Royal Numerology

According to tradition, the recognition of mathematics as a subject worthy of study in its own right, regardless of its utility for practical matters, was achieved some 2600 years ago by Thales, who lived on the Asiatic coast of the Aegean Sea. Some time later the followers of Pythagoras (who lived in one of the Greek colonies that had been established in Italy) proclaimed that the entire universe was governed by numbers. The numbers they referred to were integers (whole numbers) or ratios of integers. They considered that all things were built up from individual blocks, which they called atoms. Because atoms are distinct units, they are countable. This meant to the Pythagoreans that geometry could be looked on as a branch of arithmetic.

It was soon realized, however, that there are numbers, such as tt and V2, that could not be expressed as the ratio of integers. These numbers were therefore called irrational numbers. This caused quite a problem because it meant that many triangles could not be built up of atoms. An isosceles right triangle, for example, might have an integer number of atoms along each of its sides. Its hypotenuse would have V2 as many atoms, but this is impossible because V2 is irrational and any integer multiplied by V2 is also irrational and therefore cannot represent a whole number of atoms. According to legend, the Pythagoreans found the existence of irrational numbers rather disturbing and attempted to suppress the knowledge of their existence.1

'If V2 were exactly equal to 1.4, then V2 would equal the ratio of two whole numbers, 14 and 10, that is 14/10. If so, then an isosceles right triangle with 10 atoms on two sides would have exactly 14 atoms on its hypotenuse. But actually V2 is slightly greater than 14/10, and so the triangle should have more than 14 (but fewer than 15) atoms on the hypotenuse. It is impossible to find two whole numbers, the ratio of which is exactly equal to V2. Of course, this problem disappears if the assumption that atoms are the basis of geometry is discarded. This may be one reason that the Greeks never developed the concept of atoms significantly. The concept lay essentially dormant and undeveloped for about 2000 years.

a be

Figure 2.1. Triangular and square numbers, (a) Triangular numbers as arrangements of individual elements, (b) Square numbers as arrangements of individual elements, (c) Square numbers as combinations of two successive triangular numbers in which the first triangle is turned over and fitted against the second triangle.

a be

Figure 2.1. Triangular and square numbers, (a) Triangular numbers as arrangements of individual elements, (b) Square numbers as arrangements of individual elements, (c) Square numbers as combinations of two successive triangular numbers in which the first triangle is turned over and fitted against the second triangle.

Nevertheless, it was possible to establish connections between arithmetic and geometry through such formulas as the Pythagorean theorem relating the three sides of a right triangle (A2 + B2 = C2), or the relationship between the circumference and radius of a circle (C = 2tjt), or the relationship between area and radius of a circle (A — ttr2), and so on. It was found that some numbers could be arranged in geometric patterns, and relationships were discovered between these patterns. For example, "triangular" numbers are shown in Fig. 2.1a and "square" numbers in Fig. 2.1b. The combination of any two successive triangular numbers is a square number, as shown in Fig. 2.1c. The Pythagoreans were said to believe so strongly in the significance of mathematics that they established a religious cult based on numbers.

The assignment of significance to numbers was not, and is not, limited to the Greeks. According to the Jewish Cabala, the deeper meaning of some words can be found by adding up the numbers associated with their constituent letters. The resultant sum would be significant of a particularly subtle connotation of the word. Nowadays, there are lucky and unlucky numbers, such as 7 or 13. Card games and gambling games are based on numbers. Some buildings do not have a thirteenth floor, even though they may have a twelfth and a fourteenth floor. In nuclear physics, reference is made to magic numbers, in that nuclei having certain numbers of protons or neutrons are particularly stable, and their numbers are deemed magic. In atomic physics, at one time some effort was expended trying to find significance in the fact that the so-called fine structure constant seemed to have a value exactly equal to 1/137.

The ancient Greeks were also fascinated by the shapes of various regular figures and developed a hierarchy for ranking these shapes. For example, a square o a b c d e f

Figure 2.2. Geometrical shapes and symmetry, (a) Equilateral triangle, (b) The triangle rotated by 90°. (c) Rotation of the triangle by 120° from its original position to a new position, which is indistinguishable from its original position, (d) Square, (e) Rotation of the square by 45°. (f) Rotation of the square by 90° from its original position to a new position, which is indistinguishable from the original position.

may be considered to show a higher order of "perfection" than an equilateral triangle. If a square is rotated by 90° about its center, its appearance is unchanged (see Fig. 2.2). An equilateral triangle, on the other hand, must be rotated by 120° before it returns to its original appearance. A hexagon needs to be rotated by only 60° to preserve its appearance. An octagon needs only a 45° rotation; a dodecagon (12 sided figure) needs only a 30° rotation.

