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This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material please acknowledge it.
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~580 B.C. to ~500 B.C. |
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Math G Dr. Ian Walton Compiled By: Nazih Malak Date: 02-May-01 |
Pythagoras is considered to be the first real mathematician. As an extremely important Greek philosopher and mathematician and founder of the Pythagorean School, but very few information is known about him and his life. The Pythagorean view of the world consisted of a belief that numbers were the keys to the various qualities of mankind and matter. In the Pythagorean view, everything was composed of numbers, the explanation for any objects existence could only be found in numbers. This was a completely novel concept, because at that time, numbers existed for practical purposes only, as a device for solving problems in calendar construction, building and commerce. Pythagoreans were the first who saw a number as an idea, important in itself. They also made a distinction between logistic (art of computation) and arithmetic (number theory).
Pythagoras was supposedly born about 569 B.C. on an island of Samos, Ionia, located in the Aegean Sea (see map below), although others will only estimate his birth between the years of 550 or 560 B.C.
Pythagoras birthplace
At first glance, there seems to be a great deal of biographical material on Pythagoras, dating from the first centuries of the Christian era; but when history is studied more closely, the information is fairly vague. Unlike many later Greek mathematicians, there are no documented writings attributed to Pythagoras. It is believed that this is due to a society which he led, half-religious and half-scientific, which followed a code of concealment and therefore caused him to remain somewhat of a mysterious figure in history. It has been stated that those facts that we do know about Pythagoras may be unreliable, because many of the accounts of this great scholar carry an almost unreliable character. A great deal of material is still, to this day, considered legend and or myth. Sometimes Pythagoras is represented as a man of science, and sometimes as a preacher of mystic doctrines. Although some claim this to be a contradiction in ideology, most historians feel that the union of mathematical genius and mysticism was actually a common bond, valid and necessary for the times in which he lived. The detailed accounts of how Pythagoras invented the musical scale, performed miracles and pronounced prophecies further enhance the romantic mystique that surrounds his name.
Pythagoras' father's name was Mnesarchus and his mother's name was Pythais.Mnesarchus was a merchant from Tyre (modern Lebanon), who supposedly brought corn to Samos at a time of famine and was granted citizenship of Samos as a mark of gratitude.As a child Pythagoras may have spent his early years in Samos but is known to have traveled widely with his father. There are accounts of Mnesarchus returning to Tyre with Pythagoras. It is then believed that in Tyre, Pythagoras was taught by Chaldaeans and the scholars of Syria. It is also documented that he was well educated, learning to play the lyre, learning poetry and reciting Homer. Although all accounts of his physical appearance are likely to ambiguous, there was one common description that is consistent in all the accounts. Pythagoras had a "striking" birthmark on his thigh.
There are conflicting viewpoints on which of his teachers had the most effect and influence on Pythagoras while he was a young man. One of the most important was Pherekydes who was widely believed to be the ultimate teacher of Pythagoras. The other two main figures who were considered to be a major influence on Pythagoras and, at some point, introduced him to mathematical ideas were Thales and his pupil Anaximander. Thales is accredited with influencing Pythagoras' interest in mathematics and astronomy, going as far as to advise him to travel to Egypt to learn more of these subjects. He was however, an old man by the time they had met and although he did contribute to Pythagoras' ultimate development, he was probably unable to do more than steer him in a right direction. However, Thales' pupil Anazximander gave lectures in Miletus, which Pythagoras attended. Anazximander's lectures on geometry and cosmology and a variety of his other ideas were certain to influence Pythagoras' own views. One popular theory states that Pythagoras became a disciple of Anaximander, his astronomy becoming a natural development of Anaximander's. His geometry is also said to have descended from the teachings in Miletus.
To better
visualize Pythagoras place on a historical timeline, the following chart serves
as a good illustration. It shows the period between 600 BC and 200 BC, which
allows us to see Pythagoras place in the sequence of other philosophers and
mathematicians of the period.
In 525 B.C., Cambyses, the King of Persia, invaded Egypt. Polycrates abandoned his alliance with Egypt and sent 40 ships to join the Persian fleet against the Egyptians. Cambyses won the Battle of Pelusium in the Nile Delta and captured Heliopolis and Memphis causing the Egyptian resistance to collapse. Pythagoras was taken prisoner and brought to Babylon. While prisoner, Pythagoras became an appropriate student in the sacred rites of the Magoi and learned their mystical worship of the gods. He also reached what was considered the epitome of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians. In approximately 520 B.C., Pythagoras left Babylon and returned to Samos.Historians were not able to find any documentation on how he was able to obtain his freedom. Pythagoras made a journey to Crete shortly after his return to Samos to study the systems of laws there. He then returned and founded a school, which was called the "Semicircle." Drawing on the notion that all the philosophers before him had ended their days on foreign soil, Pythagoras decided to make his home in Crotone, a town in Southern Italy. Once at home in Crotone, Pythagoras founded a philosophical and religious school. Pythagoras was the head of that society with an inner circle of followers known as mathematikoi. They lived permanently with Pythagoras and the Society, had no personal possessions, and were vegetarians. They obeyed strict rules:
1) At its deepest level, reality is mathematical in nature.
