Zeno of Elea | Biography

Zeno of Eleawas a Greek philosopher and logician whose development of paradoxical philosophical arguments about motion greatly influenced mathematical thought. One of the last major proponents of the Eleatic school of philosophy, he created his paradoxes to defend and confound this school of thought's detractors. His importance in early Greek philosophy and mathematics was noted by both Plato and Aristotle, who credited Zeno with the creation of dialectics.

Zeno was born in Elea, a southern Italian city, in the fifth century B.C. Although the exact date is unknown, historians place his birth at around 490 B.C., based largely on references by Plato in his book Parmenides. Plato describes Zeno as "tall and fair to look upon" and a favorite disciple of Parmenides, a Greek philosopher. He also reports that Zeno was about 40 years old when he accompanied his teacher to Athens in 449 B.C. In Athens, Zeno met and greatly impressed a young Socrates. Legend indicates that Zeno stayed for a number of years in Athens, where he made a good living by teaching philosophy. Athenian statesmen Pericles and Callias possibly studied under him.

According to Diogenes Laëritus, Zeno also created a unique cosmology in which he proposed the existence of several worlds. However, he was predominantly known for his paradoxes, which instigated centuries of philosophical debate. Little else is known about Zeno's life. He returned to live and work in Elea, where he eventually was caught up in a political intrigue that led to his death.

Develops Philosophical Paradoxes

Zeno's teacher, Parmenides, was without question the greatest influence on his life. The founder of the Eleatic school of philosophy, Parmenides believed that there is only the "One" unchangeable being who constitutes the universe and argued that people's senses were mistaken in interpreting a world of motion and change. Parmenides was attacked and ridiculed by some, including the Pythagoreans. Zeno set out to defend his mentor's view of the world by creating a series of philosophical paradoxes.

At a very young age, Zeno wrote his famous and only known work, Epicheiremata. Only a few small fragments, amounting to about 200 words, survive. Fortunately, through Plato, Aristotle, Proclus, and other ancient authors, historians have been able to piece together the fundamentals of Zeno's work, which reportedly was published without his consent.

Zeno's unique approach to defending his master does not take the usual course of trying to prove Parmenides' position. Instead, he set out to create proofs that would reveal the absurd and self-contradictory nature of his opponents' views. The result was a series of paradoxesthat profoundly influenced both philosophical and mathematical thought.

Zeno's philosophical puzzles focus on the goal of proving that our sense of motion, time, and change are based on illusion. According to Plato and Proclus, Zeno developed 40 different paradoxes (of which only eight survive) designed to disprove, confound, and agitate Parmenides' detractors. Among the most famous is the paradox of "Achilles," who was a mythological Greek hero who was the "swiftest of mortals." This paradox states that Achilles could not catch a crawling tortoise with a head start because Achilles must always reach a point at which the tortoise has started from and already passed. The "Dichotomy" and "Arrow" paradoxes similarly focus on the illusion of motion as represented by points in space or time. The paradox of the arrow, for example, states that an arrow can never reach it target because an object can only reach one point at a time and to be someplace (or at some point) an object must be at rest. If the arrow as at rest at each point, then motion forward is impossible.

Simplistically, these paradoxes may appear to be mere philosophical "brain teasers," but they represent the foundation of a type of philosophical argument called dialectic. This method of deduction allows one to work out contradictory conclusions from a single postulate or proposition that is assumed to be true. The dialectic approach to philosophy influenced the work of Plato, Aristotle, Immanuel Kant, Georg Hegel, Bertrand Russell, and even Albert Einstein , who, like Zeno, was interested in the concepts of space and time.

Influences 20th Century Mathematics

Although developed primarily as a means of defending his teacher's philosophical views, Zeno's paradoxesalso created difficulties for mathematicians. Some historians believe Zeno's paradoxes may have been directed toward the Pythagoreans (followers of the great Greek mathematician Pythagoras) and their handling of the infinitesimal in geometry. Eudoxius of Cnidus, who was a member of Plato's academy, may also have been influenced by Zeno in his development of a theory of proportions to accurately deal with the infinitesimal. However, since Greek mathematicians had not developed a strong concept of convergence or infinity, they mostly chose to ignore the mathematical dilemmas presented by the paradoxes. This early neglect was reinforced by Aristotle's declaration (without proof) that Zeno's paradoxes were merely fallacies.

Although Zeno's influence on early Greek mathematics probably was minimal, many of his paradoxes, like the "Achilles" paradox, relate to the concepts of infinity and the continuum, which would become major areas of interest in calculus. But it would take 19th- and 20th-century mathematicians to clearly understand the mathematics of Zeno's paradox, which is based on the distinction between measures of intervals and the number of points they contain. With the development of this concept, Zeno's paradoxes established an important influence on the work 19th-century German mathematician Karl Weierstrass, who helped start the "mathematical renaissance" and developed the notion of uniform convergence. The dilemmas presented by the paradoxes continued to exert a strong hold on mathematical thought through the work of Bertrand Russell, German mathematician Lewis Carroll, who introduced the concept of infinite sets, and many others.

Despite being dubbed the first "great doubter" and "man of destiny" in mathematics, Zeno was primarily a philosopher and not a mathematician. It is a testament to Zeno's keen intellect that later mathematicians went on to form numerous mathematical theories and problems from his work. While Cantor's theories on infinite setsled to explaining some of Zeno's paradoxes, the philosophical problems he presented have remained of interest and debate.

As reported by Plato, Zeno admitted that his book developed from the "pugnacity" of a younger man defending his mentor. Zeno, who was reported to be an active participant in politics, exhibited a similar contentiousness later in life when he "defended" good government by joining a plot against the tyrant Nearchus. The plot failed, and Zeno was arrested. According to legend, Zeno was tortured to reveal the names of his co-conspirators. Instead, the unyielding Zeno named the tyrant's own friends. Another legend says that Zeno bit off his tongue and spit it at his interrogators rather than betray his friends. He was eventually executed around 440 B.C.