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December 22, 2011

Zeno's Arrow, the GAP Trail and the Speed of Christmas

Once upon a time there was a Greek guy named Zeno of Elea. Some pretentious people were making ridiculous arguments wrapped in very fancy structures and speeches and he wanted to show them they were being silly, so he took their thinking to its logical conclusion and described the problem of an arrow reaching its target.

So you've got the archer at distance=0, and the target at distance=100 feet. (We'll use feet because we're Americans and we don't like the metric system.) The archer wants the arrow to hit the target so she aims, draws the bowstring, and lets loose.

An unsophisticated person might think the arrow would fly to the target, but Zeno explained that if one was to follow his colleague's arguments, before the arrow could get to the target it would have to reach the halfway point (50 feet). And so, at some intermediate point the arrow is halfway there.

But then, before the arrow can go the rest of the way, it has to go with the halfway thing again. So at another intermediate point, the arrow is at 75 feet.

Almost there, right? But no, before it can hit the target it has to reach halfway, 87.5 feet.And again and again and again: 93.75 feet, 96.975 feet, 98.4875 feet, all of a sudden these numbers look like the damned metric system. The absurdity is, Zeno explains, that at any moment there's always a halfway distance to be reached first, and so truly reaching the destination is logically impossible.

Big silence; dead air. Zeno reformatted the story in terms of walking across the room, and explained why you could move most of the way across the room but you could never really get there, if you followed their thinking. The legend says that Diogenes, who was getting tired of the exercise, then walked across the room and sat down, rolling his eyes at to Zeno to show that he understood the fallacy in a way that was lost to the others.

It seems like an esoteric paradox that might have been amusing a few centuries ago, but we have better things to consider these days (like figuring out how many wars we're in) except that it keeps popping up.

The progress of the Great Allegheny Passage also seems to follow Zeno's Paradox. Each year, half of the remaining trail between Pittsburgh and DC is completed and introduced; each year the gaps are reduced, and yet in spite of the incremental progress it seems like true completion is never achieved.

Or say you're driving to Grandma's with your kids in the back seat. Are we there yet? No. Are we there yet? No. Are we there yet? No. Are we there yet? No. Are we there yet? No. Are we there yet? No. Are we there yet? No.

It may be that Zeno just figured out quantum theory before his time. If you're comfortable with location as a probability distribution rather than a fixed certainty, you're probably good with Zeno's arrow eventually being mostly there.

Today's XKCD works the same theme:

We are grateful for the XKCD-symbiotic website, Explain which draws out the cleverness of the work. Today they point out the nuanced fallacy of Zeno having an Advent calendar because he lived B.C.

Merry Christmas. I hope it gets here, and/or that we get there.


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