Pythagoras
Pythagoras of Samos is often described as
the first pure mathematician. He is an
extremely important figure in the
development of mathematics yet we know
relatively little about his
mathematical achievements. Unlike many later Greek
mathematicians, where at
least we have some of the books which they wrote, we
have nothing of
Pythagoras's writings. The society which he led, half religious
and half
scientific, followed a code of secrecy which certainly means that
today
Pythagoras is a mysterious figure. We do have details of
Pythagoras's life from
early biographies which use important original sources
yet are written by
authors who attribute divine powers to him, and whose aim
was to present him as
a god-like figure. What we present below is an attempt
to collect together the
most reliable sources to reconstruct an account of
Pythagoras's life. There is
fairly good agreement on the main events of his
life but most of the dates are
disputed with different scholars giving dates
which differ by 20 years. Some
historians treat all this information as
merely legends but, even if the reader
treats it in this way, being such an
early record it is of historical
importance. Pythagoras's father was
Mnesarchus ([12] and [13]), while his mother
was Pythais [8] and she was a
native of Samos. Mnesarchus was a merchant who
came from Tyre, and there is a
story ([12] and [13]) that he brought corn to
Samos at a time of famine
and was granted citizenship of Samos as a mark of
gratitude. As a child
Pythagoras spent his early years in Samos but travelled
widely with his
father. There are accounts of Mnesarchus returning to Tyre
with
Pythagoras and that he was taught there by the Chaldaeans and the
learned men of
Syria. It seems that he also visited Italy with his
father. Little is known of
Pythagoras's childhood. All accounts of his
physical appearance are likely to be
fictitious except the description of a
striking birthmark which Pythagoras had
on his thigh. It is probable that he
had two brothers although some sources say
that he had three. Certainly he
was well educated, learning to play the lyre,
learning poetry and to recite
Homer. There were, among his teachers, three
philosophers who were to
influence Pythagoras while he was a young man. One of
the most important was
Pherekydes who many describe as the teacher of
Pythagoras. The other two
philosophers who were to influence Pythagoras, and to
introduce him to
mathematical ideas, were Thales and his pupil Anaximander who
both lived on
Miletus. In [8] it is said that Pythagoras visited Thales in
Miletus when
he was between 18 and 20 years old. By this time Thales was an old
man and,
although he created a strong impression on Pythagoras, he probably did
not
teach him a great deal. However he did contribute to Pythagoras's interest
in
mathematics and astronomy, and advised him to travel to Egypt to learn
more
of these subjects. Thales's pupil, Anaximander, lectured on Miletus
and
Pythagoras attended these lectures. Anaximander certainly was
interested in
geometry and cosmology and many of his ideas would influence
Pythagoras's own
views. In about 535 BC Pythagoras went to Egypt. This
happened a few years after
the tyrant Polycrates seized control of the city
of Samos. There is some
evidence to suggest that Pythagoras and Polycrates
were friendly at first and it
is claimed [5] that Pythagoras went to Egypt
with a letter of introduction
written by Polycrates. In fact Polycrates had
an alliance with Egypt and there
were therefore strong links between Samos
and Egypt at this time. The accounts
of Pythagoras's time in Egypt suggest
that he visited many of the temples and
took part in many discussions with
the priests. According to Porphyry ([12] and
[13]) Pythagoras was refused
admission to all the temples except the one at
Diospolis where he was
accepted into the priesthood after completing the rites
necessary for
admission. It is not difficult to relate many of Pythagoras's
beliefs, ones
he would later impose on the society that he set up in Italy, to
the customs
that he came across in Egypt. For example the secrecy of the
Egyptian
priests, their refusal to eat beans, their refusal to wear even cloths
made
from animal skins, and their striving for purity were all customs
that
Pythagoras would later adopt. Porphyry in [12] and [13] says that
Pythagoras
learnt geometry from the Egyptians but it is likely that he was
already
acquainted with geometry, certainly after teachings from Thales and
Anaximander.
In 525 BC Cambyses II, the king of Persia, invaded Egypt.
Polycrates abandoned
his alliance with Egypt and sent 40 ships to join the
Persian fleet against the
Egyptians. After Cambyses had won the Battle of
Pelusium in the Nile Delta and
had captured Heliopolis and Memphis, Egyptian
resistance collapsed. Pythagoras
was taken prisoner and taken to Babylon.
Iamblichus writes that Pythagoras (see
[8]):- ... was transported by the
followers of Cambyses as a prisoner of war.
