Myth-buster at work.
My recent foray into logical fallacies reminded me of one that has irked me for many years.
According to Aristotle (restating Zeno):
In a race, the quickest runner [Achilles] can never overtake the slowest [Tortoise], since the pursuer must first reach the point whence the pursued started [i.e. the pursued has a head start], so that the slower must always hold a lead.
Ridiculous, of course.
To show what’s wrong with Aristotle’s analysis, I begin with an example that adopts his “logic”:
Achilles (A), a quasi-god with a tricky tendon, runs at a mortal speed of 15 miles an hour (a 4-minute miler, he).
Tortoise (T) plods at a speed of 1 mile an hour. (I exaggerate for simplicity of illustration.)
If A gives T a 15-mile lead, A reaches T’s starting point in 1 hour. T has, in that hour, moved ahead by 1 mile.
A covers that mile in 1/15 of an hour, in which time T has moved ahead by 1/15 of a mile.
A runs the 1/15 of a mile in 16 seconds, in which time T has moved ahead by another 23.47 feet.
And so on.
Therefore, A can never catch T.
What’s the catch? It’s verbal sleight-of-hand, much like the “proof” that 1 = 2 (“proof” here; fallacy explained here), or the “proof” that a boost in government spending causes GDP to rise by a “multiplier” (fallacy exposed here).
We know that A must be able to catch T, but we are trapped in a fallacious argument which seems to prove that A can’t catch T. Let’s break out of the trap.
The verbal sleight-of-hand in the Zeno/Aristotle argument is that A’s and T’s movements involve distance but not time. Velocity (distance/time) is ignored. This allows Zeno/Aristotle to imply (a nonsensical) sequence of events: T proceeds to a certain point; A reaches that point and waits for T to proceed to the next point; and so on.
In fact (if a fable may be called a fact) A catches up with T by covering a greater distance than T in the same length of time — that is, A proceeds at a greater velocity than T. Along the way, A passes points already passed by T, but A doesn’t pause at any of those points and allow T to move a bit farther ahead. A keeps on moving and catches up with T.
Going back to the example (A runs 15 miles an hour, etc.), we can determine when and where A catches T simply by describing events correctly. To begin:
A’s time (in hours) x A’s velocity (in miles per hour) = A’s distance (in miles).
T’s time x T’s velocity + T’s head start = T’s distance.
When A catches up with T, A’s time in motion will equal T’s time in motion and A’s distance in motion will equal T’s distance in motion + T’s head start.*
Example:
T has a head start of 15 miles.
T and A start plodding/running from their respective positions at the same time.
When A runs for 15/14 hours at 15 miles an hour he travels a distance of 225/14 miles (16 and 1/14 miles).
In that same 15/14 hours, T (plodding at generous 1 mile an hour) travels a distance of 1-1/14 mile.
Adding the distance T travels in 15/14 hours to T’s head start of 15 miles, we see that T is exactly 16-1/4 miles from A’s starting point after plodding for 15/14 hours.
In sum, A catches up with T when both have been moving for 15/14 hours, at a distance of 16-1/4 miles from A’s starting point.
Moreover, once A catches up with T, A then moves farther ahead of T with each stride because A is running at 15 miles an hour, whereas T is moving at only 1 mile and hour.
There is a variant of the Achilles-Tortoise “paradox” which says that Achilles never reaches a goal because he gets halfway there, then half of the remaining half, and so on; that is, he gets infinitesimally close to the goal but never reaches it. It would be fair to point out that Achilles is able to get halfway to the goal, and halfway might have been chosen as the goal. But let’s proceed as if the Achilles must reach the original goal.
Why can’t he get there? Zeno assumes (without realizing or admitting it) that the goal keeps receding from Achilles, even as he runs toward it. That’s the only explanation that makes sense. Otherwise, if the goal is 15 miles from Achilles and Achilles runs at 15 miles an hour, he’ll be halfway to the goal in 30 minutes, three-fourths of the way to it in 45 minutes, and at it in 1 hour.
It’s true that Achilles will reach the halfway point, the three-fourths point, etc. But it’s not true that Achilles won’t reach the goal — unless, like the mechanical rabbit in dog racing — the goal keeps moving away from Achilles.
Travel involves distance and velocity. Aristotle/Zeno ignored the latter. They were either clever or stupid. Take your pick.
* Mathematically:
tA is A’s time in motion and tT is T’s time in motion.
tA = tT = t (the duration of the race) when A catches up with T, both having started at the same time.
dA is A’s distance from his starting point and dT is T’s distance from A’s starting point, which includes T’s head start: h.
dA = dT = d (the distance A travels) when A catches up with T.
dA = (vA)(t), where vA is A’s speed
dT = h + (vT)(t), where vT is T’s speed
Substituting into dA = dT, to find the duration (time) of the race:
(vA)(t) = h + (vT)(t)
(vA)(t) – (vT)(t) = h
t(vA – vT) = h
t = h/(vA – vT)
Given t, vA, and vT, it is trivial to compute d, the distance traveled by A when he catches up with T.
Politics and Prosperity 