Was Zeno Right About the Impossibility of Motion?

On Zeno and the paradoxes of motion

From the painting Zeno of Elea shows Youths the Doors to Truth and False, by Pellegrini Tiballdi

One bearded sage concluded: there's no motion.
Without a word, another walked before him.
He couldn’t answer better; all adored him
And all agreed that he disproved that notion.
But one can see it all in a different light,
For me, another funny thought comes into play:
We watch the sun move all throughout the day
And yet the stubborn Galileo had it right.

~ Motion, Alexander Pushkin

Around the year 500 BC, the bearded sage Zeno proved the impossibility of motion. His premise was simple: imagine an arrow in flight, travelling from bow to target. Then examine the arrow at a single instant of time. What does it look like?

Obviously, Zeno said, the arrow will appear as though it were stationary. It will have its usual length and occupy its normal amount of space. It will not move, nor will it show any sign of future movement. From such an instantaneous examination, indeed, it would be impossible to deduce whether the arrow was at rest or in motion.

This same observation must hold true at every moment of the arrow’s flight. So when does it move? If at every instant the arrow is stationary, and time is simply made up of a succession of instants, then how can its motion be possible? When do things move, if not in any of the moments that make up time?

But things do move, and so this is a clearly paradoxical result. Zeno used it to argue his case that all motion was an illusion, something he did in support of the ideas of his master, Parmenides, who believed the cosmos was a single and unchanging reality. Any appearances to the opposition - that of motion, or of change - were deceptions that reason and logic could dispel.

Of course, many took that explanation as equally absurd. Motion is obviously possible. It is said Diogenes the Cynic contradicted Zeno by simply getting up and walking away after hearing his argument.

Even so, Zeno’s paradoxes have proved hard to put fully aside. The demonstration that we can move does not by itself contradict the underlying logic of his argument. And even Zeno himself could hardly have believed that walking was impossible, or that the arrow would never reach its target.

Instead he thought the paradox proved our conception of reality was incorrect, and that the deeper, unchanging world described by Parmenides was the true reality. Many of his other paradoxes attempted to prove the same idea, and often focused on the logical difficulty in moving from the one - the static single instant of time - to the many - the continuous flow of instants of time we actually perceive and in which change can occur.

Solving this difficulty - and therefore putting our conceptions of time, space, and movement, on a surer footing - took centuries. Indeed, it was not really until the work of Newton, Leibniz, and Cantor that we got a proper mathematical framework to solve Zeno’s paradox. And even today, as we probe the strange realities of quantum theory and relativity, the questions raised by his paradoxes still find relevance.

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A Question of Infinity

Zeno’s paradox follows a form of argument we now call reductio ad absurdum. It works by first taking a set of assumptions we hold as true - that, for example, the arrow moves. Then, through logical argument, those assumptions are shown to lead to an absurd conclusion - that the movement must somehow occur outside of time. Since the logic cannot be denied and the result is clearly absurd, the original assumptions must be flawed.

Often, however, there are more assumptions in play than are immediately obvious. Zeno believed his paradox showed that our common beliefs about motion are wrong. But those who attack it often argue he had overlooked some hidden assumptions about the nature of time and of infinity.

Indeed, in another of his paradoxes, one often called the dichotomy, Zeno made an argument explicitly based on the nature of infinity. Think again, it says, of an arrow flying towards its target. Over its flight it must cover the distance from bow to target in some finite amount of time.

Before the arrow reaches the target, it must first reach the point that lies halfway between the bow and target, a step that takes about half the time of the overall journey. After this the arrow must then fly half the remaining distance, again in half the remaining time. And then a half of what is left again, and so on, with the distance and time getting shorter with each successive step.

Although this seems like an odd way to frame the flight, there is nothing wrong with the logic. And yet again the conclusion seems absurd: the distance can be divided in half an infinite number of times, and so the arrow must complete an infinite number of steps in the finite time available. That is impossible, Zeno argued. Nothing could possibly do an infinite number of tasks in a finite time. Once again, the concept of motion seems to produce a paradox.

