{"product_id":"the-infinite-book-isbn-9781400032242","title":"The Infinite Book","description":"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.  \u003cbr\u003e\u003cbr\u003eNow 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. \u003ci\u003eThe Infinite Book\u003c\/i\u003e is a thoroughly entertaining and completely accessible account of the biggest subject of them all–infinity.\u003ci\u003ePreface \u003c\/i\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003e1 Much Ado about Everything\u003c\/b\u003e\u003cbr\u003eThe Rough Guide to Infinity \u003cbr\u003eIntimations of the Infinite \u003cbr\u003eZeno Hour \u003cbr\u003e\u003cbr\u003e\u003cb\u003e2 Infinity, Almost and Actual, Fictitious and Factual\u003c\/b\u003e\u003cbr\u003eDarkness at Noon \u003cbr\u003eA Purely Aristotelian Relationship \u003cbr\u003eInfinity and God \u003cbr\u003eA Little Kant\u003cbr\u003e\u003cbr\u003e\u003cb\u003e3 Welcome to the Hotel Infinity\u003c\/b\u003e\u003cbr\u003eHotels \u003cbr\u003eExperiences of the Hotel Infinity \u003cbr\u003eThe Hotel Infinity’s Accounts \u003cbr\u003e\u003cbr\u003e\u003cb\u003e4 Infinity Is Not a Big Number\u003c\/b\u003e\u003cbr\u003eAn Immaculate Misconception \u003cbr\u003eAlbert of Saxony’s Paradox\u003cbr\u003eGalileo’s Paradox\u003cbr\u003eCadmus and Harmonia\u003cbr\u003eTerminator 0, 1⁄2, and 1 \u003cbr\u003eCountable Infinities \u003cbr\u003eUncountable Infinities \u003cbr\u003eThe Towering Infinito \u003cbr\u003e\u003cb\u003e\u003cbr\u003e5 The Madness of Georg Cantor\u003c\/b\u003e\u003cbr\u003eCantor and Son \u003cbr\u003eThe Chronicle of Kronecker \u003cbr\u003eCantor, God, and Infinity – the Trinity with Affinity \u003cbr\u003eAll’s Sad that Ends Bad \u003cbr\u003e\u003cbr\u003e\u003cb\u003e6 Infinity Comes in Three Flavours\u003c\/b\u003e\u003cbr\u003eTriple Top \u003cbr\u003eLet’s Get Physical \u003cbr\u003eNaked Infinities \u003cbr\u003eThe Great Blue Yonder \u003cbr\u003eInfinity on the Back Foot \u003cbr\u003e\u003cbr\u003e\u003cb\u003e7 Is the Universe Infinite?\u003c\/b\u003e\u003cbr\u003eEverything That Is \u003cbr\u003eCosmology Goes Underground \u003cbr\u003eBent Universes \u003cbr\u003eThe Problem of Topology \u003cbr\u003eThe Problem of Uniformity \u003cbr\u003eThe Problem of Acceleration \u003cbr\u003eWhere Does This Leave Us?\u003cbr\u003eThe Shining \u003cbr\u003e\u003cbr\u003e\u003cb\u003e8 The Infinite Replication Paradox\u003c\/b\u003e\u003cbr\u003eA Universe Where Nothing Is Original \u003cbr\u003eThe Great Escape \u003cbr\u003eThe Temporal Version – Been There, Done That \u003cbr\u003eThe Never-ending Story \u003cbr\u003eThe Ethics of the Infinite \u003cbr\u003e\u003cbr\u003e\u003cb\u003e9 Worlds Without End\u003c\/b\u003e\u003cbr\u003eOther Worlds in History \u003cbr\u003eOut of This World \u003cbr\u003eInflation – Here, There, and Everywhere \u003cbr\u003eConscious Interventions – Men in Black \u003cbr\u003eSimulated Universes \u003cbr\u003eHow Should We Then Live? \u003cbr\u003e\u003cbr\u003e\u003cb\u003e10 Making Infinity Machines\u003c\/b\u003e\u003cbr\u003eSuper-tasks \u003cbr\u003eRubbing Thomson’s Lamp \u003cbr\u003eSome Norse Code \u003cbr\u003eThe End-game Problem \u003cbr\u003eRelativity and the Amazing Shrinking Man \u003cbr\u003eA Matter of Timing \u003cbr\u003eNewtonian Super-tasks \u003cbr\u003eRelativistic Super-tasks \u003cbr\u003eBig Bangs and Big Crunches \u003cbr\u003e\u003cbr\u003e\u003cb\u003e11 Living Forever\u003c\/b\u003e\u003cbr\u003eChildhood’s End \u003cbr\u003eThe Sociology of Eternity \u003cbr\u003eThe Problem-Page of the Unending Future \u003cbr\u003eThe Strange, Familiar, and Forgotten \u003cbr\u003eIncestuous Time Travel \u003cbr\u003eThe Grandmother Paradox \u003cbr\u003eConsistent Histories \u003cbr\u003eTourists From the Future Time Travellers in the Financial World:\u003cbr\u003e      Perpetual Money Machines \u003cbr\u003eWhy You Can’t Change the Past \u003cbr\u003eInfinity – Where Will It All End? \u003cbr\u003e\u003cbr\u003e\u003ci\u003eNotes \u003cbr\u003eIndex\u003c\/i\u003e“Highly engaging. . . .  [Barrow] brings his charm and wit to bear. . . .  [He] introduces novel twists and turns, and presents [the] material in refreshing ways.”–\u003ci\u003eNature\u003c\/i\u003e\"Eloquent. . . .  Succinct. . . .  Barrow [has the] remarkable ability to provide clear, concise, engaging and distinctly finite explanations–even when describing some fairly advanced concepts. . . .  [An] engaging read.\"–\u003ci\u003eSan Francisco Chronicle\u003c\/i\u003e \"Clever and insightful. . . .  [A] lively history of infinity through the ages.\"–\u003ci\u003eEntertainment Weekly\u003c\/i\u003e“Entertaining. . . . Remarkably lucid and not the least mind-boggling. . . . His clear, engaging style manages to illuminate abstruse matters.... This is a useful guide to an endlessly fascinating subject.” –\u003ci\u003eAmerican Scientist\u003c\/i\u003eJohn D. Barrow is Research Professor of Mathematical Sciences at the University of Cambridge. He is the author of several bestselling books, including Theories of Everything and Impossibility.chapter one\u003cbr\u003e\u003cbr\u003e    Much Ado about Everything\u003cbr\u003e\u003cbr\u003e    ‘On a clear day you can see forever.’\u003cbr\u003e    –Alan Lerner\u003cbr\u003e\u003cbr\u003e    THE ROUGH GUIDE TO INFINITY\u003cbr\u003e\u003cbr\u003e    ‘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?\u003cbr\u003e   – Miguel de Unamuno\u003cbr\u003e\u003cbr\u003e    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?\u003cbr\u003e\u003cbr\u003e    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.\u003cbr\u003e\u003cbr\u003e    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?\u003cbr\u003e\u003cbr\u003e    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.\u003cbr\u003e\u003cbr\u003e    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?\u003cbr\u003e\u003cbr\u003e    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?\u003cbr\u003e\u003cbr\u003e    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.\u003cbr\u003e\u003cbr\u003e    INTIMATIONS OF THE INFINITE\u003cbr\u003e\u003cbr\u003e    ‘think globally but act locally’\u003cbr\u003e     Activist bumper sticker’\u003cbr\u003e\u003cbr\u003e    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.\u003cbr\u003e\u003cbr\u003e    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?\u003cbr\u003e\u003cbr\u003e    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.\u003cbr\u003e\u003cbr\u003e    \u003ci\u003eThe idea that time has no end\u003c\/i\u003e\u003cbr\u003e\u003cbr\u003e    ‘Eternity’s a terrible thought. I mean, where’s it all going to end?’– Tom Stoppard\u003cbr\u003e\u003cbr\u003e    ‘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.\u003cbr\u003e\u003cbr\u003e    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 grown up with the   naturalness of a world with a beginning and an end and do not worry   about the mind-stretching problems of a world with neither beginning   nor end – that just always is. But it is surely the finite world that   seems strangest. It needs someone or something on the outside to bring   it about in order to provide it with a context and a reason to be. Take   away our religious heritage and it may have been more natural to assume   that earthly things go on without end. But, paradoxically, it is our   Christian religious heritage that reinforces an expectation that things   go on forever, with or without us . . .\u003cbr\u003e\u003cbr\u003e    ‘World without end.\u003cbr\u003e\u003cbr\u003e    Amen’\u003cbr\u003e\u003cbr\u003e    \u003ci\u003eCycles\u003c\/i\u003e\u003cbr\u003e\u003cbr\u003e    ‘Like a circle in a spiral\u003cbr\u003e\u003cbr\u003e    Like a wheel within a wheel\u003cbr\u003e\u003cbr\u003e    Never ending or beginning,\u003cbr\u003e\u003cbr\u003e    On an ever spinning wheel\u003cbr\u003e\u003cbr\u003e    As the images unwind\u003cbr\u003e\u003cbr\u003e    Like the circles that you find\u003cbr\u003e\u003cbr\u003e    In the windmills of your mind.’\u003cbr\u003e    –Alan Bergman and Michel Jean Legrand, Windmills of Your Mind\u003cbr\u003e\u003cbr\u003eIn many   cultures there was a strong belief that all change is cyclic. There is   good reason to think so. Everyday life witnesses to it. Birth, life and   death lead to rebirth; night follows day as day follows night and the   seasons recur with metronomic regularity. Sleeping and waking, our   lives are a continually repeating cycle. What better place to look for   a picture of the ultimate pulse of the Universe?\u003cbr\u003e\u003cbr\u003e    Some believed in a more specific form of cyclicity in which all living   things were reincarnated on Earth in the guise of other creatures.   