Is the universe infinite? Will it go on for ever? Might we go on for ever? These are just some of the questions that John Barrow considers in this immensely readable and thought-provoking book. And he does it all with a light touch and a refreshing sense of humour. At one point, an impressive achievement of cosmological theory is followed immediately by a quote from film director Woody Allen: "Can we actually 'know' the Universe? My God, it's hard enough finding your way around in Chinatown."

But the book is not solely about cosmology. Barrow also considers the role of infinity in mathematics, philosophy, theology and even sociology. He says: "For something that you can't buy on the internet, 'infinity' is strangely ubiquitous. It turns up in church sermons, mathematics lectures at all the best universities, popular science books about 'Life, the Universe and Everything', and mysticism the world over, while historians remind us that people have been burnt at the stake for talking about it."

Historically speaking, mathematicians seem to have been the first to "tame" infinity. And, yet, just when they thought they had some kind of grip on it, infinity had more surprises in store.

Consider, for instance, the series 1 + 1/2 + 1/4 + 1/8 + 1/16 +..., where the dots indicate that we are to keep on adding extra terms, in the obvious way, for ever. Even though the number of terms is infinite, it is not too difficult to see that the series has a finite "sum", and that sum is 2. But now look at the series 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 +... This again turns out to have a finite sum, which happens to be 0.693... But suppose now that we decide to add up the terms of this particular series * in a different order * , by rearranging them so that each positive term is followed by two negative terms. Note that this procedure ensures that all the terms will still be "there"; we are not dropping any, nor are we smuggling in any new ones. Yet the sum of the series now turns out to be 0.346... that is, only half what it was before.

Barrow discusses well the mathematical mystery "Hilbert's Hotel", with its infinite number of rooms and its infinite capacity to absorb ever more guests, even when it is "full". And after this, the author moves on to the groundbreaking work of Georg Cantor, which revealed a whole hierarchy of different infinities in mathematics.

Next, Barrow turns to cosmological questions, such as whether the universe will last for ever, whether it is infinite in size, whether it might be one of many and whether it might even be merely part of some enormous computer simulation. And here some fundamental questions of scientific approach arise, for when a mathematical infinity turns up in a physical theory it is usually taken as a sign that the physical theory in question has something wrong with it and needs to be modified. But when we are inquiring into the "frontiers of physical reality", as Barrow puts it, how can we be quite sure that actual infinities cannot occur?

The last chapter, "Living forever", is as engagingly written as the others and as generously sprinkled with quotations, yet as I read it I found myself being drawn back instead to G.H. Hardy's thoughts on the subject in his classic essay "A mathematician's apology" (1940). There, in one startlingly candid moment, Hardy wrote: "'Immortality' may be a silly word, but probably a mathematician has the best chance of whatever it may mean."

As a mathematician myself, I have always found this remark both exhilarating and embarrassing. I want it to be true, of course - the idea that some mathematical discoveries might, by the very nature of that particular subject, have an infinite lifetime, unlike virtually anything else - and yet...

Time, perhaps, for Woody Allen again, who once remarked: "I don't want to achieve immortality through my work. I want to achieve it through not dying." Barrow uses this wonderful quote to lighten the tone just before finishing the book on a serious note. His final sentence reads: "The quest to understand the nature of... the universe... may come to rely uniquely and completely upon the promptings or the avoidance of the infinite. We will need to know it better."

For what it's worth, I wholly agree.

David Acheson is fellow in mathematics, Jesus College, Oxford.

## The Infinite Book: A Short Guide to the Boundless, Timeless and Endless

Author - John D. Barrow

Publisher - Cape

Pages - 328

Price - £17.99

ISBN - 0 224 06917 9

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