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*PYTHAGORAS AND THE MATHESIS OF CHAOS*

A talk by Frater Choronzon first delivered on 24th June 1989 to 'The Society" at The Plough, Museum Street

The title of this paper evolved as a marketing ploy for the concept of
Mathesis. That science has the same relationship to Mathematics as does
Alchemy to Chemistry and Physics.

Mathesis is one of the few words which can be found in dictionaries of
the Greek, Latin and English languages, with a near identical definition
in all three; for example, Liddel & Scott's Greek-English Lexicon
gives "learning or knowledge, especially mathematical sciences and
astrology". Chaos is also a proper word in all three languages, and
most dictionaries will tell you who Pythagoras was.

Chaos and Pythagoras are familiar in some sense to most people, but Mathesis,
as a concept, has been even more thoroughly buried by the academic establishment
than has Alchemy.

Both Pythagoras and Chaos are completely relevant however. Pythagoras invented the concept of Mathesis, which is more than can certainly be said of the Geometrical Theorem which bears his name; and Chaos Mathematics, or Non-Linear Dynamics, may well be the vehicle by which Mathesis might make an attempt at being rehabilitated to academic respectability, or at least through which its hypotheses could be seen to relate to real world problems.

MATHESIS AND MATHEMATICS

The difference between Mathesis and Mathematics is plainly illustrated by an example drawn from the work of Agrippa, the German renaissance philosopher. Following page 148 (CXLVIII) in the 1533 edition of his major work 'De Occulta Philosophia' are a series of diagrams with which many people will be familiar (see Appendix). These are the so-called Magic Squares of the Planets for which Agrippa suggests a cabbalistic origin. Each square consists of an arrangement of the natural counting numbers such that the Real Integers in each Column and Row of any of the squares add to the same Total, as do the diagonals.

[Note (1997): Research by Prof David Singmaster suggests that the Magic Squares are of Chinese origin and that designs incorporating them can be dated to the 1st Century BC. Singmaster points out that the 1510 draft of 'De Occulta Philosophia' did not include the diagrams, and that the ordering of the squares to the planets is different to that previously applied. He suggests that Agrippa may have followed the work of the Italian mathematician Lucas Pacioli.]

The squares themselves are a phenomenon within conventional and respectable Number Theory. For any of the squares it can be shown that there is more than one arrangement of the natural counting numbers which yield the same Row/Column/Diagonal addition properties. It can be proved that for any given square any such arrangemet of the numbers will sum to the same Total. For example, for the 4 by 4 Jupiter square, it is possible to show that any arrangement of the numbers 1 to 16, such that the Row/Column/Diagonal additions are the same, will always yield the same Total. To attempt such a proof would be conventional Mathematics, and such an exercise was recently set to students at an Open University Mathematics Summer School.

Agrippa's analysis of the squares is rather different (and he doesn't appear thoroughly to have proof-read the blocks from which his book was printed). He uses procedures which are outlined in foregoing chapters of his book, and which, in their way, are no less rigorous than any of the other mathematical techniques of the time. This was more than a century before Newton and Leibniz published their first works on Calculus, and nearly two centuries before the development of a consensus of rigorous Analysis; and that came into being partly as a result of those protagonists and their adherents attempts to rubbish each others work. The squares are assigned to the planets and 'lights' in descending order of their earth relative orbital periods. Special significance is assigned to certain numbers associated with the squares, these are as follows:

Base Number | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Base Number Squared | 9 | 16 | 25 | 36 | 49 | 64 | 81 |

Row/Column/Diagonal Sum | 15 | 34 | 65 | 111 | 175 | 260 | 369 |

Base * Row/Col/Diag Sum | 45 | 136 | 325 | 666 | 1225 | 2080 | 3321 |

The value 'Base * Row/Column/Diagonal Sum' is, of course, the total obtained
by summing all the numbers from 1 to Base Squared.

Agrippa then demonstrates that various divine, angelic and demonic names
drawn from the Hebrew Cabbala, and corresponding to the Planets and 'Lights',
sum to the same significant numbers using Gematria. That process was then
considered quite conventional.

These techniques and reasoning procedures are Pure Mathesis.

The subsequent procedures that Agrippa uses to derive Planetary Sigils
edge towards what might be termed Applied Mathesis. The instructions he
gives for engraving the squares, together with their related sigils, on
a metallic medium appropriate to the planet comprise the basis for a neat
Magical System well worth consideration by those who appreciate that sort
of thing. The system was summarised by Francis Barrett in his book 'The
Magus' published in 1801, where Agrippa's diagrams were reproduced, though
incompletely.

Mathesis is therefore the metaphysical counterpart of Mathematics. Among
20th Century Mathematicians there is a tendency to neglect that aspect
of the subject. The notable exception is Kurt Godel. His major theorem,
in a loose nutshell, says that within any mathematical system there are
things which will always defy proof; something which makes many boffins
uncomfortable.