Other religions believed in rebirth by transformation into a new body   and soul. In essence all these religious ideas of resurrection and   rebirth look to a future without end but with change. Like a ball   bouncing forever, so they look to a future that has no end and draw   from a past that had no beginning. Invariably, human beings had a part   to play in that never-ending cycle of existence. Life is a process, a   flow, in which we emerge temporarily but are subsumed and replaced by   other living things. A beginning or an end would be a singularity, a   disruption of the natural order of things. Such a hiatus would be   unnatural, inexplicable without the invention of other forces at work   in the Universe. Psychologically, having a place in an infinite process   endows the believing participant with a part to play in the infinite   scheme of things, a sense of community with all living things, and a   personal trajectory that is ever renewed.\u003cbr\u003e\u003cbr\u003e    \u003ci\u003eThe Supreme Being\u003c\/i\u003e\u003cbr\u003e\u003cbr\u003e    ‘God is more truly imagined than expressed, and He exists more truly than He is imagined.’\u003cbr\u003e    –St Augustine\u003cbr\u003e\u003cbr\u003e    Many cultures had a conception of a Supreme Being who controlled the   Universe. In most cases this Being was the first among many, the leader   of the gods. In others he was unique in certain respects, all powerful   and all knowing. If such a Deity controls everything, even space and   time, He cannot be limited by them and so must be eternal or transcend   time entirely. Again, we see how one is led to entertain an idea of   what we would call the infinite. It is a necessary attribute of a   certain type of Deity.\u003cbr\u003e\u003cbr\u003e    This type of search for the infinite is also closely linked to a human   desire for something transcendent, something beyond what is seen and   immediately experienced. Some would argue that this inclination arises   because there is something transcending our immediate experience. This   is the stance of the great religious traditions. Others argue that this   is a by-product of the unusual development of the human mind. At some   stage in our evolution our minds developed an ability for   self-reflection. This enabled us to imagine what would happen if we   took certain actions. This is a remarkable ability. Other animals don’t   seem to have it. They learn by direct experience rather than by   imagined experience. This type of human consciousness has all sorts of   by-products, and creates fears and psychological problems from which   simpler minds will not suffer. Is our tendency to extrapolate from the   known to the unknown and on to the unknowable a by-product of the   mind’s ceaseless attempts to correlate what we know?\u003cbr\u003e\u003cbr\u003e    \u003ci\u003eUnending space\u003c\/i\u003e\u003cbr\u003e\u003cbr\u003e    ‘The eternal silence of these infinite spaces terrifies me’\u003cbr\u003e    –Blaise Pascal, Pensées\u003cbr\u003e   \u003cbr\u003e    The greatest shared experience of human beings throughout their history   has been the appearance of the night sky. The darkness of the night   sky, studded with bright celestial objects, was a remarkable feature of   ancient life. It inspired stories, provided the means to navigate, and   elicited worship. It gave humanity a sense of place in the greater   Universe – and that place was a humble one. We appeared as an   insignificant dot amidst the star-spangled blackness of the night. That   blackness went on and on, perhaps forever. How could it end? Again, the   idea of a cosmic edge is harder to grasp than that of its absence. What   world lies beyond such an edge and where would it be? The dark night   sky might be a great dark shell that surrounds us, like a celestial   cave wall – with lights upon its ceiling. Or if you live on an island   or a continent that is partially bordered by the sea, you will have   seen that there can exist a complete change of environment. There could   be an edge to space in the way that there is an edge to land at the   coast. What lies beyond need not be nothingness, merely something   different, something that we choose not to call space.","brand":"Vintage","offers":[{"title":"Default Title","offer_id":46301785456869,"sku":"NP9781400032242","price":23.0,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781400032242.jpg?v=1767739934","url":"https:\/\/k12savings.com\/products\/the-infinite-book-isbn-9781400032242","provider":"K12savings","version":"1.0","type":"link"}