After a lifetime dedicated to mathematics, Mandelbrot became the oldest professor at Yale. One of the great things about Mandelbrot’s work was the way in which his fractals extend throughout nature. His demonstration of them appearing in natural environments in so many ways prompts the thoughtful reader to wonder about the extent to which fractals are numerological as well as scientifically mathematical.
Andrew Wiles, working at Princeton University in the 1990s, finally succeeded in proving Fermat’s Last Theorem from the seventeenth century, stating that the equation an+bn=cn will only be possible if n is not greater than 2. The gap of 300 years between its formulation and its proof emphasize the sequential nature of mathematical history. Contemporary mathematicians so often depend upon, and build upon, the work of their founding fathers.
3
History of Numerology
When we were distinguishing between mathematics and numerology in the first chapter, an important difference was that numerology was concerned with magical or mystical bonds between numbers and their environment. Certain significant numbers are believed by numerologists to have the power to make things happen or to predict what will happen to particular people in given circumstances at specific dates and times. The point was that numerologists believe that numbers have powers over, above, and beyond what might be termed their everyday use to calculate solutions using normal scientific mathematics. The earliest numerologists felt that, as potent as numbers were for solving calculation problems such as which army was likely to win a battle, how to build a stable pyramid, how much food was needed to feed a given population for a prescribed period, or how many days it would take to cover a given distance on foot, they could do other, stranger, more powerful things as well. Numerology parts company with scientific mathematics when numerologists argue that certain numbers are mysteriously influential, dominant, and predictive. Numbers, to a numerologist, are what spells and incantations are to a magician.
But how and when did these numerological beliefs begin?
There are expert historians and pre-historians who would argue that the history of numerology goes back to the ancient carvings found on bones and antlers as well as to the ancient drawings and paintings on cave walls. It is widely agreed by pre-historians that ancient cave drawings and paintings were often intended to act as a form of sympathetic magic: draw an edible animal being slain, and such an animal will be influenced by the magic in the painting to be available and vulnerable to the huntsmen on whose skill the tribal food supply depended. Were primitive attempts to use numbers cut into sticks, bones, and antlers an allied form of sympathetic magic? Did 3 notches mean that 3 animals were needed to feed the huntsman’s family? Did 4 notches mean that they would be found in 4 days’ time? Did 5 notches mean that the prey would be encountered after a 5 days’ journey? This is pure speculation at this distance in time and culture from the hunter-gatherers who carved the notches, but it’s a real possibility all the same.
The ancient systems of everyday mathematics — what we might term “ordinary” mathematics, used for simple, basic calculating — were examined in detail in the previous chapter. Very probably, those earliest counting and calculating systems had mysterious numerological purposes as well as scientific ones. Religion and magic seem to have played an integral part in human culture from the earliest prehistoric times, and on through the Stone Age, Bronze Age, and Iron Age. As early civilizations dawned there is evidence that something akin to modern numerology, something that might well have been the ancestor of contemporary numerology, originated in ancient Babylon and in ancient Egypt. The earliest Hebrew systems began in the same Chaldean area. Different versions of numerology were also being practised in Japan, China, and India. In addition, it was growing up and developing more complex and sophisticated forms in ancient Greece and Rome.
On the far side of the Atlantic, indigenous Americans like the Hopi people were exploring numerology for themselves and allocating significant values to numbers. It is particularly interesting that wise old Hopi elders, like Floyd Red Crow Westerman, who are expert numerologists, translate the 2012 date as “5” by adding its digits together, and in their system “5” signifies momentous change: the end of one era and the start of another. It is another interesting aspect of ancient Hopi numerology that there are 4 elements: earth, air, fire, and water, but there is another element — spirit — that transcends the other 4 and completes them.
In Norse mythology, numerology is recognizable because of the focus on magical numbers “3” and “9,” and their product, “27,” which is the cube of 3. There are, for example, 3 very different giant races: fire giants, frost giants, and mountain giants. The Norse universe began with 3 entities: the great cow, known as Audhumia; the primordial giant, Ymir; and Odin’s grandfather, Buri, who was the first of the gods. A giant named Hrungnir had a stone heart that was 3-sided, and it is noteworthy in this context that the valknut symbol consists of 3 interlocking triangles with 9 corners. The name valknut means “The Knot of the Dead” and the symbol was carried by warriors as a talisman — those devoted to Odin who died bravely in battle would be taken to Valhalla, the paradise of Norse warriors.
The great world tree Yggdrasil has 3 roots and joins 9 worlds together. There are 3 holy wells under Yggdrasil’s roots. The dreaded wolf, Fenrir (or Fenris), was secured with 3 chains, but only the final one held. Gullveig is killed 3 times and reborn 3 times. Ragnarok is heralded by the crowing of 3 cockerels: 1 for the gods, 1 for the dead, and 1 for the giants. The rainbow bridge, known as Bifrost, has 3 colours and 3 names. Nine magical charms were given to Svipdag by his mother, the enchantress Groa, and 9 beautiful maidens sit at the knees of Menglod. It is also significant that Aegir has 9 daughters. There are 9 locks securing the chest that belongs to Laegjarn. In another piece of Norse mythology, one of the magical fires can be lit only if 9 different types of wood are used. Thrivaldi was a giant with 9 heads.
The ancient Egyptians had a profound influence on the history of numerology. The Egyptian goddess Seshat had 2 major attributes. She was revered firstly as the inspirer of writings, and seen from this perspective as a sort of divine archivist. She was also honoured as a kind of instructor-goddess of building and construction. Numbers were another vital part of her work, which led to her being given the title of “the Enumerator.”
To get inside ancient Egyptian thought, it is necessary to look at everything in the universe from the point of view that it is alive and animated. Objective twenty-first-century science has the underlying assumption that when we work in a laboratory with sulphur, lithium, magnesium, bromine, and chlorine, we are working with something dead or inanimate. Egyptian thinkers did not look at their environment that way 5,000 years ago. They might not have been familiar with the term animism, as such, but they still subscribed to the idea that everything in their environment was impregnated with living forces, and that such forces energized it and gave it its characteristics.
On this basic concept of an animated universe, the Egyptians based their numerology — their magical and mysterious ideas about numbers. For them, numbers were not mere quantities, or units of fruit, vegetables, and meat. Numbers were more than the length of a line or the area of a pyramid’s base. Numbers were more than measurements of the volume of corn stored in a granary (thanks to Joseph’s inspiration and prudence). Numbers were expressions of the world around them and the animated spirit that it contained. If a waterfall was seen as alive and powerful, numbers expressed its power in terms of the volume of water that roared down it. The plunging of a war-horse; the spinning wheels of a chariot; the great block of stone being dragged by sweating slaves: these were all alive. These were animated. Numbers could clothe objects and events and make them more comprehensible. Numbers — in the hands of a skilled numerologist — could actually control the environment.
For the ancient Egyptians, numbers had personalities. They were as alive and as powerful as the objects they measured and quantified. They were male and female, not neutral.
Plutarch (45–120 AD) had very interesting comments to make on this idea of gender in the natural and mathematical universes. Born at Chaeronea in Boeotia, in the centre of Greece, he studied in Athens and then moved to Rome as a teacher of philosophy. Both Trajan and Hadrian liked and admired him. In his work, Moralia: Volume Five, Plutarch referred to the genders of the parts of a 90-degree triangle. The “3, 4, 5” triangle was an essential