2) Philosophy can be used for spiritual purification.
3) The soul can rise to union with the divine.
4) Certain symbols have a mystical significance.
5)
All brothers of the order should observe strict loyalty and
secrecy.
Throughout all of
his teachings, Pythagoras proclaimed himself a philosopher, rather than a
mathematician. The circumstances of his death are unclear but his legacy lives
on. Pythagoras and the early Order initially treated numbers concretely, as
patterns with pebbles, but over time the Pythagoreans developed and refined
their concept of number into the same abstract entity, which still exists
today. Though it is difficult to separate fact from fancy in some of the surviving
references to the Pythagoreans, it is generally conceded that they began number
theory, and were responsible for the introduction and development of number
mysticism in Western Society.
To the Pythagoreans, each number possessed its own special attributes. See for example the table below.
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Number |
Property of the number |
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1 |
monad(unity) generator of
numbers, the number of reason |
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2 |
dyad(diversity, opinion)
first true female number |
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3 |
triad(harmony = unity +
diversity) first true male number |
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4 |
(justice, retribution)
squaring of accounts |
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5 |
(marriage) = first female +
first male |
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6 |
(creation) = first female +
first male + 1 ? |
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10 |
(Universe) tetractys |
significance in that their
sum accounted for all the possible dimensions:
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Geometric property |
Geometric shape |
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1 |
generator of dimension |
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2 |
line of dimension 1 |
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3 |
triangle of dimension 2 |
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4 |
tetrahedron of dimension 3 |
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The Pythagoreans are also often thought by various historians to have discovered amicable numbers. Two numbers are amicable if each is the sum of the proper divisors (that is all the divisors except the number itself) of the other.
For example 220 and 284 are amicable since the sum of the proper divisors of 220 are 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284 and the sum of the proper divisors of 284 are 1 + 2 + 4 + 71 + 142 = 220. Superstitious people believed that two talismans bearing this pair of numbers would seal a perfect friendship between the wearers. The pair also became significant in magic, sorcery, and astrology.
Other numbers that were assigned mystical powers by the Pythagoreans were the perfect, deficient, and abundant numbers.
A number is perfect if it is equal to the
sum of its proper divisors, is deficient if its sum falls short of the number,
and is abundant if the sum exceeds the number. 6 is a perfect number (1 + 2 +
3), 8 is deficient (1 + 2 + 4), and 12 is abundant (1 + 2 + 3 + 4 + 6).
Fortunately, in developing their number mysticism, the Pythagoreans also valued proof. To this end they searched for the essential properties and definitions of many numbers. The following is a brief description of their ideas about numbers:
- The monad is described as limiting quantity, or as the common part or beginning of how many.
- The Pythagorean definition for a number was essentially that number is a collection of units.
- The Pythagoreans made the distinction between odd and even numbers.
Many superstitions became associated with the odd and even numbers. For example the odds were generally regarded as masculine and divine, while the evens were considered feminine and thus earthly and human. An interesting fact is that due to the influence of Pythagoras, the Pythagoreans welcomed women into their Society, and that Pythagoras' wife, Theano, was considered an accomplished mathematician in her time. Theano and her daughters are believed to have carried on Pythagoras work after his death.
- Early Pythagoreans described what we would call prime numbers as "prime and incomposite numbers" and it was generally accepted that a prime number was "measured by no number, but by a unit only", though there was some disagreement about whether 2 was prime. The composite numbers were those which were "measured by any less number". In keeping with the early Pythagorean view of number as a pattern of stones, prime numbers were also called rectilinear because they can be represented as a line of stones only, compared to the composites which can also be arranged into equal size groups of stones. Composites were further distinguished as plane or solid.
- Another legacy bequeathed unto us by Pythagoreans are polygonal or figurate numbers. In fact, when we refer to numbers as figures in modern-day English, we are using the jargon developed by the Pythagoreans. Pythagoras believed that each number is formed by a collection of units it consists of. He used pebbles to illustrate the composition and figure of any given number:
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Figurate Number |
Geometric shape |
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triangular numbers |
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square numbers |
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pentagonal numbers |
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oblong numbers |
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Let us put the Pythagorean
theorem to the test.In a right triangle ABC (C is the right angle), a = 12 cm,
b = 7 cm.Find the length of the hypotenuse, c.
= 
2) 
=
3) 
Although it is difficult to separate Pythagoras personal discoveries from those of the Society of his disciples, who tended to attribute their own discoveries to him, the fact remains that Pythagoras was responsible for a substantial amount of revolutionary ideas and concepts that helped channel the areas of Mathematics, Geometry and Music in the general direction being followed today. To give credit where it is due, we must acknowledge, that although there are indications that some of the concepts attributed to Pythagoras were known to ancient Babylonians prior to his time, the clear formulation of these concepts and the cohesive force needed to integrate and synthesize theory from those concepts must be attributed to Pythagoras and his followers.Considering the pervasive influence of Mathematics on nearly every aspect of modern life, it is doubtless that Pythagorean ideas played a major part in shaping the face of our modern world!
Bibliography