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on 10 May 2006
I was a little disappointed that works and discoveries about infinity was not treated in more detail. Instead, many of the pages are used for biographies; The life and doings of a lot of mathematicians are covered from childhood to death. This of course can be (or is) very interesting, but was not what I excpected. I also got a feeling that this was done in part to avoid writing more about infinity, which of course is a much more difficult topic. The book is intended for a reader with little mathematical background, and this may be the reason why the author avoids difficult questions. There are good and readable presentations of some of the wellknown paradoxes, which should make everyone wonder about the strange behavior when we move away from the finite experience.
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on 5 January 2008
If I'd not read Rucker's work on the subject (Infinity and The Mind), I might have thought this was pretty cool. But having said that, we may be reaching saturation point on the books about mathematicians (which this seems to be) - we need more about the *maths*.

I think there's a perception that to keep it readable it needs to be dumbed down. There's a lot of that going on. It's possible to explain *everything* in simple terms if you try hard enough. Maybe Clegg hasn't tried all that hard, or maybe he's scared of alienating the casual reader. Whatever, he doesn't do much for the mathematically literate who want to get something out of this. There's not actually all that much.
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on 8 January 2004
Sorry to be a wet blanket among all these favourable reviews, but none are from readers who know the maths. And I am afraid Mr Clegg gets some of it wrong (or at least, so simplifies it that it becomes positively misleading). There is much in the book to enjoy but I think it regrettable that a book which deals with one of the most fascinating areas of mathematics should mislead its readers on some of the key issues.
This is particularly annoying in two of the most important parts of the modern theory. Firstly, his explanation of the higher alephs is so inadequate that all subsequent discussion of the continuum hypothesis is meaningless. Even more annoying is his simplification of Godel's Incompleteness Theorem to the extent that he gets both the interpretation and the methodology wrong. You cannot discuss Godel's results without first introducing the notion of Consistency; a mathematical system is of no value unless it's axioms are consistent. Mr Clegg quotes the sentence 'This system of mathematics can't prove this statement is true.' He comments "If the system proves this statement, then it can't prove it." That's not the case. If the system proves the statement then the system has proved something which isn't true. So the system is inconsistent and therefore worthless. If, as we hope, the system is consistent then it cannot prove the statement. But that's precisely what the statement says, so the system has been unable to prove something which is nonetheless true! This is the real beauty and brilliance of Godel's insight.
I would not want these comments to discourage anyone from reading this book but I would like the reader to realise that the underlying maths is much richer and more beautiful than is conveyed here.
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VINE VOICEon 1 April 2016
* Physical

This book is well bound for a paperback in standard quality paper. It's mostly text with a few graphs to illustrate a point here and there.

* Target audience, A - level / H.N.D, Undergraduate, Graduate, Postgraduate?

This book (i.m.h.o) is designed for A-Level, or first year degree students that need a bit of background fleshing - out of some fundamental terms in some supporting math topics. The author makes it very easy to get on with and reading this book is a gentle slope in difficulties and in some related and associated topics. This book (i.m.h.o) did not 'get going' until the middle and later chapters, as it described the backgrounds of the math characters in mostly historical descriptions up to this general point. After progressing, more was made of the features of infinity which is why i bought this book and it attracted me to this volume. There is nothing here in this book i didn't know or have read about already from other sources. But this is not to say the book is not beneficial to other readers with less studies under their belts and want to learn about these topics in mostly historical terms.

* Summary

If you're new to math and enjoy recreational historical explorations of mathematical history, this is a book to consider but don't expect to delve deeply and broadly within this area of mathematical studies.
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on 30 October 2003
This excellent book helps readers get their minds around one of the most difficult concepts in the world if not the universe. The author approaches the subject of INFINITY from a number of fascinating angles and takes us through a historical journey to demonstrate how philosophers and mathematicians from Zeno, Plato and Aristotle through Galileo to Einstein, Leibniz and Hilbert have grappled with this most unthinkable of problems. I found this book thoroughly thought-provoking, highly stimulating and immensely rewarding. It enriched my knowledge and helped me push back the boundaries of my own thought processes.
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on 25 January 2005
This is a very interesting and informative description of the history of infinity.
Infinity is a fascinating (and complex) subject but Brian Clegg does an extremely good job of presenting it in a highly readable and essentially non-mathematical way. I have a mathematical background but this book should be accessible to all.
I tend to agree with the previous reviewer who criticised presentation of some of the more complex mathematics in particular the higher alephs and Godel. However, I do not believe that these issues significantly detracted from my enjoyment of the book (mainly because I had not expectation that they would be covered well).
An excellent popular science/mathematics book - highly recommended to all
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on 21 June 2012
I am a fairly incessant reader of popular books about quantum physics, relativity and related issues and the pure mathematics that go with them. But this is the first book (apart from biographies) that I have read cover to cover without once being totally baffled by some apparently fundamental point that has left a significant part of the book more or less incomprehensible.

