Skeptoid - Skeptoid #267: Zeno's Paradoxes
Aired: 2011-07-19
Duration: 16:19


=== Zeno's Paradoxes Explained (08:04) ===

[00:00:03]  A paradox is a statement that seems to be inherently self-contradictory.
[00:00:08]  That is, when we observe that something appears to be some way, even though its very nature makes it impossible to be like that.
[00:00:16]  Now, if that isn't already confusing enough, we're going to deconstruct the most famous paradoxes in history.
[00:00:22]  And they come from a Greek philosopher named Zeno.
[00:00:27]  That's coming up next on Skeptoid.
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[00:01:09]  You're listening to Skeptoid.
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[00:01:14]  Zeno's Paradoxes.
[00:01:17]  Even if you think you haven't heard of them by name, you'll recognize them.
[00:01:21]  The most familiar of Zeno's paradoxes states that I can't walk over to you because I first have to get halfway there.
[00:01:28]  And once I do, I still have to cover half the remaining distance.
[00:01:31]  And once I get there, I have to cover half of that remaining distance.
[00:01:35]  Ad infinitum.
[00:01:36]  There are an infinite number of halfway points.
[00:01:39]  And so according to logic, I'll never be able to get there.
[00:01:43]  But it's easy to prove this false by simply doing it, which we can all do.
[00:01:48]  So we have a paradox, a contradiction, something that must be true, but which clearly is not.
[00:01:55]  Does there exist a solution which adequately addresses the contradicting phenomena?
[00:02:00]  Some say there is, some say there is not.
[00:02:05]  Zeno of Elea was a Greek philosopher born about 490 BCE and was a devotee of Parmenides, founder of the Eleatic School of Thought in what is now southern Italy.
[00:02:16]  Zeno survives as a character in Plato's dialogue titled Parmenides.
[00:02:21]  And from this we know what the Eleatic school was about and where Zeno was coming from with his paradoxes.
[00:02:27]  Parmenides taught, in part, that the physical world as we perceive it is an illusion and that the only thing that actually exists is a perpetual unchanging whole that he called one being.
[00:02:39]  What we perceive as movement is not physical movement at all, just different interpretations or appearances of the one being.
[00:02:48]  Personally, I think they smoked a lot of weed at the Eleatic school, but Zeno was into this and came up with his paradoxes in order to support Parmenides' view of the world.
[00:02:58]  Zeno's paradoxes were intended to prove that movement must be impossible.
[00:03:03]  Therefore, Parmenides must be right.
[00:03:08]  He's believed to have developed a total of about nine such paradoxes, but they were never published.
[00:03:13]  The most famous and interesting are his three paradoxes of motion.
[00:03:19]  First is the paradox of Achilles and the tortoise, who contrived to have a foot race.
[00:03:24]  Achilles, knowing he was the swifter, gave the tortoise a 100-meter head start.
[00:03:30]  In the time that it took Achilles to travel the 100 meters, the tortoise moved 10, so that when Achilles got there, he found the tortoise still had a lead.
[00:03:39]  In the time it took Achilles to run those 10, the tortoise moved another meter.
[00:03:44]  No matter how many times Achilles advanced to the tortoise's last position, the tortoise had crept forward a bit more by the time he got there.
[00:03:53]  Even though Achilles would seem to be the faster runner, it was impossible for him to ever catch the tortoise.
[00:04:00]  Second and most famous is the so-called dichotomy paradox, in which we repeatedly rend in twain every distance to be traveled.
[00:04:09]  For Homer to walk to the bus stop, he must get halfway there.
[00:04:13]  Once arrived, he must travel half the remaining distance, and so on and so on.
[00:04:18]  With one-eighth of the distance remaining, then one-sixteenth, then one-thirty-second, then one-sixty-fourth, he will have an infinite supply of remaining distances to travel, and thus can never arrive at the stop.
[00:04:32]  The third is the paradox of the Fletcher, who finds that all of his arrows are unable to move at all.
[00:04:37]  At any given instant in time, the arrow is motionless in flight.
[00:04:42]  During that frozen moment, the arrow cannot move at all, since it has no time in which to do it.
[00:04:48]  Time consists of an infinite succession of moments, in each of which the arrow is unable to move.
[00:04:55]  Nowhere can we find a given instant in which the arrow has time to move, and so no matter how many such instants we have, the arrow can neither fly nor fall to the ground.
[00:05:07]  Zeno's paradoxes are often touted by some people as evidence that physics or science are wrong.
[00:05:13]  If an ancient Greek philosopher can describe a simple situation, which our intuition tells us is obviously correct, it's easy for us to assign it more significance than we do the confusing jumble that is modern science.
[00:05:26]  Why should we listen to Einstein, who gives us a lot of unfathomable equations, when Zeno's elegant fables prove that the physical world is not as science tells us it should be?
[00:05:38]  Given this line of reasoning, it's hardly surprising that Zeno has become something of a darling to some New Age supporters of a spiritual, not a physical, universe.
[00:05:49]  Famously, upon hearing the paradoxes, a fellow philosopher named Diogenes the Cynic simply stood up, walked around, and sat back down again.
[00:05:59]  My kind of guy.
[00:06:00]  His response may have been glib, but it elegantly refuted Zeno's claim.
[00:06:05]  At least it refuted the physical implications of the claim.
[00:06:08]  It did not address the philosophical aspects, nor did it provide the mathematical solutions.
[00:06:16]  Zeno's paradoxes are an interesting intersection between mathematics and philosophy.
[00:06:21]  Mathematically, it's trivial to calculate exactly when and where Achilles will overtake the tortoise, but the philosophical argument remains, apparently, intractable.
[00:06:32]  Bertrand Russell described the paradoxes as immeasurably subtle and profound.
[00:06:37]  So philosophers have come up with some pretty interesting efforts to try and resolve this.
[00:06:43]  One such tactic concerns the Planck length, which is the smallest possible unit of length within the Planck system.
[00:06:51]  Planck units are all based on universal physical constants, such as the speed of light and the gravitational constant.
[00:06:58]  Philosophically, it's reasonably accurate to describe the Planck length as a quanta of distance, the smallest possible unit.
[00:07:07]  This means that there are a finite number of Planck lengths, albeit a staggeringly large number of them, along the racetrack of Achilles and the tortoise, and between Homer and the bus stop.
[00:07:20]  There cannot be an infinite number of points, and so Homer will eventually be able to arrive.
[00:07:26]  However, while this sounds like it might elegantly solve the paradox, it doesn't.
[00:07:31]  It's not possible to force a quantum solution onto a geometric problem.
[00:07:37]  A simple illustration of why this is so is to imagine a very small right triangle with its two equal sides each of one Planck length.
[00:07:46]  The hypotenuse would have to be the square root of two Planck lengths, which is not possible.
[00:07:53]  Planck doesn't apply here.
[00:07:55]  Despite efforts to conclude otherwise, we are dealing with infinities here.
[00:08:01]  Or are we?

