The never-ending story

A guide to the biggest idea in the Universe: infinity.

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

Jonathan Cape: 2005. 328 pp. £17.99. To be published in the US by Pantheon Books ($26). 0375422277 0224069179 | ISBN: 0-375-42227-7
Seeing the big picture: the patterns in Islamic tile designs can be repeated indefinitely. Credit: PATRICK SYDER IMAGES

John Barrow is a wide-ranging author. A few years ago he wrote The Book of Nothing (Jonathan Cape, 2000), an exploration of zero, and now he has published The Infinite Book. Again he tackles a subject that has been previously explored by other popularizers of science, but again he brings his charm and wit to bear to provide an account that is highly engaging and enriched with numerous literary references and dozens of illustrations and photographs.

The opening chapters introduce readers to the concept of infinity through the eyes of the various philosophers who tackled the subject through the centuries. Their conflicting views about the nature and existence of infinity are slightly baffling, but the situation is rescued by the nineteenth-century mathematicians who were able to make sense of the infinite realm. What emerges is a series of staggeringly brilliant and beautiful ideas, whose logic and power seem to defy common sense.

For example, are all infinities equal? There are an infinite number of numbers and an infinite number of even numbers, but is the former infinity double the size of the latter, or does infinity have hidden subtleties? The Infinite Hotel is the classic device for addressing this question. Attributed to the German mathematician David Hilbert, the Infinite Hotel is a highly successful enterprise and its infinite number of rooms are fully booked. The situation seems bleak when a new guest turns up at reception, but the resourceful hotelier, comes up with a solution. He asks everybody to move up to the next room — the occupant of room 1 moves to room 2, the occupant of room 2 moves to room 3, and so on. This means that everybody still has a room, but room 1 is now empty and is available for the new guest.

This merely shows that infinity plus one equals infinity. But what if an infinite number of new guests arrive at reception? The hotelier asks the current guests to move to the room with the number that is double their current room — the occupant of room 1 moves to room 2, the occupant of room 2 moves to room 4, and so on. This means that everybody still has a room, but all the odd-numbered rooms are now empty and available for the new guests. So infinity plus infinity equals infinity, and hence the number of even numbers is equal to the number of all numbers.

Barrow explains other arguments that show that the number of whole numbers is equal to the number of fractions, but that the number of decimals is larger and represents a different scale of infinity. He also tells the tragic story of Georg Cantor, a German mathematician who developed many of the fundamental ideas relating to infinity. Cantor was heavily criticized by the ‘finitists’, such as the influential Leopold Kronecker, who successfully vetoed Cantor's applications for senior academic posts and blocked the publication of his papers.

This constant persecution led to a series of nervous breakdowns and spells of severe depression for Cantor, who gradually moved away from mathematics and spent increasing amounts of time with philosophers and theologians. Eventually Cantor did receive some acclaim from mathematicians around Europe, but in his German homeland he was largely ignored. In 1908 he complained that his German colleagues “do not seem to know me, even though I have lived and worked among them for 52 years”.

Some chapters move beyond mathematics and deal with the infinitely large in the context of the Universe and the infinitely small with respect to particle physics. Barrow also ventures even further afield, touching on theological issues and examining eternity and the notion of elixirs for immortality.

Occasionally the material feels familiar, but Barrow is generally able to introduce novel twists and turns, and presents standard material in refreshing ways. For example, it is well known that the density at the centre of a black hole is supposed to be infinite, but few people realize that the average density of a large black hole, such as the one at the centre of a galaxy, could be less than the density of air. This is because the Schwarzschild radius of a giant black hole, which defines the point of no return, is so vast.

Familiar characters, such as Hilbert, are also fleshed out in new ways. We learn that one of Hilbert's students committed suicide when he failed to solve a particular mathematical problem. Hilbert was asked to speak at the funeral, so he stood at the graveside and matter-of-factly explained that the problem was not particularly difficult and that the young man had merely failed to look at it in the right way.

Even an old chestnut like the Infinite Hotel feels fresh. This results in an index that lists Fawlty Towers next to Fermat's last theorem — surely the only time that these two items have been neighbours in the same book — and this serves to indicate the book's quirkiness.

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  1. Simon Singh is the author of Big Bang (Fourth Estate, 2004), a history of cosmology.

    • Simon Singh
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Singh, S. The never-ending story. Nature 434, 437–438 (2005). https://doi.org/10.1038/434437a

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