By: A.C. Grayling
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.
Diogenes says 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.
One might think these picturesquely gory details are intended to enliven what anyone would think is the otherwise staid tale of people whose greatest excitement consists in thinking; but in fact philosophers have had a lively time, as their biographical details often show – for ideas can be dangerous things, demanding courage to express them or live by them. Diogenes wrote a tribute to Zeno as follows: ‘You wished, Zeno, and noble was your wish, to slay the tyrant and set Elea free from bondage. But you were crushed; for, as all know, the tyrant caught you and beat you in a mortar. But what is this that I say? It was your body that he beat, not you.’
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.
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. 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.
Again, consider Achilles racing a tortoise. If the tortoise is given a head start, however small, Achilles 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 Achilles must therefore reach that next point. But by the time he does so … and so on.
A third argument is this. Consider an arrow fired at a target. 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.
Some answers are suggested by Aristotle himself. 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.
As to the arrow argument: Aristotle says that it depends on the assumption ‘that time is composed of “nows” [that is, discrete intervals]. If this is not conceded, the deduction will not go through.
’ 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 dividings can never stop.
The Pythagoreans also appear to be the target in Zeno’s argument against space, given their doctrine about air coming into the cosmos from outside the cosmos. ‘If there is space, it will be in something, for all that is, is in something, and what is in something is in space. So space will be in space, and this goes on ad infinitum; therefore there is no space.’ Leaving aside the assumption that space is regarded as a container in something like Newton’s sense of absolute space, rather than (say) a set of relationships between objects, and whether there are fallacies of equivocation (that is, multiple senses in the same word) in the words ‘something’ and ‘in’, there is the question why the concept of infinite space should be intrinsically incoherent, as Zeno assumes.
This raises the question of Zeno’s deployment of the concept of infinity. What has come to be called ‘the standard solution’ to Zeno’s paradoxes of motion invokes calculus, invented independently by Newton and Leibniz in the seventeenth century, and his talk of infinity prompts discussions about actual and potential infinities, the concept of the former only receiving a full formal defence in the work of the mathematicians Richard Dedekind and Georg Cantor at the turn of the twentieth century. Ideas variously to the effect that the elements of physical reality cannot be infinitely divisible, that the notion of space, or of perceived reality as a whole, is contradictory, that there is a need to construct paraconsistent logics in which both arms of contradictions can be held to be true, are just some of the outcomes that reflection on Zeno’s paradoxes has prompted.
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.
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.