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The society was both a mystical religion and one of the
most productive schools of mathematics in history. A fundamental
mystical belief of the Pythagorean society was that
“all is number,” which is to say that the entire universe can be
explained in terms of numbers. The Pythagoreans had other
mystical beliefs as well, such as an aversion to beans and a
belief in the transmigration of souls (that is, that people’s
souls after death occupy the bodies of animals and vice versa).
But the principal ideas of the Pythagoreans that have been
handed down concern numbers. The Pythagoreans assigned
gender to numbers, saying that odd numbers are male and
even numbers female. They also worked with figurate numbers,
numbers that can be identified by a shape when dots
are used. Square numbers of dots can be formed into squares,
triangular numbers form dot triangles, and so forth.
Although the name Pythagoras is most closely associated
with the Pythagorean theorem (the sum of the squares on the
legs of a right triangle equals the square on the hypotenuse),
this theorem was known to the Chinese and probably to the
Babylonians before the time of Pythagoras.
When Pythagoreans talked about numbers, they meant the
natural numbers, 1, 2, 3, and so forth. Fractions are simply
ratios of natural numbers, so fractions posed no problem. But
the Pythagoreans proved that objects in the real world cannot
always be measured completely with natural numbers.
For example, if the unit chosen to be one is the side of a
square, the diagonal of the square cannot be measured exactly
using that unit.