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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  engaging
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.
Elsewhere in infinity plus:
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