Introduce all Zeno’s paradoxes

Zeno of Elea

-Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides.-Best known for his paradoxes.

A. Paradoxes of Motion

1. The Achilles In a race, the quickest runner can never overtake the slowest, since the

pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. ( Aristotle Physics 239b15).

Achilles, who moves very fast, is in a race with a tortoise. Achilles has given tortoise a head start. Before Achilles can catch the tortoise he must reach the point where the tortoise started. But in the time he takes to do this the tortoise has moved to a new point. So next Achilles must reach this new point. Every time that Achilles reaches the point where the tortoise was, the tortoise has move to a new point, so Achilles never catches the tortoise.

Space is composed of points, which have no extension. Space is continuous in the sense that, between any two points, there is a

third. Space is infinitely divisible in the sense that a line can be cut in half and

then into quarters and so on. In order to catch the tortoise, Achilles would have to complete an infinite

number of tasks in a finite amount of time. Achilles can’t complete an infinite number of tasks in a finite amount of time, so Achilles can’t catch the tortoise.

2. The Dichotomy (The Racetrack)

The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics, 239b11)

This paradox is known as the ‘dichotomy’ because it involves repeated division into two.

It is the one about a thing's not moving because what is traveling must arrive at the half way point before the end.

Before the runner can travel a given distance d, he must travel the distance d/2. In order to travel d/2,he must travel d/4, etc. Since this sequence goes on forever. The distance d can’t be traveled (at least not in a finite time).

Another problem is:

The runner cannot even take a first step. As there is no first point in this series, one can never really get started.

So motion is illusory.

3. Arrow paradox

The third argument is that the flying arrow is at rest, which result follows from the assumption that time is composed of moments …. (Aristotle Physics, 239b.30)

First, Zeno assumes that it travels no distance during that moment—‘it occupies an equal space’ for the whole instant. But the entire period of its motion contains only instants, all of which contain an arrow at rest, and so, the arrow cannot be moving.

At every instant during its flight, a moving arrow is at some points in its circle and occupied a portion of space equal to itself. Thus it does not moving during this instant. But, then, it does not move during any instant of its flight.

Time is composed of instants, which have no extension. Time is continuous in the sense that, between any two times, there is

another. At no instants during the flight is the arrow in the motion, if it happened,

then the arrow is not in the motion during the flight, so the arrow is not in the motion during the flight.

4.The Stadium

It is about bodies moving in opposite directions with equal speed, and Zeno seems to think that twice the speed is the same as half the speed.

Two bodies B and C race round the stadium at the same speed but in opposite directions. A third body A is at rest. Suppose at some atomic instant , B racing left passes a unit length of A. Then in the same time, B and C pass two unit lengths of each other. But then they pass one unit in half the time, which is indivisible.

B. Zeno's Plurality Paradoxes

If something is divisible, then it is infinitely divisible. Now if each part has zero size, then the total has zero size, for an infinite number of zero lengths add up to zero. If on the other hand each part has some finite size, then the total is infinite, for an infinite number of finite lengths, however minuscule, must add up to an infinite total. So something divisible is either infinite or else has no size at all. Thus something finite is not divisible.

- The total number of things is both finite and infinite. It is finite because, if there are many things, then there must be as many as there are "neither more nor less". And in that case their number is limited, hence finite. But on the other hand if there are many things, they must be infinite in number, for between any two there must always be others, and between those others still, and so on. (This paradox apparently is meant to apply to spatial points,

rather than to physical objects.)

Limited and Unlimited

This paradox is also called the Paradox of Denseness. Suppose there exist many things. Then there will be a definite or fixed

number of those many things, and so they will be “limited.” But if there are many things, say two things, then they must be distinct, and to keep them distinct there must be a third thing separating them. So, there are three things. But between these, …. In other words, things are dense and there is no definite or fixed number of them, so they will be “unlimited.” This is a contradiction, because the plurality would be both limited and unlimited.

The paradox of place

"… if everything that exists has a place, place too will have a place, and so on ad infinitum”. (Aristotle Physics 209a25).

Given an object, we may assume that there is a single, correct answer to the question, “What is its place?” Because everything that exists has a place, and because place itself exists, so it also must have a place, and so on forever. That’s too many places, so there is a contradiction. (Aristotle’s Physics 209a23-25 and 210b22-24).

The paradox of millet

… Zeno's reasoning is false when he argues that there is no part of the millet that does not make a sound; for there is no reason why any part should not in any length of time fail to move the air that the whole bushel moves in falling. (Aristotle Physics, 250a19).

If you drop a sack of millet on the floor, it makes a sound. The sack is composed of individual grains, so they make an audible sound. But if you drop an individual millet grain or a small part of one, then eventually your hearing detects no sound, even though there is one. Therefore, you cannot trust your sense of hearing.

Large and Small

Suppose there exist many things. These things must be composed of parts which are not themselves pluralities. Yet things that are not pluralities cannot have a size or else they’d be divisible into parts and thus be pluralities themselves.

But the parts of pluralities are so large as to be infinite. The parts cannot be so small as to have no size since adding such things together would never contribute anything to the whole so far as size is concerned. So, the parts have some non-zero size. If so, then each of these parts will have two spatially distinct sub-parts, one in front of the other. Each of these sub-parts also will have a size. The front part, being a thing, will have its own two spatially distinct sub-parts, one in front of the other; and these two sub-parts will have sizes. Ditto for the back part. And so on without end. A sum of all these sub-parts would be infinite. Therefore, each part of a plurality will be so large as to be infinite. Thus every part of any plurality is both so small as to have no size but also so large as to be infinite.

Infinite Divisibility

Imagine cutting an object into two non-overlapping parts, then similarly cutting these parts into parts, and so on until the process is complete. Assuming the hypothetical division is “exhaustive” or does comes to an end, then at the end we reach what Zeno calls “the elements.” Here there is a problem about reassembly. There are three possibilities. (1) The elements are nothing. In that case the original objects will be a composite of nothing, and so the whole object will be a mere appearance, which is absurd. (2) The elements are something, but they have zero size. So, the original object is composed of elements of zero size. Adding an infinity of zeros yields a zero sum, so the original object had no size, which is absurd. (3) The elements are something, but they do not have zero size. If so, these can be further divided, and the process of division was not complete after all, which contradicts our assumption that the process was already complete. In summary, there were three possibilities, but all three possibilities lead to absurdity. So, objects are not divisible into a plurality of parts.

Alike and Unlike

If things are many, . . . they must be both like and unlike. But that is impossible; unlike things cannot be like, nor like things unlike” (Hamilton and Cairns (1961), 922).

Consider a plurality of things, such as some people and some mountains. These things have in common the property of being heavy. But if they all have the property of being heavy, then they really are all the same kind of thing, and so are not a plurality. They are a one. By this reasoning, Zeno believes it has been shown that the plurality is one (or the many is not many), which is a contradiction. Therefore, by reductio ad absurdum, there is no plurality.

The paradox is solved by Plato and it is now rarely discussed.