Greeks, who had no concrete understanding of convergence or infinity, found such reasoning incomprehensible. Aristotle even discarded them as “fallacies” (Salmon 13). These paradoxes were hidden in the “math closet” for some 2500 years and just referred to as “novelties of philosophy.”
In the 19th century, they were brought out again by such people as Bertrand Russel and Lewis Carrol. But today Zeno’s paradoxes can be merely explained in minor detail. Nonetheless, controversies still arise of such a prevalent issue even today. Although many of Zeno’s Paradoxes have been continuously refuted, many even proven wrong, they are still very interesting to try and analyze to understand.
Background
Even though he was not a mathematician, Zeno, better known as Zeno of Elea, was the first prominent doubter of the mathematical world. Born around 495 B.C. in the Greek colony of Elea in southern Italy, Zeno worked as a student under the philosopher Parmenides. While Zeno accompanied Parmenides on a trip to Athens, he met the infamous Socrates. After leaving quite a lasting impression, Socrates characterized one of his characters in one of his books of Plato, entitled Parmenides, after Zeno.
Meanwhile, back in Elea, Zeno became engaged in politics. Due to his position in a conspiracy of plotting against the city’s tyrant, Nearchus, Zeno was tortured to death. There are several known stories about how the interrogation proceeded. “One anecdote claims that when his captors tried to force him to reveal the other conspirators, he named the tyrant’s friends. Other stories state that he bit off his tongue and spit it at the tyrant or that he bit off the Nearchus’ ear or nose” (Wachsmuth 3). The majority of information the scientific and mathematical world have today regarding Zeno and his paradoxes are accredited mainly to Aristotle and then by word of mouth.
Zeno wrote only one book, and from it only 200 pages have survived. Around 40 different paradoxes were believed to be written by Zeno. However, only eight have made notice. He composed his paradoxes to disprove pre-conceived theories of space and time. This essay presents four of them.
Has Anyone Figured This Out Yet?
Problems and Solutions of the Dichotomy:
The Dichotomy Paradox:
Paradox: “Motion cannot exist” (Paradox of Motion, 2). Huh? Okay, let’s go deeper: Before that which is in motion can reach its destination, it must reach the midpoint of its course. Before it can reach its midpoint, it must reach its quarterpoint. And before it reaches its quarterpoint, it must first reach its eighthpoint, etc. Hence, motion can never start. There’s never a beginning of a halfway point. Thus, motion cannot exist.
Zeno’s argument can be explained in more terms as well. Consider a ray of light reflecting between an infinite sequence of mirrors as displayed in Figure 1:
[ILLUSTRATION GOES HERE]
“While assuming that matter, time, and space are continuous and infinitely divisible, we conceive of a point-like particle traveling at a constant speed through a series of mirrors” (Paradox of Motion, 3). Next, this goes on to describe how “the sizes and separations would decrease geometrically at each step by a factor of two. The reflection would infinitely continue to decrease in size and therefore never hit a last mirror.”
Seems very sensible. Or does Zeno’s paradox merely go on to illustrate, flawlessly, how Greek philosophers of the yesteryear had the aptitude of explaining any mathematical theorem until it contradicts itself?
Refutation: Zeno argues that an infinite sum for any given set of infinite sums exists. This is indeed wrong, as any calculus textbook would clearly explain. A finite sum follows any set of infinite numbers.
This ties in with the refutation of his next paradox, The Achilles Paradox (which also deals with motion), to be explained in greater detail. Let’s see the next paradox of his and conclude how this repeal ties in…
Problems and Solutions of the Achilles
The Achilles Paradox:
Paradox: Assume you have a race between Achilles and a tortoise. Then suppose that Achilles runs 10 times as fast as the tortoise, and that the tortoise has a 10-meter lead at the beginning of the race. Zeno argued that in such a situation, it would take the Achilles an infinite amount of time to catch the tortoise.
By the time Achilles has run the 10 meters, the tortoise has traveled one meter, therefore placing him one meter ahead of Achilles. By the time Achilles has covered the distance of one meter, the tortoise has traveled 1/10 of a meter. The tortoise is still ahead of the Achilles. After Achilles has accomplished a distance, the tortoise has as well gained distance and is that much ahead of Achilles (source: Paradox of Motion 3).
