Mathematical mysteries: Zeno's Paradoxes

Rachel Thomas Share this page
Nov 2001

The paradoxes of the philosopher Zeno, born approximately 490 BC in southern Italy, have puzzled mathematicians, scientists and philosophers for millennia. Although none of his work survives today, over 40 paradoxes are attributed to him which appeared in a book he wrote as a defense of the philosophies of his teacher Parmenides. Parmenides believed in monism, that reality was a single, constant, unchanging thing that he called 'Being'. In defending this radical belief, Zeno fashioned 40 arguments to show that change (motion) and plurality are impossible.

The most famous of Zeno's arguments is the Achilles:

'The slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead.'

This is usually put in the context of a race between Achilles (the legendary Greek warrior) and the Tortoise. Achilles gives the Tortoise a head start of, say 10 m, since he runs at 10 ms-1 and the Tortoise moves at only 1 ms-1. Then by the time Achilles has reached the point where the Tortoise started (T0 = 10 m), the slow but steady individual will have moved on 1 m to T1 = 11 m. When Achilles reaches T1, the labouring Tortoise will have moved on 0.1 m (to T2 = 11.1 m). When Achilles reaches T2, the Tortoise will still be ahead by 0.01 m, and so on. Each time Achilles reaches the point where the Tortoise was, the cunning reptile will always have moved a little way ahead.

This seems very peculiar. We know that Achilles should pass the Tortoise after 1.11 seconds when they have both run just over 11 m, so Achilles will win any race longer than 11.11m. But why in Zeno's argument does it seem that Achilles will never catch the tortoise?

If you think of the distances Achilles has to travel, first 10 m to T0, then 1 m to T1, then 0.1 m to T2 etc., we can write it as a sum of a geometric series:

10 + 1 + 0.1 + .... + 10(2-n) + ...

Now it is a little clearer. As the distance that Achilles travels to catch the tortoise is the sum of a geometric series where the multiplier is less than one (read more), we know that the distance is finite (and equal to 11.11m) as the series converges. And as he only has to travel a finite distance, Achilles will obviously cover that distance in a finite time if he is traveling at a constant speed.

So how did Zeno manage to confuse us? Zeno's argument is based on the assumption that you can infinitely divide space (the race track) and time (how long it takes to run). By dividing the race track into an infinite number of pieces, Zeno's argument turned the race into an infinite number of steps that seemed as if they would never end. However, each step is decreasing, and so dividing space and therefore time into smaller and smaller pieces implies that the passage of time is 'slowing down' and can never reach the moment where Achilles passes the Tortoise. We know that time doesn't slow down in this way. The assumption that space (and time) is infinitely divisible is wrong (more on the physical implications of the limiting process).

There are ways to rephrase the Achilles argument that can take our brains in a slightly different direction. In one example, known as Thomson's Lamp, we suspend our disbelief once again and consider a lamp with a switch that we press to turn on, and press again to turn off.

Now, the lamp is initially off and I switch it on. After 1 minute I switch it off. After half a minute I switch it back on. After a quarter of a minute I switch it off. After one eighth of a minute I switch is back on and so on, each time halving the length of time I wait before I switch the lamp on or off as appropriate (I have very quick reflexes). After 2 minutes, (the sum of the infinite series 1 + 1/2 + 1/4 + ...), I will have finished this infinite sequence of actions. So at this point, is the lamp on or off? And will it have made a difference if the lamp was initially on rather than off?

As with Zeno's original version of Achilles, these arguments are based on the infinite divisibility of time, and the paradox that results can be seen to illustrating that time is not infinitely divisible in this way.

Interestingly, as mentioned above, the Achilles paradox was only one of 40 arguments Zeno is thought to have produced, and in another of his arguments called the Arrow, Zeno also shows that the assumption that the universe consists of finite, indivisible elements is apparently incorrect. So, here is where the real paradox of Zeno lies. In his arguments, he manages to show that the universe can neither be continuous (infinitely divisible) nor discrete (discontinuous, that is made up of finite,indivisible parts).

This seeming contradiction in the nature of reality is echoed by concepts from an area developed over 2000 years after Zeno lived, the Theory of Relativity. For example, light is now thought of as having a dual nature, behaving sometimes as a particle or photon (discrete), and at other times like a wave (continuous). In fact even Zeno's belief in monism - in a static, unchanging reality - which was the basis for his producing the arguments in the first place, seems oddly similar to cosmologists ideas about 'worldlines' (the 'history' of a particle in spacetime) where 'the entire history of each worldline already exists as a completed entity in the plenum of space time' (read more).

