The rules include: reality is mathematical in nature, philosophy can be used for spiritual purification, that one's soul can rise to union with the divine, that certain symbols have a mystical significance and that all followers should obey and observe strict loyalty and secrecy. .
One of the great Greek contributions in mathematics was the recognition that mathematics dealt with abstractions. This was attributed to the Pythagoreans. Numbers were separated completely from physical objects or pictures. The Egyptians had thought that a line was no more than a stretched rope or the edge of a field and that a rectangle was the boundary of a field.
The Pythagreans regarded numbers as the ultimate parts of real material objects. They stated all objects were made up of or composed of whole numbers or that numbers were the substance of the universe. Numbers to them were like atoms to us. They didn't distinguish numbers from geometrical dots either. So a number was an extended point or like a very small sphere. They depicted numbers as dots in sand or pebbles; and classified numbers according to the shapes made by the dots or pebbles. For example the numbers 1, 3, 6, and 10 were triangular because the corresponding dots would form a triangle. Also the numbers 1, 4, 9, and 16 were called square numbers because those dots could be arranged in the shape of a square. To pass from square number to the next, Pythagoreans formed what is called a gnomon. Originally in Babylonian times, it probably was just an upright stick, which cast a shadow and helped them tell time. In Pythagoras? time it meant a carpenter's square. It was the remains from a square when a smaller square was cut out of one corner. .
According to the Pythagoreans a number that equaled the sum of its divisors including 1 but not the number itself was a perfect number such as 6, 28 and 496. Those, which exceeded the sum of the divisors, were known as excessive and those that were less were called defective.