Pythagoras (approximately 569 BCE - 475 BCE, ) was an Ionian mathematician and philosopher, known best for the Pythagorean theorem which bears his name.

Known as "the father of numbers", Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. Because legend and obfuscation cloud his work even more than with the other pre-Socratics, one can say little with confidence about his life and teachings. Pythagoras and his students believed that everything was related to mathematics, and thought that everything could be predicted and measured in rhythmic patterns or cycles.

Biography

Pythagoras was born on the island of Samos, off the coast of Asia Minor. He was born to Pythais (a native of Samos) and Mnesarchus (a merchant from Tyre). As a young man he left his native city for Crotona in Southern Italy, to escape the tyrannical government of Polycrates. Many writers credit him with visits to the sages of Egypt and Babylon before going west; but such visits feature stereotypically in the biographies of many Greek wise men, and are likely more legend than fact.

Upon his migration from Samos to Crotona, Pythagoras established a secret religious society very similar to, and possibly influenced by, the earlier Orphic cult.

Pythagoras undertook a reform of the cultural life of Crotona, urging the citizens to follow virtue and form an elite circle of followers around himself. Very strict rules of conduct governed this cultural center. He opened his school to men and women students alike. They called themselves the Mathematikoi.According to Iamblichus, the Pythagoreans followed a structured life of religious teaching, common meals, exercise, reading and philosophical study. Music featured as an essential organizing factor of this life: the disciples would sing hymns to Apollo together regularly; they used the lyre to cure illness of the soul or body; poetry recitations occurred before and after sleep to aid the memory.

The Pythagorean theorem that bears his name was known earlier in Mesopotamia, Egypt and India. For a chronology of the theorem and its proofs, see the article on the Pythagorean theorem. Whether Pythagoras himself proved this theorem is not known, as it was common in the ancient world to credit to a famous teacher the discoveries of his students. The earliest known mention of Pythogoras's name in connection with the theorem came five centuries after his death, in the writings of Cicero and Plutarch.

Pythagoras' followers were commonly called "Pythagoreans." For the most part we remember them as philosophical mathematicians who had an influence on the beginning of axiomatic geometry, which after two hundred years of development was written down by Euclid in The Elements.

The Pythagoreans are known for their theory of the transmigration of souls, and also for their theory that numbers constitute the true nature of things. They performed purification rites and followed ascetic, dietary and moral rules which they believed would enable their soul to achieve a higher rank among the gods. Consequently, they expected they would be set free from the wheel of birth.

The Pythagoreans also believed that the sexes are equal. Pythagoras started a school, together with his wife Theano, built on this principle. After he died, his wife and daughters ran and taught at the school. Theano herself discovered a formula to derive the golden rectangle.

All slaves were treated humanely, and animals were respected as creatures with souls. The highest purification of the soul was "philosophy", and Pythagoras has been credited with the first use of the term.

The Jains in India follow similar beliefs and practices, which leads some neo-Pythagoreans and neo-Platonic authors to believe that Pythagoras had visited India, and studied under the Jains.It was the Pythagoreans who discovered that the relationship between musical notes could be expressed in numerical ratios of small whole numbers.

The Pythagoreans elaborated on a theory of numbers the exact meaning of which is still debated among scholars. They taught that all things were numbers, that the essence of everything is a number, and that all relationships can be expressed numerically.

No texts by Pythagoras survive, although forgeries under his name - a few of which remain extant - did circulate in antiquity. Critical ancient sources like Aristotle and Aristoxenus cast doubt on these writings. And ancient Pythagoreans usually quoted their master's doctrines with the phrase autos ephe ("he himself said") - emphasizing the essentially oral nature of his teaching. Pythagoras appears as a character in the last book of Ovid's Metamorphoses, where Ovid has him expound upon his philosophical viewpoints.

Some consider Pythagoras the pupil of Anaximander and some ancient sources tell of his visiting, in his twenties, the philosopher Thales, just before the death of the latter. No account exists of the specifics of the meeting, other than the report that Thales recommended that Pythagoras travel to Egypt in order to further his philosophical and mathematical training.

In astronomy, the Pythagoreans were well aware of the periodic numerical relations of the planets, moon, and sun. The celestial spheres of the planets were thought to produce a harmony called the music of the spheres. These ideas, as well as the ideas of the Platonic solids, would later be used by Johannes Kepler in his attempt to formulate a model of the solar system in his work The Harmony of the Worlds.

Pythagoreans also believed that the earth itself was in motion and that the laws of nature could be derived from pure mathematics. They may have coined the term cosmos, a term implying a universe with orderly movements and events.It is sometimes difficult to determine which ideas Pythagoras taught originally, as opposed to the ideas his followers later added. While he clearly attached great importance to geometry, classical Greek writers tended to cite Thales as the great pioneer of this science rather than Pythagoras.

The later tradition of Pythagoras as the inventor of mathematics stems largely from the Roman period.Whether or not we attribute the Pythagorean theorem to Pythagoras, it seems fairly certain that he had the pioneering insight into the numerical ratios which determine the musical scale, since this plays a key role in many other areas of the Pythagorean tradition, and since no evidence remains of earlier Greek or Egyptian musical theories.

Another important discovery of this school - which upset Greek mathematics, as well as the Pythagoreans' own belief that whole numbers and their ratios could account for everything in nature - was the incommensurability of the diagonal of a square with its side. This result showed the existence of irrational numbers.

The influence of Pythagoras has transcended the field of mathematics, and the Hippocratic Oath - with its central commitment to First do no harm - has its roots in the oath of the Pythagorean Brotherhood.

Pythagoreans

Pythagoreanism

Pythagorean Comma

Pythagorean Numerology - Crystalinks


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