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aristarchus writes:

Quartz

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Almost 2,500 years ago, the philosopher Zeno of Elea set out to challenge the way we understand the physical world through a set of brain teasers that have stuck with us for millennia. The most powerful of Zeno's paradoxes grapple with the concept of infinity while pitting observable reality against the scientific language we use to describe that reality, suggesting that elements of the everyday, like motion and speed, are actually illusory.

Example paradoxes are:

The millet paradox, which states that one falling grain of millet makes no sound but a ton of falling millet makes a big one, is more of a stoner observation than a profound question about the physical world. His paradoxes of motion and space, on the other hand, are legendary. Four of the more than 40 thought experiments he is said to have devised are most often employed as vivid introductions to the intersection of math and philosophy, where something readily apparent is a challenge to definitively prove.

Dichotomy paradox: If you want to walk across the room, you have to first walk half that distance, then half the remaining distance, ad infinitum, so how do you ever get there?

Achilles paradox: If a turtle gets a head start in a race against Achilles, Achilles has to cover half the distance between himself and the turtle in order to catch up. Then half that. And half again. And again. In an upset, the turtle wins!

Arrow paradox: At any given instant, an arrow in flight occupies a certain space, no more and no less. At the next instant, it occupies a different space. If you assume an instant is indivisible, the arrow is not in motion. So how does it move? "It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever," as Bertrand Russell put it.

Stadium paradox: Imagine three sets of three bodies in stadium rows: three As, three Bs, three Cs. The As are stationary; the Bs are moving right; the Cs are moving left at the same speed. In the same timeframe, the Cs will pass just one of the As, but two of the Bs. Crazy, right? (It doesn't seem like it, but if you think of space and time atomistically, they pass without passing.)

It took more than 2,000 years to break the dichotomy and Achilles paradoxes, and the people to do it were the French mathematical prodigy Augustin-Louis Cauchy and the German Karl Weierstrass. The mathematical answer can be summed up by the intuitive answer: Eventually, you get there.

In mathematical terms, one way of putting it is "the limit of an infinite sequence of ever-improving approximations is the precise value" (pdf). By going from one side of the room to another, you go 100% of the way across. You can chop that 100% up into infinite pieces, but those pieces converge on a limit of 100, and the sum of those pieces is the value—the infinite number of increasingly small pieces adds up to a finite number. ½ + ½ = 1, of course. ½ + ¼ + ¼ also equals 1. And so forth: the numbers you add up to get to 1 can expand to infinity, but it's not changing the end result. Not all infinite geometric series converge to a limit, but some do (pdf), predictably: "All those (and only those) in which the ratio of consecutive terms is greater than –1 and less than +1, so that the absolute values of the terms get progressively smaller."

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