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Exploring Zeno of Elea's Theory of Paradoxes

philosopher Zeno of Elea  (490-430) BCE

The Ancient Greek philosopher Zeno of Elea left an indelible mark on Western philosophy with his theories on paradoxes, motion and time. His writings challenged prevailing beliefs concerning the universe and sought to undermine belief in the physical world as a whole. Learn more about Zeno's philosophical works in this post.

zeno of elea contributions in philosophy

Who is Elea Zeno

Plato’s Parmenides and Diogenes Laertius’ Lives of Eminent Philosophers are almost the only sources of information we have about Zeno’s life. If Plato’s account is correct, Zeno was born in 490 BCE, and accompanied Parmenides to Athens in about 450 BCE where the young Socrates met them.

Zeno was said to be not just Parmenides’ pupil but his adopted son and his lover. He was a tall and handsome man, Plato says; and Diogenes says that his books ‘are brimful of intellect’. Aristotle said that Zeno invented ‘dialectic’, the form of philosophical argument aimed at arriving at truth (as opposed to ‘eristic’, argument conducted merely for the sake of argument or for point-scoring), in part by starting from the views of an opponent and demonstrating that they lead to unacceptable conclusions.

Zeno of Elea death

Zeno was a man of ‘noble character, both as a philosopher and as a politician’, for when his attempt to overthrow the tyrant Nearchus failed he was arrested and tortured before being killed, but did not betray his friends. His death produced a multiplicity of legends. Saying to Nearchus that he had something private to whisper in his ear, Zeno ‘laid hold of it [the ear] with his teeth, and did not let go until stabbed to death’. 

Another version says it was the tyrant’s nose, not his ear, that he bit off. A third says that he bit off his own tongue and spat it at the tyrant rather than reveal any secrets, and this so roused the citizens that they stoned the tyrant to death. When Nearchus told him to reveal who was behind the coup attempt, Zeno said, ‘You, the curse of the city!’ whereupon Nearchus had him thrown into a giant mortar and pounded to death.

Zeno of Elea philosophy

He composed a prose treatise in which he developed his philosophy. Zeno’s contribution to Eleaticism is, in a sense, entirely negative. He did not add anything positive to the teachings of Parmenides. He supports Parmenides in the doctrine of Being. But it is not the conclusions of Zeno that are novel, it is rather the reasons which he gave for them. In attempting to support the Parmenidean doctrine from a new point of view he developed certain ideas about the ultimate character of space and time which have since been of the utmost importance in philosophy. Parmenides had taught that the world of sense is illusory and false. The essentials of that world are two— multiplicity and change. True Being is absolutely one; there is in it no plurality or multiplicity. Being, moreover, is absolutely static and unchangeable. There is in it no motion. Multiplicity and motion are the two characteristics of the false world of sense. Against multiplicity and motion, therefore, Zeno directed his  arguments, and attempted indirectly to support the conclusions of Parmenides by showing that multiplicity and motion are impossible. He attempted to force multiplicity and motion to refute themselves by showing that, if we assume them as real, contradictory propositions follow from that assumption. Two propositions which contradict each other cannot both be true. Therefore the assumptions from which both follow, namely, multiplicity and motion, cannot be real things.

 Zeno of Elea's Theory of Paradoxes

Zeno of  Elea the master of paradoxes.

In Plato’s Parmenides Zeno is reported as saying that his arguments about the impossibility of motion and plurality are offered as a defence of the Parmenidean thesis that reality is One and unchanging: ‘[my arguments are] a defence of Parmenides’ argument against those who try to make fun of it, saying that if What Is is One, the argument has many ridiculous consequences which contradict it. 

Now my treatise opposes the advocates of plurality and pays them back the same and more, aiming to prove that their hypothesis “that there are many things” suffers still more ridiculous consequences than the hypothesis that there is One.’ In other words, Zeno’s arguments have the form of a reductio ad absurdum of an initial hypothesis, by showing that contradictions can be deduced from it.

What are the Paradoxes of Zeno?

The most renowned of Zeno's philosophical works are the paradoxes, which he described as logical problems meant to show that the reasoning of his opponents was contradictory. One such example is his famous Paradox of Achilles and the Tortoise – where he proposed that a tortoise given a head start in a race could never be beaten by the fastest runner, no matter how fast they ran. These paradoxes often take on an impossible nature, leaving us questioning our accepted beliefs about the world.

Zeno of Elea theory of arguments against multiplicity.

(1) If the many is, it must be both infinitely small and infinitely large. The many must be infinitely small. For it is composed of units. This is what we mean by saying that it is many. It is many parts or units. These units must be indivisible. For if they are further divisible, then they are not units. Since they are indivisible they can have no magnitude, for that which has magnitude is divisible. The many, therefore, is composed of units which have no magnitude. But if none of the parts of the many have magnitude, the many as a whole has none. Therefore, the many is infinitely small. But the many must also be infinitely large.

For the many has magnitude, and as such, is divisible into parts. These parts still have magnitude, and are therefore further divisible. However far we proceed with the division the parts still have magnitude and are still divisible. Hence the many is divisible ad infinitum. It must therefore be composed of an infinite number of parts, each having magnitude. But the smallest magnitude, multiplied by infinity, becomes an infinite magnitude. Therefore the many is infinitely large. (2) The many must be, in number, both limited and unlimited. It must be limited because it is just as many as it is, no more, no less. It is, therefore, a definite number. But a definite number is a finite or limited number. But the many must be also unlimited in number. For it is infinitely divisible, or composed of an infinite number of parts.

Zeno of Elea paradox of motion

Zeno created about forty paradoxes, of which ten are known. Aristotle’s Physics is the chief source for Zeno’s arguments against motion. They can be described as follows. 

