Zero - Charles Seife [32]
The number of rabbits goes as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…; the number of rabbits you have in any given month is the sum of the rabbits that you had in each of the two previous months. Mathematicians instantly realized the importance of this series. Take any term and divide it by its previous term. For instance, 8/5 = 1.6; 13/8 = 1.625; 21/13 =1.61538…. These ratios approach a particularly interesting number: the golden ratio, which is 1.61803….
Pythagoras had noticed that nature seemed to be governed by the golden ratio. Fibonacci discovered the sequence that is responsible. The size of the chambers of the nautilus and the number of clockwise grooves to counterclockwise grooves in the pineapple are governed by this sequence. This is why their ratios approach the golden ratio.
Though this sequence is the source of Fibonacci’s fame, Fibonacci’s Liber Abaci had a much more important purpose than animal husbandry. Fibonacci had learned his mathematics from the Muslims, so he knew about Arabic numerals, including zero. He included the new system in Liber Abaci, finally introducing Europe to zero. The book showed how useful Arabic numerals were for doing complex calculations, and the Italian merchants and bankers quickly seized upon the new system, zero included.
Before Arabic numerals came around, money counters had to make do with an abacus or a counting board. The Germans called the counting board a Rechenbank, which is why we call moneylenders banks. At that time, banking methods were primitive. Not only did they use counting boards, they used tally sticks to record loans: a money value was written along the stick’s side, and it was split in two (Figure 16). The lender kept the biggest piece, the stock. After all, he was the stockholder.*
Figure 16: A tally stick
Italian merchants loved the Arabic numbers. They allowed the bankers to get rid of their counting boards. However, while businessmen saw their usefulness, the local governments hated them. In 1299, Florence banned Arabic numerals. The ostensible reason was that the numbers were easily changed and falsified. (A 0 could be turned into a 6 with a simple flourish of a pen, for instance.) But the advantages of zero and the other Arabic numerals were not so easily dispensed with; Italian merchants continued to use them, and even used them to send encrypted messages—which is how the word cipher came to mean “secret code.”
In the end the governments had to relent in the face of commercial pressure. The Arabic notation was allowed into Italy and soon spread throughout Europe. Zero had arrived—as had the void. The Aristotelian wall was crumbling, thanks to the influence of the Muslims and the Hindus, and by the 1400s even the staunchest European supporters of Aristotelianism had their doubts. Thomas Bradwardine, who was to become archbishop of Canterbury, tried to disprove atomism, Aristotle’s old nemesis. At the same time, he wondered whether his own logic was faulty, since he based his arguments on geometry, whose infinitely divisible lines automatically reject atomism. However, the battle against Aristotle was far from over. If Aristotle were to fall, the proof of God—a bulwark of the church—was no longer valid. A new proof was needed.
Worse yet, if the universe were infinite, then there could be no center. How could Earth, then, be the center of the universe? The answer was found in zero.
Chapter 4
The Infinite God of Nothing
[ THE THEOLOGY OF ZERO ]
And new philosophy calls all in doubt,
The element of fire is quite put out;
The sun is lost, and th’ earth, and no man’s wit
Can well direct him where to look for it….’
Tis all in pieces, all coherence gone;
All just supply, and all relation:
Prince, subject, Father, Son, are things forgot.
—JOHN DONNE, “AN ANATOMY OF THE WORLD”
Zero and infinity were at the very center of the Renaissance. As Europe slowly awakened from the Dark Ages, the void and the infinite—nothing and everything—would destroy the