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The concept of infinity is awe-inspiring. What does it mean? Can some infinities be bigger than others? Robert and Ellen Kaplan reveal the wonder of the art of the infinite
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The Independent Online

"If the doors of perception were cleansed," wrote William Blake, "every thing would appear to man as it is, infinite." Just as our own bustling lives are lived out under the enormous canopy of stars, so our thoughts move against an infinite background. Infinity oppresses, puzzles and exhilarates us by turns. It touches to what ultimately matters. The Renaissance philosopher Giordano Bruno died at the stake for his vision of a universe made up of infinite worlds. Many of those who have near-death experiences speak of a blissful opening out toward infinity.

What does it mean? What lies behind the language and even behind the thoughts? Is "infinity" a synonym for "boundless", and "boundless" no more than "vague"? Is yearning all we can know of it, thrones piled on dominions all we can hope, bottomless pits all we can fear? Or has infinity a structure we can get our minds around, through the oldest and newest of the arts?

This art is mathematics - long mislabelled as a science. It thrives on invention, demands great daring, and (as Shakespeare said of lovers, madmen and poets) is of imagination all compact. Above all, mathematics is freedom. So wrote the mathematician Georg Cantor, who, a century ago, threw open for us the gates to the infinite paradise. He knew, as we all do, that the counting numbers go on forever (if in doubt, just ask yourself what the last one could be), but wondered whether there might be collections of numbers larger still: collections with more things in them even than this infinite collection that begins so humbly with one, two and three. Fractions looked like a good candidate, since you can endlessly split up the space between any two numbers, such as zero and one, into fraction after fraction (1/2, 1/4, 1/8 and so on - and this endless halving won't even include 1/3 and 17/19 and all that rabble).

When he found that he was wrong - that miraculously there are as many fractions as integers - he didn't give up, but looked elsewhere, and discovered at last, to his astonishment and ours, that there are more decimals than integers or fractions: a new, larger size of infinity! And beyond decimals? He went on to find yet larger sizes of the infinite, larger and larger still, with no end in sight. His finite mind had encompassed worlds more infinite than the universe that we live in.

You might expect that Cantor's discoveries would also have taken him beyond any reach of ours; but his tools for discovery are those that any eight-year-old can easily use. They amount to no more than pairing up: if you can make couples by matching all of the numbers in one set (like the counting numbers) and all those in another (like the fractions), then the two sets are the same size. If they can't be paired, one set must be larger. At last we have a way of thinking clearly about the infinitely large, when before all we could do was shrink from it. And the infinitely small, too. Blake again: "To see the world in a grain of sand..."

In the 17th century the German diplomat, philosopher and mathematician, Gottfried Leibniz, claimed that in the least cell of the least leaf on the least shrub in the least garden lay another garden, and in the least cells of its least leaves lay gardens again, and so shrinking down forever, worlds without end. The first science-fiction fantasy? A flight of post-prandial fancy? In another part of the garden that is mathematics another mathematician, Abram Robinson, came up 50 years ago with supersmall numbers that fulfil Leibniz's vision. They sift through the smallest cracks in our real numbers without leaving a trace. His infinitesimals give us a way to understand calculus without the cumbersome machinery and slippery limits that we usually have to master.

It isn't only at the largest and smallest extremes that infinity comes into play: the whole fabric of mathematical thought is woven with it. When we say that the area of any circle with radius "r" is pi "r" squared, or that the Pythagorean theorem is true for all right triangles, "any" and "all" really do mean each and every one of the mind-bogglingly infinite number of possible circles and right triangles, not just the few we've drawn. In other walks of life when we say all ("All men are created equal"), we mean at most the finite few there have been or will be. How have we the hubris to make or the skill to prove such defiant mathematical claims? By the one thread that ties our time-bound minds to the timeless: logic. The Greeks knew that even the gods had to obey its iron (or is it golden?) necessity.

What is so peculiar about mathematics, this art of the infinite, is that its artists freely invent what turns out to be real. Just as the way in which we now see is affected by how Rembrandt and Monet painted, and how we size up others by what Austen and Woolf wrote, so the way in which we understand the world is shaped by the inventions of mathematical artists as radical as Sterne and Joyce, though this is far less well known, since their language hides their thoughts from most of us. Why do their inventions come to be read as discoveries? Does art in fact dictate to life? Or did Spinoza have the right explanation: the order of the world and the order of the mind are the same.

Science begins with wonder and ends with awe. Mathematics begins with awe and ends with wonder. Our first encounters with the infinite are awesome: nightmarish, even, when you think of agoraphobia, as in Ford Madox Ford's overwhelming novel, The Good Soldier: "Upon an immense plain, suspended in mid-air, I seem to see three figures, two of them clasped in an intense embrace, and one intolerably solitary. It is in black and white, my picture of that judgement, an etching perhaps; only I cannot tell an etching from a photographic reproduction. And the immense plain is the hand of God, stretching out for miles and miles, with great spaces above it and below it."

At the other extreme, awe of the infinite can still life's tumult. This is John Heath-Stubbs's translation of "The Infinite", by the 19-century Italian poet Giacomo Leopardi:

This lonely hill was always dear to me,

And this hedgerow, that hides so

large a part

Of the far sky-line from my view.

Sitting and gazing,

I fashion in my mind what lie beyond -

Unearthly silences, and endless space,

And very deepest quiet; then for a while

The heart is not afraid. And when

I hear

The wind come blustering among

the trees

I set that voice against this

infinite silence:

And then I call to mind Eternity,

The ages that are dead, and the

living present

And all the noise of it. And thus it is

In that immensity my thought

is drowned:

And sweet to me the foundering in

that sea.

Cantor's great inventions - or revelations - have have taken our awe and transformed it into wonder. We probe at the infinite now with his instruments and are just beginning to understand its structure. But as Newton said, looking back on his life's work, "I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

Robert and Ellen Kaplan's new book, 'The Art of the Infinite: Our Lost Language of Numbers', is available from tomorrow, price £20 (Allen Lane)