Zeno's Paradoxes (Achilles, the Dichotomy, the Arrow)

Achilles runs 10 meters per second, ten times as fast as a tortoise, and gives it a 100-meter head start. When he reaches the tortoise’s starting point, it has crawled 10 meters further. When he covers those 10 meters, it is 1 meter ahead, and when he covers that meter, 0.1 meters.

Every time Achilles reaches the place where the tortoise was, the tortoise has moved on. According to Aristotle, Zeno of Elea concluded that the quickest runner can never overtake the slowest in a race^[1].

Adding up the stages gives a definite point where Achilles draws level, a little over 111 meters from his start.

$$100+10+1+0.1+\cdots =\frac{1000}{9}$$

The argument depends on whether a runner can complete infinitely many stages, one after another, in a finite time.

Zeno and Parmenides

Zeno was born around 490 BCE in Elea, a Greek city in southern Italy^[2]. He was a friend and student of Parmenides, an older philosopher from the same city^[3].

Parmenides set out his views in a poem. In its central argument, what is cannot come into being or perish, is whole and uniform, and does not move. Read strictly, this rules out change, motion and plurality altogether, and interpreters disagree about whether Parmenides meant it that strictly^[4].

Plato’s dialogue Parmenides has the two visit Athens for the Great Panathenaea, Parmenides about 65 and Zeno nearly 40, while Socrates is still very young. There Zeno says that his book was written to protect the arguments of Parmenides against people who made fun of him^[5].

The book worked indirectly. It took the opposing view as a premise and set out to derive absurd conclusions from it^[3,6].

Proclus reports that the book contained forty paradoxes^[2], and about ten of Zeno’s paradoxes are known today^[3]. Four are about motion, and Aristotle’s Physics reports all four, calling them arguments that “cause so much disquietude” to those who try to solve them^[1].

The Dichotomy

The first of the four, as Aristotle states it, denies that motion exists, because a moving thing must reach the halfway point before it reaches the goal^[1]. From the halfway point the same demand applies to the remaining half, and then to half of that.

For a run of 1 meter, the first stages are these.

No row reaches 1. Each row leaves a remainder and the next stage covers only half of it, so if the run is a list of stages to be completed one by one, the list has no last stage.

The argument also runs in the other direction. Before the halfway point comes the quarter point, and before that the eighth.

Every distance has a shorter one that must be covered first. There is no first distance, and on this version the runner cannot even take a first step^[3].

Achilles and the tortoise

The second argument, called the Achilles, says that the pursuer must first reach the point the pursued started from, so the slower runner always holds a lead^[1]. With the numbers from the opening, the first four stages look like this.

Each stage is one tenth of the one before, in distance and in time, and the lead is not zero at the end of any stage.

The distance left to the point where the two draw level shrinks the same way, and every stage covers nine tenths of it. The Achilles has the same form as the Dichotomy, with nine tenths in place of a half^[6].

Adding up the stages

After $n$ stages, the distance Achilles has run, in meters, is

$$100+10+\cdots +\frac{100}{10^{n-1}}=\frac{1000}{9}\left(1-\frac{1}{10^n}\right)$$

The factor in parentheses falls short of 1 by $\frac{1}{10^n}$, which becomes smaller than any positive number as $n$ grows. The sum of the whole series is defined as the number the partial sums approach, here $\frac{1000}{9}$ meters.

The times form a series of the same kind, $10+1+0.1+\cdots =\frac{100}{9}$ seconds, about 11.1 seconds.

The meeting point can also be found without a series. After $t$ seconds, Achilles is $10t$ meters from his start and the tortoise is $100+t$ meters from it, so they are level when

$$10t=100+t$$

This gives $t=\frac{100}{9}$, and by then Achilles has run $\frac{1000}{9}$ meters.

The stages of the Dichotomy add up in the same way, $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots =1$.

The partial sums never reach 1^[2], and the definition does not say they do. The sum of the series is their limit, which is not itself one of the partial sums^[3].

A theory of infinite sums that settles such cases was not fully worked out until Cauchy, in the nineteenth century^[6]. The treatment of motion built on this mathematics, which also draws on Weierstrass, Dedekind, Cantor and Lebesgue, is sometimes called the Standard Solution^[3].

Aristotle’s two replies

Aristotle’s first reply, in Book VI of the Physics, does not add anything up. Zeno assumes that nothing can pass over infinitely many things in a finite time, and Aristotle answers that a finite time is itself infinite in respect of divisibility^[1].

If a steady runner covers the whole distance in one minute, the first half takes 30 seconds, the next quarter 15 seconds, and each later stage half the time of the one before. The time divides exactly as the distance does^[6].

In Book VIII he returned to the argument and judged that this could not be the end of the matter. The reply divides the time into infinitely many pieces, and a time made of infinitely many pieces seems to be infinite^[6].

His second reply separates two ways in which the halves can be present in a run^[6].

Achilles, on this view, covers only a potential infinity of distances, never an actual one^[3].

The Arrow

The third argument, in Aristotle’s report, says that if everything is at rest when it occupies a space equal to itself, and a moving thing is always in a “now”, then the flying arrow is motionless. Reconstructed step by step, it has three parts^[6].

Aristotle rejects the second step. Time, he says, is not composed of indivisible moments, any more than any other magnitude is composed of indivisibles^[2,3].

The Standard Solution accepts the first two steps and denies that the third follows from them. On what is called the at-at theory, motion is being at different places at different times, and nothing has to happen within a single instant^[3].

Speed at an instant is also defined without any motion during the instant. It is the derivative of position with respect to time^[3], the limit of distance over time for shorter and shorter intervals.

$$v(t)=\underset{h\to 0}{\lim }\frac{x(t+h)-x(t)}{h}$$

Within a single instant the arrow would cover 0 meters in 0 seconds, and $0/0$ is not a number^[6]. The limit never divides by an interval of zero length.

Henri Bergson, who thought this picture absurd, described it with the words “movement is composed of immobilities”^[6].

The Stadium

The fourth argument, the Stadium, uses rows of equal bodies moving past each other^[1]. In the usual reconstruction one row stands still while the other two move in opposite directions at the same speed. In one interval, the front body of a moving row passes two bodies of the other moving row but only one body of the row at rest^[3,6].

Aristotle says the argument rests on a false assumption^[1]. If space and time come in smallest units, the argument raises a problem of its own. The rows then jump from one position to the next, and a body that passes two others within one smallest time is never level with the first of the two, since no instant lies between neighboring smallest times^[6].

Infinitely many tasks

The sums show that the stages of a run have a finite total length and a finite total time. Whether anything can complete infinitely many separate acts in a finite time is a further question, discussed today under the name of supertasks^[6].

James Thomson’s lamp, from 1954, makes the question concrete. The lamp is switched on for half a minute, off for a quarter minute, on for an eighth, and so on, and one can ask whether it is lit or dark at the end of the minute^[3].

Thomson took the lamp to show that such tasks are impossible. The series of switchings has no last member and its limit is not one of its terms, so the setup does not fix the state of the lamp at the end, and the argument does not establish the impossibility. There is still no agreement among philosophers on what the lamp shows^[3].

References