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The Infinite Book: A Short Guide to the Boundless, Timeless and Endless
 
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The Infinite Book: A Short Guide to the Boundless, Timeless and Endless [Format Kindle]

John Barrow

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chapter one

Much Ado about Everything

‘On a clear day you can see forever.’
–Alan Lerner

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...

Booklist

As prolific science writer and physicist Barrow regularly remarks, infinity is not merely the smallest or biggest thing, or the longest time imaginable: it's a quality that is unimaginable. It's thus a paradox that mathematicians, physicists, and philosophers have discovered quite a bit about infinity, albeit with different degrees of certitude. As also related in David Foster Wallace's Everything and More: A Compact History of Infinity (2003), Barrow recounts the career of German mathematician Georg Cantor, whose explorations of set theory resulted in fundamental proofs about infinities (some are bigger than others, for example). However joyous such discoveries are to the numbers masters, physicists' encounters with infinities are less rapturous because they hint at deficiencies in general relativity; hence their joy over string theory, which eliminates infinities that arise in calculations about the big bang and black holes. Performing with his customary fluency and accessibility, Barrow imparts for general readers a feeling for the nub of thought about the mathematical, cosmic, ethical, and theological implications of infinity. Gilbert Taylor
Copyright © American Library Association. All rights reserved

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Amazon.com: 4.5 étoiles sur 5  12 commentaires
56 internautes sur 59 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 A Broad Look at Infinity 15 août 2005
Par R. Hardy - Publié sur Amazon.com
Format:Relié
When we get the capacity to look closer and closer into molecules, atoms, and subatomic particles, will we always be able to find something smaller? When our telescopes or probes look deeper into space, will we always find something larger? Is there a limit to the shortness of an instant, or the duration of eternity? In _The Infinite Book: A Short Guide to the Boundless, Timeless, and Endless_, John D. Barrow has invited us to look at infinities in many ways. He's competent to do so, as a professor of mathematical sciences at Cambridge, and as the author of previous books which successfully explained such concepts as nothing or impossibility. Like his previous efforts, this is highly readable stuff, but extraordinarily mysterious. The topic is something that everyone has pondered in some way; who has not, looking into the stars, wondered how far they go? It has a universal appeal, and a history within religion, philosophy, mathematics, and physics, all of which Barrow goes into here, in an entertaining summary. There are answers here, but plenty of mysteries.

Consider a universe that is infinite in space; this is a possibility, for no one knows that space is not infinite. In a universe of infinite size anything that can happen does happen, and does so infinitely often. In such a universe, not only are you here, but somewhere out there is another you doing exactly what you are doing; in fact, there are an infinite number of you. There is also somewhere out there another you who has done everything you ever did, but on the day after his sixteenth birthday, wore black socks instead of brown. This has to be the case in an infinite universe, Barrow shows; it is enough to make us uncomfortable, but discomfort is not an argument that an infinite universe cannot exist. Mathematicians have had fun and frustration with infinity ever since Zeno, who "proved" that because to walk a mile, you first had to walk a half mile, and then a quarter mile more, and an eighth mile more after that, and so on forever, that you never were able to finish that mile, and if you are under the impression that you have accomplished a mile journey at some time, you are just deluded. The world as it perceived by us, with all its journeys, is an illusion. Georg Cantor solved the problems of dealing with infinities mathematically, but his work was viciously attacked and blocked from publication, but surprisingly, Catholic theologians welcomed his ideas as a way of understanding the infinite, the infinite that included God, of course. Today, mathematicians take Cantor's work for granted, and its religious implications are not the common stuff of sermons.

