Historical Backgrounds

Numbers & Geometry - "Irrational Proportions"

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Pythagoras' Pentagram

There is some historical uncertainty regarding the pentagram. It is a fact that the pentagram was the "secret" recognition symbol of the Pythagoreans. At the same time it is said that the same Pythagoreans were completely surprised to find out about the existence of irrational numbers, suggesting that they were incapable of calculating anything more difficult than 1+1=2. That leaves the question why the Pythagoreans selected a symbol that supposedly contained all their secret knowledge, if they were not even capable of constructing it? It seems obvious that every mathematical group that chooses a complex geometric figure as it's secret symbol, obviously knows the method to construct it, otherwise the figure would merely be a symbol of their stupidity. Pythagoras of Samos supposedly lived from about 580BC to 496BC. His universal knowledge was unmatched at that time. He is supposed to have acquired his knowledge during long journeys that initiated him in the Egyptian-, Hebrew and Babylonian temple cults. About 300BC, Euclid describes the golden section in his Elements, a series of thirteen mathematics books that compiled most available knowledge at that time. About 2000 years before Pythagoras, his Egyptian temple teachers already knew and used the golden section proportions in their constructions.

At the highpoint of Hellenistic culture, the Greeks were openly aware of the golden section and it's mathematical qualities. Euclid's Elements were no secret writings, but proclaimed a public mathematical tradition. Since Euclid lived and worked in Alexandria it becomes immediately clear that also the extensive Jewish (and later Christian) intellectual community that lived there, must have had open access to this information. Euclid's books were so popular that up to the nineteenth century they became the most widely spread books after the Bible. The connection between the division in extreme and mean ratio and √5 was open source knowledge. Long before Euclid, Theodorus, Plato's math teacher, proved that the square roots of the numbers 3 to 17 are irrational (with exception off course of 4, 9 and 16). The fact that he stopped at 17 (and subsequently didn't seem to have the prove that all natural numbers are either cubes of irrational- or natural numbers), is irrelevant with regard to the golden section. The famous Egyptian papyrus Rhind from 1500BC talks about a sacred proportion. No information is provided about this proportion but the fact that the Egyptians used √5 and possibly even golden section proportions in their pyramid of Cheops underlines their extensive knowledge of geometry and their specific interest in sacred proportions. As mentioned above, the only missing link in the knowledge of irrational numbers and their corresponding geometric proportions is Pythagoras himself. A widely spread story claims that initially Pythagoras and his followers had no idea of irrational numbers leaving them completely surprised when they finally found out about their existence. Since this story involves Pythagoras' followers it must have taken place during the final period when Pythagoras settled at Crotone in the South of Italy and founded his community. According to Jamblichus, Pythagoras by then was already sixty years old, leaving him very little time until his death to develop and establish his teachings. The irrational number discovery story is extremely unlikely because it suggests that Pythagoras, who studied for years in foreign countries that definitely knew of the existence of irrational numbers and geometrical proportions, did not learn about their existence, only to find them out all over again at the age of over sixty years. The story becomes even more unlikely if one considerers that Pythagoras was apparently the "inventor" of the right angled Pythagoras triangles in which the two right angle sides relate to the oblique side as a²+b²=c². Pythagoras new about the existence of triangles with lengths 3, 4 and 5 (9+16=25), but he surely also has been looking for simpler triangles with base lengths 1. In this case the formula is 1²+1²=c² with outcome √2. The next triangle that he tried must have had base lengths 1 and 2. In this case the formula is 1²+2²=c² which makes that c is √5. The confrontation with these two irrational numbers is as such the inevitable result of the first experiments with the axiom of Pythagoras. In fact, most scholars agree on the fact that Euclid most likely obtained his knowledge on the golden section from the Pythagoreans.  It is as such impossible that Pythagoras and his Pythagoreans did not know of the existence of complex geometrical proportions such as the golden section proportion.

A possible explanation for the strange misunderstanding regarding the Pythagoras irrational number story, might be found in the fact that early Pythagorean knowledge was not publicly accessible. Contrary to later mathematic teachings, the school of Pythagoras had a highly closed character in which divulgence of the group's knowledge was considered a mortal sin. Traces of this veil of secrecy can be found in later Greek mathematics as well. In his famous dialogue "Timaeus", Plato talks very "cryptically" about a certain mathematical phenomenon. Regarding this text different explanations exist. The mystic atmosphere of secrecy that surrounded mathematical knowledge throughout antiquity and early Christianity most likely has its origin in the religious / scientific tradition of the temple cults.