=== Why Quantum Solutions Fail (06:44) ===

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[00:09:44]  Intuitively, we understand 0.99999 repeating to be a value that forever approaches 1 but never quite gets there.
[00:09:55]  This is fine as a concept and a thought experiment, but it is mathematically wrong.
[00:10:00]  0.9 does in fact equal 1.
[00:10:05]  They are simply two different ways of writing the same value.
[00:10:08]  It's easy to prove this to most people's satisfaction by dividing both values by 3.
[00:10:14]  Both 1 divided by 3 and point repeating 9 divided by 3 equal point repeating 3.
[00:10:22]  Therefore, both are equal to each other.
[00:10:25]  Another way of looking at it is to consider the fraction 1 9th, which is equal to 0.Repeating 1.
[00:10:32]  2 9ths is equal to 0.Repeating 2, and so on, all the way up to 8 ninths equals point repeating 8, and 9 9ths equals 0.Repeating 9.
[00:10:43]  And we all know that 9 9ths equals 1.
[00:10:46]  When we divide the number 1 into 9 equal slices, that top slice goes all the way up to exactly 1, a finite and reachable number.
[00:10:57]  If this spins your brain inside its skull, realize that you already accept many other interpretations of the same idea.
[00:11:04]  Consider any other number whose decimal value is an infinite repeating series, say 3 sevenths.
[00:11:12]  It equals point repeating 428571.
[00:11:16]  And we all accept that it equals 3 sevenths, not a number approaching 3 sevenths, but that never quite gets there.
[00:11:24]  It's two ways of writing the same thing.
[00:11:28]  It's the same concept when Homer takes his final step and places his foot down, completing his journey to the bus stop.
[00:11:35]  He did not take a journey of infinite length.
[00:11:39]  we can write an equation that describes how his final step consists of an infinitely reiterating series of smaller and smaller fractions, just as Zeno said.
[00:11:50]  And that equation would be described as the summation of n from 1 to infinity of 1 over 2 to the nth power.
[00:11:59]  We in the Brotherhood call this an absolutely converging series.
[00:12:03]  And contrary to Zeno's understanding, it equals 1.
[00:12:09]  Another popularly proposed solution, particularly for the Fletcher's paradox, involves time and speed.
[00:12:16]  Zeno, charges his critics, only considered the distances and geometry involved, and since he left time out of his paradoxes completely, he also excluded speed, since speed is a function of distance and time.
[00:12:31]  When a body is in motion, its position is always changing.
[00:12:35]  Motion is fluid.
[00:12:36]  It is not a ratcheted series of jumping from point to point.
[00:12:40]  Consequently, at any given moment in time, a moving body has no single exact position.
[00:12:48]  Zeno's conjecture that the arrow is always frozen at some point cannot be observed, reproduced, or computed, since that's not the way things move.
[00:12:58]  Imagine taking a photograph of a moving object.
[00:13:01]  There will always be some motion blur.
[00:13:03]  No matter how fast is the shutter of your camera, even infinitesimally fast, there will always be some tiny amount of blur.
[00:13:12]  There is no such thing as a moving arrow frozen in time.
[00:13:18]  Similarly, Zeno's computation that Achilles will never catch the tortoise also omits time.
[00:13:24]  Zeno's premise assumes that each segment of the race, wherein Achilles advances to the tortoise's previous position, takes some amount of time.
[00:13:33]  And since there's an infinite number of such segments, it will take Achilles an infinite amount of time.
[00:13:39]  This is also wrong.
[00:13:40]  As the physical length of each segment decreases exponentially in a converging series, so does the time it takes Achilles to traverse it.
[00:13:49]  Achilles' time to catch the tortoise is represented by a converging series that equals a finite number.
[00:13:58]  Achilles will catch the tortoise because the very succession of segments proposed by Zeno add up to a finite distance that Achilles will cover in a finite amount of time.
[00:14:10]  Homer will reach the bus stop because all of those infinitely compounding fractional segments are an absolutely converging series equal to a finite distance.
[00:14:22]  The Fletcher's arrow is always in motion once it is shot.
[00:14:26]  At no instant in time is it ever frozen with a fixed position from which it has no time to move.
[00:14:32]  So to summarize Zeno's paradoxes, they're basically word games that play upon an easily misunderstood mathematical concept.
[00:14:41]  There is no paradox because Zeno's math was wrong.

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