What Are Zeno's Paradoxes, Including Achilles, Dichotomy, Stadium, and Arrow?
To be rather brief, Zeno was an ancient Greek philosopher, and he thought up many paradoxes. He was a founding member of the Eleatic Movement, which, along with Parmenides and Melissus, came up with a basic approach to life: Don’t rely on your five senses to get a full understanding of the world. Only logic and math can fully lift the veil on life’s mysteries. Sounds promising and reasonable, right? As we shall see, such caveats are only wise to use when one fully understand the discipline, something Zeno couldn’t do, for reasons we shall uncover (Al 22).
Sadly, Zeno’s original work has been lost to time, but Aristotle wrote of four of the paradoxes we attribute to Zeno. Each one deals with our “misperception” of time and how it reveals some striking examples of impossible motion (23).
All the time we see people run races and complete them. They have a starting point and an ending point. But what if we thought of the race as a series of halves? The runner finished half of a race, then a half-of-a-half (a fourth) more, or three-fourths. Then a half-of-a-half-of-a-half more (an eighth) for a total of seven-eighths more. We can keep going and going on but according to this method the runner never finished the race. But even worse, the time the runner moves in is also halved so they reach a point of immobility as well! But we all know he does, so how can we reconcile the two viewpoints? (Al 27-8, Barrow 22)
Turns out this solution is similar to the Achilles Paradox, with summations and proper rates to be considered. If we think about the rate in each segment, then we would see that no matter how much I half each [portion the rate remains constant. Some have pointed out that for large velocities (like in relativity) and in small scales (like quantum mechanics), Zeno may have been onto something, but he had no clue about these disciplines and so his paradox remains solved (Al 28-9).
This is similar to the Dichotomy, but with a few variations. Imagine our Greek hero Achilles has accepted a race with a tortoise. Naturally, this is an unfair competition, so to give the tortoise a chance Achilles decided to let the Tortoise complete half of the race before he started. Zeno states that as Achilles completed the race, he kept cutting the distance between himself and the tortoise in half. Achilles will be ¼, then 1/8, then 1/16, and so on. But that means Achilles will never catch up with him but continue to cover half of the remaining distance as he progressively goes on, meaning we have arrived at an impossibility (Al 21, Barrow 21).
Poor Zeno never had calculus. If he had, then he could have thought of the limit of this series as it got smaller and smaller. We all know now that if we look at the overall pattern of this halving, it approaches a whole. That is because this is a geometric series with our common ratio being a half! But the Greeks did not have this tool at their disposal and thus were unable to solve this paradox. But this paradox also has a non-Calculus portion also. You see, Zeno only thought of the distance being halved, but for any rate to remain constant, as we have in the race, the time would also have to halved. We would thus arrive at the full time of the race when we look at the constant rate with the distance travelled held true (Al 24-5).
Imagine 3 wagon trains moving inside a stadium. One is moving to the right of the stadium, another to the left, and a third is stationary in the center. The two moving ones are doing so at a constant speed. If the one moving to the left started at the right side of the stadium and vice versa for the other wagon, then at some point all three will be at the center. From one moving wagon’s perspective, it moved a whole length when comparing itself to the stationary one but when compared to the other moving one it moved two lengths in that span of time. How can it move different lengths in the same time? (31-2).
For anyone acquainted with Einstein, this one is an easy solution: reference frames. From one trains perspective, indeed it seems to be moving at different rates but that is because one is trying to equate motion of two different reference frames as one. The speed difference between wagons depends on what wagon you are located in, and of course one can see the rates are indeed the same so long as you are careful with your reference frames (32).
Imagine an arrow that is on its way to its target. We can clearly tell the arrow moves because it reaches a new destination after a certain time has passed. But if I looked at an arrow in a smaller and smaller time window, it would appear motionless. So, I have a huge number of time segments with limited motion. Zeno suggested that this could not happen, for the arrow would simply fall out of the air and hit the ground, which it clearly doesn’t so long as the flight path is short (33).
Clearly, when one considers infinitesimals this paradox falls apart. Of course, the arrow acts that way for small time frames, but if I look at the motion at that moment it is more or less the same throughout the flight path (Ibid).
Al-Khalili, Jim. Paradox: The Nine Greatest Enigmas in Physics. New York: Broadway Paperbooks, 2012: 21 -5, 27-9, 31-3. Print.
Barrow, John D. The Infinite Book. New York: Pantheon Books, 2005: 20-1. Print.
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