Mathematical Notes: Zero tolerance and Saracen magic

MATHEMATICIANS HAVE kept the secret to themselves for too long. Theirs is the most beautiful of the arts, and the most accessible, yet bad teaching has made it seem boring at best and more often frightening. How many of us still break out in a cold sweat at the mere sight of an equation?

One way to redress the balance is to take a closer look at zero. To follow the evolution of this most important of numbers from no more than a punctuation mark among its Sumerian inventors, through its role as Kingmaker - turning a humble 1 into 10, 100, 1,000 - to what we now know is its power to generate all numbers and to reveal the crests and troughs in the waves of change that carry us along. Watching the way people have struggled to understand it humanises mathematics and shows how closely allied its substance is to that of common thought.

Astonishingly, the Greeks didn't have a word for it. It seems extraordinary today that Archimedes could have tried to calculate the number of grains of sand in the universe, without having access to powers of 10 (and hence to zero).

Zero seems to us a neutral concept, but nothing can be further from the truth. Depicted by the Mayans as a tattooed man with his head thrown back, zero was crucial as an element in a counting system which involved horrific rituals intended to stave off the end of time. William of Malmesbury cursed zero and algebra and the new-fangled Arabic numerals as "dangerous Saracen magic". It both disturbed and inspired poets, philosophers, and theologians, as well as mathematicians through the centuries.

In the end, however, this least of numbers conquered our superstitions by letting us add and multiply swiftly (thanks to the positional way of writing numbers that depended on it); it let us balance our books on its fulcrum, and opened up the world of science by letting us set up and solve the equations describing nature. And the recurrent need to know when the world will end by discovering when it began itself centres on grasping the elusive nature of zero.

Mathematics - commonly pictured as inhuman and abstract - is actually shot through with vivid personalities, as in the heated battles between Newton and his German rival Leibniz. Each wanted to glorify God by revealing His handiwork in shaping the golden frame of the universe, and each thought the other to have diminished that glory by his views of how things move. Was space made up of particles so small as to count for almost nothing, or was motion too smooth to roll forward on any particles at all? After hundreds of years the battle between them has yet to be resolved.

The light shines from this oval mirror of zero on distant spaces and on the interior of our thought. Is there a real zero out there, a true vacuum somewhere or a beginning to time or a dead centre of the universe? And what about the scales we have made to measure everything from temperature and pressure to altitude: are the zeroes on them discovered in nature or invented by us? Why does zero sometimes dominate our imaginations with fears of loss and meaninglessness, scorn on no-accounts and longings to be anonymous, and at other times with hopes of ever-renewed beginnings or revelations that the calm centre of things is as empty of the positive as of the negative? And behind thse mathematical and psychological disguises of zero lies Zero itself, the puzzle of existence, the vanishing point of our thoughts about not just what we are, but why.

Is that nothing or everything inside its slender ring?

Robert Kaplan is the author of `The Nothing That Is: a natural history of zero' (Penguin Press, pounds 12.99)