Zeno of Elea (c.465 BCE) was a Greek philosopher of the Eleatic School and a student of the elder philosopher Parmenides (an older contemporary of Socrates). Little is known of Zeno's life outside of his association with the Eleatic School founded by Parmenides. Parmenides argued against the validity of our senses and the supposed `truth' they tell us about the world. As with all of the extant writings of the Pre-Socratic philosophers, Parmenides' work seeks to establish the underlying form of being, that essential 'stuff' from which all of life and the sensible world comes. Parmenides claimed that the previous definitions for this 'stuff' were wrong in that they posited individual elements like water (Thales of Miletus) or Air (Anaximenes) when, actually, all of reality and observable existence was One. Parmenides would argue that if a person were to place a board and a hammer and a nail on a table, sense perception would indicate three separate objects on that table. Parmenides, however, would claim that this perception would be wrong as the board, hammer, and nail are all composed of the same basic material and participate in the unity of existence and so, in spite of what we might conclude through the senses, the three objects are really one. As Parmenides had many critics, Zeno, in his famous Paradoxes, sought to prove the truth of his master's claim logically and so silence those who sought to prove him wrong.
Zeno set about to prove the unity of existence mathematically. Arguing against motion, the senses, and plurality, he wrote 40 paradoxes showing how, logically, change and motion cannot exist (of these 40, contained in one volume, less than ten exist today). His best known are The Race Course, The Achilles, The Arrow, and The Stadium, all of which prove the logical impossibility of plurality and motion. Zeno's paradoxes have fascinated mathematicians and logicians for hundreds of years and have yet to be satisfactorily solved.
The paradox of The Race Course, to use just one example, shows how motion is a lie of the senses and cannot logically exist. This paradox claims that, if a runner is to sprint 100 metres, she must first travel half that distance. In order to travel half that distance, she must first travel half that distance and, to do that, she must first travel half that distance. By this progression, Zeno showed that, no matter how small a distance was left, it was still impossible, logically, for the runner to ever meet her goal. No matter how far or near, there would always be a distance which separated the runner from the goal. In this same way, Zeno argued, all of perceived reality is One, unchanging and eternal, and the perception that human beings live in a world of plurality, of `the many' (many things, people, places) is an illusion created by the senses.
In his dialogue of the Parmenides, Plato sets down the fundamental criticism of the claims of Parmenides and Zeno when he has Socrates say,
If a person could prove the absolute like to become unlike, or the absolute unlike to become like, that, in my opinion, would indeed be a wonder; but there is nothing extraordinary, Zeno, in showing that the things which only partake of likeness and unlikeness experience both. Nor, again, if a person were to show that all is one by partaking of one, and at the same time many by partaking of many, would that be very astonishing. But if he were to show me that the absolute one was many, or the absolute many one, I should be truly amazed. And so of all the rest: I should be surprised to hear that the natures or ideas themselves had these opposite qualities; but not if a person wanted to prove of me that I was many and also one. When he wanted to show that I was many he would say that I have a right and a left side, and a front and a back, and an upper and a lower half, for I cannot deny that I partake of multitude; when, on the other hand, he wants to prove that I am one, he will say, that we who are here assembled are seven, and that I am one and partake of the one. In both instances he proves his case. So again, if a person shows that such things as wood, stones, and the like, being many are also one, we admit that he shows the coexistence the one and many, but he does not show that the many are one or the one many; he is uttering not a paradox but a truism. If however, as I just now suggested, some one were to abstract simple notions of like, unlike, one, many, rest, motion, and similar ideas, and then to show that these admit of admixture and separation in themselves, I should be very much astonished. This part of the argument appears to be treated by you, Zeno, in a very spirited manner; but, as I was saying, I should be far more amazed if any one found in the ideas themselves which are apprehended by reason, the same puzzle and entanglement which you have shown to exist in visible objects. (127E)
In this passage, Socrates is asking how the `many' can be `one' in the physical, not just the abstract, world. The board, hammer, and nail placed on the table are, clearly, three objects which do not partake in the properties of each other. The board is made of wood, the hammer of wood and metal, the nail of metal alone. These objects cannot possibly be categorized as `one' but must, of necessity, be considered `many'.
Zeno countered this argument by showing that the `many' have to be `one' because, for plurality to exist, logic could not. Since logical sequence and understanding does exist, there can be no plurality. Professor J. M. Robinson comments on this, writing, "As we can see from the first hypothesis of the first argument of Zeno's treatise, the thesis that things are a many give rise to consequences that are inconsistent even with one another; for if things are a many they must be `both like and unlike' and this is impossible not because it violates sense perception (which is, after all, fallible), but because it violates the law of contradiction, which lies at the basis of all thought" (128). One cannot, then, claim that the board, hammer, and nail are `many' in that the three objects partake of the same basic substance of the One. A person may look at the three objects and claim there are `many' objects on the table but that would only be an expression of trust in sense perception, not a valid apprehension of the truth.
Zeno maintained that trust in the senses leads to contradictory conclusions, in that something which exists and 'is' cannot not exist and not be, and yet our senses tell us that everything is always changing from what it 'is' to something it 'is not'. This assertion stands in contradiction to Heraclitus' claim that "Life is Flux" and everything is in constant motion and transformation. To Zeno, this was a faulty conclusion based upon unreliable sense perception. That which is cannot not be because it would then contain within itself the contradiction of having the qualities of `being' and `not being' and, as this defies logic, it cannot be held as true. In this, both Parmenides and Zeno were at complete odds with the philosophy of Heraclitus but, at the same time, seemed to share his belief that the majority of human beings could not, or would not, seek to understand the truth behind the apparent reality which the senses provide.