This morning, Jon Stewart was talking about how the Republican primary race is similar to Zeno's Paradox. It has been some time since I have thought about old Zeno and his paradox. It goes something like this: If you want to cross a room that is twenty feet across, in order to get to the other side of the room, you must first travel half of the distance. When you reach that point, you must again travel half of the distance left. When you reach that point, you must again travel half of the distance left, and so on. You will never get to the other side of the room. The logic of this paradox is unassailable. Yet, we know it IS possible to walk across a room and actually get to the other side of the room. So where is the flaw in the logic of Zeno's Paradox? Anyone? No fair Googling. See if you can figure it out on your own. Hint: It has to do with time. If you can't solve it after really trying, read this: Zeno's Paradox of the Tortoise and Achilles (PRIME) (And then explain it to ME.)

Zeno's paradox doesn't actually have a flaw, it is just grossly misused. When dealing with relativity and a few other aspects of physics, Zeno's paradox is actually plausible. Ex, when an object with any amount of mass tries to accelerate to the speed of light, it can never actually reach that speed. It can always get closer then what it is currently at, but at relatavistic speeds, to further increase your speed requires a growing amount of energy, to the point that to reach the speed of light, it would actually require an infinite amount of energy. Another example is that any object falling into a black hole could never actually reach the center of the black hole, from the viewpoint of an outside observer. It would literally take an infinite amount of time.

Ahhh, Zeno's paradox. That comes up in my history of math class. I won't resolve it for you there, but it's interesting because it leads to something called the "Horror Of The Infinite." The Greeks realized that they could "avoid" the "paradox" if they disallowed infinite processes. It's why Euclid's Elements included a formula for finding the sum of a finite geometric series and not for the infinite, and why Archimedes came very very close to inventing vector calculus (realizing that a planar surface could be though of as an infinite number of lines, a 3-D object as an infinite number of planar surfaces, etc), but never made the jump to a full Calculus. On topic: It has a resolution, but even today that resolution still kicks up some controversy.

The logical flaw in Zeno's paradox lies in the premise that "half the distance" is a real entity as opposed to an artificial division of a whole by human perception. The person walking travels the whole distance, not half of it; we simply are able to visualize a point halfway across the room and slap a label on it. It's a particular instance of the fallacy of confusing mathematical ideas with real ones, or the map with the territory.

Ahh, the Leno Paradox. How did an inferior late night talk show host beat a superior late night talk show host out of a plum job? What? Zeno? Never mind.

Well, actually, Zeno was trying to illustrate the fallacy of the Pythagoreans, who believed everything could be represented as a rational number. They actually discovered the square root of 2 thanks to the Pythagorean Theorem and tried to cover up it's existence calling it an "unutterable" number. The resolution is related to the continuity of the real number line. In general, even the most esoteric mathematical ideas tend to have practical applications. Group theory was seen as the very epitome of mathematical sophistication and is now a central facet of theoretical physics.

It is pretty cool. But, what I see as the part that confuses folks is the paradox talks of infinitely halving distances (which adds up to unity, anyway), but the question involves an actual motion which involves both distance and time. Sort of reminds me of riddles. The key to a good riddle is to get the audience to focus on something that will not lead to the solution.

Dragon is correct, but I also like to play the game of halfs and still explain it. Two things happen when crossing the room. One, every time the person goes half the distance, it takes half the time. When the person is mere microns from the other wall the amount of time decreases to near infinty very very quickly. Two, the person crossing the room very quickly gets the half distance to the sub-atomic level (the first part above). At that level you never really arrive at the full distance. The molecules in your fingertip, never actually make contact with the molecules in the wall. Any way you look at this paradox, it's a fun mental exercise.