Pythagoras of Samos

PYTHAGORAS of SAMOS
A Collection of Essays and Lessons for Junior and Senior High School

Contents
1. From Then to Now - a brief history of the Universe
2. The Sixth Century B.C.E.
3. Pythagoras - his life and times
4. Academic

From Then to Now - a brief history of the Universe

The beginning

It is generally accepted that the present Universe is about 20 000 000 000 years old. It is thought that the Solar System began forming 5 000 000 000 years ago and that the first traces of life appeared about 4 500 000 000 years ago. Broad estimates place the beginnings of humanity at about 5 000 000 years ago. Since then the Earth has had three Ice Ages with the last one finishing about 10 000 B.C.E.

You might like to do Activity 1 before you continue.

Middle Stone Age (Mesolithic period 10 000 - 7 000 B.C.E.)

As the Last Ice Age waned, the ice sheets left behind fertile savanna grasslands over most of Africa, the Middle East, India and China. Large herds grazed on these grasslands and not far away were the hunter/gatherers. It has been estimated that the abundance of wildlife was able to support 40 humans per square kilometre. About this time we find the first record of the domestication of dogs, sheep, goats, and the first harvesting of wild grains. The earliest record of habitation at the spring of Jerico is about this time. Due to the movement of the herds, stone age hunter/gatherers had to be nomadic, which meant that they had to travel light and keep few possessions. Therefore this was a period of limited scientific and mathematical advancement.

According to J. Bronowski [1] there is also evidence (in about 8 000 B.C.E in the Middle East) of the mutation of light husked wild wheat to a heavier husked bread wheat, coinciding with the desire to settle and start agriculture. He argues that this mutation in wheat was vital for the successful transition from hunter/gatherer to settler.

New Stone Age (Neolithic period, 7 000 - 3 000 B.C.E)

In the new stone age humanity moved from the simple nomadic hunter/gatherer culture to cultivation and settlement. We find evidence of pottery and building with mud bricks. Jerico at this time was about 4 to 5 hectares in size supporting a population of 3 000 people. In this period we find the first evidence of calendars in Babylon and Egypt, with writing being used, and extensive use of the potter's wheel.

Bronze Age (3 000 - 1 100 B.C.E.)

In the wake of the last Ice Age the fertile savanna grasslands continued to dry out and receded to the large river plains of the Nile, Tigris and Euphrates, Indus, Ganges, Chang Jiang and Huang Ho. Following these receding grass plains were the herdsmen and their herds. Population density exploded along the fertile valleys resulting in armed conflict over available land. Large settlements were formed for added security and mutual cooperation. Specialisation of jobs within these communities resulted in increased leasure time, and the need for data collection, recording, analysis, etc. - a ripe environment for Mathematical endeavour.
Some notable dates:

3 000 - horse yolked, wheel in use
2 900 - Great Piramid of Gizeh
2 200 - 400 Babylonian mathematical tablets, Io-shu
2 000 - mounted marauding tribes, cultural mixing
1 850 - Moscow papyrus - 25 mathematical problems
1 750 - Rule of Hammurabi
1 650 - Rhind papyrus - 85 mathematical problems
1 500 - largest existing obelisk
1 400 - Joshua and the Tribe of Israel storm Jericho to get a foothold in Fertile Crescent.
1 200 - Trojan War
1 167 - Harris papyrus

Iron Age (1 100 B.C.E. ->)

This period saw the continued sophistication of urban settlement, and the development of economically and militarily powerful cities and countries. Some individuals had greater leisure time and travelled widely. The caravan trade brought news of other cultures and their knowledge and technology.
Some notable dates:

776 - first Olympiad
753 - Rome founded
740 - Works of Homer
700 - Old Testament written. The Old Testament can be seen as a chronicle of a people undergoing the change from nomadic herdsman to settlers.
650 - papyrus introduced to Greece
600 - Thales of Miletus

Though this essay is investigating the influence of Pythagoras and the Pythagoreans on Western civilisation, it cannot be denied that the East, in particular Babylon and Egypt, had already established a long history of mathematics before the sixth century B.C.E. To get an overview of the possible influence of Egypt and Babylon on Greek mathematics visit the MacTutor History of Mathematics Archive .

The Sixth Century B.C.E.

The sixth century B.C.E. is considered remarkable because it was in this century that humans seemed to have asked the definitive questions "Why?" and "How?". For the first time people looked for (had time for) natural explanations where previously there had been only myth and superstition. This didn't only happen in Greece with the Ionian philosophers and Pythagoras but also in India (Buddha) and China (Confucius). Arthur Koestler [2] says of this century, "A March breeze seemed to blow across this planet from China to Samos, stirring man[kind] into awareness, like the breath in Adam's nostrils". Koestler goes on to assert that "Every philosopher of the period seems to have had his own theory regarding the nature of the universe around him" and he likened the sixth century to an orchestra "expectantly tuning up, each player absorbed in his own instrument only, deaf to the caterwaulings of the others". He then describes his vision of the importance of Pythagoras:

"Then there is a dramatic silence, the conductor enters the stage, raps three times with his baton, the harmony emerges from the chaos. The maestro is Pythagoras of Samos, whose influence on ideas, and thereby on the destiny, of the human race was probably greater than that of any single man before or after him."