Refutation: The faulty logic of Zeno’s argument is in his assumption that the sum of two infinite numbers is always infinite; quite the contrary, it’s definite. So you see, this clearly forces all around incredulity of this entire theorem: Once one apple is found to be sour, the whole bunch is dubbed spoiled. One false element disqualifies its entire theory. This is true in any scientific or mathematical postulate.
What Zeno attempted to do was divide Achilles’ race into an infinite number of pieces. He argues that an infinite number of tasks cannot be done in a determined amount of time. He then goes on to attempt to prove that while a line can be divided into an infinite number of pieces, a time interval cannot be divided infinitely since it is inconsistent.
To demonstrate this theorem using mathematical calculations, suppose the Achilles can run 10-meters per second and the tortoise runs 5-meters per second. They are running on a track that is 100-meters long. The tortoise is given a 10-meter advantage. Who’ll win?
Let’s see the difference in table 1:
…And so on.
In general, we have:
Now we want to take the limit as n travels infinitely to find out when the distance between Achilles and the tortoise becomes zero, but that’ll involve adding infinitely many numbers in the above expression to determine time. However, to define:
S n = 1 + ½ + ½ squared + ½ cubed + … + ½ n,
then dividing in half and subtracting the two expressions:
S n – ½ S n = 1 – ½ (n + 1),
Or, equivalently, solving for S n:
S n = 2 (1 – ½ n+1).
But now S n is a simple sequence, for which we now know how to take the limits. In fact, from the last expression it is clear that S n = 2 as n approaches infinity. “Hence, we have calculated that Achilles reaches the tortoise and wins the race” (Zeno’s Paradox, 2 – 3).
The Arrow Paradox
Paradox: Time is made up of instants, which are of the smallest, indivisible measure. An arrow is either in motion or at rest. For motion to occur, the arrow must be in one position at the start of the instant as well as the end. Thus, an arrow cannot move.
This would depict an instant to be divisible, and that’s just way out there. Again, this one bad apple has spoiled the bunch.
Refutation: Objects in relative motion have different concurrent planes, so not only does a moving object look different to the world, but the world as well looks different to the moving object (Wachsmuth 4). Zeno’s concern over the lack of an instantaneous difference between a moving and a motionless arrow is answered by the theory of special relativity.
The Stadium Paradox
Paradox: If a person is running towards an object from the west at maximum possible speed, while another is running the opposite direction at the same speed, they’ll approach each other at twice the maximum possible speed (Paradox of Motion, 4).
Another explanation of this paradox concerns two rows in a stadium with an equal number of bodies of identical lengths. These rows of bodies move in opposite directions toward the midpoint at equivalent speeds. Zeno concludes that half an interval of time equals its double. For example, let {A1, A2, A3, A4} be a set of stationary bodies all equal in length. Let {B1, B2, B3, B4} be another set of moving bodies of equal lengths starting from the A midpoint. And let {C1, C2, C3, C4} be the third set with speed contrary to that of B (insert diagram).
Refutation: Lorentzian invariance of time and space: a single piece of space progresses “evenly and equably” through time, to a “static representation in which the entire history of each world line already exists as a completed entity in the plenum of spacetime” (Paradox of motion 6). This representation is harmonious to the teachings of Parminides, whose arguments, ironically enough, were defended by Zeno himself.
Conclusion
Various mathematicians have disproved Zeno’s paradoxes by way of mathematical laws and equations, but the principles and questions of logic, which arise from them, cannot be dismissed. To say we have reached the conclusion of Zeno’s paradoxes, given the history of Aristotle’s Final Resolution, would be an absurd statement. Due in part to their questions of motion – or, in this case, the absence of – Zeno’s Paradoxes will always throw a “but” in many theories.
In fact, many of these paradoxes have been questioned in our everyday lives to date: The Achilles and the tortoise are now seen on Cartoon Network, and the arrow has stirred many meticulous conversations as well as jokes, for a few examples. But many more certainly exist across the world. Questions these paradoxes ask have baffled many mathematicians for centuries so far, as they will for centuries to come.