What comes to your mind when you hear about Pythagoras?
Pythagoras of Samos was a famous Greek mathematician and philosopher ( c. 570 — c. 495 BC). Pythagoras is associated with the famous geometric theorem that bears his name. The theorem states that In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
We all know about this. But back in his day, he was known as a mystic and a prophet. He was the founder of Pythagoreanism, a cult based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. In Croton he became the spiritual leader of a secret society based on a philosophical religion. Pythagoreans survived in Italy for nearly two hundred years after his death, and his doctrines continued to be taught until the sixth century CE.
The Pythagoreans were the first to study the regular solids. Pythagoras is said to have known of the cube and the tetrahedron, and there is a longstanding scholarly disagreement over whether he also knew of the icosahedron and the octahedron. [Platonic solid]
By the People, For the People
Many think very highly about democracy, a rule, of the people, by the people, and for the people. A by-product of the Ancient Athenian Civilisation. For it to work effectively, the people who vote need to have knowledge of the political know-how, at least this is what Socrates also believed in. When we let the not so rational vote, we give rise to not democracy, but demagoguery.
When we let the not so rational vote, or the ones who are uneducated about the government, we give rise to not democracy, but demagoguery
Demagoguery gives rise to cult groups. The person with the most followers wins. Socrates was to have first hand of demagoguery, the catastrophic experience of the foolishness of voters. In 399 BC, the philosopher was put on trial on trumped up charges of corrupting the youth of Athens. A jury of 500 Athenians was invited to weigh up the case and decided by a narrow margin that the philosopher was guilty. He was put to death by hemlock in a tragic painful process.
The Pythagoreans are a prime example of such an irrational cult group.
What distinguished the Pythagoreans was their means of purifying the mind. They did not achieve purity by meditating, but by studying mathematics and science
The Pythagorean brotherhood was not unlike other cults of the day. The members were chosen very carefully — they participated in an initiation, endured ritual purifications, and took a vow of secrecy. They lived by a set of strict, sometimes bizarre, rules. According to legend, they were vegetarians, but were prohibited from eating beans; they could not stir a fire with a knife; they could not wear rings; and they had to touch the earth when it thundered.
The Pythagoreans believed in the transmigration of souls — that the souls of the dead returned as animals, and entered an infinite cycle of reincarnation moving up or down through the ranks of animals and humans. The only way to escape this cycle was through purification of body and mind. As in many cults, the purification of the body was achieved through modest living, abstinence, and restraint.
What distinguished the Pythagoreans was their means of purifying the mind. They did not achieve purity by meditating, but by studying mathematics and science. The ultimate union with the divine was said to follow from an understanding of the order of the universe, and the key to understanding the universe was to understand mathematics. Pythagoras said, “Beatitude is the knowledge of the perfection of the numbers of the soul.” This belief is expressed very succinctly by the Pythagoreans’ motto, “All is number.”
The Pythagoreans believed that God ordered the universe with numbers, and that every number could be expressed as a ratio of two whole numbers. Using modern terminology, the Pythagoreans believed that all numbers are rational.
Since mathematics was part of their religion, and Pythagoras was their spiritual leader, any mathematical result obtained by his followers was “the word of the master” and was attributed to him.
The “IRRATIONAL” CULT
According to legend, one of the Pythagoreans, Hippasus of Metapontum (c. 500 BCE), did not follow this tradition of deference, and was severely disciplined. One account states that he was drowned at sea, while another contends that he was expelled from the Pythagoreans and a tombstone was erected for him as a symbolic send-off. Again, accounts differ on what Hippasus had done to deserve this harsh treatment, and there are two competing stories (both of which may be true).
One tale suggests that Hippasus discovered the dodecahedron and showed how to inscribe it in a sphere, but he failed to give credit to Pythagoras. This discovery may have been especially meaningful to the Pythagoreans because of the dodecahedron’s pentagonal faces. They adopted the pentagram, or pentagon-star, which was the Greek symbol for health, as the special symbol used to identify others in the brotherhood. The pentagram is created by connecting the vertices of a regular pentagon, and doing so creates a new regular pentagon inside it.
2. Irrational Number
The second tale states that Hippasus proved that not every number is rational, and he failed to keep this discovery a secret. That all numbers are rational was one of the supporting pillars of the Pythagoreans’ system of beliefs. The existence of an irrational number was a devastating and damaging realisation. It is in this light that we can imagine their outrage at Hippasus.
The discovery of irrational numbers is said to have been so shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from their gods for divulging this.
To Be, or Not To Be (Rational)
To be, or not to be, that is the question:
Whether ’tis nobler in the mind to suffer
The slings and arrows of outrageous fortune,
Or to take arms against a sea of troubles
And by opposing end them
Ironically, the proof of the existence of an irrational number is one of the most significant and lasting contributions of the Pythagoreans to mathematics. Funny innit? Throughout history, such dogmatism has often found itself in harsh opposition to evident reality. Bringing together a group of well educated scientists does not guarantee an utopian or an egalitarian society. We need a rational mind which is willing to accept the truth and change for the better. A mind, which does not subordinate its grasp of reality to anyone’s orders, directives, or controls; nor sacrifice its knowledge, its view of the truth, to anyone’s opinions, threats or wishes.And that I think is what helps us evolve into good human beings and form a better peaceful society that we all wish to live in.
Of all the math they knew, sadly, the pythagoreans could not understand something as simple as this.