Yet today we know Zeno was wrong. The mathematics of infinite series - developed about two millennia after he posed this paradox - show that under the right conditions you really can accomplish an infinite number of things in a finite time. The assumptions of the paradox were flawed, but it was the hidden assumption about infinity that was mistaken, not the one about motion.

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The Infinite and Infinitesimal

But what about the arrow frozen in time? Summing an infinite series does not immediately seem to help, since the arrow still does not move in any of the infinite steps.

But maybe we should question the idea of an “instant” of time. This concept plays an important role in framing the paradox. Zeno implicitly assumes time is made up of a sequence of these instants, rather like a movie is made from a series of still images.

Yet does time really function in this way? We experience time as something continuous, and we cannot stop it or freeze it. Even the fastest photograph still captures a span of time and thus a period of motion, rather than a single frame of reality. Neither can we ever subdivide time to reach a moment of zero duration or some kind of fundamental unit of time equivalent to an atom of matter.

Zeno’s paradox of the arrow, rather than ruling out the possibility of motion, might actually be telling us that time is not something that can be frozen, and that the very idea of an instant of time is flawed. All we can do is reach ever smaller spans of time - but no matter how small we make those spans, motion is still present.

Indeed, the modern mathematics of motion, developed by Newton and Leibniz, does not consider instants of time at all. In calculus we talk of infinitesimally small periods of time, of durations closer to zero than any number can be. These periods of time are not instants, they are not of duration zero, but they are so short that for all practical purposes they cannot be distinguished from zero.

In a sense they can be regarded as a logical continuation of subdividing time. If I can walk one metre in one second, I can also cover half a metre in half a second, and so on, all the way down into the billionths and trillionths of a second. Eventually, at the very bottom, I can cover an infinitesimally small distance in an infinitesimally small time.

This, then, is the modern answer to Zeno. Instants of time in which motion is frozen do not exist. But subdivisions of time so small that they might as well be regarded as instants do exist, and within them movement continues. The journey of the arrow may be made of an infinite number of infinitesimals¹ - but that is okay, since an infinite sum of motions can add up to a finite journey.

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The Quantum Zeno Paradox

Except, perhaps it is not really okay at all. There is a second, more subtle problem with Zeno’s assumptions: the idea that we can make measurements of time and position at all. As Zeno goes on endlessly subdividing the arrows flight, he will at some point cross the border separating the classical world from the quantum one - and once he does, things get really strange.

In the quantum world objects are defined by their wave functions, a quantity that evolves over time according to the laws of quantum mechanics. The wave function does not tell us precisely the state an object is in - its position, for example - but rather the probabilities we have of finding it in any possible state.

Whenever we make a measurement, the wave function collapses. The object is forced to choose between its possible states, and does so in accordance with the probabilities in its wavefunction. After each measurement the wave function begins evolving again, and once more the probabilities of all the possible states begin to shift.

It is possible, however, to trap the wave function. If someone measures an object at high frequency, its wave function can never evolve away from its initial state. Every measurement finds the object in the same place, resets the wavefunction, and so, if done fast enough, renders motion impossible. A watched quantum pot, it is sometimes said, never boils.

In a sense, then, Zeno was right: if we break time into small enough chunks and insist on checking the position of the arrow at every one, its motion will cease. This is not a theoretical paradox, either: scientists have really done this, and have trapped atoms using what physicists call the Quantum Zeno Effect.

What all that means for the nature of reality is still uncertain. Yet one thing is for sure: physics is not done with Zeno yet. For all our cleverness with infinities and the nature of time, the deep questions his paradoxes of motion raised about space, time, and reality itself still linger on.

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1. You could ask whether infinitesimals have any more physical reality than instants. The question troubled mathematicians for centuries and although the concept is generally accepted today, we still lack a real physical understanding of spacetime at the smallest scales.