1. philosophy of Zeno of Elea Achilles and tortoise

Let's take Milkha Singh as Achilles.  If the tortoise is given a head start, however small,  Milkha Singh can never overtake it. For to do so he must reach the point from which the tortoise started; but by the time he does so, the tortoise will have moved on, and Milkha Singh  must therefore reach that next point. But by the time he does so … and so on.

2. Dichotomy Paradox

Example- Suppose you are walking from one end of a stadium to the other. To do this you must get to the halfway point. But to get there, you have to get to the place halfway to the halfway point. Indeed to get to any point you have to get halfway to it, but first you have to get halfway to that halfway, and before that halfway  and so on ad infinitum. But one cannot traverse an infinite number of points in a finite time; therefore motion is an illusion.

The Dichotomy Paradox, examines motion and the idea that any movement requires covering an infinite variety of distances. To illustrate this point, Zeno postulates that if something wants to travel a distance, it must first travel half the distance; and then half of that remaining distance; then half of the next remaining distance and so on infinitely. But since infinity is an unachievable concept, Zeno concluded that motion was impossible. This paradox has proven difficult to resolve as questions surrounding both time and geometry remain unanswered – making Zeno’s theory one of physics’ most irresolvable mysteries.

3. Arrow fired at a target

 The Arrow Paradox, also known as the Flight of the Arrow paradox. At any point in its flight the arrow occupies exactly the space that is its length. It is therefore motionless in that space, for (says Zeno) all things are at rest when occupying a space equal to their own size. But then because the arrow occupies its own exact space at every point on its flight, it is motionless at every point in its flight.

Illuminates a deep insight into our perceptions of time. It challenges the concept that everything occurs in moments by claiming the arrow is always at rest. If a single arrow is thrown from one point A to point B, and Zeno proposes that we observe it throughout its flight, how does one explain how something can be in two places at once?

''If everything is motionless at every instant and time is entirely composed of instants, then motion is impossible''. -  Zeno 

One day Zeno told his philosophy to Diogenes. Diogenes was refuted his philosophy by walking. Example he get up and walking from one place to another.  

One relevant consideration for paradoxes such as the ‘Stadium’ and ‘Achilles’ is that if you sum ½ + ¼ + ⅛ … you get 1 for intervals of both space and time. So if you sum the distances that one must traverse to get to each halfway point (halfway across the stadium, halfway to that halfway point, and so on) you get the finite distance between the two ends of the stadium. The same applies to the time that elapses for each successive act of getting to a given point, then to a next given point, and so on. Once again, the conclusion is that one can traverse an infinitely divisible space in a finite time.

Zeno’s argument assumes that it is impossible to traverse an infinite number of points in a finite time. But this is to fail to distinguish infinite divisibility and infinite extension. One cannot traverse an infinite extension in a finite time, but one can an infinitely divisible space, for time itself is infinitely divisible; so one is traversing an infinitely divisible space in an infinitely divisible time.

Zeno’s arguments are so framed as to suggest that he principally had the Pythagoreans in mind. In arguing that number is the basis of reality they correlatively held that things are sums of units. Zeno is reported to have said, ‘If anyone can explain to me what a unit is, I can say what things are.’  He here offers a classic case of deducing a contradiction from the premise ‘that there are many things’, as follows: ‘If things are a many [a plurality], they must be just as many as they are, and neither more nor less.

 Now, if they are as many as they are, they will be finite in number. But: if things are a many, they will be infinite in number, for there will always be other things between them, and others again between those. And so things are infinite in number.’

Another argument against plurality turns on the supposition that things can be divided into parts. You have to assume that the parts themselves have to be something, because if the divisions of things finally reach nothing, how can something be composed out of nothing? Suppose you argue that the parts are not nothing, but have no size; how then can the thing they compose have size, given that no number of things without size can constitute a thing with size? So you are left with the assumption that the elements of things have to be something, and with a size. But then they are not the elements of things, because they can be further divided, and if their parts in turn have size they are therefore divisible, and their parts likewise and so on; so the dividing can never stop.

A suggestive result of reflection on the paradoxes is that they arise from conflicts between the conceptual conveniences we put to work to organize our experience. For example: when we are thinking of motion as a continuous event that occurs over an interval of time, we are thinking of an object travelling from one position to another against a background of fixed reference points, and from this perspective we do not, and arguably cannot, think of the object as being successively and determinately at given points in space different from immediately neighbouring points at discrete instants of time. 

But when we think of the object from this second and different perspective, namely the perspective of it being at a given point in its journey, we do not and arguably cannot think of it in the way we think of it from the first perspective, that is, as passing through that point in a way unspecifiable as ‘a place at a time’, given that this is exactly what we are doing from the second perspective. The problem therefore lies in us; sometimes our ways of describing the same things for different purposes from different perspectives are inconsistent with each other. This does not entail that motion itself is illusory.

Whatever the merits of Zeno’s arguments individually, and however well the counterarguments to them fare, the fact is that they further provoke reflection on the Parmenidean idea that so influenced Plato and a great deal of subsequent philosophy: the idea, namely, that appearance is not reality.


A History of Western Philosophy Book by Bertrand Russell
Graham, D. W. (ed.), The Texts of Early Greek Philosophy: The Complete Fragments and Selected Testimonies of the Major Presocratics, 2 vols., Cambridge: Cambridge University Press, 2010
Barnes, J., The Presocratic Philosophers, 2nd edn, London: Routledge & Kegan Paul, 1982
Hussey, E., The Presocratics, London: Duckworth, 1995 Kirk, G. S., J. E. Raven and M. Schofield, The Presocratic Philosophers,
2nd edn, Cambridge: Cambridge University Press, 1984 Osborne, C., Presocratic Philosophy: A Very Short Introduction, Oxford: Oxford University Press, 2004


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