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Griechische Mathematik: Pythagoras


The Theorem of Pythagoras, Greek Stamp 1955 ( Scott 583, Michel 633), actually his greatest achievement is the idea that everything can be expressed by numbers. See also Pythagoras and his Theorem, San Marino 1983, Michel 1275, Scott 1045. Euclid's Elements, Book I, Proposition 47

It is to this gentleman that we owe pure mathematics. The contemplative ideal -- since it led to pure mathematics -- was the source of a useful activity. This increased it's prestige and gave it a success in theology, in ethics, and in philosophy. Bertrand Russell

When Pythagoras of Samos (Πυθαγόρας ο Σάμιος)^(569/70 – c. 475 BC) was asked by Leon a tyrant of Phlius who are you? He replied: I am a philosopher. A friend of wisdom. And here we have the first philosopher of history who used this name for his profession. Pythagoras was the first true mathematician. He too, was the first person, who gave the name of cosmos (or kosmos, kosmema i.e. a jewel) to the universe. Who was Pythagoras? According to him an encarnation of Aethalides a son of the God Hermes who when Hermes asked him what he wants, except eternal life that he could not give to him, decided an eternal memory to remember after death all his previous lifes. Then he was born as Euphorbus who was injured by Menelaus in Troja. He was also born as Hermotimus and as Pyrrhus a poor fisherman in Delos. He was also in Hades having seen Homer hanged there on a a tree punished together with Hesiod by the gods. Some say that he was also born as a women, Alco.

Pythagoras, a son of Mnesarchos and Pythais, was born in Milet on the Island of Samos about 569-570 B.C. His teacher was Pherecydes (according to Diogenes Laertius), and Pythagoras studied thanks to an advice of his uncle Zoilos. Pythagoras using a recommendation for the Pharao Amasis written by the tyrant Polycrates and enough money goes to Egypt to obtain the knowledge of the priests.

Thaletes made Pythagoras sail to Egypt and meet with the priests at Memphis and Diospolis, because they had been the ones who had instructed him in those disciplines, through which he was called a learned man among the people.
Porphyrius, Vita Pythagorae 12

Some say that Thales gave to him the advice to go to Egypt to further his mathematical knowledge. The Priests tried to avoid teaching him their secret knowledge but Pythagoras resisted until he was considered their brother. He continued his studies in astronomy as a student of the Chaldeans. He learned logistics and geometry from the Phoenicians. It was said that he discussed also with Zarathustra about good and evil. Pythagoras divided people in two categories. The “Mathematicians”, those who had the right to acquire “Mathema” i.e. knowledge. The other group were the “acusmaticer” those who only were allowed to “akouein” i.e hear. Like in the school where some students even if teached will remain acusmaticer. Life for Pythagoras students was very hard. They had to just hear Pythagoras in total silence. They were supposed to hear after this time the harmony of the spheres. Otherwise they had to repeat the study for another five years!

Δοκεῖ γάρ τισιν ἀναγκαῖον εἶναι τηλικούτων φερομένων σωμάτων γίγνεσθαι ψόφον, ἐπεὶ καὶ τῶν παρ' ἡμῖν οὔτε τοὺς ὄγκους ἐχόντων ἴσους οὔτε τοιούτῳ τάχει φερομένων· ἡλίου δὲ καὶ σελήνης, ἔτι τε τοσούτων τὸ πλῆθος ἄστρων καὶ τὸ μέγεθος φερομένων τῷ τάχει τοιαύτην φορὰν ἀδύνατον μὴ γίγνεσθαι ψόφον ἀμήχανόν τινα τὸ μέγεθος. Ὑποθέμενοι δὲ ταῦτα καὶ τὰς ταχυτῆτας ἐκ τῶν ἀποστάσεων ἔχειν τοὺς τῶν συμφωνιῶν λόγους, ἐναρμόνιον γίγνεσθαί φασι τὴν φωνὴν φερομένων κύκλῳ τῶν ἄστρων. Ἐπεὶ δ ἄλογον δοκεῖ τὸ μὴ συνακούειν ἡμᾶς τῆς φωνῆς ταύτης, αἴτιον τούτου φασὶν εἶναι τὸ γιγνομένων εὐθὺς ὑπάρχειν τὸν ψόφον, ὥστε μὴ διάδηλον εἶναι πρὸς τὴν ἐναντίαν σιγήν· πρὸς ἄλληλα γὰρ φωνῆς καὶ σιγῆς εἶναι τὴν διάγνωσιν· ὥστε καθάπερ τοῖς χαλκοτύποις διὰ συνήθειαν οὐθὲν δοκεῖ διαφέρειν, καὶ τοῖς ἀνθρώποις ταὐτὸ συμβαίνειν.

