Zeno of Elea

Zeno (490 – 430 BCE), also known as Zeno of Elea, is the most famous and influential of the Eleatic school after Parmenides.  In Plato’s dialogue Parmenides, a major source for the life and thought of Parmenides and Zeno, Plato says Parmenides was 65 and Zeno 40 when the two visited Athens to spread their teachings, and Socrates is very young.  Plato says Zeno was good looking and loved by Parmenides, which many believe meant that the two had a physical relationship that does not disturb Plato or Socrates in the slightest.

Aristotle says Zeno invented dialectic, reasoning back and forth from both sides to find the truth.  Assuming the principle of noncontradiction, which Aristotle endorses, if one side contains a contradiction, you can eliminate it.  This is also known as ‘reductio ad absurdum’, reduction to absurdity.  To eliminate a position, we would start, “Suppose you are right, and X is Y…but this leads to a contradiction, so we know that X can not be Y”.  However, Zeno used dialectic differently, as he assumed all human arguments and concepts contain contradictions.  This is similar to the goddess of Parmenides arguing for Pythagoras’ view and then saying that it is illusion.  For Parmenides and Zeno, as well as Heraclitus, all human concepts are incapable of grasping the truth in itself, apart from anything.


Diogenes Laertius wrote that Zeno was very skilled at arguing all sides of a position, and that he was imprisoned and killed by King Demylus of Elea, possibly for arguing against his position and contradicting the king.  Plutarch, the Roman historian, wrote that Zeno tried to kill Demylus, and when he failed and was captured, he bit off his own tongue and spit it in Demylus’ face, killing himself.

While none of Zeno’s writings have survived, he is famous for his paradoxes, discussed by Aristotle and many others.  There were supposedly forty arguments that resulted in contradictions, but only nine survive today found in the writings of others, and some seem to be alternate versions of the same arguments.  While some have thought that Zeno’s paradoxes are meant to show that reality is paradoxical, Zeno intended his arguments to show that human beings are incapable of knowing true reality with human thought, and that any attempt to know the whole or a part of the whole results in contradictions.  Thus, while things seem to clearly move in front of our eyes, when we try to argue how they move our thinking becomes entangled in contradiction and paradox.


Remember that Xenophanes said no one will know the truth of the things about which he speaks, the One and the supreme order of reality.  If true, this means that Zeno was not trying to eliminate one side of a dialectic, but show that all sides of an argument result in contradictions as no differentiated human argument or conception can be the supreme undifferentiated unity itself.  While mathematicians have supplied further conceptions to make sense of Zeno’s paradoxes, there is still no general agreement on the paradoxes, how they can be resolved or even if they are indeed paradoxes and genuine examples of contradiction.


For the his paradox of nonbeing, Zeno argues if being is many and not one, then being contains nonbeing and what is alike is not alike, which is impossible, as the similar cannot be different and the different cannot be similar.  This is entirely in line with Parmenides and antithetical to Heraclitus.  If being is absolute, not relative, then nonbeing cannot exist at all, in any form or to any degree.  Thus the paradox: nonbeing seems quite real, but cannot really be.


For his paradox of place, Zeno argues if things have a place, then a thing’s place has a place, and this place has a place, ad infinitum.  This would be true of Anaxagoras’ infinite cosmos, unlike Hesiod’s cosmos in which it takes an anvil nine days to fall from the height of the heavens to the earth.  Because this results in an infinite regress, we have a paradox: finite places are not infinite, but there can’t be a finite place without an infinite regress.

Zeno’s most famous paradox is the paradox or impossibility of motion, which Aristotle attacks extensively in his Physics.  There are three versions of this argument that are nearly identical: the Tortoise and Achilles, the dichotomy argument, and the arrow in flight.

Achilles Tortoise

In the Tortoise and Achilles version, Zeno argues that the quickest runner cannot overtake the slowest given a head start, as there are an infinite number of points to cross and this leads to an infinite regress.  Say that a tortoise is given a ten meter head start, and Achilles attempts to catch up.  First Achilles must reach the point from which the tortoise started, to find that the tortoise has gained a bit of ground.  Then Achilles must reach this next point, only to find that the tortoise continues to crawl ahead.  There seem to be an infinite number of points ahead of Achilles that he must cross, as each point reached spawns a new point of destination.  Therefore, though we all know Achilles can catch the tortoise, it seems that he must cross an infinite number of points to do it.

The dichotomy argument works similarly, but with one mover moving towards a fixed position.  To get to the destination, the mover must cross half the distance, then cross half the remaining distance, and again and again until the destination is reached, which seems impossible given that there is a never-ending set, an infinite regress, of half-segments to be crossed.  The path is finite and infinite at the same time, which cannot be.

