ZERO - THE LENS

If you look at zero you see nothing; but look through it and you will see the world. For zero brings into focus the great, organic sprawl of mathematics, and mathematics in turn the complex nature of things. From counting to calculating, from estimating the odds to knowing exactly when the tides in our affairs will crest, the shining tools of mathematics let us follow the tacking course everything takes through everything else - and all of their parts swing on the smallest of pivots, zero. With these mental devices we make visible the hidden laws controlling the objects around us in their cycles and swerves.

Even the mind itself is mirrored in mathematics, its endless reflections now confusing, now clarifying insight. Zero's path through time and thought has been as full of intrigue, disguise and mistaken identity as were the careers of the travellers who first brought it to the West. In the history book you will see it appear in Sumerian days almost as an afterthought, then in the coming centuries casually alter and almost as casually disappear, to rise again transformed. Its power will seem divine to some, diabolic to others. It will just tease and flit away from the Greeks, live at careless ease in India, suffer our Western crises of identity and emerge this side of Newton with all the subtlety and complexity of our times.

My approach to these adventures will in part be that of a naturalist, collecting the wonderful variety of forms zero takes on - not only as a number but as a metaphor of despair or delight; as a nothing that is an actual something; as the progenitor of us all and as the riddle of riddles. But we, who are more than magpies, feather our nests with bits of time. I will therefore join the naturalist to the historian at the outset, to relish the stories of those who juggled with gigantic numbers as if they were tennis balls; of people who saw their lives hanging on the thread of a calculation; of events sweeping inexorably from East to West and bearing zero along with them - and the way those events were deflected by powerful personalities, such as a brilliant Italian called Blockhead or eccentrics like the Scotsman who fancied himself a warlock.

As we follow the meanderings of zero's symbols and meanings we'll see along with it the making and doing of mathematics — by humans, for humans. No god gave it to us. Its muse speaks only to those who ardently pursue her. And what is that pursuit? A mixture of tinkering and inspiration; an idea that someone strikes on, which then might lie dormant for centuries, only to sprout all at once, here and there, in changed climates of thought; an on-going conversation between guessing and justifying, between imagination and logic. Why should zero, that O without a figure, as Shakespeare called it, play so crucial a role in shaping the gigantic fabric of expressions which is mathematics? Why do most mathematicians give it pride of place in any list of the most important numbers? How could anyone have claimed that since 0x0 = 0, therefore numbers are real? We will see the answers develop as zero evolves.

And as we watch this maturing of zero and mathematics together, deeper motions in our minds will come into focus. Our curious need, for example, to give names to what we create and then to wonder whether creatures exist apart from their names. Our equally compelling, opposite need to distance ourselves ever further from individuals and instances, lunging always toward generalities and abbreviating the singularity of

things to an Escher array, an orchard seen from the air rather than this gnarled tree and that.

Below these currents of thought we will glimpse in successive chapters the yet deeper, slower swells that bear us now toward looking at the world, now toward looking beyond it. The disquieting question of whether zero is out there or a fiction will call up the perennial puzzle of whether we invent or discover the way of things, hence the yet deeper issue of where we are in the hierarchy. Are we creatures or creators, less than - or only a little less than - the angels in our power to appraise?

Mathematics is an activity about activity. It hasn't ended has hardly in fact begun, although the polish of its works might give them the look of monuments, and a history of zero mark it as complete. But zero stands not for the closing of a ring: it is rather a gateway. One of the most visionary mathematicians of our time, Alexander Grothendieck, whose results have changed our very way of looking at mathematics, worked for years on his magnum opus, revising, extending - and with it the preface and overview, his Chapter Zero. But neither now will ever be finished. Always beckoning, approached but never achieved: perhaps this comes closest to the nature of zero.