The more sides a regular figure has, the smaller the amount by which it must be rotated in order to restore its original appearance. In this sense, increasing the number of sides of a regular figure increases its perfection. As the number of sides of such a figure increases, it comes closer and closer in appearance to a circle, and so it is natural to look upon a circle as being the most perfect flat (or two-dimensional) figure that can be drawn. No matter how much or how little a perfect circle is rotated about its center, it maintains its original appearance. Its appearance is constant. It is important to note that perfection is identified with constancy—something that is perfect cannot be improved, so it must remain constant.2

The Greek concern with perfection and geometry set the tone for their approach to science. This is particularly illustrated in Plato's Allegory of the Cave. In this allegory, Plato (427-347 B.C.) envisioned humans as being like slaves chained together in a deep and dark cave, which is'dimly illuminated by a fire burning some distance behind and above them. Thereis a wall behind them and a wall in front of them. Their chains and shackles are such that they cannot turn around to see what is going on behind them. There are people moving back and forth on the other side of the wall behind them, holding various objects over their heads and making unintelligible (to the slaves) sounds and noises. The slaves can see only the shadows of these objects, as cast by the firelight on the wall in front of them, and can hear only the muffled sounds reflected from the wall in front of

2E. A. Abbott's charming little book, Flatland, describes a fictional two-dimensional world, peopled by two-dimensional figures, whose characters are determined by their shape. Women, for example, the deadlier of the species, are straight lines, and are therefore able to inflict mortal wounds, just like a thin sharp rapier. The wisest male, in accordance with the Greek ideal of perfection is, of course, a perfect circle.

Figure 2.2. Geometrical shapes and symmetry, (a) Equilateral triangle, (b) The triangle rotated by 90°. (c) Rotation of the triangle by 120° from its original position to a new position, which is indistinguishable from its original position, (d) Square, (e) Rotation of the square by 45°. (f) Rotation of the square by 90° from its original position to a new position, which is indistinguishable from the original position.

them. The slaves have seen these shadows all their lives, and thus know nothing else.

One day, one of the slaves (who later becomes a philosopher) is released from the shackles and brought out of the cave into the real world, which is bright and beautiful, with green grass, trees, blue sky, and so on. He is initially unable to comprehend what he sees because his eyes are unaccustomed to the bright light. Indeed, it is painful at first to contemplate the real world, but ultimately the former slave becomes accustomed to this new freedom. He has no desire to return to his former wretched state, but duty forces him to return to try to enlighten his fellow human beings. This is no easy task because he must become accustomed to the dark again, and must explain the shadows in terms of things the slaves have never seen. They reject his efforts, and those of anyone else like him, threatening such people with death if they persist (as indeed was decreed for Socrates, Plato's teacher). Plato insisted, however, that duty requires that the philosopher persist, despite any threats or consequences.

The first task of the philosopher is to determine the Reality, or the Truth, behind the way things appear. As a particular example, Plato pointed to the appearance of the heavens, which is the subject matter of astronomy. The Sun rises and sets daily, as does the Moon; and the Moon goes through phases on approximately a monthly schedule. The Sun is higher in the summer than in the winter; morning stars and evening stars appear and disappear. These are the "Appearances" of the heavens, but the philosopher (nowadays the scientist) must discover the true reality underlying these appearances—what is it that accounts for the "courses of the stars in the sky"? According to Plato, the true reality must be Perfect or Ideal, and the philosopher must look to mathematics, in particular, to geometry, to find the true reality of astronomy.

Most objects in the sky, such as the stars, appear to move in circular paths about the Earth as a center; and it is tempting to conclude that the truly essential nature of the motion of the heavenly objects must be circular, because the circle is the perfect geometrical figure. It does not matter whether all heavenly objects seem to move in circular paths or not; after all, human beings can perceive only the shadows of the true reality. The task of the philosopher (or scientist) is to show how the truly perfect nature of heavenly motion is distorted by human perceptions. Plato set forth as the task for astronomy the discovery of the way in which the motions of heavenly objects could be described in terms of circular motion. This task was called "saving the appearances." As will be seen, the goal of discovering the true reality is one of the major goals of science, even though the definition of true reality is now somewhat different from Plato's definition. This goal has strongly affected the way in which scientific problems are approached. In some cases it has led to major insights; in other cases, when too literally sought, it has been a severe handicap to the progress of science.

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