in the universe since the Pythagoreans believed that all matter was fundamentally composed of integers). In the arrow paradox, Zeno was essentially trying to show that it was logically absurd to start with a unity and go to a multiple as the Pythagoreans held. And in the Achilles and stadium paradoxes Zeno was trying to show that it was absurd to start with a multiple and go to a unity.
But what surprised me was that Zeno was not fundamentally interested in the mathematical implications of his paradoxes. He was more interested in their philosophical implications. To give a bit of context, Zeno was a student of Parmenides, who was perhaps the purest of the monists. Parmenides held that all things are a single unity and that any change is an illusion. Most other Greek philosophers found this idea absurd since it seems pretty self-evident that things are changing around us --- arrows fly through the air, runners race around a track.
Zeno's purpose in devising these paradoxes was to show that it wasn't so self-evident that things were actually moving at all. And it was a counterpoint to the Pythagoreans, who believed that all things began in the number one, but then proceeded in multiples of the number one (the number two, three, four, etc., and through the numbers, all things in the universe since the Pythagoreans believed that all matter was fundamentally composed of integers). In the arrow paradox, Zeno was essentially trying to show that it was logically absurd to start with a unity and go to a multiple as the Pythagoreans held. And in the Achilles and stadium paradoxes Zeno was trying to show that it was absurd to start with a multiple and go to a unity.