Essays, Philosophy, Science.

Zeno’s Paradox? Not so much

-Oregon Scribbler,

Oregon Scribbler.

Zeno of Elea, an ancient Greek mathematician and philosopher, was well-known for his logical paradoxes.  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.

If you are out on a limb, why not go all the way? Zeno went further, in the grand tradition of reductio ad absurdum, oft employed by not just ancient philosophers but modern day zealots, and suggested that in fact motion is impossible.  The paradox is strikingly silly, but where is the logical flaw?

If we ignore the logical trap, the answer is 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!  (Not surprisingly, 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 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 

-Oregon Scribbler,

Oregon Scribbler.

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

-PD-US, from Raphael

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.

Perhaps not surprisingly, summable infinite series were first explored by Newton, as it turned out one of the steps towards his construction of the first methods of calculus, which he invented to help characterize and quantify:  Motion!   As Newton was groping for 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 3)


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 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.



1. A similar explanation along these lines, which starts with Zeno’s aforementioned Achilles and the Tortoise motion paradox, can be found here. A more complex discussion of Zeno’s paradoxes can be found here.

2. My father first described to me the motion paradoxes of Zeno as a bemusing example of the limits of logic. These memories of such moments with my father, which were relatively rare, I still treasure. It was usually after a dinner meal, still around the table, that he would grab a random section of the day’s newspaper, call for a pencil, and tell a story of science or math, with occasional cramped diagrams in the margins of the newspaper, intense and intent.

3. 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.

2 thoughts on “Zeno’s Paradox? Not so much

  1. Seeing outside consciousness would lead to have no perception about anything in this world. Besides this, the argument that author provides is perfectly ok, but in my opinion you misunderstood it. What matters here is not the division of space only, but the division of time too. You can repeatedly divide a distance as much as you wish, but time is not slowing down in any way – that’s what the author says, so in fact there is no paradox here whatsoever.

    • As the author of this piece, I disagree with your contention that what Zeno said was OK or that it is not a paradox. Based on his argument, Zeno insisted that motion was impossible. Yet obviously it is not impossible, so Zeno’s “logic” is clearly flawed. Even Zeno himself called it a paradox. On another note, I am not sure what you are saying about “outside consciousness”, perhaps you could elaborate.

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