Zeno, not to be mistaken for the Stoic Zeno, is a Greek mathematician and philosopher of the Eleatic School in Ancient Elea, southwestern Italy (now known as Castellammare de Veila, Italy). Born in the same place, he went to Athens to accompany his teacher – Parmenides at the age of 40.
Very little of Zeno’s works remain, but the writings of Plato and Aristotle clearly provide reference to Zeno’s paradoxes. Philosophically speaking, Zeno believed Parmenides’s claim that the “universe, or being, is a single, undifferentiated substance, a oneness, although it may appear diversified to the senses.”
Zeno’s intention was to discredit the senses and to challenge our understanding of motion against immobility, which he sought to do through a brilliant series of arguments, or paradoxes, on time and space that have remained complex intellectual puzzles up to this day.
The Dichotomy and the Flying Arrow
A typical paradox called the Dichotomy asserts that a runner cannot reach a goal because, in order to do so, he must traverse a distance; but he cannot traverse that distance without first traversing half of it, half of the half, and so on, ad infinitum. Because an infinite number of bisections exist in a spatial distance, and because every interval, no matter how slight, will require a certain amount of time, one cannot travel any distance in finite time, however short the distance or great the speed.
Another of Zeno’s paradox is that if a thing moves, it must move either where it is or where it is not, but it cannot move where it is because there it is at rest, nor can it move where it is not because it does not even exist in that specific location in time. This argument is more commonly known as the Arrow Paradox. It is somewhat connected to the previous one mentioned because like the Dichotomy, it also leads to a conclusion that motion does not exist and what we perceive as motion is only an illusion.
This paradox disproves the existence of motion by the assumption that time is composed of “moments” (and nothing else), and that everything that is in locomotion is always in a “now,” hence the motionlessness of the flying arrow.
The first thing we must notice about Zeno’s framework of disproving the concept of motion is his explicit use of infinity, which almost all of his paradoxes are dependent on.
Observable infinity can be illustrated as an infinite progression between two distinct points, assuming that for every space in between lies a series of numbers that are infinitely divisible. Although it is safe to say that there exists an infinity inside a finite line, it is still impossible to deny that there exists an end or a last term in the set of infinite numbers.
1, ½, ¼, ⅛… ∞…2
It can be seen in the illustration above that between points 1 and 2, there exists an infinite amount of divisible numbers, yet it only exists, in a perverse irony, within a finite line composed of the last term 2. The existence of a last term primarily contradicts the very essence of infinity. Infinity, by its very definition, naturally disposes itself of its “last term” having said that it cannot end.
At this point, our concept of infinity must be expanded to allow room in disproving the Dichotomy paradox. The Dichotomy is naturally fallacious according to Aristotle as he drew a sharp distinction between what he termed a ‘continuous’ line and a line divided into parts (up to infinity).
Consider a simple division of a line into two: on the one hand there is the undivided line, and on the other the line with a mid-point selected as the boundary of the two halves. Aristotle claims that these are two distinct things: and that the latter is only ‘potentially’ derivable from the former. In other words, the infinitely divisible line (one which cannot be traversed) is only potentially derivable from the undivided line.
While it is possible to take forever in traversing a certain distance, it is not required however to travel half the distance at the most, meaning a runner can possibly traverse a line completely by going more than half of the distance to avoid the possibility of Zeno’s infinite half-runs.
The Arrow Paradox presupposes that the past and the future does not exist, leaving only a series of “nows” all of which contain an arrow at rest, thereby giving Zeno the reason to conclude that the flying arrow is motionless.
To assume that an object is at rest in any instant, one must accept that an instant lasts 0 seconds because motion is directly relative to time; whatever speed the arrow has, it will get nowhere if it has no time at all.
In order to maintain the consistency of this argument, Zeno proposed that an instant is indivisible. Then suppose that an arrow ‘apparently’ moved during an instant. It would be at different locations at the start and end of the instant, which implies that the instant has a ‘start’ and an ‘end’, which in turn implies that it has at least two parts, and so is divisible, and so is not an indivisible moment at all.
The most important lesson we can derive from the likes of Parmenides and Zeno is their view that our senses cannot be trusted because it is incoherent with reason as illustrated by Zeno’s paradoxes. And it is in Zeno’s observation of this incoherence that his paradoxes were founded, all aimed to mark a line between the senses and reason.
While some may question the value of these paradoxes to modern times, it must be pointed out that ancient philosophers like Zeno and Parmenides opened the doors for the development of modern science. Zeno and Parmenides’s intent of differentiating perception from reasonable sense paved the way for the development of rationalism and empiricism. Efforts to disprove Zeno’s paradoxes also brought about significant developments in modern calculus especially for 19th century mathematics.
Philosophically disproving (as opposed to mathematical methods) these ancient philosophical puzzles reminds us that the value of philosophy lies not just in metaphysical exercise, but also in its potential to open new doors for scientific development, and in its contribution to modern social discourse.