The Philosophy Book

28

as self-evident, as if god-given, and
worked out mathematical proofs that
had the impact of divine revelation.
Because these mathematical
discoveries were a product of pure
reasoning, Pythagoras believes
they are more valuable than mere
observations. For example, the
Egyptians had discovered that a
triangle whose sides have ratios of
3:4:5 always has a right angle, and
this was useful in practice, such as
in architecture. But Pythagoras
uncovered the underlying principle
behind all right-angled triangles
(that the square of the hypotenuse
equals the sum of the squares of the
other two sides) and found it to be
universally true. This discovery was
so extraordinary, and held such
potential, that the Pythagoreans
took it to be divine revelation.
Pythagoras concludes that the
whole cosmos must be governed
by mathematical rules. He says

PYTHAGORAS

Pythagoras’s Theorem showed that shapes
and ratios are governed by principles that
can be discovered. This suggested that it
might be possible, in time, to work out the
structure of the entire cosmos.

that number (numerical ratios and

mathematical axioms) can be used

to explain the very structure of the

cosmos. He does not totally dismiss

the Milesian idea that the universe

is made up of one fundamental

substance, but he shifts the enquiry

from substance to form.

This was such a profound change

in the way of looking at the world,

that we should probably forgive

Pythagoras and his disciples for

getting somewhat carried away,

and giving numbers a mystical

significance. Through exploring the

relationship between numbers and

geometry, they discoved the square

numbers and cube numbers that

we speak of today, but they also

attributed characteristics to them,

such as “good” to the even numbers

and “evil” to the odd ones, and even

specifics such as “justice” to the

number four, and so on. The number

ten, in the form of the tetractys (a

triangular shape made up of rows of

dots) had a particular significance

in Pythagorean ritual. Less

contentiously, they saw the number

one as a single point, a unity, from

which other things could be derived.

The number two, in this way of

thinking, was a line, number three a

surface or plane, and four a solid; the

correspondence with our modern

concept of dimensions is obvious.

The Pythagorean explanation of

the creation of the universe followed

a mathematical pattern: on the

Unlimited (the infinite that existed

before the universe), God imposed a

Limit, so that all that exists came to

have an actual size. In this way God

created a measurable unity from

which everything else was formed.

Numerical harmonies

Pythagoras’s most important

discovery was the relationships

between numbers: the ratios and

proportions. This was reinforced by

his investigations into music, and

in particular into the relationships

between notes that sounded pleasant

together. The story goes that he

first stumbled onto this idea when

listening to blacksmiths at work. One

had an anvil half the size of the other,

and the sounds they made when

There is geometry in

the humming of the strings,

there is music in the

spacing of the spheres.

Pythagoras

a

2 b

2 c

2

a

2

a

b

2

b c

c

2

=

+