56 of 59 people found the following review helpful:
5.0 out of 5 stars
A Broad Look at Infinity, August 15, 2005
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."
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29 of 31 people found the following review helpful:
5.0 out of 5 stars
World without end--AMEN!, February 3, 2006
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.
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30 of 34 people found the following review helpful:
5.0 out of 5 stars
Infinitely Great Book, August 29, 2005
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.
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