Zeno was a Greek philosopher living in Southern Italy at about
350 BC. He was engaged in an argument between schools of philosophers
regarding motion: 1) Was motion continuous or discontinuous? 2) Was
motion real or unreal? Zeno invented his paradoxes to support his side
of this argument. They are not truly paradoxes, but they are very clever
and have been faking people out for over two-thousand years. Let us
examine Zeno's Paradoxes.
There are nine surviving paradoxes, but all share a similar error, and you can easily generate more by the paradox-load. The author's favorite is The Achilles and the Tortoise Foot Race. Achilles, famous Greek warrior, hero of the Trojan War, runs a foot race against Torty, the World's most famous Tortoise. One can set this up an infinite number of ways. Let us endow Achilles with 10 times the speed of Torty, set the race at 1.25 mile, and give Torty a one mile lead. They are off and Achilles runs the mile lead in 0.1 unit of time (remember - ten times as fast). But during that 0.1 time unit, Torty runs 0.1 mile and so is still ahead. Well, Achilles runs the 0.1 mile in 0.01 time units, but Torty uses the 0.01 to run another 0.01 mile. This can be repeated forever, but with each repetition the time gets shorter. An infinite repetition gets to 0.111111... time units, where "..." is math speak for "repeats forever". The sum of these times, 0.111111..., is exactly 1/9 time unit. It will take Achilles 0.1250 time units to run the mile and one-quarter. Zeno's reporting system stops at 0.111111..., so it never gets to the end of the race. Clever Zeno offers no paradox, just a flawed reporting system.
If you have been hit by a car, a bicycle, someone on a skateboard, you probably have decided motion is real. Is it continuous? In year 2011, we still do not know. It has been suggested that space may "be foamy" or "have a structure" at a very very minute distance called the Planck length. The Planck length is 1.6 x 10^-35 meter. This is a whole lot smaller than a single electron or a single proton, so neither Achilles, Torty, nor we could ever be aware of it. It is probably just right for philosophical debate.
These 350 BC Greeks used a reasoning step that we now know to be false. Since at every given instant Achilles, Torty, or an Arrow shot at someone, (from the Arrow Paradox), had to be somewhere, occupying a space its own size: an Achilles size space for Achilles, Torty size for Torty, Arrow size for the Arrow. They saw this as evidence that at any given instant the item (Achilles, Torty, or Arrow) was motionless. Therefore, they argued, at any step on its journey, a moving item is motionless. Bolstered by the Calculus invented late 1600 s, by Gottfried Leibniz and Isaac Newton, we now know that an object in motion is always in motion and its motion can be computed even for each single point on its path. There is no "motionless". The false motionless argument, and the erroneous data reporting together accounted for the "paradoxes".
There are nine surviving paradoxes, but all share a similar error, and you can easily generate more by the paradox-load. The author's favorite is The Achilles and the Tortoise Foot Race. Achilles, famous Greek warrior, hero of the Trojan War, runs a foot race against Torty, the World's most famous Tortoise. One can set this up an infinite number of ways. Let us endow Achilles with 10 times the speed of Torty, set the race at 1.25 mile, and give Torty a one mile lead. They are off and Achilles runs the mile lead in 0.1 unit of time (remember - ten times as fast). But during that 0.1 time unit, Torty runs 0.1 mile and so is still ahead. Well, Achilles runs the 0.1 mile in 0.01 time units, but Torty uses the 0.01 to run another 0.01 mile. This can be repeated forever, but with each repetition the time gets shorter. An infinite repetition gets to 0.111111... time units, where "..." is math speak for "repeats forever". The sum of these times, 0.111111..., is exactly 1/9 time unit. It will take Achilles 0.1250 time units to run the mile and one-quarter. Zeno's reporting system stops at 0.111111..., so it never gets to the end of the race. Clever Zeno offers no paradox, just a flawed reporting system.
If you have been hit by a car, a bicycle, someone on a skateboard, you probably have decided motion is real. Is it continuous? In year 2011, we still do not know. It has been suggested that space may "be foamy" or "have a structure" at a very very minute distance called the Planck length. The Planck length is 1.6 x 10^-35 meter. This is a whole lot smaller than a single electron or a single proton, so neither Achilles, Torty, nor we could ever be aware of it. It is probably just right for philosophical debate.
These 350 BC Greeks used a reasoning step that we now know to be false. Since at every given instant Achilles, Torty, or an Arrow shot at someone, (from the Arrow Paradox), had to be somewhere, occupying a space its own size: an Achilles size space for Achilles, Torty size for Torty, Arrow size for the Arrow. They saw this as evidence that at any given instant the item (Achilles, Torty, or Arrow) was motionless. Therefore, they argued, at any step on its journey, a moving item is motionless. Bolstered by the Calculus invented late 1600 s, by Gottfried Leibniz and Isaac Newton, we now know that an object in motion is always in motion and its motion can be computed even for each single point on its path. There is no "motionless". The false motionless argument, and the erroneous data reporting together accounted for the "paradoxes".
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