Popular Science Review - The Millennium Problems

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Review - The Millennium Problems - Keith Devlin

1n 1900 in Paris, the great German mathematician David Hilbert described 23 problems that he felt were the biggest challenges to mathematicians of the time, including some problems that have escaped from mathematical obscurity like Fermat's last theorem. One problem, the Riemann Hypothesis, joins six other newer challenges to form the Millennium problems - a set of seven problems announced in 2000 as a very conscious echo of Hilbert's challenge. The big difference is that the US-based Clay Foundation that set the millennium challenge offers $1 million for each solution.

Keith Devlin is not exactly encouraging in the introductory chapter of his book on these problems. He points out that while big breakthroughs in the sciences can typically be explained (in a rough, but understandable way) in a few paragraphs, the mathematical problems represented here are so detached from the world as normal human beings see it that it takes a whole book to explain what they are. But in a way that's the delight and fascination of these problems.

In many respects this book deserves four stars. For each of the problems we get a thumbnail sketch of the key characters involved, some background to where the problem came from and as close as we can get to an explanation of the problem itself without straying into impossibly complex maths. Yet Devlin has been, frankly, lazy. He has decided that it's all too complex to explain to someone without at least good high school/A-level maths, and because of that doesn't explain some semi-basics that would help an awful lot.

So, for example, in the first real chapter on the Riemann Hypothesis, Devlin plunges into using natural logarithms without explaining what they are or why they are used. His excuse? "The natural logarithm should be familiar to anyone who has taken a first course in calculus. Unfortunately it would be too great a digression to explain it here..." I'm sorry, this really is just laziness. A few minutes explanation here (and at a few more points in the book) would have made the whole thing more accessible to a wider range of readers. I really wonder if this reflects the fact that Devlin is an academic - perhaps a good science writer might have done this better.

Does this approach exclude anyone without Devin's "first course in calculus"? Thankfully, no. There will be parts you have to skim over and take on trust, but this still remains a book that will interest anyone who wants to know a little more about what modern mathematics really is - just be brave and ignore Devlin's attempts to put off his readership.

In case you are wondering what those other 6 problems are, there's: Yang-Mills Theory and the Mass Gap Hypothesis, The P vs. NP problem (one for the computer lovers), The Navier-Stokes Equations, The Poincaré Conjecture, The Birch & Swinnerton-Dyer Conjecture and The Hodge Conjecture. Non the wiser? You will be if you read this book!

Also in hardback:

Reviewed by Peter Spitz

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