## Tuesday, September 02, 2008

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There are actually a number of paradoxes attributed to Zeno of Elea, a Greek philosopher who lived some 2500 years ago (this is not the same Zeno who founded the Stoic school, and who lived arond 150 years later). But the one most often thought of when one mentions “Zeno’s Paradox” is the one about Achilles and the tortoise.

If the great hero Achilles should enter into a foot race with a tortoise, and should give the tortoise even a small head start, he can never catch the tortoise in the race [we assume, here, that the tortoise does, in fact, move ahead at a constant, if slow, rate]. When Achilles reaches the place where the tortoise started, the latter will have moved ahead some amount. When Achilles reaches that spot, the tortoise will have moved ahead a little more. And so on — at any point, Achilles will be catching up with where the tortoise used to be, and the tortoise will always have moved ahead.

Of course, we know that’s ridiculous: Achilles is faster, higher, stronger, and we know that in a real race he’d zoom past the tortoise before you could say “Aesop’s fables”. So what’s going on here? How do we reconcile these two things?

The fallacy here is that we’re looking at finite points in an infinite sequence. We can observe that if we use the logic in the paradox, not only does Achilles never catch up to the tortoise, but neither can the tortoise ever reach the finish line. In fact, we don’t need the tortoise at all: if Achilles runs from a start line to a finish line, at some point he’ll be halfway there. Some time later, he’ll be halfway through the rest. And then halfway and then halfway... and by always getting halfway, he’ll never arrive.

But he does, and it’s because we can’t stop at a finite point along the way; we have to look at the whole, infinite series. And some monotonically decreasing infinite sequences[1] such as this will converge on finite sums. We can write it this way:

½ + ¼ + ⅛ + ...   =   1
...or in more abstract mathematical terms, like this:
 ∞∑k=1 1—2k =  1

Understanding infinite convergent series is a basic concept in calculus. Of course, calculus wasn’t known to Zeno of Elea, and wouldn’t formally be invented for another 2000 years or so.

Nevertheless, Achilles still left the tortoise in the dust.

[1] There’s more to it than just “monotonically decreasing” — the series has to converge, in mathematical terms, and there are tests for that — but that characterization is close enough for this explanation.

#### 1 comment:

W.M. Irwin said...

Barry, it seemed to me that what made Zeno's Paradox so compelling was that it redefined the role of the observer to that of one experiencing the unnatural slowing of time. Since this is intuitively uncomfortable for most of us, I think we tend to impose our own "time frame" to the observations, in violation of the redefined role for the observer. And this may also be a clue to how flawed experiments and hoaxes can deceive others with their results (by altering the conditions of the observer). Does this make any sense?