AD) enjoyed, as his name implies, a rich mixture of Greek and Roman culture and learning. His mathematical works, principally on astronomy, were honoured with the Arabian title Almagest, meaning “The Greatest.”
Diophantus (200–284 AD) did remarkable early work on number theory, and his book Arithmetica provided a great deal of inspiration for Pierre Fermat (1601–1665). Fermat’s Last Theorem states that if we call 3 positive integers a, b, and c, then the equation an+bn=cn will only be possible if n is not greater than 2. Proving it became a leading mathematical problem for centuries, and even made its way into the Guinness Book of Records!
A brilliant Persian mathematician named Al-Khowarizmi (780–840) was also a gifted scientist and astronomer. His additional interest in astrology made him something of a numerologist — like Pythagoras — as well as a scientific mathematician. The modern word algebra was transliterated from his book Hisab al-jabr w’al-muqabala, where it
was rendered as “al-jabr.”
Francesco Pellos (1450–1500) was the inventor of the decimal point — a tremendously useful part of contemporary mathematics. The gifted Scots theologian John Napier (1550–1617) indulged in mathematics more or less as a hobby when he wanted a break from theology. He was largely responsible for creating logarithms, which were perfected by Henry Briggs. At around the same period, Sir Isaac Newton (1642–1727) published his epoch-making Principia — a masterpiece of mathematics and science. There is some justification for those who regard him as the greatest scientist who has yet lived. His work on gravitation and the 3 laws of motion are unforgettable.
Another important milestone in the history of mathematics was William Jones’s (1675–1749). He used the Greek symbol “π” to show the result of dividing the circumference of a circle by its diameter. This feat was published in his book, New Introduction to Mathematics, in 1706.
Calculus was the particular brainchild of the Italian maths genius Maria Agnesi (1718–1799). Her famous textbook on it, Istituzioni Analitiche, was an authoritative teaching aid on calculus for many years.
David Hilbert (1862–1943) was one of the most outstanding mathematical leaders in the late nineteenth and early twentieth centuries. His great contributions were to invariant theory and the axiomatization of Euclid’s geometry. One of his other theories that was essential to functional analysis was named after him as the theory of Hilbert spaces. Tragically, he and a number of other brilliant academic mathematicians at the University of Göttingen were persecuted by the Nazis.
Benoit B. Mandelbrot (1924–2010) was a superb French-American mathematician whose name is associated with the mathematical idea of fractals. The word comes from the Latin fractus, which means broken. A fractal could be described as a rough or fragmented geometric shape that can be divided repeatedly into smaller and smaller parts. These smaller parts resemble — not always exactly — the original larger whole. This characteristic is described as self-similarity. Fractals play a large part in such varied sciences as soil chemistry, seismology, and medicine. 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.
