Against the Gods. Bernstein Peter L.

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      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, СКАЧАТЬ