For the ‘arrow in flight’ argument, Zeno argues that at each point along an arrow’s path, the arrow is not moving, as it occupies a single space in a moment of time.  This means that the arrow, as it moves, is unmoving at each point.  Because a thing cannot move and stay unmoving at once, all motion is impossible.


Neo-Platonists were concerned with the infinite, the finite and the problems of the two being involved with each other.  Nicholas of Cusa, a medieval German Neo-Platonist, argued that we can conceive of the infinite, and thus our mind must be in some way infinite itself.  He uses the example of a mustard seed, asking us to imagine an infinite supply of earth, water and light for it to grow.  Clearly having the parable of Jesus in mind, Nicholas argues that a single mortal mustard seed would multiply endlessly in these conditions, giving us a real physical infinite regress in both the growth of the seed and our comprehension that this growth is infinite.

Georg Cantor

The later German philosopher Hegel, who is well aware of the work of Parmenides, Zeno and the Neo-Platonists, argued that the infinite was present as the finite, using the image of a circle as unending.  Similar work with the infinite of the German mathematician Cantor drove him temporarily insane and he was committed to an asylum after discovering his famous Cantor Set.  If you imagine a line that splits into two segments in the middle again and again in each part, after an infinite number of splits you would have twice that, twice as infinite, number of sections.

Cantor Set

If our reality is infinite, not having a beginning or end, that would mean that everything is infinite but some infinite is more or less relative to other parts, which is how things move at particular speeds.  Consider two treadmills, one running twice as fast as the other, but both moving without cease.  While both eventually result in an infinite amount of motion, one moves twice as much as the other, such that one infinite is twice another infinite.  From the perspective of the slower treadmill, even if both moved at infinite speeds, imperceptible to us, while it stays immobile the faster treadmill moves at a constant and finite speed.


Consider the arrow in flight of Zeno’s motion paradox.  The space the arrow must cross is infinite insofar as it is infinitely divisible.  To cross any space is to cross an infinite distance, which is not impossible if the motion is itself infinite, but greater relative to the space.  The quicker the arrow crosses the space, the greater it is relative to the space.  Aristotle argued that as the space between the tortoise and Achilles, between the arrow and its target, gets smaller and smaller, the time it takes to cross each division of space gets smaller and smaller.

Berkeley philosopher

Diogenes, who we will study, had his own refutation.  After hearing of Zeno’s arguments against motion, he stood up and walked away.  This is similar to Samuel Johnson’s refutation of Berkeley’s idealism.  Berkeley (for whom our town is named) believed that reality is in the mind and the world is a dream in the mind of God, our own existences being dreams within the dream.  Johnson, walking from church with a friend speaking to him of Berkeley’s thought, kicked a rock and said, “I refute it thus!”.  Zeno would reply to both that motion and material seem quite real, but these demonstrations only underscore how argument for the everyday is strangely impossible.

two slit quantum paradox

Quantum theorists have recalled Zeno’s paradox of motion in conjunction with the wave-particle paradox, calling it the ‘quantum Zeno effect’, that observation seems to change how reality behaves, either as a unified wave or disunified number of particles whether or not the system is observed.  Similarly, if we observe the infinite divisibility of a space, it seems impossible to cross, but only if we consider it’s infinite divisibility, which seems to disappear as soon as we cease to consider it as part of the problem.

Mandelbrot Set

Similarly, fractal geometry suggests that the organic patterns we see everywhere in the cosmos, as well as our human bodies, result from an infinite number of permutations.  Mandelbrot, while working at IBM, found that he could create patterns that imitate nature by letting simple math games reiterate a practically infinite number of times.  The Mandelbrot Set, the overarching pattern of these patterns, is now famous, and there are many websites and YouTube videos exploring the endless complexity of the Mandelbrot Set.


Lewis Carroll was fascinated by Zeno’s Achilles and Tortoise, created a dialogue between the two in which the Tortoise, speaking as Zeno, always outsmarts Achilles leading him into paradoxes and infinite regresses.  In Hofstadter’s book Godel, Escher, Bach, he considers Zeno’s paradoxes, recreating Lewis Carroll’s dialogue in terms of modern cognitive science and fractal geometry.  Borges, the Argentinian writer famous for his Ficciones, his short works of surreal fiction, was fascinated by Zeno, and often used infinite regresses.  In his story The Library, an unending library with every possible book, including exact copies with one mistaken letter.