Aristotle says that the motion of the planets according to the Pythagoreans produces a sound. We do not hear this sound because we are born with this “noise” background.

<Πυθαγόρας> αὐτὸς δὲ τῆς τοῦ παντὸς ἁρμονίας ἠκροᾶτο συνιεὶς τῆς καθολικῆς τῶν σφαιρῶν καὶ τῶν κατ’ αὐτὰς κινουμένων ἀστέρων ἁρμονίας, ἧς ἡμᾶς μὴ ἀκούειν διὰ σμικρότητα τῆς φύσεως.

Empedocles explains that Pythagoras was able to hear this sound due to his strong nature while we normal humans are too weak...

Pythagoras exposed his teachings to his followers either in plain words or through symbols. Because his teaching was of two kinds, those who followed him were divided into Mathematics and Acusmatics. The Mathematics were those who knew the most important and deepest part of his doctrine, Acusmatics were those who were summarily taught the rules, with no accurate explanations.
Porphyrius, Vita Pythagorae 36

The Acusmatics' philosophy is made of precepts: these are imparted without them being shown the why, or told the reason why one must act in a certain way. The Acusmatics also strive to conserve all his other sayings, and consider his words divine opinions, and of their own they say nothing and they believe nothing should be said; indeed, they consider supremely knowing those that have memorised the most precepts and teachings

Such sayings are divided in three groups. Those of the first group answer the question, what is it? Those of the second to the question, what most of all? Those of the third to the question, what must one do or not do?

Examples of the first: What are the islands of the blessed? The sun and the moon. What is the oracle of Delphi? The tetrades, which is also the harmony of mermaids. [Examples of the second] Which is the wisest thing? The number, and secondly that which gave a name to things. Which is the most beautiful thing? Harmony. The most powerful thing? Intelligence. The best thing? Happiness. What is said with most truth? That men are evil.
Iamblichus, De vita pythagorica 82

Coin from the Roman Period (Ionia Samos BMC [British Museum Catalog] 237, struck during the reign of Trajan). Pythagoras touching with his hand a globe placed on a column. He knew that the Earth is spherical (Diogenes Laertios: He too, was the first person who called the earth round). Ancient coins show they knew it was round , The globe could also represent the sky (known as the blue globe) observed in many images of Zeus

Pythagoras taught that science, was revealed to men by "celestial deities," those godlike men who were the Divine Instructors of the Third Race. It was first taught to the Greeks by Orpheus, and for centuries made known only to the "chosen few" in the Mysteries. Just before the Mysteries began to degenerate, Pythagoras instituted this teaching in his School, thus preserving under the name of "philosophy" the ancient science which, as Plato says, is "the greatest good that was ever imparted to men." In his Life of Pythagoras, Iamblichus repeats the statement of Plato that the study of the science of Numbers tends to awaken that organ in the brain which the ancients described as the "eye of wisdom", the organ now known to physiology as the pineal gland. Speaking of the mathematical disciplines, Plato says in the Republic (Book VII), "the soul through these disciplines has an organ purified and enlightened, an organ better worth saving than ten thousand corporeal eyes, since truth becomes visible through this alone."

Pythagoras invented codes so that the “mathematicians” could talk together without to be understood. Something which is common today, where scientists use complex symbols or strange words sometimes to express very simple truths.

