Books: Circular arguments and an old square

Can mere words explain maths, or music? Gilbert Adair and (right) Roz Kaveney judge efforts to sum up their mysteries

The Magical Maze

by Ian Stewart

Weidenfeld & Nicolson, pounds 18.99

The Joy of

by David Blatner

Allen Lane, pounds 12.99

Is maths becoming fashionable? Hardly. Questioned on the radio about my interest in the subject, I remarked that one of its real, albeit frivolous, attractions was that it provided a haven from the by now all- enveloping soundbite culture. No features editor, I suggested, was ever likely to commission Paul Johnson to churn out a thousand feisty words on the continuum hypothesis or request Michael Winner to air his views on the zeta function. Maths, in short, is much too time-demandingly difficult for the professional rent-a-quoter. Or, rather, it becomes very difficult very rapidly; and, as with the kind of beach that slants a tad too precipitously into the ocean, the amateur (which is all I am) may find himself, with unnerving suddenness, way out of his depth.

Yet popularising, and genuinely popular, mathematical books have been around since the year dot. One of the latest, Simon Singh's Fermat's Last Theorem, has been a completely unexpected bestseller. Singh's success must have set a few pulses racing. Was it a one-off or did it belong to a genre that would lend itself to further exploitation? Could Fermat be a format?

The answer supplied by these two works is, I'm afraid, somewhat inconclusive. Both authors overdo the vulgariser's familiar tics - in particular, a waggish, elbow-nudging style which strains to render the inherent but unavoidable abstruseness of the material more palatable by bludgeoning the reader with wow-look-at this! exclamation marks, amateurishly humorous illustrations and wacky captions ("Have I Got Hypotenuse For You"). All in a misguided attempt - which Singh's book soberly eschews - to persuade us that mathematics is fun, fun, fun! It's a little like being endlessly chivvied into enjoying oneself by a demented Butlin's Camp redcoat.

The Joy of , our curtain-raiser, is a short book, though not as short as the shrift I intend to give it. There's a lot to be said about , a measurement that has fascinated mathematicians for four millennia. As the ratio of the circumference of a circle to its diameter, it's encoded in everything from a soap bubble to a planet. The universe could not exist without it. The snag is that it's an irrational number whose infinite decimal expansion - 3.14159265358... - seems utterly random. Clearly, it cannot be, but it has defied every endeavour by mathematicians to uncover an emergent pattern.

Number-crunching, unfortunately, is all that appears to interest David Blatner. It's amusing to learn of the slow but steady advance in the computations of , from the Biblical "3" (which patently left a lot to be desired) to the recent calculation of 5 1.5 billion - billion, not million - digits on a supercomputer. Amusing, too, that the first million digits of its decimal expansion crowd the margins of the text, if in an illegibly minuscule typeface. But Blatner never explores the psychology of these mad numerologists or addresses the kind of metaphysical poser that occurs to the reader (such as: can the millionth digit of - it's 1, for the record - really be said to exist before it materialises on a computer screen?).

Ian Stewart is a phenomenon in the world of popular maths, with some 60 books under his belt, but is not one of his best. A companion volume to this year's Royal Institution Christmas Lectures, it's much more ambitious than Blatner's but beset by the same relentless jocularity. The problem is compounded, in the first half at least, by the fact that instead of focusing on a cluster of crucial and beautiful but also off- puttingly complex theorems and attempting to pare them down to their non- equational essence, Stewart has chosen to examine a few logical fallacies in a bid to demonstrate that they can be related to certain key branches of mathematics. Showing how practically everything in maths is interconnected is a laudable and necessary exercise, except that - in the event - it has drained most of the charm out of the original puzzles.

I am personally grateful to him for having at last produced a lucid exposition of the most fiendisly counter-intuitive of all such puzzles. It goes something like this: behind one of three doors (A, B and C) is a new car. Behind each of the other two, a goat. Invited to choose one door, you opt for (say) B. A mathematician, who knows where the three prizes are located, then points to C, informs you that there's a goat behind it and offers you a last-minute chance of switching from B to A.

You decline on the grounds that the odds on your winning the car are already 50/50. You would be wrong. Not only would switching doors improve those odds, it would actually double them. You don't believe it? Buy the book.

The Aesthetics of Music

by Roger Scruton

Oxford University Press, pounds 35

To read this book, taking time out to listen to the musical examples with which it is so intelligently peppered, is a disconcerting experience. While the music he admires usually moves through conflict to resolution, Roger Scruton is remorselessly combative. People who listen to music he dislikes, or to music he does like in ways he disapproves, are louts, pedants or ignoramuses. This is a bullying book which shouts about the true and beautiful.

Scruton's love for music is undoubted, and his taste worthy of respect. He has studied scores, read other philosophers and even composes, in a small way. His tastes are more eclectic than one might have feared: he likes the standard Germans, but has the occasional good word for Tchaikovsky and Stravinsky, and dabbles into decadence with Scriabin and Szymanowski. He respects the Second Viennese school, though he has nothing but scorn for their successors in the avant-garde. His discussion of the scores he likes is often illuminating and rarely daft; when he dislikes a score, though, he is patronising, crass or gibberingly vindictive.

Part of the trouble is that it is not Scruton's endless expert passages of specialist exposition so much as his basic reactions that determine whether we trust him as a guide through prickly technicalities. It is also the fact that he does not discuss performers at all until page 420; that he distrusts the "authenticity" movement to the point of believing that Bach's keyboard works should only ever be played on a piano; that he regards the drum-beats at the start of Beethoven's violin concerto as either the setting of a motive pulse or the start of the melody, but not as both, put there for the vulgar purpose of surprising the audience.

Scruton wants the music the great composers wrote to be as Platonically pure as his worship of it. Rightly, he loves Webern's orchestration of the six-part ricercar from Bach's Musical Offering. Yet he hears it as a recasting that breaks Bach's material up into little Webernian fragments, rather than one that makes the work transparent and helps us hear the details of what Bach wrote. It is, among other things, a lecture by Webern on how he thought we should hear Bach, which shocks us into new awareness with snarling trumpets precisely where we do not expect to hear them.

Yet Scruton regards his own way of listening as the only way. He elaborates unnecessary distinctions because his brain gets in the way of his ear. He spends pages on an attempt to distinguish the representation, the imitation and the conveying of emotion while talking about, among other things, the Countess's arias in The Marriage of Figaro. But the arias' "emotions" are a collaboration between Mozart, his librettist, the soprano, the orchestra and the listener. We agree to hear them as sorrow and yearning and the singer agrees to perform those emotions with conviction and musical skill.

Scruton loves music, but he is not always interested in the sound it makes. He tries to argue that music, properly considered, is an event in the brain with only a limited connection with sound. This is an aesthetics of music that neglects the sensuous pleasure of the Western tradition in favour of its intellectual content. Scruton believes that, if an audience now can hear past Schoenberg's theories of composition to gloriously orchestrated cascades of new sounds, then they are not entitled to that pleasure - are not, in his sense, listening at all.

And those who listen to, or dance to (say) Nirvana, are certainly not listening and dancing in any sense that Scruton regards as virtuous. Only the inhabitants of an organic society, by which he means one ruled by an elite, could do that. What is deeply dislikeable about this book is that Scruton has turned music, which he loves, into a secular religion in which he is to be both theologian and inquisitor.

Music is not a pleasure, but a test of good taste - good taste as a qualification to join Plato's Guardians. The driving beat behind Roger Scruton's prose is that of a smug schoolmaster's cane coming down on a human hand forever.