Let me just say before we start that I don't agree with Zeno. Motion is impossible? This is some different definition of motion than the one we use in everyday life, yes? But I also have to say that, although I don't agree with him, Zeno's famous paradoxes have always held an irresistible fascination for me.
Zeno apparently did believe that all motion is impossible, and that therefore it follows that what we see or experience as motion must necessarily be an illusion. I have no idea how Zeno died, but it's just possible he died of starvation. Since motion is impossible, a trip down to the local food store would have been an adventure that was fraught with cosmic impossibility. Therefore it would hardly have been worthwhile to make the attempt. So we can thus imagine Zeno wasting away, while the steak and fries waited a short distance from his doorstep, philosophically unreachable.
Not much is known about Zeno. None of his writings have come down to us, and we only know of his existence from the writings of other philosophers, some of whom may have been opponents. He was said to be a pupil of Parmenides, who was himself a philosopher who believed the world was stuck.
Zeno's most famous paradox concerns Achilles and the Tortoise. The two agree to have a race. Because Achilles is an athlete, he gives the tortoise 100 metres start. Achilles runs 10 times as fast as the tortoise, so you would expect him to easily overtake his opponent.
But not so, says Zeno.
BANG! goes the starting gun and both runners are off. Achilles runs 100 metres, to arrive at the place where the tortoise began. But the tortoise is now 10 metres ahead. So Achilles runs 10 metres, but the tortoise is still 1 metre ahead, so Achilles runs 1 metre, only to find the tortoise is still 0.1 metre ahead.
And so on. You get the picture. This proves the athlete can never overtake the tortoise, and so we have a paradox. Therefore, motion is impossible.
I like this paradox. The great thing about it is that you know it can't be right, but you can't really fault its logic. Many have tried. Most counter-arguments fall into some absurdity or logical trap. Or, even worse, bluster! To refute it is not as easy as you might think it would be. That's one reason why I like it so much.
Once I asked a Math professor to give a refutation of Zeno's Paradox of Achilles and the Tortoise, upon which he started to drone on about infinite seqeunces with a finite limit, and Leibnizian calculus. This may indeed have been very dull, but probably it was about as close as you will ever get to a refutation of Zeno.