Zeno’s Paradox? Not so much

Family.

 

The first time I heard the tale of Zeno’s paradox was in childhood, one night after dinner, with the family still around the table. My father grabbed a random section of the day’s newspaper, called for a pencil, and with occasional cramped diagrams in the margins of the newspaper, intense and intent, showed us something that fascinated. Such moments with my father I still treasure. His telling of Zeno’s paradox inspired me these years later to attack the subject more vigorously.

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Zeno of Elea, an ancient Greek mathematician and philosopher, is famed for his paradoxes of motion. One involves shooting an arrow at a target, which he argues first must move half-way to its target, then half-way again from the midpoint, then half-way again, never quite reaching the target; in fact, in order to get to the first half-way point, the arrow must go half-way to it, and to get to that point, it must go half-way to it, etc., requiring an infinite number of steps to reach the target. According to Zeno, then, you can never get to where you are going.

Out on a limb, Zeno went all the way. In the grand tradition of reductio ad absurdum, oft employed by not just ancient philosophers but modern day zealots, Zeno suggested that in fact motion is impossible. His paradox is strikingly silly: after all movement is ubiquitous. But where is the flaw in his ‘logic’?

Escaping the logical labyrinth

The answer lies in the simplest physics, that the motion of an object is defined by its displacement for a specific distance in a certain period of time. If we proceed at some speed, which is a measurement of displacement over time, we will move a certain distance in a given time period – period. This turns out to be the key to finding a way out of the paradox.

To directly refute Zeno’s paradox we can employ the strange and beautiful properties of infinite series. If we add up all of the counting numbers, that is the integers 1 + 2 + 3 all the way up to infinity, the sum of all counting numbers has an infinite value. This can be written 1 + 2 + 3 + . . . ∞ = ∞, where the . . . tells us to keep adding all of the consecutive values between 3 and ∞, and represents an example of an infinite series. Not surprisingly, the resultant of most infinite series is an infinite value. The first step out of Zeno’s trap is to recognize that the sum of certain infinite series are finite! (These are called summable infinite series.)

Zeno is implying that to take an infinite number of steps, each smaller than the next, requires an infinite amount of time. To test this, let’s assume that the distance we want to travel is one mile and that we want to arrive at our destination in one hour, so that we are moving at a constant one mile per hour. We can then write down Zeno’s recipe for our movement like this:

1 mile = 1/2mi + 1/4mi + 1/8mi + 1/16mi + 1/32mi + . . . 1 / ∞mi

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The sum of this infinite set of displacements, with each successive displacement smaller than the last, approaches closer and closer towards the value of 1 mile, until with infinitely many terms, it reaches exactly one mile! The graph above shows the successive sums of the first eight terms, and the eighth term brings it already to almost 1.

Similarly and critically, for each step we are moving at a constant speed of one mile an hour, so the first step of 1/2mi takes 1/2hr, the second step of 1/4mi takes 1/4hr, etc. and we find, not surprisingly at this point, that

1 hour = 1/2hr + 1/4hr + 1/8hr + 1/16hr + 1/32hr + . . . 1 / ∞hr

Zeno, motionless. Even the cherub looks skeptical.. Attrib: from Raphael's The School of Athens, PD-US.

 

And we arrive right back to basic physics: A finite speed of 1 mile per hour. So Zeno’s recipe of taking a finite distance and dividing it into infinitely many steps does not render motion paradoxical because those steps take place in a finite amount of time! The logical flaw in Zeno’s “paradox” is that each subsequently smaller step takes proportionally less time, rather than a fixed amount of time.

Armed with this argument, even someone as confused as Zeno would leave his bed and move about; or, in yet another of Zeno’s motion paradoxes, Achilles would just run past the tortoise rather than sit still arguing with him, a dazed look on his face. (How much wit has a man who stops to argue with a tortoise?) He would not be dizzied by what is ultimately a logical shell game.

Enter Newton

Isaac Newton explored summable infinite series, as it turned out one of the steps towards his construction of the first methods of the calculus, which he invented to help characterize and quantify: Motion! As Newton was groping for a means to mathematically characterize motion, he suggested:

"Of motion. That it may be knowne how motion is swifter or slower consider: that there is a least distance, a least progression in motion & a least degree of time.… In each degree of time wherein a thing moves there will be motion or else in all those degrees put together there will be none:… no motion is done in an instant or intervall of time."(Isaac Newton, Questiones quaedam philosophicae, 1665, XLIII)  (note 1)

A least progression in motion AND a least degree of time . . . In this short exposition, Newton succinctly described the argument just presented: Each term of the pair of infinite series above get smaller and smaller, approaching the infinitely small, recognizable today as the infinitesimal of the calculus, the Almost Nothing, which can be employed to describe in this case the infinitesimally small change in displacement over an equally infinitesimally small change in time: dx/dt. Newton’s characterization of summable infinite series was a vital step to his, and still our, modern physical definition of motion, which is the derivative of displacement with respect to time.

