Historical backgrounds

Antiquity

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During the twentieth century the progress of science accelerated tremendously. Much of this progress has been made possible because of global information exchange between scientists and large scale structural research by professional and academic institutions. The amount of time and money dedicated to scientific advancement is enormous. In classical times these recourses were not available, which makes it even more remarkable how highly developed the society already was at the beginning of our year counting. Apart from the artistic value of classical architecture, the Romans managed to develop sophisticated architectural methods and construction techniques based on the use of concrete, which made it possible for them to build the enormous dome of the Pantheon. Apart from the fact that this building is still intact after almost 2000 years, it is still the largest self supporting dome construction in the world! The Pantheon was build in 128AC and measures a diameter of 43,2 meters. When Michelangelo constructed the (not self supporting) dome of the St. Peter in Rome, he deliberately made it slightly smaller out of respect for the tremendous architectural achievements of his Roman predecessors.

Ground plan of the Pantheon in Rome.
(click on the image to enlarge)

Not made of concrete, but surely just as solid were the scientific achievements of the Greek scholars starting as early as the sixth century BC. Their intellectual work was not the result of large scale structural research like we are used to in these days. Most often progress was achieved through the work of a single genius individual that subsequently started a school or academia where his teachings were passed on to a larger group of followers. Practical science was almost always accompanied by philosophical an theological studies. It is this unique search for an all uniting harmony that  that sets apart classical from modern science. While modern science tends to put everything under the microscope trying to find the ever smaller details that make up our existing, classical scientists constantly tried to perceive which divine laws unite mathematics with arts, political organization, physics, religion, architecture, music and so forth. The classical philosopher-scientist was like the renaissance artist; a "uomo universalis", constantly trying to improve on himself and his teachings in the never ending quest for the universal harmony that forms the basis of beauty and existence.

With regard to the true merits of scientific and philosophical achievements in antiquity we can only estimate their true splendor since only a fraction of all written texts survived the centuries. Fortunately we can still admire some marble copies of the many beautiful bronze Greek statues that finished in a melting pot, while every now and then the earth reveals some archeological findings that have been buried for thousands of years. These findings give us a good impression of what the many beautiful classical buildings that turned into stone quarries during the medieval period must have looked like. It did however not take melting pots or stone quarries to destroy the larger part of our classical heritage. Apart from the numerous books that fished on the burning stake and the many inaccurate and subjective translations, the natural decay of the organic information carriers made sure that only a very small part of the classical intellectual heritage survived the ages. A small comfort is provided by the fact that the literary treasure must have been so enormous that we can still enjoy several classical masterpieces in our time. This fact is by no means an achievement of Christianity. Medieval monasteries did copy some classical works, but in most cases these were books that were not supposed to contradict the Christian doctrine. Some Latin writers, amongst which for instance Virgil, were seen as moral predecessors of Christianity. There works fitted reasonably well into the Christian exegeses and were as such copied frequently.

Book burning appears to be an old Christian tradition. It all started in Ephesus where St Paul preached and converted the Jews and Greeks to Christianity. In Acts 19:19 it is explained how book scrolls with a value of 50.000 drachmas were burned ( 1 drachma was one days wages). Another highpoint of drastic censorship was the burning of the library of the Serapeum in Alexandria in 391 AC under Theophilus of Alexandria. This ancient library had been founded by the Ptolomeans.

Around 600 AC, Gregory the Great burned a large library in Rome that had been founded by the emperor Augustus, while later in the middle ages the Holy Inquisition burned the priceless collection of hundreds of thousands of hand written books of the caliphs of Cordoba. This example was followed by archbishop Ximenes, who piled up the library books of Granada and the Alhambra to an enormous stake on the market place. As long as the church existed, the writings of different cultures have been its primary firewood.

It is mostly thanks to the initially much more developed Arabic culture that we still posses some reliable copies of classical masterpieces. It is ironical how scholars in the late middle ages and early Renaissance made the same study tour through the North of Africa and the Middle East as once their famous Greek heroes of classical philosophy had done when the left for Egypt and Babylonia to become initiated in the old mystical rites of these ancient cultures.

When we try to reveal the oldest traces of cultural development, it seems like every important civilization was preceded by an even older one. The more one approaches the original source, the scarcer the historical information becomes. As such it is very difficult to say anything substantial about the oldest origins of number symbolism. The only thing that is certain, is that every ancient and Classical civilization integrated number symbolism in its religious and philosophical teachings.

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Pythagoras

For several centuries, Pythagoras and his followers have attributed an enormous value to numbers. Especially the Pythagorean theological and cosmogonical theories behind the importance of numbers are of great interest because they also appear in Jewish and later also in Christian religion.

The Jews shared the Pythagoreans (and most early Greek philosophers) interest in the origin of existence. It is interesting to observe that over the centuries most Greek philosophers shifted their attention towards more practical themes like ethics and politics, while the Pythagoreans always kept their fascination with the creative principles of existence.

