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include the measurement of single linear dimensions, areas, volumes, structures, and changes. Further extensions of its definition include special fields such as algebra, trigonometry, statistics, and calculus. Mathematicians look for patterns and then base conjectures on the patterns they have observed. Mathematics lies at the heart of understanding the amazing complexities of the universe, from subatomic particles to the unimaginably vast spaces between the galaxies. Sir James Jeans once suggested that the universe is a thought in the mind of a great mathematician.

      In their earliest days, mathematics and numerology were practically indistinguishable. The power of numbers, and the power of those who understood more about numbers than most of their contemporaries, seemed close to magical. Numerology may be regarded as the fundamental idea that numbers, mathematical symbols, quantities, measurements, and statistical analyses have yet further powers than those which they display when we use them to solve mathematical problems. Inside their straightforward mathematical box, numbers and symbols answer questions that have been posed in numbers and symbols. Outside their mathematical box, numbers and symbols are believed to have powers far beyond mere calculating. Numerology suggests that certain digits can affect the universe around them and can influence human behaviour, success, and failure. To the mathematician, numbers and symbols solve problems of quantity, measurement, and numerical changes. To the numerologist, numbers and symbols are like the words of magical liturgies and the ritual movements of enchanters, sorcerers, and magicians. To the numerologist, numbers and symbols have the power to influence people and things in paranormal ways. They also have the power to predict likely outcomes.

      The Greek word μάθημα (mathema), from which the English word mathematics is derived, originally meant “knowledge, learning, or study.” Having studied patterns and derived conjectures from them, mathematicians look for proofs of their conjectures. There are several forms of mathematical proof, and a very basic example of the type called “direct proof” can be seen in the fact that the sum of any 2 even numbers will always be another even number. The term integer is used in mathematics to define a whole number with no fractional parts attached to it. Integers can be positive or negative: -3, -2, -1, 0, 1, 2, and 3 are all integers. The proof begins by calling any 2 even integers “x” and “y.” Because they are even — that is, they can be divided by 2 — it can be argued that x=2a and y=2b. Putting a number next to a letter in mathematics signifies that the number multiplies the letter adjacent to it. So, if x and y are both even, each of them will be made up of 2 other integers, which we call “a” and “b.” Suppose that we decide that x=8 and y=12. Then 8=2×a and 12=2×b. This means that a must be equal to 4 (8=2×4) and b must be equal to 6 (12=2×6). This then leads to the next step in the proof, which is the following equation:

      x+y=2a+2b or 8+12=2(4)+2(6)

      This can be written as 2(a+b), or 2(4+6). The “2” outside the bracket multiplies everything inside the bracket. What has been done so far shows that x+y has 2 as a factor, and the definition of an even integer is that it can be divided by 2. This then constitutes the mathematical proof that the sum of any 2 even integers is another even integer.

      It is an essential part of mathematics to set out to prove or disprove mathematical conjectures by using mathematical proofs like the one above. Unlike that basic example, it can sometimes take many years to establish an advanced mathematical proof.

      Two very great philosophical mathematicians, Giuseppe Peano (1858–1932) and David Hilbert (1862–1943), made outstanding contributions to mathematical proof theory and mathematical logic.

      Of even greater mathematical stature than Peano and Hilbert was the outstandingly brilliant Johann Carl Friedrich Gauss (1777–1855). He was justifiably known as Princeps mathematicorum, a Latin title meaning “Prince of Mathematicians” or “Foremost Mathematician.” Gauss was also known in his own day as the greatest mathematician since antiquity, and he himself referred to mathematics as “the queen of sciences.” He spread his academic and intellectual net widely,

       making massive contributions to astronomy, statistics, number theory, geophysics, and electrostatics. Gauss’s theories are at the forefront of the debate about whether mathematics is a science, and so much of that debate hinges on the definition of what constitutes a science as opposed to a field of knowledge.

      Sir Francis Bacon (1561–1626) combined philosophy with statesmanship and a deep knowledge of legal matters, but his contribution to science and the scientific method may be reckoned as his most worthwhile memorial.

      The Baconian method of studying science aimed to investigate the cause of a phenomenon, especially what made it different. He advised researchers and investigators to make lists of where the phenomenon occurred and where it did not occur. The lists were then ranked in order according to the strength of the influence of the phenomenon being studied. From Bacon’s underlying principles came the idea of natural science as distinct from other fields of knowledge and the various methods of making progress within them. The natural sciences were closely concerned with experimental procedures, and this gave rise to the controversy over whether mathematics could be classified as a science if it was capable of functioning without a reliance on experimentation. Einstein, who had an amazing gift for summarizing major facts succinctly, said that when mathematical laws referred to reality, they were not certain, and when they were certain, they didn’t refer to reality.

      Francis Bacon

      Sir Karl Raimund Popper (1902–1994), undoubtedly one of the greatest philosophers of science of all time, argued that mathematics was much closer to the natural sciences than it seemed. The outstanding physicist John Ziman (1925–2005) held the view that science is best defined as public knowledge, and must, therefore, include mathematics. Different philosophers and scientists hold varying views, but the weight of evidence, and the increasing importance of experimental mathematics, would favour the idea of including mathematics among the sciences. It is undeniable that it forms the solid rock foundation upon which many sciences are built, especially physics and chemistry.

      Mathematics and numerology can be thought of as originating rather like a pair of conjoined Siamese twins, but they have separated over the millennia. Just as academic mathematics can be seen to belong in the vicinity of the natural sciences, so numerology belongs in the vicinity of magic. Magic can be defined and described as what are believed to be methods of altering and manipulating the environment (people and things) by supernatural means. These so-called supernatural means can be thought to include knowledge of occult causes and effects that are unknown to science. It is an essential part of science to rely on observation and logical analysis; it is an essential part of magic to believe that there are inexplicable forces beyond logical cause and effect.

      There are ways in which magic resembles religion. Although heavily criticized by later thinkers, Sir James George Frazer (1854–1941), author of the monumental work on anthropology The Golden Bough (1890), worked out a succession in which the earliest peoples tried to control nature by using magic, which gave way first to religion and then to science. Fundamentally, Frazer argued, humanity had attempted to control the environment by using what were believed to be the inexplicable and illogical powers of magic. When these palpably failed so frequently, and were clearly seen to be ineffective, our remote ancestors turned to religion. When their magical activities — rituals, liturgies, spells, and incantations — did not control the environment, they thought that they must appeal to the gods to help them. In due course, as millennia passed, it became clear that appealing to the gods was also ineffectual. Along with the decline in religious belief came the realization that science could succeed far more often than either magic or religion. If making magical gestures and reciting curses failed to kill the enemy, and if prayers to divine beings also failed to halt the advance of the foe, then the invention and employment of superior weapons based on scientific principles would bring an opponent down. Where magic and prayer had failed to accomplish the desired results, scientifically designed Gatling and Maxim guns could cut the enemy down very effectively.

      There are contemporary magicians — numerologists among them — whose theory of magic regards it as some strange kind of bonding with largely unknown universal powers. They believe

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