Pythagoras Pythagorean School

Pythagorean Life and Work, Pythagorean Theorem and the Symbology of Numbers

It is important to start by mentioning that Pythagoras is a historically mysterious figure because he left no written record.

Bust of Pythagoras at the Capitoline Museums in Rome. Pythagorean School.
Bust of Pythagoras at the Capitoline Museums in Rome.

"Evolution is the Law of Life, Number is the Law of the Universe, Unity is the Law of God."

Pythagoras

That is why the life and work of Pythagoras and all that is known about him is due to oral tradition and to what they wrote about him, mainly Filolau (470-385 BC), about 100 years after his death, in his' Writings Pythagoreans', to whom some credibility is attributed since he is a survivor of the antipitagoric revolt that would determine the death of Pythagoras and also Architas of Tarentus (435-347 BC), Filolau's disciple.

Pythagoras is considered one of the founders of the most exact science, Mathematics, sharing this feat with Tales of Miletus.

He was the first pure mathematician in history, due to his concern to move from concrete cases to general, that is, because he was the first to make the transition from punctual practical observation to the generalization of a theorem, not by means of a mere intuitive reason but through a logical deduction, through the respective demonstration, paradigm of the current Mathematics.

Pythagoras lived between 570 and 490 BC

He was born on the Greek island of Samos, son of Pythais and Mnesarchus, however, according to legend, his mother, Pythais, became pregnant by the god Apollo, patron of the Oracle of Delphi, which means that Pythagoras was born for the benefit of all humanity; moreover, his name means to mean that, 'announced by Apolo': 'Pythios', the real name of Apolo, and 'agoreuo' which means 'to announce'.

During his youth and subsequent period of formation, Pythagoras was influenced by the thoughts of different origins: mainly from Thales, Anaximander and Ferecides of Siro, but also from Zeno, Xenophanes, Heraclitus, Parmenides, Anaxymes, Empedocles and Democritus, all of them pioneers of logical Greek thought.

Rafael's 'School of Athens', Apostolic Palace in the Vatican. They are represented: Pythagoras and the Pythagorean School and the symbolism of numbers.
Rafael's 'School of Athens', Apostolic Palace in the Vatican. They are represented: Pythagoras (in the foreground, in the lower left corner with a book in his hand and with a white robe on his legs), Zeno, Epicurus, Raphael, Anaximander, Averroes, Alcibíades, Antístenes, Hipatia, Xenofonte, Parménides, Sócrates, Heraclitus, Plato, Aristotle, Diogenes, Plotinus, Euclid, Zoroaster, Ptolemy and Protogenes.

Certainly due to this multifaceted amalgamation of acquired knowledge, Pitágoras founded a school in his hometown, the 'Semicircle', where he disseminated his knowledge. The curiosity of his activity is that, according to legend, he taught in caves, which was typical, at the time, to be attributed to mythical figures and the work of Pythagoras.

Around the age of 40, disagreeing with the tyrant governor of Samos, Polycrates, Pythagoras fled to Crotone, a Greek city-state in southern Italy where he married Theanos of Crete, formed a family and founded a spiritual school that would also maintain an intense political activity.

The religious doctrines of Pythagoreanism were based on theories of the immortality of the soul, the eternal return and the correlation between all things in the Universe.

Pythagoras' life and work became known as the 'feminist philosopher' for admitting women to his school, contrary to what was usual at the time, in which women were given a secondary and subservient role.

Of Pythagoras' work, he emphasizes his religious reform was so relevant that it would serve to support Plato's philosophical thought (428-347 BC).

The followers of Pythagoras' work were great mystics, lived in a deeply hierarchical community (acousmatic / mathematicians, exoteric / esoteric, administrators / politicians) and subjected to very strict rules that even controlled the most common aspects of their daily life. They actually lived as a religious sect, subjected to the power of their master.

The Pythagorean School gave great political brilliance to the city of Crotone, but it would implode due to a revolt led by Cílon, a wealthy Crotonian whose admission to the Pythagorean School was refused and which, in spite of it, ended up dynamizing the population of the city against all Pythagorean, to the point of chasing and killing them. However, it is not certain that Pythagoras was killed in this revolt, another version says that he went into exile in Metaponto, a city near Crotona, where he died in about 490 BC

After this episode, the Pythagorean School was eclipsed, never having been recomposed; only a few of the survivors have spread to neighboring cities continuing their work individually.

The Pythagorean School adopted the mystical pentagram as a symbol, also called pentalfa, a star formed by 5 straight lines and 5 points.

Pentagram, symbol of the Pythagorean School
Pentagram, symbol of the Pythagorean School

This figure has several geometric peculiarities, the most curious being that it can be traced, completely, by a single movement, without having to pass more than once through the same segment.

In fact it is interesting to note that the number 5 was considered a sacred number by the Pythagoreans: it resulted from the junction of the dyad (2, the female) with the triad (3, the male), symbolizing marriage (2 + 3 = 5).

For the Pythagoreans, all the numbers from the decade 1 to 10 had a mystical meaning.

