Essay, Research Paper: Pythagoras

Mathematics

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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.
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