His name was not used but instead the words autos epha (lat. ipse dixit) i.e. “He said it”. His students said that there are humans and gods and others such as Pythagoras.

Pythagoras began his talks always saying: “No, by the air I am breathing, no by the water I am drinking I do not allow any contradiction to what I say.” He talked with animals. He was seen at the same time in two distant places in Kroton and Metapont and many other strange histories exist:

Pythagoras of Mnesarchus first devoted himself to mathematics and to numbers, then started to perform miracles, like Pherecides had. One day in Metapontum, as a ship was about to enter the harbour, [...] Pythagoras appeared and said: "This ship will bring you a dead man." Another time in Caulonia, Aristotle reports, he predicted the arrival of the white she-bear. And the same Aristotle says of him many other things: among others this: that responding with a bite to the bite of a deadly serpent, he killed it. And that he had predicted the sedition against the Pythagoreans.

Once he also appeared, on the same day and at the same time, both in Kroton and in Metapontus. Aristotle tells also that once, in a theater, he stood up and showed to the spectators that his thigh was made of gold.
Apollonius, Mirabilia 6

The cosmos according to Pythagoreans. (Philolaus and others) The earth orbits around a central fire. The Anti-Earth (Antichthon) is not visible like the central fire because Greece is on the opposite side on the earth.

Pythagoras and his students are the first that moved the earth from the center of the world. The center is a fire called the mother of all gods. Then we have the earth, moon, sun, the five planets known to the ancient Greeks and the stars. And to make the world perfect a copy of the earth is assumed to be in a position where it is not visible so that we have a total of 10 different objects, because 10 is a special number of harmony.

Most people-all, in fact, who regard the whole heaven as finite-say it lies at the centre. But the Italian philosophers known as Pythagoreans take the contrary view. At the centre, they say, is fire, and the earth is one of the stars, creating night and day by its circular motion about the centre. They further construct another earth in opposition to ours to which they give the name counterearth. In all this they are not seeking for theories and causes to account for observed facts, but rather forcing their observations and trying to accommodate them to certain theories and opinions of their own. But there are many others who would agree that it is wrong to give the earth the central position, looking for confirmation rather to theory than to the facts of observation. Their view is that the most precious place befits the most precious thing: but fire, they say, is more precious than earth, and the limit than the intermediate, and the circumference and the centre are limits. Reasoning on this basis they take the view that it is not earth that lies at the centre of the sphere, but rather fire
Aristotle, On Heavens B 13 293 a 18

Pythagoras the musician with Philolaus (or Philolaos)

Contemporaneously with these philosophers and before them, the so-called Pythagoreans, who were the first to take up mathematics, not only advanced this study, but also having been brought up in it they thought its principles were the principles of all things. Since of these principles numbers are by nature the first, and in numbers they seemed to see many resemblances to the things that exist and come into being-more than in fire and earth and water (such and such a modification of numbers being justice, another being soul and reason, another being opportunity-and similarly almost all other things being numerically expressible); since, again, they saw that the modifications and the ratios of the musical scales were expressible in numbers;-since, then, all other things seemed in their whole nature to be modelled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number. And all the properties of numbers and scales which they could show to agree with the attributes and parts and the whole arrangement of the heavens, they collected and fitted into their scheme; and if there was a gap anywhere, they readily made additions so as to make their whole theory coherent. E.g. as the number 10 is thought to be perfect and to comprise the whole nature of numbers, they say that the bodies which move through the heavens are ten, but as the visible bodies are only nine, to meet this they invent a tenth—the ‘counter-earth’.Aristotle, Metaphysica A 5. 985 b 23

The Greeks rejected later the anti-Earth idea as observations could not confirm this idea:

The counter-earth was the first part of the system to be abandoned; and it is suggested that this abandonment was due to the extension of the geographical horizon. Discoveries were made both to the east and to the west. Hanno, the Carthaginian, had made his great voyage of discovery beyond the Pillars of Hercules, and on the other (the eastern) side India became part of the known world. It would naturally be expected that, if journeys were made far enough to the east and west, points would be reached from which it should be possible to see the counter-earth, but, as neither the counter-earth nor the central fire proved in fact to be visible, this portion of the Pythagorean system had to be sacrificed. Heath

Which thing is the wisest? The number. And what is the most beautiful? The harmony. In the begin of time there was the monas the number one who gave birth to the numbers. From these we have the points and the lines and with the harmony we have the right relation between the objects and so was the Cosmos born.