I am surprised at the adverse reviews. The full title is "A Brief History of Infinity: The Quest to Think the Unthinkable". So it quite obviously NOT a text book for the specialist, and it IS going to include some history!

In fact, the book is totally absorbing and the cliché "reads like a novel" really does fit here. I find the way Mr Clegg has woven the maths in with the stories of the protagonists helps to clarify the difficult issues involved because it helps to show where they were coming from - and to make a difficult topic much easier to follow.

I also like the way he has woven in some reasons for the origin of some of the rather intimidating mathematical notation we have to use. This is reassuring! Like virtually every other language, it has grown incrementally and is therefore ilogical and confusing in many ways. I would add that, IMHO, the design of standard keyboards is another factor - even the most elementary symbols are missing - yet there are there is a host of largely useless keys!

The book successfully and clearly draws parallels with other concepts such as irrational and imaginary numbers. Right at the beginning the infinite series is brought in by describing how a series in which the integer One is increased by successive halves will never quite reach the next integer, Two. Pi is also a simple but infinite series of fractions that can be calculated to as much accuracy as needed, but can never be exactly defined as a single number or fraction.

The reasoning and the examples continue through to Gabriel's Horn, a three-dimensional horn-shape of infinite length that narrows to a point based on another infinite sequence. The weirdness is that uncontentious calculations reveal the volume to be of finite value but the surface area to be infinite! Yet surely if the horn were to be filled with (a finite) volume of paint the (infinite) inside surface area would be totally covered... The author points out that in the real world this could be explained by the fact the molecules of paint would have a finite size so would not reach the end of the horn.

This is a good quandary to consider - but it left me wanting to read and study more. I can see it must be true in the digital world of physics and chemistry, where there are minimum physical dimensions of such as the Planck length and molecules, but why should it work out this way in pure maths? Maybe the explanation explains many of the differences between theoretical maths and the real world - and maybe of the enigma that is Infinity? Maybe other apparent enigmas can be understood in terms of extra dimensions? Or by the impossibility of exactly reconciling points in the three dimensions of Cartesian space (x, y & z co-ordinates) with cylindrical dimensions (think Latitude & longitude meeting the surface of the Earth with) or spherical dimensions (think of a surveyor with a theodolite)? After all, the latter involve rotations and curves not just points on straight lines. Anyway that is just my amateur hypothesizing. The point is that the book encourages the general reader to think wider and deeper about the fundamental nature of things.

Really excellent and thoroughly recommended.
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on 30 July 2010
A fascinating book on the development of the mathematical understanding of infinity, enlivened with accounts of some of the key personalities. Brian Clegg includes quite a lot of maths: he has the gift of explaining it in a way which means that you need only an elementary knowledge of maths (really just basic simple arithmetic), and a little concentration, to understand the full detail. The result is a model of how to explain sophisticated mathematical concepts to non-specialists without over-simplification. I really enjoyed the book, and thoroughly recommend it.
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on 20 October 2003
I would perhaps have once said 'infinitely enjoyable' - but, having read this book I think it would be impossible to say that - as I think (?) that that would mean that nothing else could ever truly appeal to me again.
This book was fascinating from start to finish - and cleared up some questions I've had lurking deep in my brain for many a year. And, it cleared these up using truly delightfully prose.
I'm definitely 'a fan' - and have just ordered Light Years for myself as a treat.
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on 25 October 2003
Unlike one of the other reviewers, I am a scientist, although I got over it.... But this book makes maths and matters of science enjoyable and accessible. Brian has a real knack at making the complex compelling, useful and enjoyable. A good read just for the hell of it or especially if you work in the field of getting others to accept new ideas.
Peter Cook
Director, Human Dynamics - Purging people's inner management demons through rock and roll....
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