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By Root 1084 0

them, finally: the ordered chaos that seems to guarantee the presence of meaning. It seemed like a task for mathematicians, anyway, and finally it was. They recognized geometric progressions, tables of powers, and even instructions for computing square roots and cube roots. Familiar as they were with the rise of mathematics a millennium later in ancient Greece, these scholars were astounded at the breadth and depth of mathematical knowledge that existed before in Mesopotamia. “It was assumed that the Babylonians had had some sort of number mysticism or numerology,” wrote Asger Aaboe in 1963, “but we now know how far short of the truth this assumption was.”♦ The Babylonians computed linear equations, quadratic equations, and Pythagorean numbers long before Pythagoras. In contrast to the Greek mathematics that followed, Babylonian mathematics did not emphasize geometry, except for practical problems; the Babylonians calculated areas and perimeters but did not prove theorems. Yet they could (in effect) reduce elaborate second-degree polynomials. Their mathematics seemed to value computational power above all.

That could not be appreciated until computational power began to mean something. By the time modern mathematicians turned their attention to Babylon, many important tablets had already been destroyed or scattered. Fragments retrieved from Uruk before 1914, for example, were dispersed to Berlin, Paris, and Chicago and only fifty years later were discovered to hold the beginning methods of astronomy. To demonstrate this, Otto Neugebauer, the leading twentieth-century historian of ancient mathematics, had to reassemble tablets whose fragments had made their way to opposite sides of the Atlantic Ocean. In 1949, when the number of cuneiform tablets housed in museums reached (at his rough guess) a half million, Neugebauer lamented, “Our task can therefore properly be compared with restoring the history of mathematics from a few torn pages which have accidentally survived the destruction of a great library.”♦

In 1972, Donald Knuth, an early computer scientist at Stanford, looked at the remains of an Old Babylonian tablet the size of a paperback book, half lying in the British Museum in London, one-fourth in the Staatliche Museen in Berlin, and the rest missing, and saw what he could only describe, anachronistically, as an algorithm:

A cistern.

The height is 3,20, and a volume of 27,46,40 has been excavated.

The length exceeds the width by 50.

You should take the reciprocal of the height, 3,20, obtaining 18.

Multiply this by the volume, 27,46,40, obtaining 8,20.

Take half of 50 and square it, obtaining 10,25.

Add 8,20, and you get 8,30,25.

The square root is 2,55.

Make two copies of this, adding to the one and subtracting from the other.

You find that 3,20 is the length and 2,30 is the width.

This is the procedure.♦

“This is the procedure” was a standard closing, like a benediction, and for Knuth redolent with meaning. In the Louvre he found a “procedure” that reminded him of a stack program on a Burroughs B5500. “We can commend the Babylonians for developing a nice way to explain an algorithm by example as the algorithm itself was being defined,” said Knuth. By then he himself was engrossed in the project of defining and explaining the algorithm; he was amazed by what he found on the ancient tablets. The scribes wrote instructions for placing numbers in certain locations—for making “copies” of a number, and for keeping a number “in your head.” This idea, of abstract quantities occupying abstract places, would not come back to life till much later.

Where is a symbol? What is a symbol? Even to ask such questions required a self-consciousness that did not come naturally. Once asked, the questions continued to loom. Look at these signs, philosophers implored. What are they?

“Fundamentally letters are shapes indicating voices,”♦ explained John of Salisbury in medieval England. “Hence they represent things which they bring to mind through the windows of the eyes.” John served