I wrote A Brief History of Infinity a while ago because it's a topic I've found fascinating since I was at school. To kick off the series, here's the introduction of that book, which summarises why it's a subject that intrigues so many:
The infinite is a concept so remarkable, so strange, that contemplating it has apparently 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.
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