**Pythagoras**

#### (Island of Samos, present Greece, h. 572 a.C. – Metapontus, today disappeared, present Italy, h. 497 b.C.) Greek philosopher and mathematician.

**Pythagoras** is often described as the first **pure mathematician**. He is an extremely important figure in the development of mathematics, but we know relatively little about his mathematical achievements.

Unlike many later Greek mathematicians, where at least we have some of the books they wrote, we have nothing from Pythagoras’ writings.

The society he led, half religious and half scientific, followed a code of secrecy which certainly means that Pythagoras is a mysterious figure today.

Although his name is linked to Pythagoras’ theorem and the school he founded, **Pythagoras gave an important impulse to the development of mathematics** in ancient Greece, the relevance also reaches the realm of the history of ideas: his thought, still tinged with the mysticism and esotericism of the ancient mystic and oriental religions, inaugurated a series of themes and motifs that, through Plato, would leave a profound imprint in the western tradition.

We have details of Pythagoras’ life from the first biographies that use important original sources, but they are written by authors who attribute divine powers to him, and whose objective was to present him as a divine figure.

What follows is an attempt to bring together the most reliable sources to reconstruct an account of Pythagoras’ life.

There is fairly good agreement on the major events in his life, but most dates are discussed with different scholars who give dates that differ by 20 years. Some historians treat this information as mere legends, but even if the reader treats it this way, being a record so early has historical significance.

There were, among his teachers, three philosophers who were going to influence Pythagoras when he was young. One of the most important was Pherekydes, who many describe as Pythagoras’ master. The other two philosophers who were to influence Pythagoras and present him with mathematical ideas were Thales and his disciple Anaximander, who lived in Miletus.

It is said that Pythagoras visited Thales in Miletus when he was between 18 and 20 years old. By then, Thales was an old man, and although he created a strong impression on Pythagoras, he probably did not teach him much. However, he did contribute to **Pythagoras’ interest in mathematics and astronomy**, and advised him to travel to Egypt to learn more about these subjects.

Around 535 B.C. Pythagoras went to Egypt. This happened a few years after the tyrant Polycrates took control of the city of Samos.

Pythagoras’ time accounts in Egypt suggest that he visited many of the temples and participated in many discussions with the priests.

According to Porphyry, Pythagoras was denied admission to all temples except that of Diospolis, where he was accepted into the priesthood after completing the rites necessary for admission.

It is not difficult to relate many of Pythagoras’ beliefs, which he would later impose on the society he established in Italy, to the customs he found in Egypt.

For example, the secret of Egyptian priests, their refusal to eat beans, their refusal to use even cloth made from animal skins, and their struggle for purity were all customs that Pythagoras would later adopt.

Porphyry says Pythagoras learned geometry from the Egyptians, but he was probably already familiar with geometry, no doubt after the teachings of Thales and Anaximander.

**Pythagoras’ Philosophy**

Approximately 520 B.C., Pythagoras left Babylon and returned to Samos. Polycrates had been killed around 522 B.C.

The death of the ruler may have been a factor in Pythagoras’ return to Samos, but it does not explain how Pythagoras gained his freedom. Darius of Persia had taken control of Samos after Polycrates’ death and he would have controlled the island on Pythagoras’ return.

This conflicts with the accounts of Porphyry and Diogenes Laertius who claim that Polycrates was still in control of Samos when Pythagoras returned there.

Pythagoras left Samos and went to southern Italy around 518 B.C. (some say much earlier). Pythagoras founded a philosophical and religious school in Croton (now Crotone, in the east of the heel of southern Italy) which 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 vegetarian.

They were taught by Pythagoras himself and obeyed strict rules. The beliefs Pythagoras held 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 importance, and

5.- that all the brethren of the order must observe strict loyalty and secrecy.

Both men and women were allowed to become members of the Society, in fact several later Pythagorean women became famous philosophers.

The outer circle of the Society was known as astronomy and they lived in their own houses, coming only to the Society during the day. They were allowed their own possessions and were not required to be vegetarian.

**Pythagoras’ Contributions to Mathematics**

Nothing is known about Pythagoras’ real work. His school practiced secrecy and communalism, which made it difficult to distinguish between Pythagoras’ work and that of his followers. Certainly, his school made outstanding contributions to mathematics, and it is possible to be quite sure about some of Pythagoras’ mathematical contributions.

First, we should be clear in what sense Pythagoras and mathematikoi studied mathematics. They were not acting like a mathematics research group in a modern university or other institution.

There were no ‘open problems’ for them to solve, and they were in no way interested in trying to formulate or solve mathematical problems.