Where did Zeno go wrong?

Martin Gardner notes, using another of Zeno’s paradoxes of motion, a runner rather than an arrow, that if a runner paused even briefly between each halving of distance to some finish line, they would indeed never reach the finish line, just as Zeno argued! The motion of a periodically pausing runner would be described by a discrete instead of a continuous function, in the language of modern mathematics. (notes 2,3)

Of course, runners in a race routinely reach the finish line in a continuous movement or steady rate, just as an arrow in flight, propelled ballistically, does not pause on its way to the target. Indeed, the crux of our solution using summable infinite series is that this function of displacement over time must be continuous; not coincidentally, this is also the fundamental starting point of Newton’s calculus. With a precise idea of continuity was born this explanation for the paradoxes of motion and further, the explanatory wonders of the calculus and its efficacy in modeling changes perceived in the world around us.

Perhaps Zeno became lost in his logical labyrinth by thinking in discrete rather than continuous terms.

Reaching the limits

Clearly, Zeno’s thought experiments had limits, in his case logically incongruous ones. But so do all idealized arguments have limits. Note that Newton’s calculus is based on infinities, ultimately unreachable in the real world. The calculus is a great leap forward because it allows for solutions to approach infinity, to get close enough to be immensely useful, or in engineering terms, useful enough.

Finally, it might be surprising to note that Zeno’s discrete step argument itself was false in the practical limit. Any actual runner, in attempting to repeatedly halve the distance she moved, would reach the point where she was taking a single stride, then where she was moving forward mere inches, until she reached a finite limit: the smallest step she was able to make. At that point, even pausing between each movement, she would eventually, slower perhaps than a tortoise or snail, cross the finish line.

Notes

1. Isaac Newton’s Questiones quaedam philosophicae, or Certain Philosophical Questions, even though titled in Latin, were written in English. They provide a very detailed picture of Newton’s early interests, and record his critical appraisal of contemporary issues in natural philosophy. Written predominantly in 1664–5, they give a significant perspective on Newton’s thought just prior to his annus mirabilis, 1666. I came across this thought of Newton’s in James Gleick’s biography of Isaac Newton, on page 43. Newton’s full notebook of Questions can be found in Certain Philosophical Questions: Newton’s Trinity Notebook, by J. E. McGuire and Martin Tamny, who provide a complete transcription of the Questiones, together with an ‘expansion’ into modern English, and a full editorial commentary on the content and significance of the notebook in the development of Newton’s thought.

 

Isaac Newton, by James Gleick

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Certain Philosophical Questions, by J.E. McGuire, Martin Tamny

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2. From Calculus Made Easy by Sylvanus Thompson and Martin Gardner. This wonderful book presents the basics of the calculus from an engineering point of view. It includes Gardner’s own explication of infinite series and a short discussion of Zeno’s paradox.

 

Calculus Made Easy, by Silvanus P. Thompson, Martin Gardner

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"Does the runner ever reach the finish line? It depends. We can model this with a pawn ... that you push across a table from one edge to the edge opposite. First you push the pawn half the distance, then pause for a second. You push it half the remaining distance and again pause for a second. If this procedure continues, the pawn will get closer and closer to the opposite edge but never reach it. ... Suppose, now, that instead of waiting a second after each pawn push, the pawn moved at a steady rate. Assume that the constant speed is such that the pawn goes half the distance in one second, half the remaining distance in half a second, and so on. No pauses. A discrete process has been turned into a continuous one. In two seconds the pawn has reached the table's far edge. Zeno's runner, if he goes at a steady rate (continuously), will reach the finish line in a finite time. The halving series, modeled in this fashion, converges exactly on the limit."(Martin Gardner, Calculus Made Easy, pp. 19-20) 

 

Two New Sciences, by Galileo Galilei, Stillman Drake

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3. In 1638 Galileo addressed Zeno’s paradox, anticipating Gardner’s argument by several hundred years. In his Dialogue Concerning Two New Sciences, Salviati (Galileo’s voice), addressing uniform acceleration, says: "(Zeno's paradox) would happen if a moving body were to hold itself for any time in each degree; But it merely passes there without remaining beyond an instant. And since in any finite time, however small, there are infinitely many instants, there are enough to correspond to the infinitely many degrees of velocity." (201 - p 157, Stillman Drake translation)