An examination of Infinity — in history and science — with excursions into literature, philosophy and religion, written by one of the most successful writers of popular science.
Infinity is surely 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? Or is infinity just a label for something that is never reached, no matter how long you go on counting? Can you do an infinite number of things in a finite amount of time? Is the universe infinite?
But infinity is also the place where things happen that don’t. All manner of strange paradoxes and fantasies characterize an infinite universe. So what is it like to live in a Universe where nothing is original, where you can live forever, where anything that can be done, is done, over and over again?
These are some of the deep questions that the idea of the Infinite pushes us to ask. Throughout history, the Infinite has been a dangerous idea. Many have lost their lives, their careers, or their freedom for talking about it. The Infinite Book will take you on a tour of these dangerous questions and the strange answers that scientists, mathematicians, philosophers, and theologians have come up with to deal with its threats to our sanity.
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John Barrow is Research Professor of Mathematical Sciences in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge. He is the author of several bestselling books including: Theories of Everything, Impossibility, and The Book of Nothing.Excerpt. © Reprinted by permission. All rights reserved.:
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?
Already we begin to sense that there are different sorts of infinity and you might believe in one but not another. In this book we are going to explore these infinities from different directions. We will see how human thinking came to embrace the idea of the infinite before recoiling from its implications. We will see how the argument raged about whether any true infinity ever materialised in our finite Universe; or whether infinities were artefacts of an inadequate description of events, are invariably relegated to happen in the infinite future, or are excluded from reality by a hidden principle that upholds the logical consistency of the Universe. We will find that eventually mathematicians became accustomed to dealing with infinities as if they were real entities, adding and subtracting them, cataloguing all the different infinities, determining their sizes, and finding that some were bigger than others – infinitely bigger. But we will mingle our story with tales that make the paradoxes of the infinite grow to become as large as life.
INTIMATIONS OF THE INFINITE
‘think globally but act locally’
Activist bumper sticker’
We know where the famous ‘lazy eight’ · symbol for infinity came from. The Oxford mathematician John Wallis, who was famous for writing the codes for both sides in the English Civil War, first wrote down the symbol in 1655. With a few strokes of his pen he adapted the Roman representation |... sometimes used instead of M for the (for them, large) number 1000. When written quickly it became · and it stuck. This and other uses of this evocative symbol can be seen in Figure 1.1.
Where did the idea of infinity come from? Does it bring with it some subtle survival value that favoured those with the inclination to develop it? Evolutionary psychologists would look for some way of thinking or acting which aided survival on African savannah landscapes a million years ago and had as a by-product the liking for generalisation without end. Nothing specific is immediately obvious. Primitive life was brief and immediate. Action was needed. Contemplation was not rewarded. The inclination to think about infinity is something that happens much later in the human story and it emerges from one of many responses to the Universe around us. What are the trails that might lead to forever?
There is a single pattern to many of the intuitions that have led human minds to contemplate the infinite. Human consciousness enables us to look ahead and see patterns. This enables us to compress experience into formulas or symbols that are shorter than the experience itself. We can write histories. This compressibility and pattern in the world is what ultimately makes mathematics so useful to us: we can pick out the patterns that are evident and represent them by strings of numbers or symbols. These strings generally have the property that they require no end. A list can always be added to. They naturally give credence to the idea of sequences of events that go on forever, even if there is no physical evidence that they do.
The idea that time has no end
‘Eternity’s a terrible thought. I mean, where’s it all going to end?’– Tom Stoppard
‘Immortality’, it has been said, ‘is the bravest gesture of our humanity towards the unknown.’ This is not an obvious response to the nature of everyday reality. Human beings, like other living things, are mortal. You would need to be a philosopher to distinguish clearly between time and our experience of it. The easier thought is to notice that time goes on for us when others die. The seasons may come and go, but there is a constant cycle of growth and decay and regrowth. The psychological responses to this state of affairs were various. For some, the response to human mortality was to regard it as an illusion or an antechamber to a more complete form of existence which was endless. The completeness of this higher form of existence was defined by its never-ending quality. For others, human lifecycles were like those of other living things and we would be reborn as part of a cycle of changes. Both of these ideas lead to an expectation of endless existence by extrapolating from what we see around us to create a satisfying perspective on the Universe in which we occupy a meaningful place. Ideas like these can play an important role in binding groups of people together, maintaining their morale in the face of adversity, and inspiring them to give their lives in defence of their fellows.
The idea that time has an end is at least as hard to maintain as the belief that it doesn’t. What would it mean? What would it feel like? It only made sense if there was some great cataclysm in the future that would destroy everything – but even in mythologies where such a drama was played out, something always happened next. Bringing time to an end seemed to involve having no actors, no gods to determine the fate of the world. Strangely, in the Christian world we have gro...
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