**Pythagoras studied the properties of numbers** that would be familiar to mathematicians today, such as odd and even numbers, triangular numbers, perfect numbers, etc.

However, for Pythagoras’ numbers there were personalities that we hardly recognize as mathematics today:

*«Each number had its own personality: masculine or feminine, perfect or incomplete, beautiful or ugly. This feeling of modern mathematics has been deliberately eliminated, but we still find nuances of it in fiction and poetry. Ten was the best number: it contained in itself the first four integers: one, two, three and four [1 + 2 + 3 + 4 = 10] , and these writings in dot notation formed a perfect triangle».*

Of course, today we especially remember **Pythagoras for his famous geometry theorem**. Although the theorem, now known as the Pythagorean theorem, was known to the Babylonians 1000 years earlier, it may have been the first to prove it.

Proclus, the last great Greek philosopher, who lived around 450 AD wrote:

After (Thales, etc.), Pythagoras transformed the study of geometry into a liberal education, examining the principles of science from the beginning and exploring the theorems in an immaterial and intellectual way: it was he who discovered the theory of the irrational and the construction of cosmic figures.

The historian in mathematics, Thomas Heath gives a list of theorems attributed to Pythagoras, or more generally to the Pythagoreans, they are:

**1) The sum of the angles of a triangle equals two right angles**. Pythagoreans also knew the generalization that states that a polygon with «n» sides has sum of interior angles 2n – 4 right angles and sum of exterior angles equal to four right angles.

**2) Pythagorean theorem**: for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

We should note here that for Pythagoras, the square of the hypotenuse would certainly not be considered as a number multiplied by itself, but rather as a geometric square built on the side.

To say that the sum of two squares equals a third square meant that the two squares could be cut and reassembled to form a square identical to the third square.

Of the practical use of this relationship there are testimonies from other civilizations prior to the Greek (such as the Egyptian and Babylonian), but **Pythagoras is attributed the first demonstration of the theorem**, as well as numerous other advances to his school.

**3) Construction of figures of a certain area and geometric algebra**. For example, they solved equations such as: a(a – x) = x² by geometric means.

**4) The discovery of irrationals**. This is certainly attributed to Pythagoreans, but it seems unlikely that it was due to Pythagoras himself.

This went against Pythagoras’ philosophy: all things are numbers, for by one number he meant the ratio of two whole numbers. However, because of his belief that all things are numbers, it would be a natural task to try to prove that the hypotenuse of a right-angled isosceles triangle had a length corresponding to a number.

**5) The five regular solids**. It is believed that Pythagoras himself knew how to build the first three, but it is unlikely that he would have known how to build the other two.

**6) In astronomy**, Pythagoras taught that the Earth was a sphere at the center of the Universe. He also recognized that the Moon’s orbit was tilted toward the equator of the Earth and was one of the first to realize that Venus as an evening star was the same planet as Venus as a morning star.

First of all, however, Pythagoras was a philosopher. In addition to his beliefs about numbers, geometry, and astronomy described above, Heath argued.

**Influence**

More than a century after Pythagoras’ death, during a trip to southern Italy before the foundation of the Academy, Plato became acquainted with Pythagorean philosophy through his disciples.

It has been asserted that the conception of number as the principle of all things paved the way for Platonic idealism; in any case, Pythagoras’ influence is clear at least in the Platonic doctrine of the soul (immortal and prisoner of the body), which also in Plato attains its liberation through knowledge.

Thus, through Plato, various Pythagorean conceptions would become recurrent or controversial themes of Western philosophy; still in the seventeenth century an astronomer as distinguished as Kepler, to whom the discovery of the elliptical orbits of the planets is due, continued to believe in the music of the spheres.

Other concepts of his, such as harmony and proportion, would be incorporated into music and the arts. Pythagoras has also been seen as the precursor of an aspiration that would have great predicament from the scientific revolution of Galileo: the mathematical formalization of knowledge.

**Last years**

The Pythagoras Society in Croton was not affected by political developments despite its desire to remain on the sidelines of politics. Pythagoras went to Delos in 513 B.C. to care for his dying old master Pherekydes.

He remained there for a few months until the death of his friend and teacher and then returned to Croton. In 510 B.C., Croton attacked and defeated his neighbor Sybaris and, no doubt, there are some suggestions that Pythagoras got involved in the dispute.

Then, around 508 B.C., the Pythagorean society of Croton was attacked by Cylon, a Croton nobleman. Pythagoras escaped to Metapontium and most authors say that he died there, and some claim that he committed suicide because of the attack on his Society.

Pythagoras, one of my favorites.