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The Infinite Book – John D. Barrow ****

Authors are often asked to review books on a topic they’ve written on themselves. The reasoning is sensible – they ought to know something about the subject – but there’s always that uneasy suspicion that there’s going to be a bit of bias creeping in. So I think it’s only fair to admit up front that I have written a book on infinity (of which more later).
Infinity is a wonderful subject, because it’s intimately mind-bending (if the combination sounds paradoxical, that’s what infinity is all about) and gives you the chance to pull in all sorts of different concepts and assocations along the way, something Barrow does with great gusto. There’s a surprisingly large amount of coverage here for God, and for the universe, and the book jumps around from Aristotle to Hilbert’s Infinite Hotel (explained at great length), from the paradoxes of infinite sets to the paradoxes of time travel. Overall it’s an enjoyable journey that gives plenty of opportunity to be amazed and surprised.
The only trouble I have with this book is the balance of content. Barrow spends a good half of the book on cosmology, which seems a bit of a cheat. Okay, there are links, and his thoughts on “is the universe infinite” are well worth reading, but in the end this wasn’t supposed to be a cosmology title. Because there is so much on cosmology, this only being a finite book (despite the title), he misses out important swathes of the history of infinity. There’s nothing, for instance, on the development of calculus – the first mathematical tool dependent on infinity – and the magnificent battle between Newton, Leibniz and Berkeley. Similarly, though Barrow does fill in a little bit of biography on Cantor, the man who devised the mathematics of infinity, he does so without explaining what Cantor’s set theory is about – a fundamental to understanding what Cantor was doing – and without even touching on important numerical concepts like cardinals and ordinals.
In a sense, the moan is that this is a book about infinity that isn’t really about maths or about mathematicians. As well as lots on cosmology, it has rather too much vague philosophical material (not helped by the illustrations from an obscure play, or the groan-inducing section titles).
Don’t think I’m putting the book down though – it is very good, and I would recommend reading it, but only after reading my own A Brief History of Infinity to fill in some of those gaps. If I were asked to compare the two, I would say that A Brief History of Infinity is better at providing a historical context, more of a feel for the people involved and more of the amazing mathematical consequences and paradoxes, while Barrow’s book is better at showing how infinity sits alongside our present day physical theories.

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Review by Brian Clegg

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