For a thousand years, infinity has proven to be a difficult and illuminating challenge for mathematicians and theologians. It certainly is the strangest idea that humans have ever thought. Where did it come from and what is it telling us about our Universe? Can there actually be infinities? Is matter infinitely divisible into ever-smaller pieces? But infinity is also the place where things happen that don't. All manner of strange paradoxes and fantasies characterize an infinite universe. If our Universe is infinite then an infinite number of exact copies of you are, at this very moment, reading an identical sentence on an identical planet somewhere else in the Universe.
Now Infinity is the darling of cutting edge research, the measuring stick used by physicists, cosmologists, and mathematicians to determine the accuracy of their theories. From the paradox of Zeno's arrow to string theory, Cambridge professor John Barrow takes us on a grand tour of this most elusive of ideas and describes with clarifying subtlety how this subject has shaped, and continues to shape, our very sense of the world in which we live. The Infinite Book is a thoroughly entertaining and completely accessible account of the biggest subject of them all-infinity.
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September 10, 2006
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Excerpt from The Infinite Book by John D. Barrow
Much Ado about Everything
'On a clear day you can see forever.'
THE ROUGH GUIDE TO INFINITY
'If there is a Universal and Supreme Conscience I am an idea in it. After I have died God will go on remembering me, and to be remembered by God, to have my consciousness sustained by the Supreme Conscience, is not that, perhaps, to be immortal?
- Miguel de Unamuno
There is something about infinity and books. Never-ending stories, libraries that contain all possible books, books that contain everything that has ever happened, and everything that hasn't; books that write themselves, books about themselves, books about there being no books, and books that end before they've begun. So you should be no more surprised to find yourself reading a book about infinity than I am to be writing one. But for something that you can't buy on the internet, 'infinity' is strangely ubiquitous. It turns up in church sermons, mathematics lectures at all the best universities, popular science books about 'Life, the Universe and Everything', and mysticism the world over, while historians remind us that people have been burnt at the stake for talking about it. It is at once the staple of the mystic contemplation of reality - 'make me one with everything' as the mystic said to the hamburger vendor - and the familiar territory of science fiction and fantasy. Can all these things really be connected? Is infinity really that big?
For thousands of years in the West there was no more seditious idea on Earth than that of infinity. The idea that things might go on and on forever, that they need have neither beginning nor end, neither centre nor boundary, was contrary to the wisdom of the West. It threatened to displace God Almighty from His uniquely infinite status, to demote the Earth from the centre of the Universe, and destroy the uniqueness and special meaning of every event in creation. It had the potential to make what was once merely the possible become inevitable.
Yet the temptation to think that way was strong and simple. Once you start doing something over and over again it's not too hard to imagine what it would be like never to stop. Infinity is just one thing after another. And this tantalising mixture of simplicity and sophistication remains with us today. Infinity is a subtle idea to capture precisely and easy to throw into the dustbin of wishful thinking, but for the ordinary person in the street it is less surprising and more readily intelligible than any comparable abstraction. We are immune to its subtleties; protected by a strange familiarity inbred by religious traditions, or from just staring out at the dark night sky; convinced by our method of counting that there could never be a biggest number. If in doubt just add one. Or can you?
Yet infinity remains a fascinating subject. It lies at the heart of all sorts of fundamental human questions. Can you live forever? Will the Universe have an end? Did it have a beginning? Does the Universe have an 'edge' or is it simply unbounded in size? Although it is easy to think about lists of numbers or sequences of clock 'ticks' that go on forever, there are other sorts of infinity that seem to be more challenging. What about an infinite temperature or an infinite brightness - can such physical things actually be infinite? Or is infinity just a shorthand for 'finite but awfully big'? These sorts of infinity seem more problematic than the unending futures promised to the followers of many traditional religious faiths. Eternal life doesn't need anything infinite to happen here and now. It just means that there will always be something happening - always a there and then.
The other religiously motivated infinity is that which goes loosely with the idea of a God of limitless power and knowledge, which is a key ingredient of many Western religious traditions. This is another familiar touchstone for the concept of the infinite for everyone. You don't need to be a mathematician to feel that this type of transcendental infinity is familiar. Or do you?
You do need to be something of a mathematician to appreciate the other type of infinity. Numbers go on and on. Infinity seems to be nothing more than where they would get to if counting went on forever. But surely it never does and mathematical infinity looks like a promise that is never fulfilled, a numerical Peter Pan, a shorthand for a goal that is never reached, a potential but not an actual, a number bigger than all numbers. Or is it?