Is the modern cosmological view of the world provided by modern physics not the same, with symmetry and geometry playing a central role and mathematical much more founded?

Pythagoras and his students believed that all complex phenomena can be reduced to simple ones and this is what is the driving force of fundamental scientific research.

Pythagoras theorem and associated formula; one of 10 mathematical formulas that changed the face of the world according to this Nicaraguan stamp, Bronowski refers to this theorem as ‘the most important single theorem in the whole of Mathematics’. (Nicaragua 1971, Las 10 formulas matematicas que cambiaron la faz de la tierra, Michel 1613-1622, Scott 877-881, C761-C765)

The discovery of a relation between music and numbers played probably a important role in his ideas about numbers. Although he is not to be the first known the theorem given his name he was probably the first who provided a proof.

In right-angled triangles, the square of the hypotenuse is equivalent to the sum of the squares of the sides that form the right angle. If one listens to those who concern themselves with antiquities, who attribute this theorem to Pythagoras, one will also find that some say he sacrificed an ox for this discovery. Proclus, In Euclidem I 47

It is said that 1, some say 100, animals has been sacrificed by him for his discovery of the proof but this seems to be a legend. As Porphyrius in “Vita Pythagorae 18” says: “What he said to his companions, nobody can say for sure, because they kept a great secret about this. But his most known opinions are these: he said that the soul is immortal, and that it passed to animate beings of another species, then that what has been repeats itself in regular intervals, and there is nothing that is truly new; finally that all animate beings must be considered as belonging to the same race.”

καί ποτέ μιν στυφελιζομένου σκύλακος παριόντα φασὶν ἐποικτῖραι καὶ τόδε φάσθαι ἔπος· "παῦσαι μηδὲ ῥάπιζ’, ἐπεὶ ᾖ φίλου ἀνέρος ἐστίν ψυχή, τὴν ἔγνων φθεγξαμένην ἀίων.

When he saw that a young dog was beaten he said: Stop beating the dog who has the soul of a friend that I recognized.

Was he so eccentric? At least it is known that he went to Delos when he was told that his old teacher Pherekydes is dying. He spends months in Delos until the last days of his teacher.

Pythagoras daughters (Myia or Myria), Damo and Arignote), like his wife Theano of Thurii were mathematicians.

It seems that he died when he was around 100 years (probably 475 BC) old but exact details are not known.

Pythagoras Life by Diogenes Laertios

The Pythagoras Cup Invention

Josh Kutz, Between Mathematics and Mythology: The Heroic Figure of Pythagoras, 2002, PhD Thesis, Brandeis University


The Pythagoras Foundation has an extensive collection of publications about Pythagoras and the Pythagoreans. (1500 books and about 5000 journal articles). A Newsletter can be obtained on request via this site (Summary of information published about Pythagoras and the Pythagoreans, the
work in progress and contributions from specialists).

Pythagoras and the Pythagoreans, Fragments and Commentary (Interesting Ideas of the Pythagoreans from text fragments)
Pythagoras and Harmony
Pythagoras Monument in Samos (German) , more images 1 , 2 (Samos today)

The Ko'ans of Pythagoras

A Greek "Eleftherios Argyropoulos" provided 520 different proofs of the Pythagoras Theorem (Guinness Book Records)

The Babylonian tablet Plimpton 322 with Pythagorean triplets

Zalmoxis the slave of Pythagoras ?


The Pythagoras Tree (Java Applet)

Pythagorean Theorem 44 proofs, More Proofs including an older from Babylon

G. S. Kirk, J. E. Raven, M. Schofield , The Presocratic Philosophers: A Critical History with a Selection of Texts , Cambridge University Press;

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