Robert Matthews puts on his crampons for a thrilling climb in the highest maths Mathematical Mountaintops by John Casti, Oxford, £19.95, ISBN 0195141717

Robert Matthews

Reading about the exploits of mountaineers risking life and limb to be the first up some perilous peak is one of life's great vicarious pleasures. The thrills on offer in John Casti's account of assaults on the five most famous peaks in mathematics are naturally rather more recherche'. Even so, for those seeking something a bit more strenuous than the usual trot around the foothills, Casti provides a bit of a treat. Not only does he recount the stories of the many attempts-some doomed from the outset-on these major mathematical challenges, but he also shows what vistas their conquests have opened up. He gives us some inkling as to why mathematicians try to scale such peaks, other than "because they're there.

Casti's first peak is the search for a method for showing whether any given Diophantine equation has solutions involving only rational numbers. What a singularly awful place to start his tour: an abstruse question with an even more abstruse solution. By the time I'd staggered to the end of this chapter, Casti had me feeling like a pensioner whose Alpine walking holiday had begun with an ascent of the Eiger I won't bother to explain what a Diophantine equation is, except to say that the mathematicians who scrambled to the top of this peak found that no, there isn't a general method - pretty scant reward for a hard climb.

Fortunately, it's downhill a bit to Casti's second peak, the proof of the famous 19th-century conjecture that four colours suffice to colour a map so that no two regions that share a boundary have the same colour. While the story of the failed attempts and the ultimate supercomputer-assisted success will be familiar to many, Casti finds new angles on the story, such as what exactly constitutes a "map", and philosophical aspects of proofs so computer-dependent that no human can hope to check them. Next on Casti's tour is the most thrilling climb of all: Cantor's Continuum Hypothesis. It states that there is no infinity between those of the natural numbers (1, 2, 3 and so on) and the reals (numbers like ²/₃ and p).

The very idea that there are varieties of infinity is enough to bring on an attack of vertigo. Indeed, the eponymous German mathematician seems to have succumbed to terminal altitude sickness contemplating such possibilities. Casti is at his best when taking us through the implication of the stunning result that the answer is yes and no.

Casti's fourth peak brings us back down to much more familiar territory: stacking fruit. Roughly speaking, Kepler's Conjecture states that fitting each layer of oranges into the spaces in the layer below is the tightest way to pack them-something all greengrocers know. Amazingly, it took almost 400 years and a big computer to prove this true. As Casti shows, still more amazing is how a Hungarian mathematician found the tiny set of 150 footholds up to the peak, out of the infinitude of nooks and crannies that led nowhere.

Finally, Casti leads us up Fermat's Last Theorem (FLT). As with Everest, guides of varying ability offer package tours up this peak these days. Typically, Casti's route is more demanding, but offers captivating views over landscapes usually ignored, such as the link between glamorous FLT and dowdy old right-angle triangles. Seasoned mathematical ramblers will also enjoy the tale of how one determined assault on FLT based on an entirely natural-looking property of complex numbers, worked fine for all integer powers up to 22 - then suddenly failed. It seems even numbers can be cussed.
Apart from the Diophantine misjudgement, Casti has given us something rare here: a book on higher mathematics that challenges and entertains in equal measure.

Author Robert Matthews is science correspondent for Sunday Telegraph

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