Zeno’s Paradoxes

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In Zeno’s first paradox of motion, the Race Course, he argues that because a runner must reach the halfway point before he can reach the finish line and the halfway point to the halfway point ad infinitum that it is impossible to traverse the infinite number of these halves in a finite time.

In the second paradox of motion, titled the Achilles, this swiftest of runners will never be able to catch the turtle that began the race with a head start because every time Achilles catches up to where the turtle was, the turtle will have moved forward however slightly. Zeno says that he will get increasingly closer, but will never catch the turtle.

These are just two of Zeno’s paradoxes that were intended to illustrate via reductio ad absurdum that the generally held assumptions about motion, in these instances, and change in general are logically self contradictory. Thus our common opinion of the way the world works is an illusion, and therefore Parmenides must be correct in his assertions that there is only being and it is one, singular, indivisible and absolutely unchanging.

Zeno’s arguments have apparently generated much interest in the past century. Minimally, it would be historically interesting for more of the arguments against Parmenidean thought that provide the milieu in which Zeno made his famous arguments to have survived. Perhaps he may have been more successful in defending Parmenides in directly refuting the arguments of others than I think he is in these paradoxes.

Let us begin with the Race Course. There are numerous ways in which his argument is flawed. Initially, even if we assume that his logic is perfect, there is an implicit assumption that the mathematical/geometric system maps directly onto the world. This is almost certainly not the case notwithstanding its descriptive usefulness in modeling the world. Just because we can conceive of infinite divisibility of space does not make it so; just as being able to imagine pond fairies does not mean they exist. Strictly speaking mathematics is a coherent system of thought. It has proven useful for modeling various scientific theories, theories that are increasingly accurate approximations, but are approximations nonetheless. Fundamentally it is entirely plausible that discrete or finite mathematics in which the notion of continuity is neither required nor supported more closely approximates our world, as evidenced by quantum theory. To provide a limited metaphor for the physical discovery that energy cannot take any arbitrary value, but must be a multiple of an extremely small fixed amount, if one were driving a car at planck scale, one would only be able to drive at the speed of 10 mph or 20 mph or 30 mph but nowhere in between (see Planck constant).

However one need not resort to quantum physics to refute Zeno. In the Race Course there is a fixed beginning and a fixed end. It is linguistic sleight of hand to transition from the premise of the fixed race course to saying that an infinite distance needs to be crossed. Even if space is infinitely divisible, it most certainly does not mean that a fixed distance magically becomes infinite. Explaining that the runner must reach the halfway mark of the halfway mark et cetera does not change the total distance; all the halves will always add up to the same whole. If before a walk across town on my map of the city I start writing in the halfway points as Zeno describes, all I am doing is labeling a map.

Let us have a little more fun, and out do Zeno in his own game. Why can’t we just say that Zeno—unintentionally I am sure—just left out half the problem? Let us consider time. Most people would agree that it takes someone half as long to go half the distance and a quarter as long to go a quarter the distance and an eighth as long to go an eighth the distance; similarly ad infinitum. In fact, the division of time seems to be a perfect mirror image of the division of space. Consider that each time we divide the race course in half the new halfway mark is twice as close to us as the previous mark. If we do divide the course infinitely, in the first step the runner will not only have crossed the vast majority of halfway points, she/he will actually have crossed an infinite number of them. The first step, unsurprisingly, did not take an infinite amount of time.

When we consider Zeno’s second paradox of motion, the Achilles, we see a similar distortion of time to say the least. Simple mathematics of rates shows that an object in motion at a given rate in a given direction will be overtaken by an object in motion in the same direction at a greater rate. Maybe we can cut Zeno some slack for not knowing that distance equals rate times time. Let us take Zeno’s word, and assume that he continues to get closer and closer without overtaking him. If we consider after enough time that Achilles has gone 0.99999999999 (repeating) the total distance; let’s call this distance x.

If x = 0.99999999999 (repeating) and multiply both sides of this equation by 10

we get 10x = 9. 9999999999 (repeating), now we subtract x from the left, and its given equivalent from the right

and 9x = 9 and when we solve for x here,

x = 1

This is not a trick. The value 0.9 (repeating) is mathematically equivalent to 1. So unfortunately for Zeno even according to his own argument Achilles does catch the turtle.

What is really at stake here? Zeno is trying to convince us the world is not as it seems; that basically we are all deluded. Is it even possible for him to succeed, if we consider that he has to try to garner evidence to convince us that we cannot trust evidence. He seems to be in an impossible and inherently contradictory position. One might ask how he and any Parmenideans could come to their conclusions without necessarily having relied on the world they dismiss as illusion. How could Parmenides be strictly correct in his assertion of a divisionless whole if there are different people to disagree about it in the first place? Does the Parmenidean necessarily become a de facto solipsist?

If we are generous we might say that it is possible for Parmenides to have been correct about being and non-being, but he just got the scale wrong; that what he really meant—even if he did not know he meant it—was something more along the lines of recent cosmological theory regarding the entire universe and that even though it has a limit there is nothing outside of it, but to deny any change even of an internal variety still seems unjustifiable.

Parmenides certainly seems to have been influential. Pythagoras and Plato both seem to believe in some eternal plane of form that is somehow more real than the existence that we can know. I think it is a tragedy that Plato abandoned the Socratic beauty and simplicity of knowing that we do not know. The world would likely be a different place if more of Democritus’ works had survived as it is possible that the enlightenment could have begun with him if his arguments had held sway. Of course, tautologically, if things were different, they would be different. At the end of the day most limbs philosophers have leaned out on sooner or later have broken. Zeno certainly made some interesting arguments that are fun to this day, but I do not expect a resurgence of Parmenideanism any time soon.