This vision of Pythagoras so appealingly painted by Koestler is the one which most of us identify with, and yet there is little evidence to support it. Much of what we know about Pythagoras has had to be inferred from texts written many centuries after his death and much of what has been passed off as Pythagorean is pure fancy.

John Burnet [3] gives a good description of the historical references to Pythagoras and the Pythagoreans. He asserts that there is only one (sketchy) reference to Pythagoras which is contemporary to his generation, some credible references from the next generation, and some credible (but reserved) discussions in the next century by Plato, Aristotle, and Timaeus. Burnet viewed the essays on Pythagoras by Porphry, Iamblichus, and Diogenes Laertius in the third century AD as a "mass of incredible fiction" in that they dwelled on the mystical cultish nature of the Order. Interestingly, in the earlier references, particularly in Plato and Aristotle, there appears to be a tension between a respect for the philosophy and science of the Pythagoreans and caution not to appear sympathetic to the role of the Pythagorean Order in Ionian politics and religion. As in all historical study, one has to know the "political correctness" of the times to fully appreciate the opinions expressed. To view translated excerpts from original sources click this link to Hanover University

Pythagoras (580?-501? B.C.E.)- his life and times

Burnet claims that we may be reasonably certain that Pythagoras passed his early manhood on the island of Samos, and was the son of Mnesarchus in the reign of Polycrates (532 B.C.E.) Burnet is sceptical about the extensive travels attributed to Pythagoras by late writers, even his much quoted visit to Egypt. He cites credible sources that say Pythagoras left Samos in his late fifties (in order to escape from the tyranny of Polycrates), that he came to Italy in 529 B.C.E. and remained at Croton for twenty years. He retired to Metapontum, when the Crotoniates rose in revolt against the authority of the Pythagorean Order.

Pythagorean Order

Burnet sees the Pythagorean Order as essentially , a "religious fraternity, and not, as has been maintained, a political league." He asserts that the main purpose of the Order was the cultivation of holiness and that it resembled an Orphic society.

For a time the Pythagorean Order held supreme power in the Achaean cities, but the surge of democracy brought many purges against it resulting in the death or exile of members of the Order. At some time after this the Pythagorean Order was able to return to Italy, and once more acquired great influence for the next two centuries.

The teachings of Pythagoras and The Pythagoreans

Burnet states: "Pythagoras apparently preferred oral instruction to the dissemination of his opinions by writing, and it was not till Alexandrian times that anyone ventured to forge books in his name. The writings ascribed to the first Pythagoreans were also forgeries of the same period. The early history of Pythagoreanism is, therefore, wholly conjectural; but we may still make an attempt to understand, in a very general way, what the position of Pythagoras in the history of Greek thought must have been. "

Teaching a way of life.

We get some insight into the lives of the Pythagoreans when we realise that they taught the transmigration of souls (the belief that on death we return as another) and that they had many taboos. See for example the following practices:

1. To abstain from beans.
2. Not to break bread.
3. Not to stir the fire with iron.
4. Not to eat from a whole loaf.
5. Not to eat the heart.
6. Not to walk on highways.
7. Do not look in a mirror beside a light.
8. When you rise from the bedclothes, roll them together and smooth out the impress of the body.

The Pythagoreans taught that the purpose of life was to purify the soul and body. They expanded on their Orphic beginnings to include "purification" through science and knowing. To reach purification they taught that one had to discover the "harmonies" of the cosmos - and scientific (mathematical) enquiry was the vehicle with which to find them.
Arguably the greatest scientific achievement of the Order was the discovery (attributed to "The Master" as Pythagoras was called) of the mathematical order in the musical scale and the harmonies so produced. It is not difficult to appreciate how the Pythagoreans would extrapolate from this success to the belief that, in the quest for the secrets of the cosmos "all is number".

Mathematical Teaching.

Tradition has it that most of the first two books of Euclid's Elements can be attributed to the Pythagoreans. If this is the case, then much of what is still taught to adolescents around the world today as geometry, number , space and algebra is essentially Pythagorean. Since the initial discovery of these concepts, text book writers over the following generations have developed and abstracted these ideas to the point where it is difficult for us to get a clear view of how the Pythagoreans actually conceptualised about their cosmos. The way they visualised number was different from the way we think of number. To the Pythagoreans number was a living thing - it was real, a substance of the landscape. It was there all around them, but hidden; and the way to enlightenment was in the discovery of its secrets.

The challenge I have undertaken for the rest of this essay is to give the reader an insight into how the Pythagoreans may have viewed their mathematics, and the best place to start is with arguably the best known mathematical formula - Pythagoras' Theorem.

On to Pythagoras' Theorem

References:

[1] Bronowski, J., The Ascent of Man. London: British Broadcasting Corporation., 1973

[2] Koestler, Arthur, The Sleepwakers - A History of Man's Changing Vision of the Universe. London: Hutchinson & Co., 1959

[3] Burnet, John, Early Greek Philosophy, 3rd Ed., London: Adam and Charles Black, 1920.