Whilst he was there he gladly
associated with the Magoi ... and was instructed
in their sacred rites and
learnt about a very mystical worship of the gods. He
also reached the acme of
perfection in arithmetic and music and the other
mathematical sciences taught
by the Babylonians... In about 520 BC Pythagoras
left Babylon and returned to
Samos. Polycrates had been killed in about 522 BC
and Cambyses died in the
summer of 522 BC, either by committing suicide or as
the result of an
accident. The deaths of these rulers may have been a factor
in
Pythagoras's return to Samos but it is nowhere explained how
Pythagoras obtained
his freedom. Darius of Persia had taken control of Samos
after Polycrates' death
and he would have controlled the island on
Pythagoras's return. This conflicts
with the accounts of Porphyry and
Diogenes Laertius who state that Polycrates
was still in control of Samos
when Pythagoras returned there. Pythagoras made a
journey to Crete shortly
after his return to Samos to study the system of laws
there. Back in Samos he
founded a school which was called the semicircle.
Iamblichus [8] writes
in the third century AD that:- ... he formed a school in
the city [of Samos],
the 'semicircle' of Pythagoras, which is known by that name
even today, in
which the Samians hold political meetings. They do this because
they think
one should discuss questions about goodness, justice and expediency
in this
place which was founded by the man who made all these subjects his
business.
Outside the city he made a cave the private site of his own
philosophical
teaching, spending most of the night and daytime there and doing
research
into the uses of mathematics... Pythagoras left Samos and went to
southern
Italy in about 518 BC (some say much earlier). Iamblichus gives some
reasons
for him leaving. First he comments on the Samian response to his
teaching
methods. Pythagoras founded a philosophical and religious school
in
Croton (now Crotone, on the east of the heal of southern Italy) that
had many
followers. Pythagoras was the head of the society with an inner
circle of
followers known as mathematikoi. The mathematikoi lived permanently
with the
Society, had no personal possessions and were vegetarians. They
were taught by
Pythagoras himself and obeyed strict rules. The beliefs
that Pythagoras held
were [2]:- (1) that at its deepest level, reality is
mathematical in nature, (2)
that philosophy can be used for spiritual
purification, (3) that the soul can
rise to union with the divine, (4) that
certain symbols have a mystical
significance, and (5) that all brothers of
the order should observe strict
loyalty and secrecy. Both men and women were
permitted to become members of the
Society, in fact several later women
Pythagoreans became famous philosophers.
The outer circle of the Society
were known as the akousmatics and they lived in
their own houses, only coming
to the Society during the day. They were allowed
their own possessions and
were not required to be vegetarians. Of Pythagoras's
actual work nothing is
known. His school practised secrecy and communalism
making it hard to
distinguish between the work of Pythagoras and that of his
followers.
Certainly his school made outstanding contributions to mathematics,
and it is
possible to be fairly certain about some of Pythagoras's
mathematical
contributions. First we should be clear in what sense Pythagoras
and the
mathematikoi were studying mathematics. They were not acting as a
mathematics
research group does in a modern university or other institution.
There were no'open problems' for them to solve, and they were not in any sense
interested in
trying to formulate or solve mathematical problems. Rather
Pythagoras was
interested in the principles of mathematics, the concept of
number, the concept
of a triangle or other mathematical figure and the
abstract idea of a proof. As
Brumbaugh writes in [3]:- It is hard for us
today, familiar as we are with pure
mathematical abstraction and with the
mental act of generalisation, to
appreciate the originality of this
Pythagorean contribution. In fact today we
have become so mathematically
sophisticated that we fail even to recognise 2 as
an abstract quantity. There
is a remarkable step from 2 ships + 2 ships = 4
ships, to the abstract result
2 + 2 = 4, which applies not only to ships but to
pens, people, houses etc.
There is another step to see that the abstract notion
of 2 is itself a thing,
in some sense every bit as real as a ship or a house.
Pythagoras believed
that all relations could be reduced to number relations. As
Aristotle
wrote:- The Pythagorean ... having been brought up in the study
of
mathematics, thought that things are numbers ... and that the whole cosmos
is a
scale and a number. This generalisation stemmed from Pythagoras's
observations
in music, mathematics and astronomy. Pythagoras noticed that
vibrating strings
produce harmonious tones when the ratios of the lengths of
the strings are whole
numbers, and that these ratios could be extended to
other instruments. In fact
Pythagoras made remarkable contributions to
the mathematical theory of music. He
was a fine musician, playing the lyre,
and he used music as a means to help
those who were ill. Pythagoras studied
properties of numbers which would be
familiar to mathematicians today, such
as even and odd numbers, triangular
numbers, perfect numbers etc. However to
Pythagoras numbers had personalities
which we hardly recognise as mathematics
today [3]:- Each number had its own
personality - masculine or feminine,
perfect or incomplete, beautiful or ugly.
This feeling modern mathematics
has deliberately eliminated, but we still find
overtones of it in fiction and
poetry. Ten was the very best number: it
contained in itself the first four
integers - one, two, three, and four [1+2+3+4
= 10] - and these written in
dot notation formed a perfect triangle. Of course
today we particularly
remember Pythagoras for his famous geometry theorem.
Although the
theorem, now known as Pythagoras's theorem, was known to the
Babylonians
1000 years earlier he may have been the first to prove it.