Is God a Mathematician_ - Mario Livio [10]
The fact that someone would find numbers fascinating is perhaps not surprising in itself. After all, even the ordinary numbers encountered in everyday life have interesting properties. Take the number of days in a year—365. You can easily check that 365 is equal to the sums of three consecutive squares: 365=102 + 112 + 122. But this is not all; it is also equal to the sum of the next two squares (365 132 + 142)! Or, examine the number of days in the lunar month—28. This number is the sum of all of its divisors (the numbers that divide it with no remainder): 28= 1 + 2 + 4 + 7 + 14. Numbers with this special property are called perfect numbers (the first four perfect numbers are 6, 28, 496, 8218). Note also that 28 is the sum of the cubes of the first two odd numbers: 28 = 13 + 33. Even a number as widely used in our decimal system as 100 has its own peculiarities: 100 = 13 + 23 + 33 + 43.
OK, so numbers can be intriguing. Still, one may wonder what was the origin of the Pythagorean doctrine of numbers? How did the idea arise that not only do all things possess number, but that all things are numbers? Since Pythagoras either wrote nothing down or his writings have been destroyed, it is not easy to answer this question. The surviving impression of Pythagoras’s reasoning is based on a small number of pre-Platonic fragments and on much later, less reliable discussions, mostly by Platonic and Aristotelian philosophers. The picture that emerges from assembling the different clues suggests that the explanation of the obsession with numbers may be found in the preoccupation of the Pythagoreans with two apparently unrelated activities: experiments in music and observations of the heavens.
To understand how those mysterious connections among numbers, the heavens, and music materialized, we have to start from the interesting observation that the Pythagoreans had a way of figuring numbers by means of pebbles or dots. For instance, they arranged the natural numbers 1, 2, 3, 4,…as collections of pebbles to form triangles (as in figure 1). In particular, the triangle constructed out of the first four integers (arranged in a triangle of ten pebbles) was called the Tetraktys (meaning quaternary, or “fourness”), and was taken by the Pythagoreans to symbolize perfection and the elements that comprise it. This fact was documented in a story about Pythagoras by the Greek satirical author Lucian (ca. AD 120–80). Pythagoras asks someone to count. As the man counts “1, 2, 3, 4,” Pythagoras interrupts him, “Do you see? What you take for 4 is 10, a perfect triangle and our oath.” The Neoplatonic philosopher Iamblichus (ca. AD 250–325) tells us that the oath of the Pythagoreans was indeed:
Figure 1
I swear by the discoverer of the Tetraktys,
Which is the spring of all our wisdom,
The perennial root of Nature’s fount.
Why was the Tetraktys so revered? Because to the eyes of the sixth century BC Pythagoreans, it seemed to outline the entire nature of the universe. In geometry—the springboard to the Greeks’ epochal revolution in thought—the number 1 represented a point •, 2 represented a line , 3 represented a surface , and 4 represented a three-dimensional tetrahedral solid . The Tetraktys therefore appeared to encompass all the perceived dimensions of space.
But that was only the beginning. The Tetraktys made an unexpected appearance even in the scientific approach to music. Pythagoras and the Pythagoreans are generally credited with the discovery that dividing a string by simple consecutive integers produces harmonious and consonant intervals—a fact figuring in any performance by a string quartet. When two similar strings are plucked simultaneously, the resulting sound is pleasing when the lengths of the strings are in simple proportions. For instance, strings of equal length (1:1 ratio) produce a unison; a