Zeno of Elea - Biography

Zeno of Elea was a Greek thinker, born around 490 B.C., about whom very little is known. Plato described him as a fair and tall man that was beloved by Parmenides of Elea. Zeno became famous for writing a series of paradoxes that have intrigued, puzzled and irritated thinkers from his time until the present day. Most of what is known about his life and purposes comes from either Plato or Aristotle, the first accusing him of being a mere defender of his mentor Parmenides of Elea while the second has described Zeno as the inventor of dialectics in the lost work Sophist. Zeno's propositions are quite literally para-doxes, from the Greek against, or contrary to, beliefs or opinions. His paradoxes challenged very basic common conceptions of plurality, space and motion. From its mathematical terms, Zeno's paradoxes are believed to have been influenced by the efforts of Pythagoras, to apply mathematical concepts to the natural world. The ultimate reasons for Zeno’s paradoxes are most likely forever lost. The number of paradoxes written is also subject to dispute, mostly around 40, and none of them survive to our times in their original form. Most of the information we have on Zeno's paradoxes comes from Aristotle's Physics, where Aristotle's develops on, and refutes some of, the paradoxes. Zeno is also known as one of the first examples of anti-logic or reductio ad absurdum (proof by contradiction). The paradoxes that most propelled thinkers from the past two millennium are the ones of the tortoise and Achilles, of the flying arrow, and of the dichotomy argument. Philosophers and mathematicians have argued extensively about the nature of the paradoxes, if metaphysical or mathematical.

Plato's account of Zeno and his paradoxes can be found in the dialogue Parmenides of Elea, written on the occasion of the Athenian visit of Parmenides of Elea and his Pupil Zeno, both from Elea. From Plato's descriptions of Parmenides of Elea, Zeno, and Socrates, it is assumed that Zeno was born around 490 B.C. In the Parmenides of Elea dialogue by Plato, Socrates appears as a young listener to Zeno's first readings of his paradoxes in Athens. There Zeno is said to have read the following paradox: "If the things that are are many, that then they must be both like and unlike, but this is impossible. For neither can unlike things be like, nor like things unlike." Zeno further seems to argue that his overall aim with his paradoxes was to demonstrate the inconsistency of the common belief that there are many things and, although the passage is out of a fictional work by Plato, it is nevertheless taken as an evidence of Zeno's overall project. Plato also claimed that all Zeno does is to repeat Parmenides of Elea but twisting the form so as to fool the others into thinking he is saying something altogether different. He claims that while Parmenides of Elea states that all is one, Zeno claims that there are not many things, which are virtually the same statement.

Aristotle, for his part, claimed that Zeno's paradoxes were constantly reworked and rewritten by those who edited them so that it is hard to say what is the original and what are reworked versions by other authors. The paradoxes of motion, present in Aristotles Physics, do not have a direct connection with the supposed thesis that Zeno's overall project was to question the common belief that there are many things. It may be said, based on the paradoxes as described by Aristotle that if Zeno were to have an overall project, it would be to question both plurality and motion.

One of the best know paradoxes attributed to Zeno is Achilles and the Tortoise, preserved in the following form in Aristotle's Physics: "In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead." So if Achilles is running 1000m behind the tortoise, when he completes the distance, the tortoise will have already advanced to another point in space, say 100m ahead. When Achilles covers the following 100m, the tortoise will be already another 10m ahead and so on. However small the distance between Achilles and the tortoise, following Zeno's logic, there will always remain a gap, the gap which corresponds to the distance the tortoise managed to advance while Achilles was busy reaching the previous position of the tortoise.

In another paradox classified as the paradox of motion, and named The Dichotomy Paradox, Zeno challenges the possibility of travelling from one point to another. As from Aristotle's Physics: "That which is in locomotion must arrive at the half-way stage before it arrives at the goal." So if a car is to cover a distance of 100m, it will first have to cover 50m. Before covering 50m, it will have to cover 25m. This logic is then pushed to infinity making it impossible to start moving as well as impossible to define a first distance, as it could always be divided in half. Zeno concludes that all motion must therefore be a mere illusion. The paradox is called The Dichotomy Paradox because of its repeatedly split by two of the distance. Aristotle saw in this paradox a variation of the Achilles and the Tortoise paradox. It is know as the Race Course Paradox as well.

The last of the motion paradoxes is The Arrow Paradox. Once more, from Aristotle's Physics: "If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless." If in an instant of time, an object is occupying a specific place in space, it means that this object is still. But, even in the case of a flying arrow, it can always be said that, in a single, duration-less moment, it occupies a specific place in space and is therefore motionless. In the next moment in time, the arrow may well be in a different place, but at that moment, as in every moment, it is motionless. The conclusion is that if time is composed of instants and if in every instant everything occupies a place in space (motionless), then motion is impossible. Different then the Achilles and the Tortoise paradox and the Dichotomy Paradox, where time was always divided in slots, in The Arrow Paradox, time is divided into points, instants.

The paradox that is believed to have reached the present day closest to its original form is a paradox named The Antinomy of Limited and Unlimited, provided by Simplicius in his commentary on Aristotle's Physics. It says: "If there are many things, it is necessary that they be just so many as they are and neither greater than themselves nor fewer. But if they are just as many as they are, they will be limited. If there are many things, the things that are are unlimited; for there are always others between these entities, and again others between those. And thus the things that are are unlimited". The argument seems to develop from the postulate that if two things are separate, then there must be a third thing in between them, and so on ad infinitum. It suggest a contradiction between things being definable and therefore countable, finite, and things being separate, having a third thing in between, and therefore logically infinite.

Thomas Aquinas, in the 13th Century, commented on Aristotle's comments of Zeno's paradoxes, arguing that time is not made of instants. Bertrand Arthur William Russel for his part, agreed with Zeno's proposal that in a duration-less instant, an object can only be still in space, but argued that what happens in between two such moments, given that the object has travelled in space, is motion. Peter Lynds simply dismisses Zeno's paradoxes by claiming that they are founded on the assumption that such things as an instant exist, which Lynds objects. The early discoveries of quantum mechanics at the end of the 1970s, where it was noted that a quantum system can be inhibited or hindered through observation, were named "quantum Zeno effect" for its similarity with Zeno's The Arrow Paradox. His paradoxes have also populated many novels, most notably works by Leo Tolstoy, Lewis Carroll, and Jorge Luis Borges. Slavoj Žižek also discusses some of them in his Looking Awry: an introduction to Jacques Lacan through popular culture.