Zeno’s paradoxes

Zeno of Elea (c. 490 – c. 430 BC) is one of the most enigmatic pre-Socratic philosophers. Though none of his own works have survived, there are fragmentary mentions of his on the classics like Aristotle and Plato. He was a member of the Eleatic School and, according to Plato at least, aimed to reinfoced Parmenides’s arguments (Parmenides being the founder of the school). While we know very little of Zeno himself, other than some hearsay (Plato says that he was tall and handsome, for instance, and that he was Parmenides’ lover),2 we do have two very interesting paradoxes. I had written these two years back, but for some reason thought that they deserved to be together (as only one of them really gets enough traction, while the other is ignored). And they are also still incomplete, as two more paradoxes remain: one called the Millet Seed (which seems to be a variation of sorites paradox) and the paradox of place – as we have almost nothing to work with though, analysing these paradoxes would be a mere speculation and detraction. So what follows below are two variations of the famous paradoxes of Zeno: Achilles and the tortoise, and the arrow paradox.

Zeno's Achilles and tortoise paradox
Zeno’s paradox of motion – Achilles and the tortoise

Zeno’s paradox of motion – Achilles and the tortoise

The most famous of Zeno’s paradoxes, and also the one with amusing historical examples: Zeno’s paradox of motion. In one version of the paradox Zeno proposes that there is no such thing as motion. There are many variations, and Aristotle recounts four of them, though essentially one can call them variations of two paradoxes of motion. One concerning time and the other space. Let us focus on space and recount the Achilles and tortoise paradox:

in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead”; and “the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal (Aristotle’s Physics, Book VI.9).

Though intuitively illogical, there is some sense in this. Right?

The description is more complicated than the paradox actually is. Zeno’s point is simply that space is divisible, and because it is divisible one cannot reach a specific point in space when another has moved from that point further. Let us take an example:

The distance is 1000x (where x is the measure unit for distance – mile, meter, whatever).
Let us also say that Achilles give 100x head start to the tortoise.
Whenever Achilles reaches 100x in time t1, the tortoise would have moved further (for instance, 150x).
Thus, when Achilles reaches 150x in time t2, the tortoise would have moved even further to 175x.

Zeno’s point is that given these conditions, Achilles cannot catch up with the tortoise because space can be infinitely divided into smaller units still – where the tortoise will always be a fraction of space ahead.

There are by now numerous ‘solutions’ to this paradox, (some have written a 272 pages long book on Zeno’s paradoxes). Then again, as Aristotle pointed out already, this is not really a paradox, but poor physics. Anyone with high-school level of physics will see the problem: both Achilles and tortoise will have stopped moving as such at some point in time.

Perhaps by today’s standards we can say that this paradox is a challenge to conventional physics. But what if that is Zeno’s point with the paradox? – A challenge to all of Ancient Greek thought that everything is in motion, always – a challenge to Heraclitus ‘everything is flux’ view pointed out in the Ship of Theseus paradox (fragment DK B12). For this, it is best to look at Zeno’s point on motion in relation to time.

Zeno's arrow paradox
Zeno’s paradox of motion – The Arrow

Zeno’s paradox of motion – The arrow

So let us focus on this other aspect of the paradox of motion in. With Achilles and the tortoise, Zeno’s paradox points towards motion being inconceivable due to infinite divisibility of space (Aristotle calls it “bisection”). Here, let’s refer to time. Zeno’s paradox is best explained through his example of a flying arrow. As Aristotle describes the paradox:

if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless”, or simply “the flying arrow is at rest (Aristotle’s Physics, Book VI.9).

Something can fly, while being at rest – makes sense right? Aristotle certainly thought it didn’t. He has a quite simple answer to the paradox:

This is false, for time is not composed of indivisible moments (Aristotle’s Physics, Book VI.9).

Complex explanation by Aristotle, so let us take another quote that explains this one [sidenote: a quote that explains a quote that explains a quote, etc. – does that count as infinite regress?]:

So while it is true to say that that which is in motion is at a moment not in motion and is opposite some particular thing, it cannot in a period of time be over against that which is at rest: for that would involve the conclusion that that which is in locomotion is at rest (Aristotle’s Physics, Book VI.8).

Then again, this is not really an answer to the paradox, but a difference of opinion – a different view on cosmology. As pointed out in the example of Achilles and the tortoise, Zeno wanted to challenge the traditional view that everything is in constant movement. Positing, as Aristotle does (or later Bergson), that time cannot be thought of as being composed of ‘indivisibles’ [sidenote: what strange language English is – indivisibles], does not really help us with the paradox. Bergson thought of motion as we do in contemporary physics: motion must involve both time and space. Thus, to disprove motion as such, both paradoxes should involve both time and space (and not individually at each occasion).


Footnotes

  1. There are also rumours of Zeno conspiring to overthrow several tyrants, of biting an ear of one of them, and a nose of another.
  2. There are also rumours of Zeno conspiring to overthrow several tyrants, of biting an ear of one of them, and a nose of another.

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