Against the Gods. Bernstein Peter L.

in the future.

      We can appreciate the full measure of Fibonacci’s achievement only by looking back to the era before he explained how to tell the difference between 10 and 100. Yet even there we shall discover some remarkable innovators.

      Primitive people like the Neanderthals knew how to tally, but they had few things that required tallying. They marked the passage of days on a stone or a log and kept track of the number of animals they killed. The sun kept time for them, and five minutes or a half-hour either way hardly mattered.

      The first systematic efforts to measure and count were undertaken some ten thousand years before the birth of Christ.27 It was then that humans settled down to grow food in the valleys washed by such great rivers as the Tigris and the Euphrates, the Nile, the Indus, the Yangtse, the Mississippi, and the Amazon. The rivers soon became highways for trade and travel, eventually leading the more venturesome people to the oceans and seas into which the rivers emptied. To travelers ranging over longer and longer distances, calendar time, navigation, and geography mattered a great deal and these factors required ever more precise computations.

      Priests were the first astronomers, and from astronomy came mathematics. When people recognized that nicks on stones and sticks no longer sufficed, they began to group numbers into tens or twenties, which were easy to count on fingers and toes.

      Although the Egyptians became experts in astronomy and in predicting the times when the Nile would flood or withdraw, managing or influencing the future probably never entered their minds. Change was not part of their mental processes, which were dominated by habit, seasonality, and respect for the past.

      About 450 BC, the Greeks devised an alphabetic numbering system that used the 24 letters of the Greek alphabet and three letters that subsequently became obsolete. Each number from 1 to 9 had its own letter, and the multiples of ten each had a letter. For example, the symbol “pi” comes from the first letter of the Greek word “penta,” which represented 5; delta, the first letter of “deca,” the word for 10, represented 10; alpha, the first letter of the alphabet, represented 1, and rho represented 100. Thus, 115 was written rho–deca–penta, or ρδπ. The Hebrews, although Bemitic rather than Indo-European, used the same kind of cipher-alphabet system.28

      Handy as these letter–numbers were in helping people to build stronger structures, travel longer distances, and keep more accurate time, the system had serious limitations. You could use letters only with great difficulty – and almost never in your head – for adding or subtracting or multiplying or dividing. These substitutes for numbers provided nothing more than a means of recording the results of calculations performed by other methods, most often on a counting frame or abacus. The abacus – the oldest counting device in history – ruled the world of mathematics until the Hindu-Arabic numbering system arrived on the scene between about 1000 and 1200 AD.

      The abacus works by specifying an upper limit for the number of counters in each column; in adding, as the furthest right column fills up, the excess counters move one column to the left, and so on. Our concepts of “borrow one” or “carry over three” date back to the abacus.29

      Despite the limitations of these early forms of mathematics, they made possible great advances in knowledge, particularly in geometry – the language of shape – and its many applications in astronomy, navigation, and mechanics. Here the most impressive advances were made by the Greeks and by their colleagues in Alexandria. Only the Bible has appeared in more editions and printings than Euclid’s most famous book, Elements.

      Still, the greatest contribution of the Greeks was not in scientific innovation. After all, the temple priests of Egypt and Babylonia had learned a good bit about geometry long before Euclid came along. Even the famous theorem of Pythagoras – the square of the hypotenuse of a right triangle is equal to the sum of the square of the other two sides – was in use in the Tigris-Euphrates valley as early as 2000 BC.

      The unique quality of the Greek spirit was the insistence on proof. “Why?” mattered more to them than “What?” The Greeks were able to reframe the ultimate questions because theirs was the first civilization in history to be free of the intellectual straitjacket imposed by an all-powerful priesthood. This same set of attitudes led the Greeks to become the world’s first tourists and colonizers as they made the Mediterranean basin their private preserve.

      More worldly as a consequence, the Greeks refused to accept at face value the rules of thumb that older societies passed on to them. They were not interested in samples; their goal was to find concepts that would apply everywhere, in every case. For example, mere measurement would confirm that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. But the Greeks asked why that should be so, in all right triangles, great and small, without a single exception to the rule. Proof is what Euclidean geometry is all about. And proof, not calculation, would dominate the theory of mathematics forever after.

      This radical break with the analytical methodologies of other civilizations makes us wonder again why it was that the Greeks failed to discover the laws of probability, and calculus, and even simple algebra. Perhaps, despite all they achieved, it was because they had to depend on a clumsy numbering system based on their alphabet. The Romans suffered from the same handicap. As simple a number as 9 required two letters: IX. The Romans could not write 32 as III II, because people would have no way of knowing whether it meant 32, 302, 3020, or some larger combination of 3, 2, and 0. Calculations based on such a system were impossible.

      But the discovery of a superior numbering system would not occur until about 500 AD, when the Hindus developed the numbering system we use today. Who contrived this miraculous invention, and what circumstances led to its spread throughout the Indian subcontinent, remain mysteries. The Arabs encountered the new numbers for the first time some ninety years after Mohammed established Islam as a proselytizing religion in 622 and his followers, united into a powerful nation, swept into India and beyond.

      The new system of numbering had a galvanizing effect on intellectual activity in lands to the west. Baghdad, already a great center of learning, emerged as a hub of mathematical research and activity, and the Caliph retained Jewish scholars to translate the works of such pioneers of mathematics as Ptolemy and Euclid. The major works of mathematics were soon circulating throughout the Arab empire and by the ninth and tenth centuries were in use as far west as Spain.

      Actually, one westerner had suggested a numbering system at least two centuries earlier than the Hindus. About 250 AD, an Alexandrian mathematician named Diophantus wrote a treatise setting forth the advantages of a system of true numbers to replace letters substituting for numbers.30

      Not much is known about Diophantus, but the little we do know is amusing. According to Herbert Warren Turnbull, a historian of mathematics, a Greek epigram about Diophantus states that “his boyhood lasted l/6th of his life; his beard grew after l/12th more; he married after l/7th more, and his son was born five years later; the son lived to half his father’s age, and the father died four years after his son.” How old was Diophantus when he died?31 Algebra enthusiasts will find the answer at the end of this chapter.

      Diophantus carried the idea of symbolic algebra – the use of symbols to stand for numbers – a long way, but he could not quite make it all the way. He comments on “the impossible solution of the absurd equation 4 = 4x + 20. “32 Impossible? Absurd? The equation requires x to be a negative number: −4. Without the concept of zero, which Diophantus lacked, a negative number is a logical impossibility.

      Diophantus’s remarkable innovations seem to have been ignored. Almost a millennium and a half passed before anyone took note of his work. At last his achievements received their due: his treatise

27

The background material presented here comes primarily from Hogben, 1968, Chapter I.

28

See Hogben, 1968, p. 35; also Eves, 1983, Chapter I.

29

See Hogben, 1968, p. 36 and pp. 246–250.

30

The background material on Diophantus is from Turnbull, 1951, p. 113.

31

Ibid., p. 110.

32

Ibid., p. 111.