In modern times religion and science seem to grow ever further apart. This makes it difficult to imagine that until some centuries ago clerical institutions and monasteries were the most important promoters (and just as often obstructionists), of arts, literature and science. In the early Middle Ages, monasteries were the only channel through which Classical literature remained available. Classical literature had a revival during the reign of Charlemagne. Until that time most attention was drawn to the translations and copying and editing of Bible books and some classical texts that contained Christian values, such as the popular neo-Platonism of Plotinus. The first proto renaissance of Charlemagne came almost five hundred years after the edict of Milan. The first centuries of Christianity passed without any intellectual achievement and very little interest in Classical literature. The decline of the Roman empire was happily assisted by a gratefully censoring clerical apparatus that moved like a broom-car behind the funeral procession of the Classical civilization. What remained in Charlemagne's days must have been a watery infusion of what was produced during the heydays of the Hellenistic and Roman culture. Adapting itself to the harsh conditions of every day medieval life, the Christian religion had been transformed into a tableau vivant for the illiterate in which shepherds and donkeys played the main characters. Completely stripped of its original Jewish / Greek mystical tradition, the Christian religion was smoothly integrated in the existing North European polytheistic tradition. An innumerable amount of saints had replaced the former nature gods, without showing any consideration or respect for the original Hebrew Christian teachings.

Plato's Timaeus dialogue has been interpreted in several different ways. Some think that Plato talks about the golden section while the Danish scholar Tons Brunés believes that the text discusses his "sacred cut", the proportion by which a square or a cube doubles in surface or volume through the use of √2. It is known that the Greek were fascinated by this problem and that especially the Romans made vast scale use of the √2 proportion in their architecture. As such Brunés interpretation of the Timaeus dialogue is highly plausible. In this case however Plato's Timaeus is used as an example of how the concept of mystical mathematical teachings for the "initiated" seeped trough in Hellenistic days. During the time of Pythagoras, the secret aspect of his teachings was much stronger, which is easily explained by the tradition of scientific temple initiation of the foreign cultures that contributed to Pythagoras knowledge. In Hellenistic republic, education became more public and the secret aspect of mathematics became less important. From this tradition the public mathematical tradition of Euclid's Elements evolved.

Many people do however believe that next to public mainstream mathematics, the old secret/mystical Pythagorean tradition kept on existing as some sort of elitist initiated community with its own mystical knowledge and secret mathematical axioms.

There was some reason for the existence of a secret Pythagorean community, because it was not only mathematics that occupied their time. Especially religious mysteries-, medical science and politics had their strong interest. The original school of Pythagoras was completely sectarian but later the Pythagoreans became a secret society and their members spread throughout the vast Hellenistic territory. Many claimed to be Pythagoreans, amongst which also Plato. Whoever claimed to be a Pythagorean obviously nurtured the concept of secret knowledge for the initiated.    

It will always remain difficult to make a reconstruction of events that did not only take place thousands of years ago, but on top of that were already hidden by a veil of secrecy in their time. As such all information regarding the Pythagoreans is obtained through people that had no part in the secret teachings or through people that betrayed the secret. In case of the Pythagoreans betrayal seems highly unlikely because they supposedly preferred to be killed rather than to give away their secrets. Besides that, the original nucleus of Pythagoreans is supposed to have been murdered completely.

It is a fact that throughout the entire antiquity, from Hellenistic through Roman times, the Pythagorean continued to inspire every philosopher and scientists as the perfect spiritual communion. In almost all cultures that came under Hellenistic influence, groups originated that claimed to be part of the Pythagorean tradition. Besides people like Plato, who claimed to be a Pythagorean, also the much discussed Essenes, described in the books of Josephus, claimed to be a Pythagorean group.

It is not exaggerated to say that Pythagoras became the most acclaimed-, praised and revered philosopher of antiquity, which by itself is quite an achievement for someone that didn't leave us a single written word. It almost seems as if not publishing was the classical key to success. Besides Pythagoras, also Socrates and of course Jesus became famous through the words that others wrote about them instead of through the words that they wrote themselves. The only thing that distinguishes Socrates from Pythagoras and Jesus, is that Socrates never claimed secret teachings.

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