Pythagorean School

Summary: Religious devotees of mathematics, the Pythagoreans could not accept irrational numbers but made lasting contributions.

The Pythagorean School is the name given to a number of mathematicians and followers of Pythagoras. Pythagoras founded the school in the sixth century b.c.e. in what is now southern Italy. It appears to have been a religious sect built around the proposition that reality was revealed through numbers. It was one of the earliest philosophical schools, and at the time there were no rigid boundaries between philosophy, religion, and politics.

The school had aspects of all three and was a major political force in some Greek cities. To some extent, it was thought of as a secret society. Initiates are said to have taken a vow of silence. This fact and many others about the school are difficult to verify because of a lack of sources from this time. Most of what is known about the Pythagoreans comes in fragments from later philosophers like Plato or Aristotle. Much of the detail about Pythagoras’s life is revealed from even later sources, in the works of Diogenes Laertius, Iamblichus, and Porphyry, who wrote many centuries after his death. As a result, much information about the school is ancient hearsay that embellishes what was already a peculiar belief system. The Pythagorean habit of attributing discoveries to Pythagoras, as well as the silence, also makes it hard to distinguish the discoveries of the man from his school. Nevertheless, the influence of his school and mathematical philosophy can still be felt in the twenty-first century concept of “the liberal arts.”

Pythagoras and the Foundation of the School

Pythagoras lived from around 580 to 500 b.c.e., but the exact dates are uncertain. He was the son of a leading citizen of Samos (an island in the Aegean), and it is possible that his political significance led Pythagoras to leave the city during the rule of Polykrates the Tyrant. He does not seem to have become prominent until around 530 b.c.e. in the city of Croton, on the southern shore of Italy. The ancient authors account for his life before then by a journey gathering the wisdom of other cultures, such as the Egyptians and Babylonians.

The wisdom he gained is said to have given him many powers. For example, he claimed to recall his previous incarnations, such as his life as the Trojan hero Euphorbus. He is also said to have appeared talking to friends at Metapontium in southern Italy and Tauromenium, on Sicily, on the same day, despite this being impossible with the transport of the day. The same chapter of Porphyry’s Life of Pythagoras also recounts that a river spoke very clearly to say “Hail Pythagoras!” as he crossed it. Some ancient authors, such as the philosopher Heraclitus, were unconvinced. These and similar tales show not only that he was seen as a divine figure in the ancient world but also that the ancient sources are not wholly reliable. Some modern historians go so far as to discount any mathematical achievements being the work of the actual Pythagoras. Instead, they argue that the achievements of Pythagoreans were attributed to Pythagoras to add luster to his memory.

The school was extraordinarily egalitarian for its era. It admitted both men and women at a time when women were not considered citizens and were usually treated in the same manner as children. The school spread as a society throughout southern Italy and seems to have become a potent political force. Eventually, the power of the school was challenged by the non-Pythagoreans, and violence ensued. Polybius, writing in the second century b.c.e., described the chaos as being a maelstrom of murder, sedition, and “every kind of disturbance.” There are several conflicting stories of the death of Pythagoras, but the oddest is that it occurred because he refused to cross a field of beans when an angry mob was chasing him. This behavior was eccentric even by the standards of ancient Greece and only makes sense in light of the Pythagorean beliefs taught at the school.

Pythagorean Beliefs

Pythagoreans believed that numbers were a fundamental property of the universe and that the cosmos operated in harmonies that could be represented as ratios of whole numbers. The purpose of life was to achieve harmony with the universe through a process of purification to counter the corrupting influence of the body. One of the features of this purification was that Pythagoreans were vegetarian—a strong political statement. At this time, one of the duties of a citizen was to participate in civic religious events. Avoiding such events or refusing to perform them properly could draw the ire of the gods. Almost all festivals required the sacrifice of an animal, usually an ox or a goat. The fat and bones would be offered to the gods on the altar and meat would be part of a communal meal. A vegetarian was therefore separating himself from the community.

As for the material that made the cosmos, Pythagoras thought it was governed by numbers. He is said to have come to this conclusion after discovering that musical harmonies can be represented as ratios of whole numbers. The connection between two such different practices such as music and mathematics led Pythagoras to believe that there must be something cosmically significant about numbers. These ratios are embedded in the tetractys symbol—a triangle of 10 dots in four rows, one dot at the top, then two dots, then three, and finally four. The ratios of the motions of the planets were also assumed to be harmonious, and it is said that Pythagoras claimed to be able to hear “the music of the spheres,” the harmonies generated by these motions. Numbers that could not be represented by ratios of whole numbers were therefore a serious problem in Pythagorean cosmology.

The Pythagorean Legacy

It is hard to be sure that the theorem that bears his name was actually a Pythagorean concept. While 3-4-5 triangles were used before Pythagoras’s time, he may have been the first to prove the Pythagorean theorem, or this might be a later proof attributed to the inspiration of the school. However, there is reason to consider the interest in irrational numbers to be a Pythagorean innovation. Quite how this was discovered is uncertain. The Pythagorean theorem can be used to prove that √2 is irrational, but irrationality can also be found in the “pentalpha,” a five-pointed star more commonly known as “pentagram,” adopted as a symbol by the Pythagoreans. The discovery of irrational numbers is sometimes credited to Hippasus of Metapontum. Usually in these tales, Hippasus meets a grisly end at the hands of Pythagoras who resents the existence of irrational numbers. While this might be a fantastical tale, it is believed that the Pythagoreans were sworn to secrecy concerning the existence of irrational numbers because it was a significant threat to their belief system.

A celebrated legacy of the Pythagorean school is that its approach to applying mathematics to the natural world led to the establishment of the quadrivium: arithmetic, astronomy, geometry, and music that, with grammar, logic, and rhetoric, formed the “liberal arts” that were the foundation of medieval university courses. While the philosophy of liberal arts has changed in modern times, mathematics remains an important feature, as it can be found in many areas in higher education.

Bibliography

Burkert, W. Lore and Science in Ancient Pythagoreanism. Translated by Hans Carl Verlag. Cambridge, MA: Harvard University Press, 1972.

Kahn, Charles H. Pythagoras and the Pythagoreans. Indianapolis, IN: Hackett Publishing, 2001.

Riedweg, Christoph. Pythagoras: His Life, Teaching and Influence. Translated by Steven Rendall. Ithaca, NY: Cornell University Press, 2005.