Zeno of Elea (l.c. 465 BCE) was a student of the famous Eleatic philosopher Parmenides (l.c.485 BCE), founder of the Monist school of philosophy. As the name suggests, Monism believed in the single (mono), unified nature of reality and rejected plurality - the belief in different objects and entities in the world - as an illusion. To Parmenides, the nature - the essence - of reality was One and anything which appeared to contradict this view was an error in perception caused by trusting in one's senses.
Parmenides claimed, "There is a way which is and a way which is not" (a way of truth and a way of opinion) and, further, that:
There is not, nor will there be, anything other than what is since indeed Destiny has fettered it to remain whole and immovable. Therefore those things which mortals have established, believing them to be true, will be mere names: "'coming into being and passing away,' 'being and not being,' 'change of place'...(Robinson, 116)
In other words, Parmenides argues that we may think the world we live in is comprised of multiples but, in reality, it is One. Nothing is capable of inherently changing in any significant fashion because the very substance of reality is unchangeable and 'nothingness' cannot be comprehended.
Parmenides seems to have had a hard time convincing others of his vision, however, because it seemed obvious to his listeners that there were significant differences between one city and another or between a horse and a goat. Zeno's paradoxes were created to mathematically prove to these skeptics that Parmenides' philosophy was sound and that they had allowed themselve to be fooled into error by trusting their senses.
From Aristotle, primarily, it is known that once there were 40 paradoxes penned by Zeno of which there exist less than 10 today. The most famous of these are The Race Course, The Achilles, The Arrow, and The Stadium. For a comprehensive treatment of all of Zeno's Paradoxes one should consult the Stanford Encyclopedia entry listed in bibliography below; this article will only treat of three of the famous four.
The Race Course argues that, in order for someone to move from Point A to Point Z one must first get to the halfway point between A and Z. However, in order to reach that halfway point one must get to the halfway point between Point A and that halfway point and, to reach that spot, one must travel halfway again. Since there will always be a space, even an infinitesimal one, between one point and another, and since one must always move halfway before one may reach one's destination, motion is impossible. Our senses, which tell us that motion is possible, must then be flawed and should not be trusted to apprehend truth.
The Achilles argument claims that the swiftest runner will never overtake the slowest who starts the race first, "inasmuch as, reckoning from any given instant, the pursuer, before he can catch the pursued, must reach the point from which the pursued started at that instant, and so the slower will always be some distance in advance of the swifter" (Baird/Kaufmann, 28). In other words, because the slower runner (usually depicted as a tortoise while the swifter is known as 'Achilles') is in advance of the swifter, the swifter must first arrive at that point where the slower runner began and, as the slower will always be ahead of the swifter by virtue of having started earlier, the swifter must always be in pursuit of the slower without ever having hope of catching him.
The Arrow, another argument against the reality of motion, shows how an arrow in flight cannot possibly move as "everything is at rest when it is in a place equal to itself, and if the moving object is always in the present [and therefore in a place equal to itself] then the moving arrow is motionless" (Robinson, 134). Since all we live in, at any moment, is the present and since the arrow is motionless at any given moment in its flight, the arrow cannot actually be moving.
The Purpose of the Paradoxes & Their Result
The intended aim of Zeno's paradoxes was to prove the vision of Parmenides, that there was a way of truth (logic) and a way of opinion (the senses) and how, clearly, the senses could not be trusted to lead one to an apprehension of Truth. Far from being simple word-puzzles, however, the paradoxes gave rise to a method of analysis and discussion of a question, known as dialectic, which proved a most valuable tool in philosophical inquiry. Professor J. M. Robinson comments:
Dialectic involves two people: a questioner and an answerer. The questioner invites the answerer to take up a position which he thinks he can defend, and proceeds by a series of well-placed questions to drive him from it. He does this by getting the answerer to agree to a series of further propositions which, taken together, imply the contradictory of the original thesis. (Robinson, 137)
It is obvious the effect this had on the young writer, Plato. Plato's Dialogues are virtual handbooks on the art of dialectic. In Republic dialectic is used to find out what justice is, in Euthyphro, Piety, in Ion it is 'What is Art' and in Meno, what constitutes Virtue is sought through the dialectic method. Whether he intended to, Zeno gave to ancient Greece the art of dialectic and handed to Plato the means by which the latter would then lay the groundwork for Western philosophy.