It is a pleasure to puzzle through these matters with Barrow as a guide, at least partially because this is a general overview which skims through details in order to provide a larger picture. String theory, for instance, takes a couple of pages, and cosmology not much longer. If one flavor of infinity is just too much for you to consider, another will soon present itself. Thus Barrow is able to give an accessible guide to such mathematical chestnuts as the Hotel Infinity, which although it has an infinite number of rooms and they are all occupied, can take on new guests, even an infinite number of guests, even if it has to take occupants when an infinite number of the other inns in the Hotel Infinity chain are closed. There is an examination of why the sky is dark at night, if the stars are infinite in number and there must be one out there no matter where you look. Barrow demonstrates that non-zero interest rates are evidence that time travel does not happen. He speculates about computers which could do an infinite number of tasks in a finite time; such computers might exist, and might be able to calculate the infinite digits of pi - but then how would you print it out? He shows that although we think of infinity sometimes as the biggest of numbers, infinity is not a big number, but something entirely different, and no big number ever provided such a degree of interest and research in so many fields. And as in so many of his discussions of aspects of the infinite, paradox always holds sway. We are still at the beginning of trying to find the answers to much of the material here. After all, as Barrow points out, "You can discover whether the Universe is infinite, but the learning will take an infinite time."
30 internautes sur 32 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 World without end--AMEN! 4 février 2006
Par Joanna Daneman - Publié sur Amazon.com
Format:Relié
The value of "The Infinite Book" is definitely summed up in the chapter "The Madness of Georg Cantor." Believe it or not, "new math", that strange evolution of math teaching that stumped homework for a generation in the 60's was a direct result of Cantor's theories about sets, and the supposition that some infinite sets could be larger than others--which is the first thing you REALLY learn about infinity in mathematics.

The other great part of this book is the coupling of mathematics theory with physics. The assertion by Einstein that a singularity would be a breakdown of the laws of physics, and that any theory involving singularities would thereby have in it, the "seeds of its own destruction." Then author Barrow moves on to a very good explanation of string theory (imagine a particle that stretches like a tube in a warm enviroment, but contracts to a single point in a cool environment.) The explanations, illustrations are so clearly written in this book. It's a valuable reference for students of physics and mathematics and a great read for the curious about these subjects. Recommended.
30 internautes sur 34 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 Infinitely Great Book 30 août 2005
Par John Matlock - Publié sur Amazon.com
Format:Relié
The infinite is one of those concepts that takes just a bit of understanding. Well, maybe more than just a bit. Infinity doesn't necessarily mean a bunch, it means, mathematically, that a number cannot be determined.

Infinity is not a new idea. Mathematicians have been working on them for hundreds of years. Physicists really got involved when Einstein published the Theory of Relativity in 1905. He talked about all kinds of things happening when you approached the speed of light. Then when you actually got to the speed of light his equations went to infinity. This was a bit disconcerting. One of the real reasons for the willingness of physicists to believe in string theory is that that the equations still show valid values at and above the speed of light.

But enough on infinity. If you want to know more, here's the book for you. It discusses just about everything there is to know about infinity. It would be great for the high school math/physics teacher to use for examples. Or, it's just plain fun reading to anyone that's interested.
25 internautes sur 30 ont trouvé ce commentaire utile 
3.0 étoiles sur 5 Light infinite 3 août 2006
Par Enrique Perez de Vargas - Publié sur Amazon.com
Format:Relié|Achat vérifié
If you are interested in infinity and you are not familar with Cantor or Borges' "The Library of Babel", then you may be amazed by this book. Otherwise, you can find it too light. Probably good as a light summer reading.

Infinity is a fascinating subject, and I thought that this book would contain a lot of interesting information in its 300 pages. I have found many quotations, a lot of superficial theology and ethics, and little information on the concept itself. I missed more depth in handling the mathematical concepts.

Anyway, there is a very good part of the book (from my point of view) devoted to eternal inflation and simulated universes, especially for how the theories are introduced and chained. Even if it is not strictly related to infinity, it is the best part of the book. The chapter that describes Cantor's works is worth reading too.
5 internautes sur 5 ont trouvé ce commentaire utile 
4.0 étoiles sur 5 Infinite questions. 2 décembre 2006
Par Regnal - Publié sur Amazon.com
Format:Relié|Achat vérifié
I have not been disappointed by any of John Barrow's book so far. He has a unique gift of writing with exceptional clarity about difficult topics. This is not a typical cosmology book, but large portion is devoted to beginning, shape and future of The Universe.
Like in his previous "Book of Nothing", author mixes philosophical and scientific musings about infinities (big and small) affecting theology, mathematics, cosmology, physics (TOE) and our existence.
I found Georg Cantor's life and his quest for understanding "absolute infinity" (God?) quite interesting and emotional. And check how Blaise Pascal argued about believing (or not) in God, because of infinite gain (or loss!!).
One truth emanates from "The Infinite Book": we are far, infinitely far from knowing the truth about everything (Immanuel Kant's rings the bell!). The more we learn the bigger infinite number of questions surface in front of us. Are we nearing the limits of knowledge? Professor John Barrow does not suggest it has come to this, but read about them and enjoy stretching your mind.
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