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A Brief History of Infinity: The Quest to Think the Unthinkable

by Brian Clegg

(Robinson, £8.99, 243 pages, paperback, published September 2003.)

Infinity: a concept particularly hard to grasp and yet one familiar to us all. Early in this cover scanengaging history of the Big Subject, Brian Clegg illustrates infinity's familiarity by describing children's counting games: you can always count higher, and children will delight in doing so until one has to trump another with a cry of "infinity!" And yet, despite its familiarity, the very idea of infinity (and the various shades of infinity -- the more infinite, the differently infinite, and even the countable infinite) has been a highly controversial one to generations of mathematicians, philosophers and even theologians. Why the controversy? Well, there are many reasons, but one of the fundamentals is that "when you lose the concept of quantity you also lose the mechanics of arithmetic" -- the rules just fall apart.

Clegg rather cleverly begins his history by looking at the infinitely small rather than the infinitely, well, infinite. A sequence starting at 1 and then dividing into ever-smaller fractions -- 1 + 1/2 + 1/4 +1/8 + ... -- has no obvious end; it is infinitely long despite having a finite upper limit; it will never, quite, reach two. Aristotle pointed out that the sequence of whole numbers has no limit: in his terms it was potentially infinite even though he would not accept that infinity itself exists. Potential infinity, for Aristotle, was "a direction towards which real numbers could head" without being a thing of itself. This question of the nature of infinity is a recurring one: is there, indeed, such a thing as infinity, or is it merely a handy concept, a tool for mathematicians? Ultimately, do we really need even a concept of infinity, when we could just as easily substitute the concept of that number that is greater than any number we could conceivably require for the current purpose (which is, crucially, still a finite number)?

Given the subject, it was perhaps inevitable that I would be left wanting more... While the author does admirably in bringing in non-Western perspectives -- Egyptian scholars, Indian mythology, and so on -- it is still very much a Western history of infinity. The inclusion of other perspectives left me wondering about other cultures: what would, for example, a Chinese history of infinity be like? Perhaps a companion piece to this book would be A Brief Anthropology of Infinity.

Clegg has an amiable, easy style; he chats us through complex ideas rather than lecturing to us. Like other good science populists (Isaac Asimov springs immediately to mind), he explores his subject through narrative and anecdote: hanging discussions of infinity onto their historical development puts flesh onto what might easily be dry bones in other writers' hands, and so we have a tour of fascinating personalities and slices of history, a subject breathed into life.

Brian Clegg has just had me revisiting areas of mathematics I haven't even considered since university. And I enjoyed it.

Review by Keith Brooke.

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