PYTHAGORAS

Pythagoras can rightly be regarded as the originator of Mathematics. By the standards of the ancient world he was an unusual individual. Regarded by many followers as an incarnation of the Hyperborean Apollo, he was a formidable intellect, a gifted musician, an adept of several different mystery schools, and allegedly possessed of miraculous powers.

He was bom in the fifty-fourth Olympiad, that is 569 BC in conventional
notation, on the Island of Samos. His parents appear to have been Phoenician,
and the lad was fortunate to have his education entrusted to Pherikides
of Syros. He appears to have lived on Mainland Phoenicia and is traditionally
numbered among the Seven Sages of Antiquity (the intellectual equivalent
of the Seven Wonders of the World). Pythagoras almost certainly formed
views on cosmology and re-incarnation early in his life with influence
from Pherekides.

As a young man Pythagoras travelled to Egypt, apparently with a letter
of introduction from Polycrates, the Tyrant of Samos, seeking admission
as a neophyte to the priesthood of Memphis. They tumed him down, quite
probably for no reason other than dislike of foreigners, for which the
Egyptians were notorious. They suggested however that he try the temples
at Diospolis (now in Isreal) where, after some arduous tests, he was admitted.
Pythagoras learned Heiroglyphic script, he was the first Greek to do so,
and stayed there for many years. Iamblichus writing in the 3rd century
EV states that Pythagoras was an Egyptian Priest for twenty years. It
was probably during this period that he acquired some eccentric personal
habits, among them an abhorrence for eating beans, and other dietary fads.

Pythagoras' sojourn in Egypt ended abruptly in 525 BC when Cambyses, King
of Persia, invaded, and took him as a prisoner to Babylon. He was there
for a further ten years, and studied with Zaratas, a Magus of the Zoroastrian
and Chaldean tradition. It seems probable that he may have been Zaratas'
slave.

By the time he was freed (or escaped) to return to his native Samos,
Pythagoras had acquired a considerable depth of learning in the major
philosophical traditions of the Middle East. He was also an Adept in the
mystery schools of Phoenicia, Egypt and Babylon.

He lived on Samos in a hermit's cave, doing odd bits of teaching work,
and made journeys to every oracle in Greece and Crete. During this time
he became quite well known in Greece, but his fellow Samians found his
personality eccentric and his method of teaching by riddles impenetrable.
For example he wore trousers supposedly to conceal a 'golden thigh', which
he would occasionally display. After some years and aged around 60, he
fell foul of the local politics and joined the emigration to Magna Graecia,
as the Greek colonies in Southem Italy were called.

There is some inconsistency in the ancient biographies on exact dates, but it seems likely that Pythagoras was established in Croton, present day Crotone on the 'instep' of Italy, by 508 BC. There he set up a scholastic community which was to have a profound influence on the development of Greek philosophy.

One of the problems facing any present day researcher attempting to piece together the detail of Pythagoras' own original work lies in the lack of contemporary written material. Pythagoras was insistent on strict secrecy, and nothing committed to permanent record has survived, other than some poetry - 'The Golden Verses' - the origins of which are suspect.

Many discoveries in the fields of acoustics, harmonics, geometry, and astronomy are directly attributable to the Pythagorean School, and one of the most durable concepts was that of Numbers as divine entities which were able to exert influence on terrestrial events through the relative movements of their correspondent planets. Much of the theory is written up by Agrippa (op cit), who also published comprehensive tables of correspondences which are in part a precursor to Crowley's book 777.

Active Pythagorean schools were in existence until the sixth century. The last, in Alexandria, was suppressed by the Byzantine Emperor Justinian in the course of a Christist pogrom against philosophy and paganism. The refugees fled East to Persia, where Zoroastrianism continued to flourish until its suppression by Islam between the 8th and 10th centuries.

There is little doubt that from those times forward, science, mathematics and philosophy were dangerous fringe occupations throughout the Christist Empire. It can be argued that Christist intolerance was responsible for a hiatus of nearly 1000 years in the development of human knowledge, aided and abetted by Pythagorean secrecy. For example, it is now generally accepted that the earth is spherical, and that it orbits the sun along with the other planets. It seems certain that the Pythagoreans understood this, both Copernicus and Kepler acknowledge it in their writings, but the development of calculus, as a means of modelling the mechanics of the solar system, had to wait until the 17th century.

Christist intellectual fascism was the main reason for the delay, in my view. Copernicus strikes a distinctly paranoid tone in the introduction to his major work 'De Revolutionibus Orbium Coelestium' (1543), and there is an element of Russian Roulette in the careers of many serious philosophers of that period. Followers of Copernicus, like Galileo, were subjected to the inquisition, and Giordano Bruno was burned at the stake in Rome by Papal order of Clement VIII in 1600, with his tongue in a gag. Newton was fortunate to be working in the more open minded climate of Cambridge.

Those who have developed the concepts of Mathesis and its applications have, arguably, fared even worse. Any resurgence of the Gnostic tradition has been suppressed, sometimes with exceptional brutality, as in the cases of the pogroms against the Cathars and the Templars. Nor is this something only of the dim and distant past.

