PYTHAGORAS OF SAMOS

 Pythagoras of Samos (ca. 580–500 BCE) was a great Greek mathematician. As a young man, he was coached by the aging Thales, and he would continue the Greek quest to turn the mathematics of the Egyptians, Babylonians, and early Indians from a practical computational discipline into a beautiful, abstract philosophy. It was Pythagoras who gave us the ubiquitous Pythagorean theorem, which allows us to determine the length of a right triangle’s hypotenuse. Today GPS and maps use this theorem—as well as our very early understanding of numbers and of geometry—to compute distances between two locations.

Pythagoras was born on the Greek island of Samos, a stone’s throw from the Anatolian Plateau of Asia Minor, which at that time was also part of greater Greece. The island is home to the Temple of Hera, one of the Seven Wonders of the Ancient World (although, unlike the almost-intact Great Pyramid, this temple has only one marble column still standing). Today the main town on the island is called Pythagoreion in honor of the island’s native son.

Pythagoras began his life as a precocious intellectual adventurer, curious about nature, life, philosophy, religion, and mathematics. As a young man, he traveled extensively. In Egypt he met with priests in temples to learn about their religion, their knowledge of the world, and their mathematics. In Mesopotamia he visited astronomers to learn how they observed celestial bodies, and he studied their mathematical and scientific methods. Did he learn about the theorem he is now credited with developing, or did he simply absorb related concepts in Mesopotamia? This we do not know. Because mathematics had roots in India as well, and because some Pythagorean ideas appear to be related to Indian mathematical principles, some historians have surmised that Pythagoras may have traveled as far as India. We have no confirmation of this conjecture, however.

Neither do we know how the great Thales met the young Pythagoras. We do know that the two men knew each other and that Thales recognized Pythagoras’s budding intellect and encouraged him to expand his horizons. According to the third-century philosopher Iamblichus, who wrote a biography of Pythagoras, “Thales, admiring his remarkable ability, communicated to him all that he knew, but, pleading his own age and failing strength, advised him for his better instruction to go and study with the Egyptian priests.”

Pythagoras wanted to see much more than Egypt, so he first traveled east to Phoenicia, visiting Byblos, Tyre, and Sidon, where he met with priests and learned about Phoenician rites and customs. Pythagoras suspected that the rites and rituals he was observing and learning in Phoenicia had Egyptian roots, and he proceeded to Egypt to find their origin, just as Thales had encouraged him to do.

There, he studied with the priests and prophets and instructed himself on every possible topic … and so he spent 22 years in the shrines throughout Egypt, pursuing astronomy and geometry and, of set purpose and not by fits and starts or casually, entering into all the rites of divine worship, until he was taken captive by Cambyses’ force and carried off to Babylon, where again he consorted with the Magi, a willing pupil of willing masters. By them he was fully instructed in their solemn rites and religious worship, and in their midst he attained to the highest eminence in arithmetic, music, and the other branches of learning. After twelve years more thus spent he returned to Samos, being then about 56 years old.

When he returned to his native island, Pythagoras was steeped in exotic ideas that he had absorbed during his travels. He developed a religious belief that the soul never dies but rather transmigrates to other living things. Hence, if a person kills another living thing—even a small insect—he could be killing a being with the soul of a deceased friend. This idea, which bears a strong resemblance to the Indian notion of reincarnation, led Pythagoras to a strictly vegetarian lifestyle. He also developed an aversion to eating beans—perhaps another fetish acquired as a result of his travels.

Pythagoras began to think about how he could combine the science of numbers and measurement that he absorbed in Egypt and Mesopotamia with the theorems of his Greek predecessor, Thales. Numbers fascinated him, so much so that eventually he and his followers would come to believe that “God is number.” Further, Pythagoras transformed mathematics into the abstract philosophical discipline we see in pure mathematics today.

Pythagoras’s notion that numbers held powers led to a kind of number mysticism, and he became a sort of guru. A growing group of disciples who adhered to his strict lifestyle principles and devoted their time to studying the abstract concepts of the new discipline of mathematics gathered around him. At some point a fearful island leader who worried that the group might someday vie for political power and unseat him applied political pressure on the Pythagoreans, and they were forced to leave Samos. Pythagoras and his followers moved to a place called Crotona, in the center of the bottom of the Italian “boot,” which was also part of Magna Graecea (greater Greece). Isolated from the surrounding population, members of the secret society dedicated themselves to their religion—number mysticism—and the study of mathematics.

Pythagoras continued the work of Thales in pure mathematics and is seen to have transformed the discipline into “a liberal form of education, examining its principles from the beginning and probing the theorems in an immaterial and intellectual manner.” The “educational” aspect of mathematics was pursued in lectures that Pythagoras delivered to the members of his sect. These talks consisted of theorems, results, and discoveries about numbers and their meaning. As a form of public service to the outside community that surrounded the sect’s compound—and, perhaps, to avoid being chased away, as had happened at Samos—Pythagoras gave public lectures to the entire community living in the area. The talks within the sect were strictly confidential, however. Most of the discoveries Pythagoras and his followers made about numbers were kept secret, with only select facts released to the outside world.

Pythagoras and his followers also understood fractions, such as 2/7 or 31/77. We call such numbers rational numbers, perhaps because they make sense to us. But when the Pythagoreans went further in their mathematical and mystical exploration of numbers and their properties, they ran into a conundrum that stunned them and perhaps even brought on their demise. This paradox—which came about in the interface between geometry and arithmetic—would come to a head with the work of Georg Cantor in the nineteenth century, and it continues to haunt us even today.