Pythagoras of Samos (Aprx 570-480 BCE)
“First Pure Mathematician”
“Truth is so great a perfection,
that if God would render himself
visible to men, he would choose
light for his body and truth for his
soul”
Charles Wu
Life
-Born in Samos, an island in Ancient Greece.
- Like Socrates, he wrote nothing, so all accounts
of his life are based on others.
-Some details from Pythagoras's life are from
original sources, but they are written by authors
who present him as a 'god-like figure'.
-Some historians treat all this information as
merely legends but, even so, this is of historical
importance because of the early record date.
Life & Historical Period
- Spent his early years in Samos but travelled widely with his father, and his two
brothers.
-He was well educated, learning to play the lyre, learning poetry and to recite
Homer.
-He had three main teachers: Pherecydes, Thales, and his pupil Anaximander.
-Anaximander certainly was interested in geometry and cosmology and many
of his ideas influenced Pythagoras's own views.
-Around 545 BCE, Samos was governed by a tyrant named Polycrates
-In about 535 BCE Pythagoras went to Egypt, possibly because of disputes
with authorities
-He adopted many of the customs of Egpyt
-During time of Persia expansion, so they invaded Egypt in 525BCE, forcing
him to flee
Life & Historical Period
-Returned to Samos around 520, and founded a school called
“semicircle”
-Left Samos because other philosophers before him spent their
lives on foreign soil
-Went to Crete to study law,and founded another school.
-Pythagoras was the head of the society with an inner circle of
followers known as mathematikoi.
Ideas
-Pythagoras mainly had mathematical contributions,
but also taught of philosophy and ethics:
a) Transmigration of souls, that is, by successive incarnations
of the soul in the bodies of different animals a system by
which certain vices and virtues were to be punished and
rewarded after death.
b) Mathematics contains the key to all philosophical
knowledge. This was later developed into an elaborate
number-theory by his followers
c) Virtue is a harmony, and may be cultivated not only by
contemplation and meditation but also by the practice of
gymnastics and music.
Ideas
Pythagoras believed in metempsychosis(reincarnation) and thought that eating meat
was an abominable thing. Saying that the souls of all animals enter other animals
after death.
He imposed on his followers (the mathematikoi) that they were to have no personal
possessions and were to live as vegetarians. They were taught by Pythagoras
himself and obeyed strict rules. Their beliefs were:
(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.
Ideas
-His school practised secrecy and communalism (meaning the belief in or practice of
communal ownership, as of goods and property) making it hard to distinguish between
the work of Pythagoras and that of his followers.
-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.
-Pythagoras studied properties of numbers which would be familiar to mathematicians
today. However Pythagoras believed numbers had personalities which we hardly
recognize as mathematics today:
“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] ...”
Pythagorean Theorem
-Of course today we remember Pythagoras for his famous geometry theorem.
Although the theorem was known to the Babylonians 1000 years earlier he may have
been the first to prove it.
-It is hard for us today to appreciate the originality of
this Pythagorean contribution since we are so familiar
with the pure mathematical abstraction and with the
mental act of generalization.
-Pythagoras believed that all relations could be
reduced to number relations.
Ideas
A list of theorems attributed to Pythagoras, or rather more generally to the
Pythagoreans.
(i) The sum of the angles of a triangle is equal to 180 degrees.
(ii) The theorem of Pythagoras - a²+b²=c²
Note that to Pythagoras, the square on the hypotenuse would certainly not be thought
of as a number multiplied by itself, but rather as a geometrical square constructed on
the side. To say that the sum of two squares is equal to a third square meant that the
two squares could be cut up and reassembled to form a square identical to the third
square.
(iii) Finding areas, inventing basic algebra.
(iv) The discovery of irrationals. (ie, pi, sqrt2)
(v) The five regular solids.
continued....
Ideas
(vi) In astronomy Pythagoras taught that the Earth was a sphere at the
centre of the Universe. He recognised that the orbit of the Moon was
inclined to the equator of the Earth and he was one of the first to realize
that Venus as an evening star was the same planet as Venus as a morning
star (he discovered the planet).
-He taught on the dependence on pairs of opposites
-He viewed the soul as a successive reincarnation in different species until
its eventual purification and ultimate understanding
-He thought that that all existing objects were fundamentally composed of
form and not of material substance.
Pros/Cons
Pros
-The Pythagorean were known
for their mutual friendship, Cons
unselfishness, and honesty.
-Very Secretive
-Invention of abstract math
-Strict life
-Numbers have personalities?