Symbology of Numbers

Let us see the symbology of numbers through the numbers and meanings adopted by the Pitagórica School:

1- the ´monad´, the source of all beings, from it all numbers can be generated by repeated addition;

2- the 'dyad', represents the indefinite, the variety, symbolizes matter and imperfection;

3- the 'triad', symbol of perfection,, representing the masculine character;

4- it represented cause and effect, perfection, nature and man;

5- as we have seen, it symbolized the meeting of the feminine with the masculine, marriage;

6- 2 by 3 product (dyad x triad, female x male), symbolized marriage and procreation;

7- represented light, health and was considered a geometrically unusual number because it is not possible to divide a circle into seven equal parts with some geometric construction known at the time;

8- symbol of friendship for being the sum of two perfect squares (4 + 4);

9- representative of love because it is the number of months of human pregnancy and equality and justice due to the fact that their factors are equal (3 × 3);

10- represented God and the Universe, as expressed in what is considered the 'Pythagorean oath':

"I swear by the One who engraved the tetraktys on our soul, source and origin of eternal nature".

PS: 'Tetraktys' means tetrada, that is, the sum of the four numbers of the decade that reproduce 10: 1 + 2 + 3 + 4 = 10.

This symbology of numbers remains today.

The Pythagoreans were deeply dedicated to the study of different disciplines, philosophy, politics, medicine, religion, astronomy, music and, mainly, geometry and arithmetic.

Well, I believe that, at this moment, my readers will be surprised that I have not yet told you about the most famous theorem on the planet, of course, the 'Pythagorean Theorem'!

So here goes ...

It is fair to begin by mentioning that some particular cases of the Pythagorean Theorem were known long before the work of Pythagoras, in Egypt, Mesopotamia (Babylon), India and China. The particular merit of the Pythagorean School was that it was able to demonstrate its generality through deductive reasoning.

Note that I wrote 'merit of the Pythagorean School', not Pythagoras himself, and it was not a mere lapse; in fact, great doubts persist that this generalization is due personally to Pitágoras himself, what is known is that, surely, it is due to some member or members of his School.

Pythagorean Theorem

One of the great geometric discoveries of the Pythagoreans is the relationship between the sides of any right triangle.

It is what we know today by Pythagorean Theorem: in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

The right triangle of Pythagoras. Pythagorean School and the symbolism of numbers.
The Pythagorean right triangle

Let's look at an example for the particular case of a right triangle whose sides measure 3, 4 and 5 centimeters.

Geometric verification of the Pythagorean Theorem; in this case: 9 + 16 = 25.
Geometric verification of the Pythagorean Theorem; in this case: 9 + 16 = 25.

The Pythagorean theorem, on the one hand, turns out to be the ultimate symbol of the Pythagorean School, would also become the beginning of the end of its doctrine and its School.

For the Pythagorean School, numbers were the essence of all life and of all things, everything being able to be measured from the unit using multiples or parts of that unit, that is, the Pythagoreans only conceived the existence of 'rational numbers' , that is, of whole numbers and numbers that can be represented by fractions formed from whole numbers.

Now, as the story goes, Hypaso Metaponto, born around 500 BC and a member of the Pythagorean School, applied the Pythagorean Theorem to determine the diagonal measurement of a square on side 1, having discovered that the measurement of that diagonal (d) could not be represented by a number rational.

Square root hitting 2

According to the Pythagorean theorem, it must be:

Equation 1
Equation 1
Equation 2
Equation 2
Equation 3
Equation 3

Now, it is easy to see that there is no whole or fractional number that, raised to the square, is equal to 2, thus showing that there would have to be another type of numbers besides the integers and fractionals exclusively admitted by the Pythagoreans, as we saw in Countless Numbers.

In possession of this discovery, which appeared to be able to make him famous, turned out to be his death sentence after all. Hipaso divulged his discovery to the whole community, thus evidencing, with the breaking of the oath of silence about new discoveries to which he was subject in the Pythagorean community, that the Pythagorean School assumption was wrong; as a result, he was murdered, with some records stating that it was Pythagoras himself who carried out the sentence. Pythagoras' life and work thus ended with a mixed sense of failure and glory.

It is now demonstrated that the value of that diagonal is represented by an 'irrational number' (that type of real numbers that cannot be represented by a fraction).

Hipaso disappeared but there was nothing more to be done, his discovery of the existence of irrational numbers remained for posterity. This, however, does not erase the scientific achievement of the Pythagorean School that represents the statement and the demonstration of the Pythagorean Theorem.

Great work of Pythagoras!

We hope you enjoyed the information about the life and work of Pythagoras and the symbolism of numbers.

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5 thoughts on “The Life and Work of Pythagoras, The Pythagorean Theorem and the Symbology of Numbers”

  1. Hello I would like to know if in addition to the channel content, there is a specific book that I can buy and that talks about the life of Pitágoras?
    Thank you very much

    1. Hi Josias,
      You can read in the National Geographic magazine, a special edition, Pythagorean Theorem. Another alternative, from a series made by Paul Strathern, Pythagoras in 90 minutes.
      There are no books published with Pythagorean writings, so some information that is published may be contradictory. The information about Pythagoras came to us indirectly.
      Thanks and big hug.

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