Extract from A Brief History of
Infinity
by Brian Clegg
1. To Infinity and beyond
In this unbelievable universe in which we live there are no absolutes.
Even parallel lines, reaching into infinity, meet somewhere yonder.
Pearl S. Buck, A Bridge for Passing
The infinite is a concept so  remarkable,
so strange, that contemplating it has driven at least two great mathematicians
over the edge into insanity.
In the Hitch-hiker's Guide to the Galaxy, Douglas Adams described
how the writers of his imaginary guidebook got carried away in devising
its introduction:
'Space', it says, 'is big. Really big. You just won't believe
how vastly, hugely, mind-bogglingly big it is. I mean, you may think
it's a long way down the street to the chemist, but that's just peanuts
to space. Listen ... ' and so on. After a while the style settles
down a bit and it starts telling you things you actually need to know
...
Infinity makes space seem small.
Yet this apparently unmanageable concept is also with us every day.
My daughters were no older than six when they first began to count quicker
and quicker, ending with a blur of words and a triumphant cry of 'infinity!'
And though infinity may in truth make space seem small, when we try
to think of something as vast as the universe, infinite is about the
best label our minds can apply.
Anyone who has broken through the bounds of basic maths will have found
the little 
symbol creeping into their work (though we will discover that this drunken
number eight that has fallen into the gutter is not the real infinity,
but a ghostly impostor). Physicists, with a typical carelessness that
would make any mathematician wince, are cavalier with the concept. When
I was at school, studying A-level (high school) physics, a common saying
was 'the toast rack is at infinity.' This referred to a nearby building,
part of Manchester Catering College, built in the shape of a giant toast
rack. (The resemblance is intentional, a rare example of humour in architecture.
The companion building across the road, when seen from the air, looks
like a fried egg.) We used the bricks on this imaginative structure
to focus optical instruments. What we really meant by infinity was that
the building was 'far enough away to pretend that it is infinitely distant'.
Infinity fascinates because it gives us the opportunity to think beyond
our everyday concerns, beyond everything to something more --
as a subject it is quite literally mind-stretching. As soon as infinity
enters the stage it seems as if common sense sanity leaves. Here is
a quantity that turns arithmetic on its head, making it seem entirely
feasible that 1 = 0. Here is a quantity that enables us to cram as many
extra guests as we like into an already full hotel. Most bizarrely of
all, it is quite easy to show that there must be something that is bigger
than infinity -- which surely should be the biggest thing there could
possibly be.
Although there is no science more abstract than mathematics, when it
comes to infinity, it has proved hard to keep spiritual considerations
out of the equation. When human beings contemplate the infinite, it
is almost impossible to avoid things theological, whether in an attempt
to disprove or prove the existence of something more, something greater
than the physical universe. Infinity has this strange ability to be
many things at once. It is both practical and mysterious. Mathematicians,
scientists and engineers use it quite happily because it works -- but
they consider it a black box, having the same relationship with it that
most of us do with a computer or a mobile phone, something that does
the job even though we don't quite understand how.
The position of mathematicians is rather different. For them, modern
considerations of infinity shake up the comfortable, traditional world
in the same way that physicists suffered after quantum mechanics shattered
the neat, classical view of the way the world operated. Reluctant scientists
have found themselves having to handle such concepts as particles travelling
backwards in time, or being in two opposite states at the same time.
As human beings, they don't understand why things should be like this,
but as scientists they know that if they accept the picture it helps
predict what actually happens. As the great twentieth century physicist
Richard Feynman said in a lecture to a non-technical audience:
It is my task to convince you not to turn away because you don't
understand it. You see, my physics students don't understand it either.
That is because I don't understand it. Nobody does.
Infinity provides a similar tantalising mix of the normal and the counter-intuitive.
All of this makes infinity a fascinating, elusive topic. It can be
like a deer, spotted in the depths of a thick wood. You will catch a
glimpse of beauty that stops you in your tracks, but moments later you
are not sure if you saw anything at all. Then, quite unexpectedly, the
magnificent animal stalks out into full view for a few, fleeting seconds.
A real problem with infinity has always been getting though the dense
undergrowth of symbols and jargon that mathematicians throw up. The
jargon is there for a very good reason. It's not practical to handle
the subject without some use of these near-magical incantations. But
it is very possible to make them transparent enough that they don't
get in the way. To open up clear views on this most remarkable of mathematical
creatures, a concept that goes far beyond sheer numbers, forcing us
to question our understanding of reality.
Welcome to the world of infinity.
Brian Clegg has a degree in Natural Sciences and
an MA in Operational Research. He spent 17 years working at British
Airways, much of it finding applications of new technology. In 1994
he left the airline to set up his own company, Creativity Unleashed
Limited, which provides training on creativity to business. He has
contributed to many magazines and written a range of books, most recently
specializing in popular science. He now lives in a Wiltshire village
with his family and golden retriever.
© Brian Clegg 2003;
reproduced with permission; all rights reserved.
A Brief History of Infinity: the quest to think the unthinkable
was published by Robinson in the UK in September 2003; ISBN: 1841196509.
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