Nontheless, in the past decade the practitioners of Mathematics and Mathesis alike have been making some progress in an area which both have shunned for centuries.

CHAOS!

At around the time in the late 1970s when Peter Carroll first published
his Rites of Chaos, a mathematician at the IBM Research Centre at Yorktown
Heights, Professor Benoit Mandlebrot, was working to devise a performance
test for new computer designs. Mandlebrot dusted off some work on iteration
theory by the French mathematician Gaston Julia which had been published
in 1918. This had been consigned to obscurity as an oddball involving
horrendous calculation - just what Mandlebrot was looking for. He appears
to have adapted Julia's work to his own task, set the computer to work;
he was rewarded with mankind's first sight of an exquisitely intricate
geometrical pattem - the Set of Points which bears his name:

It is comparatively easy to generate the set, even with a home computer;
but the number crunching involved is horrendous and, depending on processor
speeds the job can take from minutes to hours, though it can be reduced
to seconds with modern parallel processors.

The figure drawn by the machine intrigued Mandlebrot, he set it to work
magnifying some of the boundary regions. It seemed that the deeper he
looked into the structure of the set, the more intricate it became. Typical
illustrations are to be found in his published work, and that of the many
computer graphic artists who have built on his techniques.

Mandlebrot found himself staring at tendrils and whorls, at sea-horses and bottomless holes, and at minatures of the whole Mandlebrot Set itself. The set had the peculiar property of self similarity.

Self similar sets fall within the domain of Fractal Geometry. Loosely put, a self similar pattern is one which looks more or less the same however much you magnify it, for example a snow-flake, and self similar structures are generally referred to as being Fractal in character. Examples of fractal forms are to be found all around us and in us. Clouds, flowing water, mountain landscapes, the flames of a fire, our own cardio-vascular system: all exhibit fractal characteristics.

Fractal geometry is the geometry of the chaos of natural form, and Mandlebrot had stumbled on the means of modelling its pattems mathematically.

Through this decade Chaos Mathematics has been generating debate in the scientific community in exactly the same way as Chaos Magic has challenged traditional thinking in occult circles. This is, without doubt, one of those co-incidences which frequently spring from any formless array, such as the set of all human thoughts.

Mathematics has, from Newton's time, been able to model a subset of real world mechanical situations. For example, the motion of a pendulum can be concisely described; but make the pendulum bob of steel and put a couple of magnets beneath it and the behaviour becomes non-linear or chaotic; at that point conventional mathematics gives up. It also gives up on fluid dynamics, that includes water flowing through pipes and weather modelling. Non-linearities arising from the interaction of more than two particles are responsible for the failure of quantum mechanics to be precise about the configuration of any atom other than Hydrogen.

Barely a week goes past without a piece in Nature or New Scientist on
some freshly discovered application of Chaos Mathematics (or Non-Linear
Dynamics as they prefer, more respectably, to call the subject). Geologists
use it to model seismic pattems; meteorologists to model atmospheric movements;
biologists to model animal populations; astro-physicists to model the
structure and evolution of galaxies.

The power of Chaos Mathematics as a modeling tool is twofold. Firstly
it can account for the existence of isolated ordered structures in an
disordered or chaotic matrix; for example, the existence of a large Red
Spot for many hundreds of years amid the swirling turmoil of Jupiter's
atmosphere, or a smoke ring. Secondly it can make limited predictions
of the future behaviour of chaotic phenomena. The predictions are limited
because of the so called Butterfly Effect.

The future behaviour of any non-linear function is highly sensitive to the initial conditions. This extreme sensitivity establishes the process by which the tiny eddies set up by a butterfly, flapping its wings at some critical location in the Carribean, can be translated within a week into a hurricane in London. The mathematics describing the process are unassailable, but the implications of the results are closer to Mathesis than to conventional Mathematics.

When Agrippa gives instructions for manipulating a sigil to effect some
action at a distance, it could be said that he is doing no more than specifying
a starting point for some 'Butterfly' process. The Dalai Lama's Oracle,
and most Rain-dancers would probably feel happy with that concept as well.
Thus Chaos re-unites Mathematics and Mathesis, but not necessarily in
a way that Pythagoras would have expected.

REFERENCES & RECOMMENDED FURTHER READING:

Agrippa von Netysheim, H. Cornelius De Occulta Philosophia 1533

Barrett, Francis The Magus or Celestial Intelligencer 1801

Carroll, Peter Liber Null 1978

Copemicus, Nicolaus De Revolutionibus Orbium Coelestium English Translation
by Charles Glenn Wallis 1543 1952

Crowley, Aleister Liber 777 1909

Gleik, James Chaos - Making a New Science 1987

Gorman, Peter Pythagoras - A Life 1979

Hofstadter, Douglas R Godel, Escher, Bach 1979

Mandlebrot, Benoit The Fractal Geometry of Nature 1982

Peitgen, H-O & Richter, P H The Beauty of Fractals 1986

Peitgen, H-O & Saupe, D The Science of Fractal Images 1988

Singmaster, David Sources in Recreational Mathematics (sixth edition)
1993