Definition:Pythagoreans

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Definition

The Pythagoreans were a semi-mystical cult which dated from around $550$ B.C.E., founded by Pythagoras of Samos.

Can claim to be the world's first university.

It is feasible to suggest that their initial work in the field of geometry may have formed the basis of at least the first two books of Euclid's The Elements.


Attendees of the school were divided into two classes:

the Probationers (or listeners)
the Pythagoreans.

A student was a listener for $3$ years, after which he was allowed to be initiated into the class of Pythagoreans, who were allowed to learn what was considered to be the deeper secrets.


Pythagoreans were a closely-knit brotherhood, who held all their worldly goods in common, and were bound by oath never to reveal the secrets of the Founder.

There exists a legend that one of the Pythagoreans was thrown overboard to drown after having revealed the existence of the regular dodecahedron.


For some considerable time they dominated the political life of Croton, where they were based, but in $501$ B.C.E. there was a popular revolt in which a number of the leaders of the school were murdered.

Pythagoras himself was murdered soon after.

Some sources state that the reasons for this were based on the fact that their puritanical philosophy was at odds with the contemporary way of thinking. Others suggest that there was a reaction against their autocratic rule.

Whatever the truth of the matter, between $501$ and about $460$ B.C.E. the political influence of the cult was destroyed.


Its survivors scattered, many of them fleeing to Thebes in Upper Egypt, where they continued to exist more as a philosophical and mathematical society for another couple of centuries, secretive and ascetic to the end, publishing nothing, ascribing all their discoveries to the Master, Pythagoras himself.


Quadrivium

The quadrivium was the medieval name of the required course of study of the Pythagoreans, which had been adopted by the educational establishments in Europe.

The required bodies of knowledge were divided into discrete and continuous:

Thus:

Arithmetic: study of the absolute discrete
Music: study of the relative discrete
Geometry: study of the stable continuous
Astronomy: study of the moving continuous.


Trivium

The trivium was the medieval name of the supplementary course of study of the Pythagoreans, adopted by the educational establishments in Europe.

These supplementary bodies of knowledge were:

Grammar
Rhetoric
Logic.


Notable Quotes

Number rules the universe.
-- Quoted in 1937: Eric Temple Bell: Men of Mathematics: They Say: What Say They? : Let Them Say


Everything is Number.
-- Quoted in:
1992: George F. Simmons: Calculus Gems: Chapter $\text {A}.2$: Pythagoras (ca. $580$ – $500$ B.C.)
1980: David M. Burton: Elementary Number Theory (revised ed.): Chapter $1$: Some Preliminary Considerations: $1.3$ Early Number Theory


Critical View

The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but, saturated with it, they fancied that the principles of mathematics were the principles of all things.
-- Aristotle in Metaphysics (Book $\text I$, Chapter $5$, ca. $300$ B.C.E.)


Sources