Mathematical Mountaintops by John Casti, Oxford, 拢19.95, ISBN
0195141717
READING about the exploits of mountaineers risking life and limb to be the
first up some perilous peak is one of life鈥檚 great vicarious pleasures. The
thrills on offer in John Casti鈥檚 account of assaults on the five most famous
peaks in mathematics are naturally rather more recherch茅. 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鈥攕ome doomed from the outset鈥攐n 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 鈥渂ecause they鈥檙e there鈥�.
Casti鈥檚 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鈥檇 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鈥檛 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鈥檛 a general method鈥攑retty scant reward for a hard
climb.
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Fortunately, it鈥檚 downhill a bit to Casti鈥檚 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 鈥渕ap鈥�, and philosophical aspects of proofs so computer-dependent
that no human can hope to check them.
Next on Casti鈥檚 tour is the most thrilling climb of all: Cantor鈥檚 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 鈪� and &pgr;).
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鈥檚 fourth peak brings us back down to much more familiar territory:
stacking fruit. Roughly speaking, Kepler鈥檚 Conjecture states that fitting each
layer of oranges into the spaces in the layer below is the tightest way to pack
them鈥攕omething 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鈥檚 Last Theorem (FLT). As with Everest,
guides of varying ability offer package tours up this peak these days.
Typically, Casti鈥檚 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鈥攖hen 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.