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The Joy of x: A guided tour of mathematics, from one to infinity

Kester Brewin

RJoyofX.jpgSteven Strogatz
Atlantic Books

There is a joke concerning a sociologist, a physicist and a mathematician on a train crossing the border into Scotland which ends with the following punchline (a word I use guardedly) 'No,' the mathematician said. 'All we know that there exists at least one sheep in Scotland, at least one side of which is black.' The 'joke' as it stands turns on the over-nerdish insistence of the mathematician for rigorous statements about the observable world. Yet it's for precisely this reason that, to the mathematician, the joke isn't funny at all, for this is what mathematicians actually do. Beginning in our own dimension but quickly stretching out in many more than we can easily perceive, those who study maths are the astronauts of a vast universe of their own making. Their craft is a sleek one: pure logic drives them forward. Any step outside of this is into non-existence because what they cannot deduce by logical induction is shot down.

This presents something of a problem for Steven Strogatz in The Joy of x. On the first page he refers to an artist friend of his who he says is 'not sure what mathematicians do all day, or what they mean when they say a proof is elegant' and that 'the strange symbols keep him out - he doesn't even know how to pronounce them.' In recent years there have been whole slews of books popularising science and other disciplines, but mathematics presents a tougher problem because the only way to adequately describe mathematics is ... with mathematics. For Strogatz to 'x up' mathematics for the non-mathematician must inevitably mean adding layers of explanation, but to move from the purity of algebra into the vagaries of English is dirty and difficult work.

Mathematically rigorous statements do not make a great read, and mathematicians have long been Stylites of their own making, climbing far above us onto narrow pillars, often refusing to come down to ground to explain what possible use their altitude has. Yet Strogatz does mostly manage to reach down to ground and give the reader a leg up from the very basics of 1 + 1 and onwards towards infinity. He has an easy and comfortable style which does a very good job of communicating his joy in numbers - and joy at its best has no 'use' at all. This is not a book designed to convince you that mathematics is incredibly useful, an invaluable tool that you should all go out and become more proficient at. Pure mathematics - the geometry, number theory and algebra that form the bulk of the chapters - never sets out to be useful, and yet, as Strogatz shows so well, it is a highly creative field. As he notes, 'maths always involves both invention and discovery: we invent the concepts but discover their consequences.'

The consequences - and subsequent uses that can result from mathematical discoveries - pepper the book with pleasing frequency without becoming too laboured. Strogatz variously shows how Larry Page and Sergey Brin used mathematical algorithms to make their 'Google' search engine come up with better page results, and why Michael Jordan appeared to hang in the air while doing a slam-dunk. At times - though this is perhaps because of my own background teaching the subject - I cried out for more detail. In order to keep the book pacy and avoid getting bogged down in complexities the chapters are very brief, though there are nearly 40 pages of notes at the back which add some depth. But by following the basic outline of a high school curriculum Strogatz sustains interest with points of departure that will be familiar to the vast majority of readers.

It was GH Hardy who said 'immortality may be a silly word, but probably a mathematician has the best chance of whatever it might mean.' Why? Because, he says, 'languages die and mathematical ideas do not.' Very few people speak classical Greek; all students who sit a GCSE in mathematics are still required to know the mathematical idea attributed to that most famous of ancient Greeks, Pythagoras. Strogatz won't achieve immortality with this book, though he should be applauded for taking on such a wide brief and making it accessible to people of a wide variety of knowledge of the subject. His joy is unquestioned, but, for me, he didn't quite leave readers with a genuine sense of the elegance of x.

Because of its essential sterile syntax, mathematics can be said to be the only language within which poetry is impossible. The beauty that maths does possess is a hard-edged one. Not poetic, but almost infinitely symmetrical. It is this 'almost' that I finally ached for. In The Joy of x Strogatz wants to offer a picture of a subject that is totally internally consistent. And yet, the surprising thing about mathematics is that, despite this projection of a subject that wants to be 100% proof, it has been shown to fail at the final hurdle. Gödel's 'incompleteness theorem' gave an algebraic backbone to verbal loops like 'this statement is a lie,' and thus showed that any system that tried to be totally consistent within itself would be inconsistent.

It is a shame Strogatz didn't draw Gödel into the conclusion of this fine book, as to do so could have drawn his readers into the true 'joy of x' - that this subject that appears to offer a way of interrogating our world with purely rational tools, ends up, at its extremes, approaching mystery.


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