Zeno of Elea

Greek philosopher (c. 490 bc–c. 430 bc)

Zeno was born at Elea (now Velia in Italy) and in about 450 bc accompanied his teacher, Parmenides, to Athens. There he propounded the theories of the Eleatic school and became famous for his series of paradoxes and his invention of dialectic.

Little survives of Zeno's written work and this only in other authors' writings. He proposed that motion and multiplicity are unreal (thus supporting Parmenides's theories) since assumption of their existence gave rise to contradictory propositions. One of the most famous arguments against plurality and motion is that of Achilles and the tortoise: if the tortoise is given a start in a race against Achilles, when Achilles reaches the tortoise's starting position, the tortoise will have advanced a small way to a new position. Endless repetition of this argument means that Achilles can never overtake the tortoise.

Zeno's paradoxes remained unresolved for about 20 centuries, in fact until the advances in rigor of mathematical analysis (to the development of which these paradoxes may be said to have contributed). These advances included the study of convergent series (infinite series with a finite sum), the invention by Gottfried Leibniz and Isaac Newton of calculus, and Georg Cantor's theory of the infinite in the 19th century.

Following his return to Elea Zeno died while joining a coup against the tyrant Nearchus.

Zeno of Elea (born ca. 490 B.C.) was a Greek philosopher and logician. A member of the Eleatic school of philosophy, he was famous throughout antiquity for the rigorously logical and devastating arguments which he used to show the absurdities and contradictions of his opponents.

Zeno was born in the southern Italian city of Elea. Plato says that Zeno and Parmenides visited Athens about 449 B.C., where the young Socrates made their acquaintance and where Zeno made a striking impression. Ancient authorities asserted that, like Parmenides, Zeno was a Pythagorean, that he engaged in political activity in his native city, and that he was put to death for plotting against a tyrant. An oft-repeated story tells of his bravery under torture and the painful death which he endured.

It is possible that Zeno wrote more than one work, but he is best known for a single volume of epicheiremata (attacks) on the postulates of Parmenides's opponents. Only fragments of this work have survived, but a fairly clear idea of his methods may be found in the summaries given by Aristotle and the 6th-century A.D. Neoplatonist Simplicius. Zeno seems to have had no constructive theories of his own to set forth, and some of his destructive arguments seem to apply equally well to the conclusions drawn by Parmenides.

Zeno's original contribution to thought was the method of deduction which he developed to work out two sets of contradictory conclusions from a given postulate. From his argument against the pluralists, the following example is typical. Let it be postulated that everything has developed from an originally plural source (as opposed to Parmenides's One). If things are many, then they are finite since they are the number that they are. But if things are many, they are infinite since there must always be other things between them, and others between those ad infinitum.

From Zeno's arguments on motion comes the famous example which shows the impossibility of ever reaching the end of a given line. A moving body cannot reach the end because it must constantly reach the midpoint, and then another midpoint, again ad infinitum. It is unclear from the available evidence if Zeno's purpose was to offer a serious defense of the One or if he simply wanted to show that other arguments were no better. Aristotle called him the inventor of dialectic, and Zeno was undoubtedly one of the important early Greek logicians.

Further Reading

Selected passages from Zeno's work in English translation and with commentary are in Geoffrey S. Kirk and John E. Raven, The Presocratic Philosophers: A Critical History with a Selection of Texts (1957). Excellent discussions of Zeno's importance and place in the development of Greek thought are in John Burnet, Early Greek Philosophy (1892; 4th ed. 1930), and Kathleen Freeman, The Pre-Socratic Philosophers: A Companion to Diels, Fragmente der Vorsokratiker (1940; 3d ed. 1953). More generalized discussions of the Pre-Socratics and their place in Greek literature and thought appear in the standard histories of Greek literature, such as that by Albin Lesky, A History of Greek Literature (1958; trans. 1966).

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(fl. early 5th c. bc) The pupil and principal defender of Parmenides, Zeno was called the inventor of dialectic by Aristotle. His one book, of which we possess only fragments, contained many arguments for the unreality of the pluralistic world that we take ourselves to inhabit. The most famous of these are the four arguments against motion, known as Zeno's paradoxes. But Zeno also proposed many other antinomies, showing that objects must be both limited and unlimited in number, like and unlike, one and many, infinitesimally small and infinitely large. Zeno's own attitude to these antinomies, as to his arguments against motion, has been disputed, but he is consistently described as a convinced Parmenidean.

Zeno of Elea (zē'nō, ē'lēə) , c.490–c.430 B.C., Greek philosopher of the Eleatic school. He undertook to support in his only known work, fragments of which are extant, the doctrine of Parmenides by demonstrating that motion and multiplicity are logically impossible. The substance of his argument against multiplicity was that a whole must be composed of ultimate indivisible units, or it must be divisible ad infinitum. If the whole is divisible ad infinitum, there is a contradiction involved in the assumption that an infinite number of parts can be added up to a finite total. The essence of his argument against motion was that a moving body can never come to the end of a line, as it must first cover half the line, then half the remainder, and so on ad infinitum. The thrust of these arguments was to demonstrate, through logical reasoning, the error of common-sense notions of time and space. According to Aristotle, Zeno was the first to employ the dialectical method. Contemporary philosophers and mathematicians have taken renewed interest in Zeno's problems.

Bibliography

See A. Grunbaum, Modern Science and Zeno's Paradoxes (1967).

(5th century bc). Greek philosopher, celebrated for his paradoxes of motion. These are: Achilles and the Tortoise, the Flying Arrow, the Stadium, and the Row of Solids. They are well described by J. Burnet in Early Greek Philosophy (4th edn. 1930).

Zeno introduced the philosophical technique of trying to establish characteristics of reality — or at least of what cannot be real — from paradoxes.

The basis of the paradoxes is that if a distance between A and B can be subdivided an infinite number of times, and an object travels from A to B by crossing successive points (subdivisions), then it must cross an infinite number of points — which would take an infinite time. Therefore, motion is impossible! The paradoxes derive from the manner of expressing the problem. This is itself interesting, indicating the extreme importance of finding appropriate mathematical and logical kinds of description, as well as appropriate conceptual models for describing or explaining the physical world and mind.

(Published 1987)

— Richard L. Gregory

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