The reason why the Jews shared this interest must reside in the fact that they had been exposed to the same cultural influences that were common in North Africa, the Middle East and the rest of the Mediterranean area. Originally the Jewish religion was a very simple monotheistic religion, but then we are talking of a period of about 2500 years BC. From the time in which Pythagoras lived (about 500 years BC) the Jews had- and continued being subdued to several deportations and occupations that might not have left only bad experiences. The Egyptians and the Babylonians had highly developed cultures that were far out superior to the tribal Jewish civilization in these days. As such it is logical that within the Jewish religion, certain groups started to develop a more abstract intellectual image of their God, more like an indescribable and unreachable phenomenon that can be found in the basis of the entire existence. They shared the piousness with which they approached this mystical basic creative principle of existence with the Pythagoreans. The Pythagoreans may not have been monotheists, but their religious approach of the early cosmogonical theme was rather "un-Greek".

The well known Ionic philosopher Thales (a predecessor of Pythagoras) searched for the origin of the cosmos (Greek word for order) in an original substance (Greek: archè) which he thought to be water (not that stupid). Other Greek philosophers continued to rationalize in the same quasi physical direction and even came up with the incredibly "modern" theory of the atoms. In this theory of the philosopher Democritus (ca. 425 BC), everything is build up from atoma (undividable), that can cluster together in various ways, forming physical objects.    

The Pythagoreans searched for the harmony that forms the basis of existence in something higher, namely the number. Untouchable and completely abstract but in its simplicity also easy to rationalize. Actually, numbers are excellent phenomenon to meditate about mystical themes. Even without knowing the zero, the Pythagoreans managed to develop very advanced mathematical and scientific theories. The parallel with the Jewish religion that was mentioned above, resides in their quest for the unity, the unreachable and unpronounceable "Ain Sof" like the Jews describe this abstract religious image of their God. According to these hard to date Jewish mystical teachings, this indescribable phenomenon created the universe through a primal contraction with nothingness. The Jewish mystics called this "Zim Zum", a beautiful thought that even in our time hasn't lost it's meaning.

Irrational numbers

A well known story tells that the Pythagoreans mentally dropped from their chairs when the found out that not all geometrical proportions can be brought back to whole numbers. This could be partly true because the phenomenon of irrational numbers remained hard to handle for a long time. At the same time it is highly unlikely that mathematicians of their caliber did not discover that certain proportions could not be related to whole numbers. An illustration for the possible awareness of Pythagoreans of irrational numbers is their choice of the pentagram as the (secret) symbol of their community. The five pointed star or pentagram is based on the proportions of the golden section, a proportion that can not be related to whole numbers because it is based upon the square root of five.

The Golden section relates to a length 1 as (√5+1)/2 being approximately: 1,618.033.988.749.894.848.204.586.834.365..

Many special characteristics of the Golden Section will be described in the next chapter, where also the architectural qualities of the √2 proportion will be discussed.

Regarding the historical dating of the Pythagorean teachings we have to consider that many discoveries of his followers were attributed to the great maestro himself. "Autos epha", or "he said it himself" is the name by which this practice became known. Through the many qualities of his followers that were attributed to him, Pythagoras obtained an almost divine status that claimed him to have universal knowledge of the entire existence. This phenomenon was not unique in antiquity. Actually it was common practice for disciples to blow up the status of their teachers beyond any reasonable proportion. As such it is fair to put the originality of some of Pythagoras' "inventions" in a historical perspective. Regarding the famous Pythagoras triangles; there actually exists a Babylonian clay tablet called "Plimpton 322" dating from about 2000 BC, on which fifteen three-number combinations are mentioned that all form Pythagoras triangles. The writer of this tablet apparently new almost 1500 years before Pythagoras that r=√(p²+q²) calculates the sides of any straight angled ("Pythagoras") triangle. It is almost certain that Pythagoras obtained this and other knowledge from oriental cultures. In itself this is nothing strange. It happens very often that common mathematical knowledge obtains a name only when a certain scientists by describing or even proving the fact puts it in the spotlight. It is rather strange that despite the exalted credits that the followers of Pythagoras attributed to their master, they never made a secret about the origin of his wisdom and knowledge. The books on Pythagoras by Jamblichus and Porphyrius from the third century AC (quoting many older sources that have gone lost), describe his lengthy stay in the Middle East. Quoting Porphyrius' Life of Pythagoras: "6 Regarding his training, most people say that he learned the so called mathematical sciences from the Egyptians, Chaldeans and Phoenicians, since the Egyptians traditionally excelled in geometry, the Phoenicians in arithmetic and the Chaldeans in astronomy." ; "11..Pythagoras visited the Egyptians, according to Diogenes, as well as the Arabians, the Chaldeans and the Hebrews, where he obtained accurate knowledge about dreams..".. 

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