So Zeno's paradoxes still challenge our understanding of space and time, and these ancient arguments have surprising resonance with some of the most modern concepts in science.

Zeno and the Paradox of Motion
Zeno's race course, part 2 - Lecture notes from the University of Washington
Zeno at St Andrews site

About the author

Rachel Thomas is an assistant editor of Plus.



The argument the author proposes doesn't really solve Zeno's paradox. "We know that the distance is finite" is not a true statement; it is a non-evidentiary assumption, based on appearance in consciousness. Seeing outside consciousness is impossible. Probably Nobel quantum physicists Max Planck & Erwin Schrodinger would defend Zeno's simple paradox? They were both convinced that consciousness itself is the matrix for space/time/world - i.e., multiplicity is an illusion in consciousness or mind? So, imho, Zeno lives!

Permalink In reply to by Anonymous (not verified)

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.


I agree with the other comment that arguing against the infinite divisibility of space and time doesn't really get to grips with the paradox. My reason though is that it doesn't really matter whether or not you can go on dividing and subdividing the journey forever, you just don't want to anyway. Right from the start it seems pointless and irrelevant as a means of describing motion in this context.

Normally it's of practical use to say where a moving object is, even though you're talking as if it's stationary, provided it's not moving so fast that it's gone before you can find it, and/or the area under consideration is big enough (like a submarine in the Atlantic) for the information to save you time and trouble otherwise spent in searching for it. When this is no longer so, as with very fast objects over very small distances then quantum notions like superposition and probability clouds help out.

Nevertheless Zeno exploits the pedant in us which recognises an essential contradiction between motion and position, the fact or feeling that strictly speaking we can't really say where any moving object is, however practically useful it may be to do so. That once specified, a position is really a stop even if the object continues moving, and stops can go on being specified forever. The antidote to pedantry is humourous exaggeration, so here goes:

You're at the station and race for your train. You jump on at the last minute without giving yourself time to look at the departure board. "Does this train go to Cambridge?" you ask the conductor. "Yes" comes the reply, so you settle in your seat. An hour or so later you look up to see that the train is rushing through Cambridge station without even slowing down. "But you said it goes to Cambridge" you protest. "And so it does" comes the reply, "It just doesn't stop there".


So, here's my take on both forms of the paradox - Zeno's original, of Achilles and the Tortoise (converging on a distance), and Thomson's variation, the Lamp (converging on a time). In both of them, assuming that the runner / lamp-switcher in question CAN infinitely continue to cover ever smaller increments of distance / time (which, as the article points out, modern physics dictates that they CAN'T), they will never reach the point of convergence.

So, using the example figures from the article: assuming that they both continue along the race-track infinitely, neither Achilles nor the Tortoise will ever reach the 11.111111...m mark. I.e. the real point, in my opinion - apart from Zeno's conclusion that Achilles can never overtake the Tortoise - the real point is that neither of them will even finish the race (or to be more precise, assuming that the race is 100m long, neither of them will finish more than 11.11111...% of the race). They're "infinitely recursing" in their movement, as they approach but never reach the point where one would be a tied winner with the other.

Similarly: assuming that someone continues switching it infinitely, the "state of the lamp" will never reach the 2min mark. I.e. apart from Thomson's conclusion that we can't say whether the lamp will be on or off after 2 mins, the real point is that it's impossible for "the lamp as something that's on or off" to even EXIST when time is gte 2 mins. The lamp will continue "infinitely recursing" in its temporal state, as it approaches but never reaches the point where that dude with a tired index finger could stop and take a break.

In conclusion: Zeno's Paradox / Thomson's Lamp are solvable, if we assume that the universe doesn't exist. :P


This paradox is intrinsically flawed. It assumes a "finish line". Remove "finish line" and "time". Then Achilles blows past Tortoise. Change reality. Infinity = nothing = everything.


A question my geometry teacher could not answer - If two points determine distance, and two points on a line can always be divided by another point, then are there an infinite number of points within a finite space? Assuming/expecting that a 'point' inhabits some measurable amount of space, how can an infinite number of points exist within a finite space?

According to the definition of point in the Elements of Euclid, Book 1, Def 1, . ''A point is that of which there is no part'' and Def.2 ''And a line is a length without breadth''. If that be so, within two points ( rational or irrationals), infinite points, rational and irrational may lie.